Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm...
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![Page 1: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/1.jpg)
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
![Page 2: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/2.jpg)
Cartesian mesh MPDATA
![Page 3: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/3.jpg)
Cartesian mesh MPDATA
![Page 4: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/4.jpg)
MPDATA BASIC SCHEME
![Page 5: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/5.jpg)
MPDATA BASIC SCHEME
![Page 6: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/6.jpg)
MPDATA BASIC SCHEME
![Page 7: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/7.jpg)
EDGE BASED FORMULATION
![Page 8: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/8.jpg)
CONVERGENCE OF FINITE-VOLUME MPDATA ON UNSTRUCTURED MESH
![Page 9: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/9.jpg)
CONVERGENCE OF FINITE-VOLUME MPDATA ON UNSTRUCTURED SKEWED MESH
![Page 10: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/10.jpg)
ROTATING CYLINDER
BASIC MPDATA MPDATA+FTC
![Page 11: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/11.jpg)
FCT
![Page 12: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/12.jpg)
REVOLUTION OF A SPHERE AROUND THE DIAGONAL OF A DOMAIN
INITIAL MPDATA GAGE AFTER 1 REVOLUTION
![Page 13: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/13.jpg)
INITIAL MPDATA GAGE
UPWIND LEAPFROG
![Page 14: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/14.jpg)
EULER EQUATIONS – CONSERVATIVE FORM
![Page 15: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/15.jpg)
NONOSCILLATORY FORWARD IN TIME FLOW SOLVERS
![Page 16: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/16.jpg)
FLOW SOLVER
![Page 17: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/17.jpg)
CONVERGENCE STUDY
MPDATA - NFT EULER SOLVER
M=0.5
MPDATA UPWIND
![Page 18: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/18.jpg)
NACA 0012 COMPUTATIONAL MESH
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AGARD MPDATA + FCT
M = 0.8 α = 1.25
![Page 20: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/20.jpg)
MPDATA v AGARD SOLUTION
![Page 21: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/21.jpg)
THE SAME MESH
MPDATA v R-K SOLUTION
![Page 22: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/22.jpg)
EFFECT OF FCT
![Page 23: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/23.jpg)
EFFECT OF PRESSURE SWITCH
![Page 24: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/24.jpg)
ADAPTIVITY
REFINEMENT INDICATORS MESHING TECHNIQUES
![Page 25: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/25.jpg)
REFINEMENT INDICATORS
From gradient of dependent variable Based on MPDATA lead error In the spirit of Richardson extrapolation Driven by an objective functional
![Page 26: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/26.jpg)
LEAD ERROR
![Page 27: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/27.jpg)
MPDATA ERROR INDICATOR
![Page 28: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/28.jpg)
Remeshing Mesh movement Mesh enrichment P-refinement Combinations
MESHING TECHNIQUES
![Page 29: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/29.jpg)
M = 2.5 α = 0
![Page 30: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/30.jpg)
M = 2.5
Cp theoretical = 0.329
Cp computed <0.3334,0.3339>
![Page 31: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/31.jpg)
M = 5
M = 15
![Page 32: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/32.jpg)
Comparison of theoretical and computed shock angles for 15deg wedge
![Page 33: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/33.jpg)
NACA64A010 OSCILLATING
AEROFOIL
M=0.796 k=0.2002
αm =1.01deg c=0.5m
![Page 34: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/34.jpg)
Mesh movement
![Page 35: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/35.jpg)
RAE 2822 M = 0.75 α = 3
![Page 36: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/36.jpg)
RAE 2822 M = 0.75 α = 3
MPDATA 7523 pointsAGARD 20580 points
![Page 37: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/37.jpg)
MPDATA fine mesh 16101 points
enrichment 11915 points
![Page 38: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/38.jpg)
M = 0.8 α = 1.25
Pressure Contours
![Page 39: Error Indicator based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) Joanna Szmelter Piotr K. Smolarkiewicz Cranfield.](https://reader036.fdocuments.in/reader036/viewer/2022062318/5515f930550346cf6f8b57dd/html5/thumbnails/39.jpg)
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