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Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield University NCAR Royal Military College of Science Boulder Shrivenham Colorado

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Cartesian mesh MPDATA

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MPDATA BASIC SCHEME

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EDGE BASED FORMULATION

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CONVERGENCE OF FINITE-VOLUME MPDATA ON UNSTRUCTURED MESH

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CONVERGENCE OF FINITE-VOLUME MPDATA ON UNSTRUCTURED SKEWED MESH

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ROTATING CYLINDER BASIC MPDATA MPDATA+FTC

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FCT

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REVOLUTION OF A SPHERE AROUND THE DIAGONAL OF A DOMAIN INITIAL MPDATA GAGE AFTER 1 REVOLUTION

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INITIAL MPDATA GAGE UPWIND LEAPFROG

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EULER EQUATIONS – CONSERVATIVE FORM

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NONOSCILLATORY FORWARD IN TIME FLOW SOLVERS

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FLOW SOLVER

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CONVERGENCE STUDY MPDATA - NFT EULER SOLVER M=0.5 MPDATA UPWIND

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NACA 0012 COMPUTATIONAL MESH

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AGARD MPDATA + FCT M = 0.8 α = 1.25

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MPDATA v AGARD SOLUTION

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THE SAME MESH MPDATA v R-K SOLUTION

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EFFECT OF FCT

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EFFECT OF PRESSURE SWITCH

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ADAPTIVITY REFINEMENT INDICATORS MESHING TECHNIQUES

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REFINEMENT INDICATORS From gradient of dependent variable Based on MPDATA lead error In the spirit of Richardson extrapolation Driven by an objective functional

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LEAD ERROR

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MPDATA ERROR INDICATOR

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Remeshing Mesh movement Mesh enrichment P-refinement Combinations MESHING TECHNIQUES

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M = 2.5 α = 0

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M = 2.5 Cp theoretical = 0.329 Cp computed

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M = 5 M = 15

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Comparison of theoretical and computed shock angles for 15deg wedge

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NACA64A010 OSCILLATING AEROFOIL M=0.796 k=0.2002 α m = 1.01deg c=0.5m

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Mesh movement

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RAE 2822 M = 0.75 α = 3

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MPDATA 7523 points AGARD 20580 points

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MPDATA fine mesh 16101 points enrichment 11915 points

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M = 0.8 α = 1.25 Pressure Contours

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CONCLUSIONS MPDATA evinces properties useful for construction of refinement indicators. Edge-based data structure enables the use of MPDATA in conjunction with all standard adaptive meshing techniques known for unstructured meshes. NFT MPDATA edge-based Euler solver has low implicit diffusion and remains accurate for a broad range of flow speeds. Present work extends utility of MPDATA to new applications

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