i. Center for Turbulence Research _, , -- 113 Annual Research Briefs 199_ 7,'-'_ /_" N94-12292 Similarity states of homogeneous stably-stratified turbulence at infinite Froude number By Jeffrey R. Chasnov 1. Motivation and objectives Turbulent flow in stably-stratified fluids is commonly encountered in geophysical settings, and an improved understanding of these flows may result in better ocean and environmental turbulence models. Much of the fundamental physics of stably- stratified turbulence can be studied under the assumption of statistical homogeneity, leading to a considerable simplification of the problem. A study of homogeneous stably-stratified turbulence may also be useful as a vehicle for the more general study of turbulence in the presence of additional sources and sinks of energy. Our main purpose here is to report on recent progress in an ongoing study of asymptotically long-time similarity states of stably-stratified homogeneous turbu- lence which may develop at high Reynolds numbers. A similarity state is char- acterized by the predictability of future flow statistics from current values by a simple rescaling of the statistics. The rescaling is typically based on a dimensional invariant of the flow. Knowledge of the existence of an asymptotic similarity state allows a prediction of the ultimate statistical evolution of a turbulent flow with- out detailed knowledge of the very complicated and not well-understood non-linear transfer processes. We present in this report evidence of similarity states which may develop in homo- geneous stably-stratified flows if a dimensionless group in addition to the Reynolds number, the so-called Froude number, is sufficiently large. Here, we define the Froude number as the ratio of the internal wave time-scale to the turbulence time- scale; its precise definition will be given below. In this report, we will examine three different similarity states which may develop depending on the initial conditions of the velocity and density fields. Theoretical arguments and results of large-eddy simulations will be presented. We will conclude this report with some speculative thoughts on similarity states which may develop in stably-stratified turbulence at arbitrary Froude number as well as our future research plans in this area. 2. The governing equations Choosing our co-ordinate system such that the z-axis is pointed vertically up- wards, we assume a stable density distribution p = p0 - flz + p', where p0 is a constant, uniform reference density, fl > 0 is a constant, uniform density gradient along z, and p' is the density deviation from the horizontal aver- age. The kinematic viscosity u and molecular diffusivity D of the fluid are assumed II"L PRECEDING PAGE BLANX NOT FILMED 114 J. R. Cha_nov constant and uniform. After application of the Boussinesq approximation, the gov- erning equations for the fluid velocity u and the density fluctuation p' are V.u=O, (2.1) Ou p'g V(p + pogz) + vV2u ' (2.2) _-+u. Vu= Po Po Op' q- u. Vp _= flu3 + DV2p _, (2.3) where g = -jg with g > 0, j is the vertical (upwards) unit vector, and p is the fluid pressure. We will consider three limiting flows which may occur in a stably-stratified fluid. Firstly, we will consider decaying isotropic turbulence with an isotropic passive scalar, whose governing equations are obtained from (2.1) - (2.3) when g, fl = 0. Secondly, we will consider decaying isotropic turbulence in a mean passive scalar gra- dient, obtained when g = 0 only, and; thirdly, we will consider buoyancy-generated turbulence (Batchelor, Canuto & Chasnov, 1992), obtained when fl = 0 only. The conditions under which these limiting flows may develop in a stably-stratified fluid where both g and fl are nonzero are most easily determined after a transforma- tion of the equations to dimensionless variables. First, to make the equations more symmetric in the velocity and density fields, we define following Cambon (private communication) a normalized density fluctuation 8 such that it has units of velocity, (2.4) Vp0Z " Use of 8 instead of p' in (2.2) - (2.3) modifies the terms proportional to g and fl into terms proportional to N, where (2.5) is the Brunt-Vaisala frequency associated with the internal waves of the stably stratified flow. Furthermore, ½(u 2) is the kinetic energy and 1(8)2 is the potential energy of the fluid per unit mass, and the equations of motion conserve the total energy (kinetic + potential) in the absence of viscous and diffusive dissipation. Now, defining dimensionless variables as uo x u (p + pogZ) T=t-_- ° X=-- U=--, P= O=_0, (2.5) ' lO' UO PoUo 2 ' where/0, u0, and _0, are as yet unspecified length, velocity, and normalized density scales, the equations of motion become v. u = 0, (2.6) Homogeneous turbulence at infinite Froude number 115 00U---T+ U. VU = -J_.0010o_00,_-, - VP + V2U, (2.7) O0 +u. vo = 1 0Uoo.U + _Ro v o, (2.8) where U0 U0I 0 V FO=Nl----_, Ro- v , a=_. (2.9) F0 and R0 can be regarded as an initial Froude number and Reynolds number of the flow, respectively, although their precise definition is yet dependent on our specification of 10,u0, and 00; a is the Schmidt (or Prandtl) number of the fluid. 2.1 Isotropic turbulence with an isotropic passive scalar This limiting flow may be obtained by initializing the flow with an isotropic veloc- ity and density field with given kinetic and potential energy spectrum of comparable integral scales. The unspecified dimensional parameter l0 may be taken equal to the initial integral scale of the flow, and u0 and 00 may be taken equal to the initial root-mean-square values of the velocity and normalized density fluctuations. The non-dimensional variables of (2.5) ensure that the maximum values of U and O and the non-dimensional integral length scale of the flow is of order unity at the initial instant, and, provided that u0 is of order 00, implying comparable amounts of kinetic and potential energy in the initial flow field, and F0 >> 1, both of the terms multiplied by lifo in (2.7) and (2.8) are small initially. Over times in which these terms remain small, the resulting equations govern the evolution of a decaying isotropic turbulence convecting a decaying isotropic passive scalar field. 2.2 Isotropic turbulence in a passive scalar gradient Here, the flow is initialized with an isotropic velocity field with given kinetic en- ergy spectrum and no initial density fluctuations. Again we take the dimensional parameter 10to be the initial integral scale of the flow and u0 equal to the initial root-mean-square value of the velocity field. The maximum value of U and the non-dimensional integral length scale of the flow are then of order unity. However, the initial conditions introduce no intrinsic density scale, and such a scale needs to be constructed from other dimensional parameters in the problem. If at some time in the flow-evolution not too far from the initial instant the maximum of the dimen- sionless density fluctuation O is also to be of order unity, then the dimensionless group multiplying Ua in equation (2.8) must necessarily be of order unity. Setting this group exactly equal to unity yields an equation for 00 with solution Oo= Nlo. Thus defining 00, we find that the dimensionless group multiplying 19in equation (2.7) is equal to 1/F_ so that, in the limit of F0 >> 1, this term is small at the initial instant and may be neglected for some as yet to be determined period of time. The resulting equations then govern the evolution of decaying isotropic turbulence in the presence of a mean passive scalar gradient over this period of time. 116 J. R. Cha_nov g.$ Buoyancy-generated turbulence Here, the flow is initialized with an isotropic density field with given potential energy spectrum and no initial velocity fluctuations. Similarly as above, we take the dimensional parameter l0 to be the initial integral scale of the density field and 00 to be equal to the initial root-mean-square value of the 0-field. The maximum value of 0 and the dimensionless integral scale of the flow is then of order unity. However, here the initial conditions introduce no intrinsic velocity scale. If at a time in the flow-evolution not too far from the initial instant we wish the maximum of the dimensionless velocity fluctuation U to also be of order unity, then the dimensionless group multiplying O in equation (2.7) must necessarily be of order unity. Setting this group exactly equal to unity yields a simple quadratic equation for u0, with solution u0 = v/-NT_000,or equivalently, u0 = _/gloPo/!Po, where p_ is the value of (p,2)1/2 at the initial instant. We note that this is the same velocity scale chosen previously by Batchelor et al. (1992) in their study of buoyancy-generated turbulence. Upon use of the identity Oo= ug/Nlo, we find that the dimensionless group multiplying U3 in (2.8) is exactly equal to 1/F 2 so that, in the limit of F0 >> 1, this term is small at the initial instant. Using the definition of u0, the initial Froude number here is seen to be equal to F0 = p_o/fllo. For times over which the term multiplied by Fo 2 may be neglected, the resulting equations then govern the evolution of buoyancy-generated turbulence. 3. Asymptotic similarity states 3.1. Final period of decay Exact analytical treatment of (2.1) - (2.3) is rendered difficult because of the quadratic terms. Under conditions of a final period of decay (Batchelor, 1948), these terms may be neglected and an exact analytical solution of (2.1)- (2.3) may be determined. Although most of the results concerning the final period are well-known or easily found, we recall them here since the ideas which arise in a consideration of the final period are relevant to our high Reynolds number analysis. During tim final period, viscous and diffusive effects dissipate the high wavenum- her components of the energy and scalar-variance spectra, and, at late times, the only relevant part of the spectra are their forms at small wavenumbers at an earlier time. Defining the kinetic energy spectrum E(k, t) and the density-variance spec- trum G(k, t) to be the spherically-integrated three dimensional Fourier transform of the co-variances ½(ui(x, t)ui(x + r, t)) and (p'(x, t)p_(x + r, t)), an expansion of the spectra near k = 0 can be written as E(k,t) = 27rk_(B0 + B2k 2+...) (3.1) e(k, t) = 4 2(c0 + 2+...), (3.2) where B0, B2,..., and Co, C2,... are the Taylor series coefficients of the expansion. In a consideration of isotropic turbulence, Batchelor and Proudman (1956) assumed the spectral tensor of the velocity correlation (ui(x)uj(x + r)) to be analytic at Homogeneous turbulence at infinite Froude number 117 k = 0 and determined that B0 = 0 and that non-linear interactions (which are important during the initial period) necessarily result in a time-dependent non- zero value of B2. Saffman (1967a) later showed that it is physically possible for turbulence to be initially created with a non-zero value of B0 and that, for decaying isotropic turbulence, B0 isinvariant in time throughout the evolution of the flow. By analogous arguments, it can be shown that the spectrum of the density correlation is itself analytic at k = 0 when Co # 0, and, for an isotropic decaying density (scalar) field, Co is invariant in time (Corrsin, 1951). Here, rather than present an exact derivation of the final period results, we will demonstrate how a simple dimensional analysis can recover the correct decay laws. We consider separately the three different limiting flows envisioned above. Isotropic turbulence with an isotropic passive scalar The evolution of the mean-square veIocity may be found by dimensional analysis assuming the only relevant dimensional quantities are the low wavenumber invari- ant of the energy spectrum B0, if non-zero initially, viscosity v, and time t. The equations of motion are assumed to be linear in the velocity field during the final period so that (u2> must linearly depend on B0, and we find {u2) o¢Bov-]t-_, (3.3) as determined by Saffman (1967a). If B0 is initially zero, then B2 is necessarily non-zero and is also invariant during the final period when nonlinear interactions are negligible. A corresponding dimensional analysis based on B2 instead of B0 yields 5 5 (u2>o(B2v-_t-_, (3.4) as originallydeterminedby Batchelor(1948). Analogous arguments appliedto theisotropicpassivedensity(scalar)field,which isseentobe uncoupled from the velocityfieldduringthefinalperiod,impliesadependenceon Co,necessarilylinear, the diffusivity D, and time t, yielding (p,2) _ CoD-]t-], (3.5) as originally determined by Corrsin (1951), and, if Co is initially zero, (/2) (x C2D-}t-}. (3.6) Isotropic turbulence with passive scalar gradient The passive density (scalar) field for this flow is driven by velocity fluctuations, and the low wavenumber coefficient of the density-variance spectrum is no longer invariant in time. In fact, an exact relation, valid even when nonlinear terms are non-negligible, holds between Co and B0 and is Co(t) = l a_D ,2 _. _o_ , (3.7) 118 Y. R. Chasnov indicating that there is now only one invariant, namely B0, which is relevant to our dimensional analysis. Making use of this invariant, we find in the final period the decay law (p,2)cx(cid:0)32Boy-It½, (3.8) which is most simply found by substitution of (3.7) directly into (3.5). If B0 is zero, then B2 is invariant during the final period, and C2 is related to B2 (exact only when nonlinear terms are negligible) by Cz(t) = 1oZn ,2 5_, --z_• (3.9) The decay law in the final period is then B t (pa)cxt32Bzu-_t-_. (3.10) We have thus found the interesting result that the density-variance may either increase or decrease during the final period depending on the form of the low- wavenumber energy spectrum. This result may be of use to researchers intercsted in determining the form of the low wavenumber energy spectrum in homogeneous turbulence under experimental conditions. Buoyancy-driven flow Here, it is the velocity field which is driven by density fluctuations, and the low wavenumber coefficient B0 of the kinetic energy spectrum is no longer an invariant. The relevant invariant here is the low wavenumber coefficient Co of the density- variance spectrum. As before, an exact relation holds between B0 and Co, valid even when nonlinear terms are non-negligible, and is Bo(t) - 2 gZCo t2" (3.11) 3 p_ Making use of the invariant Co, we find another form for the mean-square velocity fluctuation under an assumption of a final period in which non-linear terms are negligible g2Co. -_,½ (u2) c<_, _ . (3.12) P0 The mean-square density fluctuations decay as for the isotropic passive scalar flow. Clearly, an increase in (u2l during a "final period" contradicts the very existence of a final period since the Reynolds number of the flow is increasing in time. If Co is initially zero, then C2 is necessarily non-zero and is invariant when nonlinear terms are negligible. B2 is now related to C2 by B2(t) = 2 gZC__t_2__£_ (3.13) 3 p_ ' Homogeneous turbulence at infinite Froude number 119 and the mean-square velocity follows g2C2 I/-] t-_ (3.14) <u p-T Although the mean-square velocity decays in this case, the integral scale grows like t1/_ so that the Reynolds number increases in time, again contradicting the existence of a final period. 3._. Exact high Reynolds number similarity states At high Reynolds numbers, direct effects of viscosity and diffusivity occur at much larger wavenumber magnitudes than those scales which contain most of the energy and density-variance so that the asymptotic forms of (u2) and (/2) can be expected to be independent of u and D. Viscous and diffusive smoothing of the energy and density-variance containing components of the spectra are now replaced by nonlinear transfer processes so that one can still reasonably expect the asymptotic scaling of (u2) and (p,2) to be on the form of the spectra at low wavenumbers. The low wavenumber coefficient B0 is an invariant, even at high Reynolds numbers, for decaying isotropic turbulence and so is Co for a decaying isotropic passive scalar. If there is a mean passive scalar gradient, then Co is asymptotically related to B0 by (3.7). For buoyancy-driven flows, Co is an invariant and B0 is asymptotically related to Co by (3.11). Based on dimensional analysis, we can now determine the high Reynolds number, long-time evolution of the energy and density-variance when B0 and Co are non-zero for our three limiting flows. Isotropic turbulence with an isotropic passive scalar The low wavenumber coefficients B0 and Co are separately invariant and the high-Reynolds number asymptotic results are the Saffman (1967b) decay law (u2)_ B_ot-_, (3.15) and its analogous law for the passive density-variance <p,z)o<CoBo_t-_. (3.16) The nonlinearity of the governing equations is reflected by the nonlinear dependence of (u 2} and (/2) on B0, in contrast with the results of the final period. Note, however, that the linearity of the density equation in p' results in a linear dependence of </2) on C0. Dimensional arguments can also determine the asymptotic behavior of the veloc- ity and density integral scales, and one finds l 2 L,,,Lo (x B_ t'. (3.17) 120 J. R. Ghasnov lsotropic turbulence with passive scalar gradient Here Co is no longer invariant, but depends on B0 asymptotically as (3.7) so that the density-variance now evolves as (p,2) 4 (3.18) when B0 is non-zero. Buoyancy-driven flow Here Co is invariant and B0 isnot, and B0 depends on Co asymptotically as (3.11). By dimensional arguments (Batchelor el al., 1992), the mean-square velocity and density-variance evolve as {u2} cx(g2Co/P2o)_t-i , (3.19) 0,(g2Colp2o)2,t _t2 (3.20) and the integral scales evolve as L,, Lp, oc(g2Co/p2o)}t_. (3.21) A computation of a Reynolds number based on the root-mean square velocity fluc- tuation and the integral scale shows that Re oct _/s increases asymptotically, pre- cluding the development of a final period in this flow. 3.3. Approzimatc high Reynolds number similarity states When either B0 or Co are zero, there are no longer strictly invariant quantities on which to base asymptotic similarity states. The coefficients B2 and C2 are affected by nonlinear transfer processes, and exact results as found above become unobtainable. Nevertheless, if we make an additional assumption, which needs to be verified by numerical or experimental data, that the tlme-variation of B2 or C2 due to nonlinear processes are small compared to the rate of change of the energy or density-variance, then approximate asymptotic similarity states may still be based on the nearly-invariant low wavenumber coefficients. The analysis proceeds in exact analogy to that above, and, for use in comparison to the numerical simulation data, we state the results below. Isotropic turbulence with an isotropic passive scalar For B0 = 0, we have the Kolmogorov (1941) decay law 2 10 (us} cxBit -r. (3.22) Three additional approximate similarity states exist for the decaying isotropic pas- sive scalar depending on which of B0 or Co are zero (Lesieur, 1990): (p,2} cxC2Bolt -_ (3.23) Homogeneous turbulence at infinite Froude number 121 (p,2) o¢CoBalt -} (3.24) (p,2) o¢C2B_} t-a_, (3.25) Isotropic turbulence with passive scalar gradient For B0 = 0, we assume that B2 is approximately invariant. The low wavenumber scalar-variance spectrum coefficient C2 is approximately related to B2 by (3.9). The density-variance is found to evolve as o,Z'B oO. (3.26) Buoyancy-driven flow For Co = 0, we assume that C2 is approximately invariant. The low wavenumber energy spectrum coefficient B2 is approximately related to C2 by (3.13). The mean- square velocity and density-variance evolve approximately as (u2)_ (g2Co/P2o)4t-_ , (3.27) g2(p'2/p2o) (x (g2Co/p2o)_t-_ , (3.28) and the integral scales evolve as L_, Lp, o( (g2Co/P_o)_ta'. (3.29) A computation of a Reynolds number based on the root-mean square velocity fluc- tuation and the integral scale shows that Re again increases asymptotically, but now as t1/7. 4. Large-eddy simulations The high Reynolds numbers required to test the asymptotic scaling determined above may be obtained by a large-eddy simulation (LES) of Eqs. (2.1)- (2.3) using a pseudo-spectral code for homogeneous turbulence (Rogallo, 1981). For the subgrid scale model, we employ a spectral eddy-viscosity and eddy-diffusivity (Kraichnan 1976; Chollet and Lesieur 1981) parametrized by [ ,e(klkn,,t)= O.145+5.01exp - [ km J ' (4.1) and D_(klkm,t) = u,(k]km,t), (4.2) O"e where km is the maximum wavenumber magnitude of the simulation and at is an eddy Schmidt number, assumed here to be constant and equal to 0.6. We take the initial energy spectrum to be E(k,O) = A.k; a(k/kp)"exp (-(n/2)(k/kp)2) , (4.3) 122 J. R. Chasnov where n is equal to 2 or 4, An is chosen so that (u 2) = 1, and kp is the wavenumber at which the initial energy spectrum is maximum. The case n = 2 corresponds to B0 ¢ 0, and the case n = 4 corresponds to B0 = 0. In the 2563 numerical simula- tions presented here, the minimum computational wavenumber is 1, the maximum wavenumber is about 120, and we take kp = 100. The initial energy spectrum is set to zero for wavenumbers greater than 118 to allow the subgrid scale eddy-viscosity and eddy-diffusivity to build up from zero values. The relatively large value of kp chosen here allows an attainment of an asymptotic similarity state before the inte- gral scales of the flow become comparable to the periodicity length. A velocity field with initial energy spectrum given by (4.3) is realized in the simulation by requir- ing the spectral energy content at each wavenumber to satisfy (4.3) but randomly generating the phase and velocity component distributions (Rogallo 1981). In the simulations of decaying isotropic turbulence with a decaying isotropic pas- sive scalar, the passive scalar-variance spectrum is also initialized with the spectrum given by (4.3) with An chosen so that (p'2/ = 1. We present the results of two simu- lations for this flow: the first with an initial energy spectrum with n = 2 convecting two passive scalar fields with initial spectra with n = 2 and n = 4, and the second with an energy spectrum with n = 4 convecting two passive scalar fields with n = 2 and n = 4. Computations of these two velocity fields and four scalar fields are sufficient to test the theoretical scaling discussed in §3. In the simulations of decaying isotropic turbulence in a passive scalar gradient, the initial fluctuating passive density field is taken identically equal to zero, and two simulations are presented with an initial energy spectrum with n = 2 and n = 4. The exact value of/3 is inconsequential provided it is non-zero, and we choose/3 = 1. In the simulations of buoyancy-generated turbulence, the initial fluctuating ve- locity field is taken identically equal to zero, and two simulations are presented with an initial density spectrum with n = 2 and n = 4. Here, the exact value of g is inconsequential provided it is non-zero, and we choose units such that g/po = 1. 4.1. Results In the interest of brevity, we present here only results from the large-eddy simula- tions pertaining to the power-law predictions of §3. More detailed results concerning decaying isotropic turbulence with and without a passive scalar gradient will be pub- lished in Chasnov (1993) and Chasnov & Lesieur (1993) - slightly lower-resolution simulations (1283 ) of buoyancy-generated turbulence have already been published in Batehelor, Canuto & Chasnov (1992). In figures 1 and 2, we plot the instan- taneous power-law exponents (logarithmic derivatives) versus time normalized by the initial large-eddy turnover time T(0), of the mean-square velocity decay and the passive density-variance decay when/3, g = 0 in (2.2) and (2.3), appropriate for the study of decaying isotropic turbulence with a decaying isotropic passive scalar. In figure 3, we plot the time-evolution of the power-law exponent of the passive density-variance when /3 = 1 and g = 0, appropriate for the study of decaying isotropic turbulence in the presence of a passive scalar gradient, and, in figures 4 and 5 we plot the time-evolution of the power-law exponent of the mean-square velocity and density-variance when/3 = 0 and g/Po = 1, appropriate for the study