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

Cartesian mesh MPDATA

MPDATA BASIC SCHEME

EDGE BASED FORMULATION

CONVERGENCE OF FINITE-VOLUME MPDATA ON UNSTRUCTURED MESH

CONVERGENCE OF FINITE-VOLUME MPDATA ON UNSTRUCTURED SKEWED MESH

ROTATING CYLINDER BASIC MPDATA MPDATA+FTC

FCT

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

INITIAL MPDATA GAGE UPWIND LEAPFROG

EULER EQUATIONS – CONSERVATIVE FORM

NONOSCILLATORY FORWARD IN TIME FLOW SOLVERS

FLOW SOLVER

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

NACA 0012 COMPUTATIONAL MESH

AGARD MPDATA + FCT M = 0.8 α = 1.25

MPDATA v AGARD SOLUTION

THE SAME MESH MPDATA v R-K SOLUTION

EFFECT OF FCT

EFFECT OF PRESSURE SWITCH

ADAPTIVITY REFINEMENT INDICATORS MESHING TECHNIQUES

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

LEAD ERROR

MPDATA ERROR INDICATOR

Remeshing Mesh movement Mesh enrichment P-refinement Combinations MESHING TECHNIQUES

M = 2.5 α = 0

M = 2.5 Cp theoretical = Cp computed

M = 5 M = 15

Comparison of theoretical and computed shock angles for 15deg wedge

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

Mesh movement

RAE 2822 M = 0.75 α = 3

MPDATA 7523 points AGARD points

MPDATA fine mesh points enrichment points

M = 0.8 α = 1.25 Pressure Contours

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