衝突噴流熱伝達
Theory and Physics
Flow Structure of Impinging Jets
Professor, what's the difference between an impinging jet and ordinary convection?
An impinging jet (jet impingement) is a flow where a jet ejected from a nozzle collides with a wall surface, achieving a very high heat transfer coefficient near the stagnation point. It can achieve heat transfer rates 2 to 3 times higher than typical forced convection in pipes, making it widely used in applications such as internal cooling of gas turbine blades, steel plate quenching, and electronic component cooling.
The flow is divided into three regions. (1) Free jet region: The area from the nozzle towards the wall, containing a potential core. (2) Impingement zone: The region near the wall where the flow changes direction. (3) Wall jet region: The area where the flow spreads radially along the wall.
Nusselt Number Correlation
Is there a correlation for the Nusselt number of impinging jets?
Yes. Martin's (1977) correlation is widely used. The stagnation point Nusselt number for a single circular nozzle is
where $Re_D = u_j D / \nu$ is the Reynolds number based on nozzle diameter $D$ and jet exit velocity $u_j$. A more general form including the effect of nozzle-to-wall distance $H$ is
Many experimental results show the stagnation point Nu number is maximum at $H/D \approx 6$ to $8$.
What happens if $H/D$ is too large?
Beyond the potential core length (typically 4 to 6D), the jet diffuses and the velocity at impingement decreases, reducing the Nu number. Conversely, for $H/D < 4$, in a confined geometry, cross-flow effects (where spent flow interferes with fresh jets) can also cause performance degradation.
Effect of Jet Arrays
In actual gas turbine cooling, there are multiple rows of jet holes, right?
Correct. In array impingement, the interference between jets and the influence of cross-flow become important. A smaller hole spacing $S/D$ increases the area-averaged Nu number, but also strengthens cross-flow, degrading the cooling performance of downstream jets. Typically, designs use $S/D = 4$ to $8$. The correlation by Florschuetz et al. (1981) is the standard reference data for array jets.
The Origin of Jet Impingement Cooling—A Thermal Control Technology Born from NASA Space Development
Jet impingement cooling was systematically engineered during the NASA Apollo program in the 1960s. It was adopted for cooling the capsule surface during atmospheric re-entry and the engine nozzles of the Saturn V rocket, using impingement cooling from multiple cooling holes. The correlation for impinging jet heat transfer compiled by Martin (1977) (Nu = f(Re, Pr, H/D, x/D)) is still widely used today as an initial estimation formula in design. Later, it was adapted for internal cooling of gas turbine blades, and its applications have since expanded to include local cooling of electronic devices and medical equipment (endoscope tip cooling).
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 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 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 due to convection, where the air, the "carrier," transports heat. 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 are 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 added milk to 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 in a "thick" manner. In low Reynolds number flows (slow, viscous), diffusion is dominant. Conversely, in high Re 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, and 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 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: In CFD, "pressure" 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 $S_\phi$: Warmed air rises—why? Because it becomes lighter (lower density) than its surroundings and is 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 applied to 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, like turning on a heater in a winter room but the warm air doesn't rise.
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 (viscosity model needed for non-Newtonian fluids)
- 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 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 | $Re = \rho u L / \mu$. Criterion for Laminar/turbulent transition. |
| CFL Number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability. |
Numerical Methods and Implementation
Turbulence Model Selection is Critically Important
I've heard that turbulence model selection is critical for CFD of impinging jets.
That's correct. Impinging jets are famous as a benchmark problem for turbulence models, with many models overpredicting the Nu number in the impingement zone. The main cause is the overestimation of turbulent kinetic energy production terms in the impingement region.
Specifically, the standard k-ε model can overpredict the stagnation point Nu number by more than double the experimental value. The v2f model and SST k-ω show significant improvement, but a 20-30% overprediction can still remain. Models based on the $\omega$-equation generally perform better than k-ε based ones.
What is the most reliable RANS model?
According to literature reviews, the v2f model (Fluent: $\overline{v^2}$-$f$ model) tends to produce predictions closest to experimental values for impinging jets. Next is SST k-ω. However, v2f is available in Fluent and OpenFOAM (v2f turbulence model) but is not standard in STAR-CCM+.
Mesh Requirements
How fine does the mesh need to be?
In the impingement zone, ensure $y^+ < 1$ in the wall-normal direction, and a resolution of $\Delta r / D \approx 0.02$ to $0.05$ relative to nozzle diameter $D$ is needed in the wall-parallel direction. Sufficient cells must also be placed between the nozzle exit and the wall to resolve jet development. A typical 2D axisymmetric calculation requires 50,000 to 200,000 cells, while a 3D array jet may require millions of cells.
If axisymmetric calculation is possible, 2D is more efficient, right?
For a single circular nozzle, an axisymmetric (Axisymmetric) model is very efficient. Both Fluent and STAR-CCM+ have axisymmetric solvers. However, for rectangular nozzles or cases with cross-flow, 3D is mandatory.
Consideration of LES/DES
Is LES also used for impinging jets?
It is actively used at the research level. Kelvin-Helmholtz instabilities generated in the jet shear layer and large-scale vortex structures in the impingement region cause unsteady fluctuations in the Nu number. LES can directly resolve these vortex structures, so the time-averaged Nu number distribution is closer to experiments than RANS. DES (Detached Eddy Simulation) and SBES are also useful as intermediate options.
Coffee Break Yomoyama Talk
Numerical Schemes for Jet Impingement Heat Transfer—Mesh Resolution at the Stagnation Point is Crucial
The most difficult aspect of jet impingement CFD is predicting the heat transfer at the Stagnation Point. Here, the velocity gradient is maximum, so standard k-ε models have a tendency to overestimate turbulent kinetic energy production, leading to a 20-30% overprediction of the Nu number. The v2-f model and SST-ω, which are strong in stagnation point accuracy, are recommended, with y+<1 for the first cell being a mandatory requirement. Also, the experimental rule states that heat transfer is maximum when the nozzle-to-plate distance ratio H/D (H: distance, D: hole diameter) is between 4 and 8; parameter studies in CFD must always check the sensitivity to this ratio. In practice, it's a golden rule to set mesh resolution based on both H and D as reference lengths.
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 (Central Differencing)
Second-order accurate, but numerical oscillations occur for Pe > 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 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. LES: CFL ≈ 1 is recommended. Physical meaning: Information should not travel more than one cell per timestep.
Residual Monitoring
Convergence is typically judged when residuals for the Continuity Equation, momentum, and energy 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 iterations 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 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 cannot tell the source of the water by looking downstream—this discretization method reflects the physics that upstream information determines downstream conditions. Although first-order accurate, it is highly stable because it correctly captures flow direction.
Is LES also used for impinging jets?
It is actively used at the research level. Kelvin-Helmholtz instabilities generated in the jet shear layer and large-scale vortex structures in the impingement region cause unsteady fluctuations in the Nu number. LES can directly resolve these vortex structures, so the time-averaged Nu number distribution is closer to experiments than RANS. DES (Detached Eddy Simulation) and SBES are also useful as intermediate options.
Numerical Schemes for Jet Impingement Heat Transfer—Mesh Resolution at the Stagnation Point is Crucial
The most difficult aspect of jet impingement CFD is predicting the heat transfer at the Stagnation Point. Here, the velocity gradient is maximum, so standard k-ε models have a tendency to overestimate turbulent kinetic energy production, leading to a 20-30% overprediction of the Nu number. The v2-f model and SST-ω, which are strong in stagnation point accuracy, are recommended, with y+<1 for the first cell being a mandatory requirement. Also, the experimental rule states that heat transfer is maximum when the nozzle-to-plate distance ratio H/D (H: distance, D: hole diameter) is between 4 and 8; parameter studies in CFD must always check the sensitivity to this ratio. In practice, it's a golden rule to set mesh resolution based on both H and D as reference lengths.
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 (Central Differencing)
Second-order accurate, but numerical oscillations occur for Pe > 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 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. LES: CFL ≈ 1 is recommended. Physical meaning: Information should not travel more than one cell per timestep.
Residual Monitoring
Convergence is typically judged when residuals for the Continuity Equation, momentum, and energy 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 iterations 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 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 cannot tell the source of the water by looking downstream—this discretization method reflects the physics that upstream information determines downstream conditions. Although first-order accurate, it is highly stable because it correctly captures flow direction.
Practical Guide
Industrial Application Examples
Please tell me about industrial applications where impinging jets are actually used.
| Industry Sector | Specific Application | Typical Conditions |
|---|---|---|
| Gas Turbine | Internal blade impingement cooling | $Re_D = 5000$ to $40000$, $H/D = 1$ to $3$ |
| Steel | Secondary cooling in continuous casting | Water spray impingement, $H/D = 10$ to $50$ |
| Electronics | Server chip cooling | Microjet arrays, $D = 0.5$ to $2$ mm |
| Glass Industry | Glass plate tempering | Air array jets, uniform cooling is critical |
| Drying | Paper/film drying | High-temperature air jets, includes evaporative heat transfer |
Microjets are used in electronics cooling?
For high-density data centers and next-generation power semiconductors, liquid microjets are being considered to surpass air-cooling limits. Research is advancing on arrays of 0.5mm diameter nozzles directly impinging on chip surfaces, achieving $Nu \sim 100$ to $500$.
Actual CFD Verification
How is CFD verification done in practice?
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