What you are considering is nothing else that an LES model (with no explicit SGS model). Therefore, you can compute the TKE taking into account that you have a locally filtered (induced by the grid and discretization of the equation) variable. The so called LES no-model is often used to test the consistence of the SGS model.

Thanks Filippo for your answer.

Exactly, this is the case. But what I was asking is that is it necessary in this case to have a balance in the TKE equation? Or the equation can be un-balanced due to the coarse grid and thus smaller dissipation contribution ?

First, the balance between production and dissipation depends on the physics of the problem. If you have turbulence not in equilibrium also in a fully resolved DNS this balance does not physically apply.

Second, the coarse DNS can or can not have a suitable dissipation. That depends on the type of discretization of the equations not only on the grid.

1. Grid sizes of 3~4 times of Kolmogorov length scale is good enough for DNS.

2. My personal experiences: The budget balance of TKE (or Reynolds stresses) equation is dependent on your numerical scheme. For 2nd order FVM or FDM, I cannot get the transport equation perfectly balanced. In contrast, spectral accurate methods can always give balanced budget terms. I think the excessive truncation error induced by low-order schemes cannot ensure the energy conservation. I have also tried with so-called energy-conserving FDM, the budget balance is never good enough to publish. In CFD, only momentum equations are solved, the transport equation of TKE requires some manipulation of the momentum equations. Those mathematical manipulation cannot be well represented with low-order schemes. For example, the chain rule is almost impossible for low-order scheme to obey.

Hope this helps.

I agree that the balance of TKE is somehow, by a numerical point of view, a different issue. I often doubt about the real meaning of assessing to satisfy an equation that is not resolved but induced by the solution of other equations. However, that appears also someway a traditional task in the turbulence field and it is generally required by some reviewers. For results with a coarse DNS have also a look here https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20000039436.pdf

Furhermore, I would highlight that there is no physical constraint about the conservation of energy for the filtered fields.

Thanks for your comments. Thats my case. I don’t see a parfect balance and i am using a second order FVM.

thanks Filippo for the article

The TKE equation is an induced equation. Its balance depends on the discretization used for the momentum equation. If you employ energy-conserving schemes both in space and time, the (discrete) TKE balance over the entire domain is perfectly satisfied regardless of grid size and scheme order.

Of course, the average terms of the TKE will look different from a reference DNS solution due to under-resolution and truncation errors.

Thanks Francesco for your comment.

Can you please emphasize what do you mean exactly by an energy conservation scheme in a DNS (maybe coarse DNS) calculation.

Best regards

@Fransecso

at least for a second order (time and space) schemes

Try

1. F. E. Ham, F. S. Lien, and A. B. Strong, “A fully conservative second-order finite difference scheme for incompressible flow on nonuniform grids,” J. Comp. Phys., vol. 177, pp. 117–133, 2002.

2. Y. Morinishi, T. S. Lund, O. V. Vasilyev, and P. Moin, “Fully conservative higher order finite difference schemes for incompressible flow,” J. Comp. Phys., vol. 143, pp. 90–124, 1998.

My experience is that: the so-called "energy conservative scheme" does not help to give balanced budget for TKE, after all even the definition of energy conservation is debatable in the frame work of FDM/FVM.

thanks Xingjun

Hi Elie,

please have a look at my JCP papers and references therein. You might find particularly useful the latest one (2017). There it is shown that (at least for a no-forced case and periodic boundary conditions) a fully conservative method gives an energy balance satisfied up to machine precision.

First, "kinetic energy" conservation (total energy is conserved by a physical principle) is a principle that exists for INVISCID flows and specific BC case since in general the kinetic energy equation is not a conservative equation. It involves the pointwise velocity therefore the kinetic energy equation built from the filtered velocity has not a strict conservation constraint.

Hystorically, Arakawa ( Computational design for long-term numerical integration of the equations of fluid motion: Two-dimensional incompressible flow. Part I - Journal of Computational Physics, 1966 ) introduced a discretization that, based on the induced conservation of kinetic energy, provided numerically stable results at moderately high Re number flows.

From a theoretical point of view, a coarse DNS is an unresolved (filtered) simulation so that this issue must be analysed under careful hypothese.

http://www.dca.iag.usp.br/www/material/adwgandu/DESATIVADOS/agm-5801_2011/MATERIAL_BIBLIOGRAFICO/ARTIGOS/Arakawa_JCP_1966.pdf

@Francesco, could you please provide us with the reference you mentioned? thanks

Sure, please find it attached. You might want to look at Eqs. (9)-(10). Please do not hesitate to ask should you have any further questions.

Also, please note that the paper refers to incompressible flow; for compressible flows, things are more subtle due to the physical exchange between kinetic energy and internal energy. Still, numerical methods exist for the mass and momentum equations that are able to eliminate spurious errors in the induced energy equations.