The coupled steady state problem

From Lemmata 8.2, 8.3 8.6, and 8.7 we can infer the following Corollary:

Corollary 8.1

Assume that the assumptions of Lemmata 8.3 and 8.7 hold. Denote V²_def ≡ { w∈V₂ |w satisfies (8.55) } . (8.56)

Then the following operators are well-defined:

T_{N S→LES} : V¹_0,W −→ V₂ , u−→w, w solution of (8.17)−(8.19) , (8.57)

T_{LES→N S} : V²_def −→ V¹_0,W , w−→u, u solution of (8.20)−(8.22).

(8.58) Proof:

The statement follows immediately from Lemmata 8.2, 8.3, 8.6, and 8.7. ♦

An outline of this section reads as follows: Given that ||f||−1 is ”sufficiently small” orν is

”sufficiently large”, we can show:

• T_{LES→N S} is continuous on a suitable subset V²_cont ⊂V²_def.

• TN S→LES is continuous and compact on V¹_0,W, if ΓW is piecewise smooth.

• Using theSchauderFixed-Point Theorem, we show that T_{LES→N S}◦T_{N S→LES} and TN S→LES ◦TLES→N S have a fixed point; the fixed point is a solution of the coupled problem (8.11)-(8.16).

• The fixed point is uniquely determined, if an additional assumption regarding||f||−1

and ν holds.

Lemma 8.8

Suppose that the assumptions of Corollary 8.1 hold. We define V²_cont ≡ { w ∈ V²_def | w satisfies (8.60) } (8.59)

with

3ν

8 > C_b2

νC_P,i² K_a(f, ν,w) + C_bC₁(d,Ω_i)C_P,i||w||1/2,2,Γi . (8.60)

Then the operator T_{LES→N S} |V²cont

is continuous.

Proof:

Requiring the assumptions of Lemma 8.7, TLES→N S is well-defined. Let a sequence wn

in V²_def be given with w_n −→ w in H¹(Ω). Denote u_n ≡ T_{LES→N S}(w_n) and u ≡ T_{LES→N S}(w). Then we have to show that u_n−→u inH¹(Ω_i).

8.3. The coupled steady state problem

Based on (8.44) with Lemma 8.4, we write u_n = ˜u_n+W_n and u = ˜u+W, where W_n andW satisfy (8.49). Since

||un−u||1,Ωi ≤ ||u˜n−u˜||1,Ωi+||Wn−W||1,Ωi ≤ ||u˜n−u˜||1,Ωi+C1(d,Ωi)||wn−w||1/2,2,Γi

≤ ||u˜_n−u˜||1,Ωi+C₁(d,Ω_i)C_tr||w_n−w||1,Ωi

(8.61)

we can complete the proof by showing ||u˜_n−u˜||1,Ωi → 0 as n → ∞. Due to (8.51) the following equations hold

ν(∇u˜n,∇v) +b_Ωi( ˜un,u˜n,v) +b_Ωi( ˜un,Wn,v) +b_Ωi(Wn,u˜n,v)

= (f,v)−ν(∇Wn,∇v)−bΩi(Wn,Wn,v) ν(∇u,˜ ∇v) +bΩi( ˜u,u,˜ v) +bΩi( ˜u,W,v) +bΩi(W,u,˜ v)

= (f,v)−ν(∇W,∇v)−b_Ωi(W,W,v) .

Denoteφ_n≡u˜n−u˜ and Ψn≡Wn−W. Subtracting the second equation from the first one and settingv =φ_n gives

ν(∇φ_n,∇φ_n) +b_Ωi( ˜u_n,u˜_n,φ_n)−b_Ωi( ˜u,u,˜ φ_n) +b_Ωi( ˜u_n,W_n,φ_n)

−bΩi( ˜u,W,φ_n) +bΩi(Wn,u˜n,φ_n)−bΩi(W,u,˜ φ_n)

= −ν(∇Ψn,∇φ_n)−bΩi(Wn,Wn,φ_n) +bΩi(W,W,φ_n). Now we proceed using the following transformations:

bΩi( ˜un,u˜n,φ_n)−bΩi( ˜u,u,˜ φ_n) = bΩi(φ_n,u˜n,φ_n) ,

bΩi( ˜un,Wn,φ_n)−bΩi( ˜u,W,φ_n) = bΩi(φ_n,Wn,φ_n) +bΩi( ˜u,Ψn,φ_n) , b_Ωi(W_n,u˜_n,φ_n)−b_Ωi(W,u,˜ φ_n) = b_Ωi(Ψ_n,u˜_n,φ_n) ,

b_Ωi(W_n,W_n,φ_n)−b_Ωi(W,W,φ_n) = b_Ωi(Ψ_n,W_n,φ_n) +b_Ωi(W,Ψ_n,φ_n) . Then the following equation holds

ν(∇φ_n,∇φ_n) = −b_Ωi(φ_n,u˜_n,φ_n)−b_Ωi(φ_n,W_n,φ_n)−b_Ωi( ˜u,Ψ_n,φ_n)−b_Ωi(Ψ_n,u˜_n,φ_n)

−ν(∇Ψ_n,∇φ_n)−b_Ωi(Ψ_n,W_n,φ_n)−b_Ωi(W,Ψ_n,φ_n) . This yields the following inequality

ν(∇φ_n,∇φ_n) ≤ |bΩi(φ_n,u˜n,φ_n)|+|bΩi(φ_n,Wn,φ_n)|+|bΩi( ˜u,Ψn,φ_n)|+|bΩi(Ψn,u˜n,φ_n)| +|ν(∇Ψ_n,∇φ_n)|+|b_Ωi(Ψ_n,W_n,φ_n)|+|b_Ωi(W,Ψ_n,φ_n)|.

8. Some analytical results for LES with near wall modelling

The following estimates hold

|bΩi(φ_n,u˜n,φ_n)| ≤ C_b||φ_n||²1,Ωi||∇u˜n||L²(Ωi)

≤ C_b2

νC_P,i² K_a(f, ν,w_n)||∇φ_n||²_L²_(Ωi)

|b_Ωi(φ_n,W_n,φ_n)| = |b_Ωi(φ_n,φ_n,W_n)| ≤ C_b||φ_n||1,Ωi||∇φ_n||L²(Ωi)||W_n||1,Ωi

≤ C_bC₁(d,Ω_i)C_P,i||w_n||1/2,2,Γi||∇φ_n||²_L²_(Ωi)

|bΩi( ˜u,Ψn,φ_n)| = |bΩi( ˜u,φ_n,Ψn)| ≤ Cb||u˜||1,Ωi||Ψn||1,Ωi||∇φ_n||L²(Ωi)

≤ Cb

2

νKa(f, ν,w)CP,iC1(d,Ωi)||wn−w||1/2,2,Γi||∇φ_n||L²(Ωi)

≤ 8

ν³Ka(f, ν,w)²C_b²C₁²(d,Ωi)C_P,i² ||wn−w||²1/2,2,Γi+ ν

8||∇φ_n||²_L²_(Ωi)

|b_Ωi(Ψ_n,u˜_n,φ_n)| ≤ 8

ν³K_a(f, ν,w_n)²C_b²C₁²(d,Ω_i)C_P,i² ||w_n−w||²_1/2,2,Γi+ ν

8||∇φ_n||²_L²_(Ωi) ν|(∇Ψn,∇φ_n)| ≤ √

4ν||∇Ψn||L²(Ωi)

rν

4||∇φ_n||L²(Ωi)≤2ν||∇Ψn||²_L²_(Ωi)+ν

8||∇φ_n||²_L²_(Ωi)

≤ 2νC₁²(d,Ωi)||wn−w||²_1/2,2,Γi+ν

8||∇φ_n||²_L²_(Ωi)

|b_Ωi(Ψ_n,W_n,φ_n)| ≤ C_b||Ψ_n||1,Ωi||W_n||1,ΩiC_P,i||∇φ_n||L²(Ωi)

≤ C_bC1(d,Ωi)CP,i||wn−w||1/2,2,Γi||∇φ_n||L²(Ωi)C1(d,Ωi)||wn||1/2,2,Γi

≤ 2

νC_b²C_P,i² C₁⁴(d,Ωi)||wn−w||²_1/2,2,Γi||wn||²_1/2,2,Γi+ν

8||∇φ_n||²_L²_(Ωi)

|bΩi(W,Ψn,φ_n)| ≤ 2

νC_b²C_P,i² C₁⁴(d,Ωi)||wn−w||²_1/2,1,Γi||w||²_1/2,2,Γi+ν

8||∇φ_n||²_L²_(Ωi) . Putting it all together we arrive at the following inequality for φ_n, viz.,

ν||∇φ_n||²_L²_(Ωi)≤[5ν 8 +Cb

2

νC_P,i² Ka(f, ν,wn) +CbC1(d,Ωi)CP,i||wn||1/2,2,Γi]||∇φ_n||²_L²_(Ωi) + 8

ν³C_b²C₁²(d,Ωi)C_P,i² [Ka(f, ν,w)²+Ka(f, ν,wn)²]||wn−w||²1/2,2,Γi

+ 2νC₁²(d,Ω_i)||w_n−w||²_1/2,2,Γi + 2

νC_b²C_P,i² C₁⁴(d,Ωi)(||wn||²_1/2,2,Γi+||w||²_1/2,2,Γi)||wn−w||²_1/2,2,Γi . (8.62)

This can be rearranged to ( 3ν

8 −C_b2

νC_P,i² K_a(f, ν,w_n)−C_bC₁(d,Ω_i)C_P,i||w_n||1/2,2,Γi ) ||∇φ_n||²_L²_(Ωi)

≤ 8

ν³C_b²C₁²(d,Ωi)C_P,i² (Ka(f, ν,w)²+Ka(f, ν,wn)²)||wn−w||²_1/2,2,Γi + 2νC₁²(d,Ωi)||wn−w||²_1/2,2,Γi

+ 2

νC_b²C_P,i² C₁⁴(d,Ω_i)(||w_n||²_1/2,2,Γi +||w||²_1/2,2,Γi)||w_n−w||²_1/2,2,Γi . (8.63)

8.3. The coupled steady state problem

Since w_n −→ w, for each > 0 there exists N₀ ∈ N s.t. (||w_n−w||1/2,2,Γi) < for all n > N0. If (8.60) is satisfied then there exists N1∈Ns.t. the l.h.s. term in (. . .) in (8.63) is strictly positive for alln≥N1. Therefore∇φ_n→0 in L²(Ωi) and Poincare’s inequality impliesφ_n→0 inH¹(Ω_i) as n→ ∞. Together with (8.61) this gives the assertion. ♦

Remark 8.8

It is worthily comparing (8.55) and (8.60). Taking into account (8.55), (8.60) can be rewritten as

3ν

8 > C_b2

νC_P,i² C_u,1^ap||f||−1,Ωi+C_b2

νC_P,i² νC1(d,Ωi)||w||1/2,2,Γi

+Cb

2

νC_P,i² C_u,3^ap||w||²_1/2,2,Γi+CbC1(d,Ωi)CP,i||w||1/2,2,Γi

= C_b2

νC_P,i² C_u,1^ap||f||−1,Ωi+C_bC_P,i(2C_P,i+ 1)C₁(d,Ω_i)||w||1/2,2,Γi

+C_b2

νC_P,i² C_u,3^ap||w||²_1/2,2,Γi .

Typically, c_poi,0(Ω_i) = O(y_layer) and y_layer = O(ν^1/2) in the laminar case and y_layer = O(ν^1/5) in the turbulent case. Hence, in a first approximation CP,i≈1, and thus (8.60) is not an essentially stronger ”small data” restriction than (8.55). ♦ Lemma 8.9

Assume thatΓ_W is piecewiseC¹ smooth. Under the assumptions of Corollary 8.1 and that νe−C_bC_M² K_b(f, νe)>0 , with K_b(f, νe) ≡

r2

ν_eC_LES^ap ||f||−1,Ω

(8.64)

the operator T_{N S→LES} is continuous on V¹_0,W ={ v ∈H¹(Ωi) | ∇ ·v = 0 in Ωi, v·n= 0 onΓ_W }.

Proof:

Given a sequence u_n in V¹_0,W with u_n −→ u in H¹(Ω_i) as n → ∞. Denote w_n ≡ T_{N S→LES}(u_n) andw≡T_{N S→LES}(u). Then we have to show thatw_n−→winH¹(Ω) as n→ ∞. The proof takes two steps.

(1) Let δ be fixed. Given un−→uinH¹(Ωi), then βj(un)−→βj(u) in L^∞(ΓW).

(2) Given βj(un)−→βj(u) in L^∞(ΓW), then wn−→w inH¹(Ω).

So let u_n −→ u in H¹(Ω_i) be given. First we show that u_n ≡ g_δ∗u_n −→ u ≡ g_δ∗u (with g_δ being defined in (4.10)) in C^m(Ωi), for each fixed m. Denote α= (α1, . . . , α_d) a multiindex with|α|=m. First we extendu_nand uby zero onto R^d\Ω_i. Thenu_n and u are at least inE⁰(R^d). ThusDα(g_δ∗u_n) = (Dαg_δ)∗u_n,Dα(g_δ∗u) = (Dαg_δ)∗u, and

8. Some analytical results for LES with near wall modelling

Subtracting (8.66) from (8.65) gives (for all v inV₂) bΩ(wn,wn,v)−bΩ(w,w,v) + (νe∇(wn−w)),∇v) After expanding this can be rewritten as

b_Ω(Φ_n,w_n,v) +b_Ω(w,Φ_n,v) + (ν_e∇Φ_n,∇v)

8.3. The coupled steady state problem

Now we need estimates for the right hand side terms. First, we have

|b_Ω(Φ_n,w_n,Φ_n)| ≤ C_b||∇w_n||L²(Ω)||Φ_n||²1,Ω ≤ C_bC_M² K_b(f, ν_e)||∇Φ_n||²_L²_(Ω) , where we used Lemma 8.1 and (8.64) in the last step. Next we have for eachK >0

| Lemma 8.1 in the last step. Substituting the last two inequalities into (8.67) gives

ν_e−C_bC_M² K_b(f, ν_e)

We assume that Ω is sufficiently smooth s.t. there exists a continuous linear prolongation operatorΠ : W^m,p(Ω)−→ W^m,p(R^d), d= 2,3. Then, under the assumptions of Lemma 8.9 the operator TN S→LES is compact.

Proof:

We have to show that for each given bounded sequence (un) in V¹_0,W, i.e. ||un||1,Ωi < C for all n, the sequence (T_{N S→LES}(u_n)) contains a subsequence, that converges to some w inV2 in the norm ofH¹(Ω).

8. Some analytical results for LES with near wall modelling

Before proceeding it is worthwhile recalling theSchauder Fixed-Point Theorem, cf. [ZeiI], p.57: Let M be a nonempty, closed, bounded, convex subset of a Banach space X, and suppose T : M −→M is a compact operator. Then T has a fixed-point.

Now we can state a theorem concerning the existence of at least one solution of the steady state model for coupling LES and DNS. In all the preceding steps, all inequalities have been handled very carefully. In contrast, this theorem and its corresponding proof will be presented slightly lax. As will be seen in the proof, figuring out all inequalities involved in the proof is a sisyphus-like task without being necessary.

Theorem 8.1

If ν and νe are ”sufficiently large” and if ||f||−1,Ω and ||f||−1,Ωi are ”sufficiently small”, then T_{N S→LES}◦ T_{LES→N S} and T_{LES→N S}◦ T_{N S→LES} have at least one fixed-point.

Proof:

First we show the existence of a fixed-point of the operator

S₁ ≡ T_{N S→LES}◦ T_{LES→N S} : V¹_0,W ⊃ M₁ 7→ M₁ (8.68)

withM1 to be determined within the proof. The proof takes three steps.

I Show that there is a closed and bounded ball M₁⁰ ⊂V¹_0,W, s.t. S1|M₁⁰ is well-defined and continuous.

II Show that there is a closed and bounded ballM₁ ⊂M₁⁰, s.t. S₁(M₁)⊂M₁. III Show that S₁ is compact.

Then M₁ is nonempty, closed, bounded and convex. Thus the Schauder Fixed-Point The-orem ensures that there isu∈M1, s.t. S1(u) =u.

Ad I: T_{N S→LES} is well-defined on V¹_0,W. As a consequence of Lemmata 8.8 and 8.9, S₁ is well-defined and continuous on M₁⁰ ⊂V¹_0,W provided TN S→LES(M₁⁰) ⊂V²_cont. Figure 8.1 provides some illustration:

Therefore we have to ensure that w ∈ M₁⁰ satisfies the following conditions: First, from (8.55) we have to ensure(i)

ν

2 > CbC_P,i² 2

νKa(f, ν,w) . Second, from (8.60) we need(ii)

3ν

8 > C_b2

νC_P,i² K_a(f, ν,w) +C_bC₁(d,Ω_i)C_P,i||w||1/2,2,Γi .

8.3. The coupled steady state problem

T

_{NS LES}

T

_{LES NS}

M’

V

²

1

T

_{NS LES}

(

₁

)

cont

S (M )

1 ₁

V

_0,W¹

V

_0,W¹

M

₁

M

₁

M’

Figure 8.1.: Schematic representation of subspaces.

Due to the priori estimate in Lemma 8.1, we have

||w||1/2,2,Γi ≤CtrCM||∇w||L²(Ωi)≤CtrCM||∇w||L²(Ω)≤Ctr

C_M²

ν_e ||f||−1,Ω . (8.69)

Thus if ||f||²_−1,Ω is ”sufficiently small” and νe is ”sufficiently large”, then we can make

||w||1/2,2,Γi sufficiently small such that (i) and (ii) are satisfied. Therefore we can ensure existence of a closed and bounded ballM₁⁰ ⊂V¹_0,W, s.t. T_{N S→LES}(M₁⁰)⊂V²_cont, i.e.,S₁|M₁⁰

is well-defined.

Ad II: Now we have to show that there is a closed and bounded ballM1 ⊂M₁⁰ s.t. S1(M1)⊂ M1. AsM₁⁰ is a closed and bounded ball, there is ρ >0 s.t. B(0, ρ) ⊂ M₁⁰, with B(0, ρ) being the ball with radiusρ around the origin. From Lemma 8.5 we know that

||∇S1(u)||²L²(Ωi) ≤ 2

νKa(f, ν,w) . (8.70)

We can combine this with (8.69) giving the following estimate

||∇S1(u)||²L²(Ωi)≤ 2

νC_u,1^ap||f||−1,Ωi+C_u,2^apCtr

C_M²

ν_e ||f||−1,Ω+C_u,3^apC_tr² C_M⁴

ν_e² ||f||²_−1,Ω . (8.71)

Then we simply require for f and ν_e that the r.h.s. in (8.71) is smaller thanρ. Therefore there is a closed and bounded ballM1 ⊂M₁⁰ s.t. S1(M1)⊂M1.

Ad III: S₁ is a compact operator as T_{LES→N S} is continuous and T_{N S→LES} is continuous and compact.

From I-III we can infer that the operatorS₁ has at least one fixed-point.

8. Some analytical results for LES with near wall modelling

Secondly, we show the existence of a fixed-point of the operator S₂ ≡ T_{LES→N S} ◦ T_{N S→LES} : V²_cont 7→ V² .

This operator is well-defined, continuous and compact according to Lemmata 8.8, 8.9, and 8.10. From Lemma 8.1 we know the following a priori estimate for S₂(w), viz.,

||S₂(w)||²1,Ω ≤ C_M⁴

ν_e² ||f||²₋1,Ω .

Then we require thatν andf are such that the right hand side is smaller thanρ, satifying B(0, ρ)⊂V²_cont. Now we can apply the Schauder Fixed-Point Theorem. ♦ Remark 8.9

For proving Lemmata 8.9 and 8.10 we need that β(·,·) ≥β₀ >0 and β(·,·) is continuous

on R². ♦

Before giving the proof regarding uniqueness, we need a further result regardingβj(·).

Lemma 8.11

Assume ΓW is piecewise smooth. Then for all u1, u2 in H¹(Ωi)

||β_j(u₁)−β_j(u₂)||L^∞(ΓW) ≤ C_β(δ)||u₁−u₂||H¹(Ωi) , C_β(δ) =C_lcCδ^−5/2 . Proof:

For given u inC^∞(R^d) and fixed j = 1, . . . , d−1 we set y(u) ≡T r|ΓWu·tj and z(u) ≡ n^TT r|ΓWD(u)t_j. As Γ_W is piecewise smooth, y(u) and z(u) are in L^∞(Γ_W). So let u₁, u2 inV¹_0,W be given. Denote yi≡y(ui),zi≡z(ui), i= 1,2. Then for eachx in ΓW

|β(y₁(x), z₁(x))−β(y₂(x), z₂(x))| ≤ C_lcmax[|y₁(x)−y₂(x)|;|z₁(x)−z₂(x)|]. We will use the following embedding result, see [Gri85], p. 27, i.e.,

W^k+d/p,p(R^d) ⊂ C^k−1,α(R^d) , ∀ α ∈ [0,1[, k∈N, k≥1 .

Moreover we use the following estimate, see [JL01], p.271: Forf ∈L²(Ω),f = 0 onR^d\Ω, and f ≡g_δ∗f ∈C^∞(R^d) we have

||gδ∗f||W^k,2(R^d) ≤ C||gδ||W^k,1(R^d)||f||L²(Ω) ≤ C3δ^−k||f||L²(Ω) . Therefore we obtain

||β_j(u₁)−β_j(u₂)||L^∞(ΓW) = max

x∈ΓW |β(u₁)(x)−β(u₂)(x)|

≤ C_lc max x∈ΓW

max[ |y₁(x)−y₂(x)|;|z₁(x)−z₂(x)|]

= C_lcmax[||T r|ΓW(u1−u2)·tj||L^∞(ΓW),

||n^TT r|ΓW∇(u₁−u₂)t_j||L^∞(ΓW)]

≤ Clcmax[||u1−u2||L^∞(R^d),||∇(u1−u2)||L^∞(R^d)]

= C_lcmax[||u1−u2||C⁰(R^d),||∇(u1−u2)||C⁰(R^d)]

≤ C_lcCmax[||u₁−u₂||_W^5/2,2₍R^d),||∇(u₁−u₂)||_W^5/2,2₍R^d)]

≤ C_lcCδ^−5/2||u₁−u₂||1,Ωi .

8.3. The coupled steady state problem

♦

Theorem 8.2

Suppose that the assumptions of Theorem 8.1 hold. Moreover we assume that there exists >0 s.t. β₀−/2>0 and

Then the solution(w,u) of (8.11)-(8.16)is unique.

Proof: Subtracting the second equation from the first one, expanding, and settingv =w gives

ν_e(∇w,∇w) + Therefore we arrive at the following estimate

νe(∇w,∇w) +

8. Some analytical results for LES with near wall modelling

Now the goal is to bound both r.h.s. terms. Regarding the former we have

|b_Ω(w,w1,w)| ≤ C_bC_M³

ν_e ||f||−1,Ω||∇w||²_L²_(Ω) . (8.75)

Concerning the latter the following estimate holds:

d−1

X

j=1

|h[βj(TLES→N S(w1))−βj(TLES→N S(w2))]w1·tj,w·tjiΓW|

≤

d−1

X

j=1

||βj(TLES→N S(w1))−βj(TLES→N S(w2))||L^∞(ΓW)||w1·tj||L²(ΓW)||w·tj||L²(ΓW)

≤ 1 2

d−1

X

j=1

||βj(TLES→N S(w1))−βj(TLES→N S(w2))||²_L∞(ΓW)||w1·tj||²_L²_(ΓW)

+

2

d−1

X

j=1

||w·tj||²_L²_(ΓW)

≤ 1 2

d−1

X

j=1

||βj(T_{LES→N S}(w1))−βj(T_{LES→N S}(w2))||²_L∞(ΓW)C_tr² C_M⁴

ν_e² ||f||²_−1,Ω

+

2

d−1

X

j=1

||w·tj||²_L²_(ΓW) ,

where we bounded theL²(Γ_W) norm by theW^1/2,2,ΓW norm, cf. [Otto99], p.159 in the last step. From Lemma 8.11 we know that (for each j= 1, . . . , d−1)

||βj(TLES→N S(w1))−βj(TLES→N S(w2))||²L^∞(Γ_W)

≤C_β²(δ)||TLES→N S(w1)−TLES→N S(w2)||²1,Ωi . Thus we need an estimate for the term ||T_{LES→N S}(w₁)−T_{LES→N S}(w₂)||1,Ωi. We write T_{LES→N S}(wk) =u˜k+Wk (k = 1,2), with Wk denoting the Hopf extension. Moreover, we write ˜u≡u˜1−u˜2. Then

||T_{LES→N S}(w₁)−T_{LES→N S}(w₂)||²1,Ωi ≤ 2||W₁−W₂||²1,Ωi+ 2||u˜₁−u˜₂||²1,Ωi

≤ 2C₁²(d,Ω_i)||w₁−w₂||²_1/2,2,Γi+ 2||u˜₁−u˜₂||²1,Ωi

≤ 2C₁²(d,Ωi)||w1−w2||²1/2,2,Γi+ 2C_P,i² ||∇u˜||²_L²_(Ωi) .

8.3. The coupled steady state problem

Combining the last five inequalities, we arrive at ν_e||∇w||²_L²_(Ω)+

Combining trace inequality and the a priori estimate forw_i (cf. (8.27)), we obtain

||wi||²_1/2,2,Γi ≤ C_tr²||wi||²_1,(Ω\Ωi) ≤ C_tr²||wi||²1,Ω ≤ C_tr²C_M⁴

ν_e² ||f||²_−1,Ω . Inserting this into (8.77) gives (using (8.73))

3ν

8. Some analytical results for LES with near wall modelling

Finally we apply the following estimate, videlicet,

||w||²_1/2,2,Γi ≤ C_tr²||w||²_1,(Ω\Ωi) ≤ C_tr²C_M² ||∇w||²_L²_(Ω) . (8.81)

Substituting this, (8.80) can be rearranged to (νe−Kc) ||∇w||²_L²_(Ω) + withK₂ being defined in (8.79) and

K_c ≡ C_bC_M³

In this section the following issues are discussed, viz., the slip with resistance boundary condition for LES, the matching condition on Γ_i the corresponding steady state problem and its simplification, and finally how to use this coupling scheme in a computational model.

8.4.1. The slip with resistance boundary condition for LES

Traditionally, boundary condition (7.5) has been used in LES with near wall modelling.

In [GL00] Galdi and Layton proposed the following slip with linear friction and no

Im Dokument Finite-element simulation of buoyancy-driven turbulent flows (Seite 94-106)