Test case C1.3: Flow over Naca0012 airfoil

Aravind Balan, Michael Woopen, Jochen Schutz and Georg May

AICES Graduate School, RWTH Aachen University, Germany

2nd International Workshop on High-Order CFD Methods

May 27, 2013

Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 1 / 14

Outline

1 Solver Description

2 Discontinuous Galerkin

3 Hybridized Discontinous Galerkin

4 Numerical Results

Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 2 / 14

Solver Description

Hybridized Discontinuous Galerkin method for convection-diffusion equations

∇ · (fc(u)− fv(u,∇u)) = S

Hybridization to reduce the globally coupled degrees of freedom

λ ≈ u|ΓLocal solvers to solve for u from λ

Adjoint-based error estimator

Netgen mesh generator, Ngsolve FE library

Nonlinear equations solved by Damped Newton procedure

GMRES with ILU(n), using PETSc

Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 3 / 14

Discontinous GalerkinGeneral convection-diffusion equation

σ = ∇u∇ · (fc(u)− fv(u, σ)) = s(u, σ)

The solution spaces : uh ∈ Vh, σh ∈ Hh

Vh := ϕ ∈ L2(Ω) : ϕ|Ωk ∈ Pm(Ωk)

Hh := τ ∈ L2(Ω)× L2(Ω) : τ |Ωk ∈ Pm(Ωk)× Pm(Ωk)

Discontinuous Galerkin method

∑k

∫Ωk

σh · τ +

∫Ωk

(∇ · τ )uh −∫∂Ωk

(τ · n)uh = 0

∑k

∫Ωk

−(fc − fv) · ∇ϕ+

∫∂Ωk

ϕ(fc − fv)−∫

Ωk

(∇ · σh)ϕ =∑k

∫Ωk

sϕ

Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 4 / 14

Discontinous GalerkinGeneral convection-diffusion equation

σ = ∇u∇ · (fc(u)− fv(u, σ)) = s(u, σ)

The solution spaces : uh ∈ Vh, σh ∈ Hh

Vh := ϕ ∈ L2(Ω) : ϕ|Ωk ∈ Pm(Ωk)

Hh := τ ∈ L2(Ω)× L2(Ω) : τ |Ωk ∈ Pm(Ωk)× Pm(Ωk)

Discontinuous Galerkin method

∑k

∫Ωk

σh · τ +

∫Ωk

(∇ · τ )uh −∫∂Ωk

(τ · n)uh = 0

∑k

∫Ωk

−(fc − fv) · ∇ϕ+

∫∂Ωk

ϕ(fc − fv)−∫

Ωk

(∇ · σh)ϕ =∑k

∫Ωk

sϕ

Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 4 / 14

Hybridizing...

The solution spaces : uh ∈ Vh, σh ∈ Hh, λh ∈Mh

Vh := ϕ ∈ L2(Ω) : ϕ|Ωk ∈ Pm(Ωk)

Hh := τ ∈ L2(Ω)× L2(Ω) : τ |Ωk ∈ Pm(Ωk)× Pm(Ωk)

Mh := µ ∈ L2(Γ) : µ|Γk ∈ Pm(Γk)

Hybridized Discontinuous Galerkin method

∑k

∫Ωk

σh · τ +

∫Ωk

(∇ · τ )uh −∫∂Ωk

(τ · n)λh = 0

∑k

∫Ωk

−(fc − fv) · ∇ϕ+

∫∂Ωk

ϕ(fc − fv)−∫

Ωk

(∇ · σh)ϕ =∑k

∫Ωk

sϕ

∑k

∫∂Ωk

(fc − fv)µ = 0

Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 5 / 14

Hybridizing...

The solution spaces : uh ∈ Vh, σh ∈ Hh, λh ∈Mh

Vh := ϕ ∈ L2(Ω) : ϕ|Ωk ∈ Pm(Ωk)

Hh := τ ∈ L2(Ω)× L2(Ω) : τ |Ωk ∈ Pm(Ωk)× Pm(Ωk)

Mh := µ ∈ L2(Γ) : µ|Γk ∈ Pm(Γk)

Hybridized Discontinuous Galerkin method

∑k

∫Ωk

σh · τ +

∫Ωk

(∇ · τ )uh −∫∂Ωk

(τ · n)λh = 0

∑k

∫Ωk

−(fc − fv) · ∇ϕ+

∫∂Ωk

ϕ(fc − fv)−∫

Ωk

(∇ · σh)ϕ =∑k

∫Ωk

sϕ

∑k

∫∂Ωk

(fc − fv)µ = 0

Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 5 / 14

Naca0012 Subsonic: M=0.5, α = 2

Figure: Mach number

Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 6 / 14

Naca0012 Subsonic: M=0.5, α = 2

10−3 10−2 10−1

10−10

10−8

10−6

10−4

10−2

1/√

ndof

cder

ror

p = 1

p = 2

p = 3

p = 4

(a) Drag coefficient

10−3 10−2 10−110−5

10−4

10−3

10−2

1/√

ndofcl

erro

r

p = 1

p = 2

p = 3

p = 4

(b) Lift coefficient

Figure: Error Vs Dofs

Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 7 / 14

Naca0012 Subsonic: M=0.5, α = 2

10−3 10−2 10−1

10−10

10−8

10−6

10−4

10−2

1/√

ndof

cder

ror

UniformResidualAdjoint

Error estimateCorrected

Figure: Error Vs Dofs, p = 2

Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 8 / 14

Naca0012 Subsonic : Work units

10−2 10−1 100 101 102 103

10−11

10−9

10−7

10−5

10−3

work units

cder

ror

p = 1

p = 2

p = 3

p = 4

Figure: Error Vs Work units

Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 9 / 14

Naca0012 Transonic: M=0.8, α = 1.25

10−3 10−2 10−110−6

10−5

10−4

10−3

10−2

1/√

ndof

cder

ror

p = 1

p = 2

p = 3

p = 4

(a) Drag coefficient

10−3 10−2 10−110−4

10−3

10−2

10−1

1/√

ndofcl

erro

r

p = 1

p = 2

p2newp = 3

(b) Lift coefficient

Figure: Error Vs Dofs

Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 10 / 14

Naca0012 Transonic: M=0.8, α = 1.25

(a) Initial mesh (b) Adjoint adapted mesh on Drag, p2

Figure: Mesh

Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 11 / 14

Naca0012 Transonic: M=0.8, α = 1.25

10−3 10−2 10−110−5

10−4

10−3

10−2

1/√

ndof

cder

ror

ResidualAdjoint

Corrected

Figure: Error Vs Dofs, p = 2

Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 12 / 14

Naca0012 Laminar: M=0.5, α = 1, Re=5000

10−3 10−2 10−110−6

10−5

10−4

10−3

10−2

1/√

ndof

cder

ror

p = 1

p = 2

p = 3

p = 4

(a) Drag coefficient

10−3 10−2 10−110−6

10−5

10−4

10−3

10−2

10−1

1/√

ndofcl

erro

r

p = 1

p = 2

p = 3

p = 4

(b) Lift coefficient

Figure: Error Vs Dofs

Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 13 / 14

Acknowledgement

Financial support from the Deutsche Forschungsgemeinschaft (GermanResearch Association) through grant GSC 111 is gratefully acknowledged

Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 14 / 14

Adjoint Equations

Error in targeteh = J(w)− J(wh)

Adjoint EquationN ′[wh](dw; zh) = J ′[wh](dw)

Global error estimateeh ≈ Nh(wh, zh)

Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 15 / 14