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NASA Technical Memorandum 81315

_IOMOGEN_OUS "LIBUULhNCJ_ (L_ASA) 9_ p

hC A05/t1_ AOI CS_L 23U

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Numerical Experiments inHomogeneous Turbulence

Robert S. Rogallo

September !981

National Aeronautics ancl

Space AOm;n_strat_on

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,"'_ NASA Technical Memorandum 81315

Numerical Experiments inHomogeneous Turbulence

Robert S. Rogallo

September 1981

N/EWNaltonal Aeronaul_cs and

Space Aclnl_nnstrat_on

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NUMERICAL EXPERIMENTS IN HOMOGENEOUS TURBULENCE

Robert S, Rogallo

Ames Research 3enter

SUM_I,ARY

The direct simulation methods developed by Orszag and Patterson (1972) for

IsotropJc turbulence have been extended to homogeneous turbulence in an

incom_rL, sslble fluid subjected to unifo_ deformation or rotation. The results of

simulations for irrotational strain (plane and axisymmetrlc), shear, rotation, and

relaxation toward isotropy following axisymmetric strain are compared with linear

theory and experimental data. Emphasis is placed on the shear flow because of its

importance and because of the availability of accurate and detailed experimental

data. The computed results are used to assess the accuracy of two popular models

used in the closure of the Reynolds-stress equations. Data from a variety of the

computed fields and the details of the numerical methods used in the simulation are

also presented.

INTRODUCTION

The turbulence problem has remained the greatest challenge to fluid dynamiclsts

since its definition by Reynolds I00 years ago. The nonlinearity of the

Navier-Stokes equations prevents statistical analysis of the dynamics of energetic

turbulent fields, and as a result no theory has yet been devised that is devoid of

assumptions beyond that of the validity of the Navier-Stokes equations themselves.

Althou_h statistical, tensorial, and dimensional analysis combined with the

continuity condition provide constraints on the statistics of a turbulent field, our

knowledge of the statistics themselves has been gained primarily through experiment.

The turbulent flows of interest in engineering, !;oophysics, and meteorology are

complex even at the statistical level, but there are interesting flows with

spatially homogeneous statistics that have been examined both theoretically and

experimentally. Although these homogeneous flows do not have net diffusion of

turbulence momentum or energy, they can be a_isotropi¢ and extract energy from the

mean motion ; therefore, they provide a case for study intermediate in complexity

between !sotropic flows and general inhomogcneous flow_. Although homogeneous flows

are mort, difficult to achieve _.x_,erlmen_ ill,,' than luhomogeneous flows, such as Jets,

wakes, ;_nd pipe flow, they are considerably easier to simul,',te on a computer becauseof the absence of flow boundaries.

The method of direct numerical simulation of homogeneous turbulence in general

use today is that of Orszag and Patterson (1972). The turbulence field i_

represented by the coefficients of a truncated three-dimensional Fourier series, or

'r

L

equivalently by spatially periodic discrete values on a uniform mesh in physical

space. Trans£or_atlon between the two spaces (physlcal and wave) is accomplished byfast Fourier transforu. Differentiation is performed in wave space asmultlpllcstlon by the wave number, and velocity products are carried out In physicalspace to avoid the expensive convolutlon sums required for that operstlon in wave

space. The field can be tlme-advanced in either space by any algorithm appropriatefor systems of ordln_ry dlff_rentlal equations.

There are two basic requirements that a direct slmulatlon must meet if it is torepresent turbulence. First, it must represent a solution of the Navler-Stokes

equations. This means that all scales of motion must be adequately resolved, in thedeterministic sense, by the computatlonal mesh. In partlcular, the Reynolds number

should be small enough to allow the mesh to accurately capture the viscousdissipation scales. The explicit use of Fourier series then implies a spatlallyperiodic solution of the Navler-Stokes equation. The second requirement is that this

periodic solution be sufficiently compllcated, that is, it must provide adequate

statistical resolution (large sample) of the set of all possible fluid motionsallowed by the Navler-Stokes equations. The computatlonal sample of a scale of

motion is inversely proportional to the volume of the scale. Thus scales of motioncomparable to the computatlonal period have a very small sample (-i), and scalescomparable to the mesh spacing have a large sample (-i05 for a 1283 mesh). The two

requirements for a turbulence simulation conflict; the sample improves as the energy

moves to smaller scales but the viscous resolution is degraded.

The range of scales (the ratio of the wavelength of the fundamental to that ofthe highest wave number carried) in a given direction is M/2, where M is the number

of nodes in that direction. The largest computers available today allow 128 mesh

cells in each direction, which limits the useful range of scales to about i0 or so.

That is, the bulk of the total turbulent energy must be contained within one decade

of scale. This is a severe constralnt and it limits completely resolved direct

simulation to very low Reynolds numbers. In practice we usually accept some error in

order to obtain a higher Reynolds number. In some cases we sacrifice sample and

shift the computed scale range toward the small scales , and in others we sacrifice

resolution of the dissipation process and shift the computed range toward the large

scales. For the current study we are most interested in the anigotropy of the

Reynolds-stress tensor, and less interested in the details (such as intermlttency) of

the small scales, and we generally accept incomplete resolution of the dissipation

scales. It appears, however, that the error in total dissipation is much less

affected than is its spectral distribution because the total is determined primarilyby energy transfer down-scale within the energetic scales; this in turn depends onhow well the energetlc scales themselves are resolved and not on how well the actual

dissipative scales are resolved.

Computation and physical experiment complewent each other rather well for

homogeneous turbulence. The experimental difficulties are associated with setting up

the mean motion, achieving a homogeneous turbulence field, and in the actual

measurement of the various spectra, correlations, etc. There is no doubt that the

physical experiment produces result8 at all scales in accordance with the

Nsvler-Stokea equations, although frequency response (in both space and time) Of the

instrumentation can limit their measurement. The computation on the other hand has

no dl/flculty settlns up the mean flow or making measurements. The difficulty is

that because of limited resolution (statistical accuracy at the large scales,nuaerlcal accuracy at the small scales), It is not always simple to relate computed

quan_itte_ to those of the experiment at much higher Reynolds number. The goal ofthis study is to use experimental data to validate simulated turbulence fields, which

7

'i

in turn provide measurements that are not available from experiment. The quantities

measured here are primarily those volume averages appearing in the Reynolds-stress

equations,

In the sections that follow we consider four cases of homogeneous turbulence,

each being the evolution from an initially Isotropic state caused by a mean velocity

gradient uniform in space and time. The deformations considered are (I) irrotatlonal

plane strain, (2) Irrotational axlsymmetric strain, (3) uniform shear, and (4)

uniform rotation. In the case of axlsymmetrlc strain We allow the turbulence to

relax back toward Isotropy when the straining ceases. The only theory available to

describe the evolution of these flows is a linear theory valid in the limit in which

the time scale of all turbulence scales is much longer than the time scale given by

the mean velocity gradient. This theory, due originally to Taylor (1935), has been

developed for each case by various authors and is used, togeth±r with available

experimental data, to establish (we hope) some credibility for the simulations.

Then, as an example of how the detailed measurements of the simulated fields might be

used to aid the turbulence-model maker, we have compared measured values from the

fields with modeled values following recommendations of Rotta (1951) and Launder,

Reece, and Rodl (1975).

Finally, more details are given in the appendixes, Appendix A presents in

tabular form various raw data measured in each field, including the Reynolds-stress

budget; Appendix B provides a more detailed exposition of the equations and numerical

methods used in the simulations.

PLANE STRAIN

Nearly Isotroplc turbulence, subjected to a uniform strain in two directions,

has been investigated experimentally by Townsend (1951) and by Tucker end Reynolds

(1968) by passing grld-generated turbulence through a channel of changing cross

section. Townsend imposed a total strain of 4 in directions transverse to the

stream; Tucker and Reynolds imposed total strains of 6 in two different

configurations, the first, like that of Townsend, was strained in the transverse

directions and the second had one strain imposed in the flow dlrectioD. We fo]low

the nomenclature of Townsend and take the flov direction to be x, imposing positive

strain in the z direction and negative strain in the y direction. The results of

Tucker and Reynolds indicate no significant difference between the two types of

strain. The longitudinal strain does have the advantage, like the axisymmetric

strain discussed in the next section, that the strain is giwen by direct measurement

of the variation of mean flow velocity down the channel.

Results of four computed cases, all evolving from the same isotropic initial

state are presented. The initial state itself is the rest, It of an isotropic

simulation so that the spectral transfer is establlshed. The mlcroscale Reynolds

number q\It/_ of the initial field is -35, and dimensionless strain rates aq2/_ - .5,

I, 2, and 4 _where a - d_/dz = -d_/dy, q2 = UlUi ' c = 2_Ui,JUl,J)

are imposed in cases B2DI, B2D2, B2D3, and B2D4, respectively. The total strain

achieved is 4; higher strains result in inadequate resolution because the

computational mesh moves with the mean flow and becomes excessively strained. At the

initial state, the computational cell has Av = 2Az, and at a strain of 4, Az = 2Ay.

W

Three independent dimensionless parameters can be formed from the time t, strain

rate a, viscosity v, and turbulent velocity and length scales q and L respectively.

These are taken to be at, aL/q, qL/v, the dimensionless time, ratio of

turbulence and strain time scales, and a turbulence macroscale Reynolds number. No

information on macroscales is directly available from the experiments, so we have

instead estimated L by assumlug ¢ - q3/L in the Isotroplc initial state. The iu!tlal

conditions are then scaled by the dimensionless groups aq2/c and q_/_u. Estimates

of the quantities for uhe experiments are as follows:

o4_v _

Townsend .5 in. mesh .45 95

l in. mesh .45 230

Tucker-Reynolds .3 475

Present results .5,1,2,4 129

Here the isotroplc relation e -, 10vq2/A_l has been used. For the Townsend

experiment, the microscales are given and these are used to estimate c; for the

Tucker and Reynolds experiment we have estimated _ by fitting a power law, with

exponent of -1.2, to the plotted energy history; these estimates are very rough

indeed.

The linear theory of "sudden distortion" was originated by Taylor (1935) who

considered the inviscid irrotational strain of a typical Fourier component of the

velocity field. The effect of viscosity can be included by use of an integrating

factor (Pearson (1959)). Batchelor and Proudman (1954) extended the inviscld theory

to determine the effect of rapid irrotatlonal strains on the energy tensor of an

initially isotroplc turbulence field, and the result is independent of the form of

the energy spectrum E(k). Although viscosity can in principle be ::_dled exactly,

the necessary integrals over Fo_rler space contain exponential factors _nd probably

cannot be evaluated in closed form. The result would depend on the initial, spectrum.

A comparison of the linear theory, experimental, and computational results is

presented in figures 1 through 3. The evolution of the normalized energy tensor is

shown in figure i, and the structur_ parameters introduced by Townsend are shown in

figures 2 and 3. The computational results approach the linear theory consistently

with increasing strain rate but, although they follow the trends of the experiment,

quantitative agreement (particularly for K I and K[ ) is lacking.

The differences between the two experiments are not consistent with our

estimates of their dimensionless time scales and Reynolds numbers. At a given level

of strain, the structure parameter K I should increase with strain rate but decrease

with increasing Reynolds number. Our estimates of the dimensionless strain rates and

Reynolds numbers of the experiments would cause us to expect higher K I values from

the the Townsend experiment than from the Tucker and Reynolds experiment, but that is

not the case.

,1 ¸

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AXISY_IETRIC STRAIN

Turbulence in an axisymmetric strain field is the simplest anisotropic flow.

Because it occurs often in engineering applicatio,_s, it has been studied i_ a number

of experiments. We will compare the computational results with the experimental data

of Uberoi (1956) and Hussain and RamJee (1976), who achieved total strains up to 16

at a wide range of Reynolds numbers. Unfortunately, _nformatlon concerning the

dissipation rate and length scales was not presented, and we are unable to determine

with any confidence the turbulence Reynolds numbers or time scales of the turbulence

prior to straining.

Simulations were made with four constant strain rates applied at each of two

viscosity values. The dimensionless strain rates aq2/¢, where

a - d_/dx = -2d_/dy = -2d_/dz, were 0.3, 0.6, 1.2, and 2.4, and the £nitial

Reynolds numbers q4/v_ were 15 and 56. The evolution of the longitudinal and

transverse component energies, normalized by their initial values for five of the

eight simulated cases, the inviscid linear theory, and the experimental results of

Uberoi and Hussaln-RamJee, are shown in figure 4. The computational results approach

rapld-distortion theory consistently with increasing strain rate and Reynolds number,

and the differences between computation and experiment seem to indicate a higher

experimental strain rate. Beyond a strain of 4, the experimental component

energy u 2 grows because of the redistribution of energy by the pressure-straln

correlation, and the computational data (appendix A) indicate that _his is a result

primarily of the growth of the "slow-pressure-strain" term with increasing strain.

:q

i!

HOMOGENEOUS SHEAR

Homogeneous shear turbulence is the closest structurally to flows of engineering

interest and has been extensively investigated by experimenters. In a series of

papers from Johns Hopkins University, Rose (1966), Champagne, Harris and Corrsln

(1970), Harris, Graham and Corcsln (1977), aud Tavoularis and Corrsin (1981)

(hereafter TC) have provided extensive documentation of the development of turbulence

subjected to uniform shear. The results of Mulhearn and Luxton (1975) are in

substantial agreement with the earlier work of the Johns Hopkins group. On the other

hand, the linear theory of hoeoge_eous shear turbulence has been worked out by

Delssler (1961, 1970, 1972), and Fox (1964); their results are consistent with the

initial stage of development of the experimentally observed turbulence.

The linear theory can be worked out in a somewhat simplex way than that of

Delssler, who worked directly with the equations for the spectrum tensor after

dropping out the triple correlations. Written in a coordinate system moving with the

mean shear, the ilnear_zod Navier-Stokes equations are

u t + sV + PK = _(_u

vt - stpx + py = _v

Wr + Pz = v_w

ux - stv.¢ + Vy + w z " 0

rI........

L_

i!!i

where

d_s -- and

dy @ = T_x2.+ - st + _z--_

Upon taking the Fourier transform of these equations and solving the remaining

ordinary differential equations for the time variation, we obtain

F_ = -Ak3 + B[k3 2 tan-lrl - k12 1"+'-_ }

i

F9 = Bk I /k12 + k32 1 + q2

{F_ = Ak I - Bklk 3 tan-lq + 1 + _2where F is th_ integrating factor

-_ 3/2(q 1 3)log F = _s (k12 + k32) ": _ q

k2 - klst

q =

Jkl2 + k32

and A(k), B(k) are determined from the initial conditions. The evolution of the

spectrum tensor from an iso

....... -7

the experimental evidence at small times (st < 4)and predicts (surprisingly well)

the magnitude of the shear stress correlation "_/u'v" and the ordering

u" > w" > v" of the component energies. It has been determined experimentally,

however, that the v component eventually grows, gaining energy through the

pressure-strain correlation caused by the nonlinear transport terms in the

Navler-Stokee equations; this in turn leads to growth in uv and the energy. The

ultimate fate of the turbulence is currently a speculative issue, but there is

evidence of a self-similar evolution. The energy is still growing at the largest

times observed in both the experiments and computation, and the ratios of component

energies and integral scales are still varying, although rather slowly. If the

turbulence approaches a self-slmilar state with slngle velocity scale q and single

length scale L, its attributes depend only upon t, q, L, s, and u. Dimensional

analysis then requires

q

If the velocity and length scales both grow monotonically at large

times, sL/q must approach a finite nonzero constant, because if sL/q_=,li-_r

theory is valid and predicts the ultimate decay of q; similarly, if sL/q÷O, the

flow approaches isotropy and q decays. With nonzero s the flow can not become

isotropic in any case because for isotropic flow there is strong experimental

evidence that L" q3/¢ and q- t-n from which it follows that d(sL/q)/d(st) > O.

Four _hear flows were simulated, three of them (BSH9, BSHIO, and BSHll) from the

same initial conditions. The initial energy spectrum for these runs was simply a

square pulse E(k) - 1 for 16 < k < 32, a wave-number band containing roughly a

quarter million Fourier modes. The inltial spectrum for the fourth case (BSHI2) was

a square pulse at I0 < k < 20, the lower wave number allowing a higher Reynolds

number to be attained.

The shear rates and viscosity used are as follows:

Case Spectrum s

(E(k)-[)

BSH_ 16

w.q__ _¸_¸ ..... ,v,,_..... -.....

statlsti_al sample, and although the simulation is still a periodic solution of the

_avier-Stokes equations, it is not representatlve_ at the large scales, of

turbulence. However the smaller scales still retain a good sample, and stati_tics

from them (e.g., microscale_) show no anomalous behavior. As shown in figure ii, the

dissipation scales are adequately sampled throughout the run, but for st~12 the

spectrum does not decay with wave number rapidly enough to allow complete resolution

of the dissipation scales. All four cases appear to be adequately resolve_ at

st - 8; BSH9 and BSHI2 are marginal at st - I0 ; and BSHIO and BSHII are marginal atst - 12.

The primary dimensionless flow parameters are presented in figures 12 and 13.

We have used the longitudinal integral scale LII,I in the streamwise direction(x) asthe length scale and q . (_)I/2 as _he velocity scale. The mean shea_ s

provides the time scale of the shear. In figure 12, the experimental data of TC

indicate that the growth of velocity and length scales has equilibrated at st-8; this

is prevented in the computation by the peculiar growth of the integral scale. The

turbulence Reynolds number shown i_ figure 13 indicates that the experimental data

are consistent with exponential growth in beth velocity and length scales, having

roughly the same growth rate as the computation. The expei lmental Reynolds numbers

are simply not attainable with the available computational resolution but we hope

that the Reynolds numbers attained are high enough for the statlstics of =he

energetic scales to achieve nearly asymptotic values. The classical[ mixing length

(fig. 14) is formed from velocity scales of the turbulence and the (constant for each

run) time scale of the shear; it appears to grow exponentially at _arller times than

does q (fig. 5(a)).

The distribution of energy among the velocity components is presented in figsre

15; i= suggests the possibility that the component energies approach fiKed ratios.

If such a structural equilforlum occurs between the tendency of the shear to produce

anisotropy and the tendency of the turbulence to become '.sotropic, the anisotropy

measure of the equilibrium should depend on sL/q and (probably weakly)

on qL/u. This _s shown more clerrly in figure 16 where an adhoc linear scaling

of the anisotropy is shown which collapses the data reasonably well. The discrepancy

between computation and experiment could be a Reynolds-number effect similar to tha:

in the shear-stress torte]scion shown in figure 17. The Reynolds number shown there

was chosen because it is the only dimensionless group independent of L and t.

Although it seems physically reasonable to eliminate explicit dependence on time, the

elimination of L simply allows us to avoid the use of the measured integral scales.

The rapid rise of the shear-stress correlation from its initial isotropic value of

zero is evident. The shear-stress correlation is equally well collapsed by

mlcroscale Reynolds number. The reduction of the shear-stress correlation with

increasing Reynolds number is primarily due to the fact that the contribution to u-_

of the small scales increases less rapidly with the increase of small-scale energy

than does the contribution to the normalizing factor (u'v_); this is because of the

tendency toward isotropy of small scales. The explicit use of Reynolds-number to

collapse correlations for medium and large scales was first used by Stewart and

Townsend (1951) in isotroplc turbulence. In the shear flow, unlike Isotroplc flow,

we have a direct energy measure, u--v, th_c is rela_Ivel3, Reynolds number insensitive,

and this can be used to advantage to normalize velocity autocorrelations.

The_au_ocorrelations presented in flgttre 18 are normalized by t.he sin81e energy

measure uv, and plotted against the separation, normalized by the _Ingle length scale

LII,I, the integral scale of Rll(r,O,O). The experimental points are from the

Tavoularls and Corrsin data at X/H m Ii. If the turbulence were i_ exact structural

equilibrium this scaling would collapse all of the data, except near r - 0 where

8

i/' --7 ._

r

_:!_cous effects appear. At infinite Reynolds number the collapse would be complete.

Perfect collapse requires that ratios of the component energies and ratios of the

macroscales be universal constants. This is too much _o expect here, however, for we

have already s_n that ratios of the component energies in both the experiment and

computation are still varying slowly with sL/q and qL/v; moreover the

integral scales of the computation are not as reliable as we would llke because of

the small sample of the largest energetic scales.

The mlcroscales measured by TC did not vary after st - 8.65, but the computed

mlcroscales are growing slowly at st - 8, as shown in figure 19, with the growth rate

diminishing with increasing Reynolds number.

TABLE I. - COMPARISON OF EXPERIMENTAL AND COMPUTED TURBULENCE FIELDS

QuantityTavoularis- Case Case

Corrsln BSH9 BSHI2

X/H=11 st=lO st-lO

Z/q 2 (Component energlee) .535 .508 .485

Uu__32/q2 .186 .185 .209

/q2 .279 .307 .306

-'_/u'v" (Shear-stress .45 .49 .49

correlation)

LII,2/LII,I (Integral scale .33 .44 .59

LII,3/LII,I ratios) .25 .17 .14

L22,1/LII, I .23 .41 .33

L33,1/LII, 1 .34 .23 .17

LI2,1/LII, I .90 .88 .88

LI2,2/LII, 1 .40 .69 .74

XI2/AII (Microscale .67 .65 .76

_13/_ii ratios) .68 .69 .79

121/_i I .68 .86 .80

k3]/_l I .79 .88 .85

Oa/O b (Principal stress 4.3 4.8 4.0

Oa/O c ratios) 2.1 1.9 1.8

aa( °) (Principal axis 20 22 24orientation)

VT/V _ (-Uv)Ivs (Dimensionless 179 16 38

RI . u'_ll/v measures) 266 76 104

SLll,i/q 2.83 2.23 1.963581 219 472

qLll,I/_

£/Ltl, I = (__-_)/sLll,lv .I16 .167 .174

A summary of the TC data at X/H = Ii (st-12.6) and of the computational results

at st = i0 from cases BSH9 and BSHI2 (our highest Reynolds numbers) is presented in

table I. The results of the computation agree well with the experimental data,

!

except for the integral scale ratios (and quantities containing viscosity

explicitly). Part of this discrepancy is due to the fact that TC measured the scales

by integrating to the first zero of the correlations and that we integrated over the

negative portions as well; but this affects only Lil,3 in the experimental data. The

main cause of the discrepancy is that the integral scale, as stated by Batchelor

(_.953), is not as representative of the _cales of motion containing the energy as we

would llke; it overemphasizes the largest scales where our sample Is poor. In the

experiment, a large sample of the largest eddies is swept past the probes'by the mean

flow, but in the computation we move with the mean flow and always observe the same

large scales.

There is some evidence in both the computational and experimental results that

the turbulence approaches a structural equilibrium, with length and velocity scales

growing exponentially. The experimental evidence, stated by Tavoularis and Corrsln

themselves, although they seem to expect linear _ather than exponential growth, is

that the measured downstream transport Dq2/Dt and the production -s_-_ of the

turbulence are in a fixed ratio, within the experimental error, for 7.5 < X/H < 11.

This implies that the dissipation rate is also in a fixed ratio to the tra--nsport? and

this relationship was checked independently by measurement of the velocity derivative

variances. This is equivalent to the fact that both the shear-stress correlations

and mlcroscales are constant. The energy growth rate is then proportional to the

energy, and the growth is exponential. In table 2 we present a check of the

experimental data for consistency with linear and exponential growth by us%ng the

data at X/H - 7.5 and X/H - 9.5 to predict the data at X/H = II. Although either

growth form extrapolated to X/H - ii probably lles within the experimental error, the

exponential form is consistently closer to the published values. The computations

clearly indicate that the energy of periodic perturbations of a uniform shear grows

exponentially, but the turbulence macroscale is bounded by the computational pcrlod.

Unfortunately, by the time the computational velocity is clearly growing

exponentially, the integral scales indicate an insufficient sample at the largest

scales, and the relevance of the exponential growth to turbulence (as opposed to

periodic solutions of the Navler-Stokes equations) is suspect.

TABLE 2. - DATA PBEDICTED AT X/H " ii

BASED ON DATA AT X/H = 7.5 AND X/H - 9.5

Quantity

Data predicted at X/H - II

Measured Linear growth Exponential growth

u 2 .475 .465 .478

v___2 .165 .163 .167

w 2 .248 .242 .247

q2 .888 .870 .892

Lll,I 57 56 573.42 3.26 3.35

10

UNIFORM ROTATION

The decay of turbulence in a region in uniform (solid-body) rotation presents a

rather special problem to the turbulence modeler because in the Reynolds stress

equations the net production term (production plus fast-pressure-straln) vanishes

when the turbulence is isotropic. This is an artificial situation in the sense that

it is probably impossible physically to generate isot_opic turbulence in a fluid

rotating as a whole, but mathematically it is simply a matter of specifying isotropicinitial conditions. At the Reynolds-stress level it is not possible to determine

wh_ther initially isotropic turbulence will remai_ Isotropic. The experimental and

compatational results indicate that the turbulence does become slightly anlsotropic,

and this is consistent with the linear theory valid _.t low Rossby number.

At small Rossby numbers (q/_}L) the mean rotation (_) caoses plane wavesei_'_ to propagate with phase speed 2_.k/k, giving the Fourier modes

some physical "wave" significance (unlike most turbulent fields in which the wave

space is purely a mathematical convenience). The nonlinear t_rms can then be viewed

as direct wave interactions. The billnear convection term in the Navier-Stokes

equation is interpreted in wave space as the interaction between wave vectors k" and

k_" to give the convective contribution to the time derivative at wave vecto_ k'+k"

so that the time derivative at k is determined by the convolution sum over all--wav--e

vectors _', k" satisfying k'+k"--= _. Taking the y-axis as the axis of rotation, theFourier mode_ at low Rossby number have the form

This is a classical "two-time-scale" problem at low Rossby numbers; the"long-tlme" is qt/L where q and L are representative of the turbulence and the

"short-time" is _t. We are interested in the iong-tlme variation of the spectrum

tensor, and this requires that we integrate the wave interactions over clmes (T)much larger than the short-time scale but much smaller than the long-time scale. The

correlation over this intermediate time between _(k) and the contribution to its

tlme-derJvative due to the interaction between O(k') and _(k") is proportional to

sin _T

a_Twhere k2

For very small Rossby numbers, only interactions between waves k', k" havingII I|

k2/k ± k_/k" _+k2/k

//,

velocity components suffer the reduced energy flow, However the experimental data of

Wigeland and Nagib (1978) indicate that the energy is redistributed among the

velocity components with the ratio of energies reaching an equilibrium when the

transverse ma¢:roscale begins to grow at a faster rate than the longitudinal

macroscale.

We have simulated the flow at five rotation rates, but with a microscale

Reynolds number of only about 3, because of a program input error. As a result,

dissipation dominates the Reynolds-stress equations. The dissipation rate is

nevertheless raduced slightly (-25 percent at the highest notation rate) at early

times, as shown in figure 20. A structure parameter indicating component energy

anlsotropy is shown in figure 21; the longitudinal component v--Zgalns energy through

the slow-pressure-straln term. As the tables in appendix A indicate, the

contribution of the fast-pressure-straln term varies widely in magnitude and takes

both signs, indicating that it is oscillating. The slow-pressure term, on the other

hand, varies slowly and alwaTs works to enhance v-_/. The experimental data show

higher anlsotropy, but this could well be a result of the anlsotroplc _nltlal

conditions of the experiment or to the Reynolds number disparity.

In figure 22 the anisotropy at the dissipation scale is also shown to reach an

equilibrium, and the correlation with _t, rather than _t, indicates tha_ this is

not entirely a viscous effect.

TURBULENCE MODELS

The results of the numerical simulations can be used to aid the construction of

turbulence models at any level, from simple algebraic models of the Reynolds stresses

to spectral models such as the quasl-normal approximation. Appendix A contains

information at the Reynolds-stress level for a sample of the computed fields

discussed in the previous sections. In this section we illustrate how these data

might be used to test proposed models or to determine "model constants" for some of

the modeled terms in the Reynolds-stress equations.

THE REYNOLDS-STRESS EQUATIONS

When the turbulence field is homogeneous, the Reynolds-stress equations become

dd-{. (u-T_f) = -Ui,kUjU k - Uj,kUiU k + 2psij - 2vUi,kUj, k

production pressure- disslnatlonstrain

where the mean flow gradient Ui, j depend_ only on time, and slj is the turbulent

straln-rate tensor sij = (ui,j+uj,_)/2. The over-bar d_:notes an average over physical

space. Pressure Is decomposed into i_s so-called "fast" and "slow" parts given by

12

I̧ ,,

i

V2P (1) " -ul,juj,i

V2p(2) . _2Ui,ju j,|

(slow pressure)

(fast pressure)

The terms "fast" and "slow" refer to the fact that p(t) depends directly on mean flow

gradients and can therefore have its time scale Imposed directly by the mean strain

rate. The slow term, on the other hand, responds indirectly through changes in

turbulence structure. Fur homogeneous flows we have exactly

where

k_k I

- k

The sum indicated is over all Fourier modes. The fourth rank tensor _ has the

ohvlous symmetry

= [m= _mi

and the continuity condition, ki_ i = 0, implies

a mi = 0

In addition, the contraction a_If - 2ulu m provides a convenient scaling factor,

and as a factor of the mean deformation rate it emphasizes the close relation between

the production and fast-pressure-straln terms in the stress equations. In essence,

the production and fast-pressure terms sum to produce the "net" production. Tables

of the tensor _, shortened by use of its symmetry, are included for each recordedfield in appendix A, with the budget of the stress equations above.

In the Reynolds-stress equations only the production term can be computed

directly from the stress tensor. The dissipation and fast-pressure terms are

second-order velocity moments like the stress; however, they contain velocity

derivat Ires and thus depend on the spectral distribution of stress (the spectrum

tensor) and not merely o_ its average value. They must be modeled along with the

slow-pressure term, which is a thlrd-order velocity moment.

We consider first the slow-pressure-strain and dissipation terms, following the

analysis of Lumley and Newman (1977) who combine thfi slow-pressure-strain tensor with

13

the deviator of the dissipation tensor. Lumley writes

22P(1)slj - 2vUl,kUJ:k + _ _61J " -_'$1J

where _ is a traceless tensor and E is the dissipation rate of kinetic energy

•c = vui,jui, j

which must be modeled. The approximate.on (Rotta (1951)) that _ depends only on the

tensor _,

blj " q2 - _ 6iJ

and on scalar attributes of the flow, requires for proper tensor invariance

that _ be an isotropic function of _,

t II_lj)_ij = Bbij + _(bikbkJ +

The coefficients 8 and y are functions of scalar invariants of the field such as

Reynolds number, q4/v_, II, III,

where-211 = bijbji , 3111 = bljbjkbki

are the invariants of _ (the third independent invariant, bii is zero by the

definition of _). Lumley and Newman show that for hlgh Reynolds _umbers , y-_O, if

slightly anlsotropic flows are to return to isotropy.

We will retain for our purposes here only the linear term of the model. Some

information about the coefficient of the linear term 8 can be determined by

considering limiting cases. In the absence of a mean flow (for example the "return

to isotropy" following an applied strain), _ > 2 if the anlsotropy is to decay. For

any anlsotropy _2 as the Reynolds number vanishes since in that case the velocity

components all decay in proportion to their magnitudes and the anisotropy _ is

constant. Lumley and Newman fit the slight!y anisotropic data of Comte-Bellot and

Corrsin (1966) and find that, again, _-_2 as the anisotropy vanishes at high Reynolds

number in the absence of a mean flow. In later papers, Lumley (1978, 1979) shows

that in order to force the energy tonsor predicted by the model to be realizable

(have positive definite component energies in principle axes) 8+2 as a component

energy vanishes. Lumley found a function of the invarian_s

14

%

b'_"L

1 _

G(_I,HI) = l/g + 3III + II

that vanishes if and only if a component energy vanishes, and he suggests that this

function be explicitly included in the model to ensure reallzabillty. For example,

8 = 2 + _'_(ll,llI)

where _" is a function of scalar laver/ants.

The relaxation toward Isotropy of turbulence subjected to axisymmetrlc strain

has been simulated, zaktng as initial conditions fields at total strains 2 I/3 , 2, and

4 from the set of axisymmetrlc strain runs discussed earlier. In figure 23 _e show

the variation of _ and its pressure-straln and dissipation terms with the anlsotropy

_. In this flow only the normal stresses are significant, and two of them are nearly

equal, tht deviation being a result of tile fl;_Ite sample. The two families of polnts

in the pressure-straln plot correspond to the two values of viscosity in _he

simulat&ons, indicating the Reynolds-number dependence of the coefficient 8. This

Reynolds-number effect is more apparent in the dissipation and pressure-straln parts

than in their sum _, indicating the advantage of treating these terms together.

This is done by default when experi_ental data are used to determine model constants,

because the dissipation is assumed to be isotroplc in order to find the

pressure-strain (plus dissipation devlater) from the Reynolds-stress equations, The

model coefficient _ for each field was found by least-square fitting the

measured _ to the measured _. The fit is nearly perfect for each field. This is

exp._cted since _ and _ are _th diagonal, traceless, and axlsymmetrlc and thus can

be .ulated exactly by a single coefficient. The resulting coefficients for all of

the fields are plotted against a Reynolds number qq/9_ and G(II,III) in figures 24

and 25. Although the sca_ter in the coe_flclents is large, the trends are consistent

with Lumley's predictions. The Rotta model applied to the axlsymmetric strain runs,

during the straining period, gives essentially the same results.

In figure 25 we present the data from the plane-straln cases discussed earlier,

the groups of four points being associated with the four strain rates applied. The

single degree of freedom of the linear model does not fit the two degrees of freedom

of the data well, but theflt improves as the strain rate decreases. The nonlinear

model, having two degrees of freedom, would fit the data exactly.

The anlsotropy of the shear cases is plotted in figure 27. The fit achieved by

the Rotta model is shown in figure 28 as a plot of the measured _ for each field

against the _ predicted for that field by the linear model, using the measured

and the coefficient _ giving the least-square fit for that field. Although this

would appear to be a more difficult case than the two-dimensional strain, since four

elements of the stress tensor are active, the model seems to fit the data fairly

well. The coefficient grows with Reynolds number (fig. 29) as predicted by Lumley,

but its variation with C(II,llI) is uu_'r.tai _ (fig. 30).

The modeled terms for the unlfo[m rotation cases are presented in figure 31.

The dissipation anlsotropy is correlat_',_ with the stress anisotropy, but _he pressure

strain is negatively correlated wlch it. The pressure-straln term is evidently the

source of the anisotropy in this case, contrary to its usual role as the "return to

Isotropy" mechanism.

The performance of the R_tta model must be Judged by how well it fits the

elements of the tensor _ at each field and how well the coefficient for each field

15

k'.,_

r"

can be fitted to the scalar lnvariants of the field, With the exception of theplane-strain runs, which seem to be sensitive to strain rate, the linear model

provides a fairly good fit to the data with coefficients that vary (qualitatively)

with Reynolds number and anlsotropy level in accordance with Lumley*s predictions.

The results do suggest, as has Lumley, that the tlme-scale ratio between theturbulence and mean deformation be included in the list of invariants determining theRotta coefficient.

It would seem that modeling the tensor

a£j k2 _ _K

would be an easier task than modeling the slow-pressure term which is a third-order

velocity moment; however, this has not proved to be the case. The most widely used

model of the tensor _ is that of Launder, Reece, and Rodl (1975) (LRR hereafter)which has the form

mi

where C is a coefficient to be determined (again, a function of Invarlants) and Aand B are fourth-rank tensors of known form that depend linearly on th_

Reynolds-stress tensor, satisfy the symmetry required of _, and vanish when

contracted over lJ . The contraction over lJ gives

mi = 2Tin m Bml = 0Ajj , jj

When the turbulence is isotropic

= 2Euiu j " _ u_u£ dij -3- 6ij

in which case

mi= 0B£j

and

2E

- _mi61j -

which gives the exact result for an Isotroplc flow regardless of the value uf C .The model has certain failings, as do all models. Some of these have been pointed

16

V

_r

out by Lumley (1978), who shows that the model can not meet the reallzabilitycondition, and by Leslie (1980), who demonstrates that the model is too simple in

form to represent the general fourth-rank tensors _ that might occur. This latterargument is easily verified by noting that regardless of the coefficient C , themodel produces the following spurious symmetries (no summation implied)

akj = a i _ J # k

i£ . aJJaij Ji

i#jj£ lj

ajj " aii

The addition of nonlinear terms (higher-order products of the tensor) would remove

these spurious symmetries and introduce more coefficients, and as noted by Lumley,

might allow the model to meet the realizability constraint. The Justification for

nonlinear dependence on the stress-tensor, when a is linear (formally) in the

spectrum tensor, is that _ depends also on the distribution of energy over wave

space, a dependence lost when integrating the spectrum tensor to the stress tensor.

The idea that the anisotropy of the spectrum tensor with respect to k might be

expressible in terms of the anisotropy of the stress itself leads to the--possibility

of nonlinear dependence on the stress tensor, and, as suggested by Lumley, the

connection might be made using the theory of rapid-distortion. In view of thedifficulties associated with the LRR model, several authors (e.g. Leslie (1980)) have

suggested modeling the rapid-pressure-straln term, or even the total pressure-strain,

as a unit. This avoids the detailed constraints on the model imposed by the formal

definition of _, but the ability tc handle flows with large variations of imposedmean strain would be lost.

We compare the results of the LRR model with the computational data for the

shear runs in figure 32. The 36 different elements of the tensor _ are eachnormalized by q2. The constant for each field, found by least-square fit to the

data, is plotted against Reynolds number in figure 33, and against G(II,II) in figure

34. The model succeeds rather well in fitting the 36 different elements with a

single constant for each field, and the constant appears to be related to the

anisotropy level, if not to the Reynolds number.

The least-square procedure, which was used simply to illustrate how well the

models fit the data, indicates that there is room for improvement. The use of the

data to determine the coefficients of a postulated model can be done in a number of

ways. For example, rather than finding coefficients for the Rotta and LRR models

separately, they can be found simultaneously by fitting the sum of the individually

modeled terms. This is how experimental data are used for the shear flow (Leslie

(1980)). When this procedure is used on the computed results, model coefficients

very close to the "recommended values" are found.

The real value of the data of appendix A how_ver, is not that it allows model

coefficients to be determined in a more precise way than does experimental data, but

that it provide# the detail needed to determine the range of validity of a model and,

it is hoped, to suggest how the model might be modified to increase that range.

17

CONCLUSIONS

The large-scale direct simulation of homogeneous turbulence begun by Orszag and

Patterson a decade ago, has advanced wi_h increases in computer power to the point

where today it is possible to capture the statistics of the energetic scales of

motion at low Reynolds number for moderate imposed mean deformations. Further

advances in hardware and method will permit higher resolution in the future, but the

resolution available today is adequate to allow the extraction of information useful

to both model builders and "pure" theoreticlans.

The relevance of the simulation to real turbulence has been established, in this

author's opinion, to the point that the results can be used to fill in the gaps in

the experimental database, and it is hoped that the information included for this

purpose in appendix A will lead to closer scrutiny of the fundamental assumptions

implicit in turbulence models.

t8

b •/

iL,

L

i

.......... ----..mmm_Wmm_ -

APPENDIX A4

LISTING OF DATA FROM A VARIETY OF COMPUTED TURBULENCE FIELDS

The "raw data" of a variety of computed turbulence Fields is presented here in

tabular form. Fields from the following (deform!_g mean) runs are included.

Run Deformation Rate Viscosity-- (T--=_) (L2T -I )

B2DI (Plane strain) 4.5 .01/_

B2D2 _(d_/dy)t 9 .01/4

B2D3 d_/dz J 18 .01/_B2D4 36 .01/_

Initial Condition

(Developed

isotropie

field)

CDII

CDI2

CDI3

CDI4

CD21

CD22

CD23

CD24

(Axisymmetriccontraction)

d_/dx

-2(d_/dy)

-2(dw/dz)

I0 .01/_ (Developed

20 .01/_ isotropic

40 .01//2 field)5 .ol/_I0 .02/_ (Developed

20 .02//2 isotroplc

40 .02//2 field)

5 .02/_

BSH9 (Shear) 20/2 .01//2 (Undeveloped

BSHI0 20,'_ .02/,_ isotropic field,

BSHll d_/dy 40_ .02//2 square spectrum)

BSHI2 20 .005

BRI (Rotation) 2.5 .01/_'2 (DevelopedBR2 5 .Ol/r_ isotroplc

BR3 d_/dz I tO .01/_72 field)

BR4 -(dO/dx) I 20 .01/_BR6 40 .o1/,:_

Fields from these runs are named using the run name above followed by a letter

designating the individual field. The plane strain fields are given at nominal

strains of ,_, 2, 2/2, and 4, with suffixes A, B, C, and D, respectively. The

axisymmetric-strain fields are given for nominal strains of 3_, 2, and 4, with

suffixes A, C, and F, respectively. For the shear runs, the suffixes are notconsistent from run to run, but for all runs the fields are given for st = 2, 4, 6,

g... The rotation fields sre taken more or less evenly in time and are labeled

sequentially A, B_ C, D, E, and F for each run.

Ruus named RX... are relaxation-to-isotropy cases for each of which two fields

are glve_; for example, RX24A2 and RX24A4 are fields from the run using as inltislconditions the turbulence field CD24A but setting the strain rate to zero. Thus the

data of field CD24A with net production set to zero provide a third field of the

relaxation run.

19

. ¢. ,... ,,t

I_

.Ill.:

i

k'

t

The following data are presented for each field:

T = time

VISC = v

RII - ulu-----__ q2

PII " 2Ui,ju--_ (Net production of RII)

DII = 2e= 2vui,jui, j (Net dissipation of RII)

_Kinematic viscosity)

,(Velocity varianue =

2*kinetic energy)

with units [T]

[L2T -I ]

[L2T-2]

II = bijbJi (Invarlant of anisotropy tensor)

III = biJb_kbki (Invariant of anisotropy tensor)

MEAN DEFORMATION DU/DX etc. [T-l ]

INTEGRAL SCALES

Lll,l = Rll(r,O,O)dr [L]

MICROSCALES

Xll " (_u/_x) 2[L]

(I.]

(Note the absence of 2's in these length definitions.)

REYNOLDS-STRESS BUDGET

IJ = lJ

RIJ/RII " uiuj_k

I d (u-_)RArE = 2_ dt

-iPROD = _ (Ui,kUjU k "" II3,k_).uk )

P2S = _I (,)sij¢ P

PIS = i_.p(1)si j

DISS = -- ui _u_,J k

Stress component

Component energy ratio

Net time rate

Production

Fast-pressure-strain

Slow-pressure-strain

Dissipation

20

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