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Simulation and Optimization of Electromagnetohydrodynamic
FlowsBrian Dennis
Ph.D. Candidate
Aerospace Engineering Dept.
Penn State University
George DulikravichProfessor
Mechanical and Aerospace Dept.
University of Texas at Arlington
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
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)
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
Introduction cont.
• Numerical simulation is necessary for performing optimization involving EMHD flows
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
Fully Consistent Model of EMHD
Conservation of Linear Momentum
30 i]ΤΤαρg[1Dt
vDρ me ppp
)vv(μ t
v )EE(σ 2
TTT
κ 2
sTEσT
κ5
5
Eq e BEσ1 BEdσ 2 BTσ 4
BTdσ 5 B)BE( σ 7 B)BT( T
κ10
EEε p BBμ1m
BEvε p
)BE(εDt
Dp
Conservation of Energy
EκΤdκΤκQDt
TDρC 421hp
BEκBΤκEdκ 1075
ΤdΤΤ
κΤEσEEσ 2
41
BΤEΤ
κΤdE
Τ
κ 105
Dt
BDEVε
μ
B
Dt
EεDE p
m
p
.
Conservation of Mass
0v
Maxwell’s Equations
ep qBvεEε ,
0B ,
t
BE
,
vqBvεEεt
)Evε(μ
Bepp
ΤdσΤσEdσEσ 5421
BΤκTBEσ 10-1
7 .
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
Least-Squares Finite Element Method (LSFEM)
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
• 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)
h and p-Type Finite Element Solver
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
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
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
LSFEM CodeA 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•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 SolversDirect Solver:
•Sparse LU factorization from PETSc library
Iterative Solvers:
•Jacobi preconditioned conjugate gradient method
•Multi-p multilevel method
LSFEM for EMHD
Nondimensional Form for EMHD Equations
Nondimensional Form for EMHD Equations
First Order Form for EMHD Equations
Iterative Solution Algorithm
General First Order System for 2-D
General First Order System for 2-D
General First Order System for 2-D
Code Validation
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
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
Mesh for analytic test cases
Comparison between computed and analytic solution for Couette-Poiseuille
flow
Mesh for driven cavity benchmark
Comparison with benchmark results for driven cavity at
Re = 1000
Computed streamlinesRe = 1000
Computed pressureRe = 1000
Computed vorticityRe = 1000
Comparison with benchmark results for driven cavity at
Re = 10000
Computed streamlinesRe = 10000
Computed pressureRe = 10000
Computed vorticityRe = 10000
Convergence history for driven cavity benchmark
Backward Facing Step Problem
Mesh for Backward Facing Step Problem
Computed Streamlines
Re = 100
Re = 400
Re = 500
Comparison of Computed and Measured Reattachment Lengths
Comparison of Present Results with Results from Commercial
CFD Codes
Mesh for Buoyancy-Driven Cavity
Computed Static Temperature Distribution
Computed Heat Flux on Vertical Walls
Computed Streamlines
Computed Streamline in Upper Left Corner
Comparison of Present Computations with Other Published Results
Test Cases for MHD and EMHD
• Test against analytical solutions for MHD and EMHD
LSFEM for MHD
First order system for MHD
First order system for 2-D MHD
First order system for 2-D MHD cont.
Mesh for Hartmann Flow Test Case
Hartmann FlowMHD LSFEM code was compared with the analytic solution to Poisuille-Hartmann flow
Comparison of LSFEM and analytic solution for Hartmann flow
Comparison of LSFEM and analytic solution for Hartmann flow
Comparison of LSFEM and analytic solution for Hartmann flow
Computed and analytic velocity profile
Comparison of LSFEM and analytic solution for Hartmann flow
Computed and analytic induced magnetic field
Comparison of h and p-type methods for Hartmann flow
Comparison of LSFEM and analytic solution for Hartmann flow with applied electric and
magnetic field
Simulation and Optimization of Magneto-hydrodynamic Flows
with LSFEM
Optimization of Magneto-Hydrodynamic Control of Diffuser
Flows Using Micro-Genetic Algorithms and Least-
Squares Finite Elements
Goal
• 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
• 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
• 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
Given diffuser shape
Convergence History for a Typical Design
Parallel Genetic Algorithm• 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
• 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• 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.• 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
• Population size of 15
• 100 generations
• 9-bit strings for each design variable
• elitism
• tournament selection
• uniform crossover
• parallel micro-GA
GA Convergence History
Run 1 Run 2
ResultsTwo 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
With optimized applied magnetic field
Simulation of Magneto-hydrodynamic Flows with Conjugate Heat Transfer
with LSFEM
Objective
• 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• 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
• 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
Solidification with MHD
1.49E-04
2.97E-04
5.94E-04
1.34E-032.
97E-04
8.91E-04
1.34E-037.43E-04
2.23E-03
2.08E-03
1.04E-03
X
Y
0 0.05 0.10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
X
Y
0 0.05 0.10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Figure 14: Velocity magnitude for no applied magnetic field.
Figure 15: Streamlines for no applied magnetic field.
Optimized MHD Solidification
1.49E-04
5.94E-041.19E-03
8.91E-04
1.34E-03
2.23E-031.34E-03
7.43E-04
1.34E-03
2.97E-04
X
Y
0 0.05 0.10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
X
Y0 0.05 0.10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Figure 16: Velocity magnitude for optimized case 1.
Figure 17: Streamlines for optimized case 1.
Optimizer MHD Solidification
1.87E-05
3.11E-06
9.33E-06
2.18E-05
4.67E-05
2.80E-05
2.80E-05
2.18E-05
1.87E-05
3.11E-06
1.56E-05
1.56E-05
X
Y
0 0.05 0.10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
X
Y
0 0.05 0.10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Figure 21: Velocity magnitude for optimized case 2.
Figure 22: Streamlines for optimized case 2.
Solidification with Optimized Magnetic Boundary Conditions
Figure 10. Isotherms and streamlines without and with an optimized magnetic field (test case 2 with six sensors)
(2D) 24 Oct 2002 MHD SOLIDIFICATION PROBLEM
-0.44-0.4
4
-0.37-0.3
7-0.37
-0.31-0.31
-0.25
-0.25
-0.19
-0.19
-0.12
-0.12
-0.12
-0.06
-0.06
0.00
0.00
0.06
0.06
0.13
0.13
0.19
0.19
0.19
0.25
0.25
0.25
0.31
0.31
0.31
0.38
0.38
0.38
0.44
0.44
0.44
(2D) 24 Oct 2002 MHD SOLIDIFICATION PROBLEM
(2D) 24 Oct 2002 MHD SOLIDIFICATION PROBLEM
- 0. 44
-0.44
-0. 44
-0.38
-0.38
-0.31
-0.31
-0.25
-0. 25
-0.25
-0.19
-0.19
-0.19
-0.13
-0.13
-0.13
-0.06
-0.06
0.00
0.00
0.06
0.06
0.13
0.13
0.13
0.19
0.19
0.19
0.25
0.25
0.25
0.31
0.31
0.38
0.38
0.44
0.44
(2D) 24 Oct 2002 MHD SOLIDIFICATION PROBLEM
Simulation of Electro-magneto-hydrodynamic Channel Flows
Objective
• Simulation of a steady state EMHD blood pump
• Both electric and magnetic fields are required to produce the driving force
Governing Equations
Boundary conditions and geometry
•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
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
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35X
0
0.05
0.1
0.15
0.2
0.25
0.3
YPot21.87518.7515.62512.59.3756.253.1250-3.125-6.25-9.375-12.5-15.625-18.75-21.875
Computed Velocity Vectors
0.35 0.36 0.37 0.38 0.39 0.4 0.41X
0
0.01
0.02
0.03
0.04
0.05Y
Computed Static Pressure
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35X
0
0.05
0.1
0.15
0.2
0.25
0.3
Y
P-1.1625-3.32499-5.48749-7.64999-9.81249-11.975-14.1375-16.3-18.4625-20.625-22.7875-24.95-27.1125-29.275-31.4375
Computed Static Temperature
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35X
0
0.05
0.1
0.15
0.2
0.25
0.3
Y
T318.734317.525316.316315.107313.898312.689311.48310.271309.062307.853306.645305.436304.227303.018301.809
EMHD Channel Electrode Configuration
+
-
Mesh for EMHD channel problem
0 5 10 15 20X
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Y
Computed Electric Potential Distribution
0 5 10 15 20X
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Y
POT437.5375312.5250187.512562.50-62.5-125-187.5-250-312.5-375-437.5
Computed Electric Potential on Lower Wall
0 5 10 15 20X
0
100
200
300
400
500
PO
T
Compute Static Pressure
0 5 10 15 20X
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Y
P-67.4512-249.846-432.242-614.637-797.032-979.427-1161.82-1344.22-1526.61-1709.01-1891.4-2073.8-2256.19-2438.59-2620.98
Compute Static Pressure
0 5 10 15 20X
-2500
-2000
-1500
-1000
-500
0
P
P centerP wall
Computed Axial Velocity Distribution
0
50
U
0
5
10
15
20
X00.2
0.40.6
0.81
Y
X Y
Z
EMHD Channel Electrode Configuration
+ - + -
- + - +
Computed Electric Potential
1 2 3 4 5X
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Y
POT43.7537.531.252518.7512.56.250-6.25-12.5-18.75-25-31.25-37.5-43.75
Computed Electric Field Lines
1 2 3 4 5X
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Y
Computed Static Pressure
1 2 3 4 5X
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Y
P17.210815.017512.824110.63078.437396.244024.050661.8573-0.336063-2.52943-4.72279-6.91615-9.10951-11.3029-13.4962
Computed Streamlines
1 2 3 4 5X
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Y
Conclusions• 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• 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• 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
• Better preconditioning of linear systems
• Closer look at scaling of the equations
• Extension to 3-D
• Better boundary conditions
• Unsteady
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