Journal of Fluid Science and Technology

Science and Technology Effect of Molecular Diffusivities on Countergradient Scalar Transfer in a Strong Stable Stratified Flow (Study on the Linear and Nonlinear Processes by using RDT) Kouji NAGATA, Takashi SATO and Satoru KOMORI Department of Mechanical Science and Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464 8603, Japan E-mail: nagata@nagoya-u.jp Department of Mechanical Engineering, and Advanced Research Institute of Fluid Science and Engineering, Kyoto University, Yoshidahonmachi, Sakyo-ku, Kyoto 606-8501, Japan Abstract Linear rapid distortion theory (RDT) is applied to unsteady, unsheared, stable, thermally stratified air (Pr = 0.7), thermally stratified water (Pr = 6), and salt-stratified liquid (Sc 600) flows. The effects of diffusivity and viscosity are included in the analysis and turbulence quantities such as turbulent scalar fluxes and their cospectra are obtained. The results are compared with previous laboratory measurements and direct numerical simulations (DNS). The results showthat countergradient scalartransfer (CGST), which transports the scalar counter to the mean gradient (i.e., negative eddy diffusivity), can be predicted by linear RDT, as shown in the previous studies. However, the small-scale persistentdowngradient scalar transfer (P-DGST) in air flows and the small-scale persistent CGST (P-CGST) in water flows cannot be predicted by RDT. In a linear process, small-scale CGST occurs first and then it spreads on a large-scale regardless the values of Pr or Sc; then, the small-scale fluxes change their signs to become downgradient flux again (i.e., small-scale flux oscillates with time). The results suggest that the small-scale turbulent scalar transfer in a strong, stable stratified flow is dominated by nonlinear processes, and only the large-scale wave-like motions are controlled by the linear processes. Key words : Stratified Flow, Heat Transfer, Diffusion, Rapid Distortion Theory 1. Introduction Received 30 Nov., 2007 (No.T2-04-0127) Japanese Original: Trans. Jpn. Soc. Mech. Eng., Vol.70, No.698, B (2004), pp.2598 2603 (Received 16 Feb., 2004) [DOI: 10.1299/jfst.3.232] Density-stratified flows often occur in the oceans, atmospheric boundary layers, and many industrial operations. The diffusion of scalar variables such as heat and mass in stratified flows is strongly affected by buoyancy. It is, therefore, of great importance to investigate the effects of buoyancy on heat and mass transfer to predict the turbulent diffusion of scalar quantities in environmental and industrial flows. In strong, stable density stratifications, scalar quantities such as heat and mass are sometimes transported against the time-averaged gradient of heat and mass, which is known as countergradient scalar transfer (hereafter referred to as CGST). The authors have conducted precise measurements of the countergradient scalar fluxes in strong, stable thermally (Pr = 6) and salt-stratified (Sc 600) water flows with and without mean shear (1) and also performed three-dimensional direct numerical simulation (DNS) of stratified water (Pr = 6) and air (Pr = 0.7) flows (2). Their results indicate the difference between the small-scale scalar trans- 232

fer in stratified water and air flows for different values of Pr or Sc. In stratified water flows (Pr, Sc 1), small-scale motions first contribute to CGST, and the small-scale CGST in the salt-stratified flow at very high value of Sc (Sc 600) persists throughout the decaying process with temporal large-scale flux oscillations (i.e., persistent countergradient: P-CGST). On the other hand, in unsheared, stable stratified air flows, they indicated that small-scale CGST does not occur, and the small-scale scalar transfer is persistently downgradient (P-DGST). For the precise mechanism of CGST in decaying grid turbulence, the reader should refer to our previous paper (1). In short, CGST can be explained by the imbalance between the turbulence kinetic energy and available potential energy. On the other hand, the rapid distortion theory (RDT), which is a linear analysis, has also been applied to the strong, stable stratified air and water flows. The previous RDT analysis suggested that the CGST can be explained by a linear process (3) (5). However, the question remains whether the P-CGST in water flows and P-DGST in air flows are dominated by either linear or nonlinear processes. In order to clarify the CGST mechanism and to construct a precise turbulence model for heat and mass transfer in stratified water and air flows, it is of great importance to investigate the small-scale CGST and its dependence on Pr and Sc. The purpose of this study is, therefore, to apply the linear RDT to unsteady, unsheared, strong stable (with Froude number Fr = u /NL < 1), thermally stratified water (Pr = 6), thermally stratified air (Pr = 0.7), and salt-stratified liquid (Sc 600) flows, and to investigate the linear process in the small-scale scalar transfer and its dependence on Pr or Sc in strong, stable stratified flows. The effects of diffusivity and viscosity are included in the analysis, and turbulence quantities such as turbulent scalar fluxes and their cospectra are obtained. It should be noted that the RDT analysis of a strong salt-stratified flow (1) with a high Sc (Sc 600) has not been reported previously. By comparing the results of the RDT with the previous measurements (1), (6), (7) and DNS (2), the linear and nonlinear processes in small-scale scalar transfer in strong, stable stratified flows are investigated. 2. Nomenclature g : gravitational acceleration k 0k : initial peak wavenumber of a spectrum of turbulence kinetic energy k 0p : initial peak wavenumber of a spectrum of potential energy KE 0 : initial value of turbulence kinetic energy k : wavenumber vector k : magnitude of k (= (k1 2 + k2 2 + k2 3 )1/2 ) L : integral length scale in the initial (undistorted) flow N : Brunt-Väisälä frequency (= { (g/ρ 0 )(dρ/dy)} 1/2 ) PE 0 : initial value of potential energy Pr : Prandtl number (= ν/κ) Sc : Schmidt number (= ν/κ = Pr in RDT) t : time u : streamwise rms velocity u i : streamwise fluctuating velocity U : mean velocity ρ : fluctuating density ρ 0 : bulk-averaged density κ : molecular diffusivity of scalar ν : kinematic viscosity Θ ρ3 : one-dimensional spectrum of vertical density flux ρu 3 3. Rapid Distortion Theory (RDT) 3.1. Outline of the Analytical Procedure Unsteady, unsheared, strong, and stable stratified flows (Fig. 1) are analyzed by using 233

Fig. 1 Schematic of flow condition. A shear-free, stable stratified flow with a constant negative density gradient is considered. The initial (t < 0) turbulence is assumed to be isotropic the linear RDT. The vertical scalar gradient dρ/dx 3 is assumed to be constant, which is a good approximation of the previous measurement (1). The solutions to this problem have been reported by Hanazaki & Hunt (4), and the reader should refer to Hanazaki & Hunt (4) for details. Under the assumption of strong stratification (Fr < 1), which applies to our previous measurement (1), the nonlinear terms in the Navier-Stokes equations can be neglected and the linear RDT equations are obtained in the frame of reference moving with a uniform mean flow: ( ) ( ) d dt + ki k 3 νk2 û i = δ k 2 i3 ˆρ, (1) ( ) d dt + κk2 ˆρ = N 2 uˆ 3. (2) The Fourier coefficients û = (ˆ u 1, uˆ 2, uˆ 3 )andˆρ are defined in terms of velocity and density fluctuations as u i = k û i (k, t)e ik x, (3) g ρ = ˆρ(k, t)e ik x. (4) ρ 0 k Note that ρ in the above equation acts as an active scalar (with a molecular diffusivity κ); therefore, hereafter, ρ is referred to as a scalar for the better comparison with the previous experiment and DNS. By solving Eqs. (1) and (2) together with dk i /dt = 0, û and ˆρ can be obtained analytically. Then, we can calculate the three-dimensional spectral functions Φ; further, by integrating Φ, we obtain the turbulence intensities and turbulent fluxes. For example, the three-dimensional spectral function for the vertical turbulent scalar flux is calculated by Φ ρ3 (k, t) = 1 2 ˆρ uˆ 3 + ˆρ uˆ 3. (5) Here, * denotes the complex conjugate. The one-dimensional power spectrum of the vertical turbulent scalar flux is given by Θ ρ3 (k 1, t) = Φ ρ3(k, t)dk 2 dk 3. (6) The vertical turbulent scalar flux is then calculated by ρu 3 (t) = Θ ρ3(k 1, t)dk 1 = Φ ρ3(k, t)dk. (7) 234

Fig. 2 Time variations of the vertical turbulent scalar flux in thermally stratified air (Pr = 0.7), thermally stratified water (Pr = 6), and salt-stratified liquid (Sc = 600) flows. The bulk Richardson number and Reynolds number are the same for all the cases 3.2. Analytical Conditions A homogeneous isotropic turbulence with an integral length scale L was assumed for the initial condition. We used the initial velocity and scalar spectra adopted by Hanazaki & Hunt (4) : ) 1 2 ( 2 ( 2 E(k) = KE 0 9π k 0k ( ) 1 ( 2 2 2 S (k) = PE 0 9π k 0p ) 5 k 4 e 2k2 k 2 0k, (8) ) 5 k 4 e 2k2 k 2 0p. (9) The values of KE 0, PE 0, k 0k,andk 0p are estimated from the previous experiment (1). The Brunt-Väisälä frequency N is selectedto have the same value (=1.7rad/s for the water flows) as that in the previous experiment (1). It should be noted that N for the air flow is greater than that for the water flow for the same bulk Richardson and Reynolds numbers, because the mean velocity of the air flow should be greater by a factor of ν water /ν air for the same Reynolds number, where ν water and ν air denote the kinematic viscosity of water and air, respectively. The bulk Reynolds number (= UL /ν) and the bulk Richardson number (=N 2 L 2 /U 2 ) are fixed in all the analysis. The RDT analysis is conducted for thermally stratified air (Pr = 0.7), thermally stratified water (Pr = 6), and salt-stratified liquid (Sc = 600) flows. 4. Results and Discussion 4.1. Effects of Pr and Sc on Turbulent Scalar Flux Figure 2 shows the time variations of the normalized vertical turbulent scalar flux ρu 3 in the thermally stratified air (Pr = 0.7), thermally stratified water (Pr = 6), and salt-stratified liquid (Sc = 600) flows. Here, the initial energy partition PE 0 /KE 0 is 0.11, and the initial peak wavenumbers are k 0k = k 0p = 200 m 1. Figure 2 indicates that the vertical turbulent scalar flux oscillates and temporally becomes negative, which indicates a net CGST. This means that the CGST can be explained by a linear process, as shown previously (4). However, some important discrepancies, including qualitative discrepancies, are observed and must be addressed. First, the predicted CGST is significant in the stratified air flow for the same bulk Richardson and Reynolds numbers, whereas the previous measurements (1) and DNS (2) clearly indicate that the CGST is significant in the stratified water flow for the same bulk Richardson and Reynolds numbers. Second, the RDT predicts a very small difference between the thermally stratified water and salt-stratified liquid flows. However, our previous measurement (1) has shown that 235

Fig. 3 Time variations of the cospectra of the vertical turbulent scalar flux in (a) thermally stratified water (Pr = 6) and salt-stratified liquid (Sc = 600) flows and (b) thermally stratified air (Pr = 0.7) flow. The cospectra are pre-multiplied by k 1 and the positive values of k 1 Θ ρ3 indicate CGST. The numbers indicate the dimensionless time, Nt/2π the CGST is significant in salt-stratified liquid flows at a very high Sc value of 600, even when the bulk Richardson number (or initial Brunt-Väisälä frequency N) and Reynolds number are identical. Note that ρu 3 0whenNt (4). These results suggest that the linear RDT analysis fails to predict the effect of Pr and Sc on CGST. Figure 3 shows the time variations of the cospectra of the vertical turbulent scalar flux in the thermally stratified water (Pr = 6), salt-stratified liquid (Sc = 600), and thermally stratified air (Pr = 0.7) flows. The cospectra are pre-multiplied by k 1, and the negative and positive values of k 1 Θ ρ3 indicate DGST and CGST, respectively. The numbers correspond to the dimensionless time, Nt/2π. Figure 3 shows that the small-scale CGST occurs first and subsequently spreads on a large scale irrespective of the values of Pr or Sc in a linear system. These results disagree with the previous measurements of air flows (6), (7) in which small-scale CGST was not observed. 4.1.1. Effects of Initial Conditions on Small-Scale CGST To examine the effects of the initial conditions on small-scale scalar transfer (in a linear system), calculations were performed for different values of PE 0 /KE 0, k 0k,andk 0p in thermally stratified air flows. Figure 4 shows the time variations of the cospectra of vertical turbulent scalar flux in the thermally stratified air flow with an initial energy partition of PE 0 /KE 0 = 0.3. The values of k 0k and k 0p are the same as those used in Figs. 2 and 3. It can be observed that no qualitative difference exists for different values of PE 0 /KE 0. For a considerably larger value of PE 0 /KE 0 (not shown), CGST occurs from the beginning (at t = 0). However, a small-scale flux first becomes downgradient, and P-CGST or P-DGST is not observed. It should be noted that in the previ- 236

Fig. 4 Time variations of the cospectra of the vertical turbulent scalar flux in thermally stratified air flow. PE 0 /KE 0 = 0.3. The values of k 0k and k 0p are the same as those in Fig.3 Fig. 5 Time variations of the cospectra of the vertical turbulent scalar flux in thermally stratified air flow. k 0k = 200 m 1 and k 0p = 300 m 1.ThevalueofPE 0 /KE 0 is the same as that in Fig.3 ous measurements of the stratified air flows (6), (7), such cospectrum was not measured because of the different initial condition (i.e., PE 0 /KE 0 is not considerably large in grid turbulence). Figure 5 shows the time variations of the cospectrum of the vertical turbulent scalar flux in the thermally stratified air flow with the initial peak wavenumbers of k 0k = 200 m 1 and k 0p = 300 m 1.ThevalueofPE 0 /KE 0 is the same as that used in Figs. 2 and 3. Figure 5 shows that no qualitative difference is observed for different initial peak wavenumbers. The same results were obtained for other values of k 0k and k 0p. Further, it should be noted that the values of PE 0 /KE 0, k 0k,andk 0p are estimated from the laboratory measurements, and it is unrealistic to significantly decrease or increase the values. A qualitative change is not observed (not shown) in the evolution of the cospectra in stratified water flows, as in stratified air flows, due to the changes in PE 0 /KE 0, k 0k,andk 0p within the possible range in the experiments. 4.2. Small-Scale P-CGST and P-DGST in Stratified Water and Air Flows It is found that small-scale P-DGST observed in the previous measurements in air flows cannot be predicted by the linear RDT; small-scale CGST first occurs both in water and air flows. The results are not consistent with those in the previous experiment (1), (6), (7) and DNS (2). Thus, P-DGST in strongly stratified air flows may be the result of the nonlinear effect. On the other hand, in strongly stratified water flows, small-scale P-CGST has been observed, as showninfig.6 (1). Figure 7 shows the time variations of the cospectra of the vertical turbulent 237

Fig. 6 Time variation of the cospectra of the vertical turbulent scalar (mass) flux in the salt-stratified flow obtained in the previous laboratory measurements (1). f denotes the frequency, and the cospectra are pre-multiplied by f. The cospectra are normalized by the mean velocity U and the difference between the initial concentrations of the upper and lower streams in the mixing-layers type grid turbulence, ΔC Fig. 7 Time variation of the cospectra of the vertical turbulent scalar flux in the saltstratified liquid flow predicted by RDT scalar flux in the salt-stratified liquid flow at 0.487 (CGST) < Nt/2π <0.649 (DGST). Figure 7 shows that the small-scale flux first changes its sign and becomes DGST. This result qualitatively disagrees with that in the previous measurement (1) and DNS (8). As shown in Fig. 6, our previous measurement indicates that the small-scale flux always contributes to the CGST (i.e., P-CGST), and only the large-scale motions oscillate temporally. This result is consistent with the explanation for P-CGST provided by Gerz & Schumann (8) ; the difference in the efficiency of the nonlinear transfer of kinetic and potential energy from large scales to small scales induces small-scale P-CGST. It should be noted that there is no energy production by mean shear in the present unsheared flows; therefore, the turbulent kinetic energy (which may drive the P-CGST) should be fed in small scales from large scales by the nonlinear energy cascade process. On the other hand, in the RDT analysis, energy transfer does not occur across the wavenumbers; therefore, P-CGST cannot be predicted by the RDT. Note that the explanations for small-scale CGST (but not for small-scale P-CGST) provided by Komori & Nagata (1) and Holloway (9) are also based on the nonlinear mechanism. Gerz & Schumann (8) also suggested that P-CGST mechanism operates even in a stratified air flow with a low Pr value of 0.7 if the Reynolds number is considerably large. Note again that the RDT result for the stratified air flow shown in Fig. 3 is not a P-CGST. When we 238

consider the violent mixing at small scales at significantly higher turbulent Reynolds number flows (or sheared flows), together with the generation mechanism of the countergradient scalar transfer based on the nonlinear effect (1), (8), the small-scale P-CGST in a stratified air flow can be expected. It would be interesting to investigate P-CGST in a wind tunnel. The above results suggest that in strong stable stratifications, small-scale turbulent scalar transfers are strongly affected by the nonlinear effect; therefore, the RDT analysis is valid only for wave-like large-scale fluid motions in strong stable stratified flows. 5. Conclusions The linear RDT is applied to stable thermally stratified air (Pr = 0.7), thermally stratified water (Pr = 6), and salt-stratified liquid (Sc = 600) flows without mean shear, and the smallscale CGST and its dependence on the molecular diffusivity (i.e., Pr or Sc) are investigated. The results are compared with the previous measurements and DNS in stratified water and air flows. The result shows that CGST can be explained by a linear process. However, the effects of Pr and Sc on the CGST cannot be predicted correctly by using the linear RDT; small-scale P-DGST in air flows and small-scale P-CGST in water flows, which have been observed in the previous measurements and DNS, cannot be predicted by the linear RDT. In a linear process, small-scale CGST first occurs and subsequently spreads on a large scale irrespective of the values of Pr or Sc. As Nt/2π increases, the small-scale flux changes its sign prior to the large-scale flux (i.e., small-scale flux oscillates with time) in a linear system. This result is consistent with the explanation for P-CGST provided by Gerz & Schumann (1991) based on the nonlinear transfer of kinetic and potential energy from large to small scales. The present results show that the turbulent small-scale scalar transfer in a strong stable stratified flow is dominated by nonlinear processes, and only large-scale wave-like motions are controlled by linear processes. Acknowledgments This study was partially supported by the Japanese Ministry of Education, Culture, Sports, through Grants-in-Aid (Nos.16760126 and 14102016) and the Center of Excellence for Research and Education on Complex Functional Mechanical Systems (COE program of the Ministry of Education, Culture, Sports,, Japan). The part of discussion is based on the DNS carried out by using the supercomputer NEC SX-6 at the Center for Global Environmental Research, National Institute for Environmental Studies, Environmental Agency of Japan. References ( 1 ) Komori, S. and Nagata, K., Effects of Molecular Diffusivities on Counter-Gradient Scalar and Momentum Transfer in Strongly Stable Stratification, Journal of Fluid Mechanics, Vol. 326, (1996), pp.205 237. ( 2 ) Nagata, K. and Komori, S., Direct Numerical Simulation of the Prandtl Number Effects on the Countergradient Scalar Transfer in Strong Stable Stratification, Transactions of the Japan Society of Mechanical Engineers, Series B (in Japanese), Vol.62, No.600 (1996), pp.3142 3148. ( 3 ) Hunt, J. C. R., Stretch, D. D., and Britter, R. E., Length Scales in Stably Stratified Turbulent Flows and Their Use in Turbulence Models, In Stably Stratified Flow and Dense Gas Dispersion (ed. J. S. Puttock), Clarendon Press, (1988), pp.285 321. ( 4 ) Hanazaki, H. and Hunt, J. C. R., Linear Processes in Unsteady Stably Stratified Turbulence, Journal of Fluid Mechanics, Vol. 318, (1996), pp.303 337. ( 5 ) Galmiche, M. and Hunt, J. C. R., The Formation of Shear and Density Layers in Stably Stratified Turbulent Flows: Linear Processes, Journal of Fluid Mechanics, Vol. 455, (2002), pp.243 262. ( 6 ) Lienhard V, J. H. and Van Atta, C. W., The Decay of Turbulence in Thermally Stratified Flow, Journal of Fluid Mechanics, Vol. 210, (1990), pp.57 112. 239

( 7 ) Yoon, K. and Warhaft, Z., The Evolution of Grid-Generated Turbulence Under Conditions of Stable Thermal Stratification, Journal of Fluid Mechanics, Vol. 215, (1990), pp.601 638. ( 8 ) Gerz, T. and Schumann, U., Direct Simulation of Homogeneous Turbulence and Gravity Waves in Sheared and Unsheared Stratified Flows, Turbulent Shear Flows 7, Springer- Verlag, (1991), pp.27 45. ( 9 ) Holloway, G, The Buoyancy Flux from Internal Gravity Wave Breaking, Dynamics of Atmospheres and Oceans, Vol. 12, (1988), pp.107 125. 240