Generation of Large-Scale Vorticity in a Homogeneous Turbulence with a Mean Velocity Shear

PHYSICAL REVIEW E 68, 016311 ͑2003͒

Generation of large-scale vorticity in a homogeneous turbulence with a mean velocity shear

Tov Elperin,* Nathan Kleeorin,† and Igor Rogachevskii‡ The Pearlstone Center for Aeronautical Engineering Studies, Department of Mechanical Engineering, Ben-Gurion University of the Negev, Beer-Sheva 84105, P.O. Box 653, Israel ͑Received 14 January 2003; revised manuscript received 6 May 2003; published 25 July 2003͒ An effect of a mean velocity shear on a turbulence and on the effective force which is determined by the gradient of the Reynolds stresses is studied. Generation of a mean vorticity in a homogeneous incompressible nonhelical turbulent flow with an imposed mean velocity shear due to an excitation of a large-scale instability is found. The instability is caused by a combined effect of the large-scale shear motions ͑‘‘skew-induced’’ deflection of equilibrium mean vorticity͒ and ‘‘Reynolds stress-induced’’ generation of perturbations of mean vorticity. Spatial characteristics of the instability, such as the minimum size of the growing perturbations and the size of perturbations with the maximum growth rate, are determined. This instability and the dynamics of the mean vorticity are associated with Prandtl’s turbulent secondary flows.

DOI: 10.1103/PhysRevE.68.016311 PACS number͑s͒: 47.27.Nz

I. INTRODUCTION a large-scale instability can be excited which results in a mean-vorticity production. This instability is caused by a Vorticity generation in turbulent and laminar flows was combined effect of the large-scale shear motions ͑‘‘skew- studied experimentally, theoretically, and numerically in a induced’’ deflection of equilibrium mean vorticity͒ and number of publications ͑see, e.g., Refs. ͓1–16͔͒. For in- ‘‘Reynolds stress-induced’’ generation of perturbations of stance, a mechanism of the vorticity production in laminar mean vorticity. The skew-induced deflection of the equilib- compressible fluid flows consists in the misalignment of rium mean vorticity W¯ (s) is determined by (W¯ (s) )U˜ term ͓ ͔ •“ pressure and density gradients 12,13 . It was shown in Ref. in the equation for the mean vorticity, where U˜ are perturba- ͓ ͔ 13 that the vorticity generation represents a generic prop- tions of the mean velocity ͑see below͒. The Reynolds stress- erty of any slow nonadiabatic laminar gas flow. In incom- induced generation of a mean vorticity is determined by pressible nonhelical flows, this effect does not occur. The ϫF F “ , where is an effective force caused by a gradient of role of small-scale vorticity production in incompressible the Reynolds stresses. ͓ ͔ turbulent flows was discussed in Ref. 10 . This paper is organized as follows. In Sec. II, the govern- On the other hand, generation and dynamics of the mean ing equations are formulated. In Sec. III, the general form of ͑ vorticity are associated with turbulent secondary flows see, the Reynolds stresses in a homogeneous nonhelical turbu- ͓ ͔͒ e.g., Refs. 1,3–8 . These flows, e.g., arise at the lateral lence with an imposed mean-velocity shear is found using boundaries of three-dimensional thin shear layers whereby simple symmetry reasoning, and the mechanism for the ͑ ͒ longitudinal streamwise mean vorticity is generated by a large-scale instability caused by a combined effect of the ͓ ͔ lateral deflection or ’’skewing‘‘ of an existing shear layer 3 . large-scale shear motions and Reynolds stress-induced gen- The skew-induced streamwise mean vorticity generation cor- eration of perturbations of the mean vorticity is discussed. In responds to Prandtl’s first kind of secondary flows. In turbu- Sec. IV, the equation for the second moment of velocity fluc- lent flows, e.g., in straight noncircular ducts, streamwise tuations in a homogeneous turbulence with an imposed mean mean vorticity can be generated by the Reynolds stresses. velocity shear is derived. This allows us to study an effect of The latter is Prandtl’s second kind of turbulent secondary a mean velocity shear on a nonhelical turbulence and to cal- flow, and it ‘‘has no counterpart in laminar flow and cannot culate the effective force determined by the gradient of the be described by any turbulence model with an isotropic eddy Reynolds stresses. Using the derived mean-field equation for ͓ ͔ viscocity’’ 3 . vorticity, in Sec. IV we studied the large-scale instability Note that the effect of the generation of large-scale vor- which causes the mean vorticity production. ticity in the helical turbulence due to hydrodynamical ␣ effect was considered in Refs. ͓14–16͔, where ␣ is determined II. THE GOVERNING EQUATIONS by the hydrodynamical helicity of the turbulent flow. In the present study, we demonstrated that in a homoge- Our goal is to study an effect of the mean velocity shear neous incompressible nonhelical turbulent flow with an im- on a nonhelical turbulence and on the dynamics of a mean posed mean velocity shear ͑when the ␣ effect does not exist͒, vorticity. The equation for the evolution of vorticity W ϵ ϫ “ v reads Wץ ;Electronic address: [email protected]* ϭ ϫ͑vϫWϪ␯ ϫW͒, ͑1͒ “ “ tץ URL:http://www.bgu.ac.il/˜elperin †Electronic address: [email protected] ‡ ϭ ␯ Electronic address: [email protected]; where v is the fluid velocity with “•v 0 and is the kine- URL:http://www.bgu.ac.il/˜gary matic viscosity. This equation follows from the Navier-

Stokes equation. In this study, we use a mean-field approach whereby the velocity and vorticity are separated into the mean and fluctuating parts: vϭU¯ ϩu and WϭW¯ ϩw, the fluctuating parts have zero mean values, and U¯ ϭ͗v͘, W¯ ϭ͗W͘. Averaging Eq. ͑1͒ over an ensemble of fluctuations, we obtain an equation for the mean vorticity W¯ :

¯Wץ ϭ ϫ͑U¯ ϫW¯ ϩ͗uϫw͘Ϫ␯ ϫW¯ ͒. ͑2͒ “ “ tץ

Note that the effect of turbulence on the mean vorticity is determined by the Reynolds stresses ͗uiu j͘, because ͗ ϫ ͘ ϭϪ ͗ ͘ϩ 1 ͗ 2͘ ͑ ͒ u w i “ j uiu j 2 “i u , 3 and the curl of the last term in Eq. ͑3͒ vanishes. We consider a turbulent flow with an imposed mean ve- ¯ (s) ¯ (s) locity shear “iU , where U is a steady-state solution of the Navier-Stokes equation for the mean-velocity field. In FIG. 1. Mechanism for the tangling turbulence: generation of order to study a stability of this equilibrium, we consider anisotropic velocity fluctuations by tangling of the mean velocity gradients with the background turbulence. perturbations U˜ of the mean velocity, i.e., the total mean (s) velocity is U¯ ϭU¯ ϩU˜ . Similarly, the total mean vorticity is of the mean magnetic field with the velocity fluctuations. The ¯ ϭ ¯ (s)ϩ ˜ ¯ (s)ϭ ϫ ¯ (s) ˜ ϭ ϫ ˜ W W W, where W “ U and W “ U. Thus, Reynolds stresses in a turbulent flow with a weak mean ve- the linearized equation for the small perturbations of the locity shear form another example of a tangling turbulence. mean vorticity, W˜ ϭW¯ ϪW¯ (s), is given by Indeed, they are strongly anisotropic in the presence of shear and have a steeper spectrum (ϰkϪ7/3) than a Kolmogorov ͑ ͓ ͔͒ -W˜ background turbulence see, e.g., Refs. 17,22–25 . The anץ ϭ ϫ͑U¯ (s)ϫW˜ ϩU˜ ϫW¯ (s)ϩFϪ␯ ϫW˜ ͒, ͑4͒ isotropic velocity fluctuations of tangling turbulence were “ “ tץ studied first by Lumley ͓22͔. Different properties of anisotropic turbulence, such as anisotropic scaling behavior and F ϭϪ ͓ ¯ Ϫ ¯ (s) ͔ where i j f ij(U) f ij(U ) is the effective force and anomalous scalings, were studied recently in ͓26–28͔. ϭ͗ ͘ “ ͑ ͒ ͑ ͒ f ij uiu j . Equation 4 is derived by subtracting Eq. 2 We assumed that the generated tangling turbulence does written for W¯ (s) from the corresponding equation for the not affect the background turbulence. This implies that we mean vorticity W¯ . In order to obtain a closed system of considered one-way coupling due to a weak large-scale ve- equations, in Sec. IV we derived an equation for the effective locity shear. force F. Equation ͑4͒ determines the dynamics of perturba- The general form of the Reynolds stresses in a turbulent tions of the mean vorticity. In the following sections, we will flow with a mean-velocity shear can be obtained from simple show that under certain conditions the large-scale instability symmetry reasoning. Indeed, the Reynolds stresses ͗uiu j͘ can be excited which causes the mean vorticity production. form a symmetric true tensor. In a turbulent flow with an imposed mean-velocity shear, the Reynolds stresses depend ¯ III. THE QUALITATIVE DESCRIPTION on the true tensor “ jUi , which can be written as a sum of ( Ûץ)the symmetric and antisymmetric parts, i.e., U¯ ϭ In this section, we discuss the mechanism of the large- “ j i ij ¯ ϭ ¯ ϩ ˆ ץ ¯ Ϫ ␧ scale instability. The mean velocity shear can affect a turbu- (1/2) ijkWk , where ( U)ij (“iU j “ jUi)/2 is the true ¯ ϭ ϫ ¯ ͑ ͒ lence. The reason is that additional strongly anisotropic ve- tensor and W “ U is the mean vorticity pseudo-vector . locity fluctuations can be generated by tangling of the mean We take into account the effect which is linear in perturba- ˜ ϭ ˜ ϩ ˜ ץ ˜ ˜ ץ -velocity gradients with the isotropic and homogeneous back tions ( U)ij and W, where ( U)ij (“iU j “ jUi)/2. Thus, ground turbulence ͑see Fig. 1͒. The source of energy of the general form of the Reynolds stresses can be found using , U˜ ) , M , N , H , and Gץ) :this ‘‘tangling turbulence’’ is the energy of the background the following true tensors ͓ ͔ ij ij ij ij ij turbulence 17 . where The tangling turbulence is a universal phenomenon, e.g., it was introduced by Wheelon ͓18͔ and Batchelor et al. ͓19͔ ͒ ͑ ͒ ˜ ץ͑ ͒(s) ¯ ץϩ͑ ͒ ˜ ץ͑ ͒(s) ¯ ץϭ͑ for a passive scalar and by Golitsyn ͓20͔ and Moffatt ͓21͔ for M ij U im U mj U jm U mi , 5 a passive vector ͑magnetic field͒. Anisotropic fluctuations of ͒ ͑ ͒(s) ¯ ץ s)͒ ϩ␧) ¯ ץ a passive scalar ͑e.g., the number density of particles or tem- ϭ ˜ ␧ Nij Wn͓ nim͑ U mj njm͑ U mi͔, 6 perature͒ are produced by tangling of gradients of the mean passive scalar field with a random velocity field. Similarly, ͒ ͑ ͔ ͒ ˜ ץ͑ ϩ␧ ͒ ˜ ץ͑ ϭ ¯ (s)͓␧ anisotropic magnetic fluctuations are generated by tangling Hij Wn nim U mj njm U mi , 7

016311-2 GENERATION OF LARGE-SCALE VORTICITY IN A . . . PHYSICAL REVIEW E 68, 016311 ͑2003͒

ϭ ¯ (s) ˜ ϩ ¯ (s) ˜ ͑ ͒ Gij Wi W j W j Wi , 8 s) ϭ ¯ (s)ϩ ¯ (s) ␧) ¯ ץ ( U )ij (“iU j “ jUi )/2 and ijk is the fully antisymmetric Levi-Civita` tensor ͑pseudotensor͒. Therefore, the Reynolds stresses have the following general form:

,͒ U˜ ͒ Ϫl2͑B M ϩB N ϩB H ϩB Gץ͑ f ͑U˜ ͒ϭϪ2␯ ij T ij 0 1 ij 2 ij 3 ij 4 ij ͑9͒ where Bk are the unknown coefficients, l0 is the maximum scale of turbulent motions, ␯ ϭl u /a is the turbulent vis- T 0 0 * cosity with the factor a Ϸ3 –6, and u is the characteristic * 0 turbulent velocity in the maximum scale of turbulent motions 2 ͑ ͒ l0. The parameter l0 in Eq. 9 was introduced using dimensional arguments. The first term on the right hand side of Eq. FIG. 2. Mechanism for ‘‘skew-induced’’ generation of perturba- ˜ ͑9͒ describes the standard isotropic turbulent viscosity, tions of the mean vorticity Wx by quasi-inviscid deflection of the ¯ (s) whereas the other terms are determined by fluctuations equilibrium mean vorticity W ez . In particular, the mean vorticity ˜ ¯ (s) Wxex is generated by an interaction of perturbations of the mean caused by the imposed velocity shear iU . “ j ˜ ¯ (s)ϭ Let us study the evolution of the mean vorticity using vorticity Wyey and the equilibrium mean vorticity W Sez , i.e., ˜ ϰ ¯ (s) ˜ ϰ ˜ ϫ ¯ (s) ͑ ͒ ͑ ͒ F ϭϪ ˜ Wxex (W )Uxex Wyey W . Eqs. 4 and 9 , where i “ j f ij(U) is the effective •“ force. We consider a homogeneous divergence-free turbu- particular, the mean vorticity W˜ e is generated from W˜ e lence with a mean-velocity shear, e.g., U¯ (s)ϭ(0,Sx,0) and x x y y by equilibrium shear motions with the mean vorticity W¯ (s) W¯ (s)ϭ(0,0,S). For simplicity, we use perturbations of the ϭSe , i.e., W˜ e ϰ(W¯ (s) )U˜ e ϰW˜ e ϫW¯ (s) ͑see Fig. 2͒. ˜ ϭ ˜ ˜ z x x •“ x x y y mean vorticity in the form W Wx(z),Wy(z),0 . Then Eq. Here e , e , and e are the unit vectors along x, y, and z axis, ͑ ͒ „ … x y z 4 can be written as Ϫ␤ 2 ˜ Љ respectively. On the other hand, the first term, Sl0Wx in Eq. ͑11͒ determines a Reynolds stress-induced generation of ˜Wץ x ˜ ϭSW˜ ϩ␯ W˜ Љ , ͑10͒ perturbations of the mean vorticity Wy by turbulent Reynolds y T x ץ t stresses ͑see Fig. 3͒. In particular, this term is determined by ( ϫF) , where F is a gradient of the Reynolds y “ ˜ ץ Wy ˜ ϭϪ␤Sl2W˜ Љϩ␯ W˜ Љ , ͑11͒ stresses. This implies that the mean vorticity Wyey is t 0 x T y gener-ated by an effective anisotropic viscous termץ ϰϪl2⌬(W˜ e )U¯ (s)(x)e ϰϪl2SW˜ Љe . This mechanism ϭ͓ ϩ ϩ Ϫ ͔ 0 x x•“ y 0 x y␤ 2 ץ ˜ 2ץЉϭ ˜ where Wx Wx / z , B1 2(B2 B3) 4B4 /4. In of the generation of perturbations of the mean vorticity W˜ e ͑ ͒ 2 ˜ ЉӶ ˜ y y Eq. 10 , we took into account that l0Wy Wy , i.e., the char- can be interpreted as a stretching of the perturbations of the acteristic scale L of the mean vorticity variations is much ˜ ¯ (s) W mean vorticity Wxex by the equilibrium shear motions U larger than the maximum scale of turbulent motions l . This ϭ ͑ 0 Sxey during the turnover time of turbulent eddies see Fig. assumption corresponds to the mean-field approach. For 3͒. The growth rate of this instability is caused by a com- derivation of Eqs. ͑10͒ and ͑11͒, we used the identities pre- bined effect of the sheared motions ͑skew-induced genera- sented in Appendix A. tion͒ and the Reynolds stress-induced generation of pertur- We seek for a solution of Eqs. ͑10͒ and ͑11͒ in the form bations of the mean vorticity. ϰexp(␥tϩiKz). Thus when ␤Ͼ0, perturbations of the mean This large-scale instability is similar to a mean-field mag- vorticity can grow in time and the growth rate of the insta- netic dynamo instability caused by the ⍀ϫJ effect ͑see, e.g., bility is given by Refs. ͓29,30͔͒ or the shear-current effect ͓31͔, where J is the electric current and ⍀ is the angular velocity. Indeed, the ␥ϭͱ␤Sl KϪ␯ K2. ͑12͒ first term in Eq. ͑10͒ is similar to the differential rotation ͑or 0 T large-scale shear motions͒ which causes a generation of a The maximum growth rate of perturbations of the mean toroidal mean magnetic field by a stretching of the poloidal ␥ ϭ␤ 2 ␯ ϭ ͑ vorticity, max (Sl0) /4 T , is attained at K Km mean magnetic field with the differential rotation or by ϭͱ␤ ␯ ␥Ͼ ͒ Sl0/2 T . The sufficient condition 0 for the excita- large-scale shear motions . On the other hand, the first term Ͼ ␲ ͱ␤␶ ͑ ͒ ⍀ϫ ͑ tion of the instability reads LW /l0 2 /(a 0S), where in Eq. 11 is similar to the J effect see, e.g., Refs. ϵ ␲ * ␶ Ӷ ͓ ͔͒ ͓ ͔ LW 2 /K and we consider a weak velocity shear 0S 1. 29,30 , or to the shear-current effect 31 . These effects Now let us discuss the mechanism of this instability using result in the generation of a poloidal mean magnetic field ⍀ϫ a terminology from Ref. ͓3͔. The first term SW˜ from the toroidal mean magnetic field. The J effect is y related to an interaction of the mean rotation and electric ϭ(W¯ (s) )U˜ in Eq. ͑10͒ describes a skew-induced genera- •“ x current in a homogeneous turbulent flow, while the shear- ˜ tion of perturbations of the mean vorticity Wx by quasi- current effect occurs due to an interaction of the mean vor- inviscid deflection of the equilibrium mean vorticity W¯ (s).In ticity and electric current in a homogeneous aniso-

016311-3 ELPERIN, KLEEORIN, AND ROGACHEVSKII PHYSICAL REVIEW E 68, 016311 ͑2003͒

where p are the pressure fluctuations, ␳ is the fluid density, F(st) is an external stirring force with a zero-mean value, and ϭ Ϫ ϩ␯⌬ UN ͗(u )u͘ (u )u u. We consider a turbulent •“ •“ ϭ ␯ӷ flow with large Reynolds numbers (Re l0u0 1). We assumed that there is a separation of scales, i.e., the maximum scale of turbulent motions l0 is much smaller than the characteristic scale of the inhomogeneities of the mean fields. Using Eq. ͑13͒, we derived an equation for the second mo- ϵ͐ ment of the turbulent velocity field f ij(k,R) ͗ui(k ϩ Ϫ ϩ ͘ K/2)u j( k K/2) exp(iK•R)dK: ͒ ץ f ij͑k,R ϭI ͑U¯ ͒f ϩF ϩ f (N) ͑14͒ t ijmn mn ij ijץ

͑see Appendix B͒, where

͑ ¯ ͒ϭͩ ␦ ␦ ϩ ␦ ␦ Ϫ␦ ␦ ␦ Iijmn U 2kiq mp jn 2k jq im pn im jq np

ץ FIG. 3. Mechanism for a ‘‘Reynolds stress-induced’’ generation Ϫ␦ ␦ ␦ ϩ␦ ␦ ͪ ¯ ͑ ͒ pUq , 15 “ ץ iq jn mp im jnkq ˜ k p of perturbations of the mean vorticity Wyey by turbulent Reynolds ˜ stresses. In particular, the mean vorticity Wyey is generated by an and R and K correspond to the large scales, and r and k to effective anisotropic viscous term ϰϪl2⌬(W˜ e )U¯ (s)(x)e ͑ ͒ ϭ 2 (N) 0 x x•“ y the small scales see Appendix B , kij kik j /k , f ij (k,R) ϰϪ 2 ˜ Љ l0SWx ey . This mechanism of the generation of perturbations of is the third moment appearing due to the non- ץ ץthe mean vorticity W˜ e can be interpreted as a stretching of the ϭ y y linear term, “ / R, ˜ perturbations of the mean vorticity Wxex by the equilibrium shear ¯ (s)ϭ F ͑k,R͒ϭ͗˜F ͑k,R͒u ͑Ϫk,R͒͘ϩ͗u ͑k,R͒˜F ͑Ϫk,R͒͘ motions U Sxey during the turnover time of turbulent eddies. ij i j i j and tropic turbulent flow of a conducting fluid. The magnetic dynamo instability is a combined effect of the nonuniform ˜F͑k,R,t͒ϭϪkϫ͓kϫF(st)͑k,R͔͒/k2. mean flow ͑differential rotation or large-scale shear motions͒ and turbulence effects ͑anisotropic turbulent motions which Equation ͑14͒ is written in a frame moving with a local ve- cause the ⍀ϫJ effect and the shear-current effect͒. locity U¯ of the mean flow. In Eqs. ͑14͒ and ͑15͒, we ne- .(͉ ¯On the other hand, the kinematic magnetic dynamo insta- glected small terms which are of the order of O(ٌ͉3U bility is different from the instability of the mean vorticity Note that Eqs. ͑14͒ and ͑15͒ do not contain terms propor- ͉ ¯ although they are governed by similar equations. The mean ٌ͉2 ¯ ϭ ϫ ¯ tional to O( U ). vorticity W “ U is directly determined by the velocity ¯ ϭ ¯ (s)ϩ ˜ ¯ The total mean-velocity is U U U, where we consid- field U, while the magnetic field depends on the velocity ered a turbulent flow with an imposed mean-velocity shear field through the induction equation. ¯ (s) “iU . Now let us introduce a background turbulence with ¯ ϭ zero gradients of the mean fluid velocity “iU j 0. The IV. EFFECT OF A MEAN-VELOCITY SHEAR ON A background turbulence is determined by the equation (ϭ ϩ (N0 ץ (0) ץ TURBULENCE AND LARGE-SCALE INSTABILITY f ij / t Fij f ij , where the superscript (0) corresponds to the background turbulence, and we assumed that the ten- In this section, we study quantitatively an effect of a sor Fij(k,R), which is determined by a stirring force, is mean-velocity shear on a nonhelical turbulence. This allows independent of the mean-velocity. Equation for the devia- F us to derive an equation for the effective force and to tions f Ϫ f (0) from the background turbulence is given by study the dynamics of the mean vorticity. ij ij ͒(f Ϫˆf (0ˆ ͑ץ ϭ͓Iˆ͑U¯ (s)͒ϩIˆ͑U˜ ͔͒ˆf ϩˆf (N)Ϫˆf (N0), ͑16͒ tץ A. Method of derivations

To study an effect of a mean-velocity shear on a turbu- ˆϵ ˆ (N) lence, we used an equation for fluctuations u(t,r) that is where we used the following notations: f f ij(k,R), f ϵ (N) ˆ (N0)ϵ (N0) ˆ (0)ϵ (0) obtained by subtracting equation for the mean field from the f ij (k,R), f f ij (k,R), f f ij (k,R), and ˆ ¯ ˆϵ ¯ corresponding equation for the total field: I(U) f Iijmn(U) f mn(k,R). Equation ͑16͒ for the deviations of the second moments in u pץ ϭϪ͑U¯ ͒uϪ͑u ͒U¯ Ϫ “ ϩF(st)ϩU , ͑13͒ k space contains the deviations of the third moments and a t •“ •“ ␳ N problem of closing the equations for the higher momentsץ

016311-4 GENERATION OF LARGE-SCALE VORTICITY IN A . . . PHYSICAL REVIEW E 68, 016311 ͑2003͒ arises. Various approximate methods have been proposed for Lˆ ͑ ˆf Ϫˆf (0)͒ϭ␶͑k͓͒Iˆ͑U¯ (s)͒ϩIˆ͑U˜ ͔͒ˆf (0), ͑19͒ the solution of problems of this type ͑see, e.g., Refs. ͓32– 34͔͒. The simplest procedure is the ␶ approximation which ˆ ϵ ϭ␦ ␦ Ϫ␶ ¯ (s) ϩ ˜ where L Lijmn im jn (k)͓Iijmn(U ) Iijmn(U)͔, and was widely used for the study of different problems of tur- we used Eq. ͑17͒. The solution of Eq. ͑19͒ yields the expres- bulent transport ͑see, e.g., Refs. ͓32,35–37͔͒. It allows us to sion for the second moment ˆf ϵ f (k,R): express the deviations of the third moments ˆf (N)Ϫˆf (N0) in k ij space in terms of those for the second moments ˆf Ϫˆf (0) by ˆϷˆ ¯ (s)͒ϩ␶ ͒ ˆ ˜ ͒ϩˆ ¯ (s)͒␶ ͒ˆ ˜ ͒ assuming that f f ͑U ͑k ͓I͑U I͑U ͑k I͑U ϩIˆ͑U˜ ͒␶͑k͒Iˆ͑U¯ (s)͔͒ˆf (0), ͑20͒ ˆf Ϫˆf (0) ˆf (N)Ϫˆf (N0)ϭϪ , ͑17͒ ␶͑k͒ where where ␶(k) is the correlation time of the turbulent velocity ˆf ͑U¯ (s)͒ϭˆf (0)ϩ␶͑k͓͒Iˆ͑U¯ (s)͒ϩIˆ͑U¯ (s)͒␶͑k͒Iˆ͑U¯ (s)͔͒ˆf (0). field. Here we assumed that the time ␶(k) is independent of the gradients of the mean fluid velocity because in the frame- In Eq. ͑20͒, we neglected terms which are of the order of work of the mean-field approach we may only consider a ͉ ˜ ͉2 ͉ ¯ (s)͉2͉ ˜ ͉ O( “U ) and O( “U “U ). The first term in the equa- ␶ ͉ ¯ ͉Ӷ ␶ ϭ (s) weak shear: 0 “U 1, where 0 l0 /u0. tion for ˆf (U¯ ) is independent of the mean-velocity shear The ␶ approximation is, in general, similar to the eddy and it describes the background turbulence, while the second damped Quasinormal Markowian ͑EDQNM͒ approximation. and the third terms in this equation determine an effect of the However, there is a principlal difference between these two mean-velocity shear on turbulence. approaches ͑see Refs. ͓32,34͔͒. The EDQNM closures do not relax to the equilibrium, and do not describe properly the C. Effective force motions in the equilibrium state. In the EDQNM theory, there is no dynamically determined relaxation time, and no Equation ͑20͒ allows us to determine the effective force: ͓ ͔ F ϭϪ ͐˜ ˜ϭˆϪˆ ¯ (s) slightly perturbed steady state can be approached 32 .Inthe i “ j f ij(k,R)dk, where f f f (U ), and we used ␶ ˜ϵ˜ approximation, the relaxation time for small departures the notation f f ij(k,R). The integration in k space yields from the equilibrium is determined by the random motions in the second moment ˜f (R)ϭ͐˜f (k,R)dk: the equilibrium state, but not by the departure from the equi- ij ij librium ͓32͔. Analysis performed in Ref. ͓32͔ showed that Ϫ 2 ͒ ˜ ץ͑ ϭϪ ␯͒ ͑ ˜ the ␶ approximation describes the relaxation to the equilib- f ij R 2 T U ij l0 rium state ͑the background turbulence͒ more accurately than ϫ͓4C M ϩC ͑N ϩH ͒ϩC G ͔, the EDQNM approach. 1 ij 2 ij ij 3 ij ͑21͒ ␶ ͑ ͒ Note that we applied the approximation 17 only to where C ϭ8(q2Ϫ13qϩ40)/315, C ϭ2(6Ϫ7q)/45, C study the deviations from the background turbulence which 1 2 3 ϭϪ2(qϩ2)/45, and the tensors M , N , H , and G are are caused by the spatial derivatives of the mean-velocity. ij ij ij ij determined by Eqs. ͑5͒–͑8͒. In Eq. ͑21͒, we omitted terms The background turbulence is assumed to be known. In this ϰ␦ because they do not contribute to ϫF ͓see Eq. ͑4͒ for study, we used the model of an isotropic, homogeneous, and ij “ ˜ ͑ ͒ nonhelical background turbulence: perturbations of the mean vorticity W]. To derive Eq. 21 , we used the identities presented in Appendix A. Equations ͑ ͒ ͑ ͒ ϭ ϭ ϭ ϭ 9 and 21 yield B1 4C1 , B2 B3 C2 , and B4 C3. u2 Note that the mean velocity gradient U¯ (s) causes gen- (0)͑ ͒ϭ 0 ͑ ͒E͑ ͒ ͑ ͒ “i f ij k,R Pij k k , 18 eration of anisotropic velocity fluctuations ͑tangling turbu- 8␲k2 lence͒. Inhomogeneities of perturbations of the mean veloc- ˜ ϭ␦ Ϫ ␦ ␶ ity U produce additional velocity fluctuations, so that the where Pij(k) ij kij , ij is the Kronecker tensor, (k) Ϫ Reynolds stresses ˜f (R) are the result of a combined effect ϭ2␶ ¯␶(k), E(k)ϭϪd¯␶(k)/dk, ¯␶(k)ϭ(k/k )1 q,1ϽqϽ3 ij 0 0 of two types of velocity fluctuations produced by the tan- is the exponent of the kinetic energy spectrum ͑e.g., qϭ5/3 ¯ (s) ˜ for Kolmogorov spectrum͒, and k ϭ1/l . We assumed that gling of mean gradients iU and iU by a small-scale 0 0 “ ͑ “͒ the generated tangling turbulence does not affect the back- Kolmogorov turbulence. Equation 21 allows us to deter- F ϭϪ ˜ ground turbulence. This implies that we considered a one- mine the effective force i “ j f ij(R). way coupling due to a weak large-scale velocity shear. D. The large-scale instability in a homogeneous turbulence B. Equation for the second moment of velocity fluctuations with a mean-velocity shear We assume that the characteristic time of variation of the Let us study the evolution of the mean vorticity using Eq. ͑ ͒ second moment f (k,R) is substantially larger than the cor- 21 for the Reynolds stresses. Consider a homogeneous non- ij (s) relation time ␶(k) for all turbulence scales. Thus in a steady helical turbulence with a mean-velocity shear, e.g., U¯ state, Eq. ͑16͒ reads ϭ(0,Sx,0) and W¯ (s)ϭ(0,0,S). For simplicity, we consider

016311-5 ELPERIN, KLEEORIN, AND ROGACHEVSKII PHYSICAL REVIEW E 68, 016311 ͑2003͒ perturbations of the mean vorticity in the form W˜ stability, such as the minimum size of the growing pertur- ϭ ˜ ˜ ͑ ͒ ͑ ͒ bations and the size of perturbations with the maximum „Wx(z),Wy(z),0…. Then Eq. 4 reduces to Eqs. 10 and growth rate. ͑ ͒ ␤ϭ ϩ Ϫ ϭ 2Ϫ ϩ 11 , where C1 C2 C3 4(2q 47q 108)/315. We The results obtained in this study are different from those ͑ ͒ ͑ ͒ seek for a solution of Eqs. 10 and 11 in the form discussed in Refs. ͓14–16͔, whereby the generation of large- ϰexp(␥tϩiKz). Thus, the growth rate of perturbations of the scale vorticity in the helical turbulence occurs due to hydro- ␥ϭͱ␤ Ϫ␯ 2 mean vorticity is given by Sl0K TK . The maxi- dynamic ␣ effect. The latter effect is associated with the term mum growth rate of perturbations of the mean vorticity, ␣W˜ in the equation for the mean vorticity, where ␣ is deter- ␥ ϭ␤ 2 ␯ Ϸ 2␶ ϭ max (Sl0) /4 T 0.1a S 0, is attained at K Km mined by the hydrodynamic helicity of the turbulent flow. ϭͱ␤ ␯ * Sl0/2 T. Here we used that for a Kolmogorov spectrum We considered a nonhelical background homogeneous turbu- ϭ ␤Ϸ (q 5/3) of the background turbulence, the factor 0.45. lence which implies that Eq. ͑4͒ for the mean vorticity W˜ ␥Ͼ The sufficient condition 0 for the excitation of the insta- does not have the term ␣W˜ . We studied a linear stage of the Ͼ ␲ ͱ␤␶ ␶ Ӷ ͑ bility reads LW /l0 2 /(a 0S). Since 0S 1 we con- large-scale instability which is saturated by nonlinear effects, * ͒ ӷ sidered a weak velocity shear , the scale LW l0 and, there- but not a finite time growth of large-scale vorticity as de- fore, there is a separation of scales as required in our model. scribed in Ref. ͓16͔. Now we consider perturbations of the mean vorticity The analyzed effect of the mean vorticity production may ˜ ϭ ˜ ˜ be of relevance in different turbulent industrial, environmen- which depend on x and z, i.e., W „Wx(x,z),Wy(x,z),0…. Then Eq. ͑4͒ reduces to tal, and astrophysical flows ͑see, e.g., Refs. ͓3–8,38–41͔͒. Thus, e.g., the suggested mechanism can be used in the analysis of the flows associated with Prandtl’s turbulent sec- ˜2Wץ ˜Wץ ⌬ x ϭ y ϩ␯ ⌬2 ˜ ͑ ͒ ondary flows ͑see, e.g., Refs. ͓3–8͔͒. These flows, e.g., arise S Wx , 22 z2 T in straight noncircular ducts, at the lateral boundaries ofץ tץ three-dimensional thin shear layers, etc. However, the results obtained in this study cannot be valid 2ץ 2ץ 2ץ ˜ ץ 2ץ Wy in the most general cases. We made the following assump- ϭ⌬ͫ␤Sl2ͩ ␤ Ϫ ͪ W˜ ϩ␯ W˜ ͬ, ,z2 y tions about the turbulence. We considered a homogeneousץ z2 x Tץ x2ץ * t 0ץ z2ץ ͑23͒ isotropic, incompressible, and nonhelical background turbulence ͑i.e., the turbulence without the large-scale shear͒. The where ␤ ϭ2(C ϪC )/␤. For 9ϭ5/3, the parameter * 1 2 weak mean velocity shear affects the background turbulence, ␤ Ϸ3.5. Therefore, the growth rate of perturbations of the * i.e., it causes generation of the additional strongly aniso- mean vorticity is given by tropic velocity fluctuations ͑tangling turbulence͒ by tangling of the mean-velocity gradients with the background turbu- ␥ϭ ͱ␤ Ϫ ϩ␤ ͒ 2␪ Ϫ␯ 2 ͑ ͒ lence. We assumed that the generated tangling turbulence Sl0K ͓1 ͑1 sin ͔ K . 24 * T does not affect the background turbulence. This implies that The maximum growth rate of perturbations of the mean vor- we considered a one-way coupling due to a weak large-scale velocity shear. In Secs. III and IV D, we considered a linear ticity, ␥ ϭ␤(Sl )2͓1Ϫ(1ϩ␤ )sin2 ␪͔/4␯ , is attained at K max 0 * T velocity shear to derive the specific results for the large-scale ϭK ϭSl ͱ␤͓1Ϫ(1ϩ␤ )sin2 ␪͔/2␯ . The large-scale in- m 0 * T instability. Thus, we studied simple physical mechanisms to stability may occur when ͉␪͉Ͻ28°. In the case of ␪ϭ0, the describe an initial stage of the mean vorticity generation. The large-scale instability is more effective, i.e., the mean vortic- simple model considered in our paper can only mimic the ity grows with a maximum possible rate. The mechanism of flows associated with the turbulent secondary flows. Clearly, this instability is discussed in Sec. III and it is associated the comprehensive theoretical and numerical studies are re- with a combined effect of the skew-induced deflection of quired for a quantitative description of the secondary flows. equilibrium mean vorticity due to the sheared motions and The obtained results may be also important in astrophys- the Reynolds stress-induced generation of perturbations of ics, e.g., in extragalactic clusters and in interstellar clouds. mean vorticity. The extragalactic clusters are nonrotating objects with a homogeneous turbulence in the center of an extragalactic clus- V. CONCLUSIONS AND APPLICATIONS ter. Sheared motions between interacting clusters can cause We discussed an effect of a mean-velocity shear on a ho- an excitation of the large-scale instability which results in the mogeneous nonhelical turbulence and on the effective force mean vorticity production and formation of large-scale vor- which is determined by the gradient of the Reynolds stresses. tices. Dust particles can be trapped by these vortices to en- hance agglomeration of material and formation of particle We demonstrated that in a homogeneous incompressible ͓ ͔ nonhelical turbulent flow with an imposed mean velocity inhomogeneities 39–41 . The sheared motions can also oc- shear, a large-scale instability can be excited which results in cur between interacting interstellar clouds, whereby the dy- a mean vorticity production. This instability is caused by a namics of the mean vorticity is important. combined effect of the large-scale shear motions ͑skew- ACKNOWLEDGMENTS induced deflection of equilibrium mean vorticity͒ and Rey- nolds stress-induced generation of perturbations of mean We are indebted to A. Tsinober for illuminating discus- vorticity. We determined the spatial characteristics of the in- sions. This work was partially supported by The German-

016311-6 GENERATION OF LARGE-SCALE VORTICITY IN A . . . PHYSICAL REVIEW E 68, 016311 ͑2003͒

͑ ͒ ͒ ͒ Israeli Project Cooperation DIP administrated by the Fed- ͗ui͑x u j͑y ͘ eral Ministry of Education and Research ͑BMBF͒ and by INTAS Program Foundation ͑Grant Nos. 99-348 and 00- ϭ ͵ ͗u ͑k ͒u ͑k ͒͘exp͓i͑k xϩk y͔͒dk dk 0309͒. i 1 j 2 1• 2• 1 2

APPENDIX A: IDENTITIES USED FOR DERIVATION ϭ ͵ ͑ ͒ ͑ ͒ f ij k,R exp ik•r dk, OF EQS. „10…, „11…, AND „21… To derive Eqs. ͑10͒ and ͑11͒, we used the following iden- ͑ ͒ϭ ͗ ͑ ϩ ͒ ͑Ϫ ϩ ͒͘ tities: f ij k,R ͵ ui k K/2 u j k K/2

ϫ ϭ͑ 2 ͒͑ ˜ ˜ ͒ ϫexp͑iK R͒dK “ M SK /4 Wy ,Wx ,0 , • ͑ ͓ ͔͒ ϫ͑U˜ ϫW¯ (s)͒ϭS͑W˜ ,ϪW˜ ,0͒, see, e.g., Refs. 42,43 , where R and K correspond to the “ y x large scales, and r and k to the small scales, i.e., Rϭ(x ϩy)/2, rϭxϪy, Kϭk ϩk , kϭ(k Ϫk )/2. This implies ϫ͑U¯ (s)ϫW˜ ͒ϭS͑0,W˜ ,0͒, 1 2 1 2 “ x that we assumed that there exists a separation of scales, i.e., the maximum scale of turbulent motions l is much smaller ϫ ϭ͑ 2 ͒͑ ˜ ˜ ͒ 0 “ J SK /2 Wy ,Wx ,0 , than the characteristic scale of the inhomogeneities of the mean ﬁelds. ϫ͓͑ ¯ (s) ͒ ˜ ͔ϭ 2͑ ˜ Ϫ ˜ ͒ “ W •“ W SK Wy , Wx ,0 , Now we calculate ͒ u ͑kץ ͒ f ͑k ,k ץ where ij 1 2 ϵͳ ͑ ͒ n 1 ͑ ͒ʹ u j k2 ץ Pin k1 ץ ϭ t t M i “ jM ij, ͒ ץ un͑k2 s)͒ ͔ ϩͳ u ͑k ͒P ͑k ͒ ʹ , ͑B1͒) ¯ ץ͑ s)͒ ϩ␧) ¯ ץ͑ ϭ ˜ ͓␧ tץ Ji “ j„Wn nim U mj njm U mi …, i 1 jn 2 and we also took into account that where we multiplied equation of motion ͑13͒ rewritten in k space by P (k)ϭ␦ Ϫk in order to eliminate the pressure ϭ͑ ¯ (s) ͒ ˜ ij ij ij jGij W Wi , ␦ “ •“ term from the equation of motion, ij is the Kronecker tensor, and k ϭk k /k2. Thus, the equation for f (k,R)is ij i j ij ͖͔ ͒ ˜ ץ͑ ϩ␧ ͒ ˜ ץ͑ s)͓␧) ¯ ͕ “ j Wn nim U mj njm U mi given by Eq. ͑14͒. For the derivation of Eq. ͑14͒, we used the following ϭ͓ ͑ ¯ (s) ˜ ͒Ϫ͑ ¯ (s) ͒ ˜ ͔ “i W •W W •“ Wi /2. identity: To derive Eq. ͑21͒, we used the following identities for the integration over the angles in k space: ͵ ͑ Ϫ 1 Ϫ ͒¯ ͑ ͒ ͑ ͒ iki f ij k 2 Q,K Q U p Q exp iK•R dK dQ ץ 4␲ ⍀ˆ ϭ ⌬ 1 1 i f ij ͵ kijmn d ijmn , ϭϪ ¯ ϩ ¯ Ϫ ͑ ¯ ͒ͩ ͪ ץ U p i f ij f ij iU p sU p i 15 2 “ 2 “ 4 “ “ ks f ץ i ϵ ͵ ⍀ˆ ϩ ͩ ijͪ ͑ ¯ ͒ ͑ ͒ s iU p . B2 ץ Tijmnpq kijmnpqd 4 ks “ “ 4␲ ϭ ⌬ ␦ ϩ⌬ ␦ ϩ⌬ ␦ ϩ⌬ ␦ To derive Eq. ͑B2͒, we multiply the equation uϭ0, writ- ͑ mnpq ij jmnq ip imnq jp jmnp iq “• 105 Ϫ ¯ ten in k space for ui(k1 Q), by u j(k2)U p(Q)exp(iK•R), ϩ⌬ ␦ ϩ⌬ ␦ Ϫ⌬ ␦ ͒, and integrate over K and Q, and average over the ensemble imnp jq ijmn pq ijpq mn ϭ ϩ ϭϪ of velocity ﬂuctuations. Here k1 k K/2 and k2 k ϩ 4␲ K/2. This yields ͒ ␦s)ٌ͒͑ ˜ ͒ϭ ͑ ϩ) ¯ ٌ͑ Tijmnpq mUn pUq 4M ij ijM pp , 105 1 1 ͵ ͩ ϩ Ϫ ͪ ͳ ͩ ϩ Ϫ ͪ ͑Ϫ ϩ 1 ͒ʹ i ki Ki Qi ui k K Q u j k 2 K ϭ 4 ⍀ˆ ϭ ␪ ␪ ␸ ϭ 2 2 kijmn kik jkmkn /k , d sin d d , kijmnpq kijmnk pq , ⌬ ϭ␦ ␦ ϩ␦ ␦ ϩ␦ ␦ and ijmn ij mn im nj in mj . ϫ ¯ ͑ ͒ ͑ ͒ ϭ ͑ ͒ U p Q exp iK•R dK dQ 0. B3 APPENDIX B: DERIVATION OF EQ. 14 ˜ ϭ ϩ Ϫ ˜ „ … Then we introduce new variables: k1 k K/2 Q, k2 ͑ ͒ ϭϪ ϩ ˜ϭ ˜ Ϫ˜ ϭ Ϫ ˜ ϭ˜ ϩ˜ In order to derive Eq. 14 , we use a two-scale approach, k K/2, and k (k1 k2)/2 k Q/2, K k1 k2 i.e., a correlation function is written as follows: ϭKϪQ. This allows us to rewrite Eq. ͑B3͒ in the form

016311-7 ELPERIN, KLEEORIN, AND ROGACHEVSKII PHYSICAL REVIEW E 68, 016311 ͑2003͒

1 1 We also use the following identities: ͵ iͩ k ϩ K ϪQ ͪ f ͩ kϪ Q,KϪQͪ i 2 i i ij 2 ͓ f ͑k,R͒U¯ ͑R͔͒ ϭ ͵ f ͑k,KϪQ͒U¯ ͑Q͒dQ, ϫ ¯ ͑ ͒ ͑ ͒ ϭ ͑ ͒ ij p K ij p U p Q exp iK•R dK dQ 0. B4 Since ͉Q͉Ӷ͉k͉, we can use the Taylor expansion ͓ ͑ ͒¯ ͑ ͔͒ “ p f ij k,R U p R f ͑kϪQ/2,KϪQ͒ ij ϭ ͵ ͓ ͑ ͒¯ ͑ ͔͒ ͑ ͒ iKp f ij k,R U p R K exp iK•R dK. f ͑k,KϪQ͒ ץ 1 Ӎ ͑ Ϫ ͒Ϫ ij ϩ ͑ 2͒ Qs O Q . ͑B6͒ ץ f ij k,K Q 2 ks ͑B5͒ Therefore, Eqs. ͑B4͒–͑B6͒ yield Eq. ͑B2͒.

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