NASA Contractor Report 198260 p . f" ICASE Report No. 96-2 ICA INERTIAL RANGE DYNAMICS IN BOUSSINESQ TURBULENCE Robert Rubinstein NASA Contract No. NAS1-19480 January 1996 Institute for Computer Applications in Science and Engineering NASA Langley Research Center Hampton, VA 23681-0001 Operated by Universities Space Research Association National Aeronautics and Space Administration Langley Research Center Hampton, Virginia 23681-0001 INERTIAL RANGE DYNAMICS IN BOUSSINESQ TURBULENCE Robert Rubinstein* Institute for Computer Applications in Science and Engineering NASA Langley Research Center Hampton, VA 23681 ABSTRACT L'vov and Falkovich (Physica D 57) have shown that the dimensionally possible inertial range scaling laws for Boussinesq turbulence, Kolmogorov and Bolgiano scaling, describe steady states with, respectively, constant flux of kinetic energy and of entropy. Following Woodruff (Phys. Fluids 6), these scaling laws are treated as similarity solutions of the direct interaction approximation for Boussinesq turbulence. The Kolmogorov scaling solu- tion corresponds to a weak perturbation by gravity of a state in which the temperature is a passive scalar but in which a source of temperature fluctuations exists. Using standard inertial range balances, the effective viscosity and conductivity, turbulent Prandtl number, and spectral scaling law constants are computed for Bolgiano scaling. * This research was supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-19480 while the author was in residence at the Institute of Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23681-0001. I. Introduction The spectra E'_(k), Eh(k), E°(k) of buoyant turbulence are defined so that 1 /0 2 < uiui > E=(k)dk /o < 02 > = E°(k)dk /o < u30 > = Eh(k)dk where gravity acts in the 3 direction. The variables ui and 0 denote fluctuations about any mean velocity and temperature field. Dimensional analysis suggests two types of inertial range scaling for these spectra. If the gravitational coupling will be neglected, the temperature is a passive scalar and the velocity has the Kolmogorov spectrum E _, e213k -5/3 (1) where e is the kinetic energy dissipation rate. Kolmogorov scaling can be expected to apply to wavenumbers k at which nonlinearity dominates the thermal forcing, so that Re(k) >> Ra(k) for appropriately defined scale dependent Reynolds and Rayleigh num- bers. Although the limiting temperature distribution is uniform and the spectrum E ° consequently vanishes in the absence of a source of temperature fluctuations, buoyant flows generally have such a source; in this case, there is a nonzero constant temperature variance dissipation rate N and dimensional analysis gives E o _.. Nc-1/3k-5/3 (2) In Kolmogorov scaling, the inertial range is isotropic, consequently the heat transfer spec- trum E h vanishes. Bolgiano identified a second possibility I in which velocity and temperature fluctu- ations are determined by N and g, where g denotes the product of thermal expansion coef_cient and acceleration of gravity. Then dimensional analysis leads to E u ,._ g4/SN2/Sk-ll/5 E h _ g1/5N3/5k-9/5 (3) E o ,._ g-215N415k-7/5 These scaling laws remained largely a dimensional possibility until L'vov and Falkovich 2 clarified their possible dynamic significance: whereas Kolmogorov scaling corresponds to a steady state inertial range with constant energy flux _, Bolgiano scaling corresponds to a steady state inertial range with constant entropy flux N; the identification of temperature variance dissipation with entropy flux is due to L'vov. 3 This scaling applies when thermal forcing dominates the nonlinearity so that Ra(k) >> Re(k). A nonzero heat transfer spec- trum is possible for Bolgiano scaling, in which the force of gravity introduces a preferred direction. This physical picture might suggest that Kolmogorov scaling will be observed in forced convection and Bolgiano scaling will be observed in free convection; however, arguments 3 that measurements in very high Rayleigh number Rayleigh-Benard convection experiments are consistent with Bolgiano scaling cannot be considered conclusive. 4 Nev- ertheless, turbulence sustained even at small scales by buoyant forcing is an interesting theoretical possibility which deserves investigation. This paper considers Kolmogorov and Bolgiano scaling following Woodruff 5 as simi- larity solutions of the direct interaction approximation (DIA) for buoyant turbulencefi It is shown that Kolmogorov scaling can be treated as a perturbation of the passive scalar state in which the gravitational coupling g vanishes. For Bolgiano scaling, this viewpoint results in close connections with the e-expansion of Yakhot and Orszag, 7 which Woodruff inter- prets as arising from an asymptotic evaluation of the integrated DIA response equation. In Eulerian theories, the e-expansion also serves as a scheme of infrared regularization, which is required since any attempt to compute amplitudes in the spectral scaling laws using the DIA equations directly is defeated by the well-known infrared divergence, s The only fully satisfactory solution of this divergence is the formulation of a Lagrangian theory. 9'1° However, the construction of analytical solutions for the Eulerian theory requires some substantial approximations; the added complexity of a Lagrangian theory of any problem with coupled fluctuating fields may justify more or less ad hoc modifications of the Eulerian theory like the e-expansion. II. Direct interaction approximation for Boussinesq turbulence It will be convenient to write the Boussinesq equations in matrix form as Go'U(k ) = F(]c) f di5 d_ U(ib)U(_) (4) dk=p+q 2 In Eq. (4), U is the vector (5) V ____ Written explicitly, the nonlinear term on the right side of Eq. (4) is fk d#d4Um(#)Un((7) (6) rim.(k) =p+0 where 6 if i,m,n # 4 (7) if i=n=4ori=m=4 i { P+mn(k) Fimn 2 0 otherwise In Eqs. (4)-(7), ]_=(_,k) _b=(w,p) _=(f_-w,q) and Pimn(k) = kmPin(k) + knPim(k) Pij(k) = 6ij - kikjk -2 The matrix Go in Eq. (4) is defined by (s) Go(_:)-l = [ (-ift ;vok2)I (-ifi gP+a_ok2)] where the vector Pa has the components Pia and u0 and _0 are the molecular viscosity and conductivity respectively. Inverting the matrix in Eq. (8) gives the bare Green's matrix Go(k) = [G_(]co)I -G_(]c)GCg(o_:e)g(P)_3, ]j (9) where G_ and Goe are the bare propagators 6_(]¢)--(--i_ -1-P0]¢2-1) COO(=k)(--in + .Ok2)-1 (10) It should be noted that, unlike the passive scalar equation in which g = 0, the Boussinesq equations are effectively nonlinear in T. The direct interaction approximation (DIA) for Boussinesq turbulence has been de- rived by Kraichnanfi We will attempt to construct an approximate analytical solution to 3 the DIA equationsin the inertial range.Thedescriptorsofbuoyantturbulencein the DIA arethe correlation functions Q_j, Q_, QT defined by < u+(k)uj(k') > = Q'_j(k)_(k + k') < uiCk)T(k') > = QpCk)6(k + k') < T(k)T(k') > = QT(k)6(k + k') (11) and the response or Green's functions, G'_j(k) =< $uiCk)/6f_'ck) > G'_r(k) =< $u+(k)/6fT(k) > Gf"(k) =< _T(k)/_f_'(k) > G°(k) =< 6T(k)/6fT(k) > where 6f_ and 6f T denote small perturbations added to the velocity and temperature equations respectively. The analysis will be based on the Langevin equation representation of DIA which for an inviscid steady state takes the form -igtu+(k) + gT(k)Pi3(k) = -_'_(k)ur(k)- _'_T(k)T(k) + f_(k) -i_T(k) = -_T"(k)ur(k) - _r(k)T(k) + fT(k) (12) The damping factors _ are defined by dw {Pm,-s(p)G"(p)Q_s(_ ) + Pn,.s(q)G'_((I)Q_(D) rf_,.(k) = 1piton(k) [ dp dq ]j 4 Jk=p+q oo uT ^ h ^ uT ^ h ^ -P,'Gm (P)Qn(q)-q,-Gn (P)Qm(P)} (13) rl._T(k) = -1Piton(k) [ dp dq ]j doff {prGm lIT (p)^Q,.n(qu) ^ 4 Jk=p+q oo + q,.GnuT (q^)Q,-mu (P) ^ } (14) 1 dp dq dw e ^ h ^ r_T"(k) = _ =p+q _ {knp,-G (p)Qn(q) + knq,'G°((7)Qh([ _) + + - knG_T(_)QT(_t) -- knG_T(_)QT(D ) (15) -kmPn,'s(q)Gn Tu (P^)Qmsu(P)- ^ kmP.vs(P) GTnu (q^)Qmsu(q) ^ Tu ^ u ^ Tu ^ u ^ -- kmqrGs (P)Oms(P)- kmPrGs (q)Qms(q) } 1 dpdq dw { o ^ _, o ^ . ^ tiT(k) = _ =p+q _ knpmG (P)Qm.(q) + knqmG (q)Qmn(P) - k.p_G_T@)Q_(q) - knqrG_T(4)Q_(p)} (16) In Eq. (12), f/_ and fT are random forces with the correlations < f;tt (k)f;(-k) > = r+U__I (_^) < f_'(k)T(-k) > = Fi"r(k) < T(k)T(-k) > = FTT(_:) where FTT(i¢) = kmk_ d_ d_ Q_,.(p)QT(_) + Qm(p)Q,-(q) j[ h ^ h ^ =p+0 (17) FjT(k) = knPi,.s(k) _ dr) dg_Q)(P)Q_((t) =p+0 Fi_(k) = P_m_(k)Pj_(k) fk d[_d_ Q_,.(#)Q_(_) =p+0 The DIA theory is completed by the definitions Eq. (11) of the correlations in terms of the fields u, T and by the equations of motion for the G's, which are given by -ifla+5(k )+ gay"(_)P+_(k)+ ,_afu(k) + %,(k^ )a,jIt = 1 --iaG'_T(]c) + 9aT(k)Pi3(k) + rl'_r(k)ar(k) + rl_p(k)G_,r(t:) = 0 -iUGy-(k) + _Rk)a_"(k) + _T"(k)a_j(_) = o (18) -il2G°(_:) + rlr(_:)G°(k) + rl#U(k)G_T(k:) = 1 These equations are simply the conditions that the solution of the Langevin equations, Eq. (12) is given by U(]_) = G(t:) [ f"(f¢) ] (19) fr(]¢) j in the matrix notation of Eqs. (4)-(9). In Eq. (19), G is the renormalized response matrix 1 [a+j(k) a_w(k) G(k)= La_"(k) a+(k) J The notation means that GTu is a row and GuT is a column. The DIA equationsare considerablymore complicatedthan the bare equations;al- though the coupling 7"/uT can be considered a renormalized gravitational coupling, 77T" has no analog in the bare equation. Moreover, whereas the bare theory contains only the two response functions of Eq. (10), DIA adds the functions G "T and GT". Analytical progress will clearly require rational simplification of these equations. It will be convenient to solve the DIA equations by an iterative procedure, starting with an an3atz for the G's and F's. When these are known, the correlations can be computed from Eqs. (11) and (19). Know- ing G and Q, substitute in Eqs. (13)-(16) to obtain the damping factors. This procedure can now be iterated, since an updated G can be found by solving Eq. (18), and updated forces can be computed from Eqs. (17). The initial assumption for the Green's matrix will be G(k) = [G"(k)I0 -G"(k)G°(k)gP3G°(k) ]] (20) so that the bare and renormalized Green's matrices have the same structure, and are determined by two scalar Green's functions G"(]¢) and Go(]e) alone. This type of diagonal approximation is useful in treating coupled field problems. Leaving the random force general temporarily, the correlation matrix will be determined by forming correlations of the amplitudes found from Eqs. (11) and (19). The explicit expressions are QijU(k)^ = G"(k)G"(-]c)F_"(k) _ - gG"(-k)G"(k)G°(k)Pi3(k)FyT(k) + g2G"(k)Ge(]_)G"(-k)Ge(-k)Pi3(k)Pj3(k)FTT(k) Q_(k) = G"(k)G°(-k)F_T(k) - gG"(k)Ge(k)Ge(-k)Pi3(k)FTT(k) QT(_:) = Ge(_:)GO(_k)FTT(]_) (21) Now we will adopt for Kolmogorov scaling the ansatz Fi_j"(_:) = Pij(k)C_,ek -3 FuT= F TT= 0 (22) and for Bolgiano scaling Frr(k) =Cf Nk-3 F"" = F "r = 0 (23) 6