Mixing Plane Method
Mixing Plane Method: Theoretical Foundations
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.
Computational Methods for the Mixing Plane Method
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.
The Mixing Plane Method in Practice
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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