#### Large-eddy simulation of stably stratified open channel flow.

J.R. Taylor, S. Sarkar and V. Armenio.

Journal PaperPhys. Fluids., 17, 116602, 1-18, 2005./div>

#### Abstract

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.

#### Large-eddy simulation of Rayleigh-Taylor turbulence with compressible miscible fluids.

J.P. Mellado, S. Sarkar and Y. Zhou.

Journal PaperPhys. Fluids., 17, 76101-1-20, 2003.

#### Abstract

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.

#### Effects of imperfect premixing coupled with hydrodynamic instability on flame propagation.

D. Garrido-Lopez and S. Sarkar.

Journal PaperProc. Combust. Inst., 30, 621-628, 2005.

#### Abstract

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.

#### Compressibility effects and turbulence scalings in supersonic channel flow.

H. Foysi, S. Sarkar and R. Friedrich.

Journal PaperJ. Fluid Mech., 38, 207-216, 2004.

#### Abstract

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.

#### Mixing in a stably-stratified medium by horizontal shear near vertical walls.

V. Amernio and S. Sarkar.

Journal PaperTheoret. Comput. Fluid Dyn., 17, 331-349, 2004.

#### Abstract

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.

#### Reconstruction subgrid models for nonpremixed combustion.

J.P. Mellado, S. Sarkar and C. Pantano.

Journal PaperPhys. Fluids, 15, 3280-3307, 2003.

#### Abstract

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 in a turbulent reacting shear layer.

C. Pantano, S. Sarkar and F.A. Williams.

Journal PaperJ. Fluid Mech., 481, 291-360, 2003.

#### Abstract

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.

#### The Effect of Stable Stratification on Turbulence Anisotropy in Stably-Stratified Flow.

S. Sarkar.

Journal PaperComputers and Mathematics with Applications, special review issue: Turbulence Modeling and Simulation, 46, No. 5, 1229-1248, 2003.

#### Abstract

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.

#### An Investigation of Stably-Stratified Channel Flow using Large Eddy Simulation.

V. Armenio and S. Sarkar.

Journal PaperJ. Fluid Mech., 459, 1-42, 2002.

#### Abstract

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.

#### A study of Compressibility Effects in the High-Speed, Turbulent Shear Layer Using Direct Simulation.

C. Pantano and S. Sarkar.

Journal PaperJ. Fluid Mech., 450, 329-371, 2002.

#### Abstract

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.

#### A study of the Flowfield Evolution and Mixing in a Planar Turbulent Jet Using Direct Numerical Simulation.

S.A. Stanley, S. Sarkar and J.P. Mellado.

Journal PaperJ. Fluid Mech., 450, 377-407, 2002.

#### Abstract

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.

#### A Subgrid Function for Nonlinear Functions of a Scalar.

C. Pantano and S. Sarkar.

Journal PaperPhys. Fluids, 13, 3803-3819, 2001.

#### Abstract

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.

#### Validation of Acoustic-Analogy Predictions for Sound Radiated by Turbulence.

J. Whitmire and S. Sarkar.

Journal PaperPhys. Fluids, 12, 381-391, 2000.

#### Abstract

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 simulation of a plane jet.

C. Le Ribault, S. Sarkar and S. Stanley.

Journal PaperPhys. Fluids, 11, 3069-3083, 1999.

#### Abstract

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.

#### On the Shear Number Effect in Stratified Shear Flow.

F.G. Jacobitz and S. Sarkar.

Journal PaperTheor. Comput. Fluid Dyn., 13, 171-188, 1999.

#### Abstract

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.

#### On the Relationship between the Mean Flow and Subgrid Stresses in LES of Turbulent Shear Flows.

L. Shao, S. Sarkar and C. Pantano.

Journal PaperPhys. Fluids, 11, 1229-1248, 1999.

#### Abstract

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.

#### The Effect of Nonvertical Shear on Turbulence in a Stably Stratified Medium.

F.G. Jacobitz and S. Sarkar.

Journal PaperPhys. Fluids, 10, 1158-1168, 1998.

#### Abstract

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 of turbulence evolution in a uniformly sheared and stably stratified flow.

F.G. Jacobitz, S. Sarkar and C. W. Van Atta.

Journal PaperJ. Fluid Mech., 342, 231-261, 1997.

#### Abstract

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

#### Simulations of Spatially Developing Two-Dimensional Shear Layers and Jets.

S. Stanley and S. Sarkar.

Journal PaperTheor. Comput. Fluid Dyn., 9, 121-147, 1997.

#### Abstract

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.