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Simulation and Optimization of Electromagnetohydrodynamic FlowsPowerPoint Presentation

Simulation and Optimization of Electromagnetohydrodynamic Flows

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### Code Validation used to solve the system of equations

### Simulation and Optimization of Magneto-hydrodynamic Flows with applied electric and magnetic fieldwith LSFEM

### Optimization of Magneto-Hydrodynamic Control of Diffuser Flows Using Micro-Genetic Algorithms and Least-Squares Finite Elements

### Simulation of Magneto-hydrodynamic Flows with Conjugate Heat Transferwith LSFEM

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 Flows

- 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

Introduction Flows

- 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)

Introduction cont. Flows

- 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

Introduction cont. Flows

- Numerical simulation is necessary for performing optimization involving EMHD flows

Applications of EMHD Flows

- 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

Conservation of Energy Flows

Conservation of Mass

Maxwell’s Equations Flows

Sub-models Flows

- 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

Advantages of the LSFEM Flows

- 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

- 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

Least-squares finite element method (LSFEM) used to solve the system of equations

h used to solve the system of equations and p-Type Finite Element Solver

h used to solve the system of equations 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

p used to solve the system of equations-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

p used to solve the system of equations-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

LSFEM Code used to solve the system of equations

- 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

LSFEM Code used to solve the system of equations

- 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.

Sparse Linear Solvers used to solve the system of equations

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

LSFEM for EMHD used to solve the system of equations

Nondimensional Form for EMHD Equations used to solve the system of equations

Nondimensional Form for EMHD Equations used to solve the system of equations

First Order Form for EMHD Equations used to solve the system of equations

Iterative Solution Algorithm used to solve the system of equations

General First Order System for 2-D used to solve the system of equations

General First Order System for 2-D used to solve the system of equations

General First Order System for 2-D used to solve the system of equations

Verification of Accuracy used to solve the system of equations

- 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

Test Cases for NSE used to solve the system of equations

- Test against analytical solutions for NSE
- Test against driven cavity numerical benchmark solutions
- Test against experiment data for flow over backward facing step

Mesh for analytic test cases used to solve the system of equations

Comparison between computed and analytic solution for Couette-Poiseuille flow

Mesh for driven cavity benchmark Couette-Poiseuille flow

Comparison with benchmark results for driven cavity at Couette-Poiseuille flowRe = 1000

Computed streamlines Couette-Poiseuille flowRe = 1000

Computed pressure Couette-Poiseuille flowRe = 1000

Computed vorticity Couette-Poiseuille flowRe = 1000

Comparison with benchmark results for driven cavity at Couette-Poiseuille flowRe = 10000

Computed streamlines Couette-Poiseuille flowRe = 10000

Computed pressure Couette-Poiseuille flowRe = 10000

Computed vorticity Couette-Poiseuille flowRe = 10000

Convergence history for driven cavity benchmark Couette-Poiseuille flow

Backward Facing Step Problem Couette-Poiseuille flow

Mesh for Backward Facing Step Problem Couette-Poiseuille flow

Comparison of Computed and Measured Reattachment Lengths Couette-Poiseuille flow

Mesh for Buoyancy-Driven Cavity CFD Codes

Computed Static Temperature Distribution CFD Codes

Computed Heat Flux on Vertical Walls CFD Codes

Computed Streamlines CFD Codes

Computed Streamline in Upper Left Corner CFD Codes

Test Cases for MHD and EMHD Results

- Test against analytical solutions for MHD and EMHD

LSFEM for MHD Results

First order system for MHD Results

First order system for 2-D MHD Results

Mesh for Hartmann Flow Test Case Results

Hartmann Flow Results

MHD LSFEM code was compared with the analytic solution to Poisuille-Hartmann flow

Comparison of LSFEM and analytic solution for Hartmann flow Results

Computed and analytic velocity profile

Comparison of LSFEM and analytic solution for Hartmann flow Results

Computed and analytic induced magnetic field

Comparison of Resultsh and p-type methods for Hartmann flow

Comparison of LSFEM and analytic solution for Hartmann flow with applied electric and magnetic field

Goal Flows Using

- Given a fixed diffuser shape, use micro-GA and LSFEM MHD analysis to design a magnetic field distribution on the diffuser wall that will increase static pressure rise

Flow Solver Flows Using

- LSFEM solver for 2-D steady incompressible Navier-Stokes together with Maxwell’s equations for steady magnetic field
- Uses hybrid quadrilateral/triangular grid
- One analysis takes around 22 min. on a single Pentium II CPU

BC’s and Parameterization Flows Using

- Parabolic velocity specified at inlet
- Static pressure specified at outlet
- no-slip conditions on wall
- magnetic field component along wall are specified. They were parameterized with b-spline
- perfectly conducting wall bc used on all other solid surfaces

Hybrid mesh for diffuser analysis Flows Using

Given diffuser shape

Convergence History for a Typical Design Flows Using

Parallel Genetic Algorithm Flows Using

- GA is a naturally coarse grained parallel algorithm
- One node maintains the population(master) and distributes jobs to the slave nodes
- Only simple synchronous message passing is needed to implement on distibuted memory
- Population size need not match the number of slave nodes
- Asynchronous models are also being developed for use when function analysis computation times vary dramatically.

Parallel Computer Flows Using

- based on commodity hardware components and public domain software
- 16 dual Pentium II 400 MHz based PC’s
- 100 Megabits/second switched ethernet
- total of 32 processors and 8.2 GB of main memory
- Compressible NSE solver achieved 1.5 Gflop/sec with a LU SSOR solver on a 100x100x100 structured grid

Parallel Computer Flows Using

- based on commodity hardware components and public domain software
- 16 dual Pentium II 400 MHz based PC’s
- 100 Megabits/second switched ethernet

Parallel Computer cont. Flows Using

- total of 32 processors and 8.2 GB of main memory
- Compressible NSE solver achieved 1.5 Gflop/sec with a LU SSOR solver on a 100x100x100 structured grid
- GA optimization of a MHD diffuser completed in 30 hours. Same problem would take 14 days on a single CPU

Genetic Algorithm Flows Using

- Population size of 15
- 100 generations
- 9-bit strings for each design variable
- elitism
- tournament selection
- uniform crossover
- parallel micro-GA

Results Flows Using

Two optimizations were run simultaneously with 16 processors each.

Generation 100 was reached by both in about 30 hours

run 1 achieved a pressure increase of .207 Pa

run 2 achieved a pressure increase of .228 Pa

diffuser without an applied magnetic field achieved a pressure increase of .05 Pa.

With no applied magnetic field Flows Using

With optimized applied magnetic field

Objective Transfer

- Simulation of flow through channel with an applied magnetic field
- Simulation of heat transfer from the flow to a solid cold wall
- Observe the effect of applied magnetic field on flow patterns and heat transfer characteristics

Boundary Conditions Transfer

- Inlet temperature of 2000 K
- Specified outlet pressure
- Specified parabolic velocity profile at inlet
- No-slip on walls
- Symmetry boundary condition on top
- Temperature of 300 K on bottom wall
- Perfectly conducting walls except in the region 7 < x < 8 where sinusoidal magnetic field components were specified. Magnitude was varied from 0 to 5 Tesla.

Results Transfer

- Presence of magnetic field induces a large separation in the flow field close to the wall
- Size and complexity are proportional to the strength of the magnetic field
- A drop in fluid/solid interface temperature was observed in the region where the magnetic field was applied

Effect of flow on magnetic field Transfer

Figure 14: Velocity magnitude for no applied magnetic field. Transfer

Figure 15: Streamlines for no applied magnetic field.

Solidification with MHDFigure 16: Velocity magnitude for optimized case 1. Transfer

Figure 17: Streamlines for optimized case 1.

Optimized MHD SolidificationFigure 21: Velocity magnitude for optimized case 2. Transfer

Figure 22: Streamlines for optimized case 2.

Optimizer MHD SolidificationSolidification with Optimized Magnetic Boundary Conditions Transfer

Figure 10. Isotherms and streamlines without and with an optimized magnetic field (test case 2 with six sensors)

Objective Transfer

- Simulation of a steady state EMHD blood pump
- Both electric and magnetic fields are required to produce the driving force

Governing Equations Transfer

Boundary conditions and geometry Transfer

- Rectangular domain with height of 4 cm and length of 40 cm
- Triangular mesh:
- 7021 nodes
- 3422 elements
- parabolic triangles
- Specified parabolic inlet velocity profile and temperature of 310.15 K
- No slip on walls
- Wall temperature was 298.15 K
- Specified exit pressure of 1 Pa
- Positive electrode on bottom wall, negative electrode on top with 50 volts applied across them
- Uniform magnetic field of .05 Tesla specified in Z direction

Physical parameters for EMHD blood pump Transfer

- Density(kg m-3) = 1055.0
- Inlet height(cm) = 4
- Length(cm) = 40
- Inlet temp.(K) = 310
- Wall temp(K) = 298
- heat conductivity(W kg-1 K-1) = .51
- specific heat(J kg-1 K-1)= 4178
- inlet velocity(m s-1) = .05
- dynamic viscosity(kg m-1 s-1) = .004
- electric conductivity (S m-1) = 1.4
- outlet pressure (Pa) = 1

Computed Electric Potential Transfer

Computed Velocity Vectors Transfer

Computed Static Pressure Transfer

Computed Static Temperature Transfer

EMHD Channel Electrode Configuration Transfer

Mesh for EMHD channel problem Transfer

Compute Static Pressure Transfer

Compute Static Pressure Transfer

Computed Axial Velocity Distribution Transfer

EMHD Channel Electrode Configuration Transfer

Computed Electric Potential Transfer

Computed Electric Field Lines Transfer

Computed Static Pressure Transfer

Computed Streamlines Transfer

Conclusions Transfer

- A code for the simulation of MHD/EMHD was developed based on the LSFEM
- Code was tested against analytic solutions and experimental data for separate disciplines
- Code was applied to several MHD problems including a MHD diffuser optimization problem
- Code was used to simulate a EMHD pump and several channel flows with finite length electrodes

Conclusions Transfer

- LSFEM works well for NSE, MHD, and EMHD when the problem are not very nonlinear.
- The ‘squaring’ effect inherent to the method appears to enhance any nonlinearity already present in the system. Thus, as Reynolds and Hartmann numbers increase, the required number of iterations increases rapidly

Conclusions Transfer

- For LSFEM, first order forms for the PDE’s are required in order to satisfy solution continuity requirements. This leads to a relatively large number of unknowns, especially in 3D, compared to a Galerkin type method
- The linear algebraic systems can become ill-conditioned. Better preconditioners are needed.

Future Work Transfer

- Better preconditioning of linear systems
- Closer look at scaling of the equations
- Extension to 3-D
- Better boundary conditions
- Unsteady

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