磁気流体力学（MHD）
Theory and Physics
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-Herault 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 → ∞ means the flow is completely constrained by the magnetic field. In continuous casting, Ha ∼ 100-1000 is typical.
Plasma Confinement in Nuclear Fusion Reactors — 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 — 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.
Physical Meaning of Each Term
- Temporal Term $\partial(\rho\phi)/\partial t$: Imagine the moment you turn on a faucet. At first, the water comes out spluttering and unstable, but after a while, it becomes a steady flow, right? This "period of change" is described by the temporal term. The pulsation of blood flow from a heartbeat, or the flow fluctuation each time an engine valve opens and closes, are all unsteady phenomena. So what is steady-state analysis? Looking only at "after sufficient time has passed and the flow has settled down" — meaning setting this term to zero. Since computational cost is significantly reduced, solving first in steady-state is a basic CFD strategy.
- Convection Term $\nabla \cdot (\rho \mathbf{u} \phi)$: What happens if you drop a leaf into a river? It gets carried downstream by the flow, right? This is "convection" — the effect where fluid motion transports things. Warm air from a heater reaching the far corner of a room is also because the air, the "carrier," transports heat via convection. Here's the interesting part — this term contains "velocity × velocity," making it nonlinear. That is, as the flow becomes faster, this term rapidly strengthens, making control difficult. This is the root cause of turbulence. A common misconception: "Convection and conduction are similar" → They are completely different! Convection is carried by flow, conduction is transmitted by molecules. There is an order of magnitude difference in efficiency.
- Diffusion Term $\nabla \cdot (\Gamma \nabla \phi)$: Have you ever put milk in coffee and left it? Even without stirring, after a while it naturally mixes, right? That's molecular diffusion. Now, next question — honey and water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. When viscosity is high, the diffusion term becomes strong, and the fluid moves in a "thick" manner. In low Reynolds number flows (slow, viscous), diffusion is dominant. Conversely, in high Re number flows, convection overwhelms and diffusion plays a supporting role.
- Pressure Term $-\nabla p$: When you push the plunger of a syringe, liquid shoots out forcefully from the needle tip, right? Why? Because the piston side is high pressure, the needle tip is low pressure — this pressure difference provides the force pushing the fluid. Dam discharge works on the same principle. On a weather map, where isobars are densely packed? That's right, strong winds blow. "Flow is generated where there is a pressure difference" — this is the physical meaning of the pressure term in the Navier-Stokes equations. A point of confusion here: The "pressure" in CFD is often gauge pressure, not absolute pressure. When you switch to compressible analysis and suddenly get strange results, it might be due to confusing absolute/gauge pressure.
- Source Term $S_\phi$: Warmed air rises — why? Because it becomes lighter (less dense) than its surroundings, so it's pushed up by buoyancy. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat generated by a gas stove flame, Lorentz force applied to molten metal by a factory electromagnetic pump... These are all actions that "inject energy or force into the fluid from the outside," expressed by the source term. What happens if you forget the source term? In natural convection analysis, if you forget to include buoyancy, the fluid doesn't move at all — you get a physically impossible result like warm air not rising in a room with the heater on in winter.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (mean free path ≪ characteristic length)
- Newtonian Fluid Assumption: Linear relationship between shear stress and strain rate (non-Newtonian fluids require viscosity models)
- Incompressibility Assumption (for Ma < 0.3): Treat density as constant. For Mach number ≥ 0.3, consider compressibility effects
- Boussinesq Approximation (Natural Convection): Consider density variation only in the buoyancy term, using constant density in other terms
- Non-applicable Cases: Rarefied gases (Kn > 0.1), supersonic/hypersonic flow (shock capturing required), free surface flow (VOF/Level Set, etc. required)
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Velocity $u$ | m/s | When converting from volumetric flow rate for inlet conditions, pay attention to cross-sectional area units |
| Pressure $p$ | Pa | Distinguish between gauge and absolute pressure. Use absolute pressure for compressible analysis |
| Density $\rho$ | kg/m³ | Air: approx. 1.225 kg/m³@20°C, Water: approx. 998 kg/m³@20°C |
| Viscosity Coefficient $\mu$ | Pa·s | Be careful not to confuse with kinematic viscosity coefficient $\nu = \mu/\rho$ [m²/s] |
| Reynolds Number $Re$ | Dimensionless | $Re = \rho u L / \mu$. Criterion for Laminar/Turbulent Transition |
| CFL Number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability |
Numerical Methods and Implementation
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 — 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 → ∞ 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 — The Struggle to Maintain "Zero Divergence of Magnetic Field"
The most troublesome aspect of numerical implementation for MHD analysis is maintaining ∇·B=0 (magnetic flux continuity). When discretizing the magnetic field using the finite volume method, numerical errors accumulate, causing ∇·B≠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.
Upwind Differencing (Upwind)
1st-order Upwind: Large numerical diffusion but stable. 2nd-order Upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.
Central Differencing
2nd-order accuracy, but numerical oscillations occur for Pe number > 2. Suitable for low Reynolds number diffusion-dominated flows.
TVD Schemes (MUSCL, QUICK, etc.)
Suppress numerical oscillations while maintaining high accuracy using limiter functions. Effective for capturing shocks and steep gradients.
Finite Volume Method vs Finite Element Method
FVM: Naturally satisfies conservation laws. Mainstream in CFD. FEM: Advantageous for complex shapes and multiphysics. Mesh-free methods like SPH are also developing.
CFL Condition (Courant Number)
Explicit methods: CFL ≤ 1 is the stability condition. Implicit methods: Stable even for CFL > 1, but affects accuracy and iteration count. LESRelated Topics
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