混合面法
Theory and Physics
Overview
The Mixing Plane always comes up in steady-state calculations for multi-stage turbomachinery, right? What exactly is the process?
It's a method that mass-averages flow quantities circumferentially at the interface between rotating and stationary blades and passes them as boundary conditions to the downstream side. This allows blade rows with different pitch counts to be connected in a steady-state manner.
$\phi$ represents conserved variables such as total pressure, total temperature, flow angle, turbulence quantities, etc.
When you mass-average, the wake (wake) information is lost, right?
Exactly. The biggest limitation of the Mixing Plane is that circumferential variations from the upstream blade row's wake and secondary flows are not transmitted downstream. However, spanwise variations are preserved, so sufficient accuracy for stage performance prediction can be obtained.
Ensuring Conservation
Is conservation of mass and energy guaranteed with the Mixing Plane?
Conservation of mass flow rate, momentum, and total enthalpy is numerically enforced. CFX has a flux-conservative implementation where the mass flow rate upstream and downstream of the interface matches exactly. However, the increase in mixed entropy (mixing loss) is numerically unavoidable.
How much is the mixing loss?
It affects stage efficiency by about 0.1 to 0.5 points. Since mixing also occurs physically between rotating and stationary blades in real machines, this is somewhat justified, but caution is needed as it varies depending on the Mixing Plane location and method.
The Birth of the Mixing Plane Method——Denton & Dawes (1988) and the Standardization of Turbomachinery Steady-State Analysis
The Mixing Plane method was established in a 1988 paper by John Denton (University of Cambridge) and W.N. Dawes. Prior to this, multi-stage turbomachinery analysis could only analyze each stage independently and connect outputs/inputs, making it difficult to predict consistent flow fields between stages. Denton and Dawes proposed the Mixing Plane concept of "averaging and exchanging circumferential flow variables at stage boundaries," enabling continuous steady-state analysis of multiple stages. This concept was implemented in major CFD codes within 3-5 years and became a standard tool for gas turbine design in the 1990s. Denton later received the AIME Melchior Jackman Award for his contributions to turbomachinery CFD development, and the Mixing Plane method is recognized as one of the most influential inventions in the history of turbomachinery CFD.
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 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/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, 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's carried downstream by the flow, right? This is "convection"—the effect where fluid motion transports things. Warm air from a heater reaching the far side of a room is also because the "carrier," air, transports heat via convection. Here's the interesting part—this term includes "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'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. Next question—honey and water, which flows easier? 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 number flows, convection overwhelms, and diffusion plays a supporting role.
- Pressure Term $-\nabla p$: When you push a syringe plunger, 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 provides the force that pushes the fluid. Dam discharge works on the same principle. On a weather map, where isobars are densely packed? 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, 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 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 (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 numbers above 0.3, compressibility effects must be considered.
- Boussinesq Approximation (Natural Convection): Density changes are 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 (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: ~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 | $Re = \rho u L / \mu$. Indicator for laminar/turbulent transition. |
| CFL Number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability. |
Numerical Methods and Implementation
Implementation Variations
Does the Mixing Plane implementation differ between solvers?
Is it better to have more bands in Band-averaged?
Too few bands smooth out spanwise variations. CFX's default is generally sufficient, but to capture sharp variations near endwalls, increase the number of bands or set a user-defined band distribution.
Mixing Plane Placement in Multi-Stage Calculations
Where do you place Mixing Planes in a multi-stage axial turbine?
The basic approach is to place one between each rotating blade and stationary blade. The recommended Mixing Plane location is near the midpoint between the trailing edge of one blade and the leading edge of the next. Too close to the leading edge increases potential interference, and too close to the trailing edge results in insufficient wake mixing.
How do you set up the Mixing Plane in NUMECA FINE/Turbo?
When blade row boundaries are defined in AutoGrid5, a Mixing Plane is automatically placed as a Row Interface. Selecting the Non-Reflecting option suppresses pressure wave reflections at the interface, improving computational stability near surge.
Non-Reflecting Mixing Plane
What does Non-Reflecting mean?
In a standard Mixing Plane, pressure fluctuations at the interface can reflect and cause numerical oscillations. The Non-Reflecting treatment applies Giles characteristic conditions to allow waves to pass through the interface. This is particularly effective for high-load compressor stages and near surge conditions.
Numerical Implementation of the Mixing Plane Method——Circumferential Averaging Method and Radial Distribution Preservation
The Mixing Plane method exchanges circumferentially averaged flow variables at the rotor-stator boundary to couple rotors and stators in steady-state analysis. A key point in numerical implementation is "which variables to average." Simple arithmetic averaging of pressure and velocity can break energy conservation in high-enthalpy regions—instead, using mass-flux-weighted averaging is necessary for accuracy. Also, a radial-distribution-preserving setting that maintains the radial (spanwise) distribution while averaging only circumferentially is the standard for high precision. In CFX, appropriate averaging at the interface is automatically implemented, but OpenFOAM's MRFfvPatchField can sometimes use simple averaging depending on settings, so implementation specifications need to be checked.
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 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 and 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 recommended. Physical meaning: Information should not travel more than one cell per time step.
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 Factor
Typical initial values: Pressure: 0.2-0.3, Velocity: 0.5-0.7. Reduce the factor if diverging. Increase after convergence 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 cannot tell where the water comes from by looking downstream—this method reflects the physics that upstream information determines downstream conditions. Although it's first-order accurate, it is highly stable because it correctly captures the flow direction.
Practical Guide
Analysis of Multi-Stage Axial Turbines
Please teach me how to utilize the Mixing Plane when analyzing a multi-stage gas turbine with CFD.
For an HP (High Pressure) turbine with 1-2 stages and an LP (Low Pressure) turbine with 3-5 stages, you have a total of 8-12 blade rows. Place a Mixing Plane between each blade row to perform a full-stage steady-state calculation.
How many cells will that be?
About 0.5-1 million cells per blade row, so 5-10 million cells for 10 rows. With 128 cores, it takes about 12-24 hours. Thanks to the Mixing Plane, a single-pitch calculation is possible, compressing the model to 1/(number of blades) compared to a full-annulus model.
Key Points for Result Evaluation
What should I check in the CFD results for a multi-stage turbine?
| Evaluation Item | Check Method | Criteria |
|---|---|---|
| Mass Flow Conservation Between Stages | Mass flow difference before/after Mixing Plane | Within 0.01% |
| Pressure Ratio of Each Stage | Mass-averaged total pressure ratio at Mixing Plane surface | Within ±2% of 1D design value |
| Spanwise Efficiency Distribution | Adiabatic efficiency on spanwise cross-section | Natural decrease at hub/tip |
| Blade Surface Mach Number | Contours of constant Mach number on blade surface | Check shock wave location |
How does the Mixing Plane's mixing loss affect the results?
The more stages, the more Mixing Plane surfaces, and the larger the cumulative mixing loss becomes. For 10 stages, cumulative loss is about 0.5-1 po
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