The effect of nonvertical shear on turbulence in a stably stratified medium

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1 The effect of nonvertical shear on turbulence in a stably stratifie meium Frank G. Jacobitz an Sutanu Sarkar Citation: Physics of Fluis (1994-present) 10, 1158 (1998); oi: / View online: View Table of Contents: Publishe by the AIP Publishing Articles you may be intereste in Relevance of the Thorpe length scale in stably stratifie turbulence Phys. Fluis 25, (2013); / Buoyancy generate turbulence in stably stratifie flow with shear Phys. Fluis 18, (2006); / On the equilibrium states preicte by secon moment moels in rotating, stably stratifie homogeneous shear flow Phys. Fluis 16, 3540 (2004); / Anisotropy of turbulence in stably stratifie mixing layers Phys. Fluis 12, 1343 (2000); / Length scales of turbulence in stably stratifie mixing layers Phys. Fluis 12, 1327 (2000); / This article is copyrighte as inicate in the article. Reuse of AIP content is subject to the terms at: Downloae to IP:

2 PHYSICS OF FLUIDS VOLUME 10, NUMBER 5 MAY 1998 The effect of nonvertical shear on turbulence in a stably stratifie meium Frank G. Jacobitz an Sutanu Sarkar Department of Applie Mechanics an Engineering Sciences, University of California, San Diego, La Jolla, California Receive 2 January 1997; accepte 5 January 1998 Direct numerical simulations were performe in orer to investigate the evolution of turbulence in a stably stratifie flui force by nonvertical shear. Past research has been focuse on vertical shear flow, an the present work is the first systematic stuy with vertical an horizontal components of shear. The primary objective of this work was to stuy the effects of a variation of the angle between the irection of stratification an the graient of the mean streamwise velocity from 0, corresponing to the well-stuie case of purely vertical shear, to /2, corresponing to purely horizontal shear. It was observe that the turbulent kinetic energy K evolves approximately exponentially after an initial phase. The exponential growth rate of the turbulent kinetic energy K was foun to increase nonlinearly, with a strong increase for small eviations from the vertical, when the inclination angle was increase. The increase growth rate is ue to a strongly increase turbulence prouction cause by the horizontal component of the shear. The sensitivity of the flow to the shear inclination angle was observe for both low an high values of the graient Richarson number Ri, which is base on the magnitue of the shear rate. The effect of a variation of the inclination angle on the turbulence evolution was compare with the effect of a variation of the graient Richarson number Ri in the case of purely vertical shear. An effective Richarson number Ri eff was introuce in orer to parametrize the epenence of the turbulence evolution on the inclination angle with a simple moel base on mean quantities only. It was observe that the flux Richarson number Ri f epens on the graient Richarson number Ri but not on the inclination angle American Institute of Physics. S I. INTRODUCTION Stably stratifie shear flow is an ubiquitous feature of flui motion in the geophysical environment. Consequently much attention has been rawn to the turbulence evolution in vertically stably stratifie an vertically sheare flow motivate by oceanic an atmospheric applications. 1,2 The importance of the graient Richarson number Ri N 2 /S 2, where N is the Brunt Väisälä frequency, an S is the shear rate, was iscovere early by energy arguments. 1,3 The application of linear invisci stability theory by Miles 4 an Howar 5 establishe Ri 1/4 as the sufficient conition for stability in a stratifie shear flow. More recently, laboratory experiments an irect numerical simulations have been performe in orer to stuy many aspects of the turbulence evolution in vertically stratifie an vertically sheare flow. The evolution of turbulence in vertically stably stratifie an nonvertically sheare flow has receive consierably less attention. The present paper appears to be the first systematic stuy of nonvertical shear flow. A number of laboratory experiments on vertically stratifie an vertically sheare flow have been performe. Komori et al. 6 investigate the turbulence structure in a stratifie open-channel flow. Experiments on the evolution of homogeneous turbulence in stratifie shear flow were performe by Rohr et al. 7 using a salt-stratifie water channel, an by Piccirillo an Van Atta 8 using a thermally stratifie win tunnel. Itsweire et al. confirme the importance of the graient Richarson number Ri an Piccirillo an Van Atta aresse the Reynols number epenence of the turbulence evolution. The experimental investigations were complemente by irect numerical simulations by Gerz et al., 9 Holt et al., 10 an Jacobitz et al. 11 In these simulations, the epenence of the turbulence evolution on a wie range of parameters was aresse. Gerz et al. stuie the occurrence of counter graient buoyancy fluxes at high Richarson numbers. Holt et al. aresse the Reynols number epenence of the turbulence evolution. Jacobitz et al. investigate the possibility of Reynols number inepenence at high Reynols numbers an the influence of the shear number SK/, which is the ratio of a time scale of turbulence K/ to the time scale impose by the shear 1/S. Here K is the turbulent kinetic energy an the turbulence issipation rate. Kaltenbach et al. 12 performe large ey simulations of vertically stratifie an vertically sheare flow. In aition, passive scalars with linear graients in all irections were introuce. It was observe that these scalars mix more efficiently in the horizontal than in the vertical. Work on nonvertically sheare flow in a vertically stably stratifie flui is restricte to Blumen s 13 application of Howar s semicircle theorem to such a flow. Nonvertical shear flow in a stratifie flui occurs an has been stuie inirectly in experimental investigations of wakes 14,15 an jets 16 in a stratifie meium. No previous laboratory experiments or numerical investigations of the evolution of homogeneous turbulence in a nonvertical shear flow with vertical stratification are known to the authors. This is surprising as /98/10(5)/1158/11/$ American Institute of Physics This article is copyrighte as inicate in the article. Reuse of AIP content is subject to the terms at: Downloae to IP:

3 Phys. Fluis, Vol. 10, No. 5, May 1998 F. G. Jacobitz an S. Sarkar 1159 this type of flow occurs frequently in environmental an engineering applications. Examples are flow over topography, river inflow into the ocean, or effluent ischarge by power plants. The absence of previous work together with the wie range of applications are the primary motivation for the current stuy. In vertical shear flow, the graient Richarson number is usually efine with the vertical shear rate. In the case of more complex shear flows with aitional nonvertical shear components, the efinition of Ri may nee to be moifie. In nonvertical shear flow, we choose to efine the graient Richarson number as Ri N 2 /S 2, where S is the magnitue of the shear rate. This selection is motivate by the observation that N is an external frequency scale impose by the gravity acceleration an the mean stratification, while S is a istortion scale impose by the mean velocity graients. Therefore, the Richarson number Ri N 2 /S 2, the square of the ratio of the two time scales, is a measure for the competing effects of mean stratification an mean shear. In Sec. II the equations of motion are presente. In Sec. III the transport equations for secon-orer moments are iscusse. The numerical metho is summarize in Sec. IV. The results of irect numerical simulations are presente in Sec. V, an in Sec. VI the effective Richarson number is introuce. In Sec. VII the influence of aitional parameters on the turbulence evolution is iscusse. Section VIII contains a summary of the work presente here. II. EQUATIONS OF MOTION This stuy is base on the continuity equation of an incompressible flui, the three-imensional unsteay Navier Stokes equation in the Boussinesq approximation, an a transport equation for the ensity. In the following, x i enotes the ith component of an orthonormal Cartesian coorinate system, U i the ith component of the total velocity, ϱ the total ensity, an P the total pressure. The epenent variables U i, ϱ, an P are ecompose into a mean part enote by an overbar an a fluctuating part enote by small letters : U i Ū i u i, ϱ ϱ, P P p. 1 The mean streamwise velocity Ū (Ū 1,0,0) is uniirectional an has constant horizontal an vertical shear rates Ū 1 / x 2 S 2 S sin an Ū 1 / x 3 S 3 S cos, respectively. The mean ensity has a constant vertical stratification rate ϱ / x 3 S. Thus, FIG. 1. Sketch of the mean velocity with vertical shear an the mean ensity with vertical stratification. This case correspons to 0. The ecomposition of the epenent variables is introuce into the equations of motion, an the following evolution equations for the fluctuating parts are obtaine: u j x j 0, u i t u u i j S sin x x 2 S cos x 3 u i j x 1 S sin u 2 S cos u 3 i1 1 0 p x i 2 u i x j x j g 0 i3, t u j S sin x x 2 S cos x 3 S j x u x j x j Here g is the gravity acceleration, the kinematic viscosity, an the scalar iffusivity. III. TRANSPORT EQUATIONS In this section the transport equations for secon-orer moments are introuce. The overbar ā enotes the volume average of a. The transport equation for the velocity correlation R ij u i u j is erive from Eq. 5, t R ij P ij B ij ij ij, 7 P ij S sin u j u 2 i1 S cos u j u 3 i1 S sin u i u 2 j1 S cos u i u 3 j1, Ū i S sin x 2 S cos x 3 i1, ϱ 0 S x 3. 2 Therefore 0 correspons to the well-stuie case of purely vertical shear shown in Fig. 1, an /2 correspons to the case of purely horizontal shear shown in Fig. 2. It is assume that a mean pressure graient balances the mean buoyancy force: P 0 g x 0 S x FIG. 2. Sketch of the mean velocity with horizontal shear an the mean ensity with vertical stratification. This case correspons to /2. This article is copyrighte as inicate in the article. Reuse of AIP content is subject to the terms at: Downloae to IP:

4 1160 Phys. Fluis, Vol. 10, No. 5, May 1998 F. G. Jacobitz an S. Sarkar B ij g 0 u i j3 u j i3, 9 t u 3 S u 3 u 3 g p x 3 ij 1 0 p u i x j u j x i, 10 ij 2 u i u j. 11 x k x k Here P ij enotes the turbulence prouction term, B ij the buoyancy term, ij the pressure strain term, an ij the turbulence issipation term. Note that the turbulence prouction term appears only in equations for velocity correlations that contain the streamwise velocity component, an that the buoyancy term appears only in equations for velocity correlations that contain the vertical velocity component. The equations for the components of the velocity correlation tensor are: t u 1u 1 2S sin u 1 u 2 2S cos u 1 u , 12 t u 2u , t u 3u 3 2 g 0 u , t u 1u 2 S sin u 2 u 2 S cos u 2 u , 15 t u 1u 3 S sin u 2 u 3 S cos u 3 u 3 g u , t u 2u 3 g 0 u The transport equations for the ensity fluxes u i are erive from Eqs. 5 an 6 : t u i S sin u 2 i1 S cos u 3 i1 S u i u 3 g i3 1 p 1 Pr u i. 0 0 x i Pr x k x k 18 The components of the above equation are: t u 1 S sin u 2 S cos u 3 S u 1 u 3 1 p 1 Pr u 1, 0 x 1 Pr x k x k 19 t u 2 S u 2 u 3 1 p 1 Pr u 2, 20 0 x 2 Pr x k x k 1 Pr u Pr x k x k Finally a transport equation for the ensity fluctuations can by erive from Eq. 6 : t 2S u x k x k The transport equation for the turbulent kinetic energy K u i u i /2 is: t K P 2 P 3 B, P 2 S sin u 1 u 2, P 3 S cos u 1 u 3, B g 0 u 3, u i u i. 27 x k x k Here P 2 is the turbulence prouction term ue to horizontal shear Ū 1 / x 2, P 3 the turbulence prouction term ue to vertical shear Ū 1 / x 3, B the buoyancy term, an the issipation term. The total turbulence prouction is given by P P 2 P 3. The potential energy K is compute from the ensity fluctuations: K 1 2 g 0 S. IV. NUMERICAL APPROACH 28 The equations of motion are solve using a irect numerical approach. All ynamically important scales of the velocity an ensity fiels are fully resolve. A metho of lines approach is use where a spatial iscretization is first performe in orer to obtain a semiiscrete system of orinary ifferential equations. Then the system of equations is integrate to avance the solution in time. The spatial iscretization is accomplishe by a Fourier collocation metho, which yiels high accuracy but can be use only for problems with perioic bounary conitions. Following a metho originally use by Rogallo, 17 the equations of motion are transforme into a frame of reference moving with the mean flow in orer to allow perioic bounary conitions on the fluctuating parts of the epenent variables. The temporal avancement is accomplishe by a low-storage, thir-orer Runge Kutta scheme. The initial conitions are taken from a simulation of ecaying isotropic turbulence an allow for large scale growth. A computational gri with points was use for all simulations. The evolution of the largest an smallest turbulence scales was monitore to ensure a proper resolution of the simulations. This article is copyrighte as inicate in the article. Reuse of AIP content is subject to the terms at: Downloae to IP:

5 Phys. Fluis, Vol. 10, No. 5, May 1998 F. G. Jacobitz an S. Sarkar 1161 FIG. 3. Evolution of turbulent kinetic energy K as a function of the inclination angle. The ashe lines are the exponential approximation to the solution. FIG. 4. Depenence of the asymptotic value of the growth rate on the inclination angle. The coe evelope uring our previous stuy of turbulence in a stratifie flui with vertical shear only ( 0) iscusse in Jacobitz et al. 11 was moifie to account for a variable shear inclination angle. The governing equations were written in a frame of reference (x 1,x 2,x 3 ) where x 1 is coincient with the x 1 irection. The x 2 an x 3 axes lie in the same plane as the x 2 an x 3 axes but are rotate aroun the x 1 axis by the inclination angle with respect to the x 2 an x 3 axes. The avantage of this new coorinate system is that there is a single mean shear component S Ū 1 / x 3, an the stanar Rogallo transformation can be applie. In the new coorinate system, there are two gravity components g 2 g sin in the spanwise irection x 2 an g 3 g cos in the shear irection x 3 as well as two stratification components S sin an S cos, respectively. However, the aitional gravity an stratification components o not require any aitional technique in the simulation. V. RESULTS In this section the results of a series of simulations with ifferent shear inclination angles are presente. All simulations were starte from the same initial conitions taken from a simulation of ecaying isotropic turbulence without ensity fluctuations. The Richarson number Ri N 2 /S where N 2 gs / 0 is the Brunt Väisälä frequency, the Prantl number Pr / 0.72, the initial value of the Taylor microscale Reynols number Re q / where q 2K is the magnitue of the velocity, an 5 q 2 / is the Taylor microscale, an the initial value of the shear number SK/ 2.0 are fixe. The Richarson number an the shear number are base on the magnitue S (Ū 1 /x 2 ) 2 (Ū 1 /x 3 ) 2 of the shear rates. While the Richarson number an the Prantl number remain constant, the Reynols number an the shear number evolve as the simulations are avance in time. Figure 3 shows the evolution of the turbulent kinetic energy K as a function of the nonimensional time St for ifferent inclination angles. Initially K ecays in all simulations ue to the isotropic initial conitions. For 0, which correspons to the case of purely vertical shear shown in Fig. 1, the turbulent kinetic energy continues to ecay throughout the simulation. In this case the stratification influences the shear prouction of turbulence irectly. When the angle is increase, the ecay of K is less strong. For /8 the turbulent kinetic energy remains constant in time. For larger angles the evolution of the turbulent kinetic energy changes from ecay to growth. For /2, which correspons to the case of purely horizontal shear shown in Fig. 2, the turbulent kinetic energy grows the strongest. However, the growth of K is not as strong as in the unstratifie case labele Ri 0 in Fig. 3 suggesting an inirect influence of the stratification on the turbulence prouction as iscusse below. In the case of purely vertical shear ( 0), the stratification influences the shear prouction irectly. Buoyancy fluxes ecrease the vertical velocity fluctuations u 3 u 3 see Eq. 14 an the magnitue of the 1 3 velocity correlation u 1 u 3 see Eq. 16. Therefore the vertical shear prouction P 3 S cos u 1 u 3 is irectly reuce by buoyancy fluxes. It was shown in previous investigations 7,10,11 that the primary effect of stable stratification is to ecrease the shear prouction an, to a smaller extent, act as a sink for turbulent kinetic energy through the buoyancy flux. In the case of purely horizontal shear ( /2), this irect mechanism of stabilization oes not exist. The buoyancy fluxes still ecrease the vertical velocity fluctuations an the 1 3 velocity correlation. However, there is no irect influence on the 1 2 velocity correlation u 1 u 2 see Eq. 15 an the horizontal shear prouction P 2 S sin u 1 u 2. Only a reistribution through the pressure strain terms leas to an inirect influence of stratification on the turbulence evolution in the case of purely horizontal shear. It was foun that the asymptotic evolution of the turbulent kinetic energy follows approximately an exponential law. In this case the exponential growth rate obtaine from Eq. 23, 1 K SK t P 2 SK P 3 SK B SK SK, This article is copyrighte as inicate in the article. Reuse of AIP content is subject to the terms at: Downloae to IP: 29

6 1162 Phys. Fluis, Vol. 10, No. 5, May 1998 F. G. Jacobitz an S. Sarkar FIG. 5. Evolution of the normalize turbulence issipation /SK as a function of the inclination angle. FIG. 7. Evolution of the normalize vertical turbulence prouction P 3 /SK as a function of the inclination angle. reaches an approximately constant value. Then the equation can be integrate an the exponential law K K 0 exp St 30 is obtaine. This exponential assumption is shown as ashe lines in Fig. 3. It approximates the asymptotic evolution well. The constant of integration is use to fit the graphs. The epenence of the asymptotic value of the growth rate on the inclination angle is shown in Fig. 4. The growth rate increases strongly for 0 /4 an continues to increase milly for /4 /2. In the following, the epenence of each term on the right-han sie of Eq. 29 on the inclination angle is iscusse. Figure 5 shows the evolution of the normalize turbulence issipation /SK. It appears that the normalize issipation oes not epen on the inclination angle. In Fig. 6, the evolution of the normalize buoyancy flux B/SK is shown. In the asymptotic regime, the normalize buoyancy flux increases with increasing angle. The asymptotic value of the normalize buoyancy flux is about twice as large in the case of purely horizontal shear ( /2) compare to the case of purely vertical shear ( 0). Figure 7 shows the evolution of the normalize turbulence prouction P 3 /SK ue to vertical shear. It vanishes for the case of purely horizontal shear ( /2) an increases as the inclination angle is ecrease. The normalize vertical prouction P 3 /SK ecreases throughout the simulation of purely vertical shear ( 0) ue to the low Reynols number of this strongly ecaying case. The other cases reach an asymptotically approximately constant vertical prouction P 3 /SK. Figure 8 shows the evolution of the normalize turbulence prouction P 2 /SK ue to horizontal shear. It vanishes for the case of purely vertical shear ( 0), which inclues the simulation of unstratifie turbulence (Ri 0). As expecte, P 2 /SK increases as the angle is increase. Due to the effect of buoyancy that exists at all shear inclination angles, the total normalize prouction rate P/SK where P P 2 P 3 of the stratifie cases is always smaller than the total normalize prouction rate of the unstratifie case as shown in Fig. 9. Furthermore, since the effect of buoyancy acts irectly in the vertical irection, the normalize turbulence prouction P/SK in the case of purely FIG. 6. Evolution of the normalize buoyancy flux B/SK as a function of the inclination angle. FIG. 8. Evolution of the normalize horizontal turbulence prouction P 2 /SK as a function of the inclination angle. This article is copyrighte as inicate in the article. Reuse of AIP content is subject to the terms at: Downloae to IP:

7 Phys. Fluis, Vol. 10, No. 5, May 1998 F. G. Jacobitz an S. Sarkar 1163 FIG. 9. Evolution of the normalize turbulence prouction P/SK as a function of the inclination angle. FIG. 10. Evolution of the Reynols stress anisotropy b 13 as a function of the inclination angle. vertical shear is smaller than the normalize turbulence prouction in the case of purely horizontal shear. The observation of a larger stabilizing effect of stable vertical stratification in the case of vertical shear relative to horizontal shear is consistent with the transport equations for the secon-orer moments. A buoyancy term appears irectly in Eq. 14 for the vertical velocity variance u 3 u 3, reucing u 3 u 3. A buoyancy term also appears in Eq. 16 for the 1 3 velocity correlation u 1 u 3, an it reuces the magnitue of u 1 u 3 in aition to the reuction from u 3 u 3. The ecrease u 1 u 3 reuces the vertical prouction rate P 3 S cos u 1 u 3. A similar irect influence of buoyancy on the horizontal prouction rate P 2 S sin u 1 u 2 oes not exist, because a buoyancy term neither appears in Eq. 13 for u 2 u 2 nor in Eq. 15 for u 1 u 2. The 2 3 velocity correlation u 2 u 3 remains small compare to u 1 u 2 an u 1 u 3. However, there is an inirect effect of gravity through the pressure strain terms on the horizontal turbulence prouction as shown in Fig. 8 by the reuce P 2 /SK at /2 with respect to the unstratifie case. The anisotropic action of buoyancy in the vertical irection is evient in the evolution of the Reynols stress anisotropy tensor. In Fig. 10 the anisotropy b 13 is shown, on which the normalize vertical prouction rate P 3 /SK epens. The magnitue of the anisotropy b 13 ecreases with increasing angle. On the other han, the magnitue of the anisotropy b 12 increases with increasing angle as shown in Fig. 11. The anisotropy b 12 etermines the normalize horizontal prouction rate P 2. of simulations with constant graient Richarson numbers Ri but ifferent inclination angles iscusse in Sec. V. Again, an exponential evolution of the turbulent kinetic energy K was observe as shown in Fig. 12. The turbulent kinetic energy K ecays for large Ri an grows for small Ri. The value of the critical Richarson number, for which K remains constant in time, is about Ri cr in this series of simulations. The epenence of the asymptotic value of the growth rate on the graient Richarson number Ri is shown in Fig. 13. In agreement with Jacobitz et al. 11 the growth rate ecreases approximately linearly with increasing Richarson number Ri. The following relationship was obtaine from linear regression: Ri. 31 In the oceanic an atmospheric environment, the flux Richarson number Ri f B/P is frequently use instea of the graient Richarson number Ri. The flux Richarson number Ri f can be relate to the graient Richarson number Ri, if an ey iffusivity moel is use: VI. THE EFFECTIVE RICHARDSON NUMBER In this section, the effective Richarson number Ri eff is introuce in orer to parametrize the epenence of the turbulence evolution on the shear inclination angle. Inaition, the relationship between the effective Richarson number Ri eff, the flux Richarson number Ri f, an the graient Richarson number Ri is iscusse. A series of simulations with ifferent graient Richarson numbers Ri but purely vertical shear was performe. These simulations matche the initial conitions of the series FIG. 11. Evolution of the Reynols stress anisotropy b 12 as a function of the inclination angle. This article is copyrighte as inicate in the article. Reuse of AIP content is subject to the terms at: Downloae to IP:

8 1164 Phys. Fluis, Vol. 10, No. 5, May 1998 F. G. Jacobitz an S. Sarkar FIG. 12. Evolution of the turbulent kinetic energy K for ifferent graient Richarson numbers Ri in the case of purely vertical shear ( 0). The ashe lines are the exponential approximation to the solution. FIG. 14. Evolution of the flux Richarson number Ri f for ifferent graient Richarson numbers Ri in the case of purely vertical shear ( 0). Ri f B P g u 3 0 Su 1 u 3 t g S Ri t 0 S Pr t Here u 1 u 1 an u 3 u 2 sin u 3 cos are the velocity components in the plane of shear, t u 3 /S is the ey iffusivity of the ensity fiel, t u 1 u 3 /S is the ey viscosity of the velocity fiel, an Pr t t / t is the turbulent Prantl number. Therefore the flux Richarson number Ri f coincies with the graient Richarson number Ri, if the turbulent Prantl number Pr t is equal to one. The evolution of the flux Richarson number Ri f is shown in Fig. 14 for ifferent graient Richarson numbers Ri in the case of purely vertical shear. For all graient Richarson numbers Ri, the flux Richarson number Ri f evolves to a constant asymptotic value that is close to the corresponing value of the graient Richarson number Ri. Therefore, in the parameter range consiere here, the turbulent Prantl number Pr t remains always close to one. This agrees with previous simulations by Schumann an Gerz 18 where an increase of the turbulent Prantl number Pr t with increasing Richarson number Ri is observe only at larger Richarson numbers Ri 0.25, beyon the scope of the current work. Figure 15 shows the evolution of the flux Richarson number Ri f for ifferent inclination angles an constant graient Richarson number Ri 0.2. The asymptotic value of Ri f remains very close to the value of Ri 0.2. Therefore the turbulent Prantl number Pr t is again close to one. In the case of nonvertical shear it was shown in Fig. 4 that the growth rate epens on the inclination angle for a constant graient Richarson number Ri. Using the linear relationship 31 between the growth rate an the Richarson number Ri, an effective Richarson number Ri eff can be compute from the growth rates observe at an angle. Therefore the effective Richarson number Ri eff of a nonvertical shear flow is efine to be equal to the graient Richarson number Ri of a purely vertical shear flow with the same growth rate. The epenence of the effective Richarson number Ri eff on the inclination angle is shown in Fig. 16. The effective Richarson number Ri eff ecreases FIG. 13. Depenence of the asymptotic value of the growth rate on the graient Richarson number Ri for the case of purely vertical shear ( 0). FIG. 15. Evolution of the flux Richarson number Ri f for ifferent inclination angles. This article is copyrighte as inicate in the article. Reuse of AIP content is subject to the terms at: Downloae to IP:

9 Phys. Fluis, Vol. 10, No. 5, May 1998 F. G. Jacobitz an S. Sarkar 1165 FIG. 16. Depenence of the effective Richarson number Ri eff on the inclination angle. The iamons represent the irect numerical results, the ashe line represents Eq. 33 an the soli line represents Eq. 34. FIG. 17. Evolution of the turbulent kinetic energy K for ifferent shear inclination angles in the case of strong stratification with Ri 2.0. with increasing angle, because the stabilizing influence of stable stratification ecreases as the shear inclination angle is increase from 0 vertical shear to /2 horizontal shear. In the following paragraphs two possible moels for the observe epenence are iscusse. First consier the plane of shear efine by the velocity components u 1 an u 2 sin u 3 cos. This plane is the (x 1,x 3 ) plane iscusse in Sec. IV on the numerical approach. The mean shear S acts only in this plane. The components g cos an S cos of the gravity constant g an the stratification S act also in this plane. Uner the assumption that the components g sin an S sin outsie the plane of shear o not influence the evolution of turbulence, the effective Richarson number can be written as: Ri eff Ri cos This relation is shown as a ashe line in Fig. 16. It oes not fit the numerical ata well an leas to large errors at large values of the inclination angle, because it oes not take the effect of buoyancy outsie the plane of shear into account. In a secon approach, the shear S cos in the irection of gravity an S sin perpenicular to the irection of gravity are weighte ifferently. Since the horizontal component S sin is not irectly influence by gravity, it is given a larger weight than the vertical component S cos. Then the effective Richarson number Ri eff can be written as: Ri Ri eff cos 2 a sin Here a 2.7 is the weight given to the horizontal component of shear. Equation 34 is shown as a soli line in Fig. 16. It fits the numerical ata relatively well for a simple moel base on mean parameters of the flow. VII. INFLUENCE OF ADDITIONAL PARAMETERS The primary aim of this work was to stuy the influence of the shear inclination angle on the ynamics of turbulence in stratifie shear flow, keeping other parameters constant. However, more simulations have been performe to stuy the influence of aitional parameters. First, the persistence of the shear inclination angle effect in strongly stratifie, high Richarson number flow is stuie. Then the influence of a high initial shear number is iscusse. In the first series of simulations iscusse in Sec. V, the shear inclination angle was varie from 0 to /2 to stuy the influence of the inclination angle on the turbulence evolution. All other parameters were fixe in these simulations. In the secon series of simulations iscusse in Sec. VI, the graient Richarson number Ri was varie from Ri 0 tori 0.2 in orer to compare its influence on the turbulence evolution in vertically stratifie an vertically sheare flow with the effects of a variation of the inclination angle in vertically stratifie an nonvertically sheare flow. Both parameters influence the growth rate of the turbulent kinetic energy K. Since stratification effects are often parametrize by the graient Richarson number Ri, the shear inclination angle effect was parametrize by the introuction of the effective Richarson number Ri eff. The secon series of simulations covers only a limite range of Richarson numbers that result in growth rates similar to those observe in the first series of simulations. An aitional series of simulations performe to stuy the influence of the shear inclination angle on the turbulence evolution at a large graient Richarson number is now iscusse. The objective of these simulations is to confirm that the sensitivity to the inclination angle, as seen in the results presente in Sec. V with Ri 0.2, persists in a strongly stratifie meium with Ri 2.0. All simulations were starte from the same initial conitions with Ri 2.0, Pr 0.72, initial Re 33.54, an initial SK/ 2.0. Figure 17 shows the evolution of the turbulent kinetic energy K as a function of the nonimensional time St for ifferent angles. For all cases the turbulent kinetic energy K ecays ue to the strong stable stratification. However, the growth rates vary for ifferent angles. Figure 18 shows the epenence of the growth rate as a function of the angle for Ri 2.0 open iamons an Ri 0.2 fille iamons. Both series show a qualitatively similar increase of the growth rate with increasing angle. In agreement with the Ri 0.2 case, the asymptotic This article is copyrighte as inicate in the article. Reuse of AIP content is subject to the terms at: Downloae to IP:

10 1166 Phys. Fluis, Vol. 10, No. 5, May 1998 F. G. Jacobitz an S. Sarkar FIG. 18. Depenence of the growth rate on the shear inclination angle in the case Ri 0.2 fille iamons an Ri 2.0 open iamons. FIG. 20. Evolution of the turbulent kinetic energy K for ifferent shear numbers SK/ in the case of purely horizontal shear ( /2). value of the flux Richarson number Ri f was foun to be relatively insensitive to the angle variation in the strongly stratifie Ri 2.0 case. The large value of the turbulent Prantl number Pr t 5 7 agrees with the results of Schumann an Gerz. 18 Now consier only the case of purely horizontal shear ( /2). The Ri 0.2 case shows strong growth of K as iscusse in Sec. V. The Ri 2.0 case, on the other han, shows a slight ecay of K. Therefore strong stable stratification is able to suppress the growth of K in the case of horizontal shear espite the absence of the irect mechanism that is responsible for the suppression of growth in the case of vertical shear. However, the value of the critical Richarson number Ri cr at which growth is suppresse is about an orer of magnitue larger in the case of horizontal shear compare to the case of vertical shear. The shear number SK/ has an important influence on the turbulence evolution in vertically sheare an stratifie flow as iscusse in Jacobitz et al. 11 Therefore aitional simulations with high initial shear numbers were performe in purely vertical shear ( 0) an in purely horizontal shear ( /2) to ascertain if the shear number effect persists in the case of purely horizontal shear. Figure 19 shows the evolution of the turbulent kinetic energy K for ifferent initial shear numbers in the case of vertical shear. The simulations were starte from the same initial conitions with Ri 0.1, Pr 0.72, an Re The turbulent kinetic energy K grows for the cases with initial SK/ 2.0 an SK/ 6.0 but ecays for the case with initial SK/ Figure 20 shows the evolution of K in the case of horizontal shear. The simulations were starte with the same initial conitions with Ri 0.2, Pr 0.72, an Re The turbulent kinetic energy grows strongly for SK/ 2.0 but grows only milly for SK/ Figure 21 shows the corresponing growth rates. The cases with high initial shear numbers finally result in an evolution with a smaller exponential growth rate than the corresponing cases with low initial shear numbers. Therefore the stabilizing effect of high shear number flow that was observe by Jacobitz et al. 11 in the case of vertical shear persists in the case of horizontal shear. FIG. 19. Evolution of the turbulent kinetic energy K for ifferent shear numbers SK/ in the case of purely vertical shear ( 0.0). FIG. 21. Depenence of the growth rate on the initial value of the shear number SK/ in the case of purely horizontal shear fille iamons an purely vertical shear open iamons. This article is copyrighte as inicate in the article. Reuse of AIP content is subject to the terms at: Downloae to IP:

11 Phys. Fluis, Vol. 10, No. 5, May 1998 F. G. Jacobitz an S. Sarkar 1167 VIII. CONCLUSIONS The effect of both vertical an horizontal shear components in a vertically stably stratifie flui has been investigate using irect numerical simulations. To the best of our knowlege, no such stuy, either experimental or numerical, has been performe previously. It was foun that not only the magnitue of the shear but also its orientation relative to the vertical irection of gravity an stratification has a significant effect on the flow ynamics. Two series of irect numerical simulations were initially performe. In the first series, the shear inclination angle between the irection of stratification an the graient of the mean streamwise velocity was varie from 0, corresponing to purely vertical shear, to /2, corresponing to purely horizontal shear. The graient Richarson number Ri 0.2 base on the magnitue of the shear rate S was fixe. In the secon series, the graient Richarson number Ri was varie from Ri 0, corresponing to unstratifie shear flow, to Ri 0.2. The angle 0 was fixe in this simulation. All simulations were starte from the same initial conitions taken from a simulation of ecaying isotropic turbulence with no ensity fluctuations. The initial value of the Taylor microscale Reynols number Re 33.54, the initial value of the shear number SK/ 2.0, an the Prantl number Pr 0.72 were fixe for all simulations. The turbulent kinetic energy was foun to evolve exponentially after an initial perio of ecay for all simulations. For the first series of simulations, the exponential growth rate was foun to increase with the inclination angle as shown in Fig. 4. This increase is ue to a strong increase of the horizontal turbulence prouction P 2 cause by the horizontal component of shear as is increase. Although the vertical turbulence prouction P 3 ecreases when is increase, the net turbulence prouction P P 2 P 3 increases. An examination of the transport equations for the seconorer moments shows that there is a irect influence of the buoyancy on the vertical turbulence prouction but no irect influence on the horizontal turbulence prouction. Therefore the turbulence prouction in the case of purely horizontal shear is stronger than the turbulence prouction in the case of purely vertical shear, but ue to the three-imensional character of turbulence, the turbulence prouction in the case of purely horizontal shear is still smaller than the turbulence prouction in unstratifie shear flow. Therefore the growth rate is smaller in the case of horizontal shear with respect to that in the unstratifie case. For the secon series of simulations, the exponential growth rate was foun to ecrease approximately linearly with increasing graient Richarson number Ri over the range 0 Ri 0.2 as shown in Fig. 13, in agreement with previous simulations. 11 In orer to incorporate the effect of the shear inclination angle variation into a single parameter, the effective Richarson number Ri eff was introuce. The effective Richarson number Ri eff of a nonvertical shear flow is efine to be equal to the graient Richarson number Ri of a purely vertical shear flow with the same growth rate. The effective Richarson number was foun to ecrease with increasing inclination angle as shown in Fig. 16. A simple moel was use to capture the epenence of the effective Richarson number Ri eff on the angle. In this moel the horizontal component of shear is weighte more strongly than the vertical component of shear, because the horizontal component of shear prouction is not irectly influence by buoyancy an therefore contributes more strongly to the net turbulence prouction than the vertical component of shear prouction. It was also observe that, for the parameter range stuie here, the flux Richarson number Ri f epens on the graient Richarson number Ri but not on the shear inclination angle. It is interesting that the normalize turbulence prouction is strongly influence by the angle, but that the flux Richarson number remains unaffecte. An aitional series of simulations was performe to stuy the influence of the shear inclination angle of the turbulence evolution in a strongly stratifie meium with Ri 2.0 It was foun that the angle effect persists in high Richarson number flow. The value of the critical Richarson number Ri cr, at which growth of the turbulent kinetic energy is suppresse, is about an orer of magnitue larger in the case of horizontal shear compare with the case of vertical shear. The flux Richarson number Ri f was again foun to be relatively insensitive to the angle variation. Aitional simulations confirme that our earlier observation of the stabilizing effect of a large initial shear number SK/ in vertically sheare flow applies in horizontally sheare flow too. ACKNOWLEDGMENTS The stuy of the turbulence evolution in a stably stratifie flui force by horizontal shear was suggeste by Charles W. Van Atta. The authors want to acknowlege his ongoing support of this work. This stuy is supporte by the Office of Naval Research, Physical Oceanography Program, through Grant No. ONR N Supercomputer time was provie by the San Diego Supercomputer Center SDSC an the US Army Corps of Engineers Waterways Experiment Station WES. 1 L. F. 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