[juliom]

Dear community, I know these two topics have been very discusses so far. However, I want to bring a discussion based on the physical constraints of these two models.

1.- We know that the dynamic model is based on the Germano Identity; that stems from filtering at two different levels. Also, Germano assumed that the constant Cs was constant at both filtering level invoking similarity scales. However, the fundamental model is unstable. Thus, an averaging is required. What is the cause of this instability that leads to a highly variable eddy viscosity field. Is this due to the assumption of Cs constant at both filtering level? or is it just because the germano identity when the Smagorinsky model is used to define [imath]\tau_ij[\imath] and [imath]T_ij[\imath] leads to an overdetermined system of equation? or due to another reason?

2.- Another interesting model was proposed by Bardina et al. assuming similarity scales at two filtering level. However, Bardina used the relation of [imath](\gamma \Delta)[\imath] but in their work they used [imath]\gamma = 1[\imath]. My question is if [imath]\gamma = 1[\imath] then the test filter lies upon the grid filter. Under Bardinas' context, the second filter coincides with the primary filter. Therefore, how can I use the scale similarity if I am sampling at the same filtering level. Unless, the model rest upon the operation of the formal definition of [imath]\tau_ij[\imath]

[FMDenaro]

Originally Posted by juliom

Dear community, I know these two topics have been very discusses so far. However, I want to bring a discussion based on the physical constraints of these two models.

1.- We know that the dynamic model is based on the Germano Identity; that stems from filtering at two different levels. Also, Germano assumed that the constant Cs was constant at both filtering level invoking similarity scales. However, the fundamental model is unstable. Thus, an averaging is required. What is the cause of this instability that leads to a highly variable eddy viscosity field. Is this due to the assumption of Cs constant at both filtering level? or is it just because the germano identity when the Smagorinsky model is used to define [imath]\tau_ij[\imath] and [imath]T_ij[\imath] leads to an overdetermined system of equation? or due to another reason?

2.- Another interesting model was proposed by Bardina et al. assuming similarity scales at two filtering level. However, Bardina used the relation of [imath](\gamma \Delta)[\imath] but in their work they used [imath]\gamma = 1[\imath]. My question is if [imath]\gamma = 1[\imath] then the test filter lies upon the grid filter. Under Bardinas' context, the second filter coincides with the primary filter. Therefore, how can I use the scale similarity if I am sampling at the same filtering level. Unless, the model rest upon the operation of the formal definition of [imath]\tau_ij[\imath]

1) generally the source of instability is more due to the numerical aspects than to physical one...however, you can see the reason in the fact that the dynamic procedure allows for negative value of the SGS eddy viscosity which is not a physical fact in terms of the diffusion of momentum. It is like a backscatter effect that, if not balanced by the physical viscosity, can produce a numerical instability.

2) The scale similar model requires that (u_bar)_bar results different from u_bar, that is the filter is not projective (idempotent). This happens with smooth filter (like the topo-hat) but not with spectral filter.

[juliom]

Thank you professor; question 1 is very important and your explanation was very illustrative thanks.

Does your second answer mena that if I use a smooth filter (top-hat or Gaussian) I can filter any field any time using the same filter width? Because, my problem understanding this model is that Bardina used second filter = 1* first filter. Thus, how can I filter the velocity field using the same filter width from the first filter??

[FMDenaro]

Originally Posted by juliom

Thank you professor; question 1 is very important and your explanation was very illustrative thanks.

Does your second answer mena that if I use a smooth filter (top-hat or Gaussian) I can filter any field any time using the same filter width? Because, my problem understanding this model is that Bardina used second filter = 1* first filter. Thus, how can I filter the velocity field using the same filter width from the first filter??

Yes, for example the top-hat has the transfer function G(k) = sin(kh)/(kh).
You can define

V_bar^n = G^n V (n=1, 2...)