11 th INTERNATIONAL CONFERENCE ON HYPERBOLIC PROBLEMS: THEORY, NUMERICS, APPLICATIONS Simulation of Field-Aligned Simulation of Field-Aligned Ideal MHD Flows Around deal MHD Flows Around Perfectly Conducting Cylinders Using An Artificial Perfectly Conducting Cylinders Using An Artificial Compressibility Approach Compressibility Approach Mehmet Sarp YALIM David Vanden ABEELE, Andrea LANI, Herman DECONINCK Department of Aeronautics and Aerospace von Karman Institute for Fluid Dynamics (VKI) 72, Chaussée de Waterloo B-1640 Rhode-Saint-Genèse BELGIUM von Karman Institute for Fluid Dynamics
OUTLINE • Ideal Magnetohydrodynamics (MHD) Equations • An Overview of Solenoidal Constraint Satisfying Techniques • Artificial Compressibility Approach (ACA) • The COOLFluiD Framework • Upwind Finite Volume Method (FVM) MHD Solver • Results • Conclusion von Karman Institute for Fluid Dynamics
IDEAL MHD EQUATIONS ρ ⎛ ⎞ u ⎜ ⎟ ρ ⎛ ⎞ ⋅ ⎛ ⎞ B B ⎜ ⎟ ⎜ ⎟ ρ + + − ⎜ ⎟ uu I p BB ρ ∂ u ⎜ ⎟ ⎜ ⎟ ⎝ 2 ⎠ + ∇ ⋅ = 0 ⎜ ⎟ ⎜ ⎟ − ∂ uB Bu t B ⎜ ⎟ ⎜ ⎟ ⋅ ⎛ ⎞ B B ⎜ ⎟ ( ) + + − ⋅ E ⎜ ⎟ ⎜ E p u B u B ⎟ ⎝ ⎠ ⎝ 2 ⎠ ⎝ ⎠ Assumptions - quasi-neutral plasma - infinite conductivity - non-relativistic plasma - negligible Hall term von Karman Institute for Fluid Dynamics
DIMENSIONLESS NUMBERS IN MHD Plasma Beta acoustic pressure p β = = 2 magnetic pressure B μ 2 0 Magnetic Reynolds Number Lu → ∞ ⎫ L R m = η ≈ for astrophysi cs plasmas so 0 ⎬ η → ∞ R ⎭ m von Karman Institute for Fluid Dynamics
MHD WAVES Euler equations � 3 waves (shock, contact surface, rarefaction) Ideal MHD equations � 7 waves MORE COMPLEX - 1 entropy wave - 2 Alfven waves (incompressible wave, strongly anisotropic) - 2 fast magnetoacoustic waves (compressible wave, strongly anisotropic) - 2 slow magnetoacoustic waves (compressible wave, strongly anisotropic) von Karman Institute for Fluid Dynamics
von Karman Institute for Fluid Dynamics
divB CONSTRAINT ? - Numerically divB can never be exactly zero. - Discretization errors accumulate at each iteration. - After a significant amount of error accumulation, the algorithm breaks down. Precaution: Errors in divB should be damped. For this purpose: - Powell source term - Balsara’s scheme - Projection scheme von Karman Institute for Fluid Dynamics
POWELL SOURCE TERM (1) Sources of Instabilities for Ideal MHD Equations - Jacobian matrix is singular (one of the modes remain undamped) - Terms proportional to divB act as extra unphysical forces Solution Proposed by Powell A source term is added to the RHS - to remove the singularity in a mathematically appropriate way - to cancel the terms proportional to divB von Karman Institute for Fluid Dynamics
POWELL SOURCE TERM (2) ρ ⎛ ⎞ u ⎜ ⎟ ρ ⎛ ⎞ ⎛ ⎞ 0 ⋅ ⎛ ⎞ B B ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ρ + + − ⎜ ⎟ uu I p BB ρ ∂ u B ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 2 ⎠ + ∇ ⋅ = −∇ ⋅ B ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ − ∂ uB Bu t B u ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⋅ ⎛ ⎞ B B ⎜ ⎟ ⎜ ⎟ ( ) ⋅ + + − ⋅ E u B ⎜ ⎜ E p ⎟ u B u B ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ 2 ⎠ ⎝ ⎠ - no change at PDE level (i.e. mathematically divB = 0 is a constraint) - non-conservative source term introduces non-conservativity into the PDE system - new divergence wave (i.e. 8. wave for ideal MHD equations) ∂ ⎛ ∇ ⋅ ⎞ ⎛ ∇ ⋅ ⎞ B B ⎜ ⎟ + ⋅ ∇ ⎜ ⎟ = u 0 ⎜ ⎟ ⎜ ⎟ ∂ ρ ρ t ⎝ ⎠ ⎝ ⎠ - errors in divB will be swept away by the flow von Karman Institute for Fluid Dynamics
POWELL SOURCE TERM (3) Problem with Non-Conservative Nature of Powell Source Term - Jump conditions across the discontinuities can be predicted wrongly in flows involving strong shocks - Consequently, the scheme can produce incorrect shock positions and speeds Problem with Divergence Wave Equation - divB errors cannot be swept away in stagnant regions of the flow von Karman Institute for Fluid Dynamics
BALSARA’S SCHEME - discrete satisfaction of the divB = 0 condition for the magnetic field - hydrodynamic state variables at the cell center, magnetic field at the cell faces and electric field at the cell edges (i.e. staggered grid approach) - divB = 0 should be imposed as an initial condition in the whole domain - once imposed, divB = 0 will automatically be satisfied in time - divB = 0 condition is satisfied up to the machine accuracy level Implementation is exceedingly complex von Karman Institute for Fluid Dynamics
PROJECTION SCHEME - magnetic field is projected onto a discrete space in which divB = 0 is satisfied in a numerical sense: ( ) ( ) ∗ + n ∇ ⋅ = ∇ ⋅ − ∇ φ = ← − ∇ φ n 1 B B 0 B B - divB = 0 should be imposed as an initial condition in the whole domain - once imposed, divB = 0 will automatically be satisfied in time - divB = 0 condition is satisfied up to the machine accuracy level Mixture of hyperbolic and elliptic numerical methods Computationally expensive (necessity of a Poisson solver) von Karman Institute for Fluid Dynamics
ARTIFICIAL COMPRESSIBILITY APPROACH (1) ⎛ ⎞ ρ ρ ⎛ ⎞ u ⎜ ⎟ ⎜ ⎟ ( ) ⎜ ⎟ ρ ρ + + ⋅ − u uu I p B B 2 BB ⎜ ⎟ ⎜ ⎟ ∂ ⎜ ⎟ + ∇ ⋅ − + φ = B uB Bu I 0 ⎜ ⎟ ⎜ ⎟ ∂ t ( ) ⎜ ⎟ + + ⋅ − ⋅ ⎜ ⎟ E ( E p B B 2 ) u B u B ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ φ 2 β ⎝ ⎠ B ⎝ ⎠ ref - Hyperbolic system of 9 equations with eigenvalues: u, u±c f , u±c s , u±c A , ± β ref φ - is a scalar potential function, β ref is a reference velocity (e.g. inlet flow velocity) - Unlike the classical projection scheme approach, in this case, the system is purely hyperbolic and can be solved with any conventional upwind finite volume numerical scheme suitable for hyperbolic PDE systems (e.g. Lax-Friedrichs scheme) von Karman Institute for Fluid Dynamics
ARTIFICIAL COMPRESSIBILITY APPROACH (2) Incompressible CFD ∂ 1 p ( ) + ∇ ⋅ ρ = u 0 β ∂ 2 t ref Incompressibility Condition Modified System of Ideal MHD Equations ∂ φ 1 + ∇ ⋅ = B 0 β ∂ 2 t ref Solenoidal Constraint von Karman Institute for Fluid Dynamics
ARTIFICIAL COMPRESSIBILITY APPROACH (3) Physical Interpretation ∂ B ( ) + ∇ φ = ∇ × × u B ∂ t taking the divergence, differentiating w.r.t. time and substituting for φ divB from the evolution of , we get Wave equation for divB ( ) [ ] ∂ ∇ ⋅ 2 B ( ) − ∇ ⋅ β ∇ ∇ ⋅ = 2 B 0 ref ∂ 2 t Unlike the divergence wave of Powell’s approach, divB errors are radiated away with ± β ref , wave speeds independent from the flow velocity, thereby guaranteeing the removal of divB errors even in stagnant regions of flow von Karman Institute for Fluid Dynamics
ARTIFICIAL COMPRESSIBILITY APPROACH (4) Consistency Check (1) Do the modified PDEs converge towards a correct solution of the original ideal MHD system ? (2) Do the discretized equations converge to a correct weak solution of the original ideal MHD sysem ? von Karman Institute for Fluid Dynamics
ARTIFICIAL COMPRESSIBILITY APPROACH (5) For smooth flows Take the divergence of the modified induction equation, at steady state: [ ] ( ) ∇ φ = ∇ ⋅ ∇ × × = 2 u B 0 Boundary conditions: φ = 0 at outlets ∂ φ ∂ = n 0 at all other boundaries By using separation of variables, the above Laplacian equation yields ∇ φ φ = 0 everywhere. Hence the artificial term in the induction equation vanishes and we get the correct solution. von Karman Institute for Fluid Dynamics
ARTIFICIAL COMPRESSIBILITY APPROACH (6) For shocks Let’s consider only steady shocks for simplicity. The jump relations then become: + φ = + φ F F L L R R The jump relations of the original ideal MHD system are satisfied by φ = φ the hyperbolized projection form of the equations: put L R Couldn’t the modified form of the equations allow for entropy- satisfying unphysical shocks ? von Karman Institute for Fluid Dynamics
ARTIFICIAL COMPRESSIBILITY APPROACH (7) η Let’s consider the modified system with non-zero resistivity, . The modified induction equation will be: ∂ B [ ] [ ] ( ) = ∇ × × − ∇ φ − ∇ × η ∇ × u B B ∂ t For smooth flows, taking the divergence of both sides of the above equation, for each value of the resistivity, with the previously defined ∇ φ boundary conditions, disappears and ACA converges towards the correct entropy-satisfying solution of the original system of ideal MHD equations. von Karman Institute for Fluid Dynamics
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