Nonlinearity of density stratification modulates buoyancy effects. We report results from a body-inclusive large eddy simulation of a wake in nonlinear stratification, specifically for a circular disk at diameter-based Reynolds number (Re) of 5000. Five density profiles are considered; the benchmark has linear stratification and the other four have hyperbolic tangent profiles of the same thickness to model a pycnocline. The disk moves inside the central core of the pycnocline in two of those four cases and, in the other two cases with a shifted density profile, the disk moves partially/completely outside the pycnocline. The maximum buoyancy frequency (Nmax) for all the profiles is the same. The first part of the study investigates the centred cases. Non-uniform stratification results in increasing wake turbulence relative to the benchmark owing to reduced suppression of turbulence production as well as wave trapping in the pycnocline. Steady lee waves are also quantified to understand the limitations of linear theory. The second part pays attention to the effect of a relative shift between the pycnocline and the disk. The wake defect velocity decays substantially faster in the cases with a shift and the wake has higher turbulence level. The effect of disk location on the Kelvin wake waves (a family of steady waves within the pycnocline) and its modal form is obtained and explained by solving the Taylor–Goldstein equation. The family of unsteady internal gravity waves that are generated by the wake is also studied and the effect of disk shift is quantified.
Vertical transport in the ocean plays a critical role in the exchange of freshwater, heat, nutrients, and other biogeochemical tracers. While there are situations where vertical fluxes are important, studying the vertical transport and displacement of material requires analysis over a finite interval of time. One such example is the subduction of fluid from the mixed layer into the pycnocline, which is known to occur near density fronts. Divergence has been used to estimate vertical velocities indicating that surface measurements, where observational data is most widely available, can be used to locate these vertical transport conduits. We evaluate the correlation between surface signatures derived from Eulerian (horizontal divergence, density gradient, and vertical velocity) and Lagrangian (dilation rate and finite time Lyapunov exponent) metrics and vertical displacement conduits. Two submesoscale resolving models of density fronts and a data-assimilative model of the western Mediterranean were analyzed. The Lagrangian surface signatures locate significantly more of the strongest displacement features and the difference in the expected displacements relative to Eulerian ones increases with the length of the time interval considered. Ensemble analysis of forecasts from the Mediterranean model demonstrates that the Lagrangian surface signatures can be used to identify regions of strongest downward vertical displacement even without knowledge of the true ocean state.
A process study using large-eddy simulations is carried out to explore the dominant 1-D processes that affect mixed layer (ML) properties during an event of summer Monsoon Intra-seasonal Oscillations (MISO) in the Bay of Bengal (BOB). These simulations use realistic air-sea fluxes and initial conditions that were collected during the summer 2018 MISO-BOB field experiment to explore the roles of thermal inversion layer (TIL) and Langmuir turbulence (LT) in modulating ML properties. The simulations span an active period with heavy rain and strong winds and a break period with strong solar heat flux and little rain. The mixed layer depth (MLD), sea surface temperature (SST) and sea surface salinity (SSS) are most affected by the presence of near-inertial oscillations, solar heating and precipitation, all of which occur at different timescales. The subsurface warming induced by the TIL reduces the SST variability at the MISO timescale when compared with the simulation without TIL. Comparison of simulations with and without LT indicates that LT enhances subsurface warming during the active phase and reduces diurnal SST modulation during the break phase. Simulations with 1-D mixing models show a wide disparity in the evolution of MLD, SST, and SSS.
Large eddy simulations are employed to investigate the role of tidal modulation strength on wake vortices and dissipation in flow past three-dimensional topography, specifically a conical abyssal hill. The barotropic current is of the form Uc + Ut sin(Ωtt) , where Uc and Ut are the mean and oscillatory components, respectively, and Ωt is the tidal frequency. A regime with strong stratification and weak rotation is considered. The velocity ratio R = Ut/Uc is varied from 0 to 1. Simulation results show that the frequency of wake vortices reduces gradually with increasing R from its natural shedding frequency at R = 0 to Ωt/2 when R ≥ 0.2. The ratio of R and the excursion number, denoted as S, controls the shift in the vortex frequency. When 0.4<=S<=2, vortices are trapped in the wake during tidal deceleration, extending the vortex shedding cycle to two tidal cycles. Elevated dissipation rates in the obstacle lee are observed in the lateral shear layer, hydraulic jet, and the near wake. The regions of strong dissipation are spatially intermittent, with values exceeding O(Uc3/D) during the maximum-velocity phase, where D is the base diameter of the hill. The maximum dissipation rate during the tidal cycle increases monotonically with R in the downstream wake. Additionally, the normalized area-integrated dissipation rate in the hydraulic response region scales with R as (1 + R)4. Results show that the wake dissipation energetically dominates the internal wave flux in this class of low-Froude number geophysical flows.
Direct numerical simulations are performed to compare the evolution of turbulent stratified shear layers with different density gradient profiles at a high Reynolds number. The density profiles include uniform stratification, two-layer hyperbolic tangent profile and a composite of these two profiles. All profiles have the same initial bulk Richardson number (Rib,0); however, the minimum gradient Richardson number and the distribution of density gradient across the shear layer are varied among the cases. The objective of the study is to provide a comparative analysis of the evolution of the shear layers in term of shear layer growth, turbulent kinetic energy as well as the mixing efficiency and its parameterization. The evolution of the shear layers in all cases shows the development of Kelvin–Helmholtz billows, the transition to turbulence by secondary instabilities followed by the decay of turbulence. Comparison among the cases reveals that the amount of turbulent mixing varies with the density gradient distribution inside the shear layer. The minimum gradient Richardson number and the initial bulk Richardson number do not correlate well with the integrated TKE production, dissipation and buoyancy flux. The bulk mixing efficiency for fixed Rib,0 is found to be highest in the case with two-layer density profile and lowest in the case with uniform stratification. However, the parameterizations of the flux coefficient based on buoyancy Reynolds number and the ratio of Ozmidov and Ellison scales show similar scaling in all cases.
Buoyant shear layers encountered in many engineering and environmental applications have been studied by researchers for decades. Often, these flows have high Reynolds and Richardson numbers, which leads to significant/intractable space–time resolution requirements for direct numerical simulation (DNS) or large eddy simulation (LES). On the other hand, many of the important physical mechanisms, such as stress anisotropy, wake stabilization, and regime transition, inherently render eddy viscosity-based Reynolds-averaged Navier–Stokes (RANS) modeling inappropriate. Accordingly, we pursue second-moment closure (SMC), i.e., full Reynolds stress/flux/variance modeling, for moderate Reynolds number nonstratified, and stratified shear layers for which DNS is possible. A range of submodel complexity is pursued for the diffusion of stresses, density fluxes and variance, pressure strain and scrambling, and dissipation. These submodels are evaluated in terms of how well they are represented by DNS in comparison to the exact Reynolds-averaged terms, and how well they impact the accuracy of full RANS closure. For the nonstratified case, SMC model predicts the shear layer growth rate and Reynolds shear stress profiles accurately. Stress anisotropy and budgets are captured only qualitatively. Comparing DNS of exact and modeled terms, inconsistencies in model performance and assumptions are observed, including inaccurate prediction of individual statistics, non-negligible pressure diffusion, and dissipation anisotropy. For the stratified case, shear layer and gradient Richardson number growth rates, and stress, flux and variance decay rates, are captured with less accuracy than corresponding flow parameters in the nonstratified case. These studies lead to several recommendations for model improvement.
Large-eddy simulations (LES) are employed to investigate the role of time-varying currents on the form drag and vortex dynamics of submerged 3D topography in a stratified rotating environment. The current is of the form Uc + Utsin(2πftt), where Uc is the mean, Ut is the tidal component, and ft is its frequency. A conical obstacle is considered in the regime of low Froude number. When tides are absent, eddies are shed at the natural shedding frequency fs,c. The relative frequency f* = fs,c/ft is varied in a parametric study, which reveals states of high time-averaged form drag coefficient. There is a twofold amplification of the form drag coefficient relative to the no-tide (Ut = 0) case when f* lies between 0.5 and 1. The spatial organization of the near-wake vortices in the high drag states is different from a Kármán vortex street. For instance, the vortex shedding from the obstacle is symmetric when f* = 5/12 and strongly asymmetric when f* = 5/6. The increase in form drag with increasing f* stems from bottom intensification of the pressure in the obstacle lee which we link to changes in flow separation and near-wake vortices.
We use spectral proper orthogonal decomposition (SPOD) to extract and analyse coherent structures in the turbulent wake of a disk at Reynolds number Re=5×104 and Froude numbers Fr=2, 10. We find that the SPOD eigenspectra of both wakes exhibit a low-rank behaviour and the relative contribution of low-rank modes to total fluctuation energy increases with x/D. The vortex shedding (VS) mechanism, which corresponds to St≈0.11−0.13 in both wakes, is active and dominant throughout the domain in both wakes. The continual downstream decay of the SPOD eigenspectrum peak at the VS mode, which is a prominent feature of the unstratified wake, is inhibited by buoyancy, particularly for Fr=2. The energy at and near the VS frequency is found to appear in the outer region of the wake when the downstream distance exceeds Nt=Nx/U=6−8. Visualizations show that unsteady internal gravity waves (IGWs) emerge at the same Nt=6−8. A causal link between the VS mechanism and the unsteady IGW generation is also established using the SPOD-based reconstruction and analysis of the pressure transport term. These IGWs are also picked up in SPOD analysis as a structural change in the shape of the leading SPOD eigenmode. The Fr=2 wake shows layering in the wake core at Nt>15 which is captured by the leading SPOD eigenmodes of the VS frequency at downstream locations x/D>30. The VS mode of the Fr=2 wake is streamwise coherent, consisting of V-shaped structures at x/D≳30. Overall, we find that the coherence of wakes, initiated by the VS mode at the body, is prolonged by buoyancy to far downstream. Also, this coherence is spatially modified by buoyancy into horizontal layers and IGWs. Low-order truncations of SPOD modes are shown to efficiently reconstruct important second-order statistics.
The interaction between upper-ocean submesoscale fronts evolving with coherent features, such as vortex filaments and eddies, and convective turbulence generated by surface cooling of varying magnitude is investigated. Here, we decompose the flow into finescale (FS) and submesoscale (SMS) fields explicitly to investigate the energy pathways and the strong interaction between them. Most of the surface cooling flux is transferred to the FS kinetic energy through the FS buoyancy flux carried by the convective plumes. Overall, the SMS strengthens due to surface cooling. The frontogenetic tendency at the submesoscale increases, which counters the enhanced horizontal diffusion by convection-induced turbulence. Downwelling/upwelling strengthens, and the peak SMS vertical buoyancy flux increases as surface cooling is increased. Furthermore, the production of FS energy by SMS velocity gradients (the interscale transfer term, which mediates forward energy cascade) is significant, up to half of the production by convection. Examination of potential vorticity reveals that surface cooling promotes higher levels of secondary symmetric instability (SI), which coexists with the persistent baroclinic instability. The forward interscale transfer is found to increase in the regions with SI.
The three-dimensional transport pathways, the time and space scales of vertical transport, and the dispersion characteristics (single-, pair- and multi-particle statistics) of submesoscale currents at an upper-ocean front are investigated using material points (tracer particles) that advect with the local fluid velocity. Coherent vortex filaments and eddies, generated and sustained during the evolution of baroclinic instability, dominate submesoscale (0.1 - 10 km) dynamics. These coherent structures play a crucial role in the particle transport and dispersion, which we quantify here. Particles in the central frontal region organize into vertically inclined lobes with anticyclonic circulation, each associated with a coherent eddy with cyclonic rotation. Furthermore, the coherent filaments associated with heavy/light edges of the front transfer particles from the edges to the lobes by downwelling/upwelling. This flux of new particles into the central region causes the particles circulating in the lobes to adjust, which leads to slumping of the front. The particle motion in the vertical shows multiple time scales – a fast time scale with O(10m) vertical displacement within an hour and a slower near-inertial time scale, comparable to the intrinsic time scale of the growing instability. The fast O(1h) time scale is also seen in the decorrelation of vertical velocity and the tapering of the initial ballistic growth of relative particle dispersion. The overall slumping process is slower than what one might anticipate from the large magnitude of vertical velocity in the filaments since it requires a sustained correlation over time between the lateral and the vertical motion. By tracking clouds of particles, we show that their centers of mass downwell/upwell over 1-2 inertial time periods, after which an adjustment follows with a sub-inertial time scale. The shape change in clusters of four particles reveals filamentogenesis, i.e. deformation into thin, needle-like structures by the coherent, anisotropic motions of the submesoscale, which occurs as a rapid process that is complete within approximately an hour.
Factors thought to influence deep cycle turbulence in the equatorial Pacific are examined statistically for their predictive capacity using a 13-yr moored record that includes microstructure measurements of the turbulent kinetic energy dissipation rate. Wind stress and mean current shear are found to be most predictive of the dissipation rate. Those variables, together with the solar buoyancy flux and the diurnal mixed layer thickness, are combined to make a pair of useful parameterizations. The uncertainty in these predictions is typically 50% greater than the uncertainty in present-day in situ measurements. To illustrate the use of these parameterizations, the record of deep cycle turbulence, measured directly since 2005, is extended back to 1990 based on historical mooring data. The extended record is used to refine our understanding of the seasonal variation of deep cycle turbulence.
Direct numerical simulations are performed to investigate a stratified shear layer at high Reynolds number (Re) in a study where the Richardson number (Ri) is varied among cases. Unlike previous work on a two-layer configuration in which the shear layer resides between two layers with constant density, an unbounded fluid with uniform stratification is considered here. The evolution of the shear layer includes a primary Kelvin–Helmholtz shear instability followed by a wide range of secondary shear and convective instabilities, similar to the two-layer configuration. During transition to turbulence, the shear layers at low Ri exhibit a period of thickness contraction (not observed at lower Re) when the momentum and buoyancy fluxes are counter-gradient. The behaviour in the turbulent regime is significantly different from the case with a two-layer density profile. The transition layers, which are zones with elevated shear and stratification that form at the shear-layer edges, are stronger and also able to support a significant internal wave flux. After the shear layer becomes turbulent, mixing in the transition layers is shown to be more efficient than that which develops in the centre of the shear layer. Overall, the cumulative mixing efficiency (EC) is larger than the often assumed value of 1/6. Also, EC is found to be smaller than that in the two-layer configuration at moderate Ri. It is relatively less sensitive to background stratification, exhibiting little variation for 0.08<=Ri<=0.2. The dependence of mixing efficiency on buoyancy Reynolds number during the turbulence phase is qualitatively similar to homogeneous sheared turbulence.
Wake vortices in tidally modulated currents past a conical hill in a stratified fluid are investigated using large-eddy-simulation. The vortex shedding frequency is altered from its natural steady-current value leading to synchronization of wake vortices with the tide. The relative frequency (f*), defined as the ratio of natural shedding frequency (fs,c) in a current without tides to the tidal frequency (ft), is varied to expose different regimes of tidal synchronization. When f* increases and approaches 0.25, vortex shedding at the body changes from a classical asymmetric Kármán vortex street. The wake evolves downstream to restore the Kármán vortex-street asymmetry but the discrete spectral peak, associated with wake vortices, is found to differ from both ft and fs,c, a novel result. The spectral peak occurs at the first subharmonic of the tidal frequency when 0.5 ≤ f* < 1 and at the second subharmonic when 0.25 ≤ f* < 0.5.
The high-Reynolds-number axisymmetric wake of a slender body with a turbulent boundary layer is investigated using a hybrid simulation. The wake generator is a 6:1 prolate spheroid and the Reynolds number based on the diameter 𝐷 is 𝑅𝑒=105. The transition of the wake to a state of complete self-similarity is investigated by looking for the first time into the far field of a slender-body wake. Unlike bluff-body wakes, here the flow is not dominated by vortex shedding in the near wake. Instead, the recirculation region is very small, the near wake is quasi-parallel and is characterised by the presence of broadband turbulence. Until 𝑥/𝐷≈20, the wake decay of a slender body with turbulent boundary layer is very similar to the classic high-𝑅𝑒 behaviour, 𝑈𝑑∼𝑥−2/3. Extrapolation of this observation to larger 𝑥/𝐷 has led to the belief that these wakes decay following the asymptotic −2/3 decay law. Our results show, however, that this is not the case and the wake transitions to a faster decay rate once complete self-similarity is achieved. In this later region (20<𝑥/𝐷<80), mean and turbulence profiles are self-similar. Furthermore, despite the high global and local Reynolds numbers, the classic hypotheses that lead to the well-known decay exponents are not fulfilled. Instead, turbulent dissipation follows a non-equilibrium scaling and a new decay rate 𝑈𝑑∼𝑥−6/5 is observed. The transition from 𝑈𝑑∼𝑥−2/3 to 𝑈𝑑∼𝑥−6/5 is preceded by the dominance of the azimuthal |𝑚|=1 mode and the emergence of a large-scale helical structure.
The coherent structures in the turbulent wake of a disk at a moderately high Reynolds number (Re) of 50000 are examined using spectral proper orthogonal decomposition (SPOD) which considers all three velocity components in a numerical database. The SPOD eigenvalues at a given streamwise (x) location are functions of azimuthal wave number (m), frequency (St), and SPOD index (n). By x/D=10, two specific modes dominate the fluctuation energy: (i) the vortex shedding (VS) mode with m=1,St=0.135,n=1, and (ii) the double helix (DH) mode with m=2,St→0,n=1. The VS mode is more energetic than the DH mode in the near wake but, in the far wake, it is the DH mode which is dominant. The DH mode, when scaled with local turbulent velocity and length scales, shows self-similarity in eigenvalues and eigenmodes while the VS mode, which is a global mode, does not exhibit strict self-similarity. Modes m=0, 3, and 4, although subdominant, also make a significant net contribution to the fluctuation energy, and their eigenspectra are evaluated. The reconstruction of turbulent kinetic energy (TKE) and Reynolds shear stress, ⟨u′xu′r⟩, is evaluated by varying (m,St,n) combinations. Higher SPOD modes contribute significantly to the TKE, especially near the centerline. In contrast, reconstruction of ⟨u′xu′r⟩ requires far fewer modes: |m|≤4, ∣∣St∣∣≤1, and n≤3. Among azimuthal modes, m=1 and 2 are the leading contributors to both TKE and ⟨u′xu′r⟩. While m=1 captures the slope of the shear-stress profile near the centerline, m=2 is important to capture ⟨u′xu′r⟩ at and near its peak. SPOD is also performed in the vicinity of the disk to describe the modal transition to the principal contributors in the wake. The leading SPOD modes show a high-frequency shear-layer peak close to the disk and the vortex shedding mode commences its initial dominance of the wake at the end of the recirculation region.
Ekman layers over a rough surface are studied using direct numerical simulation. The roughness takes the form of periodic two-dimensional bumps whose non-dimensional amplitude is fixed at a small value ( ℎ+=15 ) and whose mean slope is gentle. The neutral Ekman layer is subjected to a stabilizing cooling flux for approximately one inertial period ( 2𝜋/𝑓 ) to impose the stratification. The Ekman boundary layer is in a transitionally rough regime and, without stratification, the effect of roughness is found to be mild in contrast to the stratified case. Roughness, whose effect increases with the slope of the bumps, changes the boundary layer qualitatively from the very stable (Mahrt, Theor. Comput. Fluid Dyn., vol. 11, issue 3–4, 1998, pp. 263–279) regime, which has a strong thermal inversion and a pronounced low-level jet, in the flat case to the stable regime, which has a weaker thermal inversion and stronger surface-layer turbulence, in the rough cases. The flat case exhibits initial collapse of turbulence which eventually recovers, albeit with inertial oscillations in turbulent kinetic energy. The roughness elements interrupt the initial collapse of turbulence. In the quasi-steady state, the thickness of the turbulent stress profiles and of the near-surface region with subcritical gradient Richardson number increase in the rough cases. Analysis of the turbulent kinetic energy (TKE) budget shows that, in the surface layer, roughness counteracts the stability-induced reduction of TKE production. The flow component, coherent with the surface undulations, is extracted by a triple decomposition, and leads to a dispersive component of near-surface turbulent fluxes. The significance of the dispersive component increases in the stratified cases.
Three-dimensional (3D) obstacles on the bottom are common sites for the generation of vortices, internal waves and turbulence by ocean currents. Turbulence-resolving simulations are conducted for stratified flow past a conical hill, a canonical example of 3D obstacles. Motivated by the use of slip boundary condition (BC) and drag-law (effectively partial slip) BC in the literature on geophysical wakes, we examine the sensitivity of the flow to BCs on the obstacle surface and the flat bottom. Four BC types are examined for a non-rotating wake created by a steady current impinging on a conical obstacle, with a detailed comparison being performed between two cases, namely NOSL (no-slip BC used at all solid boundaries) and SL (slip BC used at all solid boundaries). The other two cases are as follows: Hybrid, undertaken with slip at the flat bottom and no-slip at the obstacle boundaries, and case DL wherein a quadratic drag-law BC is adopted on all solid boundaries. The no-slip BC allows the formation of a boundary layer which separates and sheds vorticity into the wake. Significant changes occur in the structure of the lee vortices and wake when the BC is changed. For instance, bottom wall friction in the no-slip case suppresses unsteadiness of flow separation leading to a steady attached lee vortex. In contrast, when the bottom wall has a slip BC (the SL and Hybrid cases) or has partial slip (DL case), unsteady separation leads to a vortex street in the near wake and the enhancement of turbulence. The recirculation region is shorter and the wake recovery is substantially faster in the case of slip or partial-slip BC. In the lee of the obstacle, turbulent kinetic energy (TKE) for case NOSL is concentrated in a shear layer between the recirculating wake and the free stream, while TKE is bottom-intensified in the other three cases. DL is the appropriate BC for high- wakes where the boundary layer cannot be resolved. The sources of lee vorticity are also examined in this study for each choice of BC. The sloping sides lead to horizontal gradients of density at the obstacle, which create vorticity through baroclinic torque. Independent of the type of BC, the baroclinic torque dominates. Vortex stretch and tilt are also substantial. An additional unstratified free-slip case (SL-UN) is simulated and the wake is found to be thin without large wake vortices. Thus, stratification is necessary for the formation of coherent lee vortices of the type seen in geophysical wakes.
The upper layer of the ocean participates directly in the exchange of momentum, heat and moisture with the atmosphere. We consider three examples of upper-ocean flow and heat transfer in the present contribution. These examples range from the canonical problem of a stratified shear layer to the surface boundary layer driven by wind and a diurnally varying heat flux to deep cycle turbulence in the Equatorial UnderCurrents (EUC). These problems illustrate stratified shear flow turbulence, wind-driven entrainment in a stratified, rotating fluid, and the communication of surface forcing to subsurface currents in the upper ocean. We discuss the three cases by including new simulations as well as some of our previous work. Direct numerical simulation (DNS) is our tool for the canonical shear layer and, for the other problems, our tool is large eddy simulation (LES) which is increasingly being used to examine turbulent transport and mixing in the ocean. We discuss how buoyancy and rotation affects the spatial structure and temporal evolution of turbulent fluxes, and thereby the distribution of surface inputs of momentum and heat in the upper ocean.
The spindown of a geostrophically balanced density front in an upper-ocean mixed layer is simulated with a large eddy simulation (LES) model that resolves O(1000) m down to O(1) m scale. Our goal is to examine the interaction between the submesoscale and the turbulent finescale, and another related goal is to use the turbulence-resolving simulation to better characterize vertical transport, frontogenesis and dissipative processes. The flow passes through symmetric and baroclinic instabilities, spawns vortex filaments of O (100) m thickness as well as larger eddies with cross-front velocity as large as the along-front velocity, and develops turbulence that is spatially localized and organized. A O(100) m physical-space filter is applied to the simulated flow so that the coherent submesoscale is separated from the finescale in a decomposition that preserves the spatial organization of the flow unlike the typical practice of a split into a frontal average and a fluctuation that obscures the coherent submesoscale. The energy spectrum exhibits a change of slope at O(100) m with an approximately −5/3 slope over a subrange of the finescale. Analysis of the submesoscale vertical velocity (as large as 5mm/s) in the upper layer of the front reveals that downwelling is limited to the thin vortex filaments and the junction of the submesoscale eddies with these filaments while upwelling occurs over spatially extensive regions in the eddies. Conditional averaging shows that heavier (lighter) fluid is preferentially downwelled (upwelled) by these coherent submesoscale structures leading to an overall buoyancy flux that is restratifying. The submesoscale is unbalanced with local Rossby number as large as 5. The kinetic energy (KE) transport equations are evaluated separately for the submesoscale and the finescale to understand energy pathways in this problem. The buoyancy flux (associated with coherent motions) transfers the potential energy of the front and acts as the primary source of submesoscale KE which is then transported across the front with a fraction transferred to the finescale. The transfer, limited to thin regions of O(100) m horizontal width, is accomplished by primarily horizontal strain in the upper 10 m and by vertical shear in the rest of the 50-m deep mixed layer. Frontogenetic mechanisms are diagnosed through analysis of the transport equation for squared buoyancy gradient. Horizontal strain is the primary frontogenetic term that is especially strong in the near-surface layer. The frontogenesis is counteracted primarily by horizontal diffusion in the top 10 m while, further below, the balance is with the horizontal gradient of vertical velocity.
The evolution of turbulence in a shallow submesoscale front is investigated using high-resolution large-eddy simulations (LES). The model front has uniquely large cross-front temperature and salinity differences motivated by recent observations in the Bay of Bengal. The temperature and salinity differences are increased in a parametric study to examine their effects on the evolution of turbulence. Shear turbulence induced by the initially geostrophic jet causes the front to slump and an ageostrophic secondary circulation (ASC) develops. The ASC is composed of two rotating currents: a surface gravity current (SGC) with cold fresh water above a counter-current (CC) jet with warm saline water. Virtual moorings on the two sides of the front are used to depict the evolution of stratification, shear, turbulent stresses, turbulent heat and salt fluxes, as well as turbulent kinetic energy (TKE) budget terms. A barrier layer with thermal inversion develops on both sides of the front while the mixed layer depth becomes shallower across the front. Turbulent production driven by vertical shear is the dominant source of TKE on both sides of the front and horizontal shear production also contributes significantly to the TKE generation. The turbulent stresses, heat and salt fluxes are enhanced on both sides of the front with subsurface warming in the thermal inversion layers. As the cross-front temperature and salinity differences increase among the cases, the ASC intensifies, and the currents and the turbulence become stronger.
Body-inclusive large-eddy simulations of disk wakes are performed for a homogeneous fluid and for different levels of stratification. The Reynolds number is 5 × 104 and the Froude number ( 𝐹𝑟 ) takes the values of ∞ , 50, 10 and 2. In the axisymmetric wake of a disk with diameter 𝐿𝑏 in a homogeneous fluid, it is found that the mean streamwise velocity deficit ( 𝑈0 ) decays in two stages: 𝑈0∝𝑥−0.9 during 10<𝑥/𝐿𝑏<65 and, subsequently, 𝑈0∝𝑥−2/3 . Consequently, none of the simulated stratified wakes is able to exhibit the classical 2/3 decay exponent of 𝑈0 in the interval before buoyancy effects set in. Stratification affects the wake within approximately one buoyancy time scale, after which, we find three regimes: weakly stratified turbulence (WST), intermediately stratified turbulence (IST) and strongly stratified turbulence (SST). WST begins when the turbulent Froude number ( 𝐹𝑟ℎ ) decreases to 𝑂(1) , spans 1≲𝑁𝑡𝑏≲5 and, while the mean flow is strongly affected by buoyancy in WST, turbulence is not. During IST, which commences at 𝑁𝑡𝑏≈5 when 𝐹𝑟ℎ=𝑂(0.1) , the mean flow has arrived into the non-equilibrium (NEQ) regime with 𝑈0∝𝑥−0.18 , but the turbulence state is still in transition, as indicated by progressively increasing turbulence anisotropy. When 𝐹𝑟ℎ∼𝑂(0.01) at 𝑁𝑡𝑏≈20 , the wake transitions into SST, where the turbulent vertical Froude number ( 𝐹𝑟𝑣 ) asymptotes to a 𝑂(1) constant. There is strong anisotropy ( 𝑢′𝑧≪𝑢′ℎ ), and both 𝑢′ℎ and 𝑈0 satisfy 𝑥−0.18 decay, signifying the arrival of the NEQ regime for both turbulence and mean flow. Turbulence is patchy and temporal spectra are broadband in the SST wake. The wake height decreases as 𝐿𝑉∼𝑂(𝑈0/𝑁) in IST/SST. Energy budgets reveal that stratification prolongs wake life during WST/early-IST by both an energy transfer from mean potential energy to mean kinetic energy and reduction of turbulent production. In the late-IST/early-SST stages, production is enhanced and, additionally, there is injection from turbulent potential energy slowing down turbulent kinetic energy (TKE) decay. Only in the SST stage, when NEQ is realized for both the mean and turbulence, does the turbulent buoyancy flux become negative again, acting as a sink of TKE.
The effect of ambient stratification on flow past a prolate spheroid is investigated using large-eddy simulation. The aspect ratio of the body is L/D=4 and the major axis (L) is aligned with the incoming flow. The Reynolds number based on the minor axis (D) and the incoming velocity (U) is Re=104. The stratification is set using a linear density background with constant buoyancy frequency (N). Three simulations with stratification levels, Fr=U/ND=0.5,1,3 and one unstratified simulation, Fr=∞, are performed. The influence of the body slenderness ratio is assessed by comparing the present results with previous work on a sphere. An overall result is that the flow past a slender body exhibits stronger buoyancy effects relative to a bluff body. At Fr∼O(1), there is a strong interaction between the body-generated internal gravity waves (IGW) and the flow at the body. This interaction gives rise to a critical Froude number, Frc=L/Dπ, which is proportional to the aspect ratio. When Fr≈Frc boundary layer separation is delayed and wake turbulence is strongly suppressed. If Fr⪆Frc, wake turbulence is only partially suppressed and, when Fr⪅Frc, boundary layer separation is promoted and turbulence reappears. To elucidate the effect of body slenderness on stratification effects, the IGW field, the boundary layer evolution and separation are studied. Analysis of the wake reveals the strong influence of the type of separation, and therefore the body shape, on the dimensions and the energetics of the near and intermediate wake. However, the decay rate of the stratified wake in the nonequilibrium region is the same as that of a sphere.
Turbulence and mixing in a near-bottom convectively driven flow are examined by numerical simulations of a model problem: a statically unstable disturbance at a slope with inclination 𝛽 in a stable background with buoyancy frequency 𝑁 . The influence of slope angle and initial disturbance amplitude are quantified in a parametric study. The flow evolution involves energy exchange between four energy reservoirs, namely the mean and turbulent components of kinetic energy (KE) and available potential energy (APE). In contrast to the zero-slope case where the mean flow is negligible, the presence of a slope leads to a current that oscillates with 𝜔=𝑁sin𝛽 and qualitatively changes the subsequent evolution of the initial density disturbance. The frequency, 𝑁sin𝛽 , and the initial speed of the current are predicted using linear theory. The energy transfer in the sloping cases is dominated by an oscillatory exchange between mean APE and mean KE with a transfer to turbulence at specific phases. In all simulated cases, the positive buoyancy flux during episodes of convective instability at the zero-velocity phase is the dominant contributor to turbulent kinetic energy (TKE) although the shear production becomes increasingly important with increasing 𝛽 . Energy that initially resides wholly in mean available potential energy is lost through conversion to turbulence and the subsequent dissipation of TKE and turbulent available potential energy. A key result is that, in contrast to the explosive loss of energy during the initial convective instability in the non-sloping case, the sloping cases exhibit a more gradual energy loss that is sustained over a long time interval. The slope-parallel oscillation introduces a new flow time scale 𝑇=2π/(𝑁sin𝛽) and, consequently, the fraction of initial APE that is converted to turbulence during convective instability progressively decreases with increasing 𝛽 . For moderate slopes with 𝛽<10∘ , most of the net energy loss takes place during an initial, short ( 𝑁𝑡≈20 ) interval with periodic convective overturns. For steeper slopes, most of the energy loss takes place during a later, long ( 𝑁𝑡>100 ) interval when both shear and convective instability occur, and the energy loss rate is approximately constant. The mixing efficiency during the initial period dominated by convectively driven turbulence is found to be substantially higher (exceeds 0.5) than the widely used value of 0.2. The mixing efficiency at long time in the present problem of a convective overturn at a boundary varies between 0.24 and 0.3.
Large-eddy simulations are performed to investigate the development of the ageostrophic secondary circulation (ASC) and associated transport in a submesoscale front. Based on the observations in the northern Bay of Bengal and in the Pacific cold tongue, the model front has a large cross-front density difference that is partially compensated with lateral temperature and salinity gradients. Vertical stratification is varied in different cases to explore its effect on the ASC. The evolution of the ASC differs with stratification. When the front is unstratified, shear instabilities, which develop from the geostrophic shear, cause the front to slump. Cold water from the light side propagates across the front on the surface, while warm water from the dense side spreads in the opposite direction at depth. In cases with stratifications, a shear layer driven by the cross-front pressure gradient forms at the surface to initiate the ASC. Shear-driven turbulence associated with the enhanced shear in the layer causes the front to slump, and the development of the ASC onward is similar to the unstratified case. Irrespective of the initial stratification of the strong fronts simulated here, the surface layer evolves into a gravity current. The ASC is composed of the surface gravity current and a countercurrent that are separated by a middle layer with enhanced stratification and a thermal inversion. Turbulent dissipation is enhanced at the nose of the gravity current and in a sheared region somewhat behind the leading edge of the countercurrent. The gravity current propagates at a speed proportional to the buoyancy difference across the front in the case with no stratification.
Direct numerical simulation in the form of a hybrid spatially-evolving model is used to simulate the turbulent wake behind a towed sphere at a subcritical Reynolds number, Re = U∞D/𝜈 = 3700 where U∞ is the free stream velocity, D is the diameter of the sphere, and ν is the viscosity. The sphere is not present in the model domain but realistic inflow conditions are obtained from a body-inclusive simulation. As such, the model is a hybrid of both body-inclusive and body-exclusive simulations which greatly reduces computational cost and allows for downstream study of wake dynamics. Inlet conditions are extracted from three locations in the streamwise, x1, direction namely x1/D = 3,6, and 10, to investigate the sensitivity of the results to extraction position. Simulations are performed considering both unstratified and stratified fluids with Froude numbers, Fr = U∞/ND = ∞, 3, and 1 where N is the buoyancy frequency of the background, to explore the model effectiveness for varying levels of stratification. It is found that a strategic choice in extraction location is required in order to accurately capture the flow physics. Simulation results show less agreement with the body-inclusive simulations at extraction location x1/D = 3 than with x1/D = 6 or x1/D = 10. Mean wake decay of the hybrid model simulations is consistent with the body-inclusive simulations. The layered flow structure of the stratified wake is captured well while phase lines indicating internal wave propagation are clearly observed in vorticity contours and are consistent with the body-inclusive simulations. Accurate turbulent kinetic energy (t.k.e.) capture is noted for all cases. Additional simulations with higher resolution grids were performed and analyzed to demonstrate the impact of grid resolution on turbulence statistics. Examination of t.k.e. budget terms indicates that improved grid resolution results in better agreement with the body-inclusive simulation near the inlet where the resolution disparity between the hybrid and body-inclusive simulations is most significant. The hybrid model is shown to be a robust tool for the study of turbulent wake dynamics.
Stratified flow in nocturnal boundary layers is studied using direct numerical simulation (DNS) of the Ekman layer, a model problem that is useful to understand atmospheric boundary-layer (ABL) turbulence. A stabilizing buoyancy flux is applied for a finite time to a neutral Ekman layer. Based on previous studies and the simulations conducted here, the choice of ( is the Obukhov length scale and is the friction velocity) provides a cooling flux that is sufficiently strong to cause the initial collapse of turbulence. The turbulent kinetic energy decays over a time scale of during the collapse. The simulations suggest that imposing on the neutral Ekman layer results in turbulence collapse during the initial transient, independent of Reynolds number, . However, the long-time state of the flow, i.e. turbulent with spatial intermittency or non-turbulent, is found to depend on the initial value of since the cooling flux and resultant stratification increase with for a given . The lower- cases have sustained turbulence with shear and stratification profiles that evolve in a manner such that the gradient Richardson number, , in the near-surface layer, including the low-level jet, remains subcritical. The highest case has supercritical in the low-level jet and turbulence does not recover. A theoretical discussion is performed to infer that the bulk Richardson number, , is more suitable than to determine the fate of stratified Ekman layers at late time. DNS results support the implications of for the effect of initial and on the flow.
A new multi-scale modeling technique, SOMAR-LES, is presented in this paper. Localized grid refinement gives SOMAR (the Stratified Ocean Model with Adaptive Resolution) access to small scales of the flow which are normally inaccessible to general circulation models (GCMs). SOMAR-LES drives a LES (Large Eddy Simulation) on SOMAR’s finest grids, forced with large scale forcing from the coarser grids. Three-dimensional simulations of internal tide generation, propagation and scattering are performed to demonstrate this multi-scale modeling technique. In the case of internal tide generation at a two-dimensional bathymetry, SOMAR-LES is able to balance the baroclinic energy budget and accurately model turbulence losses at only 10% of the computational cost required by a non-adaptive solver running at SOMAR-LES’s fine grid resolution. This relative cost is significantly reduced in situations with intermittent turbulence or where the location of the turbulence is not known a priori because SOMAR-LES does not require persistent, global, high resolution. To illustrate this point, we consider a three-dimensional bathymetry with grids adaptively refined along the tidally generated internal waves to capture remote mixing in regions of wave focusing. The computational cost in this case is found to be nearly 25 times smaller than that of a non-adaptive solver at comparable resolution. In the final test case, we consider the scattering of a mode-1 internal wave at an isolated two-dimensional and three-dimensional topography, and we compare the results with Legg (2014) numerical experiments. We find good agreement with theoretical estimates. SOMAR-LES is less dissipative than the closure scheme employed by Legg (2014) near the bathymetry. Depending on the flow configuration and resolution employed, a reduction of more than an order of magnitude in computational costs is expected, relative to traditional existing solvers.
Steep generation sites for topographic internal gravity waves can also be sites of turbulence. The present work uses turbulence-resolving Large Eddy Simulation (LES) for tidal flow over multiscale, steep bathymetry patterned after a portion of the west ridge of Luzon Strait at 20.6 N. Oceanic values of the slope criticality, Froude number, excursion number, and aspect ratio are matched. The amplitude and phasing of velocity, overturns, and dissipation are similar to observations made at a deep mooring. However a tidal ...
The seasonal cycles of the various oceanic and atmospheric factors influencing the deep cycle of turbulence in the eastern Pacific cold tongue are explored. Moored observations at 140° W have shown seasonal variability in the stratification, velocity shear, and turbulence above the Pacific Equatorial Undercurrent (EUC). In boreal spring, the thermocline and EUC shoal and turbulence decreases. Marginal instability (clustering of the local gradient Richardson number around the critical value of 1/4), evident throughout the ...
The paper presents aerodynamic and fluid–structure interaction (FSI) simulations of two back-to-back 5 MW horizontal-axis wind turbines (HAWTs) at full scale and with full geometrical complexity operating in a stably-stratified atmospheric boundary layer (ABL) flow. The numerical formulation for stratified incompressible flows is based on the ALE-VMS methodology, and is coupled to a Kirchhoff–Love thin-shell formulation employed to model the wind-turbine structure. A multi-domain method (MDM) is adopted for computational ...
We describe a simple model for turbulence in a marginally unstable, forced, stratified shear flow. The model illustrates the essential physics of marginally unstable turbulence, in particular the tendency of the mean flow to fluctuate about the marginally unstable state. Fluctuations are modelled as an oscillatory interaction between the mean shear and the turbulence. The interaction is made quantitative using empirically established properties of stratified turbulence. The model also suggests a practical way to estimate ...
Large-eddy simulation (LES) is used to investigate how turbulence in the wind- driven ocean mixed layer erodes the stratification of barrier layers. The model consists of a stratified Ekman layer that is driven by a surface wind. Simulations at a wide range of N0/f are performed to quantify the effect of turbulence and stratification on the entrainment rate. Here, N0 is the buoyancy frequency in the barrier layer and f is the Coriolis parameter. The evolution of the mixed layer follows two stages: a rapid initial deepening and a late-time ...
Direct numerical simulation of flow past a sphere in a stratified fluid is carried out at a subcritical Reynolds number of 3700 and Fr= U/ND= 1, 2 and 3 to understand the dynamics of moderately stratified flows with Fr= O (1) . Here, U is the free stream velocity, N is the background buoyancy frequency and D is the sphere diameter. The unstratified flow past the sphere consists of a separated shear layer that transitions to turbulence, a recirculation zone and a wake with a mean centreline deficit ...
Internal gravity waves are a key process linking the large-scale mechanical forcing of the oceans to small-scale turbulence and mixing. In this review, we focus on internal waves generated by barotropic tidal flow over topography. We review progress made in the past decade toward understanding the different processes that can lead to turbulence during the generation, propagation, and reflection of internal waves and how these processes affect mixing. We consider different modeling strategies and new tools that have been developed. Simulation results, the wealth of observational material collected during large-scale experiments,and new laboratory data reveal how the cascade of energy from tidal flow to turbulence occurs through a host of nonlinear processes, including intensified boundary flows, wave breaking, wave-wave interactions, and the instability of high-mode internal wavebeams. The roles of various nondimensional parameters involving the ocean state, roughness geometry, and tidal forcing are described.
Vortex dynamics in the flow past a sphere in a linearly stratified environment is investigated numerically. Simulations are carried out for a flow with Reynolds number of Re=3700 and for several Froude numbers ranging from the unstratified case with Fr=1 to a highly stratified wake with Fr=0.025. Isosurface of Q criterion is used to elucidate stratification effects on vortical structures near the sphere and in the wake. Vortical structures in the unstratified case are tube-like and show no preference in their orientation. Moderate stratification alters the orientation of vortical structures to streamwise preference but does not change their tube-like form. In strongly stratified cases with Fr=0.5, there is strong suppression in vertical motion so that isotropically oriented vortex tubes of approximately circular cross section are replaced by flattened vortex tubes that are horizontally oriented. At Fr = 0.025, pancake eddies and surfboard-like inclined structures emerge in the near wake and have a regular streamwise spacing that is associated with the frequency of vortex shedding from the sphere. Enstrophy variance budget is used to analyze the vortical structure dynamics. Increasing stratification generally decreases enstrophy variance for Fr = O(1) cases. The flow enters a new regime in strongly stratified cases with Fr=0.25: increasing the stratification increases enstrophy variance, especially near the body. Stratification distorts the cross-sectional distribution of enstrophy variance from a circular isotropic shape in the unstratified wake into different shapes, depending on Fr and distance from the sphere, that include (1) elliptical distribution, (2) twin peaks suggestive of two-dimensional vortex shedding, and (3) triple-layer distribution where a relatively low enstrophy layer is sandwiched between the upper and the lower layers with high enstrophy. In the near wake, vortex stretching by fluctuating and mean strain are both responsible for enhancing vorticity. Increasing stratification (decreasing Fr) to O(1) values tends to suppress vortex stretching. Upon further reduction of Fr below 0.25, the vortex stretching takes large values near the sphere and, consequently, enstrophy variance in the near wake increases. The increase in vortex stretching is associated with unsteady, intermittent shedding of the boundary layer from the sides of the sphere in highly stratified wakes with Fr<0.25.
Direct numerical simulations (DNS) are performed to study the behaviour of flow past a sphere in the regime of high stratification (low Froude number Fr). In contrast to previous results at lower Reynolds numbers, which suggest monotone suppression of turbulence with increasing stratification in flow past a sphere, it is found that, below a critical Fr, increasing the stratification induces unsteady vortical motion and turbulent fluctuations in the near wake. The near wake is quantified by computing the energy spectra, the turbulence energy equation, the partition of energy into horizontal and vertical components, and the buoyancy Reynolds number. These diagnostics show that the stabilizing effect of buoyancy changes flow over the sphere to flow around the sphere. This qualitative change in the flow leads to a new regime of unsteady vortex shedding in the horizontal planes and intensified horizontal shear which result in turbulence regeneration.
The Air-Sea Interactions Regional Initiative (ASIRI) aims to understand vertical fluxes of momentum and heat across the surface layer in the Bay of Bengal. As the mesoscale and submesoscale eddies redistribute freshwater input over saline water of the bay, they influence the vertical distribution of salinity and thus impact air-sea fluxes. This study reports on numerical simulations performed to investigate processes that can lead to the observed vertical structure of stratification near the ocean surface. Processes are explored at multiple lateral scales, ranging from a few meters to tens of kilometers, to elucidate how the interplay among large-scale motion, submesoscale instabilities, and small-scale turbulent motion affects the surface layer.
A numerical formulation for incompressible flows with stable stratification is developed using the framework of variational multiscale methods. In the proposed formulation, both density and temperature stratification are handled in a unified manner. The formulation is augmented with weakly-enforced essential boundary conditions and is suitable for applications involving moving domains, such as fluid–structure interaction. The methodology is tested using three numerical examples ranging from flow-physics benchmarks to a simulation of a full-scale offshore wind-turbine rotor spinning inside an atmospheric boundary layer. Good agreement is achieved with experimental and computational results reported by other researchers. The wind-turbine rotor simulation showed that flow stratification has a strong influence on the dynamic rotor thrust and torque loads.
Direct numerical simulations of an Ekman layer are performed to study flow evolution during the response of an initially neutral boundary layer to stable stratification. The Obukhov length, L, is varied among cases by imposing a range of stable buoyancy fluxes at the surface to mimic ground cooling. The imposition of constant surface buoyancy flux , i.e. constant-flux stability, leads to a buoyancy difference between the ground and background that tends to increase with time, unlike the constant-temperature stability case where a constant surface temperature is imposed. The initial collapse of turbulence in the surface layer owing to surface cooling that occurs over a time scale proportional to L/u ∗ , where u ∗ is the friction velocity, is followed by turbulence recovery. The flow accelerates, and a “low-level jet” (LLJ) with inertial oscillations forms during the turbulence collapse. Turbulence statistics and budgets are examined to understand the recovery of turbulence. Vertical turbulence exchange, primarily by pressure transport, is found to initiate fluctuations in the surface layer and there is rebirth of turbulence through enhanced turbulence production as the LLJ shear increases. The turbulence recovery is not monotonic and exhibits temporal intermittency with several collapse/rebirth episodes. The boundary layer adjusts to an increase in the surface buoyancy flux by increased super-geostrophic velocity and surface stress such that the Obukhov length becomes similar among the cases and sufficiently large to allow fluctuations with sustained momentum and heat fluxes. The eventual state of fluctuations, achieved after about two inertial periods ( f t ≈ 4π), corresponds to global intermittency with turbulent patches in an otherwise quiescent background. Our simplified configuration is sufficient to identify turbulence collapse and rebirth, global and temporal intermittency, as well as formation of low-level jets, as in observations of the stratified atmospheric boundary layer.
An observation and modeling campaign in the Bay of Bengal is aimed at studying upper ocean and lower atmosphere processes and interactions in relation to Indian Ocean Monsoons. Air-Sea Interactions in the Northern Indian Ocean (ASIRI) is an international research effort (2013-2017) aimed at understanding and quantifying coupled atmosphere-ocean dynamics of the Bay of Bengal (BoB) with relevance to Indian Ocean monsoons. Working collaboratively, more than twenty research institutions are acquiring field observations coupled with operational and high-resolution models to address scientific issues that have stymied the monsoon predictability. ASIRI combines new and mature observational technologies to resolve submesoscale to regional-scale currents and hydrophysical fields. These data reveal BoB’s sharp frontal features, submesoscale variability, low-salinity lenses and filaments, shallow mixed layers, with relatively weak turbulent mixing. Observed physical features include energetic high-frequency internal waves in the southern BoB; energetic mesoscale and submesosacle features including an intrathermocline eddy in the central BoB; and a high-resolution view of the exchange along the periphery of Sri Lanka, which includes the 100-km wide East India Coastal Current (EICC) carrying low-salinity water out of the BoB and an adjacent, broad northward flow (~ 300 km wide) that carries high-salinity water into BoB during northeast monsoon. Atmospheric boundary layer (ABL) observations during the decaying phase of the Madden Julian Oscillation (MJO) permit the study of multi-scale atmospheric processes associated with non-MJO phenomena and their impacts on the marine boundary layer. Underway analyses that integrate observations and numerical simulations shed light on how air-sea interactions control the ABL and upper ocean processes.
Internal gravity waves, the subsurface analogue of the familiar surface gravity waves that break on beaches, are ubiquitous in the ocean. Because of their strong vertical and horizontal currents, and the turbulent mixing caused by their breaking, they affect a panoply of ocean processes, such as the supply of nutrients for photosynthesis1, sediment and pollutant transport2 and acoustic transmission3; they also pose hazards for man-made structures in the ocean4. Generated primarily by the wind and the tides, internal waves can travel thousands of kilometres from their sources before breaking5, making it challenging to observe them and to include them in numerical climate models, which are sensitive to their effects6, 7. For over a decade, studies8, 9, 10, 11 have targeted the South China Sea, where the oceans’ most powerful known internal waves are generated in the Luzon Strait and steepen dramatically as they propagate west. Confusion has persisted regarding their mechanism of generation, variability and energy budget, however, owing to the lack of in situ data from the Luzon Strait, where extreme flow conditions make measurements difficult. Here we use new observations and numerical models to (1) show that the waves begin as sinusoidal disturbances rather than arising from sharp hydraulic phenomena, (2) reveal the existence of >200-metre-high breaking internal waves in the region of generation that give rise to turbulence levels >10,000 times that in the open ocean, (3) determine that the Kuroshio western boundary current noticeably refracts the internal wave field emanating from the Luzon Strait, and (4) demonstrate a factor-of-two agreement between modelled and observed energy fluxes, which allows us to produce an observationally supported energy budget of the region. Together, these findings give a cradle-to-grave picture of internal waves on a basin scale, which will support further improvements of their representation in numerical climate predictions.
Direct numerical simulation (DNS) and large-eddy simulation (LES) are employed to study the mixing brought about by convective overturns in a stratified, oscillatory bottom layer underneath internal tides. The phasing of turbulence, the onset and breakdown of convective overturns, and the pathway to irreversible mixing are quantified. Mixing efficiency shows a systematic dependence on tidal phase, and during the breakdown of large convective overturns it is approximately 0.6, a value that is substantially larger than the commonly assumed value of 0.2 used for calculating scalar mixing from the turbulent dissipation rate. Diapycnal diffusivity is calculated using the irreversible diapycnal flux and, for tall overturns of O(50) m, the diffusivity is found to be almost 1000 times higher than the molecular diffusivity. The Thorpe (overturn) length scale is often used as a proxy for the Ozmidov length scale and thus infers the turbulent dissipation rate from overturns. The accuracy of overturn-based estimates of the dissipation rate is assessed for this flow. The Ozmidov length scale LO and Thorpe length scale LT are found to behave differently during a tidal cycle: LT decreases during the convective instability, while LO increases; there is a significant phase lag between the maxima of LT and LO; and finally LT is not linearly related to LO. Thus, the Thorpe-inferred dissipation rates are quite different from the actual values. Interestingly, the ratio of their cycle-averaged values is found to be O(1), a result explained on the basis of available potential energy.
Direct numerical simulation (DNS) and large-eddy simulation (LES) are employed to study the mixing brought about by convective overturns in a stratified, oscillatory bottom layer underneath internal tides. The phasing of turbulence, the onset and breakdown of convective overturns, and the pathway to irreversible mixing are quantified. Mixing efficiency shows a systematic dependence on tidal phase, and during the breakdown of large convective overturns it is approximately 0.6, a value that is substantially larger than the commonly assumed value of 0.2 used for calculating scalar mixing from the turbulent dissipation rate. Diapycnal diffusivity is calculated using the irreversible diapycnal flux and, for tall overturns of O(50) m, the diffusivity is found to be almost 1000 times higher than the molecular diffusivity. The Thorpe (overturn) length scale is often used as a proxy for the Ozmidov length scale and thus infers the turbulent dissipation rate from overturns. The accuracy of overturn-based estimates of the dissipation rate is assessed for this flow. The Ozmidov length scale LO and Thorpe length scale LT are found to behave differently during a tidal cycle: LT decreases during the convective instability, while LO increases; there is a significant phase lag between the maxima of LT and LO; and finally LT is not linearly related to LO. Thus, the Thorpe-inferred dissipation rates are quite different from the actual values. Interestingly, the ratio of their cycle-averaged values is found to be O(1), a result explained on the basis of available potential energy.
A numerical study based on large eddy simulation (LES) is performed to investigate the nonlinear interaction of a semidiurnal (M2) internal wave beam with an upper ocean pycnocline. During refraction through the pycnocline, the wave beam undergoes parametric subharmonic instability (PSI) with formation of waves with (1/2)M2 frequency. The three-dimensional LES enables new results that quantify the route to turbulence through PSI. The subharmonic waves generated from PSI have an order of magnitude smaller vertical scale and are susceptible to wave breaking. Convective instability initiates transition to turbulence, while shear production maintains it. Turbulence at points in the subharmonic wave paths is modulated at (1/2)M2 frequency. The beam suffers substantial degradation owing to PSI, reflected harmonics and ducted waves so that only about 30% of the incoming energy is transported by the main reflected beam.
A highly resolved computation of the flow past a sphere at Reynolds number Re = 3700 using a finite element method (FEM)-based residual-based variational multiscale (RBVMS) formulation is performed. Both uniform and turbulent inflow conditions are considered with the uniform flow case validated against a previous direct numerical simulation (DNS) study. We find that, as a result of adding free-stream turbulence of moderate intensity, the drag force on the sphere is increased, the length of the recirculation bubble is reduced dramatically, and the near-wake turbulence is significantly more energetic than in case of uniform inflow. In the case of uniform inflow, we find that the solution exhibits low temporal frequency modes, which necessitate long-time simulations to obtain high-fidelity statistical averages. Subjecting the sphere to turbulent inflow removes the low-frequency modes from the solution and enables shorter-time simulations to achieve converged flow statistics.
Most turbulent coherent structures in a convectively unstable atmospheric boundary layer are caused by or manifested in ascending warm fluid and descending cold fluids. These structures not only cause ramps in the air temperature timeseries, but also imprint on the underlying solid surface as surface temperature fluctuations. The coupled flow and heat transport mechanism was examined through direct numerical simulation (DNS) of a channel flow allowing for realistic solid–fluid thermal coupling. The thermal activity ratio (TAR; the ratio of thermal inertias of fluid and solid), and the thickness of the solid domain were found to affect the solid–fluid interfacial temperature variations. The solid–fluid interface with large (small) thermal activity ration behaves as an isoflux (isothermal) boundary. For the range of parameters considered here (Grashof number, Gr=3×105--325×105; TAR=0.01--1; solid thickness normalized by heat penetration depth=0.1--10), the solid thermal properties and thickness influence the fluid temperature only in the viscous or conduction region while the convective forcing influences the turbulent flow. Flow structures influence the interfacial temperature more effectively with increasing TAR and solid thickness compared with a constant temperature boundary condition. The change of channel flow structures with increasing convective instability is examined and the concomitant change of thermal patterns is quantified. Despite large differences in friction Reynolds and Richardson number between the DNS and atmospheric observations, similarities in the flow features were observed.
Large eddy simulations are used to examine the evolution of a shear layer in a thermocline with non-uniform density stratification. Unlike previous studies, the density in the present study is continuously stratified and has stratification in the upper half different from the lower half of the shear layer. The stratification in the upper half is fixed at Ju = 0.05, while the stratification in the lower half is increased to Jd = 0.05, 0.15, 0.25 and 0.35, leading to a progressively stronger asymmetry of the Rig profile in the four cases. Here, J is the bulk Richardson number and Rig is the gradient Richardson number. The type of shear instability and the properties of the ensuing turbulence are found to depend strongly on the degree of asymmetry in stratification. The shear instability changes from a Kelvin–Helmholtz (KH) mode at Jd = 0.05 to a Holmboe (H) mode at Jd = 0.35 and exhibits characteristics of both KH and H modes at intermediate values of Jd. Differences in the evolution among the cases are quantified using density visualisations and statistics such as mean shear, mean stratification and turbulent kinetic energy.
The excursion number, Ex=U0/Ωl, is a parameter that characterizes the ratio of streamwise fluid advection during a tidal oscillation of amplitude U0 and frequency Ω to the streamwise topographic length scale l. Direct numerical simulations are performed to study how internal gravity waves and turbulence change when Ex is varied from a low value (typical of a ridge in the deep ocean) to a value of unity (corresponding to energetic tides over a small topographic feature). An isolated obstacle having a smoothed triangular shape and 20 % of the streamwise length at critical slope is considered. With increasing values of Ex, the near field of the internal waves loses its beam-like character, the wave response becomes asymmetric with respect to the ridge centre, and transient lee waves form. Analysis of the baroclinic energy balance shows significant reduction in the radiated wave flux in the cases with higher Ex owing to a substantial rise in advection and baroclinic dissipation as well as a decrease in conversion. Turbulence changes qualitatively with increasing Ex. In the situation with Ex∼0.1, turbulence is intensified at the near-critical regions of the slope, and is also significant in the internal wave beams above the ridge where there is intensified shear. At Ex=O(1), the transient lee waves overturn adjacent to the ridge flanks and, owing to convective instability, buoyancy acts as a source for turbulent kinetic energy. The size of the turbulent overturns has a non-monotonic dependence on excursion number: the largest overturns, as tall as twice the obstacle height, occur in the Ex=0.4 case, but there is a substantial decrease of overturn size at larger values of Ex simulated here.
Large eddy simulation is used to numerically simulate flow past a heated sphere at Re=10,000. A second order accurate in space and time, semi-implicit finite difference code is used with the immersed boundary to represent the sphere in a Cartesian domain. Visualizations of the vorticity field and temperature field are provided together with profiles of the temperature and velocity fields at various locations in the wake. The laminar separated shear layer was found to efficiently transport heat from the hot sphere surface to the cold fluid in the wake. The thin separated shear layers are susceptible to Kelvin–Helmholtz instability and the pronounced rollers that subsequently form promote entrainment of both cold freestream fluid and warmer fluid near the back of the sphere. Breakdown of the shear layer into turbulence and subsequent interaction with the recirculation zone results in rapid mixing of the temperature field in the lee of the sphere. The wake dimensions of the velocity field and the temperature field were found to be comparable in the developed flow behind the re-circulating region. Profiles of the mean and fluctuating temperature and velocity in the near wake are provided together with profiles of the Reynolds stresses and thermal fluxes. Similarity was observed for the mean temperature, rms temperature, rms velocity, and the Reynolds stress component ux'ur', and the thermal fluxes T'ux' and T′ur′.
The performance of the large eddy simulation (LES) approach in predicting the evolution of a shear layer in the presence of stratification is evaluated. The LES uses a dynamic procedure to compute subgrid model coefficients based on filtered velocity and density fields. Two simulations at different Reynolds numbers are simulated on the same computational grid. The fine LES simulated at a low Reynolds number produces excellent agreement with direct numerical simulations (DNS): the linear evolution of momentum thickness and bulk Richardson number followed by an asymptotic approach to constant values is correctly represented and the evolution of the integrated turbulent kinetic energy budget is well captured. The model coefficients computed from the velocity and the density fields are similar and have a value in range of 0.01−0.02. The coarse LES simulated at a higher Reynolds number Re = 50,000 shows acceptable results in terms of the bulk characteristics of the shear layer, such as momentum thickness and bulk Richardson number. Analysis of the turbulent budgets shows that, while the subgrid stress is able to remove sufficient energy from the resolved velocity fields, the subgrid scalar flux and thereby the subgrid scalar dissipation are underestimated by the model.
Dynamical processes leading to deep-cycle turbulence in the Equatorial Undercurrent (EUC) are investigated using a high-resolution large-eddy simulation (LES) model. Components of the model include a background flow similar to the observed EUC, a steady westward wind stress, and a diurnal surface buoyancy flux. An LES of a 3-night period shows the presence of narrowband isopycnal oscillations near the local buoyancy frequency N as well as nightly bursts of deep-cycle turbulence at depths well below the surface mixed layer, the two phenomena that have been widely noted in observations. The deep cycle of turbulence is initiated when the surface heating in the evening relaxes, allowing a region with enhanced shear and a gradient Richardson number Rig less than 0.2 to form below the surface mixed layer. The region with enhanced shear moves downward into the EUC and is accompanied by shear instabilities and bursts of turbulence. The dissipation rate during the turbulence bursts is elevated by up to three orders of magnitude. Each burst is preceded by westward-propagating oscillations having a frequency of 0.004–0.005 Hz and a wavelength of 314–960 m. The Rig that was marginally stable in this region decreases to less than 0.2 prior to the bursts. A downward turbulent flux of momentum increases the shear at depth and reduces Rig. Evolution of the deep-cycle turbulence includes Kelvin–Helmholtz-like billows as well as vortices that penetrate downward and are stretched by the EUC shear.
The primary focus of this study is to contrast the influence of the mean velocity profile with that of the initial turbulence on the subsequent evolution of velocity and density fluctuations in a stratified wake. Direct numerical simulation is used to simulate the following cases: (a) a self-propelled momentumless turbulent wake, case SP50 with a canonical mean velocity profile, (b) a patch of turbulence, case TP1 with the same initial energy spectrum as (a), and (c) a patch of turbulence, case TP2 with a different initial energy spectrum with higher small-scale content. The evolution of the fluctuations is found to be strongly dependent on the initial energy spectrum, e.g., in case TP2, the kinetic energy is substantially smaller, and the late-wake vortices are less organized. The effect of the mean velocity field is negligible for mean kinetic energy (MKE) of the order 10% of the total kinetic energy and the evolution in this case is similar to a turbulent patch with the same initial energy spectrum. Increasing the MKE to 50% shows significant difference from the turbulent patch with the same initial energy spectrum during the initial stages of the evolution, but at later stages the evolution of turbulence statistics is similar. Both the turbulent patch and the momentumless wake show layering and formation of pancake eddies owing to buoyancy. Another objective of the paper is to compare the spatially evolving wake with the temporally evolving approximation when the initial near-wake condition of the temporal approximation is chosen to match the inflow of the spatially evolving model. The mean and turbulent flow statistics are found to agree well between the spatial and temporal computational models under these conditions
[1] Numerical simulations are performed to investigate the interaction of a semidiurnal internal wave (IW) beam with the nonuniform stratification of an upper ocean pycnocline. During the initial stage of the interaction, higher harmonics originate after reflection of the IW beam at the caustic and are trapped in the pycnocline. In cases where the pycnocline thickness is sufficiently large (approximately larger than twice the dominant wavelength in the beam), the incoming beam undergoes a parametric subharmonic instability (PSI) during refraction through the pycnocline at later time. Consequently, there is significant energy transfer to subharmonic motions that exhibit exponential growth with a rate of 2/3 day−1. The instability is demonstrated to be a resonant triad interaction by diagnostics in both wave frequency and wave number spaces. Smaller vertical scales originate during the PSI, lead to wave steepening, and produce convective overturns. This work identifies a potential mechanism that drives IW beam degradation during its propagation through the upper ocean.
Direct numerical simulation is used to investigate the nonlinear evolution of a horizontally oriented mixing layer with uniform stable stratification and coordinate system rotation about the vertical axis. The important dimensional parameters governing inviscid dynamics are maximum shear S(t), buoyancy frequency N, angular velocity of rotation Ω and characteristic shear thickness L(t). The effect of rotation rate, Ω, on the development of fluctuations in the shear layer is systematically studied in a regime of strong stratification. An instability mechanism, qualitatively distinct from the inertial instability, is found to deform columnar vortex cores in vertical planes for a strongly stratified rotating mixing layer. This mechanism emerges when centreline absolute vertical vorticity, ⟨ω3⟩(t)+2Ω, is nearly zero as predicted by the linear stability analysis in Part 1 (J. Fluid. Mech., vol. 703, 2012, pp. 29–48). When the initial rotation rate is moderately anticyclonic, strong destabilization and a cascade to small scales is observed, consistent with prior studies involving horizontally sheared flow in the presence of rotation. Examination of enstrophy budgets in cases which are initially inertially unstable reveal the importance of baroclinic torque in maintaining lateral enstrophy fluctuations substantially beyond the time when the flow becomes inertially stable. The cyclonic stratified cases show weak nonlinearity in vortex dynamics. At high Reynolds number, despite the strong stratification, the flow exhibits three-dimensional, nonlinear dynamics and significant vertical mixing except for cases where the rotation is stabilizing.
Direct numerical simulation is performed with a focus on the characterization of nonlinear dynamics during reflection of a plane internal wave at a sloping bottom. The effect of incoming wave amplitude is assessed by varying the incoming Froude number, Fr, and the effect of off-criticality is assessed by varying the slope angle in a range of near-critical values. At low Fr, the numerical results agree well with linear inviscid theory of near-critical internal wave reflection. With increasing Fr, the reflection process becomes nonlinear with the formation of higher harmonics and the initiation of fine-scale turbulence during the evolution of the reflected wave. Later in time, the wave response becomes quasi-steady with a systematic dependence of turbulence on the temporal and spatial phase. Convective instabilities are found to play a crucial role in the formation of turbulence during each cycle. The cycle evolution of flow statistics is studied in detail and qualitative differences between off-critical and critical reflection are identified. The parametric dependence of turbulence levels on Froude number and slope angle is calculated. Interestingly, at a given value of Fr, the turbulent kinetic energy (TKE) can be higher for somewhat off-critical reflection compared to exactly critical reflection. For a fixed slope angle, as the Froude number increases in the simulated cases, the fraction of the input wave energy converted into the turbulent kinetic energy and the fraction of the input wave power dissipated by turbulence also increase.
Direct and large eddy simulations are performed to study the internal waves generated by the oscillation of a barotropic tide over a model ridge of triangular shape. The objective is to go beyond linear theory and assess the role of nonlinear interactions including turbulence in situations with low tidal excursion number. The criticality parameter, defined as the ratio of the topographic slope to the characteristic slope of the tidal rays, is varied from subcritical to supercritical values. The barotropic tidal forcing is also systematically increased. Numerical results of the energy conversion are compared with linear theory and, in laminar flow at low forcing, they agree well in subcritical and supercritical cases but not at critical slope angle. In critical and supercritical cases with higher forcing, there are convective overturns, turbulence and significant reduction (as much as 25 %) of the radiated wave flux with respect to laminar flow results. Analysis of the baroclinic energy budget and spatial modal analysis are performed to understand the reduction. The near-bottom velocity is intensified at critical angle slope leading to a radiated internal wave beam as well as an upslope bore of cold water with a thermal front. In the critical case, the entire slope has turbulence while, in the supercritical case, turbulence originates near the top of the topography where the slope angle transitions through the critical value. The phase dependence of turbulence within a tidal cycle is examined and found to differ substantially between the ridge slope and the ridge top where the beams from the two sides cross.
Direct numerical simulation (DNS) is used to investigate the role of shear instabilities in turbulent mixing in a model of the upper Equatorial Undercurrent (EUC). The background flow consists of a westward-moving surface mixed layer above a stably stratified EUC flowing to the east. An important characteristic of the eastward current is that the gradient Richardson number Rig is larger than ¼. Nevertheless, the overall flow is unstable and DNS is used to investigate the generation of intermittent bursts of turbulent motions within the EUC region where Rig > ¼. In this model, an asymmetric Holmboe instability emerges at the base of the mixed layer, moves at the speed of the local velocity, and ejects wisps of fluid from the EUC upward. At the crests of the Holmboe waves, secondary Kelvin–Helmholtz instabilities develop, leading to three-dimensional turbulent motions. Vortices formed by the Kelvin–Helmholtz instability are occasionally ejected downward and stretched by the EUC into a horseshoe configuration creating intermittent bursts of turbulence at depth. Vertically coherent oscillations, with wavelength and frequency matching those of the Holmboe waves, propagate horizontally in the EUC where the turbulent mixing by the horseshoe vortices occurs. The oscillations are able to transport momentum and energy from the mixed layer downward into the EUC. They do not overturn the isopycnals, however, and, though correlated in space and time with the turbulent bursts, are not directly responsible for their generation. These wavelike features and intermittent turbulent bursts are qualitatively similar to the near-N oscillations and the deep-cycle turbulence observed at the upper flank of the Pacific Equatorial Undercurrent.
Direct numerical simulation (DNS) is used to investigate the evolution of intermittent patches of turbulence in a background flow with the gradient Richardson number, Rig , larger than the critical value of 0.25. The base flow consists of an unstable stratified shear layer (Rig <0.25) located on top of a stable shear layer (Rig >0.25), whose shear and stratification are varied. The unstable shear layer undergoes a Kelvin–Helmholtz shear instability that develops into billows. Vortices associated with the billows are pulled into the bottom shear layer and stretched by the local shear into a horseshoe configuration. The breakdown of the horseshoe vortices generates localized patches of turbulence. Three cases with different levels of shear and stratification, but with the same Rig , in the bottom shear layer are simulated to examine the popular hypothesis that mixing is determined by local Rig . In the case with largest shear and stratification, the vortices are less likely to penetrate the bottom layer and are quickly dissipated due to the strong stratification. In the case with moderate shear and stratification, vortices penetrate across the bottom layer and generate turbulence patches with intense dissipation rate. The case with the mildest level of shear and stratification shows the largest net turbulent mixing integrated over the bottom layer. Analysis of the turbulent kinetic energy budget indicates that the mean kinetic energy in the bottom layer contributes a large amount of energy to the turbulent mixing. In all cases, the mixing efficiency is elevated during the penetration of the vortices and has a value of approximately 0.35 when the turbulence in the patches decays.
Linear stability analysis is used to investigate instability mechanisms for a horizontally oriented hyperbolic tangent mixing layer with uniform stable stratification and coordinate system rotation about the vertical axis. The important parameters governing inviscid dynamics are maximum shear $S$, buoyancy frequency $N$, angular velocity of rotation $\Omega $ and characteristic shear thickness $L$. Growth rates associated with the most unstable modes are explored as a function of stratification strength $N/ S$ and rotation strength $2\Omega / S$. In the case of strong stratification, growth rates exhibit self-similarity of the form $\sigma ({k}_{1} L, S{k}_{3} L/ N, 2\Omega / S)$. In the case of rapid rotation we also observe self-similar scaling of growth rates with respect to the vertical wavenumber and rotation rate. The unstratified cases show $\sigma ({k}_{1} L, 2\vert \tilde {\Omega } \vert {k}_{3} L/ S)$ dependence while the strongly stratified cases show $\sigma ({k}_{1} L, 2\vert \tilde {\Omega } \vert {k}_{3} L/ N)$ dependence where $\tilde {\Omega } $ represents the difference between the angular velocity of rotation and least stable anticyclonic angular velocity, $\Omega = S/ 4$. Stratification was found to stabilize the inertial instability for weak anticyclonic rotation rates. Near the zero absolute vorticity state, stratification and rotation couple in a destabilizing manner increasing the range of unstable vertical wavenumbers associated with barotropic instability. In the case of rapid rotation, stratification prevents the stabilization of low ${k}_{1} $, high ${k}_{3} $ modes that occurs in a homogeneous fluid. The structure of certain unstable eigenmodes and the coupling between horizontal vorticity and density fluctuations are explored to explain how buoyancy stabilizes or destabilizes inertial and barotropic modes.
Direct numerical simulation is used to simulate the turbulent wake behind an accelerating axisymmetric self-propelled body in a stratified fluid. Acceleration is modelled by adding a velocity profile corresponding to net thrust to a self-propelled velocity profile resulting in a wake with excess momentum. The effect of a small to moderate amount of excess momentum on the initially momentumless self-propelled wake is investigated to evaluate if the addition of excess momentum leads to a large qualitative change in wake dynamics. Both the amount and shape of excess momentum are varied. Increasing the amount of excess momentum and/or decreasing the radial extent of excess momentum was found to increase the defect velocity, mean kinetic energy, shear in the velocity gradient and the wake width. The increased shear in the mean profile resulted in increased production of turbulent kinetic energy leading to an increase in turbulent kinetic energy and its dissipation. Slightly larger vorticity structures were observed in the late wake with excess momentum although the differences between vorticity structures in the self-propelled and 40 % excess momentum cases was significantly smaller than suggested by previous experiments. Buoyancy was found to preserve the doubly inflected velocity profile in the vertical direction, and similarity for the mean velocity and turbulent kinetic energy was found to occur in both horizontal and vertical directions. While quantitative differences were observed between cases with and without excess momentum, qualitatively similar evolution was found to occur.
Three-dimensional direct numerical simulations are performed to model an internal tidal beam at near-critical slope, and the phase dependence of turbulent processes is investigated. Convective instability leads to density overturns that originate in the upper flank of the beam and span the beam width of 6 m during flow reversal from downslope to upslope boundary motion. During this flow reversal event, negative turbulent production is observed signaling energy transfer from velocity fluctuations to the mean flow. In this note, we explain the mechanism underlying negative production.
[1] A numerical study based on large eddy simulation (LES) is performed to investigate near-bottom mixing processes in an internal wave beam over a critical slope. Transition to turbulence from an initial laminar state is followed by mixing events that occur at specific phases. Maximum turbulent kinetic energy and dissipation rate are found just after the zero velocity point when flow reverses from downslope to upslope motion. At this phase, convective instability leads to density overturns that originate in the upper flank of the beam and span the beam width of 60 m. Turbulence originating at the bottom and with smaller vertical extent is also present during the phases of peak upslope and downslope flow when the boundary layer shear is large. The present numerical simulations identify and characterize a process, internal wave beam at a critical slope during generation or after propagation from a nearby generation site, that may lead to the high turbulence levels, modulated at the tidal frequency, that is observed near the bottom in oceanic sites with near-critical topography.
The fine-scale response of a subsurface stable stratified jet subject to the forcing of surface wind stress and surface cooling is investigated using direct numerical simulation. The initial velocity profile consists of a symmetric jet located below a surface layer driven by a constant wind stress. The initial density profile is well-mixed in the surface layer and linearly stratified in both upper and lower flanks of the jet. The minimum value of the gradient Richardson number in the upper flank of the jet exceeds the critical value of 0.25 for linear shear instability. Broadband finite-amplitude fluctuations are introduced to the surface layer to initiate the simulation. Turbulence is generated in the surface layer and deepens into the jet upper flank. Internal waves generated by the turbulent surface layer are observed to propagate downward across the jet. The momentum flux carried by the waves is significantly smaller than the Reynolds shear stress extracted from the background velocity. The wave energy flux is also smaller than the turbulence production by mean shear. Ejections of fluid parcels by horseshoe-like vortices cause intermittent patches of intense dissipation inside the jet upper flank where the background gradient Richardson number is larger than 0.25. Drag due to the wind stress is smaller than the drag caused by turbulent stress in the flow. Analysis of the mean and turbulent kinetic energy budgets suggests that the energy input by surface forcing is considerably smaller than the energy extracted from the initially imposed background shear in the surface layer.
A numerical study is performed to investigate nonlinear processes during internal wave generation by the oscillation of a background barotropic tide over a sloping bottom. The focus is on the near-critical case where the slope angle is equal to the natural internal wave propagation angle and, consequently, there is a resonant wave response that leads to an intense boundary flow. The resonant wave undergoes both convective and shear instabilities that lead to turbulence with a broad range of scales over the entire slope. A thermal bore is found during upslope flow. Spectra of the baroclinic velocity, both inside the boundary layer and in the external region with free wave propagation, exhibit discrete peaks at the fundamental tidal frequency, higher harmonics of the fundamental, subharmonics and inter-harmonics in addition to a significant continuous part. The internal wave flux and its distribution between the fundamental and harmonics is obtained. Turbulence statistics in the boundary layer including turbulent kinetic energy and dissipation rate are quantified. The slope length is varied with the smaller lengths examined by direct numerical simulation (DNS) and the larger with large-eddy simulation (LES). The peak value of the near-bottom velocity increases with the length of the critical region of the topography. The scaling law that is observed to link the near-bottom peak velocity to slope length is explained by an analytical boundary-layer solution that incorporates an empirically obtained turbulent viscosity. The slope length is also found to have a strong impact on quantities such as the wave energy flux, wave energy spectra, turbulent kinetic energy, turbulent production and turbulent dissipation.
Direct numerical simulations of a horizontally oriented shear layer are performed with various values of uniform vertical stratification. The effects of stratification on statistics such as turbulentReynolds stress,turbulent kinetic energy, and viscous dissipation are examined. Comparisons between turbulent kinetic energy and scalar variance budgets among the unstratified and stratified cases are presented. In the stratified cases, spatially sparse coherent structures emerge, affecting transport of momentum and density. Additional physical insight gained from quantification of the influence of the coherent structures on the flow statistics such as turbulent scalar transport is summarized. Vortex eduction revealed vortical structure similar to that observed in prior investigations of the zigzag instability.
Direct numerical simulation is employed to study the effect of the Prandtl number, Pr=ν/α with ν the molecular viscosity and α the molecular diffusivity, on a turbulent wake in a stratified fluid. Simulations were conducted at a Reynolds number of 10 000, Re=UD/ν with U the velocity of the body and D the diameter of the body, for a range of Prandtl numbers: 0.2, 1, and 7. The simulations were run from x/D=6 to x/D=1200, a range that encompasses the near, intermediate, and far wake. Mean quantities such as wake dimensions and defect velocity were found to be weakly affected by Prandtl number, the same result was observed for vorticity as well. The Prandtl number has a strong effect on the density perturbation field and this results in a number of differences in quantities such as the total energy of the wake, wave flux, scalar and turbulent dissipation, mixing efficiency, spectral distribution of energy in the density and velocity fields, and the transfer of energy between kinetic and potential modes. The approximation Pr=1 for the ocean is often used in practice. As the qualitative behavior of the large-scale features was the same for the three cases, we conclude that Pr=1 is a reasonable approximation for the Pr=7 case in stratified wake simulations, given the significantly higher computational cost required at large Prandtl number.
Three-dimensional direct numerical simulations are performed to examine nonlinear processes during the generation of internal tides on a model continental slope. An intense boundary flow is generated in the critical case where the slope angle is equal to the natural internal wave propagation angle. Wave steepening, that drives spanwise wave breaking via convective instability, occurs. Turbulence is present along the entire extent of the near-critical region of the slope. The turbulence is found to have a strong effect on the internal wave beam by distorting its near-slope structure. A complicated wave field with a broadband frequency spectrum is found. This work explains the formation of boundary turbulence during the generation of internal tides in the regime of low excursion numbers.
Direct numerical simulations (DNS) of axisymmetric wakes with canonical towed and self-propelled velocity profiles are performed at Re = 50 000 on a grid with approximately 2 billion grid points. The present study focuses on a comparison between towed and self-propelled wakes and on the elucidation of buoyancy effects. The development of the wake is characterized by the evolution of maxima, area integrals and spatial distributions of mean and turbulence statistics. Transport equations for mean and turbulent energies are utilized to help understand the observations. The mean velocity in the self-propelled wake decays more rapidly than the towed case due to higher shear and consequently a faster rate of energy transfer to turbulence. Buoyancy allows a wake to survive longer in a stratified fluid by reducing the xs3008u1′u3′xs3009 correlation responsible for the mean-to-turbulence energy transfer in the vertical direction. This buoyancy effect is especially important in the self-propelled case because it allows regions of positive and negative momentum to become decoupled in the vertical direction and decay with different rates. The vertical wake thickness is found to be larger in self-propelled wakes. The role of internal waves in the energetics is determined and it is found that, later in the evolution, they can become a dominant term in the balance of turbulent kinetic energy. The non-equilibrium stage, known to exist for towed wakes, is also shown to exist for self-propelled wakes. Both the towed and self-propelled wakes, at Re = 50000, are found to exhibit a time span when, although the turbulence is strongly stratified as indicated by small Froude number, the turbulent dissipation rate decays according to inertial scaling.
Scalar transport and mixing by active turbulence in a high Reynolds number inhomogeneous stratified shear layer are investigated using three-dimensional Direct Numerical Simulation. Two density profiles are considered: (i) two layers of homogenous fluid with different density, namely the two-layer case, and (ii) a continuously stratified background ambient, namely the Jd case. The evolution of the mixing layer includes shear instability, formation of Kelvin–Helmholtz rollers, transition to turbulence, fully developed active turbulence, and, finally, decay toward a laminar state. In the Jd case, internal gravity waves carrying momentum and energy are observed to propagate away from the shear layer. Although different during the initial evolution, the eddy diffusivity and mixing efficiency when plotted as a function of buoyancy, Reynolds number takes similar values between the two cases later in time during the stage when turbulence decays. During this stage, the mixing efficiency computed based on the buoyancy flux is approximately 0.35, while the mixing efficiency estimated from the scalar dissipation is approximately 0.4. Parameterization of the eddy diffusivity in terms of Reynolds numbers and gradient Richardson number is also discussed.
Large-eddy simulations (LES) of heated and cooled plane and round variable-density jets were conducted using a variety of density ratios s=ρj/ρcos=ρj/ρco, which relates the jet nozzle density ρjρj to the freestream density ρcoρco. The initial momentum flux was kept constant for better comparison of the resulting data. Both simulations confirm experimental results, in that the jet half-width grows linearly with streamwise coordinate x and the lighter jets decay much faster than the heavy ones. The centerline velocity decay is however different between the plane and round geometries. Whereas the round jets exhibits a decay with 1/x1/x for all density ratios s , there seem to be two self-similar scalings in plane jets, in the limit of small and large density ratios s . In the limit of small s or for incompressible flow, UcUc scales as View the MathML sourceUc∼1/x, for strongly heated jets on the other hand we find Uc∼1/xUc∼1/x. A mixed scaling is proposed and shown to work nicely for both small and large density ratios s . For the round jet simulations, on the other hand, scaling x and UcUc by s-1/4s-1/4 ( Chen and Rodi, 1980) collapses the round jet data. Furthermore, it is shown that the streamwise growth in the mean density or the decay of the velocity fluctuations in the quasi-self-similar region, is stronger for round jets. The round jet simulation with a density ratio of s=0.14s=0.14 is seen to develop under a global instability. To the author’s knowledge, this is the first LES of a globally unstable round jet at a density ratio of s=0.14s=0.14. The frequency agrees excellently with experimental data and with the new scaling proposed by Hallberg and Strykowski (2006).
Direct numerical simulations are performed to investigate the interaction between a stably stratified jet and internal gravity waves from an adjacent shear layer with mild stratification. Results from two simulations are presented: one with the jet located far from the shear layer (far jet) and the other with the shear layer right on top of the jet (near jet). The near jet problem is motivated by velocity and stratification profiles observed in equatorial undercurrents. In the far case, internal waves excited by the Kelvin–Helmholtz (K-H) rollers do not penetrate the jet. They are reflected and trapped in the region between the shear layer and the jet and lead to little dissipation. In the near case, internal waves with wavelength larger than that of the K-H rollers are found in and below the jet. Pockets of hot fluid, associated with horseshoe vortices that originate from the shear layer, penetrate into the jet region, initiate turbulence and disrupt the internal wave field. Coherent patches of enhanced dissipation moving with the mean velocity are observed. The dissipation in the stably stratified near jet is large, up to three orders of magnitude stronger than that in the propagating wave field or the jet of the far case.
A numerical study based on large eddy simulation is performed to investigate a bottom boundary layer under an oscillating tidal current. The focus is on the boundary layer response to an external stratification. The thermal field shows a mixed layer that is separated from the external stratified fluid by a thermocline. The mixed layer grows slowly in time with an oscillatory modulation by the tidal flow. Stratification strongly affects the mean velocity profiles, boundary layer thickness and turbulence levels in the outer region although the effect on the near-bottom unstratified fluid is relatively mild. The turbulence is asymmetric between the accelerating and decelerating stages. The asymmetry is more pronounced with increasing stratification. There is an overshoot of the mean velocity in the outer layer; this jet is linked to the phase asymmetry of the Reynolds shear stress gradient by using the simulation data to examine the mean momentum equation. Depending on the height above the bottom, there is a lag of the maximum turbulent kinetic energy, dissipation and production with respect to the peak external velocity and the value of the lag is found to be influenced by the stratification. Flow instabilities and turbulence in the bottom boundary layer excite internal gravity waves that propagate away into the ambient. Unlike the steady case, the phase lines of the internal waves change direction during the tidal cycle and also from near to far field. The frequency spectrum of the propagating wave field is analysed and found to span a narrow band of frequencies clustered around 45°.
This article employs LES to simulate temporal mixing layers with Mach numbers ranging from M c = 0.3 to M c = 1.2. A form of approximate deconvolution together with a dynamic Smagorinsky subgrid model are employed as subgrid models. A large computational domain is used along with relatively good resolution. The LES results regarding growth rate, turbulence levels, turbulence anisotropy, and pressure–strain correlation show excellent agreement with those available from previous experimental and DNS results of the same flow configuration, underlining the effectiveness and accuracy of properly conducted LES. Coherent structures during the transitional stage change from spanwise aligned rollers to streamwise-aligned thinner vortices at high Mach number. In the quasi-self-similar turbulent stage, the resolved-scale vorticity is more isotropic at higher M c , and its vertical correlation length scale is smaller. The ratio of the vertical integral length scale of streamwise velocity fluctuation to a characteristic isotropic estimate is found to decrease with increasing M c . Thus, compressibility leads to increased spatial decorrelation of turbulence which is one reason for the reduction in pressure–strain correlation with increasing M c . The balance of the resolved-scale fluctuating vorticity is examined, and it is observed that the linear production by mean shear becomes less important compared to nonlinear vortex stretching at high M c . A spectral decomposition of the pressure fluctuations into low- and intermediate-to-high-wave numbers is performed. The low-wave number part of the pressure field is found not to correlate with the strain field, although it does have a significant contribution to the r.m.s of the fluctuating pressure. As a consequence, the pressure–strain correlation can be analyzed using a simplified Green’s function for the Poisson equation as is demonstrated here using the LES data.
Dinoflagellate population growth is inhibited by fluid motion, which is typically characterized by some average flow property, regardless if the flow is steady or unsteady. This study compares the effect of fully characterized steady and unsteady flow on net population growth of the red tide dinoflagellate Lingulodinium polyedrum. The unsteady flow fields were generated using oscillatory laminar Couette flow and characterized analytically to provide complete knowledge of the fluid shear exposure over space and time throughout the chamber. Experimental conditions were selected so all cells experienced a similar shear exposure regardless of their position within the chamber. Unsteady flow with maximum shears of 6.4 s-1 and 6.7 s-1 and an average absolute shear of 4 s-1, comparable with levels found at the ocean surface on a windy day, resulted in higher levels of growth inhibition than for steady Couette flow with shears of 4 and 8 s-1. Over the parameter space studied, growth inhibition increased with increasing treatment duration (5-120 min) but was insensitive to oscillation period (60-600 s) or whether the unsteady flow changed in direction. These results indicate that over the parameter space studied, unsteady flow is more inhibitory to net growth than steady flow, for the same average flow conditions, and demonstrate that flow characterization on the basis only of average flow properties is inadequate for comparing population growth in unsteady and steady flows.
Direct numerical simulations (DNS) are performed to investigate the behaviour of a weakly stratified shear layer in the presence of a strongly stratified region beneath it. Both, coherent Kelvin–Helmholtz (KH) rollers and small-scale turbulence, are observed during the evolution of the shear layer. The deep stratification measured by the Richardson number Jd is varied to study its effect on the dynamics. In all cases, a pycnocline is found to develop at the edges of the shear layer. The region of maximum shear shifts downward with increasing time. Internal waves are excited, initially by KH rollers, and later by small-scale turbulence. The wave field generated by the KH rollers is narrowband and of stronger amplitude than the broadband wave field generated by turbulence. Linear theory based on Doppler-shifted frequency of the KH mode is able to predict the angle of the internal wave phase lines during the direct generation of internal waves by KH rollers. Waves generated by turbulence are relatively weaker with a broader range of excitation angles which, in the deep region, tend towards a narrower band. The linear theory that works for the internal waves excited by KH rollers does not work for the turbulence generated waves. The momentum transported by the internal waves into the interior can be large, about 10% of the initial momentum in the shear layer, when Jd xs2243 0.25. Integration of the turbulent kinetic energy budget in time and over the shear layer thickness shows that the energy flux can be up to 17% of the turbulent production, 33% of the turbulent dissipation rate and 75% of the buoyancy flux. These numbers quantify the dynamical importance of internal waves. In contrast to linear theory where the effect of deep stratification on the shear layer instabilities has been found to be weak, the present nonlinear simulations show that the evolution of the shear layer is significantly altered because of the significant momentum and energy carried away by the internal waves.
A stratified bottom Ekman layer over a nonsloping, rough surface is studied using a three-dimensional unsteady large eddy simulation to examine the effects of an outer layer stratification on the boundary layer structure. When the flow field is initialized with a linear temperature profile, a three-layer structure develops with a mixed layer near the wall separated from a uniformly stratified outer layer by a pycnocline. With the free-stream velocity fixed, the wall stress increases slightly with the imposed stratification, but the primary role of stratification is to limit the boundary layer height. Ekman transport is generally confined to the mixed layer, which leads to larger cross-stream velocities and a larger surface veering angle when the flow is stratified. The rate of turning in the mixed layer is nearly independent of stratification, so that when stratification is large and the boundary layer thickness is reduced, the rate of veering in the pycnocline becomes very large. In the pycnocline, the mean shear is larger than observed in an unstratified boundary layer, which is explained using a buoyancy length scale, u*/N(z). This length scale leads to an explicit buoyancy-related modification to the log law for the mean velocity profile. A new method for deducing the wall stress based on observed mean velocity and density profiles is proposed and shows significant improvement compared to the standard profile method. A streamwise jet is observed near the center of the pycnocline, and the shear at the top of the jet leads to local shear instabilities and enhanced mixing in that region, despite the fact that the Richardson number formed using the mean density and shear profiles is larger than unity.
A steady Ekman layer with a thermally stratified outer flow and an adiabatic boundary condition at the lower wall is studied using direct numerical simulation (DNS) and large eddy simulation (LES). An initially linear temperature profile is mixed by turbulence near the wall, and a stable thermocline forms above the mixed layer. The thickness of the mixed layer is reduced by the outer layer stratification. Observations from the DNS are used to evaluate the performance of the LES model and to examine the resolution requirements. A resolved LES and a near-wall model LES (NWM–LES) both compare reasonably well with the DNS when the thermal field is treated as a passive scalar. When buoyancy effects are included, the LES mean velocity and temperature profiles also agree well with the DNS. However, the NWM–LES does not sufficiently account for the overturning scales responsible for entrainment at the top of the mixed layer. As a result, the turbulent heat flux and the rate of change of the mixed layer temperature are significantly underestimated in the NWM–LES. In order to accurately simulate the boundary layer growth, the motions responsible for entrainment must either be resolved or more accurately represented in improved subgrid-scale models.
Direct numerical simulation is used to investigate effects of heat release and compressibility on mixing-layer turbulence during a period of self-similarity. Temporally evolving mixing layers are analysed at convective Mach numbers between 0.15 and 1.1 and in a Reynolds number range of 15000 to 35000 based on vorticity thickness. The turbulence inhibiting effects of heat release are traced back to mean density variations using an analysis of the fluctuating pressure field based on a Green's function.
Internal gravity waves excited by the turbulent motions in a bottom Ekman layer are examined using large-eddy simulation. The outer flow is steady and uniformly stratified while the density gradient is set to zero at the flat lower wall. After initializing with a linear density profile, a mixed layer forms near the wall separated from the ambient stratification by a pycnocline. Two types of internal wave are observed. Waves with frequencies larger than the free-stream buoyancy frequency are seen in the pycnocline, and vertically propagating internal waves are observed in the outer layer with characteristic frequency and wavenumber spectra. Since a signature of the pycnocline waves is observed in the frequency spectrum of the mixed layer, these waves may affect the boundary-layer turbulence. The dominant outer-layer waves have a group velocity directed 35-60° from the vertical axis, which is consistent with previous laboratory studies. The energy flux associated with the radiated waves is small compared to the integrated dissipation in the boundary layer, but is of the same order as the integrated buoyancy flux. A linear model is proposed to estimate the decay in wave amplitude owing to viscous effects. Starting from the observed wave amplitudes at the bottom of the pycnocline, the model prediction for the spectral distribution of the outer layer wave amplitude compares favourably with the simulation results.
Direct numerical simulations of a stratified shear layer are performed for several different values of Reynolds, bulk Richardson, and Prandtl numbers. Unlike previous numerical studies, the initial perturbations are turbulent. These initial broadband perturbations do not allow the formation of distinct coherent structures such as Kelvin-Helmholtz rollers and streamwise vortices found in previous studies. In the absence of stratification, the shear layer thickness grows linearly and fully developed turbulence is achieved with mean velocities, turbulence intensities, and turbulent kinetic energy budgets that agree well with previous experimental and numerical data. When buoyancy is included, the shear layer grows to an asymptotic thickness, and the corresponding bulk Richardson number, Rib, is within the range, 0.32±0.06, found in previous studies. The apparent scatter in the evolution of Rib is shown to have a systematic dependence on Reynolds and Prandtl numbers. A detailed description of buoyancy effects on turbulence energetics, transport, and mixing is presented. The Reynolds shear stress, ⟨u′1u′3⟩, is significantly reduced by buoyancy, thus decreasing the shear production of turbulence. Owing to buoyancy, gradients in the vertical direction tend to be larger than other gradients in the fluctuating velocity and density fields. However, this anisotropy of the gradients is lower when the Reynolds number increases. Coherent finger-like structures are identified in the density field at late time and their vertical extent obtained by a scaling analysis.
The influence of compressibility on the rapid pressure–strain rate tensor is investigated using the Green’s function for the wave equation governing pressure fluctuations in compressible homogeneous shear flow. The solution for the Green’s function is obtained as a combination of parabolic cylinder functions; it is oscillatory with monotonically increasing frequency and decreasing amplitude at large times, and anisotropic in wave-vector space. The Green’s function depends explicitly on the turbulent Mach number M t , given by the root mean square turbulent velocity fluctuations divided by the speed of sound, and the gradient Mach number M g , which is the mean shear rate times the transverse integral scale of the turbulence divided by the speed of sound. Assuming a form for the temporal decorrelation of velocity fluctuations brought about by the turbulence, the rapid pressure–strain rate tensor is expressed exactly in terms of the energy (or Reynolds stress) spectrum tensor and the time integral of the Green’s function times a decaying exponential. A model for the energy spectrum tensor linear in Reynolds stress anisotropies and in mean shear is assumed for closure. The expression for the rapid pressure–strain correlation is evaluated using parameters applicable to a mixing layer and a boundary layer. It is found that for the same range of M t there is a large reduction of the pressure–strain correlation in the mixing layer but not in the boundary layer. Implications for compressible turbulence modeling are also explored.
A priori and a posteriori studies for large eddy simulation of the compressible turbulent infinitely fast reacting shear layer are presented. The filtered heat release appearing in the energy equation is unclosed and the accuracy of different models for the filtered scalar dissipation rate and the conditional filtered scalar dissipation rate of the mixture fraction in closing this term is analyzed. The effect of different closures of the subgrid transport of momentum, energy and scalars on the modeling of the filtered heat release via the resolved fields is also considered. Three explicit models of these subgrid fluxes are explored, each with an increasing level of reconstruction and all of them regularized by a Smagorinsky-type term. It is observed that a major part of the error in the prediction of the conditional filtered scalar dissipation comes from the unsatisfactory modeling of the filtered dissipation itself. The error can be substantial in the turbulent fluctuation (rms) of the dissipation fields. It is encouraging that all models give good predictions of the mean and rms density in a posteriori LES of this flow with realistic heat release corresponding to large density change. Although a posteriori results show a small sensitivity to subgrid modeling errors in the current problem, extinction–reignition phenomena involving finite-rate chemistry would demand more accurate modeling of the dissipation rates. A posteriori results also show that the resolved fields obtained with the approximate reconstruction using moments (ARM) agree better with the filtered direct numerical simulation since the level of reconstruction in the modeled subfilter fluxes is increased.
The evolution of a stratified shear layer with mean shear in the horizontal direction, orthogonal to gravity, is numerically investigated with focus on the structural organization of the vorticity and density fields. Although the Reynolds number of the flow increases with time, facilitating instabilities and turbulence, the bulk Richardson number signifying the level of stratification also increases. Remarkably rich dynamics is found: turbulence; the emergence of coherent core/braid regions from turbulence; formation of a lattice of dislocated vortex cores connected by thin horizontal sheets of collapsed density and vorticity; density-driven intrusions at the edges of the shear layer; and internal wave generation and propagation. Stratification introduces significant vertical variability although it inhibits the vertical velocity. The molecular dissipation of turbulent kinetic energy and of turbulent potential energy are both found to be substantial even in the case with highest stratification, and primarily concentrated in thin horizontal sheets. The simulation data are used to help explain how buoyancy induces the emergence of columnar vortex cores from turbulence and then dislocates these cores to eventually form a lattice of ‘pancake’ eddies connected by thin sheets with large vertical shear (horizontal vorticity) and density gradient.
Large eddy simulation has been used to study flow in an open channel with stable stratification imposed at the free surface by a constant heat flux and an adiabatic bottom wall. This leads to a stable pycnocline overlying a well-mixed turbulent region near the bottom wall. The results are contrasted with studies in which the bottom heat flux is nonzero, a difference analogous to that between oceanic and atmospheric boundary layers. Increasing the friction Richardson number, a measure of the relative importance of the imposed surface stratification with respect to wall-generated turbulence, leads to a stronger, thicker pycnocline which eventually limits the impact of wall-generated turbulence on the free surface. Increasing stratification also leads to an increase in the pressure-driven mean streamwise velocity and a concomitant decrease in the skin friction coefficient, which is, however, smaller than in the previous channel flow studies where the bottom buoyancy flux was nonzero. It is found that the turbulence in any given region of the flow can be classified into three regimes (unstratified, buoyancy-affected, and buoyancy-dominated) based on the magnitude of the Ozmidov length scale relative to a vertical length characterizing the large scales of turbulence and to the Kolmogorov scale. Since stratification does not strongly influence the near-wall turbulent production in the present configuration, the behavior of the buoyancy flux, turbulent Prandtl number, and mixing efficiency is qualitatively different from that seen in stratified shear layers and in channel flow with fixed temperature walls, and, furthermore, collapse of quantities as a function of gradient Richardson number is not observed. The vertical Froude number is a better measure of stratified turbulence in the upper portion of the channel where buoyancy, by providing a potential energy barrier, primarily affects the transport of turbulent patches generated at the bottom wall. The characteristics of free-surface turbulence including the kinetic energy budget and pressure-strain correlations are examined and found to depend strongly on the surface stratification.
Turbulence developed from Rayleigh-Taylor instability between two compressible miscible fluids in an unbounded domain is addressed in this paper. It is demonstrated that the turbulentMach number in the turbulent core has an upper bound, independent of the density ratio under a broad range of initial mean configurations. The initial thermodynamic state of the system determines the amount of potential energy per unit mass involved in the turbulent mixing stage, and thus the characteristic level of turbulent fluctuations that is achievable is linked to the characteristicspeed of sound such that the turbulentMach number is limited. For the particular case of an ideal gas, this bound on the turbulentMach number is found to be between 0.25 and 0.6, depending on the particular initial thermodynamic state. Hence, intrinsic compressibility effects (those owing to large Mach number) are likely to be limited in the turbulent stage of a pure Rayleigh-Taylor problem. This result is confirmed by large-eddy simulations(LES) of systems with density jumps at the interface of 3:1, a density ratio for which there is extensive data available in the literature. The LES predictions of the mixing depth growth and overall mixing agree with results previously obtained in incompressible configurations with a negligibly small Mach number, and the data fully describing the Reynolds stresses and the budget of the (resolved) turbulent kinetic energy equation are provided.
The problem of flame propagation in imperfectly premixed mixtures—mixtures of reactants with variable composition—is considered in this numerical study. We carry out two-dimensional direct numerical simulations of a flame propagating in a globally lean fuel-oxidizer mixture with imposed velocity and composition fluctuations of various intensities. The configuration adopted is that of a flame front interacting with spatially evolving fluctuations, and the characteristic scales of the domain and of the fluctuations imposed are significantly larger than the characteristic thickness of the flame, to account for important flame dynamics such as the hydrodynamic instability. One-step chemistry and Fick’s diffusion law are considered, along with unity Lewis number assumption for all the species. It is observed, in agreement with previous results, that relatively weak fluctuations in composition alone may lead to a large increase in flame length and burning rate. The hydrodynamic instability caused by gas expansion, catalyzed by the composition fluctuations interacting with the flame, is found to be responsible for the flame length enhancement. It is observed as well that the relative importance of this effect diminishes as the velocity fluctuations present become more intense, and that composition fluctuations have a small impact on flame length for these cases. It is additionally found that, with increasing intensity of composition fluctuations, there is eventually a reduction of burning rate per unit length of flame which leads, consequently, to a weak reduction of overall burning rate for the largest velocity fluctuation intensities covered by this study.
Turbulence in supersonic channel flow is studied using direct numerical simulation. The ability of outer and inner scalings to collapse profiles of turbulent stresses onto their incompressible counterparts is investigated. Such collapse is adequate with outer scaling when sufficiently far from the wall, but not with inner scaling. Compressibility effects on the turbulent stresses, their anisotropy, and their balance equations are identified. A reduction in the near-wall pressure–strain, found responsible for the changed Reynolds-stress profiles, is explained using a Green's-function-based analysis of the pressure field.
Stratified environmental flows near boundaries can have a horizontal mean shear component, orthogonal to the vertical mean density gradient. Vertical transport, against the stabilizing force of gravity, is possible in such situations if three-dimensional turbulence is sustained by the mean shear. A model problem, water with a constant mean density gradient flowing in a channel between parallel vertical walls, is examined here using the technique of large eddy simulation (LES). It is found that, although the mean shear is horizontal, the fluctuating velocity field has significant vertical shear and horizontal vorticity, thereby causing small-scale vertical mixing of the density field. The vertical stirring is especially effective near the boundaries where the mean shear is large and, consequently, the gradient Richardson number is small. The mean stratification is systematically increased between cases in our study and, as expected, the buoyancy flux correspondingly decreases. Even so, horizontal mean shear is found to be more effective than the well-studied case of mean vertical shear in inducing vertical buoyancy transport as indicated by generally larger values of vertical eddy diffusivity and mixing efficiency.
Large-eddy simulation of combustion problems involves highly nonlinear terms that, when filtered, result in a contribution from subgrid fluctuations of scalars, Z, to the dynamics of the filtered value. This subgrid contribution requires modeling. Reconstruction models try to recover as much information as possible from the resolved field Z̄, based on a deconvolution procedure to obtain an intermediate field ZM. The approximate reconstruction using moments (ARM) method combines approximate reconstruction, a purely mathematical procedure, with additional physics-based information required to match specific scalar moments, in the simplest case, the Reynolds-averaged value of the subgrid variance. Here, results from the analysis of the ARM model in the case of a spatially evolving turbulent plane jet are presented. A priori and a posteriori evaluations using data from direct numerical simulation are carried out. The nonlinearities considered are representative of reacting flows: power functions, the dependence of the density on the mixture fraction (relevant for conserved scalar approaches) and the Arrhenius nonlinearity (very localized in Z space). Comparisons are made against the more popular beta probability density function (PDF) approach in the a priorianalysis, trying to define ranges of validity for each approach. The results show that the ARM model is able to capture the subgrid part of the variance accurately over a wide range of filter sizes and performs well for the different nonlinearities, giving uniformly better predictions than the beta PDF for the polynomial case. In the case of the density and Arrhenius nonlinearities, the relative performance of the ARM and traditional PDF approaches depends on the size of the subgrid variance with respect to a characteristic scale of each function. Furthermore, the sources of error associated with the ARM method are considered and analytical bounds on that error are obtained.
Mixing of a conserved scalar representing the mixture fraction, of primary importance in modelling non-premixed turbulent combustion, is studied by direct numerical simulation (DNS) in strongly turbulent planar shear layers both with and without heat release at a reaction sheet. For high heat release, typical of hydrocarbon combustion, the mixing is found to be substantially different than without heat release. The probability density function of the scalar and the conditional rate of scalar dissipation are affected by the heat release in such a way that the heat release substantially decreases the overall reaction rate. To help clarify implications of the assumptions underlying popular models for interaction between turbulence and chemistry, the local structure of the scalar dissipation rate at the reaction sheet is extracted from the DNS database. The applicability of flamelet models for the rate of scalar dissipation is examined. To assist in modelling, a characteristic length scale is defined, representing the distance around the reaction sheet over which the scalar field is locally linear, and statistical properties of this length scale are investigated. This length scale can be used in studying values of the rate of scalar dissipation that mark the boundary between flames that feel a constant scalar dissipation field and those that do not.
Direct numerical simulation of uniform shear flow is used to study the anisotropy of fluctuating motion in a stably stratified medium with uniform mean shear. Turbulence is found to be three dimensional over a wide range of gradient Richardson numbers in the two flows investigated here: vertical mean shear and horizontal mean shear . The role of the turbulent Froude number in establishing the regime of stratified turbulence observed here is described. The fluctuating velocity gradients are examined. The vertical of streamwise velocity is found to dominate the other components of turbulent dissipation in both horizontal and vertical shear flows.
Boundary-forced stratified turbulence is studied in the prototypical case of turbulent channel flow subject to stable stratification. The large-eddy simulation approach is used with a mixed subgrid model that involves a dynamic eddy viscosity component and a scale-similarity component. After an initial transient, the flow reaches a new balanced state corresponding to active wall-bounded turbulence with reduced vertical transport which, for the cases in our study with moderate-to-large levels of stratification, coexists with internal wave activity in the core of the channel. A systematic reduction of turbulence levels, density fluctuations and associated vertical transport with increasing stratification is observed. Countergradient buoyancy flux is observed in the outer region for sufficiently high stratification. Mixing of the density field in stratified channel flow results from turbulent events generated near the boundaries that couple with the outer, more stable flow. The vertical density structure is thus of interest for analogous boundary-forced mixing situations in geophysical flows. It is found that, with increasing stratification, the mean density profile becomes sharper in the central region between the two turbulent layers at the upper and lower walls, similar to observations in field measurements as well as laboratory experiments with analogous density-mixing situations. Channel flow is strongly inhomogeneous with alternative choices for the Richardson number. In spite of these complications, the gradient Richardson number, Rig, appears to be the important local determinant of buoyancy effects. All simulated cases show that correlation coefficients associated with vertical transport collapse from their nominal unstratified values over a narrow range, 0.15 < Rig < 0.25. The vertical turbulent Froude number, Frw, has an O(1) value across most of the channel. It is remarkable that stratified channel flow, with such a large variation of overall density difference (factor of 26) between cases, shows a relatively universal behaviour of correlation coefficients and vertical Froude number when plotted as a function of Rig. The visualizations show wavy motion in the core region where the gradient Richardson number, Rig, is large and low-speed streaks in the near-wall region, typical of unstratified channel flow, where Rig is small. It appears from the visualizations that, with increasing stratification, the region with wavy motion progressively encroaches into the zone with active turbulence; the location of Rig [simeq R: similar, equals] 0.2 roughly corresponds to the boundary between the two zones.
Direct simulations of the turbulent shear layer are performed for subsonic to supersonic Mach numbers. Fully developed turbulence is achieved with profiles of mean velocity and turbulence intensities that agree well with laboratory experiments. The thickness growth rate of the shear layer exhibits a large reduction with increasing values of the convective Mach number, Mc. In agreement with previous investigations, it is found that the normalized pressure–strain term decreases with increasing Mc, which leads to inhibited energy transfer from the streamwise to cross-stream fluctuations, to the reduced turbulence production observed in DNS, and, finally, to reduced turbulence levels as well as reduced growth rate of the shear layer. An analysis, based on the wave equation for pressure, with supporting DNS is performed with the result that the pressure–strain term decreases monotonically with increasing Mach number. The gradient Mach number, which is the ratio of the acoustic time scale to the flow distortion time scale, is shown explicitly by the analysis to be the key quantity that determines the reduction of the pressure–strain term in compressible shear flows. The physical explanation is that the finite speed of sound in compressible flow introduces a finite time delay in the transmission of pressure signals from one point to an adjacent point and the resultant increase in decorrelation leads to a reduction in the pressure–strain correlation. The dependence of turbulence intensities on the convective Mach number is investigated. It is found that all components decrease with increasing Mc and so does the shear stress. DNS is also used to study the effect of different free-stream densities parameterized by the density ratio, s = ρ2/ρ1, in the high-speed case. It is found that changes in the temporal growth rate of the vorticity thickness are smaller than the changes observed in momentum thickness growth rate. The momentum thickness growth rate decreases substantially with increasing departure from the reference case, s = 1. The peak value of the shear stress, uv, shows only small changes as a function of s. The dividing streamline of the shear layer is observed to move into the low-density stream. An analysis is performed to explain this shift and the consequent reduction in momentum thickness growth rate.
Turbulent plane jets are prototypical free shear flows of practical interest in propulsion, combustion and environmental flows. While considerable experimental research has been performed on planar jets, very few computational studies exist. To the authors' knowledge, this is the first computational study of spatially evolving three-dimensional planar turbulent jets utilizing direct numerical simulation. Jet growth rates as well as the mean velocity, mean scalar and Reynolds stress profiles compare well with experimental data. Coherency spectra, vorticity visualization and autospectra are obtained to identify inferred structures. The development of the initial shear layer instability, as well as the evolution into the jet column mode downstream is captured well. The large- and small-scale anisotropies in the jet are discussed in detail. It is shown that, while the large scales in the flow field adjust slowly to variations in the local mean velocity gradients, the small scales adjust rapidly. Near the centreline of the jet, the small scales of turbulence are more isotropic. The mixing process is studied through analysis of the probability density functions of a passive scalar. Immediately after the rollup of vortical structures in the shear layers, the mixing process is dominated by large-scale engulfing of fluid. However, small-scale mixing dominates further downstream in the turbulent core of the self-similar region of the jet and a change from non-marching to marching PDFs is observed. Near the jet edges, the effects of large-scale engulfing of coflow fluid continue to influence the PDFs and non-marching type behaviour is observed.
In applications of large eddy simulation of turbulent flows, subgrid models are often required for closure of strongly nonlinear functions of a scalar. The Arrhenius dependence of the reaction rate on temperature, T, the T4 dependence of radiationheat transfer, as well as the species mass fractions and temperature dependence on the mixture fraction in solutions of the strained laminar flamelet model are among some of the problems of interest. A moment-based reconstruction methodology is proposed here in which the scalar field is estimated by an approximate deconvolution operation but, unlike the usual deconvolution operation with given coefficients, the coefficients in the expansion are obtained by requiring that the statistical filtered moments of the scalar field up to a certain order are matched. The estimated scalar field is then used as a surrogate for the exact scalar field to directly calculate the subgrid contribution. Tests of the proposed approach are performed by using our direct numerical simulation database of scalar transport in a turbulent shear layer using two filter sizes: 12 points and 6 points per vorticity thickness. It is found that a simple moment-based model with one coefficient performs well for polynomial nonlinearities. The performance of the model in the case of an exponential Arrhenius-type nonlinearity is generally good and can be very good depending on the stoichiometric mixture fraction and the filter size.
Predicting sound radiated by turbulence is of interest in aeroacoustics, hydroacoustics, and combustionnoise. Significant improvements in computertechnology have renewed interest in applying numerical techniques to predict sound from turbulent flows. One such technique is a hybrid approach in which the turbulence is computed using a method such as direct numerical simulation (DNS) or large eddy simulation(LES), and the sound is calculated using an acoustic analogy. In this study, sound from a turbulent flow is computed using DNS, and the DNS results are compared with acoustic-analogy predictions for mutual validation. The source considered is a three-dimensional region of forced turbulence which has limited extent in one coordinate direction and is periodic in the other two directions. Sound propagates statistically as a plane wave from the turbulence to the far field. The cases considered here have a small turbulentMach number so that the source is spatially compact; that is, the turbulence integral scale is much smaller than the acoustic wavelength. The scaling of the amplitude and frequency of the far-field sound for the problem considered are derived in an analysis based on Lighthill’s acoustic analogy. The analytical results predict that the far-field sound should exhibit “dipole-type” behavior; the root-mean-square pressure in the acoustic far field should increase as the cube of the turbulentMach number. The acoustic power normalized by the turbulent dissipation rate is also predicted to scale as turbulentMach number cubed. Agreement between the DNS results and the acoustic-analogy predictions is good. This study verifies the ability of the Lighthill acoustic analogy to predict sound generated by a three-dimensional, turbulent source containing many length and time scales.
Large eddy simulations of spatially evolving planar jets have been performed using the standard Smagorinsky, the dynamic Smagorinsky, and the dynamic mixed models and model performance evaluated. Computations have been performed both at a low Reynolds number,Red=3000, in order to make comparisons with a previous DNS at the same Reynolds number, and at a higher value, Red=30 000, to compare with high Reynolds number experiments. Model predictions with respect to the evolution of jet half-width, centerline velocity decay, mean velocity profiles, and profiles of turbulence intensity are evaluated. Some key properties of the SGS models such as the eddy-viscosity constant and the subgrid dissipation are also compared. It is found that the standard Smagorsinsky model is much too dissipative and severely underpredicts the evolution of the jet half-width and centerline velocity decay. The dynamic versions of the Smagorinsky model and the mixed model allow for streamwise and transverse variation of the constant in the eddy-viscosity expression which results in much better performance and good agreement with experimental and DNS data. The mixed model has an additional scale-similarity part which, in a priori tests against filtered jet DNS data, is found to predict the subgrid shear stress profile. Although the subgrid shear stress obtained by the dynamic Smagorinsky model is substantially smaller than that obtained in the a priori tests using the jet DNS data, surprisingly, in the a posteriori computations, the dynamic Smagorinsky model performs as well as the dynamic mixed model. Analysis of the mean momentum equation gives the reason for such behavior: the resolved stress in computations with the dynamic Smagorinsky model is larger than it should be and compensates for the underprediction of the subgrid shear stress by the Smagorinsky model. The numerical discretization errors have been quantified. The error due to noncommutativity of spatial differentiation and physical space filtering on nonuniform grids is found to be small because of the relatively mild stretching used in the present LES. The modeling error is found to be generally smaller than the discretization error with the standard Smagorinsky model having the largest modeling error.
The influence of the shear number on the turbulence evolution in a stably stratified fluid is investigated using direct numerical simulations on grids with up to 512 × 256 × 256 points. The shear number SK/ε is the ratio of a turbulence time scale K/ε to the shear time scale 1/S. Simulations are performed at two initial values of the Reynolds number Re Λ= 44.72 and Re Λ= 89.44. When the shear number is increased from small to moderate values, the nondimensional growth rate γ= (1/SK)dK/dt of the turbulent kinetic energy K increases since the shear forcing and its associated turbulence production is larger. However, a further increase of the shear number from moderate to large values results in a reduction of the growth rate γ and the turbulent kinetic energy K shows long-time decay for sufficiently large values of the shear number. The inhibition of turbulence growth at large shear numbers occurs for both initial values of the Reynolds number and can be explained with the predominance of linear effects over nonlinear effects when the shear number is sufficiently high. It is found that, at the higher initial value of the Reynolds number, the reduction of the growth rate occurs at a higher value of the shear number. The shear number is found to affect spectral space dynamics. Turbulent transport coefficients decrease with increasing shear number.
The present study sheds light on the subgrid modeling problem encountered in the large eddy simulation(LES) of practical flows, where the turbulence is both inhomogeneous and anisotropic due to mean flow gradients. The subgrid scale stress (SGS) tensor, the quantity that is key to the success of LES, is studied here in such flows using both analysis and direct numerical simulation (DNS). It is shown that the SGS tensor, for the case of inhomogeneous flow, where the filtering operation is necessarily performed in physical space, contains two components: a rapid part that depends explicitly on the mean velocity gradient and a slow part that does not. The characterization, rapid and slow, is adopted by analogy to that used in the modeling of the pressure–strain in the Reynolds-averaged Navier–Stokes equations. In the absence of mean flow gradients, the slow part is the only nonzero component and has been the subject of much theoretical study. However, the rapid part can be important in the inhomogeneous flows that are often encountered in practice. An analytical estimate of the relative magnitude of the rapid and slow components is derived and the distinct role of each component in the energy transfer between the resolved grid scales and the unresolved subgrid scales is identified. Results that quantify this new decomposition are obtained from DNS data of a turbulent mixing layer. The rapid part is shown to play an important role when the turbulence is in a nonequilibrium state with turbulence production much larger than dissipation or when the filter size is not very small compared to the characteristic integral scale of the turbulence, as in the case of practical LES applications. More importantly, the SGS is observed to be highly anisotropic due to the close connection of the rapid part with the mean shear. The Smagorinsky eddyviscosity and the scale-similarity models are tested by performing a priori tests with data from DNS of the mixing layer. It is found that the scale-similarity model correctly represents the anisotropicenergy transfer between grid and subgrid scales that is associated with the rapid part, while the eddyviscositymodel captures the dissipation associated with the slow part. This may be a physical reason for the recent successes of the mixed model (Smagorinsky plus scale similarity) reported in the literature.
Direct numerical simulations were performed in order to investigate the evolution of turbulence in a stably stratified fluid forced by nonvertical shear. Past research has been focused on vertical shear flow, and the present work is the first systematic study with vertical and horizontal components of shear. The primary objective of this work was to study the effects of a variation of the angle θ between the direction of stratification and the gradient of the mean streamwise velocity from θ=0, corresponding to the well-studied case of purely vertical shear, to θ=π/2, corresponding to purely horizontal shear. It was observed that the turbulent kinetic energy K evolves approximately exponentially after an initial phase. The exponential growth rate γ of the turbulent kinetic energy K was found to increase nonlinearly, with a strong increase for small deviations from the vertical, when the inclination angle θ was increased. The increased growth rate is due to a strongly increased turbulence production caused by the horizontal component of the shear. The sensitivity of the flow to the shear inclination angle θ was observed for both low and high values of the gradient Richardson number Ri, which is based on the magnitude of the shear rate. The effect of a variation of the inclination angle θ on the turbulence evolution was compared with the effect of a variation of the gradient Richardson number Ri in the case of purely vertical shear. An effective Richardson number Rieff was introduced in order to parametrize the dependence of the turbulence evolution on the inclination angle θ with a simple model based on mean quantities only. It was observed that the flux Richardson number Rif depends on the gradient Richardson number Ri but not on the inclination angle θ.
Direct numerical simulations (DNS) are performed to investigate the evolution of turbulence in a uniformly sheared and stably stratified flow. The spatial discretization is accomplished by a spectral collocation method, and the solution is advanced in time with a third-order Runge–Kutta scheme. The turbulence evolution is found to depend strongly on at least three parameters: the gradient Richardson number Ri, the initial value of the Taylor microscale Reynolds number Reλ, and the initial value of the shear number SK/<ε. The effect of each parameter is individually studied while the remaining parameters are kept constant. The evolution of the turbulent kinetic energy K is found to follow approximately an exponential law. The shear number SK/<ε, whose effect has not been investigated in previous studies, was found to have a strong non-monotone influence on the turbulence evolution. Larger values of the shear number do not necessarily lead to a larger value of the eventual growth rate of the turbulent kinetic energy. Variation of the Reynolds number Reλ indicated that the turbulence growth rate tends to become insensitive to Reλ at the higher end of the Reλ range studied here. The dependence of the critical Richardson number Ricr, which separates asymptotic growth of the turbulent kinetic energy K from asymptotic decay, on the initial values of the Reynolds number Reλ and the shear number SK/<ε was also obtained. It was found that the critical Richardson number varied over the range 0.04
A computational study of spatially evolving two-dimensional free shear flows has been performed using direct numerical simulation of the Navier–Stokes equations in order to investigate the ability of these two-dimensional simulations to predict the overall flow-field quantities of the corresponding three-dimensional “real” turbulent flows. The effects of inflow forcing on these two-dimensional flows has also been studied. Simulations were performed of shear layers, as well as weak (large co-flow and relatively weak shear) and strong (small co-flow and relatively strong shear) jets. Several combinations of discrete forcing with and without a broadband background spectrum were used. Although spatially evolving direct simulations of shear layers have been performed in the past, no such simulations of the plane jet have been performed to the best of our knowledge. It was found that, in the two-dimensional shear layers, external forcing led to a strong increase in the initial growth of the shear-layer thickness, followed by a region of decreased growth as in physical experiments. The final downstream growth rate was essentially unaffected by forcing. The mean velocity profile and the naturally evolving growth rate of the shear layer in the case of broadband forcing compare well with experimental data. However, the total and transverse fluctuation intensities are larger in the two-dimensional simulations with respect to experimental data. In the weak-jet simulations it was found that symmetric forcing completely overwhelms the natural tendency to transition to the asymmetric jet column mode downstream. It was observed that two-dimensional simulations of “strong” jets with a low speed co-flow led to a fundamentally different flow with large differences even in mean velocity profiles with respect to experimental data for planar jets. This was a result of the dominance of the two-dimensional mechanism of vortex dipole ejection in the flow due to the lack of spanwise instabilities. Experimental studies of planar jets do not show vortex dipole formation and ejection. A three-dimensional “strong”-jet simulation showed the rapid evolution of three-dimensionality effectively preventing this two-dimensional mechanism, as expected from experimental results.