Theoretical Justification for the Choice of the LPS SU Parameter

of the LPS SU parameter τ_M^u, especially its dependence on the mean velocity on a cell uM and the globally used grid size h. Unfortunately, these considerations are not valid universally; they apply to isotropic homogeneous turbulence and give no answer to the case of wall bounded flow, for instance.

Parameters for the Smagorinsky model that are suited to represent the influence of small scales are derived in [Lil67]. We follow this argumentation and apply it to our situation:

The LPS SU stabilization is interpreted as a turbulence model and is postulated to be of the form

τ_M^u(κ^u_M(u_M· ∇u_h), κ^u_M(u_M · ∇v_h)) with τ_M^u =τ_fM h^β

|u_M|^γ

and τ_fM > 0 and some β, γ ∈ R to be determined by the following discussion. In case of isotropic turbulence, the energy spectrum is of the form

E(k, t) =K₀ε^2/3k^−5/3,

where εrepresents the turbulent dissipation rate and simultaneously the energy transfer rate across a given wave number. Assume that the turbulent kinetic energy production and dissipation are in balance on a cell, i.e.,

ε=τ_M^ukκ^u_M(uM · ∇u_h)k²_0,M. (5.8) In order to determineβ andγ, we conduct dimensional analysis for (5.8). The notation [·]

indicates the physical unit of a quantity.

m²

s³ = [ε] =^hτ_M^u kκ^u_M(uM · ∇u_h)k²_0,Mⁱ= [τ_M^u] 1 m

m² s²

!2

= [τ_M^u]m² s⁴

⇒ [τ_M^u ] =s=^hh|u_M|⁻¹ⁱ

and thus, γ = 1 andβ = 1.

Now, we desire to find a suitable constant τ_fM. Denote by kc the resolution limit wave number of the coarse spaceD^u_M with respect to the velocity and byk_f the resolution limit wave number ofV_h. For the case of (Q2/Q1)∧Q1 elements anda= 1, it holdsk_f = ³₂h⁻¹ and kc/kf = 1/2. Due to Plancherel’s theorem for the Fourier transform, we calculate from (5.8)

kκ^u_M∇u_hk²_0,M = Z kf

kc

k²E(k)dk=K₀ Z kf

kc

k^1/3ε^2/3dk

=K₀(τ_M^u )^2/3kκ^u_M(u_M · ∇u_h)k^4/3_0,M Z kf

kc

k^1/3dk

=K₀(τ_M^u )^2/3kκ^u_M(u_M · ∇u_h)k^4/3_0,Mk_f^4/3



1− kc

k_f

!4/3



=K₀(τ_M^u )^2/3kκ^u_M(u_M · ∇u_h)k^4/3_0,Mk_f^4/3 1− 1

2 4/3!

3 2

4/3

h^−4/3

=K₀ h^2/3

ku_Mk^2/3_0,Mτ^f_M^2/3kκ^u_M(u_M · ∇u_h)k^4/3_0,M 1− 1

2

4/3!3 2

8/3

h^−8/3.

This yields a formula forτ_fM. However, this would lead to a highly nonlinear stabilization term and is not feasible for our implementation. Therefore, we content ourselves with the insights from the dimensional analysis and choose τ_fM empirically. We point out that the finding τ_M^u ∼ _|u^h

M| already reduces the range of possible parameters considerably.

5.6.4. Numerical Experiments

The initial condition of the TGV problem defines the large scale structures and vortices.

For high Reynolds numbers, the kinetic energy is transported to the small scales due to vortex stretching.

Figure 5.18.: Iso-surfaces for |ω| = 1 at t= 0 (left), |ω|= 1 at t= 2 (second from left),

|ω|= 2.5 att= 4 (second from right),|ω|= 4 att= 9 (right) withh=π/8, {a= 2π, b= 1}and grad-div stabilization γM = 1.

Figure5.18 illustrates the mechanisms of the energy cascade in case ofν = 10⁻⁴ with the setup from [CBCP15]{a= 2π, b= 1} (h =π/8). At the beginning, the flow is governed by large vortices, that decay into smaller eddies with high vorticity magnitude|ω|.

In order to validate the appropriate choice of stabilization parameters, we show energy spectra at timet= 8/bforν = 10⁻⁴. In this regime, the flow can be assumed to be nearly isotropic, so we compare the spectra with Kolmogorov’s −5/3 law. Note that for a given mesh width h, the displayed spectrum cannot be resolved properly for larger frequencies thank∼h⁻¹.

Figure 5.19.: Energy spectra at t = 9 for different mesh widths with γ_M = 1 (left) and withγ_M = 1, τ_M^u = 1 (right),{a= 2π, b= 1}.

In Figure 5.19, the energy spectra with grad-div stabilization γ_M = 1 (left) and with grad-div combined with LPS SU γM = 1, τ_M^u = 1 (right) are presented for mesh widths h ∈ {π/4, π/8, π/16}. Grad-div stabilization alone does not provide enough dissipation because on all grids, one observes that the smallest resolved scales carry too much energy.

Additional LPS SU stabilization is more dissipative; the combination γM = 1, τ_M^u = 1 is therefore more suited as a turbulence model than grad-div stabilization only. Indeed, the classical Smagorinsky model (with optimized stabilization parameter) shows a similar behavior; see FigureB.16 in the appendix.

Now, we are interested if there are choices for τ_M^u that are more suited as a turbulence model than τ_M^u = 1. In order to be able to examine the influence of|u_M|and h, we scale the TGV problem such that Ω = (0,8/√

3)³ and ku₀k_∞ = 1/10; this means that we set a= 8/√

3 and b= 1/10. The viscosity of the problem isν = 10⁻⁵.

First, we consider how τ_M^u depends on |u_M|. The semi-discrete analysis yields an upper bound of τ_M^u ≤ |u_M|⁻²; the dimensional analysis of Section 5.6.3 suggests a choice as

k

10⁰ 10¹ 10²

E

10^-4 10^-3 10^-2 10^-1 10⁰

τ_M^u = 1/(2kuhk²_∞,M) τ^u

M= 1/(2kuhk∞,M) τ^u

M= 1 k^−5/3

Figure 5.20.: Energy spectra at t = 80, h = 1/(4√

3), ku₀k_∞ = 1/10 for considering the dependence of the LPS SU parameter on|u_M|.

τ_M^u ∼ h|u_M|⁻¹. Figure 5.20 shows the combination of grad-div stabilization γ_M = 1 with different LPS SU choices ku_hk^−γ_∞,M ≤ |u_M|^−γ with γ ∈ {0,1,2}. It illustrates that τ_M^u ∼ ku_hk⁻²_∞,M ≤ |u_M|⁻² is indeed too dissipative and thus unfeasible. Note that the choice τ_M^u = 1 is less dissipative on the small scales but shows more deviations from the

−5/3-law on the coarse scales than τ_M^u ∼ ku_hk⁻¹_∞,M.

Hence, we exclude τ_M^u ∼ |u_M|⁻² from our further contemplations and look at the h-dependence more closely. In Figure 5.21, stabilization as γ_M = 1, τ_M^u ∼ h^β/ku_hk_∞,M is shown for β ∈ {0,1} and for h = 1/(4√

3) (left) and h = 1/(8√

3) (right). The choice β = 1 yields much better results thanβ = 0 for both mesh sizes. So we can rule out that the improvement arises from a constant (i.e., h-independent) multiplicative factor in the LPS SU parameter. This is in good agreement with the knowledge from Section 5.6.3.

On the grid h = 1/(4√

3), we observe again that τ_M^u = 1 is less dissipative on the small scales but deviates more from the −5/3-law on the coarse scales than τ_M^u ∼h/ku_hk_∞,M. Surprisingly, for h = 1/(8√

3), these choices show very similar behavior, meaning that stabilization is needed most in cells where (8√

3)ku_hk_∞,M ≈1. Since even for the initial condition, it holds ku₀k_∞,M ≤ 0.1, this indicates cells of relatively large mean velocity within the domain. We remark that τ_M^u = 1 is not suited as a universal choice: A LPS SU parameter as 0.1 corrupts the errors in Section 5.3, where a different flow example is considered. Therefore, a parameter is desired that incorporates properties of the flow.

In summary, the numerical tests as well as the theoretical arguments show that for isotropic

k

Figure 5.21.: Energy spectra at t= 80, ku₀k_∞ = 1/10 for considering the dependence of the LPS SU parameter on h,h= 1/(4√

3) (left), h= 1/(8√

3) (right).

turbulence a choice of τ_M^u ∼ h|u_M|⁻¹ is suited to act as an implicit turbulence model;

grad-div stabilization alone is not dissipative enough.

Im Dokument Finite Element Methods with Local Projection Stabilization for Thermally Coupled Incompressible Flow (Seite 135-139)