温度境界層
Theory and Physics
Fundamental Concepts of Thermal Boundary Layer
Professor, how is the thermal boundary layer related to the velocity boundary layer?
When fluid flows over a heated (or cooled) wall surface, a thin layer where the temperature changes rapidly forms near the wall. This is the thermal boundary layer. Similar to the velocity boundary layer, it asymptotically approaches the freestream temperature as you move away from the wall. The ratio of their thicknesses is determined by the Prandtl number $Pr$.
If $Pr > 1$ (water, oil, etc.), the thermal boundary layer is thinner than the velocity boundary layer. If $Pr < 1$ (air, molten metal, etc.), the thermal boundary layer is thicker.
What is the $Pr$ of air?
For air at room temperature, $Pr \approx 0.71$, so the thermal boundary layer is slightly thicker than the velocity boundary layer. For water, $Pr \approx 7$, so the thermal boundary layer becomes about half the thickness of the velocity boundary layer. For engine oil, $Pr \sim 1000$ or more, making the thermal boundary layer extremely thin.
Theoretical Solution for Laminar Thermal Boundary Layer
Are there analytical solutions?
For laminar boundary layers on an isothermal flat plate, the Blasius solution (velocity field) and the Pohlhausen solution (temperature field) are classical theoretical solutions. The local Nusselt number is given by
(for $Pr > 0.6$). This formula is always used in basic CFD validation. You should confirm that the wall Nusselt number from CFD matches this theoretical solution within 2-3% before proceeding to more complex problems.
What about turbulent flow?
For turbulent boundary layers, $Nu_x = 0.0296 Re_x^{4/5} Pr^{1/3}$ is an empirical correlation. In turbulence, the transport of momentum and heat is dominated by eddy diffusion (turbulent diffusivity), so the thickness ratio between the thermal and velocity boundary layers is determined by the turbulent Prandtl number $Pr_t \approx 0.85$ to $0.9$.
The Foundation of Thermal Boundary Layer Theory—Pohlhausen's Integral Method (1921)
Seventeen years after Prandtl (1904) proposed the concept of the velocity boundary layer, his student E. Pohlhausen (1921) derived the "integral equation for the thermal boundary layer," providing the first analytical solution for the temperature distribution in forced convection over a flat plate. His analysis revealed the dependence Nu ∝ Re^(1/2)×Pr^(1/3) and showed that the boundary layer thickness ratio (δ_t/δ) differs by three orders of magnitude between low-Pr liquid metals (Pr≈0.01) and high-Pr oils (Pr≈1000). This simple scaling law is still used today, a century later, to check the validity of CFD results. In practice, it's common sense to regard CFD output that deviates from this analytical solution as a sign of problems with the mesh or material properties.
Physical Meaning of Each Term
- Temporal Term $\partial(\rho\phi)/\partial t$: Think of the moment you turn on a faucet. At first, the water comes out spluttering and unstable, but after a while, the flow becomes steady, right? This term describes that "period of change." 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 looks only at "after sufficient time has passed and the flow has settled down"—meaning this term is set to zero. This significantly reduces computational cost, so starting with a steady-state solution 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 the 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, it naturally mixes after a while. That's molecular diffusion. Now a question—honey or water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. Higher viscosity strengthens the diffusion term, making the fluid move "sluggishly." In low Reynolds number flows (slow, viscous), diffusion dominates. Conversely, in high Re flows, 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 piston side is high pressure, the needle tip is low pressure—this pressure difference provides the force that pushes 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 arises 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, it might be due to mixing up absolute/gauge pressure.
- Source Term $S_\phi$: Heated air rises—why? Because it becomes lighter (lower density) than its surroundings, so buoyancy pushes it upward. 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 buoyancy means the fluid doesn't move at all—a physically impossible result where warm air doesn't rise in a heated winter room.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (molecular mean free path ≪ characteristic length)
- Newtonian Fluid Assumption: Shear stress and strain rate have a linear relationship (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 (requires shock capturing), 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: ~1.225 kg/m³ @20°C, Water: ~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 | $Re = \rho u L / \mu$. Criterion for laminar/turbulent transition |
| CFL Number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to timestep stability |
Numerical Methods and Implementation
Wall Mesh Requirements
How should I design the mesh to accurately resolve the thermal boundary layer?
Wall-normal mesh design is closely related to the wall treatment of the turbulence model. For the Low-Reynolds number approach (direct resolution), a guideline is to place the first cell at $y^+ \approx 1$ and have at least 5 layers within the viscous sublayer and 5-10 layers in the buffer layer.
Do the thermal and velocity boundary layers require different mesh densities?
For fluids with $Pr > 1$, the thermal boundary layer is thinner than the velocity boundary layer, so accurately predicting heat transfer may require an even finer mesh than for the velocity field. Specifically, $y^+_{T} = y u_\tau / \alpha < 1$ should be satisfied, which corresponds to $y^+ \cdot Pr < 1$. For $Pr = 7$ (water), $y^+ < 0.14$ is ideal.
Is such a thin first cell practically achievable?
In practice, sufficient accuracy is often obtained with $y^+ \approx 0.5$ to $1$. Even for water, if $y^+ < 1$ is satisfied, the error in the Nusselt number can be kept within 5%. What's important is the growth ratio in the wall-normal direction, which should be kept within 1.1 to 1.2.
How to Check y+ Values
How do I check if $y^+$ is appropriate after the calculation?
What if $y^+$ is too large?
Reduce the thickness of the first layer of the inflation layer (prism layer) or increase the number of layers. However, when flow velocity varies (e.g., different velocities near the inlet and downstream), it can be difficult to satisfy $y^+ < 1$ over the entire wall. Fluent's Enhanced Wall Treatment and STAR-CCM+'s All y+ Wall Treatment automatically switch wall treatments based on $y^+$ values, so using these is safer in practice.
Handling High-Pr Fluids
What about high-Pr fluids like oil ($Pr > 100$)?
Standard wall functions significantly lose accuracy as $Pr$ increases. Fluent's Enhanced Thermal Wall Treatment includes a high-Pr correction, so this should be enabled. In OpenFOAM, alphatJayatillekeWallFunction contains Jayatilleke's (1969) correction. For the mesh, the golden rule is to make $y^+$ as small as possible.
Thermal Boundary Layer y+ Management—The Branching Point Between Wall Functions and Low-Re Solutions is y+=1
The accuracy of the thermal boundary layer is first determined by the first cell height (y+ value). The wall function approach (y+=30-300) assumes the logarithmic law region to estimate wall heat flux, so errors increase rapidly in regions with strong pressure gradients or separation. On the other hand, when using low-Reynolds number models (SST-ω, v2-f, etc.), y+<1 is required, and 15-20 prism layer cells must be secured to resolve the steep temperature gradient in the viscous sublayer. In practice, a staged refinement strategy—"first check the global flow with wall functions → switch to low-Re only for important heat flux surfaces"—is the standard way to balance computational cost and accuracy.
Upwind Differencing (Upwind)
First-order upwind: Large 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 shocks and steep gradients.
Finite Volume Method vs Finite Element Method
FVM: Naturally satisfies conservation laws. Mainstream in CFD. FEM: Advantageous for complex geometries 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. LES: CFL ≈ 1 recommended. Physical meaning: Information should not travel more than one cell per timestep.
Residual Monitoring
Convergence is judged when residuals for continuity, momentum, and energy equations drop by 3-4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factors
Typical initial values: Pressure: 0.2-0.3, Velocity: 0.5-0.7. Reduce factors if diverging. Increase after convergence to accelerate.
Internal Iterations for Unsteady Calculations
Iterate within each timestep until a steady solution converges. Internal iteration count: 5-20 is a guideline. If residuals fluctuate between timesteps, review the timestep 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 then velocity is revised with the corrected pressure—this back-and-forth is repeated to approach the correct solution. It's similar to 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 technique that "stands in the river flow and prioritizes upstream information." A person in the river cannot tell where the water comes from by looking downstream—this discretization method reflects the physics that upstream information determines downstream conditions. It's first-order accurate but highly stable because it correctly captures flow direction.
Practical Guide
Heat Transfer in Transitional Boundary Layers
How does heat transfer change when transition from laminar to turbulent flow occurs?
The Nusselt number increases sharply at the transition point. This is because turbulent mixing thins the thermal boundary layer and steepens the temperature gradient at the wall. For aircraft wing leading edges or wind turbine blades, the transition location determines the external surface temperature distribution, so accurate prediction is crucial.
How can I predict transition in CFD?
The Transition SST ($\gamma$-$Re_\theta$) model is the standard choice. The freestream turbulence intensity $Tu$ at the inlet greatly affects the transition location, so it must be set accurately to match experimental conditions. For turbine airfoil CFD, sensitivity analysis within the range $Tu = 1$ to $10$% is common practice.
Heat Transfer with Separation
What happens to the thermal boundary layer in separated flows like over a backward-facing step?
Immediately after the separation point, the flow velocity near the wall decreases, reducing the Nusselt number. At the reattachment point, a flow structure similar to jet impingement forms, and the Nusselt number peaks. Downstream of the reattachment point, as the boundary layer redevelops, the Nusselt number gradually approaches the fully developed value.
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