no-819 133 FINAL REPORT ON CONTRACT F49628-U5-C-U2S VOLUM 5(U) 1/1PRINCETON UNIV NJ DEPT OF MECHANICAL AM AEROSPACEENGINEERING S A ORSZAO MY 6? AFOSR-TR 67 ±349 VOL-5

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FINAL REPORT ON* I . IAFOSR CONTRACT F49620-85-C-0026

Steven A. Orszag, Principal Investigator* Department of Mechanical and Aerospace Engineering

Princeton University* . Princeton, NJ 08544

Volume 5

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DE SEB NVI IE

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D istribution -Unlimited

PRINCET.ON UNIVERSITY

E724

FINAL REPORT ONAFOSR CONTRACT F49620-85-C-0026

Steven A. Orszag, Principal Investigator

Department of Mechanical and Aerospace EngineeringPrinceton UniversityPrinceton, NJ 08544

Volume 5

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AFOSR /pM A F49620-85--C-0026WIC. AOOPESS (City. Stat. a ZIP Code) 10. SOURCE :OF SUNOING '40S.

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7Y 0V OF REPORT 32b. TIME COll/20/ FM0F-aPOAy,,Ao.Dy s.PG ON*Final Report FROtM 10/184 rol_/_/8___ v 1987

iSUPPLEME1NTAARY NOTATION

COSArt COCES I&. SUBJECT TERMS 1CantI~ 0#1 on~as it., 1VC0&4' 'Ie Ja,,, d idffnfy aly Wi~eft A4,10"bar

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Thids report consists of papers that surriarize work done on this research project. The, ajor results include: 1) The develcm'ent and acrlication of the renormal zat-ion group)-rethod to the calculation of funjdaxrental consta'nts of turbulence, the construction of'turbulence transport models, and large-eddy sim.lations; 2) The application of RIG methodsto turbulent heat transfer throuch th"e entire rance of expernnentally accessible Peynbidsn umbers; 3) The discovery that hic-h ?evnolds nunbter turbule-nt flc.,s tend to act as if they-,ad weak nonlinearities, at least 'teun vieved in- term's of suitable 'cuas i-particles';0 I The further analysis of secorjdar-. istability1 riechanisas in free shear flows, includingThe role of these instabilities in~ c!7aouc, 3-D free shear flaws; 5) The further develop--ent of numerical simulatLcns of turbirulent snots in wall bounded shear flaws; 6) The;tudy of cellular autanrata for the solu-tion of fluid mncharacai problems; 7) The clarifi-,ation of the relationship betw.eean th e !hyzerscale instabill'ty of asotropic small-scalelow structures to long Arvelers--'.n rurai and the cellular autamaton descriptionZ1ISTAI STION/AVAILASI LITY OP AOSTRAC12 ABSTRACT SEC...nry C ..ASSFICAl'ON

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VkIYMCAL MkVILW A VOLUME 34, NUMBER I JULY 1986

Positive- and negative-effective-viscosity phenomena in isotropic and anisotropic Beltrami flows

B. J. Bayly and V. YakhotProgram in Applied and Computational Mathematics, Princeton University, Fine Hall, Princeton. New Jersey 08544

(Received 23 December 1985)

A field-theoretic approach, analogous to Kraichnan's direct-interaction approximation, to the sta-bility theory of complex three-dimensional flows is developed. The long-wavelength stability of aclass of Beltrami flows in an unbounded, viscous fluid is considered. We examine two flows in de-tail, to illustrate the effects of strong isotropy versus strong anisotropy in the basic flow. The effectof the small-scale flow on the long-wavelength perturbations may be interpreted as an effectiveviscosity. Using diagrammatic techniques, we construct the first-order smoothing and direct-interaction approximations for the perturbation dynamics. It is argued that the effective viscosityfor the isotropic flow is always positive, and approaches a value independent of the molecularviscosity in the high-Reynolds-number limit; this flow is thus stable to long-wavelength distur-bances. The anisotropic flow has negative effective viscosity for some orientations of the distur-bance, and is therefore unstable, when its Reynolds number exceeds Vi.

1. INTRODUCTION the velocity). This means that the nonlinear term in theNavier-Stokes equations reduces to the gradient of a sca-

One of the most intriguing problems in the theory of lar, which can be absorbed into the pressure field. Thus,

hydrodynamic turbulence is the formation of large-scale any velocity field of the form (1.1) is a intrivial exactstructures in a liquid performing random (turbulent) steady solution of the Euler equations. Alternatively, wemotion at small scales. It is natural to explain the spon- can think of the flow (1.1) as an exact steady flow of ataneous formation of large-scale structures as a manifesta- viscous fluid maintained by an externally imposed bodytion of a long-wavelength instability of the corresponding force f= (pv/L2 )V, where v is the kinematic viscosity ofsmall-scale flow. In a theoretical approach to the descrip- the fluid.tion of this phenomenon, it is convenient to assume that From the class (1.1) of Beltrami flows, we shall selectthe small-scale flow is driven by an external force field two for explicit calculations. To observe the effects of ex-which varies with respect to the space coordinates. A pre- treme anisotropy in the basic state, we shall consider thevious paper' treated the long-wavelength stability of two- simple flow obtained when W contains only two wavedimensional, periodic eddy systems, driven by appropriate vectors ±Q, which can be chosen to be (± 1,0,0). Withsteady forces; it was shown that the large-scale instability the choice S=(0,1,O) and real amplitude A(:Q)is likely to occur only if the small-scale flow is sufficient- - Un./ 2, the flow isly anisotropic.

In this paper we shall consider small-scale flows of the V = U,.(O,cos(x/L),- sin(x/L)), (1.2)following form- which has the form of a monochromatic, circularly polar-

V(x)= A(Q)(fi+iQxfi)e 1Q z. 1L, (1.1) ized wave. For the opposite extreme of highly isotropicQE w flow, let us choose W to consist of a large number N of

unit vectors Q distributed isotropically over the unitwhere W is a finite set of unit vectors which contains - Q sphere. The complex amplitudes A(&) are all givenwhenever it contains Q. The velocity field is guaranteed modulus Un,,/v2-N; the phases of A(Q) are immatenal,

* to be real by requiring that the Fourier amplitudes A and may be chosen randomly. In either case, L',.,,, is the* satisfy A(-Q)=AO(Q) for each Q^ W. The vector B is root-mean-square fluid velocity, and is held fixed as N is

a unit vector orthogonal to Q. For our purposes it is un- varied to obtain different degrees of isotropy. Beltran,necessary to specify just which vector i is chosen out of flows of this type have been used extensively in simula-the unit circle of candidates; changing the choice of i is tions of diffusion in helical turbulence,' but the presentequivalent to changing the phase of the corresponding work appears to be the first investigation of their dynami-amplitude A, which turns out to be irrelevant to the long- cal properties.wavelength stability results. 2nrL is the fundamental The possibility that a small-scale flow with nonzeroperiod of each Fourier component. average helicity might be unstable to the growth of lone-

The flows (1.1) have the particular property that the wavelength disturbances has been discussed by a numbrvorticity vector is everywhere a constant scalar multiple of authors. Moffatt' has shown that a certain v'er sr-of the velocity vector (we shall call this the strong Bel. metric Beltrami flow with six Fourier components. knovwntrami property, as opposed to the weak Beltrami property, as the Arnol'd-Beltrami-Childress (ABC) flow, is im icidin which the vorticity is a nonconstant scalar multiple of ly unstable to large-scale disturbances with the sane sign

34 381 kC 1986 The American Physical Soctet

382 B. J. BAYLY AND V. YAKHOT 34

of helicity (i.e., the same "handedness") as the basic flow, gies are stable, and might be observed in experiments or9 and Moissev et al.6 have shown that, in a highly compres- numerical simulations.

sible fluid, any small-scale turbulent flow with nonzerohelicity can support a growing large-scale secondary flow. I. FORMULATION OF THE LONG-WAVEThe purpose of the present paper is to perform an investi- STABILITY PROBLEMgation of the possibility of long-wave instability in

" viscous, incompressible flow, which is typically the situa- The equations of motion for a viscous fluid in the pres-tion of greatest physical interest. It is our suspicion that ence of a steady, quasiperiodic force field are most con-under these conditions, most nontrivial Beltrami flows are veniently used in dimensionless form. Using L as thestable, and as such may occur and persist in a large length scale and v/L as the velocity scale, the dimension-variety of turbulent flows.7,s less Navier-Stokes equations take the form

In this paper, we consider the behavior of long-wavelength disturbances to viscous Beltrami flows, which v1 + R v-Vv = - Vp + V'v + f,

5 we assume are maintained by steady external body forces. V-(2.1)The perturbation is expanded in powers of the Reynoldsnumber of the basic flow, and we evaluate each term in wherethe long-wave limit. This expansion is essentially thesame as the expansion for the long-wave behavior of the f(x)- F(Q)e'4 ', F(Q)=A(Q)(+iQXfi),averaged impulse-response tensor in a fully turbulent owflow, for which Kraichnan originally developed the so-'." (2.2)called direct-interaction approximation (DIA). The DIA,as usually used in turbulence theory, consists of an equa- and the I A (Q) I, now dimensionless, have been scaledtion for the impulse-response tensor, coupled with an with respect to the root-mean-square fluid velocity U,,,.equation giving the evolution of the velocity-velocity R is the Reynolds number based on U,,,,, L, and v.correlation function. In this paper, the basic flow is Henceforth, unless explicitly stated, we shall work ex-known exactly, so we do not need the auxiliary equation clusively with the nondimensional quantities as definedfor the velocity autocorrelation. The resulting system is here.

K considerably simpler to analyze than most applications of If we define the four-dimensional Fourier transforms ofthe DIA. We construct both the first-order smoothing space-time-dependent quantities byand direct-interaction approximations and examine theirimplications for the two flows described above. a(k, (o)m f dXdter-4 (,)

*,: , In the long-wave approximation, the growth or decay (2ff) (2.3)rate of a large-scale perturbation is a quadratic function a(x,t)- f dk dw e (k't-a(k,w),of the wave number of the perturbation. For this reason,the effect of the small-scale flow on the evolution of then we can write the Navier-Stokes equations aslong-wave disturbances may be interpreted as the actionof a tensor effective viscosity. In the case of the simple 'Beltrami flow (1.2) with two Fourier components, only +G 0 (k, A-1iR)PI(k)the first nontrivial diagram survives in the long-wave lim-it for any finite Reynolds number, and the first-order X f dqdcrt,(q,oad 1k-q,,-o) . (2.4)smoothing approximation correctly determines the exactevolution of large-scale perturbations. The effective Here Ga(k,c)_(-io+k' is the so-called zero-orderviscosity is anisotropic, and is actually negative for some propagator, andorientations of the perturbation. It turns out that large- Put W =kj(6,-k,k /k2)+kl(8,s-k,k,/kscale perturbations of suitable orientation are capable ofgrowing on the flow (1.2) when its Reynolds number The tensor Pqt incorporates the pressure field and the in-exceeds v-. compressibility constraint, allowing us to replace the tmo

For almost isotropic Beltrami flows at low Reynolds equations (2.1) by only one (2.4).numbers, first-order smoothing theory indicates that the The Fourier transform of the force is a sum of 6 func-viscous damping of long-wave disturbances is enhanced tions:by the reaction of the basic flow, the effective viscosity f(k) Q Q) . (2Q5being isotropic and positive in this case. At finite Rey- I 8nolds number, however, all terms in the expansion are QE "

nominally of the same order in the long-wave limit, and The Beltrami property of the basic flow imphe that thefirst-order smoothing no longer remains applicable. The nonlinear termsolution of the direct-interaction approximation for theseflows indicates that the effective viscosity increases with Pj,(k) f dqf,(q)fi(k -q 1 t"the Reynolds number, eventually approaching a value in- vanishes identically, and the basic flow thereforc takt . thedependent of the molecular viscosity in the limit of infl- formnite Reynolds number. This result supports the conjecture

that a large class of Behrami flows with nontn% ial topolo- vk)-G(k,0f4k)bw)

,_, .,., ...... ,' .".. ., .. ..- . -. ,...,., -". , . -; -.....- ''-,' _ _ . , , ."'" " -''- , .. ' . " , ,- . .." .. ,, . , •. .. .: ,-. _... .,.. ., -. ..-.

Vyjwvq UTUrwT LW VT uwv V VW 'WW W NIWVKWM bI-

an infinitesimal perturbation vi(k,w) to the flow wherev,(k,w)=G0 (k,O)f 1(k)8(o)+v;(k, ) (2.7) v.(k,w)-0 if I kI > i .

and consider its evolution according to the linearized b

equation (2.10)

v.'(k,&))=G 0(k,o)(-iR)P.d(k) v).(k,w)mO if I kI <E,

X fdqGo(q,O)f.(q)v(k-q,w,). (2.8) for some long-wave cutoff 0<e<<l. We define theoperators ( ) and I- ( ) as the projections onto the

Because of the coupling between the perturbation and the long- and short-wave subspaces, respectively, of the func-small-scale basic flow, a pure long-wave perturbation is tion space we are working in.impossible, but we can formally separate the perturbation The long- and short-wave projections of the equation ofinto long-and short-wave components: motion (2.8) are then

v. (k,w)=G0(k,o.I)(-iR)P.,k(k) f dqG°(q,Ol(fblqlv,(k-q,w), (2.1 a)

v(k,wo)=G0(k,w)(-iR)P,*(k) f dq G0(q,O)fd(q)v<(k-q,w)

+G0(k,w)(-iR)Pa,(k) f dq G0(q,O)[fd(q)v,>(k-q,w)- (fd(qv>(k-q,w))] . (2.1 Ib)

The short-wave component can be expressed in terms of the long-wave component by formally iterating (2.11 b), which .'yields an expression in the form of a perturbation series in powers of the Reynolds number of the basic flow. When this %series substituted into (2.1 la), we obtain an equation involving only the long-wave component of the perturbation, which e"

* determines the evolution of the entire perturbation. The analysis of this equation occupies the rest of the paper.

i . FIRST-ORDER SMOOTHING

The first-order smoothing approximation consists of taking only one iteration of the short-wave equation (2.1 lb) be-fore substituting the result back into the long-wave equation. This amounts to taking the first term in the Reynolds-number expansion of the evolution equation, so the results will be strictly valid only for small Reynolds number. In thisregime, the molecular viscosity dominates the inertial processes, so all disturbances are damped. Nevertheless, we can ob-serve whether the small-scale flow tends to enhance or diminish the dissipation, and this information will give us someindication of the situation at higher Reynolds numbers.

The first iteration of (2.1 lb) yields

v,(k,l-,)=G0(k,)(-iR)P,(k) f dqG 0(q,Ofd(q)v,k-q,w). (3.1)

Substituting into (2.1 la) then gives an appropriate equation of motion for the long-wave component alone:; ~v. (k,w)--G°(k,col( -iR)P.U (k) f dqtGOlqt,O)fi,(q )GOlk- q',w )( -iR )P,(k-q')

X f dq2Gl(q,Ofdq 2)v,<k-q'-q)

=-RGO°(k,w,)Pk(k) I O(k-Qn o)P*(k-')Fb(Q ')Fd(Q )v,<(k-Q '- ,w) (3.2)Q1Qe W

after performing the q integrals. But v,(k-Q -2,w) is nonzero only if Q 2 -- 01+O(4), which impliesS2f _ n, because there is only a finite number of distinct wave vectors in Wand we are interested in the limit e--0.

Therefore,v.(k,w)= -R 2 G0 (k,w)P.1k(k) I G°(k-Q)P,.6(k-_) jA (Q) I (8w,-Qbd -i^b+t.Q 1 )L'<(k,), (3.3)

6w

where we have used the fact that a helical vector F(Q)=A(Q)[B+iQxfiI has the property thatF, (&)F, -6) = I A ( Q) I '(S. - 0,0,^-d --- e , (3.4) :=

regardless of the phase of A(Q) or the choice of B.Now, we can simplify (3.3) by taking the limit of small I k I and using the incompressibility conditions

.F(QW-O) , k"v,(ko)=O.

q.,,q

- - -- -.-. ,-.--------------------------------------------------------------------------------------------------------------..,.-..-,"-...".

-iw= -k'(l+ + -R) . (3.15) -13(a) C *

The small-scale flow in this case therefore enhances thedamping of all long-wave disturbances irrespective of FIG. 1. Constituents of diagrams.

wasc-vector direction or polarization, as would be antici- contains a factor of the Reynolds number, a diagram withpated for such an isotropic small-scale nlow. In particu- cnan atro h enlsnmeadarmwtlar. the isotropic nature of the small-scale flow has elim- n vertices corresponds to a quantity proportional to R'.inated the special directions in which perturbations were The short-wave perturbation may be expressed in termsamplified or only weakly damped of the long-wave part by iterating Fig. 2(b). The first two

approximations are shown in Fig. 3. Now, the term a in

IV. HIGHER-ORDER THEORY: Fig. 3 translates into the algebraic expressionDIAGRAM EXPANSION AND DIA -R 2 Pa(k)G°(Q )Fb(Q ! )GO(kt)

Because of the complexity of the algebraic manipula- QQ 2 6 Wtions, we shall use diagrammatic notation for analyzing xPck(k-Q ')G0 (Q 2 )F (Q )v(k-Q '-Qw)the perturbation equations (2.11). The procedure is stan-dard, but we shall present the derivations in detail for thesake of completeness. Let the thin line represent the zero and so the only contribution to the long-wave projection border propagator Go, the cross denote the force f, and the of term a comes from pairs Q ',Q2 that cancel. This canvertex the quantity (- iR)P, with indices to be specified be indicated diagrammatically by replacing the forcelater. Also, let the open bar represent the long-wave part branches by an arc as in Fig. 4(a).of the perturbation, and the slashed open bar the short- In general, the long-wave part of a diagram with an oddwave part. These symbols are drawn in Figs. l(a)-l(e), number of force branches vanishes, and the long-waverespectively, part of a diagram with an even number of branches is the

Equations (2.1 Ia) and (2.1 ib) can then be represented sum of all possible diagrams obtained by connecting theby the diagrams in Figs. 2(a) and 2(b). Each vertex has a branches in pairs. For example, the long-wave parts ofline exiting on the left, a line entering from the right, and the next nontrivial diagram is shown in Fig. 4(b). Long-a line entering from the upper right, whose other end con- wave diagrams that can be disconnected by severing onesists of a force term. Each of these lines possesses a wave Go line are called reducible diagrams (labeled r in Fig. 4),vector and an index, which must agree with the wave vec- and those which cannot (labeled i) are irreducible. So astor and index at the force or vertex at each of its ends. At not to count diagrams with internal wave vector k twice,each vertex, the wave vector on the left must equal the we require the following rule for evaluating irreducible di-sum of the right wave vector and the slanted force wave agrams: the sum over all wave-vector combinations mustvector. All repeated indices are summed over, and all exclude those for which one (or more) of the internal waveinternal wave vectors are integrated, resulting in wave- vectors is k. So, for instance, the diagram in Fig. 4(c)vector sums of the same form as (2.13). Since the vertex with the crossed arcs has the value

R' 2 G 0(k,w)P. (k)Fb(Qt )Go(k- Q i,c)Pd(k-Q ')Fd(Q) "

We can now extend the iteration scheme for vir(k,w) to Having found the expansion of vi> in terms of v,< we

arbitrary order in R. The next approximations are shown can now substitute it into Fig. 2(a) to find the equation ofin Fig. 5. The two reducible diagrams in Fig. 5(b) cancel, motion for v, by itself. Only the diagrams in Fig. 6 withleaving an expression for the fourth approximation that an odd number of branches contribute, and all the reduci-involves only irreducible diagrams. This cancellation of ble diagrams cancel in the long-wave projection, leavingreducible diagrams is quite general, and the full expansion the expression in Fig. 7(a). Figure 7(a) can be rewritten inof v,(k,w) is given in Fig. 6. The quantity 1,. is the sum terms of the renormalized vertex 5"b(k,w), defined as theof all irreducible diagrams with 2m vertices, minus the sum of all the legless irreducible diagrams. The resultingexternal legs. equation [Fig. 7(b)], whose algebraic translation is

/"

=+ - - + / .

FIG. 2. Equation (2.11) in diagram form. FIG 3. First two iterations of Fig. 2(h),

' . C F -_- -. -- .•"6010VI 16.% ~ * "% -. . *' .' *', %-. -.

Y a"WrIW' 7rVJWX mrkPu . ufV Irv .rd"N ,,RV PIPW P-~ l X. VV - W - U,, . Ira .VVJWMUVVJVW~ UV JUV.. V FV' r .

356 B. J. BAYLY AND V. YAKHOT 34

< + ~.d

> X

FIG. 4. Long-wave components of simple diagrams.

v,( k,w ) = G°( k,w) ( k,wlvlk,w (4.1)

can then be solved provided we can find some sensible ap-proximation for X,(k,oi). + =

Formally, therefore, we have found the exact form ofthe Reynolds-number expansion of the long-wave equa-tion of motion. Strictly speaking, the expansion pro- etc.cedure is only valid when R << 1, in which case we canget a good approximation to the dynamics by taking only FIG. 6. Full expansion of short-wave component.the first term in the series. This term, which consists ofthe single second-order diagram, gives the first-order The resummation of Fig. 7(b) has not simplified thesmoothing approximation that was discussed in the last problem of calculating I at all; rather, we have justsection. When the Reynolds number ceases to be small, transformed the problem of finding I into the problem ofthe series in its raw, untreated form is not likely to be use- finding G. But now we can perform the same resumma-ful, and we must resort to more sophisticated methods of tion procedure on the series for G itself, and obtain a con.

C series summation. solidated series for G in terms of itself, which can be ex-Consider the subseries, exhibited in Fig. 8(a), of the pressed in closed form in terms of I (Fig. 9). This equa- L

series defining the vertex function 1. The sum of the tion, whose algebraic form isinner parts of these diagrams formally defines a functionG,(k,ai) called the full propagator, which we denote by a G,,(k,w)=Golk,)lg, + G ol k , &j) b( k ,w )Gb,( k , o)) ,.

St f heavy line. Every diagram in Fig. 7(b) belongs to exactly can be inverted:f one series analogous to Fig. 8(a) that can be similarly

summed, and so the entire series for I can be rewritten in G.(k,)-G°-(k,o)-I(k,)]-. . (4.2)terms of G, as in Fig. 8(b). The diagrams remaining in With G given by (4.2), 1 is now determined as the solu-

, .4this consolidated series are those whose arcs cannot be di- tion of a nonlinear integral equation, known as the Dysonvided into two groups such that no member of one group tion fa nlin teral equation n s theintersects any member of the other. equation. Truncating the Dyson equation and solving the

= + /.A--~ - " +-o

+ ~ s 2 ,-~+ .--ii~ = +

®

- .4 ~ ++ + +.

FIG. 5. Third and fourth iterations of Fig. 2(b). FIG. 7. Dispersion relation for long-%avec omixorint

1:41 21.

34 POSITIVE- AND NEGATIVE-EFFECTIVE-VISCOSITY... 387

the first-order smoothing approximation is exact forlarge-scale disturbances to (1.2). The results of Sec. Ill, inparticular the prediction that the flow is unstable ifR > V2, are consequently valid for all finite Reynoldsnumbers.

One way to demonstrate this is by an appeal to classical(Orr-Sommerfeld) stability theory. Because the velocityfield (1.2) is independent of the y and z coordinates, the

+instability modes have the exact form

+ * * 6 Ma) v'(x,t)=e 1 (Uos,Vos,Wos)

where

m-- - + - + k0s=(0,k2,k3 ), c=(Oc 2,C3 ),

.and Uos,Vos,Wos are functions of x alone. Eliminating

+ + -- + * v•vos and w0 s from the linearized Navier-Stokes equationsyields an Orr-Sommerfeld equation for u 0 s(x):

S(014os -2 d 2 U(x)

dx 2 j dx 2 Uo5

1

FIG. 8. Resummation of subseries. ikosR - (5.1)

where kos--- tkosf, andresulting closed-integral equation gives a new approxima.

tion to the vertex, which may have useful properties that U(x)=kos-V(x)/kos, c=kos-c/kosno finite number of terms of the original expansion had.The DIA for :g is given by the lowest truncation of the are the components of V and c parallel to kos.

TIn the long-wave limit, with finite Reynolds number,0Dyson equation the quantity kosR is small, and (5.1) can be solved to give

I.f (k,w)- R 2P&(k) uos(x) as a power series in kosR. But this is exactly thesame procedure that we followed in Sec. IV, although here

x A ll( )l 2(a,-QbQ -ieC 1.Q) we performed more preliminary simplifications before ex-Qew panding. The series obtained using either formalism must

Q ' agree at each order in R; therefore the sum of all the nth-x G,.(k - Q,w)Pf (k -Q) (4.3) order diagrams in the diagram expansion must equal the

coefficient of R" in the Orr-Sommerfeld expansion, whichwith Gb given by (4.2). By reexpanding (4.3) in powers is proportional to kos. This result can also be demon-of R, it turns out that the DIA simply consists of sum- strated directly by keeping track of the orders of succes-ming up the subset of the full diagram series that consists sive approximations to v,' while iterating (2.1 Ib); we omitof all diagrams whose arcs do not intersect at all. Wheth. this calculation because the details are much more intri-er or not the DIA is a good approximation depends on the cate and no more instructive than the Orr-Sommerfeldrelative importance of the diagrams with less trivial topo. treatment.logical structure; this issue will be discusse2 in Sec. VI. The reason for the simplification of the series appears

to be the fact that in a sufficiently regular flow, the long-V. RESULTS OF HIGHER-ORDER THEORY wave perturbation modes also have regular spatial

behavior. The Fourier components of the perturbation de-For the simple Beltrami flow (1.2), the sum of all the cay algebraically with wave number at any Reynolds

diagrams of a given order n, which is of order R" by defi- number; indeed, they decay so rapidly that the componentnition, is also of order k" in the long-wave limit. In this of the disturbance at any wave vector which is the sum oflimit, therefore, only the single second-order diagram con- k and a nonzero combination of two or more basic-flowtributes to the dynamics of the perturbation, and hence wave vectors do not participate in the dynamics. There is

therefore no transfer of perturbation energy to higher andhigher wave numbers, and it is not surprising that the

.,,,, - + . + 006 long-wave evolution depends strongly on the molecularviscosity, even in the high-Reynolds-number limit

For flows with complex spatial structure, the periurti-= - + tion modes also have very invoked spatial structure. In

fact, the perturbation is likely to become singular in theFIG. 9. Dyson equation for G.1(k,w). limit of infinite Reynolds number. There is no finite set

358 B. J. BAYLY AND V. YAKHOT

of terms from the original expansion that satisfactorily g(k,o) =[k 2-a(k,co)J/D(k,w)describes perturbations to almost-isotropic flows atmoderate or high Reynolds number. Even in the long- h (k,w) =-rlk,w)/D(k,cw)wave limit, all diagrams in the series [Fig. 8(b)] are nomi- withnally of order k , which indicates that components of theperturbation at all length scales are important in the D(k,w)=[k2 -a(k,(o)] 2-ik,co).dynamics of the long-wave part. It is reasonable to expect Equations (5.2) and (5.3) can be solved using stan,that a cascade may develop at high Reynolds numbers, numerical techniques. The functions a and r are apprwith perturbation energy from the largest scales being mated by specifying their values at a suitable mesitransferred to smaller and smaller scales until it can be ef- mats on thei ales at te s Ificiently dissipated by viscosity. The damping of long- for all other points. The integrals (5.3) can thenwave disturbances would then become independent of the evaluated at the mesh points using a standard ten-pmolecular viscosity for large Reynolds numbers, and be Gaussian scheme for the c integrals. By performindetermined by the velocity and length scales of the flow many-dimensional Newton iteration, we can rapidly Ifield alone.

dAlthough the DIA does not involve anything like the the values of a and r at the mesh points which are relAlthughtheDIA oesnotinvove nyting ikethe duced by the integration. Successively finer meshes s

complexity of the full series for I in Fig. 7, it nonetheless used until the values for the eddy viscosity (see be%includes some of the structure of the perturbation at arbi-trarily small scales. If there is an energy cascade for the changed by 1% or less upon doubling the resolution.

It is easily checked that the asymptotic behaviorperturbation, we may be able to deduce some of its quali- arkO) is quadratic in k as k-0, while the asympttative characteristics by analyzing its DIA equations. The o(k,0) is cubic in k o, ltho tessence of the problem is to solve (4.3) for the renormal- behavior of r(k,0) is cubic in k. So, although the

ized vertex 1"b(k,&w). Since the basic flow is isotropic (to tisymmetric part of I is important for the solution of*an arbitrarily good approximation), the tensors G and Y integral equation, it does not enter explicitly into

airy oaohave the forms long-wave equation. The behavior of long-wave p, turbations is determined by the quantity

Gb(k,w)='-g(k,W)(6b-k.kb/k')-ih(k,o)by(k,/k), = limk_.0[-o(k,O)/k R 1, which is a nontrivial functi" of the Reynolds number R. Using Eqs. (4.1) and (5.I.b(kw)--'°(kW)(bab-k.kb/k 2 )+ir(kw)Eb,(k.,/k) , the long-wave dispersion relation is given by

* where g, h, a, and r are scalar functions that depend on i k 2[l+RrR] (5.4the wave vector k only through its modulus k.

Since we are primarily interested in the long-time, or, in the original dimensional units,large-scale behavior of the Green's function, we can sim-plify the analysis of the integral equations by setting the -ic= -k 2[+(UL)r(R)]. (5.4frequency w to zero within the integrals; the problem is Thus, for large Reynolds number, a long-wave disturban.insensitive to this approximation, and the resulting simpli- damps out as if acted on by an effective eddy viscosification is enormous. Then, by extracting the symmetric (U,,,L)F(R). A plot of F(R) as a function of Reynoi,

% and antisymmetric parts of (4.3), respectively, we can number is shown in Fig. 10; it should be noted that asreduce the Dyson equation to a pair of scalar integral increases, r tends rapidly to a finite asymptotic valuequations over the unit sphere, which can in turn be confirming the hypothesis that at high Reynolds numbetransformed into integrals over the cosine of the angle be- the exact value of the molecular viscosity is unimportattween k and Q: for the dynamics. The straight line through the origin h.

(k,)= L 1 -c 2 ) slope -L, corresponding to the first-order smoothing al-T+R f_ dc [a(k,c)g(q,0) proximation. The curve F(R) is tangent to this line at th

q origin, confirming the result that at low Reynolds num+1(k,c)h(q,O)], bers, the DIA reproduces first-order smoothing.

2) (5.2) A couple of remarks may be made about extensions or(k,O) d -c the theory beyond the case of long-wase linear pert urba

- R f_ [y(k,c)g(q,O) tions. The analysis of Sec. II does not actually depend ,t

the precise character of ( ) as the projection onto funk+,q(k,c)h(q,O), tions with support in a neighborhood of zero in %%a'e

vector space; the formalism remains applicable if (where the "kernels" of the integral equations are given by denotes the projection onto functions with support in

q=q(k,c)=(l +k 2 -2kc)I 2 , neighborhood of any desired wave vector. We can there

fore use first-order smoothing or the DIA to insestigatta(k,c)=2kc-4k c+2k, fl(k,c)=2kc-2k2 , the behavior of a perturbation of arbitrary uaie ,cctor,y(k,c)=4k 2C -4k 3, i-(k,)=2ke_2k . although it becomes very difficult to esaluatc the sums i,

k)c.2 -(3.6) once k and w, become order one. When k h-cortni

The integral equations (5.3) are "closed" by solving Eq. much larger than unity, corresponding to a %er) %mall(4.2) for g and h in terms of a and r, which yields the re- scale perturbation, the calculations again beome sImplelations As expected, such high-wave-vector perturbation% ar rap

% % . . r . " ' - " - ' " :_

34 POSITIVE- AND NEGATIVE.EFFECTIVE.VISCOSITY ... 389

smoothing and direct-interaction approximations indicate1 that the reaction of the basic flow enhances the damping

of long-wave disturbances. A t large Reynolds number%,the DIA predicts that the damping rate should be essen-tially independent of the molecular viscosity, which agreeswith an intuitive picture of the dynamics. Our results ap-

" l pear to be in some conflict with the general conclusions ofMoffatt and Moissev ef al.,6 and it is important to dis-cover whether this conflict is real or whether it vanishesupon closer inspection of the particular problemsanalyzed. It turns out that our results are completely con-sistent with those of Moissev et al.6 for highly compressi-ble flow, the essential difference being the degree of

0.R.0.0 compressibility involved; we illustrate this by repeatingthe first-order smoothing calculation for almost-isotropic

FIG. 10. Eddy viscosity factor r as a function of Reynolds compressible Beltrami flow in the Appendix.

number R. Moffatt analyzes the Arnol'd-Beltrami-Childress (ABC)flow, which is a Beltrami flow in the class (1.1) with the

idly dissipated by molecular viscosity, before the effect of explicit form

basic flow can be felt. U--B sin(y)+ C cos(z),We can also consider the effects of weak nonlinearity, U =Csin(z)+A cos(x),

for a perturbation of small but finite magnitude. By re-taining terms of second order in the perturbation ampli- 0 =A sin(x)+B cos(y),

tude in the analysis, a quadratically nonlinear equation ofmotion may be obtained. The coefficient of the quadratic where A,B,C, are real numbers. ofBC are often chosenterm can be expressed in terms of nontrivially connected to be equal to unity in studies of ABC flows; 9 this choicediagrams, and it appears that even at this order the alge- yields a flow which is intermediate in isotropy betweenbra would be too complicated to yield any information our simple flow and almost-isotropic flows. An indication

about the nonlinear dynamics. Whatever the form of the of the inviscid stability or instability of the flow may be

coefficient, however, it must be such as to maintain the obtained by calculating Arnol'd's second variation of the

Galilean invariance of the whole problem; that is, if a uni- kinetic energy with respect to isocirculational displace-

form translation V,8(k)8(g,) is added to the long-wave ments of the fluid particles. This functional is of indefi-

velocity v'(k,a), the frequency w must change to nite sign for suitably aligned long-wave perturbations with

w-k-V. This requirement alone is sufficient to deter- the same helicity as the basic flow, and indeed such per-mine that the quadratically nonlinear term has exactly the turbations grow with time in the initial stages of theirsame form as the advective term in the basic Navier- development.5 Moffatt's analytical results are supportedStokes equations; all the complicated corrections must by a recent numerical investigation of the stability ofStok s e uat ons all th e co m lic ted orr cti ns ust A B C fl ow in a viscous fluid at m od erate R eynolds n u m -cancel. Therefore, weakly nonlinear, large-scale perturba- BCfow

0 ber.'tions to almost-isotropic Beltrami flow satisfy Despite its apparent isotropy, the ABC flow has impor-

03v< +v<'Vvc_-V'p<+vef'V2v < , tant properties that distinguish it from the typical(5.5) almost-isotropic flows investigated here. The indefinite-

vdf=v+(U~sL)r(R) . ness of the Arnol'd functional, for example, is a propertyVI. DISCUSSION of the special orientation of the basic flow wave vectors

which ceases to hold when there are more than three in-We have considered the long-wave stability properties dependent Fourier components in the flow. There are also

of two viscous Beltrami flow fields, which were chosen to indications that the form of Moffatt's equation of motionexhibit the features of extreme isotropy versus anisotropy. for long-wave perturbations, which involves the third-To simplify the analysis, we assumed that the flows were order differential operator VXV 2 (hence the sensitivity tomaintained by external body forces. Although such the helicity of the disturbance) gives way to a reflection-forced flows rarely occur in the real world, this hypothesis invariant form involving only a second-order operator formakes it possible to construct the stability theory in an flows with more than three independent Fourier corn-uncomplicated manner. If it were desired, our analysis ponents. The difference between the stability propertiescould then be extended, using multiple-scale techniques, to of the ABC flow and the almost-isotropic flows may be atreat more realistic problems, such as the development of consequence of the difference between their topologiallarge-scale structures in slowly decaying viscous flows, structures. Recent investigations of the streamline %iruc-

For the simple Beltrami flow, the first-order smoothing ture of the ABC flow9 show that large regions of spaceapproximation correctly describes the behavior of long- are occupied by streamlines with highly regular structure,wave perturbations, and instability is predicted if the Rey- while in Beltrami flows with icosahedral symmetry, fornolds number of the flow exceeds 2. For the class of example, or more complex cubic symmetry, it appear%almost-isotropic Beltrami flows, both the first-order that almost all streamlines are ergodic.1 It sems hikey

'4

that the ABC flow has more in common with our aniso- ble by the DIA.23 There are also indications that hel,tropic flow than our almost-isotropic flows, as far as its in the flow introduces effects that do not appear at anstability properties are concerned. nite order in the perturbation expansion." Under thee

Although the DIA gives physically sensible results for cumstances, we cannot present our findings as quawnthis and other problems,2' 3 it is still an approximation. In tive facts, but nonetheless, we believe that this work i'fact, it is not even a rational approximation, in the sense important step in understanding the dynamical properthat there are no indications that the omitted terms are of helical flows.any smaller than those retained. The most we can hope ACKNOWLEDGMENTSwhen using this technique is that it incorporate the mostimportant physical processes sufficiently completely to While this work was in progress, we benefited gremgive answers that are qualitatively correct. In this prob- from the comments and criticisms of S, A. Orszag, u

lem, the most important feature is probably the ability of also provided the facilities for performing the numerthe basic flow to mediate the transport of perturbation en- vork. We are grateful to H. K. Moffatt for sending uergy from large scales to small scales. In addition, the copy of his work on a closely related subject, prior to pequations have the property that the nonlinear terms con- lication, and for his comments on a draft of this parserve the energy and helicity in the flow. The work was supported by the U. S. Office of Na

We can actually construct a model system with these Research under Contract No. N00014-85-K-0201 and,essential ingredients. Imagine a large number of realiza- U. S. Air Force Office of Scientific Research under C(tions of almost-isotropic flows, all chosen from an ensem- tract No. F49620-85-C0026. B. J. B. also thanks the 1ble with well-defined statistics. We can then ask what tional Science Foundation for financial support.happens if each realization is given a perturbation and

then allowed to evolve according to the linearized APPENDIX: INSTABILITYNavier-Stokes equations; in particular, we want to know IN ISOTROPIC COMPRESSIBLE FLOWwhat happens to the perturbation averaged over all reali-zations. This situation is exactly analogous to that of the Beltrami flows, in particular the family (1.1), are alrandom harmonic oscillator extensively analyzed by exact solutions of the Euler equations in an isentropi

5. Kraichnan.' 2 compressible fluid, and can be maintained by a suita"Now, consider the effect of formally coupling the external agency in a viscous compressible fluid. In ord

dynamics of the different realizations together. The to investigate the effects of compressibility on the possib,. *~ behavior will be altered, of course, with the details de- generation of large-scale motions, it is simplest at first i

,pending on the nature of the oupling. If the couplings ignore the influence of pressure forces on the dynamics

are suitably restricted, however, the total energy and heli- the short-wave part of the perturbation. This approxim;city of all the realizations taken together will still be con- tion is crucial to the analysis of Moissev et al.,6 whstants of motion. The system obtained by choosing the nonetheless retain the pressure in the long-wave corr

j couplings to be as random as possible within these restric- ponent. For consistency, we shall neglect the perturbatiot ,; tions is called the random-coupling model, and its statisti- pressure in all its manifestations and obtain essentially th

cal behavior is given exactly by the DIA.i 2 These proper- same results.ties of the random-coupling model indicate that there are If we neglect the perturbation pressure field (and drolgrounds for expecting the DIA to include at least some of the incompressibility condition, of course), the analysis othe important physical effects in other systems also. long-wave disturbances proceeds exactly as in Sec. II. Th,indeed, for passive-scalar dispersal in reflectionally invari- only difference is that the tensor through which two veloant turbulence the DIA and modifications thereof have city components interact no longer depends only on the;been highly successful in predicting the asymptotic long- sum, but on their individual wave numberstime diffusion rate.2

.' 4

,3 P.b,(k, k2)-k'6. +k'8. (A 1,Unfortunately, the DIA does not always perform so - b

successfully, particularly in the presence of helical effects. This tensor is no longer symmetric in b,c, which is theThe predictions of passive-scalar dispersal in helical tur- crucial feature allowing the long-wave instability. Usingbulence are in error by a factor of three,3 and helicity fluc- the first-order smoothing approximation we obtain thetuations in conducting fluids give rise to anomalous ef- following equation, which is analogous to (3.3), for thefects in the diffusion of magnetic fields that are inaccessi- evolution of a long-wave perturbation:

v<(k,w)=-R 2 G°(k,w)v<(k,w) + [Q A(Q) I p;k(-k-.)GO(k- ,o)(d b&-.QQd ,,)P ,(-Q,k)1 . (A21QE W

For an almost-isotropic flow, evaluating the right-hand av < ,+R 2 VXv < +0(V1) (All4side in the long-wave limit gives

J O ) (k,w)+O(k1 ) which predicts instabilities in the form of large-scale heli-, v.(k,)=tR ,G(kw)tbikbV, cal flows with helicity in the same sense as the basic

or, in the space-time domain, flow. '6

%. . . . . . .% r %

[; +' " : "+" ++ 1 ... . .+ '" + +/+: ' k + +' ' ", + . : €d ' + ..

IG. Sivasbinsky and Y. Yakhot. Phys. Fluids 23, 1040 (1985). 7R. B. Pelz, V. Yakhot, S. A. Orszag. L. Shtilman. and E. Le.2R. H. Kraichnan. 3. Fluid Mech. 75, 657 (1976); 77, 753 (1976). vich, Phys. Rev. Lett. 54, 2505 (1985).3R. H. Kraichnan, J'Fluid Mech. 81, 385 (1977). OH. K. Moffatt, J. Fluid Mech. 159, 359 (1985).41. T. Drummond, S. Duane, and R. R. Horgan, J. Fluid Mech. 9T. Dombre, U. Frisch, 3. M. Greene, M. Henon, A. Mlehr, and

138, 75 (1984). A. M. Soward, J. Fluid Mech. (to be published'5H. K. Moffatt, J. Fluid Mech. (to be published). 'tOD. Galloway and U. Frisch (unpublished).6S. S. Moissev, R. Z. Sagdeev, A. V. Tur, 0. A. Khomenko, and I"IB. J. Bayly (unpublished).

V. V. Yanovskii, Zh. Eksp. Teor. Fiz. 85, 1979 (1983) [Soy. 12R. H. Kraichnan, J. Math. Phys. 2, 124 (1960).Phys.-JETP 58, 1149 (1983)]. 13R. Phythian and W. C. Curtis, 1. Fluid Mech. 89, 241 (1978).

%" W ' 3 '3 *3 w U w -% • 3' d .:'_'7' a L',p

, . "* " -. "- "" -- - -, . * - "** ''- - ". ." - -- -"- '' '. '

• ".% " , °..t ," -% %-% % ,- -" % f .~. -,".w .. %'"% ", %% • .*%.• , ,. *,