Compressibility effects in a turbulent transport of temperature field
aa r X i v : . [ phy s i c s . f l u - dyn ] A ug Compressibility effects in a turbulent transport of temperature field
I. Rogachevskii ∗ and N. Kleeorin † Department of Mechanical Engineering, Ben-Gurion University of the Negev, P. O. B. 653, Beer-Sheva 8410530, IsraelNordita, Stockholm University and KTH Royal Institute of Technology, 10691 Stockholm, Sweden (Dated: August 4, 2020)Compressibility effects in a turbulent transport of the mean temperature field are investigatedapplying the quasi-linear approach for small P´eclet numbers and the spectral τ approach for largeP´eclet numbers. Compressibility of a fluid flow reduces the turbulent diffusivity of the mean tem-perature field similarly to that for particle number density and magnetic field. However, expressionfor turbulent diffusion coefficient for the mean temperature field in a compressible turbulence isdifferent from those for particle number density and magnetic field. Combined effect of compress-ibility and inhomogeneity of turbulence causes an increase of the mean temperature in the regionswith more intense velocity fluctuations due to a turbulent pumping. Formally, this effect is similarto a phenomenon of compressible turbophoresis found previously [J. Plasma Phys. , 735840502(2018)] for non-inertial particles or gaseous admixtures. Gradient of the mean fluid pressure resultsin an additional turbulent pumping of the mean temperature field. The later effect is similar toturbulent barodiffusion of particles and gaseous admixtures. Compressibility of a fluid flow alsocauses a turbulent cooling of the surrounding fluid due to an additional sink term in the equationfor the mean temperature field. There is no analog of this effect for particles. I. INTRODUCTION
Compressibility of a fluid flow affects turbulent trans-port of particles, temperature and magnetic fields (see,e.g., [1–4]), e.g., it causes qualitative changes in the prop-erties of both, mean passive scalar field and fluctuations.Large-scale effects of turbulence on particle concentra-tions and temperature field are described by means ofturbulent flux of particles and turbulent heat flux, re-spectively. For incompressible flow, main contribution tothe turbulent fluxes is determined by turbulent diffusionof particles and temperature field. This corresponds togradient turbulent transport of particles and tempera-ture fields, e.g., the turbulent flux of particles is directedopposite to gradient of the mean particle number den-sity, while the turbulent heat flux is directed opposite togradient of the mean fluid temperature.The compressibility of a turbulent flow results in a re-duction of turbulent diffusivity of a mean particle num-ber density at small [5] and large [6] P´eclet numbers.The P´eclet number is the ratio of nonlinear to diffusionterms in the equation for particle number density fluc-tuations. The conclusion about the reduction of turbu-lent diffusivity by the compressibility of fluid flow hasbeen also confirmed by the test-field method in directnumerical simulations for an irrotational homogeneousdeterministic flow [5]. Various aspects related to com-pressibility effects on turbulent transport have been stud-ied using different analytical approaches, e.g., the quasi-linear approach [5–7], the spectral tau approach [6], thepath-integral approach [8–10], the multiple-scale direct-interaction approximation [11, 12], etc. ∗ † [email protected] Compressibility of a turbulent flow causes additionalnon-gradient contribution to turbulent flux of particlesthat is proportional a product of the mean particle num-ber density and effective pumping velocity. In a den-sity stratified turbulence, the effective pumping velocityof particles is proportional to the gradient of the meanfluid density multiplied by turbulent diffusion coefficient[9, 10]. The pumping effect results in accumulation ofparticles in regions of maximum mean fluid density.In a temperature stratified turbulence, similar effectreferred as turbulent thermal diffusion results in a tur-bulent non-diffusive flux of particles in the direction ofthe turbulent heat flux, so that particles are accumulatedin the vicinity of the mean temperature minimum [9, 10].This phenomenon has been studied theoretically [13–17],found in direct numerical simulations [18], detected indifferent laboratory experiments [17, 19–21], and atmo-spheric turbulence with temperature inversions [22]. Thiseffect has been shown to be important for concentratingdust in protoplanetary discs [23]. Density stratificationwhich causes turbulent pumping of particles, becomesweaker with increasing compressibility, i.e., with increas-ing the Mach number [6].Compressibility of a fluid flow in inhomogeneous tur-bulence also results in a new pumping effect of particlesfrom regions of low to high turbulent intensity both forsmall and large P´eclet numbers. This effect has beeninterpreted in [6] as a compressible turbophoresis of non-inertial particles and gaseous admixtures, while the clas-sical turbophoresis effect for incompressible inhomoge-neous turbulence [24–28] exists only for inertial particlesand causes them to be pumped to regions with lower tur-bulent intensity.The compressibility of a turbulent fluid flow affects alsopassive scalar fluctuations. In particular, it results in aslow scale-dependent turbulent diffusion of a small-scalepassive scalar fluctuations for large P`eclet numbers [8]. Inaddition, the level of the passive scalar fluctuations in thepresence of a gradient of the mean passive scalar field incompressible turbulent flow can be fairly strong. On theother hand, passive scalar transport in a density stratifiedturbulent fluid flow is accompanied by formation of large-scale structures due to instability of the mean passivescalar field in inhomogeneous turbulent velocity field [8].Another interesting feature for a compressible temper-ature stratified turbulence is that turbulent flux of en-tropy is different from turbulent convective flux of fluidinternal energy [29, 30]. In particular, turbulent fluxof entropy is given by F s = ρ h s ′ u i , where ρ is themean fluid density and s ′ and u are fluctuations of en-tropy and velocity, respectively. On the other hand, theturbulent convective flux of the fluid internal energy is F c = T ρ h s ′ u i , where T is the mean fluid temperature.This turbulent convective flux is well-known in the as-trophysical and geophysical literature, and it cannot beused as a turbulent flux in the equation for the meanentropy. This is exact result for low-Mach-number tem-perature stratified turbulence and is independent of theturbulence model used [30].In spite of many studies of turbulent transport of pas-sive scalar, some large-scale (mean-field) features relatedto compressibility effects on turbulent transport of tem-perature field are not known. In the present paper, westudy compressibility effects in turbulent transport of themean temperature field, i.e., we consider here mean-fieldeffects. This paper is organized as follows. In Section IIwe outline the governing equations. Turbulent heat fluxand level of temperature fluctuations are determined forsmall P´eclet numbers in Section III and for large P´ecletnumbers in Section IV. In Sections III-IV we also out-line the method of derivations and approximations madefor study of the compressibility effects. In Section V wediscuss how a homogeneous compressible turbulence cancause a turbulent cooling of the surrounding fluid. Fi-nally, conclusions are drawn in Section VI. In AppendixA we outline the multi-scale approach used in the presentstudy. Details of the derivation of turbulent heat fluxand level of temperature fluctuations are given in Ap-pendix B for small P´eclet numbers and in Appendix Cfor large P´eclet numbers. II. GOVERNING EQUATIONS
Evolution of temperature field T ( t, r ) in a compressiblefluid velocity field U ( t, r ) is given by [31] ∂T∂t + ( U · ∇ ) T + ( γ − T ( ∇ · U ) = D ∆ T + J ν , (1)where D is the molecular thermal conductivity, γ = c p /c v is the ratio of specific heats and J ν is the heating sourcecaused, e.g., by a viscous dissipation. In a compressibleflow, Eq. (1) for the temperature field is different from equation for particle number density n ( t, r ) [32, 33] ∂n∂t + ∇ · ( n U ) = D n ∆ n, (2)where D n is the coefficient of Brownian (molecular) dif-fusion of particles.To derive equations for the turbulent heat flux and thelevel of temperature fluctuations, we apply a mean-fieldapproach and use Reynolds averaging. In particular, thefluid temperature and velocity are decomposed into meanand fluctuating parts, where the fluctuating parts havezero mean values. For example, the temperature field T = T + θ , where T = h T i is the mean fluid temperature, θ are the temperature fluctuations, and h θ i = 0. The an-gular brackets denote an ensemble averaging. Similarly, U = U + u , where U = h U i is the mean fluid velocity,and u are the velocity fluctuations and h u i = 0. For sim-plicity, we consider the case U = 0. Averaging Eq. (1)over ensemble of turbulent velocity field, we arrive atequation for the mean temperature field as ∂T∂t + ∇ · h θ u i = − ( γ − h θ ( ∇ · u ) i + D ∆ T + J ν , (3)where F = h θ u i is the turbulent heat flux, J ν is themean heating source caused by the viscous dissipation inturbulence, and I S = − ( γ − h θ ( ∇ · u ) i is the mean sinkterm resulting in a turbulent cooling due to compress-ibility effects (see Section V). Using Eqs. (1) and (3),we obtain equation for the temperature fluctuations, θ ( x , t ) = T − T : ∂θ∂t + Q − D ∇ θ = − ( u ·∇ ) T − ( γ − T ∇· u , (4)where Q = ∇· ( θ u − h u θ i ) − ( γ −
2) [ θ ∇· u − h θ ∇· u i ]are nonlinear terms and I = − ( u ·∇ ) T − ( γ − T ∇· u are the source terms of temperature fluctuations. Theratio of the nonlinear term to the diffusion term is theP´eclet number, that is estimated as Pe = u ℓ /D , where u is the characteristic turbulent velocity in the integral(energy-containing) scale ℓ of turbulence. We considera one way coupling, i.e., we take into account the effectof turbulence on the temperature field, but neglect thefeedback effect of the temperature on the turbulence.To determine the turbulent heat flux and the levelof temperature fluctuations, and to take into accountsmall-scale properties of the turbulence, e.g., the turbu-lent spectrum, we use two-point correlation functions.For fully developed turbulence, scalings for the turbulentcorrelation time and energy spectrum are related via theKolmogorov scalings [1, 2, 34–36]. We consider the caseswith small and large P´eclet and Reynolds numbers.In the framework of the mean-field approach, we as-sume that there is a separation of spatial and temporalscales, i.e., ℓ ≪ L T and τ ≪ t T , where L T and t T arethe characteristic spatial and temporal scales character-izing the variations of the mean temperature field and τ = ℓ /u . The mean fields depend on “slow” variables,while fluctuations depend on “fast” variables. Separationinto slow and fast variables is widely used in theoreticalphysics, and all calculations are reduced to the Taylorexpansions of all functions using small parameters ℓ /L T and τ /t T . The findings are further truncated to leadingorder terms. Separation to slow and fast variables is per-formed by means of a standard multi-scale approach [37]discussed in details in Appendix A. III. TURBULENT HEAT FLUX AND LEVEL OFTEMPERATURE FLUCTUATIONS FOR SMALLP´ECLET NUMBERS
In this section we derive equations for the turbulentheat flux and the level of temperature fluctuations forsmall P´eclet numbers using the quasi-linear approach.For a random flow with small P´eclet and Reynolds num-bers, there are no universal scalings for the correlationtime and energy spectrum. This is the reason why weuse non-instantaneous two-point correlation functions inthis case. In the framework of the quasi-linear approach,we neglect the nonlinear term Q , but keep the molecu-lar diffusion term in Eq. (4). We rewrite this equationin Fourier space and find the solution of this equation,given by Eq. (B1) in Appendix B. Using this solutionand applying the multi-scale approach (see [37] and Ap-pendix A), we arrive at expressions for the turbulent heatflux and the level of temperature fluctuations in Fourierspace for small P´eclet numbers as h θ u j i = − γ − (cid:20) T Z G D (cid:18) ∇ i − Dk G D k im ∇ m + 2 ik i (cid:19) f ij d k dω − (cid:0) ∇ i T (cid:1) Z G D (cid:18) γ − γ − δ im + 2 Dk G D k im + k m ∂∂k i (cid:19) f mj (cid:21) d k dω, (5) (cid:10) θ (cid:11) = γ − (cid:26) T Z G D (cid:20)(cid:18) Dk G D k jn ∇ n − ∇ j (cid:19) F (+) j + 2 ik j F ( − ) j (cid:21) d k dω + (cid:0) ∇ n T (cid:1) Z G D (cid:20) γ − γ − δ jn + 2 Dk G D k jn + k j ∂∂k n (cid:21) F (+) j (cid:27) d k dω. (6)Details of derivations of Eqs. (5) and (6) are given in Ap-pendix B. Here G D ≡ G D ( k , ω ) = ( D k + iω ) − , f ij ≡ f ij ( k , ω ) = h u i ( k , ω ) u j ( − k , − ω ) i , F ( ± ) j = F j ( k , ω ) ± F j ( − k , ω ), where F j ( k , ω ) = h θ ( k , ω ) u j ( − k , − ω ) i is theturbulent heat flux in Fourier space, δ ij is the Kroneckerunit tensor and k ij = k i k j /k . Since we consider a one way coupling, the correlation function f ij in Eqs. (5)and (6) should be replaced by f (0) ij for the backgroundrandom flow with zero turbulent heat flux.We use a statistically stationary, density-stratified, in-homogeneous, compressible and non-helical backgroundrandom flow determined by the following correlationfunction in Fourier space [6]: f (0) ij ( k , ω ) = Φ( ω )8 π k (1 + σ c ) (cid:26) E ( k ) (cid:20) δ ij − k ij + ik (cid:0) k j λ i − k i λ j (cid:1) + i k (cid:0) k i ∇ j − k j ∇ i (cid:1)(cid:21) +2 σ c E c ( k ) (cid:20) k ij + i k (cid:0) k i ∇ j − k j ∇ i (cid:1)(cid:21)(cid:27) (cid:10) u (cid:11) , (7)where λ = − ∇ ln ρ characterizes the fluid density strat-ification, p h u i is the characteristic turbulent velocityat the maximum scale ℓ of random motions, and the parameter σ c = (cid:10) ( ∇ · u ) (cid:11) h ( ∇ × u ) i (8)is the degree of compressibility of the turbulent veloc-ity field. To derive Eq. (7), we assume that ℓ ≪ H ρ and ℓ ≪ L u , where L u = (cid:12)(cid:12) ∇ ln (cid:10) u (cid:11)(cid:12)(cid:12) − is the char-acteristic scale of the inhomogeneity of turbulence, and H ρ = | λ | − is the mean density stratification scale, whichis assumed to be constant. These conditions allow us totake into account leading effects in Eq. (7), which arelinear in stratification, ∝ ℓ /H ρ , and inhomogeneity ofturbulence, ∝ ℓ /L u . We neglect in Eq. (7) high-ordereffects which are of the order of O( λ (cid:10) u (cid:11) ), O( ∇ (cid:10) u (cid:11) ),O( λ i ∇ i (cid:10) u (cid:11) ).Generally, stratification also contributes to div u , i.e.,it contributes to the parameter σ c . Since this contribu-tion is small, i.e., it is of the order of ∼ O( λ (cid:10) u (cid:11) ), weneglect this contribution in Eq. (7). This allows us to sep-arate effects of the arbitrary Mach number, characterizedby the parameter σ c , and density stratification, describedby λ . The degree of compressibility σ c should depend onthe Mach number. This dependence is not known forarbitrary Mach numbers and can be determined, e.g., indirect numerical simulations.In Eq. (7), E ( k ) and E c ( k ) are the spectrum functionsfor incompressible and compressible parts of a randomflow. We assume that the random flow have a power-lawspectrum for incompressible E ( k ) = ( q −
1) ( k/k ) − q k − and compressible E c ( k ) = ( q c −
1) ( k/k ) − q c k − parts,where the wave number varies in the range, k ≤ k ≤ k ν .Here k ν = 1 /ℓ ν is the wave number based on the viscousscale ℓ ν , and k = 1 /ℓ ≪ k ν . We assume also thatthere are no random motions for k < k . In the modelof a compressible background turbulence used in [6], theexponents q = q c . In the present study, we consider thecase when the spectrum exponents of the incompressibleand compressible parts of random motions are different,i.e., q = q c .We assume that the frequency function Φ( ω ) has aLorentz profile, Φ( ω ) = [ πτ ( ω + τ − )] − , which cor-responds to the correlation function h u i ( t ) u j ( t + τ ) i ∝ exp( − τ /τ ). Here the correlation time for small P´ecletnumbers τ ≡ ℓ /u ≫ ( Dk ) − for all turbulent scales.To derive Eq. (7) we use identities given in Appendix B.Different contributions to Eq. (7) have been discussed in[6, 8, 38].Integration in ω and k space in Eq. (5) yields an equa-tion for the turbulent heat flux for small P´eclet numbersas h θ u i = T V eff − D T ∇ T , (9) where the turbulent diffusivity D T and the effectivepumping velocity V eff are given by D T = γ ( q − q + 1) τ (cid:10) u (cid:11) Pe (cid:20) − σ c [1 + C σ (2 − /γ )]1 + σ c (cid:21) , (10) V eff = ( γ −
1) ( q − q + 1) τ (cid:10) u (cid:11) (1 + σ c ) Pe (cid:20) C σ σ c λ u + λ P (cid:21) , (11)where λ u = ∇ ln (cid:10) u (cid:11) , λ P = ∇ ln P and C σ = ( q c −
1) ( q + 1)( q c + 1) ( q − . (12)Here we take into account that the equation of stateyields λ = − λ P + ∇ ln T . Since τ Pe = ℓ /D , the tur-bulent transport coefficients given by Eqs. (10) and (11)are determined only by the microphysical diffusion timescale, ℓ /D . Equation (10) implies that for small P´ecletnumbers, compressibility effects decrease the turbulentdiffusivity. However, the total diffusivity, D + D T , cannotbe negative, because for Pe ≪
1, the molecular diffusivityis much larger than the turbulent one, D ≫ | D T | .The first term in Eq. (11) for the effective pumpingvelocity V eff of the mean temperature field describes acombined effect of compressibility of fluid flow and inho-mogeneity of turbulence. This effect increases the meantemperature field in the regions with more intense ve-locity fluctuations due to turbulent pumping. This issimilar to a phenomenon of compressible turbophoresisfound previously for non-inertial particles or gaseous ad-mixtures [6].The second term in Eq. (11) describes an additionalturbulent pumping effect due to the gradient of the meanfluid pressure. This effect is similar to turbulent barod-iffusion of particles and gaseous admixtures [10]. Thephysics of these effects is discussed in the next section.Note that the expressions for turbulent diffusion and theeffective pumping velocity for the mean temperature fieldin a compressible turbulence are different from those forparticle number density and magnetic field (see [6]), be-cause equations for particle number density or magneticfield are different from that for the fluid temperature.Integration in ω and k space in Eq. (6) yields the ex-pression for the level of temperature fluctuations for smallP´eclet numbers as (cid:10) θ (cid:11) = ( γ − (cid:18) q c − q c + 1 (cid:19) (cid:18) σ c σ c (cid:19) Pe T + q − q + 3) Pe ℓ (cid:26)(cid:0) ∇ T (cid:1) + 18 ( γ − h λ · ∇ ) + ( γ + 3) ( λ u · ∇ ) i T (cid:27) . (13)The first term in the right hand side of Eq. (13) de- termines a dominant contribution of the compressiblepart of velocity fluctuations to the level of temperaturefluctuations. Here we take into account only the dom-inant contribution and neglect much smaller contribu-tions ∼ O[ ℓ / ( L T L u )], O[ ℓ / ( L T H ρ )], O[ ℓ /L T ], causedby the compressible part of velocity fluctuations, where L T is the characteristic scale of the mean temperaturefield variations. For small σ c , the level of temperaturefluctuations is determined by other terms in Eq. (13)caused by the density stratified and inhomogeneous partof velocity fluctuations. IV. TURBULENT HEAT FLUX AND LEVEL OFTEMPERATURE FLUCTUATIONS FOR LARGEP´ECLET NUMBERS
In this section we determine the turbulent heat fluxand the level of temperature fluctuations for large P´ecletand Reynolds numbers. We consider fully developedturbulence, where the Strouhal number is of the or-der of unity and the turbulent correlation time is scale- dependent, so we apply the Fourier transformation onlyin k space.The procedure of the derivations of the expressions forthe turbulent heat flux and the level of temperature fluc-tuations includes: (i) derivation of equations for the sec-ond moments in k space using the multi-scale approach,(ii) application of the spectral τ approach (see below)which allows us to relate the deviations of the third mo-ments (appearing due to nonlinear terms) from those ofthe background turbulence with the deviations of the sec-ond moments, (iii) solution of the equations for the sec-ond moments in the k space, and (iv) inverse transfor-mation to the physical space to obtain formulas for theturbulent heat flux and the level of temperature fluctua-tions.Starting with Eq. (4) for the temperature fluctuations θ and the Navier-Stokes equation for the velocity u writ-ten in Fourier space, we derive dynamic equations for theturbulent heat flux and level of temperature fluctuationsas ∂F j ∂t = −
12 ( γ − (cid:20) T (cid:18) ik i + ∇ i (cid:19) f ij − (cid:0) ∇ i T (cid:1) (cid:18) γ − γ − δ im + k m ∂∂k i (cid:19) f mj (cid:21) + ˆ M F (III) j , (14) ∂E θ ∂t = 12 ( γ − (cid:20) T (cid:16) ik j F ( − ) j − ∇ j F (+) j (cid:17) + (cid:0) ∇ m T (cid:1) (cid:18) k j ∂∂k m + γ − γ − δ jm (cid:19) F (+) j (cid:21) + ˆ M E (III) θ . (15)Details of derivations of Eqs. (14)–(15) are given inAppendix C. Here F j ( k ) = h θ ( k ) u j ( − k ) i , E θ ( k ) = h θ ( k ) θ ( − k ) i , f ij ( k ) = h u i ( k ) u j ( − k ) i , and F ( ± ) j = F j ( k ) ± F j ( − k ), the third-order moment terms ˆ M F (III) j and ˆ M E (III) θ written in k space and appearing due tothe nonlinear terms are given by Eqs. (C8) and (C9) inAppendix C.Equations (14) and (15) for the second moment in-clude first-order spatial differential operators ˆ M appliedto the third-order moments F (III) . The problem ariseshow to close Eqs. (14) and (15), i.e., how to express thethird-order terms ˆ M F (III) through the lower moments[1, 2, 34, 39]. We use the spectral τ approach whichis a universal tool in turbulent transport for stronglynonlinear systems. The spectral τ approximation pos-tulates that the deviations of the third-moment terms,ˆ M F (III) ( k ), from the contributions to these terms af-forded by the background turbulence, ˆ M F (III , ( k ), canbe expressed through similar deviations of the second mo-ments, F (II) ( k ) − F (II , ( k ) as [39–41]ˆ M F (III) ( k ) − ˆ M F (III , ( k ) = − F (II) ( k ) − F (II , ( k ) τ r ( k ) , (16)where τ r ( k ) is the scale-dependent relaxation time, whichcan be identified with the correlation time τ ( k ) of the tur- bulent velocity field for large fluid Reynolds numbers andlarge P´eclet numbers. Here functions with superscript (0)correspond to background turbulence with zero turbu-lent heat flux. Therefore, (16) reduces to ˆ M F (III) i ( k ) = − F i ( k ) /τ ( k ) and ˆ M E (III) θ ( k ) = − E θ ( k ) /τ ( k ). Vali-dation of the τ approximation for different situationshas been performed in various numerical simulations[5, 6, 18, 42–48].The τ approximation is a sort of the high-order clo-sure and in general is similar to Eddy Damped QuasiNormal Markovian (EDQNM) approximation. Howeversome principle difference exists between these two ap-proaches [39, 40]. The EDQNM closures do not re-lax to equilibrium (the background turbulence), and theEDQNM approach does not describe properly the mo-tions in the equilibrium state in contrast to the τ ap-proximation. Within the EDQNM theory, there is nodynamically determined relaxation time, and no slightlyperturbed steady state can be approached. In the τ approximation, the relaxation time for small departuresfrom equilibrium is determined by the random motionsin the equilibrium state, but not by the departure fromthe equilibrium. As follows from the analysis in [39], the τ approximation describes the relaxation to the equilib-rium state (the background turbulence) much more ac-curately than the EDQNM approach.Next, we assume that the characteristic times of vari-ation of the second moments F i and E θ are substantiallylarger than the correlation time τ ( k ) in all turbulence scales. This allows us to get steady-state solutions ofEqs. (14) and (15) as h θ u j i = −
12 ( γ − Z τ ( k ) (cid:20) T (cid:18) ik i + ∇ i (cid:19) f ij − (cid:0) ∇ i T (cid:1) (cid:18) γ − γ − δ im + k m ∂∂k i (cid:19) f mj (cid:21) d k , (17) (cid:10) θ (cid:11) = 12 ( γ − Z τ ( k ) (cid:20) T (cid:16) ik j F ( − ) j − ∇ j F (+) j (cid:17) + (cid:0) ∇ m T (cid:1) (cid:18) k j ∂∂k m + γ − γ − δ jm (cid:19) F (+) j (cid:21) d k . (18)In Eqs. (17) and (18) we take into account a one way cou-pling, i.e., we neglect the effect of the mean temperaturegradients on the turbulent velocity field. This impliesthat we replace the correlation function f ij in Eqs. (17)and (18) by f (0) ij for the background turbulent flow with zero turbulent heat flux.We use statistically stationary, density-stratified, in-homogeneous, compressible and non-helical backgroundturbulence, which is determined by the following corre-lation function in k space: f (0) ij ( k ) = 18 π k (1 + σ c ) (cid:26) E ( k ) (cid:20) δ ij − k ij + ik (cid:0) k j λ i − k i λ j (cid:1) + i k (cid:0) k i ∇ j − k j ∇ i (cid:1)(cid:21) +2 σ c E c ( k ) (cid:20) k ij + i k (cid:0) k i ∇ j − k j ∇ i (cid:1)(cid:21)(cid:27) (cid:10) u (cid:11) . (19)We assume here that the background turbulence is ofKolmogorov type with constant energy flux over the spec-trum, i.e., the velocity fluctuations spectrum for the in-compressible part of turbulence in the range of wave num-bers k < k < k ν is E ( k ) = − d ¯ τ ( k ) /dk , where the func-tion ¯ τ ( k ) = ( k/k ) − q with 1 < q < q > q < E c ( k ) = − d ¯ τ c ( k ) /dk ,where the function ¯ τ c ( k ) = ( k/k ) − q c with 1 < q c < q = 5 / q c = 2 (like for the Burgers turbulence).The turbulent correlation time in k space is τ ( k ) = 2 τ σ c h ¯ τ ( k ) + σ c ¯ τ c ( k ) i . (20)Note that for fully developed Kolmogorov like turbulence, σ c < k -space in Eq. (17) yields the turbulentheat flux h θ u i = T V eff − D T ∇ T , where the turbulentdiffusivity D T and the effective pumping velocity V eff oftemperature field for large P´eclet numbers are given by D T = τ (cid:10) u (cid:11) (cid:26) γ −
11 + σ c (cid:20) − σ c σ c ) (cid:16) ˜ C σ q + σ c ( q c − (cid:17)(cid:21)(cid:27) , (21) V eff = ( γ − τ (cid:10) u (cid:11) σ c ) (cid:26) σ c (cid:20) C σ σ c ) (cid:21) λ u + (cid:20) − ˜ C σ σ c σ c ) (cid:21) λ P (cid:27) , (22)and ˜ C σ = 2( q c − q + q c − . (23)For irrotational flow ( σ c ≫ D T = 13 τ (cid:10) u (cid:11) (cid:20) −
12 ( γ −
1) ( q c − (cid:21) , (24) V eff = (cid:18) γ − (cid:19) τ ∇ (cid:10) u (cid:11) , (25)Equations (21) and (24) for the turbulent diffusivity D T of mean temperature field imply that for large P´ecletnumbers, compressibility decreases the turbulent diffu-sivity. Equations (22) and (25) determine effective pump-ing velocity V eff of the mean temperature field caused bythe inhomogeneity of compressible turbulence and thegradient of the fluid pressure.Let us discuss mechanisms of the turbulent pumpingeffects. The first term in Eq. (22) implies that there is anadditional contribution to the turbulent heat flux causedby the combined effect of the inhomogeneity of turbu-lence and compressibility of fluid flow. This effect resultsin increase of the mean temperature in the region withmore intense velocity fluctuations in a compressible tur-bulence. This effect can be understood using the budgetequation for the mean internal energy density E = c v T ,where c v is the specific heat at constant volume. In par-ticular, one of the sources in the budget equation for themean internal energy density is −h P ∇ · u i [31], so that ∂ ( ρ E ) /∂t ∼ −h P ∇ · u i , where P are pressure fluctua-tions. As follows from the Bernoulli law, variations of thesum δ ( P + ρ u / ≈
0, so that δP ≈ − δ ( ρ u / V effparticles ∝ σ c τ ∇ (cid:10) u (cid:11) .The second term in Eq. (22) determines an additionalcontribution to the turbulent heat flux caused by the gra-dient of the mean fluid pressure. This turbulent pump-ing increases the mean temperature in the regions withhigher mean fluid temperature. The mechanism of thiseffect is the following. Since there is an outflow of fluidfrom the turbulent regions with higher mean fluid pres-sure, the fluid density decreases in these regions and tem-perature increases. This effect is similar to turbulent bar-odiffusion [10] of particles or gaseous admixtures.Integration in k -space in Eq. (18) yields the level oftemperature fluctuations for large P´eclet numbers (cid:10) θ (cid:11) = 8 f c ( γ − (cid:18) σ c σ c (cid:19) T + 19 ℓ (cid:26) (cid:0) ∇ T (cid:1) + ( γ − h ( γ + 3) ( λ u · ∇ ) + 2 (5 − γ ) ( λ · ∇ ) i T (cid:27) . (26)where the function f c ( q, q c , σ c ) depends on the degree ofcompressibility and the exponents of spectra incompress-ible and compressible parts of the velocity fluctuations: f c = q c − q c − q c − σ c ( q + 2 q c −
5) + q c − σ c (2 q + q c − . (27)The first term in the right hand side of Eq. (26) de-termines a dominant contribution of the compressiblepart of velocity fluctuations to the level of temperaturefluctuations. Here we take into account only the dom-inant contribution and neglect much smaller contribu-tions ∼ O[ ℓ / ( L T L u )], O[ ℓ / ( L T H ρ )], O[ ℓ /L T ], causedby the compressible part of velocity fluctuations. Forsmall σ c , the level of temperature fluctuations is deter-mined by other terms in Eq. (26) which are determinedby the density stratified and inhomogeneous part of ve-locity fluctuations. V. TURBULENT COOLING
Let us discuss here how a homogeneous compressibleturbulence can cause a turbulent cooling of the surround-ing fluid. Substituting Eq. (9) for the turbulent heat fluxinto Eq. (3), we obtain the equation for the mean tem- perature field T as ∂T∂t + ∇· (cid:2) T V eff − ( D + D T ) ∇ T (cid:3) = J ν − ( γ − h θ ( ∇ · u ) i . (28)The sink term I S = − ( γ − h θ ( ∇ · u ) i in the equa-tion (28) for the mean temperature for small P´eclet num-bers is given by I S = − ( γ −
1) (2 − γ ) (cid:18) σ c σ c (cid:19) Pe Tτ , (29)and for large P´eclet numbers it reads I S = − γ −
1) (2 − γ ) σ c (1 + σ c ) Tτ h Re / + σ c i . (30)In Eq. (30) we take into account that the exponent ofthe incompressible part of the energy spectrum q = 5 / q = 2 [50, 51].For γ <
2, the term I S causes decrease of the mean tem-perature, i.e., it results in a compressible cooling.Let us consider a simple case with a uniform meantemperature field. The heating source J ν caused by theviscous dissipation in turbulence is given by J ν = νc v (cid:20)(cid:10) ( ∇× u ) (cid:11) + 43 (cid:10) ( ∇· u ) (cid:11)(cid:21) , (31)where c v is the specific heat at constant volume. Here weuse the equation for the turbulent kinetic energy densityfor compressible turbulence written as ∂E K ∂t + div Φ K = − ε K + Π K , (32)where Φ K = − ρ ν (cid:20) h u × ( ∇× u ) i + 43 h u ( ∇· u ) i (cid:21) + (cid:10) u (cid:0) ρ u / (cid:1)(cid:11) + h u P i (33)is the flux of the density of turbulent kinetic energy, ε K = ρ ν (cid:20)(cid:10) ( ∇× u ) (cid:11) + 43 (cid:10) ( ∇· u ) (cid:11)(cid:21) (34)is the dissipation rate of the density of turbulent kineticenergy, and Π K = ρ h u · f i + h p ( ∇· u ) i is the productionrate of the density of turbulent kinetic energy causedby the external force (e.g., by an external large-scaleshear). The production term includes also the correla-tions h p ( ∇· u ) i . Using Eq. (19) for the second momentof velocity fluctuations in the background turbulence, weobtain that the viscous heating source J ν is given by J ν = (cid:10) u (cid:11) τ (1 + σ c ) − (cid:20) σ c Re − / (cid:21) . (35)Turbulence can generate acoustic waves, and the rate ofthe energy radiated by the acoustic waves per unit massfor small Mach numbers is [52, 53] E w = α (cid:10) u (cid:11) τ Ma , (36)where α ∼ is numerical coefficient, Ma = u rms /c s is the Mach number, u rms = (cid:10) u (cid:11) / and c s is the soundspeed. The second term in Eq. (35) describes compress-ibility contribution to the rate of the viscous dissipation, J (c) ν = 43 (cid:10) u (cid:11) τ σ c σ c Re − / . (37)Assuming that the compressibility contribution to theviscous heating of turbulence J (c) ν is compensated by theradiative wave energy density E w , we obtain that thedegree of compressibility for small Mach numbers is givenby σ c = α Ma Re / . (38)In the equilibrium, the total viscous heating J ν is com-pensated by the compressible cooling I S , so that the in-crease of the internal thermal energy caused by the vis-cous turbulence heating is given by c v T c = (cid:10) u (cid:11) α Ma Re / . (39)Taking into account that the sound speed c s = ( γP /ρ ) / depends on the mean temperature, we obtain from Eq. (39) that the increase of the internal thermal energycaused by the viscous turbulence heating is given by c v T c = C ∗ (cid:10) u (cid:11) Re / , (40)where C ∗ = (6 α ) / / [ γ ( γ − / . Equation (40) can berewritten in terms of the Mach number Ma = u rms /c s asMa = (cid:20) γ ( γ − α (cid:21) / Re − / . (41)For example, taking parameters typical for the atmo-spheric turbulence, ℓ = 10 cm and u rms = 2 . × cm/s, we obtain that T c = 276 K. VI. CONCLUSIONS
In the present study we investigated compressibilityeffects on turbulent transport of the mean temperaturefield. We use the quasi-linear approach for study turbu-lent transport for small P´eclet numbers. When nonlin-ear effects are much stronger than the molecular diffu-sion (i.e., for large P´eclet numbers), we apply the spec-tral τ -approach. Similarly to turbulent transport of par-ticles and magnetic fields, the compressibility decreasesthe turbulent diffusivity of the mean temperature field,but the expression for turbulent diffusivity for the meantemperature field in a compressible turbulence is differentfrom those for turbulent diffusivity of the mean particlenumber density and turbulent magnetic diffusivity of themean magnetic field.We found also turbulent pumping of the mean temper-ature field due to a joint effects of the flow compressibilityand inhomogeneity of turbulence. This effect causes anincrease of the mean temperature in the regions of moreintense velocity fluctuations. Similar compressibility ef-fect referred to compressible turbophoresis [6], results ina pumping of non-inertial particles or gaseous admixturesfrom regions of low to high turbulent intensity. Turbu-lent pumping also can be due to the gradients of the meanfluid pressure resulting in increase of the mean tempera-ture in the regions with increased mean fluid pressure,similarly to phenomenon of turbulent barodiffusion ofparticles and gaseous admixtures.Due to compressibility, there is an additional sink termin the equation for the mean fluid temperature, causinga turbulent cooling in homogeneous turbulence. This im-plies that there can be an equilibrium in a compressiblehomogeneous turbulence with a uniform mean fluid tem-perature, where the heating caused by the viscous dis-sipation in turbulence can be compensated by the tur-bulent cooling caused by the fluid compressibility. Sucheffect does not exist for particles or gaseous admixtures. ACKNOWLEDGMENTS
We have benefited from stimulating discussions withMichael Liberman. This research was supported in partby Ministry of Science and Technology (grant No. 3-16516).
Appendix A: Multi-scale approach
In the framework of the multi-scale approach, the non-instantaneous two-point second-order correlation func- tions are written as follows: h θ ( x , t ) u j ( y , t ) i = Z h θ ( k , ω ) u j ( k , ω ) i exp (cid:2) i ( k · x + k · y ) + i ( ω t + ω t ) (cid:3) dω dω d k d k = Z F j ( k , ω, t, R ) exp[ i k · r + iω ˜ τ ] dω d k , (A1) h θ ( x , t ) θ ( y , t ) i = Z h θ ( k , ω ) θ ( k , ω ) i exp (cid:2) i ( k · x + k · y ) + i ( ω t + ω t ) (cid:3) dω dω d k d k = Z E θ ( k , ω, t, R ) exp[ i k · r + iω ˜ τ ] dω d k , (A2)where F j ( k , ω, R , t ) = Z h θ ( k , ω ) u j ( k , ω ) i exp[ i Ω t + i K · R ] d Ω d K . (A3) E θ ( k , ω, R , t ) = Z h θ ( k , ω ) θ ( k , ω ) i exp[ i Ω t + i K · R ] d Ω d K . (A4)Here we introduce large-scale variables: R = ( x + y ) / K = k + k , t = ( t + t ) /
2, Ω = ω + ω , and small-scale variables: r = x − y , k = ( k − k ) /
2, ˜ τ = t − t , ω = ( ω − ω ) /
2. This implies that ω = ω + Ω / ω = − ω + Ω / k = k + K /
2, and k = − k + K / f ij ( k , ω, R , t ) = Z h u i ( k , ω ) u j ( k , ω ) i exp[ i Ω t + i K · R ] d Ω d K . (A5)After separation into slow and fast variables and cal-culating the functions F j ( k , ω, R , t ) and E θ ( k , ω, R , t ),Eqs. (A1) and (A2) in the limit of r → and ˜ τ → h θ ( x , t ) u j ( x , t ) i = Z F j ( k , ω, R , t ) dω d k , (A6) h θ ( x , t ) θ ( x , t ) i = Z E θ ( k , ω, R , t ) dω d k . (A7)For brevity of notations we omit below the large-scalevariables t and R in F j ( k , ω, R , t ), E θ ( k , ω, R , t ) and T ( R , t ). In the next subsections we consider cases ofsmall and large P´eclet numbers. Appendix B: Derivation of Eqs. (5)–(7)
We rewrite Eq. (4) in Fourier space and find solutionof this equation as θ ( k , ω ) = − i (cid:20) ( γ − Z T ( Q ) ( k i − Q i ) u i ( k − Q , ω ) d Q + Z Q i T ( Q ) u i ( k − Q , ω ) d Q (cid:21) G D ( k , ω ) , (B1)where G D ( k , ω ) = ( D k + iω ) − . Using Eqs. (A6)and (B1), we determine the functions F j ( k , R ) and E θ ( k , R ) as F j ( k , R ) = − i Z (cid:20) ( γ − (cid:18) k i + K i − Q i (cid:19) + Q i (cid:21) G D ( k + K / h u i ( k + K / − Q ) u j ( − k + K / i× T ( Q ) exp( i K · R ) d K d Q , (B2) E θ ( k , R ) = − i Z (cid:26)(cid:20) ( γ − (cid:18) k i + K i − Q i (cid:19) + Q i (cid:21) G D ( k + K / h θ ( − k + K / u i ( k + K / − Q ) i (cid:20) ( γ − (cid:18) − k i + K i − Q i (cid:19) + Q i (cid:21) G D ( − k + K / h θ ( k + K / u i ( − k + K / − Q ) i (cid:27) × T ( Q ) exp( i K · R ) d K d Q (B3)where the functions F j , G D and u i depend also on ω ,and T depend on t as well. To simplify the notations,we do not show these dependencies here. To deter-mine f ij ( k , K , Q ) = h u i ( k + K / − Q ) u j ( − k + K / i ,we use the following new variables:˜ k = (˜ k − ˜ k ) / k − Q / , (B4)˜ K = ˜ k + ˜ k = K − Q , (B5)where ˜ k = k + K / − Q , ˜ k = − k + K / . (B6)Since | Q | ≪ | k | and | K | ≪ | k | , we use the Taylor expan-sion f ij ( k − Q / , K − Q ) = f ij ( k , K − Q ) − ∂f ij ∂k m Q m + O ( Q ) , (B7) G D ( k + K /
2) = G D ( k ) [1 − D ( k · K ) G D ( k )]+ O ( K ) . (B8)In the similar way we calculate other terms in Eqs. (B2)–(B3). Using Eqs. (B2)–(B8), we arrive at expressions (5)–(6) for the turbulent heat flux and the level of tempera-ture fluctuations in Fourier space for small P´eclet num-bers.To derive Eq. (7), the second rank tensor f (0) ij is con-structed as a linear combination of symmetric tensors, δ ij and k ij , with respect to the indexes i and j , and non-symmetric tensors: k i λ j , k j λ i , k i ∇ j (cid:10) u (cid:11) and k j ∇ i (cid:10) u (cid:11) .We consider here only linear effects in λ and ∇ (cid:10) u (cid:11) .To determine unknown coefficients multiplying by thesetensors, we use the following conditions in the derivationof Eq. (7): (cid:10) u (cid:11) = R f (0) ii ( k , ω, K ) exp( i K · R ) d k dω d K , f (0) ij ( k , ω, K ) = f ∗ (0) ji ( k , ω, K ) = f (0) ji ( − k , ω, K ), and D (div u ) E = Z ( k i + K i /
2) ( k j − K j / f (0) ij ( k , ω, K ) exp( i K · R ) d k dω d K . (B9)For very low Mach numbers, i.e., when the parameter σ c is very small, the continuity equation can be written inthe anelastic approximation, div ( ρ u ) = 0, which impliesthat ( ik i + iK i / − λ i ) f (0) ij ( k , ω, K ) = 0 and ( − ik j + iK j / − λ j ) f (0) ij ( k , ω, K ) = 0.For the integration over ω in Eqs. (5) and (6), we usethe following identities: Z ∞−∞ dω ( ± iω + Dk ) ( ω + τ − ) = π τ τ − + D k ≈ π τ D k , Z ∞−∞ dω ( iω + Dk ) ( − iω + Dk ) ( ω + τ − )= π τ D k (cid:0) τ − + D k (cid:1) ≈ π τ ( D k ) , which are determined in the limit when the correlationtime τ ≫ ( D ( θ ) k ) − . For the integration over angles in k space in Eqs. (5) and (6), we use the following identity: Z π dϕ Z π sin ϑ dϑ k i k j k = 4 π δ ij . For the integration over k in Eqs. (5) and (6), we use thefollowing identities: Z k d k E ( k ) k dk = q − q + 1 ℓ , Z k d k E ( k ) k dk = q − q + 3 ℓ . Appendix C: Derivation of Eqs. (14) and (15)
In this Appendix we derive Eqs. (14) and (15) for largeP´eclet and Reynolds numbers. Using Eq. (4) for the tem-perature fluctuations θ and the Navier-Stokes equationfor the velocity u written in Fourier space, we deriveequations for the following correlation functions: F j ( k , R ) = Z h θ ( k + K / u j ( − k + K / i× exp[ i K · R ] d K , (C1) E θ ( k , R ) = Z h θ ( k + K / θ ( − k + K / i× exp[ i K · R ] d K . (C2)For brevity of notations we omit the large-scale variable t in the functions F j ( k , R , t ), E θ ( k , R , t ) and the meantemperature T ( R , t ).To derive evolution equations in the Fourier space forthe turbulent heat flux F j ( k , R ) and the level of temper-ature fluctuations E θ ( k , R ), we rewrite Eq. (4) for the1temperature fluctuations in k space as ∂θ ( k ) ∂t = − i (cid:20) ( γ − Z T ( Q ) ( k i − Q i ) u i ( k − Q ) d Q + Z Q i T ( Q ) u i ( k − Q ) d Q (cid:21) + θ (N) ( k ) , (C3)where θ (N) ( k ) are the nonlinear terms written in k space.For brevity of notations we omit below the variable t inthe functions T ( Q , t ), θ ( k , t ), θ (N) ( k , t ) and u i ( k , t ).Using Eq. (C3) for the temperature fluctuations θ writ-ten in Fourier space, we derive equations for the in-stantaneous two-point correlation functions F j ( k , R ) and E θ ( k , R ) defined by Eqs. (C1) and (C2). To this end weuse the identities: ∂∂t h θ ( k , t ) u j ( k , t ) i = (cid:28) ∂θ ( k , t ) ∂t u j ( k , t ) (cid:29) + (cid:28) θ ( k , t ) ∂u j ( k , t ) ∂t (cid:29) , (C4) ∂∂t h θ ( k , t ) θ ( k , t ) i = (cid:28) ∂θ ( k , t ) ∂t θ ( k , t ) (cid:29) + (cid:28) θ ( k , t ) ∂θ ( k , t ) ∂t (cid:29) . (C5)Equations (C3)–(C5) yield the dynamic equations as ∂F j ( k , R ) ∂t = J j ( k , R ) + ˆ M F (III) j ( k , R ) , (C6) ∂E θ ( k , R ) ∂t = S ( k , R ) + ˆ M E (III) θ ( k , R ) , (C7)where ˆ M F (III) j ( k , R ) = Z (cid:20) (cid:28) θ ( k ) ∂u j ( k ) ∂t (cid:29) + D θ (N) ( k ) u j ( k ) E(cid:21) exp[ i K · R ] d K , (C8)ˆ M E (III) θ ( k , R ) = Z (cid:20) D θ ( k ) θ (N) ( k ) E + D θ (N) ( k ) θ ( k ) E(cid:21) exp[ i K · R ] d K (C9)are the third-order moment terms in k space appearingdue to the nonlinear terms, and J j ( k , R ) = − i Z h ( γ −
1) ( k i + K i / − Q i ) + Q i i h u i ( k + K / − Q ) u j ( − k + K / i T ( Q ) exp( i K · R ) d K d Q , (C10) S ( k , R ) = − i Z (cid:26)h ( γ −
1) ( k j + K j / − Q j ) + Q j i h θ ( − k + K / u j ( k + K / − Q ) i + h ( γ −
1) ( − k j + K j / − Q j ) + Q j i h θ ( k + K / u j ( − k + K / − Q ) i (cid:27) T ( Q ) exp( i K · R ) d K d Q . (C11)To derive Eq. (14), we perform calculations in Eq. (C10)which are similar to those in Eqs. (B4)–(B7). To deter-mine h θ (˜ k ) u j (˜ k ) i in Eq. (C11), we use new variables:˜ k = (˜ k − ˜ k ) / − k + Q / , (C12)˜ K = ˜ k + ˜ k = K − Q , (C13)where ˜ k = − k + K / , ˜ k = k + K / − Q . (C14)Since | Q | ≪ | k | and | K | ≪ | k | , we use the Taylor expan-sion D θ (˜ k ) u j (˜ k ) E = F j (˜ k , ˜ K ) = F j ( − k , ˜ K ) + Q m ∂F j ∂ ˜ k m + O ( Q ) = (cid:18) − Q m ∂∂k m (cid:19) F j ( − k , ˜ K ) + O ( Q ) . (C15) Similarly, D θ (˜ k ) u j (˜ k ) E = (cid:18) Q m ∂∂k m (cid:19) F j ( k , ˜ K ) + O ( Q ) , (C16)where ˜ k = k + K / , ˜ k = − k + K / − Q . (C17)Substituting Eqs. (C15) and (C16) into Eq. (C11), ne-glecting the terms O ( Q ; K ), and returning to the phys-ical space in the large-scale variables, we obtain Eqs. (14)and (15).To determine the turbulent heat flux and the level oftemperature fluctuations, we use the following identitiesfor integration over k in Eqs. (17) and (18): Z k ν k τ ( k ) [ E ( k ) + σ c E c ( k )] dk = τ (1 + σ c ) , Z k ν k τ ( k ) E ( k ) dk = τ " − ˜ C σ σ c σ c ) , Z k ν k τ ( k ) E c ( k ) dk = τ " C σ σ c ) , Z k ν k τ ( k ) k E c ( k ) dk = 6 τ ℓ (1 + σ c ) − (cid:20) Re / + σ c (cid:21) , Z k ν k τ ( k ) k E c ( k ) dk = 4 f c (cid:18) τ ℓ (cid:19) (cid:18) σ c σ c (cid:19) , Z k ν k τ ( k ) [ E ( k ) + σ c E c ( k )] dk = 43 τ (1 + σ c ) , Z k ν k τ ( k ) E c ( k ) dk = 43 τ f ∗ (cid:18) σ c σ c (cid:19) , Z k ν k τ ( k ) E ( k ) dk = 43 τ (1 + σ c ) " − f ∗ (cid:18) σ c σ c (cid:19) , where f ∗ = 1 + 6( q c − σ c ( q + 2 q c −
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