fulltext

Adiabatic isotropic turbulence in compressible fluids

V. A. PAIS(*)

Laser-Matter Interaction Laboratory, ENEA-Centro Ricerche Frascati P.O. Box 65, 00044 Frascati, Italy

(ricevuto l’1 Aprile 1997; approvato l’8 Luglio 1997)

Summary. — We present the preliminary results about the effects of compressibility

(modified through the isoentropic parameter g) on the main quantities characterizing the isotropic turbulence in ideal gases. The numerical simulations performed for this study were done by using spectral methods, in an adiabatic regime and maintaining the kinetic energy at a fixed value. In the range of values for g , Mach and Reynolds numbers tested here, no significant differences have been found in the spectra of the kinetic and internal energies and in other statistical quantities. Once the stationary regime is reached, an “inertial subrange” k(25O3)_{attempts to appear for low wave}

numbers.

PACS 47.27.Gs – Isotropic turbulence; homogeneous turbulence.

1. – Introduction

The presence of turbulent phenomena in most of the common fluid processes (airplane flight, air pollution, river erosion, etc.) maintains permanently alive the interest of researchers in this field of Physics, even for the simplest case of isotropic turbulence in incompressible Newtonian fluids (IIT). Indeed, IIT has been successfully studied for a long time and many books and papers have already been published [1-21]. Several theories and direct numerical simulations have helped to understand fundamental aspects of incompressible isotropic turbulence; yet a recent theory based on the Renormalization Group technique [22] is expected to give important contributions.

Isotropic turbulence in compressible fluids (CIT), i.e. flows with Mach numbers comparable to or greater than unity, appears as an even more attractive field since it is related with relevant aspects of combustion, supersonic flights, weather prevision and stellar atmosphere. It has also been intensively studied, both theoretically and numerically [23-31]. However, the application of Renormalization Group theory to CIT could remarkably contribute to improving the physical insight in this field too.

(*) Fellow of the TRACS Programme. E-mail: vapaisHfrascati.enea.it

Though, understanding of turbulence is far from being complete.

Anyhow, direct numerical simulations remain one of the most important tools both to directly describe the turbulence evolution and to produce statistical data on which theories as the Renormalization Group one relies on.

Thus, we have started a numerical simulation research in the framework of the Renormalization Group Project [22], looking for a physical insight about the effect of compressibility on the quantities that characterize isotropic turbulence (energy spectrum, skewness, etc.). This study could be thought of as a natural evolution of the work carried out in 1994 [19]. On that occasion, we developed a numerical code that solves the incompressible momentum conservation (Navier-Stokes) equation in the Fourier space by using a pseudo-spectral method to treat the nonlinear term (the products are computed in the physical space, then transformed into the wave number space) and an Adams-Bashforth 2nd-order method for the time integration; a stability condition for the integration time step was also deduced. This code was used to simulate the IIT, running on the EPCC Connection Machine CM-200.

In this work, we present the physical model adopted to describe adiabatic isotropic turbulence in compressible fluids, the numerical implementation and the first results obtained.

2. – The physical model

Turbulence in compressible fluids can be described through the conservation equations (mass, momentum and energy) [32]

¯r ¯t 4 2˘ Q (rv) , (1) ¯ ¯t(rv) 42˘Q (rvv)2˘p1˘Qs1 F , (2) ¯et ¯t 4 2˘ Q [ (et1 p) v] 1 k ˜ 2 T 1˘Q (vQs)1vQ F , (3)

where the symbols have standard meanings, i.e. r is the density, v the velocity vector, p the pressure, T the temperature and F the external force density vector. The viscosity stress tensor s is given by

sab4 m

u

¯va ¯xb 1 ¯vb ¯xa

v

1

g

j 2 2 3m

h

(˘ Q v) dab, and the total energy density etby

et4 ei1 ek,

with ei representing the internal energy density and ek

(

4 ( 1 O2 ) rNvN24 ( 1 O2 ) rv2

)

the kinetic energy density. Greek subscripts indicate different Cartesian coordi-nates.

viscosity m , the bulk viscosity j and the thermal conductivity k) are assumed to be constant.

For the sake of simplicity, we have chosen the perfect-gas equation of state to close the model. Thus, thermodynamics quantities are related by the following expressions: p 4 R A rT , (4) ei4 p (g 21) , (5) c2 4 g R A T , (6) M2 4 v 2 c2 4 2 g(g 21) ek ei , (7)

with R the perfect-gas constant, A the molar weight, c the sound speed and M the Mach number. Note that the parameter g 4CpOCv, the ratio between specific heats, is

closely related with the compressibility and can be used to modify it (g 41 stands for a full compressible, Q-degree-of-freedom, fluid and g4Q for an incompressible fluid).

3. – Some simplifications

The complexity of the CIT problem is significantly higher than the IIT one. We then introduce some simplifications. First we drop the bulk viscosity term (j 40), as usually done [23-31]. Actually, this term gives the lag in the adjustement between the mechanical pressure and the thermodynamical one, and leads to a volume-variation damping. Moreover, the bulk viscosity (also known as “expansion viscosity”) is claimed to be of the same order of magnitude as the dynamic one and could imply an important amount of energy dissipation. Then, future more accurate investigations have to quantify the bulk viscosity term effects. However, at this stage, for the sake of simplicity, we prefer to drop it.

We also assume local adiabaticity by choosing the thermal conductivity k to be zero. Thus, pressure, temperature and energy densities become merely diagnostic variables, no longer prognostic ones. This allows to reduce the set of governing equations to the continuity equation

(

eq. (1)

)

, the Navier-Stokes equation

(

eq. (2)

)

and the barotropic relation between pressure and density: p Prg_{. It must be pointed out that the “locally}

adiabatic” approximation is the main difference with models proposed in previous papers [23-31] about compressible turbulence. This assumption reduces significatively the complexity of our model (when compared with the Kida and Orzag one [ 24 , 25 , 27 ], for instance) at the expense of the range of applicability. Henceforth, we refer to this model as CIAT.

4. – Spectral approach

Isotropic turbulence allows a further simplification of the model by considering the fluid domain to be a cubic box with periodic boundary conditions. Therefore, the variables can be expanded in discrete Fourier series, i.e. be Fourier-transformed, according to

u(x , t) 4

!

k u ˘ (k , t) exp [ik Q x] and u ˘ (k , t) 4 1 l3 0



x

u(x , t) exp [2ikQx] dx ,

l0 being the box side length and u a generic vector (or scalar) field. Each Fourier amplitude u×(k , t) (also called spectral amplitude) corresponds to a different wave number vector defined by

k 4 (kx, ky, kz) 4k0(nx, ny, nz) ,

with k04 2 pOl0 the leading wave number and na integer numbers. From now on, the

symbol × indicates the variable (or combination of variables) in the Fourier space. Amplitudes corresponding to the same knspectral shell, defined by k 4NkN lying in

the interval (kn2 k0O2 , kn1 k0O2 ) with kn4 k0n , can be added to form the spectrum of the variable. The kinetic-energy density spectrum ek is thus given by

e×k(kn, t) 4

!

k 4kn2 k0O2

kn1 k0O2

e×k(k , t) .

On the other hand, applying the Fourier transform (represented by the i–l operator) to a vector field u and to scalar fields f and g, it is useful to recall some transformation rules:

u(x , t) i–l u˘(k , t) , ˘_{Q u(x , t) i–l ikQu}˘(k , t) , ˘ 3 u(x , t) i–l ik Q u˘(k , t) , f (x , t) i–l f×(k, t) , ˘_{f (x , t) i–l ikf}×(k, t) , f (x , t) g(x , t) i–l fg×(k, t) 4

!

p 1q4k f×(p, t) g×(q, t) .

Using these transformation rules, the governing equations for each Fourier amplitude in the k space become

¯r× ¯t 4 2 ik Q (rv×) , (8) ¯ ¯t(rv×) 42 ikQ (rvv×) 2ikr×2 1 Re k 2 v× 2 1 3 Re(k Q v×) k , (9) p× 4

y

pin

g

r rin

h

×_g

z

, (10)

where the external force term has been dropped. In these equations and henceforth, all the variables are dimensionless. The scales for the variables are: the side length of the cubic box representing the fluid domain (l0), the initial average density (r0), the root-mean-square of the velocity components

(

v04

k

( 2 O3)(ek , 0Or0) with ek , 0standing

for the initial average value of the kinetic-energy density

)

and the eddy turnover time (t04 l0Ov0). Consequently, the scale for the wave number is 1 Ol0 and the Reynolds number is given by Re 4 (r0v0l0) Om. The pin and the rin variables are the initial dimensionless spatial distributions of the pressure and the density, respectively.

The present way to solve the model looks like a “spectral Galerkin method”. Actually, as proposed by Orzag [6], the convolution theorem to transform the variables product was not used. Thus, the method could be better described as a “pseudo-spectral Galerkin method” or as a “collocation method”, where the products are computed in the physical space and then transformed into the Fourier space [33].

5. – Numerical implementation

The Direct Numerical Simulation approach implies to solve numerically the governing equations ensuring that all turbulent scales are resolved for a given Reynolds number: the higher the Reynolds number, the smaller the ultimate turbulent scale. So, such simulations become increasingly demanding, in terms of computing power and numerical accuracy, as Re increases.

For numerical-simulation purposes, the simulation box side must be discretized, i.e. divided into finite-length intervals. Using N such intervals, the dimensionless coordinate of each collocation point in the physical space is defined by

xa4 ja

N with 0 GjaG N 2 1 ,

the cubic box contains N3 _{mesh-points and the smaller dimensionless resolved scale is} 1 ON. On the other hand, N points in each direction allow to extend the Fourier series to ka4 2 p na with 2 N 2 G naG N 2 2 1 ,

for each component of the k, corresponding to NO2 harmonics wavelengths in the Fourier space.

The integration in time is based on a finite-difference Runge-Kutta-Gill scheme with error correction [34], which allows a 4th-order precision with a reduced memory requirement, even if it is process-time consuming. Writing down the resulting numerical scheme for the governing equations is quite involved but an overview of the Runge-Kutta-Gill scheme can still be useful. Given a differential equation

¯yi

¯y0

4 fi(y0, y1, y2, R) ,

with i running from 0 (representing the independent variable) to the total number of variables, and fixed a Dy0integration step, the numerical scheme becomes

dlt( 1 ) i 4 1 2rhs ( 0 ) i 2 erri( 0 ), yi( 1 )4 yi( 0 )1 dlti( 1 ), erri( 1 )4 erri( 0 )1 3 dlti( 1 )2 1 2rhsi ( 0 )_, rhsi( 1 )4 Dy0 fi(y0( 1 ), y1( 1 ), y2( 1 ), R) , dlti( 2 )4

y

1 2 k2 2

z

( rhsi ( 1 ) 2 erri( 1 )) , yi( 2 )4 yi( 1 )1 dlti( 2 ), err( 2 ) i 4 erri( 1 )1 3 dlt( 2 )i 2

y

1 2 k2 2

z

rhsi ( 1 )_, rhsi( 2 )4 Dy0 fi(y0( 2 ), y1( 2 ), y2( 2 ), R) , dlti( 3 )4

y

1 1 k2 2

z

( rhsi ( 2 ) 2 erri ( 2 )_{) ,} yi( 3 )4 yi( 2 )1 dlti( 3 ), erri( 3 )4 erri( 2 )1 3 dlti( 3 )2

y

1 1 k2 2

z

rhsi ( 2 )_, rhsi( 3 )4 Dy0 fi(y0( 3 ), y1( 3 ), y2( 3 ), R) , dlti( 4 )4 1 6( rhsi ( 3 ) 2 2 erri( 3 )) , yi( 4 )4 yi( 3 )1 dlti( 4 ), erri( 4 )4 erri( 3 )1 3 dlti( 4 )2 1 2rhsi ( 3 )_, rhsi( 4 )4 Dy0 fi(y0( 4 ), y1( 4 ), y2( 4 ), R) ,

where the superscript indicates the steps inside an integration interval (0 at the shart of the interval and 4 at the end of the interval). Here, erri represents an error

correction term, initially set to zero and successively set to erri( 0 )4 erri( 4 ), i.e. equal to

the last value in the precedent interval.

The choice of the time integration step Dt (Dy0 in the previous scheme) is done following the stability criterium found in [16], that is: Dt  3 O(14 Re ek , 0). This

fluid while we expect that perturbations in density will tend to relax this condition. In any case, using Runge-Kutta integration methods increases stability.

Bearing in mind that we are using a pseudo-spectral method, each step computing the right-hand-side terms rhsi involves the following steps, assuming r× and rv

˘ to be known:

– transform the density r× _{back into the physical space, r}×_{(k , t) l–i r(x, t),} – transform the momentum rv˘back into the physical space, rv˘_{(k , t) l–i rv(x, t),} – compute the velocity field v in the physical space, v(x , t) 4rv(x, t)Or(x, t), – build the tensor rvv in the physical space, rvv(x , t) 4rv(x, t) v(x, t), – transform the tensor rvv into the Fourier space, rvv(x , t) i–l rvv˘(k , t), – compute the pressure field p in the physical space,

p(x , t) 4pin(x , t)

(

r(x , t) Orin(x , t)

)

g,

– transform the pressure p into the Fourier space, p(x , t) i–l p×(k, t),

– build up the right-hand-side terms of the governing equations in the Fourier space.

This pseudo-spectral method introduces aliasing errors. These errors occur for any operation which combines components with different wave numbers or introduces higher frequencies, such as a convolution sum or a product of variables. Among the de-aliasing methods, as truncation and phase shifts, we chose to drop all amplitudes outside a sphere of predefined radius [33], i.e.

u_{×(k , t) 4}./ ´ u ×(k , t) , 0 , if NkNEktr if NkNFktr with ktr4 2 p N 2 k8 3 .

6. – Some useful variables

Some useful statistical quantities to be used as diagnostic of the compressibility effects are now recalled:

a) First, we introduce a new dimensionless variable w_{×(k , t) 4 krv} ˘

(k , t) in the Fourier space, which allows to generalize the definitions used for IIT. This variable, indeed, plays the same role as the velocity in the incompressible case. So, by applying the Parseval theorem, the shell-averaged kinetic-energy density spectrum can be rewritten as [ 24 , 25 , 27 ] e×k(kn, t) 4 1 2 NkN 4 k

!

n2 k0O2 kn1 k0O2 w ˘ * (k , t) Q w˘(k , t) , (11)

with kn4 k0n and the average kinetic-energy density as

e×k(t) 4

!

kn

e×k(kn, t) .

(12)

b) Similar relations are valid for the shell-averaged internal energy:

e×i(kn, t) 4 1 (g 21) NkN 4 k

!

n2 k0O2 kn1 k0O2 kp ˘ * (k , t) Qkp ˘ (k , t) , (13)

and the average internal energy density: e×i(t) 4

!

kn

e×i(kn, t) .

(14)

c) Two different Taylor microscale dimensionless lengths can be written down, the first related to the velocity v(x , t) as in IIT:

lv(t) 4

o

av2_b

a(¯vO¯x2 a¯vO¯xb)2_b , the second implicitly using the w×(kn, t) variable:

lw(t) 4

o

5

!

kne×k(kn, t)

!

knk 2 ne×k(kn, t) .

d) Whence two microscale-based Reynolds numbers (the Taylor-Reynolds number): Relv(t) 4Re arb

k

av2b lv(t) , (15) and Relw(t) 4Re arb

k

av2b lw(t) . (16)

The resolved scale has been found [1] to be related to the Taylor-Reynolds number through

Rel(t)  3 : 4 N2 O3;

(17)

this shows why the collocation points number must increase as Re increases, thus augmenting the computational requirements.

e) Straightforwardly extending the analysis of the incompressible case, we introduce, from dimensional considerations, a spectral energy transfer as

T ×(kn, t) 4 1 2 NkN 4 k

!

n2 k0O2 kn1 k0O2 w ×* (k , t) Q [2ikQ (rvv˘)(k , t) ] ,

1e-05 0.0001 0.001 0.01 0.1 1 1 10 Γ t=0 turn-over times k(-5/3) 1.4 1.8 2.5 4.0 ∞ 1e-05 0.0001 0.001 0.01 0.1 1 1 10 Γ t=3 turn-over times k(-5/3) 1.4 1.8 2.5 4.0 ∞ 1e-05 0.0001 0.001 0.01 0.1 1 1 10 wavenumber Γ t=6 turn-over times k(-5/3) 1.4 1.8 2.5 4.0 ∞

Fig. 1. – Kinetic-energy density spectra ek(k , t) for different compressibilities, for three different

1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 1 10 Γ t=0 turn-over times k(-5/3) 1.4 1.8 2.5 4.0 ∞ 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 1 10 Γ t=3 turn-over times k(-5/3) 1.4 1.8 2.5 4.0 ∞ 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 1 10 wavenumber Γ t=6 turn-over times k(-5/3) 1.4 1.8 2.5 4.0 ∞

Fig. 2. – Internal energy density spectra ei(k , t) for different compressibilities, for three different

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 1.5 3 4.5 6 turn-over times

γ

1.4 1.8 2.5 4.0

∞

Fig. 3. – Peak-to-valley density oscillations amplitude [max (r) 2min (r) ]Oarb, temporal evolution for different compressibilities.

f ) We define the skewness of the velocity v as Sv(t) 4

a(¯vO¯x2 a¯vO¯xb)3_b a(¯vO¯x2 a¯vO¯xb)2_b3 O2 , (18)

and the skewness of the variable w× as

Sw(t) 4 3 7

o

15 2

!

knk 2 T(kn, t) ]

!

knk 2 ne×k(kn, t)(3 O2 . (19)

g) Again following the IIT definitions, we introduce a kinetic-energy dissipation rate as ’(t) 4 2 Re

!

kn k2 ne×k(kn, t) . (20)

Note that this definition does not match with the one proposed by Kida and Orzag [24, 25, 27], but is an extension of the incompressible analysis to include the effects of the density variations.

h) Finally, we use the ratio between the cut-off wave number and the Kolmogorov wave number, W(t), defined as

W(t) 4 4 ktr

k

Re3

’

, (21)

which indicates when the turbulent scales are being resolved (W D1 means all scales resolved).

7. – Hardware and software

With the CIAT basic model and algorithms settled up, we turn to the computational aspects. The complexity of the CIAT physical problem is increased with respect to the IIT one, so is the computational demand. Among the high-performance facilities available at EPCC, the choice of the Cray T3D was forced since the complexity and dimension of the problem to be treated overflow the capacity of the other parallel machines, as the Connection Machine CM-200 or the Meiko T800 Computing Surface. As a measure, we mention that a typical small job (N 464, integration steps42000) needs 1.5 hours running on 128 T3D processors.

The next two steps are to transform the old CM-200 IIRT [19-21] code to a Cray T3D

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0 1.5 3 4.5 6 turn-over times

γ

1.4 1.8 2.5 4.0

∞

Fig. 4. – Maximum density to shock-wave density ratio max (r) O((g 11)O(g21)), temporal evolution for different compressibilities.

version and then, to change its equations and structure to introduce the CIAT model. In a first attempt, we tried to use the Portland Group’s High Performance Fortran looking for a more portable code, but the Fast Fourier Transform subroutine (PCCFFT3D) that Cray software provides (and the only one available) revealed serious compatibility problems with PGHPF, which we were not able to solve. Thus, we turned to CRAFT (Cray version Fortran for DATA PARALLEL). However, we still found hard difficulties to make some parts of the code efficient and more work is needed to fully optimise it.

8. – Initial conditions and energy reinjection

One important issue is the choice of the initial conditions: v×(k , t 40), rinand pin. We decided to start from “incompressible” conditions for all the simulations, to simplify the comparison among the different compressibility cases. Thus, for the velocity, we adopt a Gaussian random field, divergence-free, with a k(25O3) _{spectrum having a} kinetic-energy density average value ek , 0. It is worth mentioning that we are studying fully

developed turbulence and, therefore, the initial spectrum shape may be not too relevant because, after some turnover times, the flow should forget the initial conditions. For the density and pressure initial fields, inspired by ref. [ 23 , 28 , 29 ], we assumed a uniform density field rin4 1 and a pressure field derived from the velocity divergence-free

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 1.5 3 4.5 6 turn-over times

γ

1.4 1.8 2.5 4.0

∞

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 1.5 3 4.5 6 turn-over times

γ

1.4 1.8 2.5 4.0

∞

Fig. 6. – kr v ˘

skewness Sw, temporal evolution for different compressibilities.

condition:

p× in4 2

k Q [k Q (rvv˘) ]

k2 .

This zero-mean pressure field is fully determined but for a function of time. We then add a constant C 4min ( pin) 11 which avoids errors in computingkp and defines the average Mach number.

The fluid is kept moving by reinjecting the same amount of kinetic energy as that lost in the higher wave numbers, into the first Fourier-shell of the velocity field

(

v×(kn4 k0, t)

)

. The average kinetic-energy density thus remains constant, ek(t) 4ek , 0.

9. – Results

Once the IIT and the CIAT codes were giving convincing results, running on the Cray T3D, we proceeded with some simulations to illustrate the effect of the compressibility on the energy spectrum and other statistical variables.

To reduce unnecessary time delays, we restricted the tests to N 464, i.e. 643 collocation points. This allows to extend the simulations up to Re 490 using a time integration step Dt 40.003. Then, we simulated an incompressible case (g4Q) with the IIT code, and four compressible cases (g 41.4, 1.8, 2.5 and 4.0) with the CIAT code,

TABLEI. – Square Mach number for the different compressibility simulated cases. g M2 1.4 1.8 2.5 4.0 Q 0.41 0.32 0.23 0.14 0.00

pushing the evolution up to 6 turnover times (2000 time-steps). For all the cases ek , 0 was set to 3 O2.

Concretely, the results are

1) No substantial differences have been observed in the evolution and in the shape of the kinetic-energy density spectra e×(k , t), one for each compressibility. This can be seen in fig. 1, where three instants of the evolution are presented: 0, 3 and 6 turnover times. Actually, this fact is not surprising because the energy density is set to be the same in all cases and the density perturbations are generated consistently with the initial incompressible pressure field. On the other hand, the choice of C revealed to be not too lucky as it leads to a very low average Mach number, as shown in table I,

35 40 45 50 55 60 65 70 75 0 1.5 3 4.5 6 turn-over times

γ

1.4 1.8 2.5 4.0

∞

35 40 45 50 55 60 65 70 75 0 1.5 3 4.5 6 turn-over times

γ

1.4 1.8 2.5 4.0

∞

Fig. 8. – Taylor-Reynolds number Relw, temporal evolution for different compressibilities.

thus reducing the role of the compressibility. One further aspect to be evidenced is that the spectrum shapes reach a near stationary form (compare 3 and 6 turnover times plots), as can be expected for a fully developed turbulence. For the 6 turnover times plot, the spectra shapes fit with the expression 5912 k3 .154

exp [29.454k0 .3420_{] and an} attempt to establish an “inertial subrange” (Kolmogorov k(25O3) _{spectrum) can be} observed for wave numbers from to 2 to 6.

2) Here too, substantial differences have been neither found in the internal energy density spectra, as shown in fig. 2. Comparison at different times reveals just a slight increment in the lower kn values and a decrement in the higher wave numbers,

eventually indicating a stronger dissipative coupling between kinetic and internal energy density for lower wave numbers. We point out that the major amount of internal energy lies in the kn4 0 shell, not reported in the figures. Between wave numbers 2

and 6, a dependence near to k(25O3)_{is shown.}

3) At this point, suspects about the existence of the compressibility effects have to be removed. The peak-to-valley variation amplitude of the density, defined as [ max (r) 2min(r) ]Oarb, shows the most apparent differences between cases (see fig. 3), a good test that the compressibility is active. Moreover, the strong oscillations shown by the density grow as g is lowered. This could be interpreted as the presence of localised shock-waves. Figure 4 illustrates the evolution of the ratio between the maximum density and the density behind a shock-wave: max (r) O

(

(g 11)O(g21)

)

, evidencing the transitory appearance of shock-waves only for the g 41.8 case.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0 1.5 3 4.5 6 turn-over times

γ

1.4 1.8 2.5 4.0

∞

Fig. 9. – Kinetic-energy dissipation e , temporal evolution for different compressibilities.

4) The evolution of velocity skewness Sv(t)

(

eq. (18)

)

, plotted in fig. 5, shows to

tend to the same value (0.5) as time grows. This fact indicates that the asymmetry in the velocity field distribution tends to be the same, independently of compressibility. This variable too shows strong oscillations: the lower g is the higher they are. This trend is related with the oscillations in density (see fig. 3) and damps with time. The evolution of the kr v variable skewness Sw(t), as defined by eq. (19), shows smaller

differences and oscillations (see fig. 6). The main difference with respect to Sv(t) is that Sw(t) seems to approach slightly different values around 0.29, increasing as g

decreases. This quantitative (but not qualitative) difference may indicate that compressibility favours an increase of the asymmetry in the kinetic-energy distribution.

5) Comparison between Taylor-Reynolds numbers for different compressibilities is illustrated in fig. 7

(

Relv, eq. (15)

)

and 8

(

Relw, eq. (16)

)

. Again, the Reynolds number

related with the velocity evidences more differences than the one related with the w× variable, even if temporal behaviours are similar. The Relv values are systematically

lower than the Relwones, more apparent for lower g’s, reaching peaks around A64 and

A 70 , and approaching A 57 and A 61 , respectively, with time. Both these Taylor-Reynolds numbers satisfy the condition (17) Rel64 .

6) The kinetic-energy dissipation rate, as defined by eq. (20), is plotted in fig. 9. We recall that this definition does not match the Kida and Orzag one, but is an extension of the incompressible analysis. The variable e does not show any relevant

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 0 1.5 3 4.5 6 turn-over times

γ

1.4 1.8 2.5 4.0

∞

Fig. 10. – Truncation wave number to Kolmogorov wave number ratio (W), temporal evolution for different compressibilities.

difference for different compressibilities. So does the W variable

(

eq. (21)

)

, plotted in fig. 10, which remains always above unity, save for the initial transient, ensuring that all the turbulent scales are resolved.

7) Finally, it must be pointed out that all the variables tend to the incompressible case as g increases. This fact could be considered as a validation of the IIT and the CIAT codes.

10. – Concluding remarks

This study is not as exhaustive as previous papers on turbulence [23-31] in compressible fluids and more work is to be done. We attempt here just to get an insight into the role of compressibility on isotropic turbulence as a start to deeper investigations.

Even if the Mach numbers simulated here were low, we can summarize that compressibility does not have an important effect on the energy density spectrum shapes, both internal and kinetic. The changes in the velocity field distribution are compensated by the changes in the density field distribution, keeping the energy variables almost unaltered. Further, the presence of localised shock-waves does not seem to affect the evolution of the variables.

The choice of the kr v variable appears suitable to extend the definition of the variables and the results from the incompressible to the compressible regime.

Simulations with higher Reynolds numbers (higher grid-points numbers) have to be performed to check the formation of an “inertial-subrange”.

* * *

This work was supported by the European Community TRACS (Training and Research on Advanced Computing Systems) programme using the high performance facilities available at the Edinburgh Parallel Computing Center (EPCC), University of Edinburgh, Scotland, UK, and the ENEA (Italian National Agency for New Technology, Energy and the Environment). I here gratefully acknowledge: P. FLOHR, for patiently supporting me during all the steps of this work, A. CARUSO, for his valid suggestions, W. D. MCCOMB, for his guide and help, B. ROBOUCH, for his useful support, and last but not least the TRACS programme staff, for help with the computational woes and life supplies!

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