Hi, Everybody,

In k-ep model and RNG model there are five adjustale constants. For some special cases, you should adjust them to satisfy with experimental data. I want to know whether these constants should have a arrangement? That is to say, which arrangement is reasonable?

(1). I would say that, the first step is to read the book on turbulence modeling by David Wilcox. (2). These functions and constants are optimized based on various standard cases. And some of these constants are inter-related. (3). The area which you are interested in is called the turbulence modelling. It is the major key area in CFD.

Dear John,

Firstly, thank you very much for your information. I understand that these constants have been optimized and they are key area in the turbulence modelling. I know they should be changed for some special cases. However, I can not get the book by David Wilcox at the moment. Could you tell me some details about them?

(1). In the optimization process, not all of the constants can be changed independently. So, it is important to identify these relationship first. (2). You can ,in principle, optimize these constants to match the particular class of problms, but this requires accurate and reliable test data. (3). If you don't have Wilcox's book (which you can purchase from his website), you should read another book by the masters of turbulence modeling, the book entitled "Mathematical Models of Turbulence", by Launder,B.E and Spalding, D.B. (1972), Academic Press, London. It is a relatively thin book. You must have at least these two books in order to read papers in turbulence modeling.

If we consider the standard high-re form of the k-e model, the constants are determined as follows:

The value of C2e is obtained by considering the decay of grid turbulence, where production and diffusion are negligible in the turbulence transport equations. These equations then reduce to a simple form which permits an exact solution for turbulent decay. This solution allows C2e to be computed given data on the rate of turbulent decay.

The constant Cmu can be determined by considering local equilibrium shear layers where data indicate that the main Reynolds shearing stress is roughly 30% of the turbulence energy.

The constant C1e is obtained from:

c1e=c2e - k**2/(sige*sqrt(Cmu))

where k is von Karman's constant and sige is the turbulent Prandtl number for e which has a value the order of unity. This relation comes from consideration of the fully-turbulent region close to a wall where e convection can be neglected.

The turbulent prandtl numbers sigk and sige are optimised to give reasonable profile shapes in free shear layers.

I'm agree with your definition of c_eps_2 and c_mu, but c_eps_1 can the be obtain from the relation :

c_eps_1 = 1 - c_mu * ( c_eps_2 - 1 )/( 4*b_12 )

where b_12 is the extra-diagonal terme of the spheric part of the anisotropic tensor :

b_12 = ( u_1u_2 ) / ( 2 tke )

for a homogeneous shear stress, and where tke is the turbulent kinetic energie (ie k).

Then, the relation you give for what you call C1e and what I call c_eps_1 is used to determine the value of sigmae.

Now, the value of sigmak is obtaine from the relation :

sigmak = 6 * sigmae *( sqrt(4*c_eps_2*c_eps_2 +1) - c_eps_2 )

which comes from the following article :

S.K.Lele - a Consistency condition for Reynolds stress closure - Physics of Fluids 28(1):64-68, 1984

That's mean that only four experiments are needed to determine 5 constants....

Sylvain

Thank you for this interesting information. Please confirm the relations for c_eps_1 and sigmak, and the standard values of the constants used in these relations.

My problem is:

Presumably, b_12 = ( u_1u_2 ) / ( 2 tke ) =0.15. Then,

c_eps_1 = 1 - c_mu * ( c_eps_2 - 1 )/( 4*b_12 )

yields c_eps_1 = - 0.53 with the standard values c_mu = 0.09 and c_eps_2 = 1.92. Perhaps, the above relation should be:

c_eps_1 = [ 1 - c_mu * ( c_eps_2 - 1 )]/( 4*b_12 )

for then c_eps_1 = 1.53, which is within 7% of the standard value of c_eps_1 = 1.44.

If

sigmak = 6 * sigmae *( sqrt(4*c_eps_2*c_eps_2 +1) - c_eps_2 )

then

sigmak/sigmae = 6*(sqrt(15.7456) - 1.92)= 12.29

which seems very wrong.

Have I made a blunder or is there something not quite with the relations or constants used therein.

In my recent message, "yields c_eps_1 = - 0.53." should have read "c_eps_1 = 0.862.", i.e.

c_eps_1 = 1 - 0.09 * ( 1.92 - 1 )/( 4*0.15 ) = 0.862.

Sorry for that, but it was a study that i've done few years ago and moreover i've done some mistake in my expressions :

1) the correct expression for sigmak/sigmae is :

sigmak/sigmae = 6 *( sqrt(4*c_eps_2*c_eps_2 +1) - 2*c_eps_2 )

which leads to sigmak = 0.7692 * sigmae for the standart k-eps constants.

2) for c_eps_1 the story is more complicated and i've made a mismatch with my notes. First of all, the relation was wrong, it should be :

c_eps_1 = 1 + c_mu * ( c_eps_2 - 1 )/( 4*b_12*b_12 )

But the story is that you don't (musn't?) have to look to b_12, but to the rapidity eta, which is define by :

eta = tke * S / epsilon

where S is the shear stress. The correcte relation between c_eps_1 and c_eps_2 is then :

c_mu*eta*eta = (c_eps_2 - 1)/(c_eps_1 - 1)

which :

when eta = 5 (DNS from Rogallo) gives c_eps_1 = 1.41

when eta = 5.7 (DNS form Rogers) gives c_eps_1 = 1.32

when eta = 4.1 (Experiments from Tavoularis et Karnik) leads to c_eps_1 = 1.61.

The mismatch was that, for a k-eps model, it comes :

eta = - 2 * b_12 / c_mu

which gives, as everybody knows, a bad value for b_12 (~0.21) compare with the experiment (~0.16).

Sorry again for that,

Sylvain