Frozen rotor method
Theory and Physics
Overview
The Frozen Rotor method has a cool name, but what does it actually do?
It's a method for performing steady-state analysis while keeping the relative position between the rotating and stationary components fixed. The rotor blade coordinates are placed in a rotating frame, and the stator blade coordinates are placed in a stationary frame, but at the interface, information is passed "as is" without taking a circumferential average.
What's the difference from Mixing Plane?
| Method | Interface Treatment | Blade Count Constraint | Computational Cost | Accuracy |
|---|---|---|---|---|
| Frozen Rotor | Fixed position, direct interpolation | Pitch ratio correction needed if 1 pitch differs | Low | Position-dependent |
| Mixing Plane | Circumferential mass averaging | Any pitch ratio | Low | Smooths circumferential fluctuations |
| Sliding Mesh | Rotation with time progression | Integer pitch ratio is desirable | High | Most accurate |
When to Use It
In what situations is Frozen Rotor used?
It is suitable for the following cases.
- High-speed screening in the initial design stage: Faster than Mixing Plane when comparing many shape candidates
- Centrifugal machines with volutes: Volutes are non-axisymmetric, making Mixing Plane difficult to apply. Frozen Rotor provides an estimate of blade-volute interaction.
- Hydraulic turbines: Evaluation of interaction with the draft tube
So it's useful when Mixing Plane can't be used, like with volutes.
Yes. However, the results depend on the relative position between the blades and the volute. For accurate evaluation, you should either calculate with multiple relative positions and average, or ultimately move to Sliding Mesh.
Frozen Rotor Settings in CFX
Please tell me the steps to set up Frozen Rotor in CFX.
Simply set the domain interface type to "Frozen Rotor". If the pitch ratio is not 1:1, automatic scaling is performed via the "Pitch Change" option. As a note, the mesh pitch of the GGI surfaces should be roughly aligned. Large discrepancies increase interpolation error.
History of Turbomachinery CFD—From the 1970s Slim Slot Method to 3D Inviscid Analysis
Attempts to analyze internal flows in turbomachinery using CFD began in the 1970s. Initially, due to computer limitations, 2D analysis of blade rows was mainstream, and 3D blade shapes could not be considered. The turning point was the development of the precursor code to ANSYS Fluent in the late 1970s and the 3D inviscid (Euler equations) turbomachinery analysis achieved by Denton (1982) on an IBM mainframe. Furthermore, in the 1990s, Harvey & Denton, Arnone, and others realized viscous analysis including unsteady rotor-stator interference, establishing the prototype of modern turbomachinery CFD (RANS + sliding mesh). The Frozen Rotor method was born within this history as a "reasonable first choice for steady-state approximation" and continues to live on as a standard method in the design exploration phase even 50 years later in the modern era.
Physical Meaning of Each Term
- Temporal term $\partial(\rho\phi)/\partial t$: Imagine the moment you turn on a faucet. At first, the water comes out spluttering and unstable, but after a while, it becomes a steady flow, right? This "period of change" is described by the temporal term. The pulsation of blood flow with a heartbeat, or the flow fluctuation each time an engine valve opens and closes—all are unsteady phenomena. So what is steady-state analysis? Looking only at "after sufficient time has passed and the flow has settled down"—in other words, setting this term to zero. Since computational cost is significantly reduced, solving first with steady-state 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 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 are completely different! Convection is carried by flow, conduction is transmitted by molecules. There is 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. That's molecular diffusion. Now, next question—honey and water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. When viscosity is large, the diffusion term becomes strong, and the fluid moves in a "thick" manner. In low Reynolds number flows (slow, viscous), diffusion is dominant. Conversely, in high Re number flows, convection overwhelms, and diffusion plays a supporting role.
- Pressure term $-\nabla p$: When you push the plunger of a syringe, the 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 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. "Where there is a pressure difference, flow is generated"—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 become strange immediately after switching to compressible analysis, confusion between 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 is pushed up by buoyancy. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat generated by a gas stove flame, Lorentz force applied to molten metal by an electromagnetic pump in a factory... These are all actions that "inject energy or force into the fluid from the outside" and are expressed by source terms. What happens if you forget the source term? In natural convection analysis, if you forget to include buoyancy, the fluid doesn't move at all—a physically impossible result where warm air doesn't rise in a room with the heater on in winter.
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 (viscosity model required for non-Newtonian fluids)
- Incompressibility assumption (for Ma < 0.3): Treat density as constant. For Mach numbers above 0.3, consider compressibility effects.
- Boussinesq approximation (Natural Convection): Consider density change 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 (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: 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 coefficient $\nu = \mu/\rho$ [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
Position Dependency Issue
Is it true that Frozen Rotor results change depending on blade position?
It's true. For example, in a centrifugal pump, the head can change by 5-10% depending on whether the blade is directly in front of the volute tongue (cutoff) or offset. This is a fundamental limitation of Frozen Rotor.
So it's not reliable?
Reporting the result from a single position directly as a performance value is risky. It is recommended to calculate 3-5 phases at intervals of 1/3 to 1/2 of the blade pitch and take the average.
Pitch Ratio Correction
What is pitch ratio?
It's the ratio of the angle of one pitch between the rotor and stator. For example, if the rotor has 7 blades (pitch 51.4 degrees) and the stator has 12 blades (pitch 30 degrees), the pitch ratio is 51.4/30=1.71. In Frozen Rotor, this pitch difference on both sides of the interface must be handled in some way.
In CFX, setting the Pitch Ratio on the interface triggers scaling interpolation in the circumferential direction. However, accuracy degrades significantly if the pitch ratio exceeds 2.
Choosing Between Mixing Plane and Frozen Rotor
How do you choose in practice?
| Situation | Recommended Method |
|---|---|
| Axisymmetric diffuser / No volute | Mixing Plane |
| Centrifugal pump with volute | Frozen Rotor (multiple phases) → Sliding Mesh |
| Multi-stage axial flow | Mixing Plane |
| Parametric study in preliminary design | Frozen Rotor (fast) |
| Pressure pulsation / noise evaluation | Sliding Mesh (mandatory) |
Ultimately, it's safe to verify with Sliding Mesh.
Exactly. Frozen Rotor is positioned as "fast estimation," Mixing Plane as "stable steady-state approximation," and Sliding Mesh as "physically correct unsteady analysis."
Numerical Settings for the Frozen Rotor Method—Interface Handling and Convergence Tips
The Frozen Rotor method is a steady-state analysis method with the MRF fixed under the assumption of circumferential uniformity between the rotor and stator. In implementation, the handling of the "rotor-stator interface" determines accuracy. At this interface, coordinate transformation (from rotating to stationary frame) is performed, but when the circumferential flow is non-uniform (e.g., strong wake interference), the assumption fails, and "circumferential dependency" occurs where results change when evaluated at different circumferential positions. The countermeasure is to perform Frozen Rotor calculations at multiple circumferential positions and take the circumferential average (Pitch Average). Also, since the interpolation accuracy of flow variables at the interface affects results, comparative verification with Sliding Mesh (more accurate) is recommended for critical areas.
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 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 is recommended. Physical meaning: Information should not travel more than one cell per time step.
Residual Monitoring
Convergence is judged when residuals for continuity, momentum, and energy drop by 3-4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factors
Pressure: 0.2-0.3, Velocity: 0.5-0.7 are typical initial values. 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 with the corrected pressure—this catchball 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—it's a discretization method reflecting the physics that upstream information determines downstream. Although first-order accurate, it is highly stable because it correctly captures flow direction.
Practical Guide
Centrifugal Pump Model Configuration
How do you set up the full model for a centrifugal pump?
A typical configuration is as follows.
- Suction Pipe: Stationary domain, non-rotating
- Impeller: Rotating domain, MRF or Sliding Mesh
- Volute: Stationary domain, non-axisymmetric
- Impeller-Volute Interface: Frozen Rotor (steady) or Sliding Mesh (unsteady)
Because the volute is non-axisymmetric, Mixing Plane cannot be applied. This is why Frozen Rotor is valued.
How do you create the mesh for the volute?
The volute's cross-sectional shape changes in a spiral pattern, so it cannot be created with TurboGrid. Use unstructured mesh from Ansys Meshing or Fluent Meshing, or automatic mesh from STAR-CCM+. For quality, it's advantageous to create a hexa-dominant mesh by sweeping cross-sectional shapes along the sweep direction.
Volute Tongue (Cutoff) Handling
What's difficult about the area near the cutoff (tongue)?
The tongue is a region where the impeller outlet flow and recirculating flow collide, with steep pressure gradients. The mesh needs to be particularly refined here. Also, in Frozen Rotor, the flow field changes significantly depending on the relative position between the blade and the tongue, so Sliding Mesh is essential for pressure pulsation evaluation.
Creating a Performance Map
Can you create an H-Q curve for a centrifugal pump using Frozen Rotor?
It's possible, but it's recommended to take the average of multiple phases at each flow rate point. The procedure is as follows.
1. For the design flow rate, perform Fro
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