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Explict-scales projections of the partitioned non-linear term in direct numerical simulatio

a r X i v :p h y s i c s /9806029v 1 [p h y s i c s .f l u -d y n ] 19 J u n 1998

EXPLICIT-SCALES PROJECTIONS OF THE PARTITIONED NON-LINEAR TERM IN DIRECT NUMERICAL SIMULATION

OF THE NA VIER-STOKES EQUATION

David McComb and Alistair Young Department of Physics and Astronomy

University of Edinburgh James Clerk Maxwell Building

May?eld Road Edinburgh EH93JZ United Kingdom

ABSTRACT

In this paper we consider the properties of the inter-nal partitions of the nonlinear term,obtained when a ?lter with a sharp cuto?is introduced in wavenumber space.We see what appears to be some degree of independence of the choice of the position of the cuto?wavenumber for both instantaneous and time-integrated partitioned nonlineari-ties.We also investigate the basic idea of an eddy-viscosity model for subgrid terms and have found that while phase modelling will be very poor,amplitude modelling can be far more successful.

INTRODUCTION

As is well known,full numerical simulation of any sig-ni?cant turbulent ?ow lies far beyond the scope of cur-rent computational resources,the main problem being the large number of degrees of freedom involved in the prob-lem.As these degrees of freedom may be represented by,for instance,the number of independently excited modes in wavenumber space,the problem becomes one of eliminat-ing modes,in some statistical sense,in order to bring the reduced number of degrees of freedom within the capacity of current (or even future)computers.One such way by which we may systematically obtain such a reduction in the number of modes is by the use of a Renormalization Group (RG)calculation.A general account of the background to this work has been given in the review by McComb (1995).In this study,we are undertaking direct numerical sim-ulations (DNS)of homogeneous,isotropic,incompressible turbulence in a box with periodic boundary conditions,in order to assess the underlying feasibility of using RG to reduce the size of the computational problem.We have already reported some results on the use of conditional av-erages (McComb et al.1997,Machiels 1997)as previously formulated by McComb et al (1992)and McComb and Watt (1992).In the present paper we concentrate on the Hilbert space partitions of the nonlinear terms and their ?ltered projections in order to assess the appropriateness of the ‘eddy viscosity’concept.Results of this study should have direct relevance to large eddy simulations (LES)in general,as well as to RG.

THE PARTITIONED NONLINEAR TERM

Consider the forced Navier-Stokes equation for station-ary turbulence,

|k |2

(3)

and f (k ,t )is a forcing term used to achieve stationarity.We may rewrite equation (1)in a highly symbolic form as

L 0u =Muu +f.

(4)

The nonlinear term (Muu in our shorthand notation)may be partitioned by introducing a cuto?at k =k 1and de?ning u ?and u +such that u α(k ,t )=u ?α(k ,t )for

α(k ,t )for k 1

ε=

∞

νk 2

E (k )dk ?

k 0

νk 2E (k )dk

(5)

where εis the dissipation rate.

Equation (4)can now be expanded to give

L 0u =ψ??+ψ?++ψ+++f (6)where the partitions are de?ned by

ψ??=Mu ?u ?(7)ψ?+=2Mu ?u +(8)ψ

Mu +u

(9)

110

100

10?410?2100

05001000150005

0.00.51.0(a)

(b)

(c)

Figure 1:Simulation output.

(a)skewness ()and dissipation rate

(

)and integral

scale Reynolds number (

);

t =1t e (

);t =4t e (

);t =8t e (

Stationarity is obtained by use of a deterministic forcing term given by,

f α(k ,t )=

ε0u α(k ,t )/[2E f (t )]if 0

0otherwise,

(17)

where ε0is the desired mean dissipation rate (supplied as an input parameter to the simulation),and

E f (t )=

k f

E (k,t )d k (18)

with E (k,t )de?ned as the energy spectrum.k f is chosen to be 1.5so that the forcing is applied to only the ?rst shell of wavenumbers.With this forcing,we have observed over many simulations that after a su?cient number of time steps the velocity ?eld reaches a statistically stationary form,as desired.

Computing the φ-?elds

|a (k )

1/2 |b (k )|2 1/2

(20)

0.0

0.5

1.0 1.5

2.0

k/k 1

?0.5

0.00.5

1.0

R (k )

Figure 3:Integrated partitions.

R (φ,φ??;k )with k 1=24.5();R (φ,φ?+;k )with k 1=24.5();R (φ,φ++;k )with k 1=24.5(

).The dot-dashed line indicates k =k 1.

0.00.5

1.0 1.5

2.0

k/k 1

?10

10?8

10?6

10?4

100

102

r (k )

Figure 4:Integrated partitions.

r (φ,φ??;k )with k 1=24.5();r (φ,φ?+;k )with k 1=24.5();r (φ,φ++;k )with k 1=24.5(

).The dot-dashed line indicates k =k 1.

and plot R (u,φ;k )in Figure 2for six di?erent time steps.We see that by the ?nal time step,the level of correlation is excellent for k >5and good for k >1.The increasing quality of correlation with increasing k is to be expected as a consequence of the fact that higher wavenumbers evolve at a greater rate than lower wavenumbers —something which is borne out by looking at the correlations computed at earlier time steps.The deviation in the ?rst shell is to be expected as this is outside the valid range of equation (19).

Finally we note that in order to compute φ-?elds for di?erent cuto?wavenumbers,k 1,we must reperform the entire DNS from initial conditions up to the fully evolved state.

Experimental Details

)and k 1=16.5()and k 1=16.5()and k 1=16.5(

The results given in this section correspond to our 643simulation.We begin by presenting results for the φ-?elds with cuto?wavenumbers k 1=16.5and k 1=24.5.Throughout this work,the cuto?wavenumbers are chosen to be half-integers so that they lie between two distinct shells.For each data set,we compute the correlation be-tween φand each of its partitions using equation (20)and also a measure of their relative magnitudes,r (k ),given by

r (a,b ;k )=

|b (k )|2

10?1

k/k 1

?0.250.00

0.25

0.500.751.00R (k )

Figure 7:Low Reynolds number (R λ≈70)

R (ψ,ψ??

;k )for cuto?wavenumbers k 1=4.5,8.5,12.5,16.5,20.5,24.5,28.5with k 0=32.The dot-dashed line indicates k =k 1.

1010

k/k 1

?0.250.00

0.25

0.500.751.00R (k )

Figure 8:Low Reynolds number (R λ≈70)

R (ψ,ψ?+

;k )for cuto?wavenumbers k 1=4.5,8.5,12.5,16.5,20.5,24.5,28.5with k 0=32.The dot-dashed line indicates k =k 1.

10?1

k/k 1

?0.250.00

0.25

0.500.751.00R (k )

Figure 9:Low Reynolds number (R λ≈70)

R (ψ,ψ++

;k )for cuto?wavenumbers k 1=4.5,8.5,12.5,16.5,20.5,24.5,28.5with k 0=32.The dot-dashed line indicates k =k 1.

k/k 1

?0.250.00

0.25

0.50

0.75

1.00

R (k )

Figure 10:High Reynolds number (R λ≈190)

R (ψ,ψ??;k )for cuto?wavenumbers k 1=16.5,32.5,48.5,64.5,80.5,96.5,112.5with k 0=128.The dot-dashed line indicates k =k 1.

k/k 1

?0.250.00

0.25

0.50

0.75

1.00

R (k )

Figure 11:High Reynolds number (R λ≈190)

R (ψ,ψ?+;k )for cuto?wavenumbers k 1=16.5,32.5,48.5,64.5,80.5,96.5,112.5with k 0=128.The dot-dashed line indicates k =k 1.

k/k 1

?0.250.00

0.25

0.50

0.75

1.00

R (k )

Figure 12:High Reynolds number (R λ≈190)

R (ψ,ψ++;k )for cuto?wavenumbers k 1=16.5,32.5,48.5,64.5,80.5,96.5,112.5with k 0=128.The dot-dashed line indicates k =k 1.

behaviour observed for k/k 1<0.5in Figure 3is an e?ect caused by the

presence of

the

forcing term.

The general picture which seems to emerge from both correlation and magnitude information,is that for k k 1,it is φ?+which is dominant while φ++is broadly insigni?cant for all values of k .

It was at this point in our work that it became appar-ent that the computational cost involved in calculating the φ-?elds was too high and so attention was turned to the ψ-?elds which are far easier to calculate.For these,a sin-gle velocity ?eld realization is enough to calculate ψand its partitions for any cuto?wavenumber,k 1.We begin by duplicating the calculations performed on the φ-?elds for the same resolution grid,and for the same cuto?wavenum-bers.Results are shown in Figures 5and 6where we see a very similar picture to that presented in Figures 3and 4.Taking advantage of the reduction in computational ef-fort,we now compute R (ψ,ψ??;k )for a number of di?er-ent cuto?wavenumbers.The results are shown in Figure 7.We see ?rst of all that there is an excellent collapse of data for all cuto?s considered and that there is good correlation between ψand ψ??below the cuto?.Above the cuto?,this correlation decays rapidly away to zero.We also note that,mathematically,ψ??(k )=0for k >2k 1so that the occurrence of a non-zero correlation in this region points to the existence of small numerical and aliasing errors.This picture is reversed when we consider R (ψ,ψ?+;k )as shown in Figure 8.The collapse of data is not so good,particularly in the low-wavenumber region,and we see that as k/k 1increases,the correlation begins to tail away from unity.

A pattern is even more di?cult to discern when we com-pute R (ψ,ψ++;k ),shown in Figure 9.We see that overall,ψ++does not correlate well with ψ,although the level of correlation increases with k/k 1as we move beyond the cut-o?.

Moderate Reynolds Number

We begin by rewriting our Navier-Stokes equation (6)for

the low-wavenumber modes as

L 0u <=ψ??

<+ψ(??)

+f <(22)

where the subscript ‘<’indicates that we are only con-cerned with k

ψ(??)=ψ?++ψ++.

(23)

In a large eddy simulation,wavenumbers k >k 1will not

be available and so we introduce some model for ψ(??)

which we will denote ?ψ(??)<.A standard form for ?ψ(??)

an eddy-viscosity model,whereby

?ψ(??)<=?δν(k )k 2u <

(24)

for some viscosity increment,δν(k ).

We now consider a hypothetical large eddy simulation,based around the idea of our wavenumber cuto?s intro-duced in previous sections.

0.0

0.20.4

0.60.8 1.0

k/k 1

?1.0

?0.5

0.0

0.5

1.0

R (k )

Figure 13:R (ψ(??)<,?ψ(??)

<;k )for cuto?wavenumbers k 1=

16.5(

),48.5,64.5,80.5,96.5,112.5(

)and T (??)(k )(

E <+2νk 2E <=T ??

<+T (??)

+W <(25)

where T ??

couplings while T (??)

k/k 1

?1.0

0.01.02.03.04.05.0

6.07.08.0δν(k )/ν

Figure 15:δν(k )for cuto?wavenumbers k 1=

16.5,32.5,48.5,64.5,80.5,96.5,112.5with k 0=128.

ergy transfer functions are plotted in Figure 14.We see that T ??has a large negative value in the ?rst shell (note

that the y -axis has been truncated for this graph)corre-

sponding to transfer of energy away from the energy input (forcing).We also see that T ??is piling up energy at the

cuto?and that this is balanced by T (??)which carries it to the higher wavenumbers.

Introducing an eddy-viscosity model as de?ned in equa-tion (24)will give us an energy balance equation for LES,

2k 2E <

.(27)

For a large eddy simulation,this would ordinarily have to be estimated by use of some model,as the whole point of LES is the absence of the high-wavenumber modes neces-sary for the calculation of T (??)

<.However,with DNS data we can calculate this,and the results are plotted in Figure 15.

The general form of these eddy-viscosities appears to be in good agreement with the form obtained theoretically by,for example,Kraichnan (1976)and we know from previous work (see,for example,Lesieur and Rogallo,1989)and our own LES experiments that this particular model provides good results.We must now ask why this is,when it is in apparent contradiction with the results presented in Figure 13.

Separating out Phase and Amplitude E?ects

1We

note that,strictly,these are not PDFs as our data set in

this instance is insu?ciently large —we would expect the true PDF of the phase components to be ?at,for example.What we have instead is a measure of the distribution of that data which we do have.

1010

k/k 1

?0.250.00

0.25

0.500.751.00R (k )

Figure 16:Phase-correlation.

R (ψθ,ψ??

θ;k )

for cuto?wavenumbers k 1=16.5,32.5,48.5,

64.5,80.5,96.5,112.5with k 0=128.The dot-dashed line indicates k =k 1.

1010

k/k 1

?0.250.00

0.25

0.500.751.00R (k )

Figure 17:Phase-correlation.

R (ψθ,ψ?+

θ;k )for cuto?wavenumbers k 1=16.5,32.5,48.5,64.5,80.5,96.5,112.5with k 0=128.The dot-dashed line indicates k =k 1.

1010

k/k 1

?0.250.00

0.25

0.500.751.00R (k )

Figure 18:Phase-correlation.

R (ψθ,ψ++

θ;k )

for cuto?wavenumbers k 1=16.5,32.5,48.5,

64.5,80.5,96.5,112.5with k 0=128.The dot-dashed line indicates k =k 1.

k/k 1

?0.250.00

0.25

0.50

0.75

1.00

R (k )

Figure 19:Amplitude-correlation.

R (ψr ,ψ??

r ;k )for cuto?wavenumbers k 1=16.5,32.5,48.5,64.5,80.5,96.5,112.5with k 0=128.The dot-dashed line indicates k =k 1.

k/k 1

?0.250.00

0.25

0.50

0.75

1.00

R (k )

Figure 20:Amplitude-correlation.

R (ψr ,ψ?+

r ;k )for cuto?wavenumbers k 1=16.5,32.5,48.5,64.5,80.5,96.5,112.5with k 0=128.The dot-dashed line indicates k =k 1.

k/k 1

?0.250.00

0.25

0.50

0.75

1.00

R (k )

Figure 21:Amplitude-correlation.

R (ψr ,ψ++

r ;k )for cuto?wavenumbers k 1=16.5,32.5,48.5,64.5,80.5,96.5,112.5with k 0=128.The dot-dashed line indicates k =k 1.

1010

ψr

0.00.1

0.2

0.3

0.4

p d f

Figure 22:PDF for ψr (

);ψ?+r (

)at k =32with k 1=64.5.

?1.0

?0.5

0.00.5 1.0

ψθ/π

0.004

0.006

0.008

0.010

0.012

p d f

Figure 23:PDF for ψθ(

);ψ?+θ(

)at k =32with k 1=64.5.

0.0

0.20.4

0.60.8 1.0

k/k 1

?1.0

?0.5

0.0

0.5

1.0

R (k )

Figure 24:Phase-correlation.

R ({ψ(??)

<}θ,{?ψ(??)<}θ;k )for cuto?wavenumbers k 1=16.5(

),48.5,64.5,80.5,96.5,112.5(

),32.5(

)with

k 0=128.

computers of the Edinburgh Parallel Computing Centre.The research of A.Young is supported by the Engineering and Physical Sciences Research Council.

REFERENCES

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