Magnetohydrodynamics (MHD)
Magnetohydrodynamics (MHD): Theoretical Foundations
Fundamentals of Magnetohydrodynamics
Professor, is Magnetohydrodynamics (MHD) a combination of fluid dynamics and electromagnetism?
Exactly. MHD is a field that deals with the interaction between electromagnetic fields and fluid motion in conductive fluids (such as liquid metals, plasma, electrolyte solutions, etc.). When a magnetic field acts on a fluid carrying an electric current, a Lorentz force is generated, altering the flow. Conversely, the motion of a conductive fluid induces a magnetic field.
Familiar application examples:
- Continuous Casting: Electromagnetic braking, electromagnetic stirring of molten steel
- Aluminum Electrolytic Refining: Flow control of molten aluminum inside Hall-Heroult cells
- Nuclear Fusion Reactors: Magnetic confinement of plasma
- MHD Pumps: Transporting liquid metals without moving parts
- Space Propulsion: MPD thrusters
Governing Equations of MHD
MHD is a system formed by coupling the Navier-Stokes equations and Maxwell's equations.
Modified Navier-Stokes Equation (with added Lorentz force):
Here, $\mathbf{J} \times \mathbf{B}$ is the Lorentz force (body force).
Ohm's Law (in a moving conductive fluid):
$\sigma$ is electrical conductivity, $\mathbf{E}$ is the electric field, $\mathbf{u} \times \mathbf{B}$ is the electromotive force due to fluid motion.
Magnetic Induction Equation:
The first term on the right-hand side is magnetic field convection (freezing), the second term is magnetic field diffusion.
The fluid motion equation includes electromagnetic forces, and simultaneously, the velocity appears in the magnetic field equation. It's a fully bidirectional coupling.
MHD Dimensionless Numbers
Let's organize the dimensionless numbers specific to MHD.
| Dimensionless Number | Definition | Physical Meaning |
|---|---|---|
| Hartmann Number Ha | $BL\sqrt{\sigma/\mu}$ | Electromagnetic Force / Viscous Force |
| Magnetic Reynolds Number Rm | $\mu_0 \sigma U L$ | Magnetic Field Convection / Magnetic Field Diffusion |
| Stuart Number (Interaction Parameter) N | $\sigma B^2 L / (\rho U) = \text{Ha}^2/\text{Re}$ | Electromagnetic Force / Inertial Force |
| Magnetic Prandtl Number Pm | $\mu_0 \sigma \nu$ | Momentum Diffusion / Magnetic Field Diffusion |
For industrial liquid metals (molten steel, molten aluminum, etc.), $\text{Pm} \sim 10^{-6}$ is extremely small. This means magnetic field diffusion is much faster than fluid momentum diffusion, resulting in $\text{Rm} \ll 1$. In this case, the magnetic field is almost determined by the externally applied field, and we only need to consider its effect on the fluid (low Rm approximation).
So, the larger the Hartmann number, the stronger the influence of the magnetic field.
Correct. Ha = 0 corresponds to regular fluid dynamics, Ha โ โ means the flow is completely constrained by the magnetic field. In continuous casting, Ha โผ 100-1000 is typical.
Plasma Confinement in Nuclear Fusion Reactors โ MHD Safeguarding Humanity's Dream
In ITER (International Thermonuclear Experimental Reactor), plasma at 100 million degrees Celsius is confined by a tokamak-type magnetic field. The core of the stability analysis for this plasma is MHD theory. Whether the plasma "escapes" from the magnetic field or whether "kink instabilities" or "ballooning instabilities" occur โ by solving these as eigenvalue problems of the MHD equations, coil design and current profiles are optimized. The fact that fusion is said to be "30 years away" is a testament to the difficulty of this MHD stability analysis.
Computational Methods for Magnetohydrodynamics (MHD)
Numerical Methods for MHD
How do you solve the coupled MHD equations in CFD?
The methods differ for low Rm approximation (most industrial liquid metals) and high Rm problems (plasma, astrophysics).
Low Rm Approximation Solution Method
For $\text{Rm} \ll 1$, the induced magnetic field is negligible compared to the applied field, and the magnetic field can be treated as a known external field $\mathbf{B}_0$. The governing equation for the electric field is:
$\phi$ is the electric potential. Solve this Poisson equation to find $\mathbf{E} = -\nabla\phi$, then calculate $\mathbf{J} = \sigma(-\nabla\phi + \mathbf{u} \times \mathbf{B}_0)$, $\mathbf{F}_L = \mathbf{J} \times \mathbf{B}_0$ and add it as a source term to the N-S equation.
Calculation procedure:
1. Calculate $\mathbf{u} \times \mathbf{B}_0$ from the flow field
2. Solve the Poisson equation for electric potential
3. Calculate Current Density $\mathbf{J}$
4. Add Lorentz Force $\mathbf{F}_L$ to the N-S equation to update the flow field
5. Repeat steps 1-4 until convergence
So it's basically an extension of N-S, just adding one more equation for electric potential.
High Rm Problem Solution Method
For $\text{Rm} \gg 1$ (plasma physics, astrophysics), it's necessary to fully solve the magnetic induction equation.
Main numerical methods:
- Constrained Transport (CT) Method: Strictly maintains $\nabla \cdot \mathbf{B} = 0$ via magnetic flux conservation on cell faces
- Divergence Cleaning Method: Adds a correction equation to damp errors in $\nabla \cdot \mathbf{B}$
- Vector Potential Method: Uses $\mathbf{B} = \nabla \times \mathbf{A}$ to automatically guarantee $\nabla \cdot \mathbf{B} = 0$
Maintaining $\nabla \cdot \mathbf{B} = 0$ is a fundamental challenge in numerical MHD. If this breaks down, unphysical magnetic monopole forces appear, causing the calculation to fail.
Hartmann Flow โ Basic MHD Verification Problem
The most basic analytical solution in MHD is Hartmann flow. It involves flow between parallel plates with a magnetic field $B_0$ applied perpendicular to the walls.
Ha = 0 gives a parabolic distribution (Poiseuille flow), Ha โ โ gives a flat distribution except near the walls in the Hartmann layer (thickness $\delta_H \sim H/\text{Ha}$).
| Ha | Velocity Profile | Hartmann Layer Thickness |
|---|---|---|
| 0 | Parabolic | None |
| 10 | Slightly flattened | $H/10$ |
| 100 | Almost flat in center | $H/100$ |
| 1000 | Gradient only in Hartmann layer | $H/1000$ |
For Ha = 1000, the Hartmann layer is only 1/1000 of the channel width. The mesh requirements seem strict.
Exactly. To resolve the Hartmann layer, a number of mesh layers near the wall proportional to the Ha number is required. This is one of the major costs of MHD computation.
The Hurdle of MHD Numerical Methods โ The Struggle to Maintain "Zero Divergence of Magnetic Field"
The most troublesome aspect of numerical implementation for MHD analysis is maintaining โยทB=0 (magnetic flux continuity). When discretizing the magnetic field using the finite volume method, numerical errors accumulate, causing โยทBโ 0, which generates unphysical magnetic forces and distorts the solution. To prevent this, methods like "divergence cleaning" and "constrained transport" have been developed. Especially in large-scale MHD simulations of solar wind, if the divergence error exceeds 1%, numerical "magnetic monopoles" are created, causing plasma to fly off in the wrong direction. Zero divergence of the magnetic field is a physical requirement and a battlefield for numerical methods.