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

Cartesian mesh MPDATA

MPDATA BASIC SCHEME

MPDATA BASIC SCHEME

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 = 0.329

Cp computed <0.3334,0.3339>

M = 5

M = 15

Comparison of theoretical and computed shock angles for 15deg wedge

NACA64A010 OSCILLATING

AEROFOIL

M=0.796 k=0.2002

αm =1.01deg c=0.5m

Mesh movement

RAE 2822 M = 0.75 α = 3

RAE 2822 M = 0.75 α = 3

MPDATA 7523 pointsAGARD 20580 points

MPDATA fine mesh 16101 points

enrichment 11915 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