Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Václav Kučera vaclav.kucera@email.cz Univerzita Karlova v Praze Matematicko-fyzikální Fakulta Katedra numerické matematiky Vytvoření a rozvoj týmu pro náročné technické výpočty na paralelních počítačích na TU v Liberci, 21. června 2010 Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 1 / 39
Let Ω R 2 be a bounded domain with boundary Ω = Γ I Γ O Γ W. Find w : Q T = Ω (0, T ) R 4 such that where w t + f s (w) = w = (ρ, ρv 1, ρv 2, e) T R 4, R s (w, w) in Q T, f i (w) = (ρv i, ρv 1 v i + δ 1i p, ρv 2 v i + δ 2i p, (e + p)v i ) T, R i (w, w) = (0, τ i1, τ i2, τ i1 v 1 + τ i2 v 2 + k θ/ x i ) T, τ ij = λδ ij divv + 2µd ij (v), d ij (v) = 1 2 ( vi x j + v j x i ). Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 2 / 39
Let Ω R 2 be a bounded domain with boundary Ω = Γ I Γ O Γ W. Find w : Q T = Ω (0, T ) R 4 such that where w t + f s (w) = w = (ρ, ρv 1, ρv 2, e) T R 4, R s (w, w) in Q T, f i (w) = (ρv i, ρv 1 v i + δ 1i p, ρv 2 v i + δ 2i p, (e + p)v i ) T, R i (w, w) = (0, τ i1, τ i2, τ i1 v 1 + τ i2 v 2 + k θ/ x i ) T, τ ij = λδ ij divv + 2µd ij (v), d ij (v) = 1 2 ( vi x j + v j x i ). Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 2 / 39
Let Ω R 2 be a bounded domain with boundary Ω = Γ I Γ O Γ W. Find w : Q T = Ω (0, T ) R 4 such that where w t + f s (w) = w = (ρ, ρv 1, ρv 2, e) T R 4, R s (w, w) in Q T, f i (w) = (ρv i, ρv 1 v i + δ 1i p, ρv 2 v i + δ 2i p, (e + p)v i ) T, R i (w, w) = (0, τ i1, τ i2, τ i1 v 1 + τ i2 v 2 + k θ/ x i ) T, τ ij = λδ ij divv + 2µd ij (v), d ij (v) = 1 2 ( vi x j + v j x i ). Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 2 / 39
We add the thermodynamical relations p = (γ 1)(e ρ v 2 /2), θ = and the following set of boundary conditions: ( e ρ 1 ) 2 v 2 /c v. Case Γ I : a) ρ ΓI (0,T ) = ρ D, b) v ΓI (0,T ) = v D = (v D1, v D2 ) T, ( ) c) τ ij n i v j + k θ n = 0 on Γ I (0, T ); j=1 i=1 Case Γ W : a) v ΓW (0,T ) = 0, b) θ n = 0 on Γ W (0, T ); Case Γ O : a) τ ij n i = 0, j = 1, 2, b) θ n = 0 on Γ O (0, T ); i=1 Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 3 / 39
Let T h be a partition of the closure Ω t into a finite number of closed triangles K T h. By F h we denote the set of all edges of T h. For a given edge Γ F h we define a unit normal n Γ. Γ 8 n Γ8 K 1 Γ 6 n Γ6 Γ 5 n Γ1 K 2 n Γ5 Γ 1 K 5 Γ2 n Γ2 n Γ7 Γ 7 Γ 4 Γ 3 K 3 n Γ4 n Γ3 K 4 Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 4 / 39
Let T h be a partition of the closure Ω t into a finite number of closed triangles K T h. By F h we denote the set of all edges of T h. For a given edge Γ F h we define a unit normal n Γ. Γ 8 n Γ8 K 1 Γ 6 n Γ6 Γ 5 n Γ1 K 2 n Γ5 Γ 1 K 5 Γ2 n Γ2 n Γ7 Γ 7 Γ 4 Γ 3 K 3 n Γ4 n Γ3 K 4 Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 4 / 39
For each interior face Γ F h there exist two neighbours K (L) Γ, K (R) Γ T h. We use the convention that n Γ is the outer normal to the element K (L) Γ. v (L) = trace of v (L) K Γ v (R) = trace of v (L) K Γ [v] Γ = v (L) v (R), v Γ = 1 2( v (L) + v (R)). on Γ, on Γ, Γ n Γ K (L) Γ K (R) Γ Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 5 / 39
For each interior face Γ F h there exist two neighbours K (L) Γ, K (R) Γ T h. We use the convention that n Γ is the outer normal to the element K (L) Γ. v (L) = trace of v (L) K Γ v (R) = trace of v (L) K Γ [v] Γ = v (L) v (R), v Γ = 1 2( v (L) + v (R)). on Γ, on Γ, Γ n Γ K (L) Γ K (R) Γ Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 5 / 39
Over T h we define the broken Sobolev space H k (Ω, T h ) = {v; v K H k (K) K T ht } We discretize the continuous problem in the space of discontinuous piecewise polynomial functions S h = {v; v K P p (K) K T ht }, where P p (K) is the space of all polynomials on K of degree p. In order to derive a variational formulation, we multiply the N-S equations by a test function ϕ H 2 (Ω, T h ), apply Green s theorem on individual elements and sum over all elements. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 6 / 39
Over T h we define the broken Sobolev space H k (Ω, T h ) = {v; v K H k (K) K T ht } We discretize the continuous problem in the space of discontinuous piecewise polynomial functions S h = {v; v K P p (K) K T ht }, where P p (K) is the space of all polynomials on K of degree p. In order to derive a variational formulation, we multiply the N-S equations by a test function ϕ H 2 (Ω, T h ), apply Green s theorem on individual elements and sum over all elements. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 6 / 39
Over T h we define the broken Sobolev space H k (Ω, T h ) = {v; v K H k (K) K T ht } We discretize the continuous problem in the space of discontinuous piecewise polynomial functions S h = {v; v K P p (K) K T ht }, where P p (K) is the space of all polynomials on K of degree p. In order to derive a variational formulation, we multiply the N-S equations by a test function ϕ H 2 (Ω, T h ), apply Green s theorem on individual elements and sum over all elements. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 6 / 39
Convective terms w t + f s (w) = R s (w, w) We multiply the convective term by a test function ϕ H 2 (Ω, T h ), integrate over K, apply Green s theorem: K We sum over all K T h : K T h K f s (w) ϕ dx + K f s (w) ϕ dx + F h f s (w)n s ϕ ds, f s (w)n s [ϕ] ds, Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 7 / 39
Convective terms w t + f s (w) = R s (w, w) We multiply the convective term by a test function ϕ H 2 (Ω, T h ), integrate over K, apply Green s theorem: K We sum over all K T h : K T h K f s (w) ϕ dx + K f s (w) ϕ dx + F h f s (w)n s ϕ ds, f s (w)n s [ϕ] ds, Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 7 / 39
Convective terms w t + f s (w) = R s (w, w) In the second term, incorporate a numerical flux H: F h f s (w)n s [ϕ] ds H(w (L), w (R), n) [ϕ] ds, F h e.g. Lax-Friedrichs H(w (L), w (R), n) =. 1( fs (w (L) )n s +f s (w (R) ) ( )n s +α w (L) w R) 2 Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 8 / 39
Convective terms w t + f s (w) = R s (w, w) In the second term, incorporate a numerical flux H: F h f s (w)n s [ϕ] ds H(w (L), w (R), n) [ϕ] ds, F h e.g. Lax-Friedrichs H(w (L), w (R), n) =. 1( fs (w (L) )n s +f s (w (R) ) ( )n s +α w (L) w R) 2 Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 8 / 39
Convective terms w t + f s (w) = Finally, we define the convective form: b h (w, ϕ) = K T h K R s (w, w) f s (w) ϕ dx+ H(w (L), w (R), n) [ϕ] ds, F h If Γ Ω, then w (R) is not defined. By providing w (R), we impose boundary conditions in some weak sense. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 9 / 39
Convective terms w t + f s (w) = Finally, we define the convective form: b h (w, ϕ) = K T h K R s (w, w) f s (w) ϕ dx+ H(w (L), w (R), n) [ϕ] ds, F h If Γ Ω, then w (R) is not defined. By providing w (R), we impose boundary conditions in some weak sense. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 9 / 39
Diffusion terms w t + f s (w) = R s (w, w) Question How does one discretize second order terms using spaces of discontinuous functions? Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 10 / 39
Diffusion terms w t + f s (w) = R s (w, w) Question How does one discretize second order terms using spaces of discontinuous functions? Answer Treat the second order terms as a first order system and apply the discretization from the previous slide. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 10 / 39
Model problem R s (w, w) = g. Due to properties of R s (w, w) we can write ( K sk (w) w ) = g. x k k=1 We introduce an auxiliary variable σ k and write ( ) K sk (w)σ k = g, k=1 σ k = w x k, k = 1, 2. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 11 / 39
Model problem R s (w, w) = g. Due to properties of R s (w, w) we can write ( K sk (w) w ) = g. x k k=1 We introduce an auxiliary variable σ k and write ( ) K sk (w)σ k = g, k=1 σ k = w x k, k = 1, 2. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 11 / 39
Model problem R s (w, w) = g. Due to properties of R s (w, w) we can write ( K sk (w) w ) = g. x k k=1 We introduce an auxiliary variable σ k and write ( ) K sk (w)σ k = g, k=1 σ k = w x k, k = 1, 2. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 11 / 39
Model problem ( ) K sk (w)σ k = g, k=1 σ k = w x k, k = 1, 2. This first order system for unknowns w, σ 1, σ 2 can be discretized using the discontinuous Galerkin method. Different choices of the numerical flux for this system give different numerical schemes. If the numerical flux is appropriately chosen, it is possible to eliminate σ from the resulting numerical scheme. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 12 / 39
Model problem ( ) K sk (w)σ k = g, k=1 σ k = w x k, k = 1, 2. This first order system for unknowns w, σ 1, σ 2 can be discretized using the discontinuous Galerkin method. Different choices of the numerical flux for this system give different numerical schemes. If the numerical flux is appropriately chosen, it is possible to eliminate σ from the resulting numerical scheme. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 12 / 39
Nonsymmetric variant of the diffusion form w t + a N h (w, ϕ) = K T h Fh I + K f s (w) = R s (w, w) ϕ dx R s (w, w) n s [ϕ] ds Fh I R s (w, ϕ) n s [w] ds + Here Rk (w, ϕ) := F D h F D h R s (w, w) R s (w, w)n s ϕ ds R s (w, ϕ)n s w ds, K T sk(w) ϕ and R s (w, w) = k=1 K sk (w) w. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 13 / 39
Nonsymmetric variant of the diffusion form w t + a N h (w, ϕ) = K T h Fh I + K f s (w) = R s (w, w) ϕ dx R s (w, w) n s [ϕ] ds Fh I R s (w, ϕ) n s [w] ds + Here Rk (w, ϕ) := F D h F D h R s (w, w) R s (w, w)n s ϕ ds R s (w, ϕ)n s w ds, K T sk(w) ϕ and R s (w, w) = k=1 K sk (w) w. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 13 / 39
Nonsymmetric variant of the diffusion form w t + a N h (w, ϕ) = K T h Fh I + K f s (w) = R s (w, w) ϕ dx R s (w, w) n s [ϕ] ds Fh I R s (w, ϕ) n s [w] ds + F D h F D h R s (w, w) R s (w, w)n s ϕ ds R s (w, ϕ)n s w ds, Nonsymmetric, coercive and suboptimal convergence rate in L 2 -norm for even p. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 13 / 39
Symmetric variant of the diffusion form w t + f s (w) = R s (w, w) a N h (w, ϕ) = K T h Fh I K R s (w, w) ϕ dx R s (w, w) n s [ϕ] ds Fh I R s (w, ϕ) n s [w] ds F D h F D h R s (w, w)n s ϕ ds R s (w, ϕ)n s w ds, Symmetric, not coercive and optimal convergence rate in L 2 -norm. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 14 / 39
Symmetric variant of the diffusion form w t + f s (w) = R s (w, w) a N h (w, ϕ) = K T h Fh I K R s (w, w) ϕ dx R s (w, w) n s [ϕ] ds Fh I R s (w, ϕ) n s [w] ds F D h F D h R s (w, w)n s ϕ ds R s (w, ϕ)n s w ds, Symmetric, not coercive and optimal convergence rate in L 2 -norm. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 14 / 39
Symmetric variant of the diffusion form w t + f s (w) = R s (w, w) a N h (w, ϕ) = K T h Fh I K R s (w, w) ϕ dx R s (w, w) n s [ϕ] ds Fh I R s (w, ϕ) n s [w] ds F D h F D h R s (w, w)n s ϕ ds R s (w, ϕ)n s w ds, Red terms are a result of applying a numerical flux to the first order system. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 14 / 39
Incomplete variant of the diffusion form w t + f s (w) = R s (w, w) a N h (w, ϕ) = K T h Fh I K R s (w, w) ϕ dx R s (w, w) n s [ϕ] ds F D h R s (w, w)n s ϕ ds Not symmetric, not coercive and suboptimal convergence rate in L 2 -norm for even p. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 15 / 39
Incomplete variant of the diffusion form w t + f s (w) = R s (w, w) a N h (w, ϕ) = K T h Fh I K R s (w, w) ϕ dx R s (w, w) n s [ϕ] ds F D h R s (w, w)n s ϕ ds Not symmetric, not coercive and suboptimal convergence rate in L 2 -norm for even p. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 15 / 39
Incomplete variant of the diffusion form w t + f s (w) = R s (w, w) a N h (w, ϕ) = K T h Fh I K R s (w, w) ϕ dx R s (w, w) n s [ϕ] ds F D h R s (w, w)n s ϕ ds Simplest DG discretization of second order terms. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 15 / 39
Interior and boundary penalty In theory and in practice we need to add the interior and boundary penalty jump terms: 1 J h (w, ϕ) = C W Γ [w][ϕ] ds + C 1 W wϕ ds. Γ F I h This term ensures coercivity, when the constant C W is chosen sufficiently large. The boundary term is balanced on the right-hand side by 1 C W Γ w Bϕ ds, thus enforcing Dirichlet boundary conditions. F D h F D h Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 16 / 39
Interior and boundary penalty In theory and in practice we need to add the interior and boundary penalty jump terms: 1 J h (w, ϕ) = C W Γ [w][ϕ] ds + C 1 W wϕ ds. Γ F I h This term ensures coercivity, when the constant C W is chosen sufficiently large. The boundary term is balanced on the right-hand side by 1 C W Γ w Bϕ ds, thus enforcing Dirichlet boundary conditions. F D h F D h Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 16 / 39
Discrete Problem Definition We say that w h is a DGFE solution of the compressible Navier-Stokes equations if a) w h C 1 ([0, T ]; S h ), b) d dt (w h(t), ϕ h ) + b h (w h (t), ϕ h ) + J h (w h (t), ϕ h ) + a h (w h (t), ϕ h ) = l h (w h, ϕ h )(t), ϕ h S h, t (0, T ), c) w h (0) = w 0 h, where w 0 h is an S h approximation of the initial condition w 0. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 17 / 39
Semi-implicit d dt (w h, ϕ) + b h (w h, ϕ) + J h (w h, ϕ) + a h (w h, ϕ) = l h (w h, ϕ) A fully implicit scheme requires the solution of a nonlinear system. In the semi-implicit scheme we linearize the nonlinear terms using their specific properties. We solve only one linear system per time level. The scheme has good stability properties. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 18 / 39
Semi-implicit d dt (w h, ϕ) + b h (w h, ϕ) + J h (w h, ϕ) + a h (w h, ϕ) = l h (w h, ϕ) A fully implicit scheme requires the solution of a nonlinear system. In the semi-implicit scheme we linearize the nonlinear terms using their specific properties. We solve only one linear system per time level. The scheme has good stability properties. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 18 / 39
Semi-implicit d dt (w h, ϕ) + b h (w h, ϕ) + J h (w h, ϕ) + a h (w h, ϕ) = l h (w h, ϕ) We introduce a partition 0 = t 0 < t 1 < < t N = T and define τ n = t n+1 t n. Time derivative: d ( wh (t n+1 ), ϕ ) wn+1 h wh n dt τ n Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 18 / 39
Semi-implicit d dt (w h, ϕ) + b h (w h, ϕ) + J h (w h, ϕ) + a h (w h, ϕ) = l h (w h, ϕ) We introduce a partition 0 = t 0 < t 1 < < t N = T and define τ n = t n+1 t n. Time derivative: d ( wh (t n+1 ), ϕ ) wn+1 h wh n dt τ n Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 18 / 39
Semi-implicit d dt (w h, ϕ) + b h (w h, ϕ) + J h (w h, ϕ) + a h (w h, ϕ) = l h (w h, ϕ) Convective terms: K T h K f s (w n+1 ) ϕ dx+ H(w (L),n+1, w (R),n+1, n) ϕ ds F h Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 19 / 39
Semi-implicit d dt (w h, ϕ) + b h (w h, ϕ) + J h (w h, ϕ) + a h (w h, ϕ) = l h (w h, ϕ) Convective terms: K T h K It holds that f s (w n+1 ) ϕ dx+ H(w (L),n+1, w (R),n+1, n) ϕ ds F h f s (w) = A s (w)w, We therefore linearize where A s (w) = Df s(w) Dw. f s (w n+1 ) A s (w n )w n+1. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 19 / 39
Semi-implicit d dt (w h, ϕ) + b h (w h, ϕ) + J h (w h, ϕ) + a h (w h, ϕ) = l h (w h, ϕ) Convective terms: K T h K f s (w n+1 ) ϕ dx+ H(w (L),n+1, w (R),n+1, n) ϕ ds F h We choose the Vijayasundaram numerical flux for f H(w (L), w (R), n) = P + ( w, n) w (L) + P ( w, n) w (L) and linearize H(w n+1 (L) Γ, wn+1 (R) Γ, n Γ) P + ( w n, n Γ ) w n+1 (L) Γ +P ( w n, n Γ ) w n+1 (R) Γ. P ± = TD ± T 1, where A s n s = TDT 1. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 19 / 39
Semi-implicit d dt (w h, ϕ) + b h (w h, ϕ) + J h (w h, ϕ) + a h (w h, ϕ) = l h (w h, ϕ) Interior and boundary penalty jump terms are linear J h (w n+1, ϕ) = 1 C W Γ [wn+1 ][ϕ] ds + C W F I h F D h 1 Γ wn+1 ϕ ds. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 19 / 39
Semi-implicit d dt (w h, ϕ) + b h (w h, ϕ) + J h (w h, ϕ) + a h (w h, ϕ) = l h (w h, ϕ) Diffusion terms (for instance incomplete variant): a I h(w, ϕ) = K T h Fh I K R s (w, w) ϕ dx R s (w, w) n s [ϕ] ds It holds that R s (w, w) = We can linearize k=1 R s (w n+1, w n+1 ) F D h R s (w, w)n s ϕ ds. K sk (w) w. k=1 K sk (w n ) wn+1. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 19 / 39
Boundary Conditions - Inlet, outlet Semi-implicit BCs at Γ I, Γ O are imposed by choosing the outside boundary state w (R) in the numerical flux. Appropriate coordinate system, neglecting the tangential derivatives and linearization give: q t + f 1(q) = 0 q x 1 t + A 1(q i ) q = 0, x 1 We seek q j such that the linearized problem has sense. Eigenvectors of A 1 (q i ) form a basis and eigenvalues are real. q i = 4 α s r s, q j = 4 β s r s. Substitution into the system reduces it to four independent equations that have an analytical solution. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 20 / 39
Boundary Conditions - Inlet, outlet Semi-implicit BCs at Γ I, Γ O are imposed by choosing the outside boundary state w (R) in the numerical flux. Appropriate coordinate system, neglecting the tangential derivatives and linearization give: q t + f 1(q) = 0 q x 1 t + A 1(q i ) q = 0, x 1 We seek q j such that the linearized problem has sense. Eigenvectors of A 1 (q i ) form a basis and eigenvalues are real. q i = 4 α s r s, q j = 4 β s r s. Substitution into the system reduces it to four independent equations that have an analytical solution. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 20 / 39
Boundary Conditions - Inlet, outlet Semi-implicit BCs at Γ I, Γ O are imposed by choosing the outside boundary state w (R) in the numerical flux. Appropriate coordinate system, neglecting the tangential derivatives and linearization give: q t + f 1(q) = 0 q x 1 t + A 1(q i ) q = 0, x 1 We seek q j such that the linearized problem has sense. Eigenvectors of A 1 (q i ) form a basis and eigenvalues are real. q i = 4 α s r s, q j = 4 β s r s. Substitution into the system reduces it to four independent equations that have an analytical solution. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 20 / 39
Boundary Conditions - Inlet, outlet Semi-implicit BCs at Γ I, Γ O are imposed by choosing the outside boundary state w (R) in the numerical flux. Appropriate coordinate system, neglecting the tangential derivatives and linearization give: q t + f 1(q) = 0 q x 1 t + A 1(q i ) q = 0, x 1 We seek q j such that the linearized problem has sense. Eigenvectors of A 1 (q i ) form a basis and eigenvalues are real. q i = 4 α s r s, q j = 4 β s r s. Substitution into the system reduces it to four independent equations that have an analytical solution. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 20 / 39
Boundary Conditions - Inlet, outlet Semi-implicit BCs at Γ I, Γ O are imposed by choosing the outside boundary state w (R) in the numerical flux. Appropriate coordinate system, neglecting the tangential derivatives and linearization give: q t + f 1(q) = 0 q x 1 t + A 1(q i ) q = 0, x 1 We seek q j such that the linearized problem has sense. Eigenvectors of A 1 (q i ) form a basis and eigenvalues are real. q i = 4 α s r s, q j = 4 β s r s. Substitution into the system reduces it to four independent equations that have an analytical solution. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 20 / 39
Boundary Conditions - Inlet, outlet Semi-implicit Conclusion: depending on the sign of eigenvalues of A 1 (q i ) we either prescribe or extrapolate α s, β s When prescribing β s, we evaluate from an appropriate state (e.g. far-field). Finally q j := q i = Tα α = T 1 q i, q 0 j = Tβ β = T 1 q 0 j. 4 γ s r s = Tγ, where γ s = { α s, λ s 0, β s, λ s < 0. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 21 / 39
Boundary Conditions - Inlet, outlet Semi-implicit Conclusion: depending on the sign of eigenvalues of A 1 (q i ) we either prescribe or extrapolate α s, β s When prescribing β s, we evaluate from an appropriate state (e.g. far-field). Finally q j := q i = Tα α = T 1 q i, q 0 j = Tβ β = T 1 q 0 j. 4 γ s r s = Tγ, where γ s = { α s, λ s 0, β s, λ s < 0. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 21 / 39
Boundary Conditions - Inlet, outlet Semi-implicit Conclusion: depending on the sign of eigenvalues of A 1 (q i ) we either prescribe or extrapolate α s, β s When prescribing β s, we evaluate from an appropriate state (e.g. far-field). Finally q j := q i = Tα α = T 1 q i, q 0 j = Tβ β = T 1 q 0 j. 4 γ s r s = Tγ, where γ s = { α s, λ s 0, β s, λ s < 0. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 21 / 39
Shock Capturing Semi-implicit In transonic and supersonic flows it is common that solutions develop discontinuities. In these cases spurious under and overshoots occur on elements near the discontinuity. Especially in the semi-implicit case, it is desirable to avoid such phenomena. We therefore locally add artificial diffusion to suppress these effects. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 22 / 39
Shock Capturing Semi-implicit To the scheme we add two artificial viscosity forms. Internal diffusion: Φ 1 h (wn h, wn+1 h, ϕ) = ν 1 h K G n (K) w n+1 h ϕ dx K T K h with ν 1 = O(1) a given constant. Here G(K) is a discontinuity indicator which measures interelement jumps of the solution: { G k 1 if interelement jumps of wh n (K) = are large near K i, 0 otherwise. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 23 / 39
Shock Capturing Semi-implicit Interelement diffusion: Φ 2 h (wn h, wn+1 h, ϕ) = ν 2 G n [w n+1 Fh I h ] [ϕ] ds, with ν 2 = O(1) a given constant. This term allows to strengthen the influence of neighbouring elements and improves the behavior of the method in the case, when strongly unstructured and/or anisotropic meshes are used. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 24 / 39
Flow around cylinder, M = 10 4 Semi-implicit Figure: Velocity isolines of exact and numerical solution, respectively. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 25 / 39
Flow around cylinder, M = 10 4 Semi-implicit 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 1 0 1 2 3 4 Figure: Velocity distribution on cylinder surface. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 26 / 39
Semi-implicit Corner eddies around cylinder, M = 10 4 L.E. Fraenkel: On Corner Eddies in Plane Inviscid Shear Flow, 1961 Figure: Streamlines, exact solution. Figure: Streamlines, numerical solution. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 27 / 39
Semi-implicit Corner eddies around cylinder, M = 10 4 L.E. Fraenkel: On Corner Eddies in Plane Inviscid Shear Flow, 1961 0.8 0.6 0.4 0.2 0 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Figure: Velocity distribution on cylinder surface: exact solution of incompressible flow, numerical solution. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 28 / 39
Semi-implicit Flow around Žukovsky profile, M = 10 4 Figure: Velocity isolines: exact solution of incompressible flow, numerical solution. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 29 / 39
Semi-implicit Flow around Žukovsky profile, M = 10 4 1.4 1.2 1 0.8 0.6 0.4 0.2 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Figure: Velocity distribution on profile surface: exact solution of incompressible flow, numerical solution. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 30 / 39
Semi-implicit Flow around Žukovsky profile, M = 10 4 4 2 0 2 4 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Figure: Pressure distribution on profile surface: exact solution of incompressible flow, numerical solution. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 31 / 39
Semi-implicit Supersonic flow around Žukovsky profile, M = 2.0 Figure: M = 0.8, Mach number isolines. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 32 / 39
Semi-implicit Supersonic flow around Žukovsky profile, M = 2.0 Figure: M = 2.0, Mach number isolines. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 33 / 39
Flow in GAMM channel, M = 0.67 Semi-implicit 1 0.8 0.6 0.4 0.2 0 Figure: Mach number isolines. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 34 / 39
Flow in GAMM channel, M = 0.67 Semi-implicit 1 0.8 0.6 0.4 0.2 0 Figure: Entropy isolines. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 35 / 39
Flow in GAMM channel, M = 0.67 Semi-implicit 1 0.8 0.6 0.4 0.2 0 Figure: Entropy. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 36 / 39
Flow in GAMM channel, M = 0.67 Semi-implicit Figure: Entropy. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 37 / 39
NACA 0012 viscous flow Semi-implicit Figure: M = 0.5, Re = 5000, α = 2, Mach isolines. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 38 / 39
NACA 0012 viscous flow Semi-implicit Figure: M = 0.5, Re = 5000, α = 25, Mach isolines. Václav Kučera Nespojitá Galerkinova metoda pro řešení stlačitelného proudění Liberec 2010 39 / 39