The turbulence equations are closed by specification of initial conditions (using either a Taylor or an exponential series) and by a modified Kovasznay-type closure. Good results for large times are obtained only for the initial-conditions closure used with four or more terms of an exponential series. The evolution of all of the initially-specified spectra can be calculated rather well from the theory. From a fundamental standpoint the method thus seems to be satisfactory

Deissler, R. G.

English

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https://ntrs.nasa.gov/search.jsp?R=19790004190 2020-03-22T00:45:13+00:00Z
NASA Technical Memoi -tndum 79011
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rtCAY OF HOMOGENEOUS TURBULENCE
FROM A GWEN STATE AT HIGHER
REYNOLDS NUMBER
R. G. Deissler
Lewis Research Center
Cleveland, Ohio
d
. ^	
6 Ck L^'
TECHNICAL PAPER to be presented at the
'Ili rty-first Annual Meeting of the American Physical Society
t" ,.'-ision of Fluid Dynamics
Los Angeles, California, November 19-21, 1978
DECAY OF HOMOGENEOUS TURBULENCE FUOhl A GIVEN
STATE, AT HILAIVI? IrVYNOLDS NUMBER
by H. G. Heisler'
Lewis Research Center
National Aeronautics and Space Administration
Cleveland, 01 ► iu 44135
A BST PACT
I'he turbulence ecluat ► o^rls are closed h}' specification of initial c 'ondi-
lions (usil`16 either' a 'Taylor ur an exponential series) and by a modified
co
rn
^—	 K0 vatiZlla l'-t\'l)e CIOSUIV	 Good I'esellts te l l' a rge times are obtained only
rn
W	 for the initial-conditions closure used with forr or more reruns of an
exponential series
	
The evolution of all of the initially-specified s1wetra
tarn be calculated rather well from the theory. Front a fundamental
standpoint the method thus seems to hr satisfactory.
1. IN'1 PODUCTION
From a practical standpoint it would be advantageous to be able to
calculate llle eVOILItioll of the turbulent ent'1'gy spectrum by slx'cifying
otllN- that (Itlantit}• initially. Unl'011 lnately, hocause of the counting be-
tweell the members of rue infinite hierarchy of multipoint correlation or
spectl • al egll;tIiOIIS, it appeal's that we at , ( , not able to do that, so that a
satisfactory theory would seem to require thr initial specification of It
uuII11W1' of il:tt'ractiIIg quantilieS. The Ilrt'diction of the e% • olutioll of the
i
ent,vo, spectrum in fact r('cluires tilt' sped :cation of an i ►lfinito number
of initial luultillolnt col-1 el: nwis or Spectra (or functiollalS of these (Itlan-
titit'S). 1 Those qu:lutitieS, tol;etlu'r will the c• ol • rclatiun or Spectral eclua-
tions, can he used to calculate the initial time derivatives of the col • rchl-
tions or spectra. The evolution of the tui-hulenc•e can then be obtainctl by
uSillg al ► l #'xl)ollential tine SerieS Which is all iteratlt't' Solution of the Navier-
Stokes equations. When the problem is formul:lted in this wad• , the correla-
tion or spectral equ;t(iollS are Closed by the initial specification of the tur-
blllellee and no ;1S S UnlI)lion is nece ssary for closing those equations.
Of C!)clrse i1) 111'11l'tiCe We CM1 S pet'ify 0111y ai fillitt' 1111111NA' Of lilt' 1l)^^'(`I'-
ol'cter qualllities. This has been called the ,,al)prol)leu1. "Z It is the
l)rohlem of bridl;itil ,  the hall helween the infinite number of correlations
Whic11 WMIld 111e01 • etitally be necessary for taltulatin" the eVoltltion of
the turb:lle ►lce, and a finite specifiable nut hci^	 of correlations. Most
uorket • s have atteml)1ed to bride the :;all by alsSUmin:; that tile initial
diStl'll ► utio ll 01' tllt • 1ll IC1 1.l fluctuations is VX.lctl\' UatlSsiarl (zero odd-order.
Col't'elations). However, that is an artificial initial conditimi, prohably
never realized for Peal lul • bulence. The llllimi , tance of accurate initial
conditions is shown, for instance, by the data of Ling and Saad, 3 NvIIvre
IlIC!I.Slll't`Illt'1)tS Ni'l'e Illade doWll5tl'ealn ll'0111 :1 tul'hlllellce- 1)rc)d tltlilg
3watcrfalI - The turbulence decay law for the initial conditions produced
by the walerlall is c•onsiderahly different from that for initial conditions
produced by a grid. As will be seen later, the skewness factors foi- the
velocity gradient are also different.
Here, as ill 1, we bridge the gap simply by specifying; a suffi-
cicllt numhor of initial corl-clations or their spectral cquivalcnts to cal-
culate the et • olutioll accurately. Fort1111a1e1y, we du not ha% a to specify
the IMIltil)(Jint correlations or their sl)cctral egllivalents themselves
(tllose NVOUld be extremely difficult to p leasure) but only two-point func-	 a
tionalS of the nl Lill ilxlillt Spect ► • al qu:lntitieS. I It will be seen that the
CV0111tioll of :111 the qu:ultities which arc specified initially call 	 cal-
c111: ► ted. If, on the otller halld, a large number Of initial conditions were
specified in order to predict the eVolutie)n (11* say aru' quantity, it alight
he objected that the illitial conditions were chosen to make the theory
agree with experiment for that One gllanlity. however, that objection
cannot be made if the CVOIUtion of all of the quantities which are slx'cified
initially (-all be e:llelllated, as ill OIC I)ITSelll theory.	 FrOnl :1 flllldaIl)ental
standpoint the calculation of the evolution of those gllalltitieS is all that
might he exI)ected fr0111 a theol-y of evolVing tlll•I)Ulellce.
11. ANALYSIS
Ill Refs. 1 and 4 we gave the basic theory for closure by specification
of initial conditions and calculated reSU1(S for low and moderate tl1l'l)llle'11Ce
Reynolds nunlhel • S (PA between 3 and 70, wlicre V A is the Reynolds num-
ber based oil 	 'Taylor microscale and the root-mean-square' Ull'hlllellt
telocity fluctllatiun). lierc wt' comimi-c calculated results With tilt' higher
Rec p o ldti nu ►l+,be1 data of Ling<<)ld 5:1ac1. 3 The Reynolds 11e11111)el• s fur those
L
Bala \\'ere high enough to obtain a -5.1-1xnrel • region ill the energy spvc-
trum (R ` between 300 and 800).
The exponential-series exilression for the erlerg tiIN'(trtllll function
1' was obtained in ltel'. 1 as
.	 E(h, t)
	
B (h)vxp 21'h(t - t 1 ) +	 112(1)exll(- 2bs (h)(t - t 1 )],	 (1)
L	 t_
8
\0wre h is the wave numbel', t is tilt , (stilt,, t I is tilt , intial tllllt', and 1,
is the kinematic viscosity. Equation (1) ^Yives the evolution in tulle of the
energy spectrum from all initial state %%hich is spectfied by the B's and
Ws in the equation. The first terns is tilt ,
 well-krlo\\u ('xl)l'ession for the
decal' of E ill the final period (\kt , .tk (urkilciwe approximation). The
rest of the (crisis in Eq. (1) therefore give the contribution of inertia to E.
In lilt` presem nott` we will 1'eta111 ;t Illaxlllllll11 Of hAll' OxIT(1llelltlal (('1'1115
in Eq. (1). This is otte more term than it was necessary to retails for the
hm . and nlodcl ate l evilolds (hiller Gala cons i(tered in Refs. 1 and 4.
With four terms retained, Nve will 11,1ve to speclt y seven spectra at t 1 in
order to VVAL41te tilt' ttlnctiolls 13, 13 19 11_1 . 13 3 , and b 1 , by . and b,;.
EvidenllY we need Vlore spectra to descrihe the initial turbulence at higher
lleynolds numbers because a \ idcr r.inge of eddy sizes is excited, and the
llll't l lllellCe sll'l1('tlll'e is 11101'0 complicated 111:111 at l0\V01' lie\11(llds Illllllbel's.
The sl)ectt'u111 Cunetiolls, in addition to E(h, t), Which we will stweify itli-
tially are '1'(h, t), VG, t), h(h, t), S(h, (), L(h, t), and N1(h, t), which
are given b\.
T = OE/ O t + 2vt,2E,	 (2)
v = (''r/t't + 2vt, T,	 (3)
R = (AWN + 2vi\2V,	 (•l)
IL
5S = ('R/(It _t" 2vh2 1t,	 (5)
L	 ('S/' ('t + 2vi, 2 S,	 (G)
and
M = M/ (It + 2v1, 2 L. 	 (7)
The quantity T is the well-known energy-transfer tunc'tion, and V, R,
S, L, and AI are, respectively, two-point functionals of three-, four-,
five-, six-, and seven-point sIvetral quantttit's. I The latter five quan-
tities are somewhat similar to T, inasmuch as they contain the effects
of transfer between wave iminhers of T. V, R, S, or L. However they
differ, in that they also coat:;in ether effects, so that the areas under
these silectra are a not necessarily zero. as ill 	 case of T. Equations (2)
through (7) are obtained, respectively, front two-, th ree -, and four-, five-,
six-. anti seven-point spectral equations. I They Ci n be thought of as two-
point alternatives to the nw it ipoint spectral equations and are much easier
to Work with.
For comparing the theory with the experiment of I ing and Saad, 3 the
experimental input call be conveniently obtained from all 	 equa-
tion for E, Eq. (8) in their paper. The higher-order spectra were not
111011sured directly in (hair VXperiment. but could be calculated from their
equation for E and Eqs. (2) through (7). Except for experimental error
those values will be the same as those that might have been measured
directly. The B's and lt's in Eq. (1) can be related to the initial sprc-
tra by successive differenliatiotns of that equation with respect to time,
setting t = t l , and using Eqs. (2) through (7).
The use of Ling and Saad's Eq. (8) for obtaining the initial conditions
might be taken as ar indication that it is necessary to know the decay
flaw to predict it. However that is not the case. Thei ► • equation was
used because the initial spectra could not be measured directly in their
exllc I r1111ent. Moreover we do not need to know E for the whole decay
Period, but only at V11OL1 01 1 early times to calculate the required initial
six, ctra, the latter being a Much smaller amount of information. At am,
rate the nature of the problem seems to be such that we do not have a
reasonable alternative to the procedure followed here.
111. RESULTS AND DISCUSSION
Before giving I'eslllts obtained from Eq. (1), we will consider a
Taylor series with a maximun g
 of seven initial spectra (the same maxi-
11111111 number that will be used with Eq. (1)), and a modified Kovasznay-
type closure (mod ified to include all effect of initial T). 5 Results for
those calculations are given in F g. 1, where the initial state is speci-
fied at t^ = 0. 0048. The quantity u i is a velocity component, the over-
bar indicates an averaged value, and the stars indicate that quantities
have been nonding ensionalized by the kinematic viscosity and an experi-
mental constant. A (the proportionality constant in the power decay law
for u) having the dimensions klength) 2 (time)1.3^ 3 The turbulent
energy (1/2)u iu i is obtained by integrating E over all wa% , e numbers.
Fir,'11re 1 indicates that the Taylor-series results agree with experiment
only for small tin g es. Evidently, many more terms (and initial spectra)
would be required in order to obtain accurate results for u iu i at large
times. Note that even if enough te1 •gus were retained in the Taylor series
to give accurate results for u i u i , the decay of the higher-order spectra
which would then have to be specified initially could not be accurately
calc1.110led. 'Thus the use of a Taylor series in the present problem does
I
IL
noL give a satisfactory solution, regardless of the number of terms re-
tained.
"The modified Kovasznay-type closure is in somewhat better agree-
nient with experiment than the Taylor series, but at large times the agree-
ment is still not good. This is in contrast to its good agreement at moder-
ate and low Reynolds numbers. 5 It was introduced to Ref. 5 in an effort
to reduce the required number of initial spectra. Evidently that effort is
successful only for moderate and low Reynolds numbers. It is possible,
of course, that a more sophisticated nicth„d might be more successful
(e.g. see Ref. 6).
A comparison between theory and experiment using the exponential
series (Eq. (1)) is given in Fitts. 2 to 8, where the initi ,.:l state is attain
specified at ti = 0.0048. As in Refs. 1 and 4. unphysical singularities
occasionally occurred in the theoretical spectra. Inasmuch as the un-
physical values were localized, particularly in the higher-order approxi-
mations, they were simply omitted in plotting the curves.
Figure 2 gives a comparison between theory and experiment for the
decay of turbulent energy. Theoretical curves are shown for one, two,
three, and four exponential terms retained in Eq. (1). The curve for
four terms is in good agreement with experiment for the whole decay
period. Comparison of the curve in Fig. 1 for one term retained (weak
turbulence approximation) with the experimental curve shows the effect
of inertia on the decay process. In contrast to the results for moderate
Reynolds number, 1, 4 where inertia and viscous effects were of the
sanie order of magnitude, the inertia effe ^ts for the present higher
Reynolds number results are at least an order of magnitude greater than
I
.r• .... r. ........ 	 • ..	 ... .... • ... y.
	
•...u,._... -	 - ^,..	 ._	 ..._	 .fl
L W
8the viscous effects. "Thus roost of the decay at high Reynolds nuntlVrs
is due to inertial self-tnter'i.tion of the turbulence, rather than to viscous
effects. Figures 3 and 4 shoA , how the energy and the transfer spectra
decay with time.
Figures 5 and G compare theory and experiment at a late time for
all of the spectra which are initially specified to describe the initial tur-
bulence. The prediction of the decay of all seven of the spectra which
are specified initially is rather Rood. 'Thus tilt' present theory appears
to solve an initial-Value probl em for higher Reynolds number turbulence
in which the decay of seven initially-specified spectra is predicted.
Altliough the initial dissipation spectrum K 2 E is not specified in-
dependently, because of its irupurtance in turbulence theory it is com-
pared with experiment and with the energy spectrum at a late time in
Fit;. 7. Attain, good agreement is indicated. The separation of the
energy and dissipation spectra is good evidence that we are dealing with
High Reynolds number turbulence.
Another imix)rtant quantity is the skewness factor of the velocity
gradient. That quantity call
	 calculated from the spectra of Z and T.7
The plot in Fig. 8 indicates excellent agreement between theory and ex-
periment for the skewness factor. The difference between the trend in
Fig. 8 and that ill 	 turbulence8
 is probably due to the differ-
ence between initial conditions for grid-generated tUr'bulence and the
waterfall-generated turbulence considered liere. 3
ACKNOWLEDGMENT
I should like to acknowledge the work of Frank Molls ill
	 out
the methods used for obtaining the numerical results given here.
9REFERENCES
1 R. G. Deissler, Phys. Fluids 14, 1629 (1971).
2 P. Graff, Z. Angew. Math.	 Phys. 20.	 461 (1969).
3 S. C. Ling and A. Saad, Phys. Fluids 20,	 1796 (1977).
4 R. G. Deissler,
	
Phys.	 Fluids .5, 1353 (1972).
5 R. G. Deissler,	 Phys.	 Fluids 17, 652 (1974).
6H. D. }logge and W. C. Meecham, J. Fluid Mech. 85, 325 (1978).
7C. C. Lin and W. H. Reid, liandbuch Der Physik, edited by S. Flubbe,
and C. 'I'ruesdel (Springer-Verlag, Berlin, 1963), Vol. VIII/2, Fluid
Dynamics II, P. 484.
8 M. S. Ubvroi, Phys. Fluids 6, 1048 (191"3).
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THEORY: FOUR TERMS IN EQ. Ili
--- EXPERIMENT
8x
,— 2E'
0	 20	 40	 60	 80 100 110 140 160 ISO
K'
Figure 1. - Comparison of theoretical energy dlsslpanon
spectrum at a late time u ` - 0. nl I with experiment of
reference 3 and with energy spectra.
THEORY: FOUR EXPONENTIAL TERMS IN EQ. Ill
- -- EXPERIMENT
Figure 8. - Comparison of ewlution of theo-
retical skewness tactnr of velocity gradient
with experiment of reference 3.


Decay of homogeneous turbulence from a given state at higher Reynolds number

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