1 / 128

Simulation and Optimization of Electromagnetohydrodynamic Flows

Simulation and Optimization of Electromagnetohydrodynamic Flows. Brian Dennis Ph.D. Candidate Aerospace Engineering Dept. Penn State University George Dulikravich Professor Mechanical and Aerospace Dept. University of Texas at Arlington. Overview. Introduction

pancho
Download Presentation

Simulation and Optimization of Electromagnetohydrodynamic Flows

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Simulation and Optimization of Electromagnetohydrodynamic Flows Brian Dennis Ph.D. Candidate Aerospace Engineering Dept. Penn State University George Dulikravich Professor Mechanical and Aerospace Dept. University of Texas at Arlington

  2. Overview • Introduction • Governing equations for EMHD • General LSFEM formulation • h and p-type finite element solver • LSFEM for EMHD • Code validation cases • Applications • Conclusion and recommendations

  3. Introduction • EMHD is study of incompressible flow under the influence of electric and magnetic fields • Various simplified analytical models exist and have been used for numerical simulation(EHD,MHD)

  4. Introduction cont. • A fully consistent non-linear model for general EMHD has been recently developed • No numerical simulations of full EMHD has been report in open literature to date • A few numerical simulations of simplified or inconsistent EMHD models have been report in open literature • A computer code for numerical simulation for 2-D planar MHD/EMHD flows has been developed using LSFEM

  5. Introduction cont. • Numerical simulation is necessary for performing optimization involving EMHD flows

  6. Applications of EMHD • Manufacturing(solidification,crystal growth) • Flow control • Drag reduction/propulsion • Pumps with no moving parts(artificial heart, liquid metal pumps) • Compact heat exchangers • Shock absorbers, active damping

  7. Fully Consistent Model of EMHD

  8. Conservation of Linear Momentum

  9. Conservation of Energy Conservation of Mass

  10. Maxwell’s Equations

  11. Sub-models • The LSFEM was applied to two sub-models of the fully nonlinear Electro-magneto-hydrodynamic system • Magneto-hydrodynamics • Electro-magneto-hydrodynamics with reduced number of source terms

  12. Least-Squares Finite Element Method (LSFEM)

  13. Advantages of the LSFEM • Can use equal order basis functions for pressure and velocity • can use the first order form of PDE’s • can handle any type of equation and mixed types of equations • Can discretize convection terms without upwinding or explicit artificial dissipation • Stable and robust method • Resulting system of equations is symmetric and positive definite

  14. Simple iterative techniques such as PCG and multigrid can be used to solve the system of equations • Inclusion of divergence constraint for magnetic flux is straight forward

  15. Least-squares finite element method (LSFEM)

  16. h and p-Type Finite Element Solver

  17. h and p-Type Methods • h-type finite element methods use large numbers of low order accurate elements • p-type finite element methods use small numbers of high order accurate elements • h/p-type finite element methods use a combination of both types together with an adaptive strategy • p-type methods convergence to the exact solution is more rapid than in with h-type methods, if the underlying solution is smooth

  18. p-Type Method • element approximation function are compose of p-type expansions. These expansions are composed of a summation of P polynomials • p-type expansions can be categorized as nodal expansions or modal expansions • the unknown coefficients in nodal expansions are the values of the function at the nodes of the element and therefor have some physical meaning. Nodal expansions based on Lagrange polynomials are typically used for low order finite elements • the unknown coefficients in modal expansions are not associated with any nodes

  19. p-Type Method • Modal expansions are hierarchical, that is, that all expansion sets less than P+1 are contained within the expansion set P+1 • Nodal expansions are not hierarchical • Modal expansion typically produce finite element matrices that have a lower condition number than those produced with nodal expansions • Expansions are usually developed in one dimension. Approximation functions for multidimensional elements are constructed through tensor products of the one dimensional expansions. • Typically, the modal expansions are combined with the first order Lagrange polynomials so that continuity of the solution is satisfied between neighboring elements in a mesh

  20. LSFEM Code • A serial code was developed in C/C++ to solve general systems with LSFEM • Steady state problems only • Mixed triangular and quadrilateral meshes for h-type elements. • Quadrilateral elements for p-type elements. Element approximation functions are constructed from a modal basis derived from Jacobi polynomials • Support for multiple material domains such as in conjugate heat transfer problems for h-version • Nonlinear equations are linearized with Newton or Picard method

  21. LSFEM Code • Integrals are evaluated numerically using Legendre-Gauss quadrature. A P+2 rule was used for p-type elements. • Static condensation is used to remove the interior degrees of freedom from p-type elements.

  22. Sparse Linear Solvers • Direct Solver: • Sparse LU factorization from PETSc library • Iterative Solvers: • Jacobi preconditioned conjugate gradient method • Multi-p multilevel method

  23. LSFEM for EMHD

  24. Nondimensional Form for EMHD Equations

  25. Nondimensional Form for EMHD Equations

  26. First Order Form for EMHD Equations

  27. Iterative Solution Algorithm

  28. General First Order System for 2-D

  29. General First Order System for 2-D

  30. General First Order System for 2-D

  31. Code Validation

  32. Verification of Accuracy • Few analytic solutions for EMHD exist • Some analytic solutions for MHD exist • NSE portion of code was validated using analytic solutions for NSE and with experiment data from driven cavity flows and backward facing step • Heat transfer/Electric/Magnetic field portions were verified with analytic solutions

  33. Test Cases for NSE • Test against analytical solutions for NSE • Test against driven cavity numerical benchmark solutions • Test against experiment data for flow over backward facing step

  34. Mesh for analytic test cases

  35. Comparison between computed and analytic solution for Couette-Poiseuille flow

  36. Mesh for driven cavity benchmark

  37. Comparison with benchmark results for driven cavity at Re = 1000

  38. Computed streamlinesRe = 1000

  39. Computed pressureRe = 1000

  40. Computed vorticityRe = 1000

  41. Comparison with benchmark results for driven cavity at Re = 10000

  42. Computed streamlinesRe = 10000

  43. Computed pressureRe = 10000

  44. Computed vorticityRe = 10000

  45. Convergence history for driven cavity benchmark

  46. Backward Facing Step Problem

  47. Mesh for Backward Facing Step Problem

  48. Computed Streamlines Re = 100 Re = 400 Re = 500

  49. Comparison of Computed and Measured Reattachment Lengths

More Related