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Boundary treatment in cut cell finite difference methods for compressible gas dynamics in domain with moving boundaries Universit` a di Catania Armando Coco joint work with: Alina Chertock, Alexander Kurganov, Giovanni Russo Boundary treatment in cut cell finite difference methods for compressible gas dynamics in domain with moving boundaries 14th International Conference on Hyperbolic Problems: Theory, Numerics, Armando Coco Applications Universit` a di Padova, 28 June 2012 June 27, 2012

Boundary treatment in cut cell finite difference methods for compressible gas dynamics in domain with moving boundaries Outline 1 Euler equations 2 Finite Difference discretization 3 Boundary treatment of Euler equations 4 Discretization of the boundary conditions 5 Preliminar numerical tests

Boundary treatment in cut cell finite difference methods for compressible gas dynamics in domain with moving boundaries Euler equations Outline 1 Euler equations 2 Finite Difference discretization 3 Boundary treatment of Euler equations 4 Discretization of the boundary conditions 5 Preliminar numerical tests

Boundary treatment in cut cell finite difference methods for compressible gas dynamics in domain with moving boundaries Euler equations Euler equations The governing equations are the compressible Euler equations:       ρ ρ u ρ v ρ u 2 + p       ρ u ρ uv       + + = 0 , (1) ρ v 2 + p       ρ v ρ uv E u ( E + p ) v ( E + p ) t x y where ρ is the fluid density, u and v are the velocities, E is the total energy, and p is the pressure. This system is closed using the equation of state (EOS), which, for ideal gases, reads: γ − 1 + ρ p 2( u 2 + v 2 ) , E = γ = const . (2) � We also introduce the notation c := γ p /ρ for the speed of sound, which will be used throughout the presentation.

Boundary treatment in cut cell finite difference methods for compressible gas dynamics in domain with moving boundaries Finite Difference discretization Outline 1 Euler equations 2 Finite Difference discretization 3 Boundary treatment of Euler equations 4 Discretization of the boundary conditions 5 Preliminar numerical tests

Boundary treatment in cut cell finite difference methods for compressible gas dynamics in domain with moving boundaries Finite Difference discretization Finite Difference discretization We identifies different points: internal points (red) X jk ∈ Ω, ( j , k ) ∈ I . These are the points where we solve the problem, and for which we write the differential equation. Ghost points X jk , ( j , k ) ∈ G . Inactive points Within ghost points, we distinguish between first layer (blue) L 1 and second layer (yellow) L 2 : L 1 ∪ L 2 = G . The first layer of points is within one grid cell from the boundary (in either direction). The second layer is made of points within two grid points from the boundary, which are not in the first layer.

Boundary treatment in cut cell finite difference methods for compressible gas dynamics in domain with moving boundaries Finite Difference discretization Finite Difference discretization (cont.) Definition of the set of ODE’s ∂ U ∂ t + ∂ f ∂ x + ∂ g ∂ y = 0 The Finite difference approximation is: � 2 , k − � 2 − � f j + 1 f j − 1 g j , k + 1 � g j , k − 1 d u jk 2 , k 2 + + , ( j , k ) ∈ I . dt ∆ x ∆ y These equations require the computation of the fluxes, which are at interface between internal cells, or between internal cells and first layer cells. � � f + 2 , k + � f − f j + 1 = (3) j + 1 j + 1 2 , k 2 , k � � � � f + F + f − F − � 2 , k = � � 2 , k = � 2 , y k , 2 , y k (4) x j + 1 x j + 1 j + 1 j , k j + 1 j +1 , k g + g − � g j , k + 1 = � 2 + � (5) j , k + 1 j , k + 1 2 2 � � � � g + 2 = � G + g − 2 = � G − x j , y k + 1 , x j , y k + 1 (6) � � j , k + 1 j , k + 1 j , k j , k +1 2 2 G ± have to be reconstructed in cells The four flux functions � F ± , � ( j , k ) ∈ I ∪ L 1 from the pointwise values: f j , k = f ( u j , k ) , g j , k = g ( u j , k ) , ( j , k ) ∈ I ∪ G .

Boundary treatment in cut cell finite difference methods for compressible gas dynamics in domain with moving boundaries Boundary treatment of Euler equations Outline 1 Euler equations 2 Finite Difference discretization 3 Boundary treatment of Euler equations 4 Discretization of the boundary conditions 5 Preliminar numerical tests

Boundary treatment in cut cell finite difference methods for compressible gas dynamics in domain with moving boundaries Boundary treatment of Euler equations Boundary treatment of Euler equations: Condition on the velocity Let us denote by u n = u · n and u τ = u · τ respectively the normal and tangential velocity. The condition on the normal velocity is simply: u n = 0 on ∂ Ω (7) The condition on the tangential velocity is ∂ u τ ∂ n = u τ k (8) and it can be obtained in the following two manners. n n τ τ Ω Ω Figure: Locally convex Figure: Locally concave boundary k < 0. boundary k > 0.

Boundary treatment in cut cell finite difference methods for compressible gas dynamics in domain with moving boundaries Boundary treatment of Euler equations By imposing that the vorticity is zero. This means imposing that R+dR � u · d l = 0 Γ for each closed circuit Γ. R Supposing that the u τ is constants along the arcs, we obtain: � u · d l = ( R + ∆ R ) u τ | R +∆ R − R u τ | R = 0 . (9) Γ For Taylor we have: u τ | R +∆ R = u τ | R − ∂ u τ ∂ n ∆ R + O (∆ R 2 ) . Plugging it into (9) and neglecting O (∆ R 2 ) terms, we obtain: ∂ u τ ∂ n = 1 R u τ | R .

Boundary treatment in cut cell finite difference methods for compressible gas dynamics in domain with moving boundaries Boundary treatment of Euler equations By imposing that the normal derivative of the total enthalpy is zero on the boundary. Let us recall that the enthalpy is h = 1 2 u 2 + e + p ρ , where p e = e ( ρ, p ) = ( γ − 1) ρ is the internal energy. The condition reads: � � � ∂ p � 0 = ∂ h ∂ n = u · ∂ u 1 1 ρ − p ∂ρ ∂ n + γ − 1 + 1 . ∂ n ρ 2 ∂ n Using the boundary conditions on density and pressure (which will be explained later) and the fact that u = u τ τ on the boundary, we obtain: � 1 � � � 0 = ∂ u τ γ ∂ p = ∂ u τ γ p p γ − 1 k u 2 ∂ n u τ + ρ − ∂ n u τ − 1 − c 2 c 2 γ − 1 ∂ n s ρ 2 τ s ρ where c 2 s is the square of the speed sound. For a polytropic gas we have c 2 s = γ p /ρ . Therefore: 0 = ∂ u τ ⇒ ∂ u τ ∂ n u τ − k u 2 τ = ∂ n = k u τ .

Boundary treatment in cut cell finite difference methods for compressible gas dynamics in domain with moving boundaries Boundary treatment of Euler equations Condition on the pressure The equation of motion for a fluid particle (balance of momentum) for Euler equations reads: ρ D u Dt + ∇ p = 0 (10) where D / Dt = ∂/∂ t + u · ∇ denotes the Lagrangian derivative. Along the boundary of the domain, the velocity vector can be defined as follows: u = u τ τ It is therefore: D u Dt = Du τ Dt τ + u τ D τ Dt = a τ τ + u 2 τ k n (11) where k denotes the curvature. The sign of k is negative for locally convex regions, and positive for locally concave regions, and a τ denotes the tangential acceleration of the fluid. By projecting Eq. (10) on the normal direction, and making use of (11), one obtain the boundary condition on the pressure: ∂ p ∂ n = − ρ u 2 τ k .

Boundary treatment in cut cell finite difference methods for compressible gas dynamics in domain with moving boundaries Boundary treatment of Euler equations Condition on the density Finally, the condition on the density is given by the requirement that the boundary is adiabatic: ∂ p ∂ρ ∂ n = c 2 ∂ n . s

Boundary treatment in cut cell finite difference methods for compressible gas dynamics in domain with moving boundaries Discretization of the boundary conditions Outline 1 Euler equations 2 Finite Difference discretization 3 Boundary treatment of Euler equations 4 Discretization of the boundary conditions 5 Preliminar numerical tests

Boundary treatment in cut cell finite difference methods for compressible gas dynamics in domain with moving boundaries Discretization of the boundary conditions Discretization of the boundary conditions We write a linear equation for each unknown of the system, i.e. for each v x ( G ), v y ( G ), p ( G ), ρ ( G ), where G ∈ G is a ghost point. In details: let G be a ghost point. We compute the projection point B on the interface: � ∇ φ �� � � B ≡ ( x B , y B ) = G − φ ( G ) n G = G − φ ( G ) . � |∇ φ | G Let us define two 3 × 3 stencils: St ( I ) (blue) and St ( II ) (red). G G G B Ω