共役熱伝達（CHT）
Theory and Physics
What is Conjugate Heat Transfer?
Professor, conjugate heat transfer is about "solving for the solid and fluid simultaneously," right? Is there really a point to coupling them?
Absolutely. For example, in turbine blade cooling design, the cooling air flowing through internal cooling passages and the high-temperature combustion gas passing over the external blade surface are thermally coupled through the blade metal. To accurately obtain the temperature distribution on the solid side, it's necessary to solve the fluid temperature field and solid heat conduction simultaneously, rather than assuming a heat transfer coefficient $h$ on the fluid side.
Not having to assume a heat transfer coefficient is certainly a big advantage. Is it also used in electronics cooling and such?
Of course. Any problem where solid heat conduction and fluid convective heat transfer are tightly coupled is a candidate for CHT: combinations of heat sinks and fan cooling, cold plate design for power semiconductor modules, heat dissipation design for LED packages, etc.
Governing Equations
Specifically, what equations are solved?
In the fluid domain, the Navier-Stokes equations and the energy equation are solved. In the solid domain, the heat conduction equation is solved. Then, continuity conditions for temperature and heat flux are imposed at the interface between these two domains.
The interface conditions can be written mathematically as follows.
Temperature is continuous, and heat flux is also continuous. Essentially, energy is conserved across the interface, right?
Exactly. The steady-state heat conduction equation for the solid side is
where $\dot{q}_v$ is volumetric heat generation (like Joule heating). The energy equation on the fluid side includes the convection term. The essence of CHT is that these two are coupled through the interface conditions.
Can we ignore interface thermal resistance (contact thermal resistance)?
The above is fine for an ideal interface. For actual TIMs (Thermal Interface Materials) or bolted joint surfaces, contact thermal resistance needs to be added to the interface conditions. In Ansys Fluent or STAR-CCM+, this can be set as a thin wall or contact resistance.
Pentium 4's Thermal Runaway Made CHT Analysis an Industry Standard
In the early 2000s, Intel's Pentium 4 saw a rapid increase in heat density due to higher clock speeds, leading to frequent problems where CPUs would experience thermal throttling (performance reduction due to heat) due to cooling design failures. Until then, cooling design was often based on the rule of thumb "just attach a heat sink and it's fine," but this incident suddenly made the necessity of CHT analysis, which solves for the CPU (solid) and cooling airflow (fluid) simultaneously, widely recognized. It's no exaggeration to say that the Pentium 4's failure was the starting point for modern thermal design CAE.
Physical Meaning of Each Term
- Temporal Term $\partial(\rho\phi)/\partial t$: Think of the moment you turn on a faucet. At first, water comes out in an unstable, spluttering manner, but after a while, the flow becomes steady, 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—all are unsteady phenomena. So what is steady-state analysis? It's looking only at "after sufficient time has passed and the flow has settled down"—meaning setting this term to zero. Since computational cost drops significantly, trying a steady-state solution first 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 "carrier," air, transports heat via convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, as flow speed increases, this term rapidly strengthens, making control difficult. This is the root cause of turbulence. A common misconception: "Convection and conduction are similar" → They're completely different! Convection is carried by flow, conduction is transmitted by molecules. There's 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 they naturally mix, right? That's molecular diffusion. Now a 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 flow (slow, viscous), diffusion is dominant. Conversely, in high Re number flow, convection overwhelmingly dominates, 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 plunger side is high pressure, the needle tip is low pressure—this pressure difference becomes the force pushing the fluid. Dam discharge works on the same principle. On a weather map, where isobars are tightly 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: "Pressure" in CFD is often gauge pressure, not absolute pressure. If results go wrong immediately after switching to compressible analysis, mixing up absolute/gauge pressure might be the cause.
- Source Term $S_\phi$: Heated air rises—why? Because it becomes lighter (lower density) 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 from a gas stove flame, Lorentz force acting on molten metal in a factory's 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, forgetting to include buoyancy means the fluid doesn't move at all—you get a physically impossible result where warm air doesn't rise in a heated room in winter.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (mean free path ≪ characteristic length)
- Newtonian Fluid Assumption: Shear stress and strain rate have a linear relationship (non-Newtonian fluids require viscosity models)
- Incompressible Assumption (for Ma < 0.3): Density is treated as constant. For Mach number ≥ 0.3, compressibility effects must be considered
- Boussinesq Approximation (natural convection): Density variation is considered only in the buoyancy term; constant density is used in other terms
- Non-applicable Cases: Rarefied gas (Kn > 0.1), supersonic/hypersonic flow (shock capturing required), free surface flow (requires VOF/Level Set, etc.)
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 $\nu = \mu/\rho$ [m²/s] |
| Reynolds Number $Re$ | Dimensionless | $Re = \rho u L / \mu$. Indicator for laminar/turbulent transition |
| CFL Number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability |
Numerical Methods and Implementation
Classification of Coupling Approaches
Are there different ways to solve CHT?
There are two main types: monolithic and partitioned. Monolithic solves the solid and fluid simultaneously with a single solver. Ansys Fluent, STAR-CCM+, and OpenFOAM's chtMultiRegionFoam use this approach.
What about partitioned?
Partitioned runs separate solid and fluid solvers and exchanges interface data. For example, co-simulation between Ansys Mechanical and Fluent, or between Abaqus and STAR-CCM+, falls into this category. It's also a method used in FSI (Fluid-Structure Interaction).
Monolithic offers high interface consistency and faster convergence. Partitioned provides flexibility to combine existing solvers, but requires attention to interface interpolation accuracy and convergence stability.
Interface Mesh Design
Does it cause problems if the mesh size is completely different between solid and fluid?
It certainly can. The solid side can often be relatively coarse, but the fluid side needs to resolve the wall boundary layer. In Ansys Fluent, you choose to set the wall first layer $y^+$ to around 1 and not use wall functions (low-Re turbulence model), or use $y^+ \approx 30$ with wall functions.
For CHT, you want to accurately capture the temperature gradient at the wall, so $y^+ \approx 1$ is better, right?
Exactly. Especially if discussing local heat transfer coefficient or Nu number distribution, you should place a sufficient number of prism layers (inflation layer). In STAR-CCM+, you can specify total thickness from the wall and number of layers for automatic generation. In OpenFOAM, you set it in snappyHexMesh's addLayersControl.
Convergence Judgment Notes
Since both solid and fluid are solved simultaneously, convergence judgment seems tricky.
Monitoring temperature and heat flux at the interface is crucial, not just residuals. You need to confirm that the average or maximum temperature at the interface stops fluctuating between iterations. In Fluent, a standard practice is to track the area-weighted average temperature of the interface using a surface monitor.
Is there a guideline for the number of iterations?
For monolithic, it's similar to regular CFD calculations. For partitioned co-simulation, set the number of sub-iterations per step to around 3–10 and confirm that interface value fluctuations become sufficiently small. Adjusting the relaxation factor (under-relaxation) is also key.
"Interface Continuity Conditions"—The Simple Equations at the Core of CHT Coupling Methods
The most important aspect of conjugate heat transfer (CHT) coupling methods is matching both temperature and heat flux at the solid-fluid interface. Implementing these "two simple equations" is actually difficult; in segregated (partitioned) solutions, adjusting relaxation factors is necessary for convergence, and without using iterative methods, heat balance at the interface won't match. If you analyze with the misunderstanding that "matching temperature alone is enough," you'll get incorrect results where the heat balance is off between the solid and fluid sides. Understanding interface conditions is the starting point for CHT.
Upwind Scheme
First-order upwind: High numerical diffusion but stable. Second-order upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.
Central Differencing
Second-order accurate, but numerical oscillations occur for Pe number > 2. Suitable for low Reynolds number diffusion-dominated flows.
TVD Schemes (MUSCL, QUICK, etc.)
Maintain high accuracy while suppressing numerical oscillations via limiter functions. Effective for capturing shock waves or steep gradients.
Finite Volume Method vs Finite Element Method
FVM: Naturally satisfies conservation laws. Mainstream in CFD. FEM: Advantageous for complex shapes and multi-physics. 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. LES: CFL ≈ 1 is recommended. Physical meaning: Information should not travel more than one cell per time step.
Residual Monitoring
Convergence is typically judged when residuals for continuity, momentum, and energy drop by 3–4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factor
Typical initial values: Pressure: 0.2–0.3, Velocity: 0.5–0.7. If diverging, lower the relaxation factor. After convergence, increase to accelerate.
Internal Iterations for Unsteady Calculations
Iterate within each time step until a steady solution converges. Internal iteration count: 5–20 iterations is a guideline. If residuals fluctuate between time steps, review the time step size.
Analogy for the SIMPLE Method
The SIMPLE method is an "alternating adjustment" technique. First, velocity is tentatively determined (predictor step), then pressure is corrected so that mass conservation is satisfied with that velocity (corrector step), and velocity is revised using the corrected pressure—this back-and-forth is repeated to approach the correct solution. It resembles two people leveling a shelf: one adjusts the height, the other balances it, and they repeat this alternately.
Analogy for Upwind Differencing
Upwind differencing is a method that "stands in the river flow and prioritizes upstream information." A person in the river looking downstream can't tell where the water comes from—it's a discretization method reflecting the physics that upstream information determines downstream conditions. It's first-order accurate but highly stable because it correctly captures flow direction.
Practical Guide
Analysis Workflow
What's the typical procedure for a CHT analysis?
A typical procedure is as follows. (1) Define solid and fluid domains from CAD. (2) Generate mesh for the fluid domain, resolving the wall boundary layer with prism layers. (3) Generate mesh for the solid domain. (4) Connect the interface conformally (node-matched) or non-conformally. (5) Set material property values. (6) Set boundary and initial conditions and execute the calculation.
Related Topics
なった
詳しく
報告