numerical solutions for ideal magnetohydrodynamics - an …avijit/mhd.pdf · 2006-05-24 ·...

NumericalSolutions for

Ideal Magnetohy-drodynamics

Outline

Introduction

Plasma Dynamicsand Ideal Magne-tohydrodynamics

ComputationalMHD

Conclusion

Numerical Solutions for IdealMagnetohydrodynamics

An Introduction

24th May 2006



Outline

Introduction


ComputationalMHD

Conclusion

Outline of Presentation

1 Introduction2 Plasma Dynamics and Ideal

MagnetohydrodynamicsThe Magnetohydrodynamic ApproximationSimplifications and the Ideal MHD EquationsEffect of Magnetic FieldMHD Waves

3 Computational MHD1D Ideal MHDThe 1D MHD EigenstructureNumerical Schemes in 1DMulti-dimensional Ideal MHDNumerical Schemes in Multidimensional MHD

4 Conclusion



Outline

Introduction


ComputationalMHD

Conclusion

Introduction

Magnetohydrodynamics : combination of fluiddynamics and electromagnetics → Dynamics ofan ionized gas (“plasma" ) in the presence ofmagnetic fieldsApplications : astrophysics, hypersonicvehicles, thermonuclear fusion, plasmapropulsion drag reduction, stealthWave structure : magneto-acoustic waves +Alfven wavesIdeal Magnetohydrodynamics : obtained byneglecting dissipative mechanisms (viscosity,thermal and electric conductivity, etc)Hyperbolic (Non-strict), Non-convexconservative system of equationsNumerical Solutions : Straightforwardapplication of algorithms for Euler equations notpossible



Outline

Introduction

Plasma Dynamicsand Ideal Magne-tohydrodynamicsTheMagnetohydrodynamicApproximation

Simplifications and theIdeal MHD Equations

Effect of Magnetic Field

MHD Waves

ComputationalMHD

Conclusion

Plasma Dynamics

Ionized Gas : ionization occurs at hightemperatures due to high-energy collisions;comprises of positive ions and electrons

Dynamic process: simultaneous ionization andde-ionization; fraction of ionized gas dependson temperature

Ionization also possible throughPhoto-Ionization and Electric Discharge

Photo-ionization: Ionization caused bysubjecting gas to UV / X / Gamma raysElectric discharge: ionization caused bypresence of very strong electric fields

Plasma : conducting fluid satisfying the plasmacriterion

Macroscopic Neutrality - charge separationallowed for distances of the order of DebyelengthLarge number of charged particles inside theDebye sphereCharacteristic lengths much larger than DebyelengthCharacteristic time scales much smaller thantime between electron-neutral particle collisions



Outline

Introduction




MHD Waves

ComputationalMHD

Conclusion

Plasma Dynamics

Debye Shielding Length : Maximum distanceover which the Coulombic field of a particle canbe felt; important physical parameter“Fourth State of Matter" : solid, liquid, gas,plasma; no sharp temperature for gas → plasmaphase transitionCollective Effects : Plasma propertiesdependent on particle interactions and collectiveeffects

Each charged particle interacts simultaneouslywith large number of particles throughelectromagnetic fieldsEach particle is a source of electro-static andmagnetic fields; aside from externally appliedelectromagnetic fields

Natural Occurence of Plasma : Solar corona,Solar wind, Earth’s Ionosphere (Van AllenRadiation Belts)Engineering applications : Controlledthermonuclear fusion reactions, MHD generator,plasma rockets



Outline

Introduction




MHD Waves

ComputationalMHD

Conclusion

Earth’s Magnetosphere and the SolarWind



Outline

Introduction




MHD Waves

ComputationalMHD

Conclusion

Tokamak - Plasma Containement



Outline

Introduction




MHD Waves

ComputationalMHD

Conclusion

The MagnetohydrodynamicApproximation

Single-fluid continuum assumption →Plasma treated as a single conducting fluid (maybe composed of many species)Extension of the Navier-Stokes to includeelectromagnetic force and energy terms

Mass Conservation∂ρ

∂t+∇.(ρu) = 0 (1)

Momentum Conservation∂(ρu)

∂t= ρeE + J ×B −∇.(pI +

12ρuu) + ψ (2)

Energy Conservation

12ρ

Dv2

Dt+ ρ

DeDt

= −p∇.u + E.J + φ (3)

(I is the 3× 3 identity matrix, ψ is the viscous term and φ represents terms related to

heat conduction, diffusion and work done by viscous forces)



Outline

Introduction




MHD Waves

ComputationalMHD

Conclusion

The MagnetohydrodynamicApproximationElectrodynamics governed by Maxwell’s Equations

∂B∂t

+∇× E = 0 (4)

∂D∂t−∇× H + J = 0 (5)

∇.D = ρe (6)

∇.B = 0 (7)

supplemented by the equation for chargeconservation :

∂ρe

∂t+∇.J = 0 (8)

and the generalized Ohm’s law which can beexpressed as

J = σ(E + u × B) (9)



Outline

Introduction




MHD Waves

ComputationalMHD

Conclusion

Simplifications and the Ideal MHDEquationsSimplifying Assumptions :

Neglecting of Displacement Current - valid forlow frequencies (till microwave range)Macroscopic Neutrality ⇒ neglect electrostaticbody forces and convection currentDissipative effects (viscosity, conductivity,electrical resistivity) neglected

Ideal Magnetohydrodynamic Equations

ρt +∇.(ρu) = 0 (10)

(ρu)t +∇.(ρuu + P∗I − BBµ

) = 0 (11)

Bt +∇.(uB − Bu) = 0 (12)

Et +∇.[(E + P∗)u − 1µ

(u.B)B] = 0 (13)

P∗ = p + B.B/2µ (full pressure = gas pressure + magnetic pressure),

E = ρu.u/2 + p/(γ − 1) + B.B/2µ (total energy)

Additionally, ∇.B = 0 (Initial Condition)



Outline

Introduction




MHD Waves

ComputationalMHD

Conclusion


Magnetic force is the gradient of the Magneticstress tensor

Tm =1µ

(BB − 12

B2I) (14)

In principal coordinates with B = Bz, it isequivalent to

Tm =1µ

0 0 00 0 00 0 B2

+1

2µ

−B2 0 00 −B2 00 0 −B2

(15)

Effect of the magnetic force on fluid elements= isotropic magnetic pressure (B2/2µ) +tension (B2/µ) along field linesFreezing of magnetic field lines to the fluid ⇐conservation of magnetic flux

Unrestricted movement of fluid along the fieldMotion transverse to the field carries the fieldlines along



Outline

Introduction




MHD Waves

ComputationalMHD

Conclusion

MHD Waves

Two types of wave motion possible:longitudinal (acoustic waves) and transverse(Alfvén waves)Alfvén Waves : equivalent to vibration of elasticcords under tension (due to tensile effect of themagnetic field)

VA = (tensiondensity

)1/2 = (B2

µρ)1/2 (16)

Carries perturbations in the transversecomponents of magnetic field and velocity

Magneto-acoustic waves : similar to soundwaves in gasdynamics

Parallel to magnetic field : same asgasdynamic sound waves (magnetic field doesnot affect flow)Perpendicular to magnetic field : in addition togas pressure, existence of magnetic pressure→total pressure is p + B2/2µ

V 2m = V 2

s + V 2A where V 2

s = γp/ρ (17)

Carries perturbations in density, pressure andplanar components of velocity and magneticfields



Outline

Introduction




MHD Waves

ComputationalMHD

Conclusion

Longitudinal and Transverse Waves

Figure: Parallel to MagneticField

Figure: Perpendicular toMagnetic Field



Outline

Introduction




MHD Waves

ComputationalMHD

Conclusion

MHD Waves

Parallel to the Magnetic Field - Two waves canexist

Alfvén Waves (transverse magnetic field andvelocity components vary): wavespeed is

A =√

(B2/µ)ρ

Acoustic Waves (pure sound waves):

wavespeed is a =√

γpρ

Perpendicular to the Magnetic Field - Only themagneto-acoustic mode exists with wavespeedV =

√a2 + A2

Propagation along arbitrary direction - Smallperturbation equations give three solutions(slow MHD wave, fast MHD wave and Alfvénwave ) with wavespeeds

ca = A.k =Bcosθ√

ρµ(A = B/

√ρµ) (18)

c2f ,s =

12

[(a2 + A2)±√

(a2 + A2)2 − 4a2A2cos2θ]

(19)(θ → angle between wave propagation directionand magnetic field)



Outline

Introduction




MHD Waves

ComputationalMHD

Conclusion

Wavespeed Diagram

Figure: Alfven Speed greaterthan speed of sound

Figure: Alfven Speed lessthan speed of sound



Outline

Introduction


ComputationalMHD1D Ideal MHD

The 1D MHDEigenstructure

Numerical Schemes in 1D

Multi-dimensional IdealMHD

Numerical Schemes inMultidimensional MHD

Conclusion

1D MHD Equations

The 1D system can be obtained by assumingthat the gradients exist only along the x-direction1D MHD system in conservative form is:

ut + f(u)x = 0 (20)

where u is the conserved vector and f(u) is theflux vector.

u =

ρρuρvρwBy

Bz

E

, f(u) =

ρuρu2 + P∗

ρuv − ByBx

ρuw − BzBx

uBy − vBx

uBz − wBx

(E + P∗)u − Bx(uBx + vBy + wBz)

(21)

Zero divergence constraint ⇒ Bx = constantFlux f(u) is non-convex and non-strictlyhyperbolic



Outline

Introduction







Conclusion

1D MHD Eigenstructure

The 1D MHD system admits 7 eigenvaluesEntropy wave with wavespeed λe = utwo Alfvén waves with wavespeeds λ±a = u±ca

two fast magneto-sonic waves withwavespeeds λ±f = u ± cftwo slow magneto-sonic waves withwavespeeds λ±s = u ± cs

whereca = Bx/

√ρ (22)

c2f ,s =

12

[γp + B.B

ρ±

√(γp + B.B

ρ)2 − 4γpB2

x

ρ2]

(23)Coincidence of Eigenvalues :

Bx = 0 (Propagation perpendicular to themagnetic field ): ca = cs = 0 → u is aneigenvalue with multiplicity 5B2

y + B2z = 0 (Propagation parallel to the

magnetic field ): c2f = max(a2, c2

a),c2

s = min(a2, c2a)

B2y + B2

z = 0 and a2 = c2a (“triple umbilic" ):

multiplicities of u ± ca is 3Eigenvectors : Roe and Balsara’s Eigensystemused for the present study



Outline

Introduction







Conclusion

Numerical Scheme in 1D

Semi-discrete form isdu i

dt+

1∆x

(f i+1/2 − f i−1/2) = 0 (24)

Spatial Reconstruction : done using anupwinded schemeTime Evolution : Runge-Kutta time-steppingusually used

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 1

1.1

0100

200300

400500

600700

800

Density

Cell No.

Brio & Wu’s Shock Tube (800 G

rid Points, 400 Time Steps, CFL 0.8)

1st Order Upwind

2nd Order ENO

3rd Order ENO

5th Order W

ENO

Figure: Brio & Wu’s ShockTube

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 1

0100

200300

400500

600700

800

Density

Cell No.

Brio & Wu’s High M

ach No. Problem (800 G

rid Points, 400 Time Steps, CFL 0.8)

1st Order Upwind

2nd Order ENO

3rd Order ENO

5th Order W

ENO

Figure: High Mach NumberProblem



Outline

Introduction







Conclusion

Multidimensional Ideal MHD

The multi-dimensional MHD system can bewritten in the conservative form∂u∂t

+∇.F = 0 or∂u∂t

+∂f∂x

+∂g∂y

+∂h∂z

= 0 (25)

Non-strictly hyperbolic system with completeset of eigenvectorsEight eigenvalues corresponding to entropywave, left and right moving slow, fast and Alfvénwaves and zero2D Ideal MHD equations:

f(u) =

ρuρu2 + P∗ − B2

xρuv − ByBx

ρuw − BzBx

0uBy − vBx

uBz − wBx

(E + P∗)u − Bxu.B

, g(u) =

ρvρuv +−BxBy

ρv2 + P∗ − B2y

ρvw − BzBy

vBx − uBy

0vBz − wBy

(E + P∗)v − Byu.B

(26)



Outline

Introduction







Conclusion

Multidimensional MHD

Multi-dimensional algorithms : To ensuresolenoidal nature of magnetic field (inmulti-dimensions,

∑faces Bn.ds = 0)

Projection Scheme: Poisson equation is solvedto subtract the magnetic field with non-zerodivergenceConstrained Transport / Central Difference(CT/CD) using a staggered meshEight-wave formulation: Multi-dimensional MHDsystem modified to include source termsproportional to ∇.B (non - conservative)

Eight-Wave Formulation : derived fromgoverning equations by adding term proportionalto ∇.B as source term∂u∂t

+∇.F = S where S = −(∇.B)[0 B u u .B]T

(27)

Results in non-conservative formulation buterror very less



Outline

Introduction







Conclusion

Numerical SchemesThe governing equation, discretized in space is givenas:

du ij

dtVij +

∑faces

F.ndS = Sij Vij ⇒du ij

dt= Res(i , j) (28)

where the residual is given by (for a quadrilateral cell)

Res(i , j) =−1Vij

[4∑

l=1

F.n ldSl + s ij

4∑l=1

B.n ldSl ] (29)

where s = [0 B u u .B]T

Spatial Reconstruction : Upwindedreconstruction based on characteristicdecouplingTime Evolution : Done using Runge-Kuttafamily of schemes



Outline

Introduction







Conclusion

Some Problems and Applications

Figure: Cloud ShockInteraction

Figure: Orszag Tang Vortex



Outline

Introduction







Conclusion


Figure: Planetary Interaction with Solar Wind



Outline

Introduction







Conclusion


Figure: Blunt Body Computations - Shock Stand-OffDistances



Outline

Introduction


ComputationalMHD

Conclusion

Conclusion

Lots of unresolved issues still remain in MHDLack of Roe-type averaging for fluxcomputation

Roe-type averaging possible only for γ = 2Arithmetic averaging used by all Roe-typeschemes

Admissibility of “exotic shocks" :Non-Convex Flux function : Rankine-Hugoniotconditions admit “intermediate waves" acrosswhich only one family of characteristicsconvergeAdmissibility is debated → straightforwardextension of admissibility conditions of convexsystems does not work

MHD system highly non-linear → Higher orderschemes yield oscillatory solutions, even at lowCFL → most schemes used till now are highlyTVD

numerical solutions for ideal magnetohydrodynamics - an …avijit/mhd.pdf · 2006-05-24 ·...

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