TKE SPECTRUM & DISSIPATION
 

For validation purposes, I did a direct numberical simulation of isotropic turbulence. It's been in the framework of atmospheric microscale. The simulation domain was a cube of 256^3m with a resultion of 1m. Visocosity (nu) was chosen to be 0.02m^2/s. On the basis of the resulting tke spectrum, I derived the dissipation rate (eps) by the commonly used formula:

eps = 2 * nu * Integral (k^2 * e) dk

where k is the wavenumber and e is the wavenumber dependent turbulent kinetic energy (tke). With this estimation of eps=7.m^2/s^3, I get an Kolmogorov length scale (lk) of

lk = ( nu^3 / eps )^0.25 = 0.03m << grid size!

This would implicate, that the grid resolution is not sufficent. Unappropriate resolution would result in too much tke at high wave numbers. However, my tke spectrum looks pretty good and follows the -5/3 and -7/3 laws.

Does anyone have an idea or have had a similar experience?

Thanks. Frodo (all4frodo@hotmail.com)

Re: TKE SPECTRUM & DISSIPATION
 

If a dissipative scheme is used, the numerical dissipation might be more than the viscous dissipation. In such cases, the energy will not accumulate at the high wavenumbers even if the actual dissipation scales are not resolved. The (dissipative) numerical scheme would dissipate most of the energy within the scales you are resolving. That is to say that the scheme would create an ariticial dissipation range in the resolved wavenumber range.

If you check most of the papers on DNS, you would see that the Kol. scale is not resolved. The primary requirement in a DNS is to resolve the viscous dissipation at the small scale (high wavenumbers). Also the numerical (artificial) dissipation should be minimal (i.e., small compared to the viscous dissipation at all wavenumbers) and all dissipation should be due to the viscous term only. If you consider the Pao's inertial-dissipation spectrum (see Tennekes and Lumley), you would see that the amount of kinetic energy at the Kol. scale is very negligible suggesting that the scales at which dissipation occurs are typically larger than the Kol. scale. So, the Kol. length is a very low end estimate of a dissipation scale. If you consider the dissipation spectrum corresponding to the Pao's energy spectrum, you will see that the wavenumber where the dissipation spectrum peaks is about 6 times smaller than the wavenumber corresponding to the Kol. length scale. So, you sometimes see that the smallest resolved wavenumber is a DNS could be about a order of magnitude higher than the Kol. wavenumber. In your case, the ratio is about 30 which is a bit on the higher side.

Suggestion : i) Check you computation of eps again. ii) Dissipation rate can be computed as rate of change of total kinetic energy. This gives you an alternate way of computing eps (if the turbulence is decaying/unforced). iii) Decrease the viscosity and run the code. If the energy does not accumulate at high wavenumbers even after the viscosity is reduced a lot, you code is too dissipative for DNS or LES.

Thank you, Mr. Kalyan.
 

Within the next days, I will post the new results, I'd like to discuss.

C.H.