Satyajit Barman, Anando G. Chatterjee, 1, e) Ravi Samtaney, 2, f) 3, 4, g)

Transcription

1 Energy fluxes and spectra for turbulent and laminar flows Mahendra K. Verma, 1, a) Abhishek Kumar, 1, b) Praveen Kumar, 1, c) 1, d) Satyajit Barman, Anando G. Chatterjee, 1, e) Ravi Samtaney, 2, f) 3, 4, g) and Rodion Stepanov 1) Department of Physics, Indian Institute of Technology, Kanpur , India 2) Mechanical Engineering, Division of Physical Sciences and Engineering, King Abdullah University of Science and Technology - Thuwal , Kingdom of Saudi Arabia 3) Institute of Continuous Media Mechanics, Korolyov 1, Perm , Russia 4) Perm National Research Polytechnic University, Komsomolskii Avenue 29, Perm, Russia Two well-known turbulence models to describe the inertial and dissipative ranges simultaneously are by Pao [Y.-H. Pao, Structure of turbulent velocity and scalar fields at large wavenumbers, Phys. Fluids 8, 1063 (1965)] and Pope [S. B. Pope, Turbulent Flows. Cambridge University Press, Cambridge, 2000]. In this paper, we compute energy spectrum E(k) and energy flux Π(k) using spectral simulations on grids up to , and show consistency between the numerical results and predictions by the aforementioned models. We also construct a model for laminar flows that predicts E(k) and Π(k) to be of the form exp( k), and verify the model predictions using numerical simulations. The shell-to-shell energy transfers for the turbulent flows are forward and local for both inertial and dissipative range, but those for the laminar flows are forward and nonlocal. a) Electronic mail: mkv@iitk.ac.in b) Electronic mail: abhishek.kir@gmail.com c) Electronic mail: praveenkumar.hcdu@gmail.com; Present address: BARC, Mumbai d) Electronic mail: sbarman@iitk.ac.in e) Electronic mail: anandogc@iitk.ac.in f) Electronic mail: ravi.samtaney@kaust.edu.sa g) Electronic mail: rodion@icmm.ru 1

2 I. INTRODUCTION Turbulence is a classic problem with more unsolved issues than the solved ones. The most well-known theory of turbulence is by Kolmogorov 1. According to this theory, the energy supplied at the large scales cascades to small scales. The wavenumber range dominated by forcing is called forcing range, while that dominated by dissipation is called dissipative range. The wavenumbers between these two ranges are termed as inertial range. According to Kolmogorov 1, the energy cascade rate or the energy flux is constant in the inertial range. Quantitatively, the one-dimensional energy spectrum E(k) and the energy flux Π(k) in the inertial range are E(k) = K Ko ɛ 2/3 k 5/3, (1) Π(k) = ɛ, (2) where ɛ is the energy dissipation rate, and K Ko is the universal constant. The above law has been verified using experiments and high-resolution simulations (see Frisch 2, McComb 3, Davidson 4, Ishihara et al. 5 and references therein). There is, however, a small correction, called intermittency correction 2, to the exponent 5/3; this issue is however beyond the scope of this paper. Kolmogorov s theory of turbulence and its ramifications have been discussed in detail in several books 2 4,6 8. The energy spectrum of Eq. (1) is universal, i.e., it is independent of fluid properties, forcing and dissipative mechanisms, etc. Equation (1) has been generalised so as to include the dissipative range that includes the dependence on the kinematic viscosity ν, and it is commonly written as E(k) = K Ko ɛ 2/3 k 5/3 f(k/ ), (3) where f(k/ ) is a universal function, and (ɛ/ν 3 ) 1/4 is the dissipation wavenumber scale, also called Kolmogorov s wavenumber. Pao 9, Pope 8, and Martínez et al. 10 have modelled f(k/ ); Pao 9 proposed that f(x) exp( x 4/3 ), but according to Pope 8 f(x) exp { } [x 4 + c 4 η] 1/4 c η, (4) where c η is a constant. Pope s model 8 is in good agreement with earlier experimental results (see Saddoogchi and Veeravalli 11 and references therein). Pao 9 argued that his predictions fit well with the experimental results of Grant et al. 12. Martínez et al. 10 proposed that E(k) (k/ ) α exp[ β(k/ )], (5) 2

3 and found good agreement between their predictions and numerical results for moderate Reynolds numbers. There are only a small number of numerical simulations that investigate the dissipative spectrum of a turbulent flow. For example, Martínez et al. 10 computed E(k) for moderate Reynolds number and showed it to be consistent with the model of Eq. (5). On the other hand, Ishihara et al. 13 showed that near the dissipation range, Eq. (5) is a good approximation for high Reynolds number flows. The energy flux in the dissipative regime of a turbulent flow has not been investigated in detail, either by numerical simulation or experiments. Note that Π(k) in the dissipative range is assumed to be very small, and it is typically ignored. In the present paper, we perform numerical simulations on very high-resolution (up to grid), and discuss E(k) and Π(k) in detail, both in inertial and dissipative regime. Laminar flows are ubiquitous in nature; some examples of such flows are blood flow in arteries, micro and nano fluidics 14, mantle convection inside the Earth 15, laminar dynamo, and passive scalar turbulence with large Schmidt number 16. Hence, models of laminar flows are very useful. In this paper, we discuss a spectral model for such flows. For laminar flows, the energy flux is typically assumed to be vanishingly small citing a reason that the nonlinearity and hence Reynolds number is small for such flows. Researchers have studied the kinetic energy spectrum for such flows and predicted this to be of the form exp( k 2 ) or exp( k) 10, See for example, Martínez et al. s proposal 10 (see Eq. (5)). Martínez et al. 10 compared their model predictions with the numerical simulations for low Reynolds number as well. In the present paper, we show that E(k) exp( k/ )/k and Π(k) exp( k/ ), where is a measure of dissipation wavenumber, are the consistent solution for laminar flows. We also performed direct numerical simulations (DNS) for Re ranging from 17.6 to 49 and verify that our model predictions and numerical results are consistent with each other. According to Kolmogorov s theory of turbulence, the energy transfers among the inertialrange wavenumber shells are forward and local That is, a wavenumber shell (say m) transfers maximal energy to its nearest forward neighbour (m + 1), and receives maximal energy from its previous neighbour (m 1). The aforementioned phenomena have been verified using numerical simulations. However, there is no definitive shell-to-shell energy transfer computation for the dissipative regime. Similar investigations for laminar flows too have not been performed. In this paper, we show that the shell-to-shell energy transfers in the dissipative regime of a turbulent flow are local; however, these transfers for laminar 3

4 flows are nonlocal. There are a number of analytical works for modeling turbulence, namely quasi-normal approximation (QNA) and eddy-damped quasi-normal Markovian (EDQNM) 24, directinteraction approximation (DIA) 25, Lagrangian DIA (LDIA) and Abridged Lagrangianhistory DIA (ALHDIA) 26, strain-based ALHDIA (SBALHDIA) 27, Lagrangian renormalized approximation (LRA) These models attempt to compute the higher-order correlations and related quantities by various closure schemes. Note that the energy flux Π(k) is computed using third-order correlation function, thus it is also computed using a closure scheme in field-theoretic treatment of turbulence 2,6. The focus of the present paper is on the validation of the presently available models on the energy spectrum and fluxes, as well as on a new spectral model for laminar flows. We remark that such studies are very important for applications encountered by engineers, and geo-, astro-, and atmospheric physicists. The outline of the paper as follows: In Sec. II, we describe the turbulence models of Pao and Pope, as well as our model for laminar flows. Sec. III contains numerical results of high-resolution turbulence simulations, as well as those of laminar flows. We conclude in Sec. IV. II. MODEL DESCRIPTION The incompressible fluid flow is described by the Navier Stokes equation: u t + (u )u = 1 ρ p + ν 2 u + f, (6) u = 0, (7) where u is the velocity field, p is the pressure field, ν is the kinematic viscosity, and f is the force field. We take the density ρ to be a constant, equal to unity. In Fourier space the above equations are transformed to 7 ( ) d dt + νk2 u(k, t) = ikp(k, t) i k=p+q k u(q)u(p) + f(k), (8) k u(k) = 0, (9) where u(k), p(k), and f(k) are the Fourier transforms of u, p, and f respectively. The above equations yield the following equation for one-dimensional energy spectrum E(k) 7 : E(k, t) t = T (k, t) 2νk 2 E(k, t) + F(k, t), (10) 4

5 where T (k, t) is the energy transfer to the wavenumber shell ue to nonlinearity, F(k) is the energy feed by the force, and 2νk 2 E(k) is the dissipation spectrum. Note that T (k) = dπ(k)/dk. In Kolmogorov s model, the force feed F(k) is active at large length scales (for k < k f, where k f is the forcing wavenumber), and it is absent in the inertial and dissipative range. Therefore, E(k, t)/ t = 0 during a steady state. Hence, for a steady-state, the energy flux Π(k) varies with k as dπ(k) dk = 2νk 2 E(k). (11) The turbulence model of Pao 9 is based on the above equation. It is more convenient to work with a nondimensionalized version of the above equation. We nondimensionalize k using the Kolmogorov wavenumber, which is defined as = ( ɛ ν 3 ) 1/4. (12) The energy flux Π(k) is nondimensionalized using ɛ as its scale. Hence, following Eq. (3) we obtain Substitution of the above variables in Eq. (11) yields k = k, (13) Π( k) = Π(k), ɛ (14) Ẽ( k) = E(k) ɛ 2/3 k = K Kof 5/3 η ( k). (15) d Π( k) d k = 2K Ko k1/3 f η ( k). (16) After the above introduction, we describe the models. A. Pao s model of turbulent flow In this paper, we discuss the models of Pao 9 and Pope 8, hence we label the energy fluxes and spectra of these models differently. For Pao s model, we label the energy spectrum, flux, nondimensional dissipative function as Ẽ(1) ( k), Π (1) ( k) and f η (1) ( k) respectively; for Pope s model 8 we use the superscript (2) for the corresponding quantities. 5

6 In Eq. (16), Π( k) and f η ( k) are two unknown functions. Hence, to close the equation, Pao 9 assumed that E(k)/Π(k) is independent of ν, and depends only on ɛ and k, or Hence, Eq. (15) implies that E(k) Π(k) ɛ 1/3 k 5/3. (17) Π (1) ( k) = 1 K Ko Ẽ (1) ( k) = f η ( k). (18) In other words, the dissipative spectrum for both E(k) and Π(k) should be of the same form. Thus Π(k) = ɛf η (k) and E(k) = K Ko ɛ 2/3 k 5/3 f η (k), substitution of which in Eq. (16) yields f (1) η ( k) = exp ( 32 ) K Ko k 4/3 ( 32 ) K Ko k 4/3, (19) Π (1) ( k) = exp, (20) Ẽ (1) ( k) = K Ko exp ( 32 ) K Ko k 4/3. (21) We can rewrite Eq. (11) as dπ(k) dk = 2νK Ko k 2 k 5/3 [ɛ] 2/3 f η (k) = 2νK Ko k 2 k 5/3 [ɛf η (k)] 2/3 [f η (k)] 1/3 = 2νK Ko k 2 k 5/3 [Π(k)] 2/3 [f η (k)] 1/3. (22) In the above equation Π(k) can be interpreted as a variable energy flux. Verma 31,32 exploited this interpretation to compute energy spectrum and flux for the two-dimensional flows with Ekman friction, and for quasi-static MHD turbulence. B. Pope s model of turbulent flow Another popular model for the turbulent flow is by Pope 8. For this model, we denote the energy spectrum, flux, nondimensional dissipative function f η ( k) as Ẽ(2) ( k), Π (2) ( k) and f η (2) ( k) respectively. Pope 8 proposed that E (2) (k) = K Ko ɛ 2/3 k 5/3 f L (kl)f (2) η (k/ ) (23) 6

7 with the functions f L (kl) and f η (kη) specifying the large-scale and dissipative-scale components, respectively: ( ) 5/3+p0 kl f L (kl) =, (24) [(kl) 2 + c L ] [ 1/2 f η (2) ( k) = exp β {[ k 4 + c 4η] }] 1/4 c η, (25) where the c L, c η, p 0, β are constants. Since we focus on the inertial and dissipative ranges, we set f L (kl) = 1. In the high Reynolds number limit, c η 0.47β 1/3 /K Ko 8. We refer to Pope 8 for the detailed derivation of above relation. We keep β = 5.2 as prescribed by Pope 8. Substitution of Ẽ(2) ( k) and Π (2) ( k) in Eq. (16) yields the following solution Π (2) ( k) = Π (2) ( k 0 ) 2K Ko k k o k 1/3 f (2) η ( k )d k, (26) which is solved numerically given f (2) η ( k) of Eq. (25). We set Π (2) ( k 0 ) = 1 at small k 0. In Sec. III A, we will validate the above two models using direct numerical simulations. C. Model for the laminar flows We will show later that the models of Pao 9 and Pope 8 do not provide satisfactory description for laminar flows for which the Reynolds number of the order of unity. Earlier Martínez et al. 10 had proposed the energy spectrum in the dissipative regime of the form E(k) (k/ ) α exp[ β(k/ )], (27) where α and β are constants. The above model is empirical. In the following, we construct a new spectral model for the laminar flows. We derive this model using the evolution equation for the energy flux. In this section, we propose that the energy spectrum for the laminar flow is E(k) = u 2 rmsf L (k) 1 k exp( k/ ), (28) where u rms is the rms velocity of the flow and = 7 Re 1.3L, (29)

8 with L as the box size, and Re = u rms L/ν is the Reynolds number. Note that of a laminar flow differs from, the Kolmogorov s wavenumber defined in Eq. (12). Interestingly, the above form of follows from = ( ɛ ) ( ) 1/4 νu 2 1/4 ν 3 L 2 ν 3 Re L. (30) The spectrum E(k) of Eq. (28) is motivated by the fact that the functional forms E(k) exp( k/ )/k and Π(k) exp( k/ ) satisfy Eq. (11). Further details of the model are as follows. Substitution of Eq. (28) in the expression for the energy dissipation rate yields ɛ = 0 = 2νu 2 rms k 2 d 2νk 2 E(k)dk 0 kf L ( k) exp( k)d k = 2νu 2 rms k 2 da, (31) where k = k/, and A = kf 0 L ( k) exp( k)d k is a nondimensional constant. Since Re, for a constant ɛ, Eq. (31) implies that A Re 3. Hence, for the laminar regime, we nondimensionalize energy spectrum and energy flux as follows: Substitution of Eq. (28) in Eq. (11) yields k = k kd, (32) Ē( k) = E(k)kRe3, u 2 rms (33) Π( k) = Π(k) Π(k) = ɛ. (34) 2νAu 2 rms k 2 d d Π( k) d k = k A f L( k) exp ( k), (35) whose solution is Π( k) = Π( k 0 ) 1 A k k 0 kfl ( k) exp( k)d k, (36) where k 0 is the reference wavenumber. We compute Π( k) using the above equation. In Sec. III B, we will validate the above predictions using direct numerical simulations. For k > k f, where k f is the forcing wavenumber, and for a constant ɛ, Ē( k) and Π( k) computed using numerical data merge into their respective single curves. Note that the energy flux in laminar flows is not vanishingly small. 8

9 Note that for k > k f, Ē( k) = exp( k), (37) Π( k) = (1 + k) exp( k), (38) are solution of Eq. (11). We revisit these issues in Sec. III B. The aforementioned models are for hydrodynamic turbulence. However, similar models based on variable energy flux have been used to describe quasi-static magnetohydrodynamic turbulence 20 and steepening of energy spectrum due to Ekman friction 31. In Appendix A we present a energy-flux based model for passive-scalar turbulence. III. NUMERICAL VALIDATION OF THE MODELS We perform direct numerical simulation of the incompressible Navier-Stokes equation [see Eqs. (6, 7)] in turbulent and laminar regimes, and compute the energy spectra and fluxes for various cases. We employ pseudo-spectral code Tarang 33 for our simulation. We use the fourth-order Runge-Kutta scheme for time advancement with variable t, which is chosen using the CFL condition. The pseudo-spectral method produces aliasing error, which is overcome by setting 1/3rd Fourier modes to zero. This dealiasing procedure is referred to as 2/3 rule 34. We compute the energy spectra and fluxes for all the simulations at the steady states. The energy spectrum E(k) is computed using 35 E(k) = 4π M k 1<k k where M is the number of modes in a shell of radius k. 1 2 u(k ) 2 k 2, (39) Note that the above formula reduces noise in the energy spectrum a low wavenumbers 35. The energy flux Π(k 0 ), rate of energy transfer emanating from the wavenumber of radius k 0, is computed using the following formula 36,37 : Π(k 0 ) = k>k 0 p k 0 δ k,p+q Im([k u(q)][u (k) u(p)]). (40) We compare the numerical results with the theoretical models discussed in the earlier section. In Subsection III A we present the results for turbulent flows, while in Subsection III B we discuss the results for laminar flows. 9

10 A. Turbulent Flow We perform our turbulence simulations on 512 3, , and grids. We employ the periodic boundary conditions on all sides of a cubic box of size (2π) 3. To obtain a steady turbulent flow, we apply random forcing 38 in the wavenumber band 2 k 4 for and grids, but in the band 1 k 3 for grid. We choose random initial condition for the grid simulation. The steady-state data of was used as an initial condition for the grid run, whose steady-state data is used for grid simulation. In all the three cases, the small scales are well resolved as k max η is always greater than 1.5, where k max is the highest wavenumber represented by the grid points, and η 1/ is the Kolmogorov s length. The Reynolds numbers for the 512 3, , and grid simulations are , , and respectively. We observe that the energy flux in the inertial range, the energy dissipation rate, and the energy supply rate by the forcing match with each other within 2-4%. The energy supply rate is chosen as 0.1, but the energy dissipation rate, as well as the energy flux, vary from to Also note that ɛ u 3 rms/l. The parameters of our runs for turbulent flows are described in Table I. TABLE I. Parameters of our direct numerical simulations (DNS) for turbulent flow: grid resolution; kinematic viscosity ν, Reynolds number Re, Kolmogorov constant K Ko, Kolmogorov wavenumber, k max η, and ɛ/(u 3 rms/l). For all our runs the energy supply rate is 0.1, and the energy dissipation rate ɛ 0.1 with 2-4% error. In the Table, we report the value of ɛ/(u 3 rms/l) which is approximately unity for all three simulations. Grid ν Re K Ko k max η ɛ/(u 3 rms/l) ± ± ± Figure 1(a, b, c) exhibits the normalized spectrum Ẽ( k)/k Ko for the 512 3, , and grid simulations. Note that the gray shaded region in the figure denotes the forcing band. The figure shows that the DNS results are consistent with the predictions of both 10

11 Pao and Pope. We also compute the Kolmogorov s constant K Ko using K Ko = E(k)k5/3 ɛ 2/3 (41) in the inertial range. As shown in Table I, the values of K Ko varies from 2.2 to 1.8 with errors in the range of 3% to 9%. Note that the error is due to the noise in the compensated energy spectrum. This value is in the same range as other numerical values of K Ko reported earlier The estimate of K Ko in DNS appears to be slightly larger than its theoretical value, which is approximately ,46. The increases in the value of K Ko in DNS may be due to finite-resolutions or lower Reynolds number of DNS than the theoretical limit of Re. Similar enhancement in the value of K Ko has been reported by Yeung and Zhou 40 ; Mininni et al. 43 have shown a decrease in K Ko with the Reynolds number. Note that for model predictions by Pao and Pope, we take K Ko = 2.2 for 512 3, K Ko = 1.85 for , and K Ko = 1.75 for simulations. A careful examination of the normalized spectrum indicates a hump in Ẽ( k)/k Ko near the transition region between the inertial range and dissipation range (0.04 k 0.2), which is due to the bottleneck effect 11,40,44, The predicted values of Ẽ( k)/k Ko by the models of Pao and Pope always decrease with k; hence they do not capture the above hump. However, as we show in Appendix B, the difference between the numerical result and the predictions of Pao s model can be used to quantify the bottleneck effect. Also, note that usually the bottleneck-induced hump decreases with the Reynolds number 44. We also remark that the bottleneck effect is related to intermittency corrections, which has been studied using large-resolution numerical simulations 51,52. In Fig. 1(d, e, f) we plot the nondimensionalized energy flux Π( k) computed using the DNS data. We observe that Π( k) is approximately constant in the inertial range, consistent with the Kolmogorov s theory 1. In the same plot, we present the energy fluxes computed using the Pao s and Pope s models [Eqs. (20, 26)]. In the inertial ranges, the predictions of both the models are in good agreement with the DNS results. In the dissipation range, the predictions of Pao s model is slightly larger than the numerical values of Π( k), but the predictions of Pope s model is lower than the numerical value. The difference in Π( k) between the predictions of Pao s model and the numerical value is due to the suppression of the energy flux at the bottleneck region. The increase in E(k) at the bottleneck region leads to an enhanced viscous dissipation, and thus a lower energy flux; this feature is unfortunately 11

12 10 0 (a) (d) Ẽ( k)/kko kmax k DNS Pao Pope Π( k) k DNS Pao Pope 10 0 (b) (e) Ẽ( k)/kko 10 2 Π( k) kmax k k 10 0 (c) (f) Ẽ( k)/kko 10 2 Π( k) kmax k k FIG. 1. For the grid resolutions of 512 3, , and : (a,b,c) plots of the normalized energy spectrum Ẽ( k)/k Ko vs. k; (d,e,f) plots of normalized energy flux Π( k) vs. k. See Eqs. (14) and Eq. (15) for definitions. The plots include the spectra and fluxes computed using numerical data (thick solid line), and the model prediction of Pao (thin solid line) and Pope (dashed line). The gray shaded region shows the forcing range. not captured by Pao s model. But, we can use the difference between the numerical E(k) and Pao s predictions to quantify the bottleneck effect, and to obtain deeper insights into the bottleneck effect, as well as into intermittency and backscatter of energy. In Appendix B we discuss the aforementioned measure. 12

13 (a) (b) m m n n FIG. 2. For the turbulent simulation on grid: the shell-to-shell energy transfer rate (a) for the whole wavenumber range, (b) for the dissipative range corresponding to the boxed region of subfigure (a). Here m denotes the giver shell, while n denotes the receiver shell. Our results indicate forward and local energy transfers in the inertial as well as in the dissipative wavenumber range. Thus, the model predictions of Pao9 and Pope8 are in good agreement with the numerical results of high-resolution DNS. However, in the dissipation range, there are small differences between the model predictions and numerical values of the energy flux. Specifically, both Pao s and Pope s models are not able to capture the bottleneck hump in the energy spectrum; and Pope s model does not capture Π(k) in the dissipation range. In addition, we also study the properties of shell-to-shell energy transfers for the numerical data of grid. For the same, we divide the Fourier space into 40 shells, whose centers are at the origin k = (0, 0, 0). The inner and outer radii of the shells are kn 1 and kn respectively, where kn = {0, 2, 4, 8, 8 2s(n 3),..., 2048}, with s = (1/35) log2 (128). The shells are logarithmically binned53. Note that the 27th shell, whose wavenumber range is 194 k 223, separates the dissipative range from the inertial range. In Fig. 2(a), we exhibit the shell-to-shell energy transfers for the whole range, while Fig. 2(b) shows these transfers for the dissipative range only. As expected, shell m gives energy dominantly to shell m + 1, and it receives energy from shell m 1 in the inertial regime, hence, the shell13

14 to-shell energy transfers are forward and local Interestingly, similar behaviour, forward and local energy transfer, is observed for the wavenumber shells in the dissipative regime as well. This is essentially because the correlations induced by forcing at small wavenumbers is lost deep inside the inertial and dissipative wavenumbers. As a result, the energy transfers is the function of neighbouring Fourier modes, and hence they are local. In the next subsection, we will discuss and compare the numerical results of the laminar flows with the model predictions of Subsection II C. B. Laminar Flow We performed the direct numerical simulation of laminar flows on 64 3 grid for four sets of parameters, which are detailed in Table II. We choose random initial condition for all our simulations. To obtain steady state data, we employ random forcing in the wavenumber band 2 k 4 with the energy supply rate of unity. The Reynolds number of these simulations varies from 17.6 to 49. For the steady-state velocity field of our simulations, the energy dissipation rate ranges from to 1.002, and it matches with the energy supply rate, which is unity, within %. TABLE II. Parameters of our direct numerical simulations (DNS) for laminar flows: kinematic viscosity ν; Reynolds number Re; Kolmogorov s wavenumber ; and k max η. For all our runs the grid resolution is 64 3, the energy supply rate is unity, and the energy dissipation rate ɛ 1. ν Re kd k max η We attempt to verify whether Pope s and/or Pao s models describe the energy spectrum and flux of laminar flows. Towards this goal, for the laminar flow with Re = 49, we plot the normalized energy spectrum Ẽ( k)/k Ko (with K Ko = 1.5) and the normalized energy flux Π( k) in Fig. 3(a,b). In the figure, we also plot the model predictions of Pao and 14

15 Ẽ( k)/kko (a) DNS Pao Pope k Π( k) (b) DNS Pao Pope k FIG. 3. For the laminar flow simulation for Re = 49, plots of (a) the normalized energy spectrum Ẽ( k)/k Ko and (b) the normalized energy flux Π( k). See Eqs. (14) and Eq. (15) for definitions. In the figure, we also plot the model predictions of Pao (thin line) and Pope (dashed line). The predictions do not match with the numerical plots. Pope models. The models predictions differ significantly from the numerical results, though Pope s predictions for E(k) are somewhat close to the numerical data. Thus Pao s and Pope s models do not describe E(k) and Π(k) of the laminar flows. We will show below that the model discussed in Sec. II C describes the numerical results quite well. Ē( k) ν = 0.12 ν = 0.16 ν = 0.20 ν = k FIG. 4. For the laminar flow simulations of Table II, plots of the normalized energy spectra of Eq. (33). The data merges into a single curve for k > k f, where k f is the forcing wavenumber. In Fig. 4, we plot Ē( k) = Re 3 E(k)k/u 2 rms (see Eq. (33)) computed using the numerical 15

16 Π( k) ν = 0.12 ν = 0.16 ν = 0.20 ν = 0.24 Model k FIG. 5. For the laminar flow simulations of Table II, plots of the normalized energy fluxes of Eq. (34). The data merges into a single curve for k > k f, where k f is the forcing wavenumber. data for Re = 49, 32.4, 23.1 and We find that for k > k f, Ē( k) merges into a single curve indicates that Ē( k) is a universal function in this range. Also, Ē( k) exp( k)/k verifying the model predictions (see Sec. II C). Note that the Ē( k) for low oes not merge into a single curve. Motivated by Eq. (36), in Fig. 5, we plot the normalized energy flux Π( k) for the four set of simulations. We observe the exponential function exp( k) fits with the numerical data for k > k f, consistent with the model predictions. The aforementioned consistency between the numerical results and the model predictions of Sec. II C yield strong credence to the model for the laminar flows. We compute the shell-to-shell energy transfers using the numerical data corresponding to the Reynolds number of 49 and We divide the Fourier space into 32 shells, whose centers are at the origin k = (0, 0, 0). The inner and outer radii of the shells are k n 1 and k n respectively, where k n = {0, 2, 4, 8, 8 2 s(n 3),..., 32}, with s = 1/27. The forcing wavenumber band 2 k 4 is inside the 2nd shell. In Fig. 6(a,b), we exhibit the shell-toshell energy transfers for Re = 49 and 17.6 respectively. We observe that the most dominant energy transfers are from the 2nd shell that contains forcing wavenumber band to shells 3 to 10 for Re = 49, and to the shells 3 to 7 for Re = Thus, the energy transfers for laminar flows are nonlocal. This is because the velocity field in laminar flows appears to be 16

17 correlated with the forcing function. This issue needs further investigation. 15 (a) (b) m 10 0 m n n 2 FIG. 6. The shell-to-shell energy transfer rate for the laminar simulation with (a) Re = 49, (b) Re = Here m denotes the giver shell, while n denotes the receiver shell. The forcing wavenumbers (belonging to the 2nd shell) gives significant energy to the shells 3 to 10 for Re = 49, and to the shells 3 to 7 for Re = Thus the energy transfer in the laminar regime is forward and nonlocal. Contrast it with local energy transfers in the turbulent regime, shown in Fig. 2 IV. CONCLUSIONS Turbulence is a complex problem, hence its models are very useful. Pao 9 and Pope 8 constructed turbulence models that capture inertial and dissipative ranges of turbulence. Several experimental results on the energy spectrum have been compared with the model predictions, and they match with each other quite well. To best of our knowledge, there were no numerical verification of Pao s and Pope s models 8,9 till today; the present paper shows that the predictions of the above models and numerical results are consistent with each other, except minor discrepancies in the values of the energy flux in the dissipation regime. The aforementioned models for turbulent flows, however, have certain deficiencies. The hump in the energy spectrum near the beginning of dissipation range is related to the bottleneck effect 44,50,54 ; this hump is not captured by of Pao s and Pope s models. Also, 17

18 the numerical values of the energy flux in the dissipative regime differ from the model predictions by a small amount. Thus, the models of Pao 9 and Pope 8 need some revisions. It is also interesting to note that Pao s model 9 does not involve any free parameter (except Kolmogorov s constant K Ko ) in comparison to several free parameters in Pope s model. The parameters of Pope s model are chosen so as to fit with E(k) derived from experiments. In this paper, we also show that for turbulent flows, the shell-to-shell energy transfers are forward and local in both inertial and dissipative ranges. In this paper, we also present a new model for the energy spectrum and flux of laminar flows (Re 1). According to our model, the energy spectrum and flux exhibit exponential behaviour (exp( k)). We verify the model predictions using numerical simulations. For the laminar flows, we also show that the energy transfers are nonlocal and forward; the forcing wavenumbers supply energy to different shells. For moderate (Re 25) to large Reynolds numbers, Martínez et al. 10 argued that the energy spectrum is of the form Eq. (27), where the parameters α and β depend on the Reynolds numbers and length scales. Our model for the laminar flow is more detailed than that by Martínez et al. 10. It is important to differentiate the behaviour of laminar flows with those of highly viscous flows for which Re 0. For the highly viscous flows, the nonlinear term vanishes, and the velocity field is computed using ν 2 u = f, or u(k) = f(k)/(νk 2 ) in Fourier space. The energy flux for such flow vanishes due to the absence of the nonlinear term. Injection of weak nonlinearity in a highly viscous flow will induce a minute flux that can be computed perturbatively. In summary, in this paper, we verify the predictions of Pao s and Pope s models 8,9 for turbulent flows. We also show that the energy spectrum and flux of laminar flows are of the form exp( k). Appendix A: Energy flux based model of passive-scalar turbulence The models discussed in Secs. II A and II C can be easily generalised to passive-scalar turbulence. Dynamics of a passive scalar is described by the incompressible Navier Stokes 18

19 equation and by an advection equation for the passive scalar ζ: u t + (u )u = 1 ρ p + ν 2 u + f, ζ t + (u )ζ = κ 2 ζ + f ζ, u = 0, (A1) (A2) (A3) where κ is the thermal (or molecular) diffusivity, and f ζ is the source term for the scalar ζ. The two important nondimensional variables used for describing the passive scalar turbulence are the Schmidt number Sc = ν/κ and the Péclet number Pe = u rms L/κ. Note that Sc = Pe Re. (A4) The phenomenology of passive scalar depends crucially on the relative strength of the nonlinear term (u )ζ with relative to the diffusion term, which is quantified using Péclet number. First, we discuss the phenomenology for Pe 1. Using scaling arguments, it can be shown that the scalar flux Π ζ (k) is proportional to the scalar spectrum E ζ (k) 6,7,55. Further, the dimensional analysis yields the following formula for the scalar spectrum 56,57 : E ζ (k) = K OC ɛ ζ ɛ 1/3 k 5/3, (A5) where ɛ ζ is the diffusion rate of the scalar, and K OC is the Obukhov Corrsin constant 7. Since ζ does not affect the velocity field, the kinetic energy spectrum and flux remain the same as Eqs. (20), (21). However, for the scalar equation, the equation of the scalar flux is d dk Π ζ(k) = 2κk 2 E ζ (k). (A6) Now we extend Pao s arguments 9 to passive scalar turbulence which implies that in the inertial and dissipation range, E ζ (k)/π ζ (k) is independent of ν and κ. Under this assumption and using Eq. (A5), we obtain E ζ (k) Π ζ (k) = ɛ 1/3 u k 5/3, (A7) substitution of which in Eq. (A6) yields the following solution based on Pao s extended conjecture 9 : Π ζ (k/k c ) = Π ζ(k) = exp ( 32 ) ɛ K OC(k/k c ) 4/3, (A8) ζ E ζ (k) Ẽ ζ (k/k c ) = ɛ ζ ɛ 1/3 k = K 5/3 OC exp ( 32 ) K OC(k/k c ) 4/3, (A9) 19

20 where is the diffusion wavenumber. Therefore, k c = ( ɛu κ 3 ) 1/4 (A10) k ( c ν ) 3/4 = = Sc 3/4. (A11) κ Thus, the scalar spectrum and flux in the turbulent regime (Pe 1) are given by Eqs. (A9, A8) respectively. When the nonlinear term of the scalar equation is much smaller than the diffusion term, or when Pe 1, the scalar flux is quite small. Note however that Π ζ 0 as long as Pe is finite. Following arguments similar to those in Sec. II C, we deduce the scalar spectrum and flux for Pe 1 regime as: Ē ζ (k/ k c ) = exp( k/ k c ), Π ζ (k/ k c ) = (1 + k/ k c ) exp( k/ k c ), ɛ ζ = 2ζ 2 rmsκ k 2 cb, (A12) (A13) (A14) where Ē ζ (k/ k c ) = E ζ(k)kpe 3, (A15) ζ 2 rms Π ζ (k/ k c ) = Π ζ(k) ɛ ζ. (A16) Here B is a constant, and k c = Pe/L. In dimensional form, the scalar spectrum is E ζ (k) = ζrms 2 1 k Re exp( k/ k c ). Note that the kinetic energy spectrum and flux would be described by Eqs. (37, 38). (A17) Using these results, we can derive the spectra and fluxes for u and ζ for Sc 1, Sc 1, and Sc 1. For Sc 1, using the fact that Sc = Pe/Re, we deduce that Pe Re. Hence, in the turbulent regime, both Re, Pe 1. Hence, E(k) and E ζ (k) are given by Eq. (21) and Eq. (A9) respectively. In the laminar regime, the corresponding spectra would be described by Eq. (37) and Eq. (A12). Note that k c for this case. On the other hand for Sc 1, this case Re Pe and k c. The spectral properties are very similar to the case of Sc 1, except that k c. We can choose the appropriate formulae depending on the magnitudes of Re and Pe. 20

21 The situation is more complex for large Schmidt number. For Sc 1, Re Pe and k c. Hence the scalar spectrum extends much beyond, and the spectra in the wavenumber regime < k < k c is difficult to derive. Since u k 0 for k >, we can argue that the effective Pe u kd L/κ 1 in < k < k c. Hence we expect Eq. (A12) to describe the scalar turbulence here. Regarding the nonlinear transfers of ζ 2, in the range < k < k c, we expect u( ) to act as a mediator. Hence, ζ 2 transfers involve triads with k, p q that differs from typical local and forward transfers that involves k p q. Therefore, for Sc 1, E(k) is determined by Eq. (21) when Re 1, and by Eq. (37) when Re 1. Regarding the scalar spectrum, for k <, it is given either by Eq. (A9) or Eq. (A12) depending on Pe 1 or Pe 1, but for < k < k c, E ζ (k) is described by Eq. (A12). For Sc 1, Batchelor 58 had proposed that E ζ (k) = K Ba ɛ ζ (ν/ɛ) 1/2 k 1 exp( K Ba κk 2 (ν/ɛ u ) 1/2 ), (A18) but Kraichnan 18 argued that E ζ (k) = K Ba ɛ ζ (ν/ɛ) 1/2 (1 + 6K Ba )k 1 exp( 6K Ba (k/k c )). (A19) Here K Ba is the Batchelor constant 7. These derivations of Batchelor 58 and Kraichnan 18 are based on closure schemes. We refer the reader to Gotoh and Yeung 16 for detailed discussion. Numerical results of Gotoh et al. 59 and Yeung et al. 60 tend to be closer to the predictions of Kraichnan 18 and Eq. (A17), thus Kraichnan s model appears to provide a better description to the passive-scalar turbulence for Sc 1. Interestingly, our formula for E ζ (k), Eq. (A17), is same as that of Kraichnan 18, except the prefactor. Thus, our models based on energy flux provide good alternatives to those derived earlier using turbulence phenomenologies or using closure schemes. It will be useful to verify the aforementioned models of passive-scalar turbulence using numerical simulations. This exercise, however, is computationally intensive, and it is relegated to a future communication. We remark that there are a number of works on passive-scalar turbulence. See Gotoh and Yeung 16 for detailed references. 21

22 Appendix B: Quantification of bottleneck effect using Pao s model In this appendix, we show how Pao s model 9 can be used to estimate the magnitude of the bottleneck effect 44,50,54. In Fig. 1(a,b), we reproduce the Ẽ( k) plot of Fig. 1(c) for the grid simulation. Here the thick red curve and the thin green curves represent the numerical result and Pao s prediction respectively. The bottleneck effect is visible at the knee from where the viscous dissipation starts to dominate. The difference is better represented in the log-linear plot of Fig. 7(b). In Fig. 7(c) we plot the difference δẽ( k) between the numerical result and Pao s prediction, and it appears to be a good measure of the bottleneck effect. Thus, the phenomenological model of Pao could be useful for such quantification, and it may help us understand the physics of bottleneck effect 44,50,54, and associated intermittency and backscatter. Ẽ( k) (a) Ẽ( k) 4 2 (b) δẽ( k) (c) kmax k kmax k k FIG. 7. For the grid resolution : (a) plot of Ẽ( k) vs. k and Pao model on log-log scale; (b) plot of Ẽ( k) vs. k and Pao model on log-linear scale; (c) plot of the difference between Ẽ( k) and Pao model spectrum δẽ( k) with k. Note that Ẽ( k) peaks near the transition from the inertial range to the dissipation range. ACKNOWLEDGEMENTS We thank Mohammad Anas for a valuable feedback on. Our numerical simulations were performed on Cray XC40 Shaheen II at KAUST supercomputing laboratory, Saudi Arabia and Chaos cluster of IIT Kanpur. This work was supported by the research grants PLANEX/PHY/ from Indian Space Research Organisation, India, and by the Department of Science and Technology, India (INT/RUS/RSF/P-03) and Russian Science Foundation Russia (RSF ) for the Indo-Russian project. 22

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