圧縮機CFD解析
Theory and Physics
Overview
Is the CFD analysis approach different for axial and centrifugal compressors?
The fundamental governing equations are the same, but in centrifugal compressors the role of the diffuser is significant, while in axial compressors managing blade loading is the main theme. However, what they have in common is that CFD is required to predict pressure ratio and adiabatic efficiency with accuracy.
Pressure Ratio and Adiabatic Efficiency
How is the pressure ratio defined?
It is defined as the total pressure ratio.
$p_0$ is the total pressure (stagnation pressure). And the adiabatic efficiency is the ratio of work between an isentropic process and the actual process.
So it's calculated from temperature. In CFD, can it be derived directly from head or total pressure?
You obtain the mass-flow-averaged total pressure and total temperature at the inlet and outlet and calculate it. Using functions like massFlowAve in CFX-Post or ParaView is standard.
Compressibility Effects
The impeller tip speed in centrifugal compressors is close to the speed of sound, right?
Yes. In turbocharger centrifugal compressors, the impeller tip speed reaches 400-500 m/s, and the relative Mach number can exceed 1.2. Therefore, compressibility cannot be ignored.
So supersonic flow occurs within the blade row?
It becomes supersonic near the inlet and decelerates through shock waves in the inter-blade passage. The increased loss due to shock wave-boundary layer interaction is an important physical phenomenon that determines CFD accuracy.
Software Used
What software is strong for centrifugal compressors?
Ansys CFX + TurboGrid has the most proven track record in industry. It can automatically generate structured grids from the meridional shape of a centrifugal impeller using TurboGrid. NUMECA FINE/Turbo's AutoGrid5 is also strong for centrifugal compressors, with excellent mesh generation for splitter blades. STAR-CCM+ is easy to start with using polyhedral mesh + automatic prism layers, but the grid quality in inter-blade passages often does not match TurboGrid.
Economic Impact of a 1-Point Compressor Efficiency Improvement
For centrifugal compressors used in large LNG liquefaction plants, just a 1-point increase in adiabatic efficiency directly translates to annual electricity cost savings of several hundred million yen. The higher the pressure ratio, the more exponentially significant the effect of efficiency improvement becomes—this is why significant development funds are invested in compressor CFD. Modern industrial centrifugal compressors achieve adiabatic efficiencies of 85-90%, but LES calculations and optimization algorithms continue to be used to shave off those "last few percent."
Physical Meaning of Each Term
- Temporal Term $\partial(\rho\phi)/\partial t$: Imagine 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 "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? It looks 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 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 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'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, after a while they naturally mix. That's molecular diffusion. Now a question—honey and 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 number 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, 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: "Pressure" in CFD is often gauge pressure, not absolute pressure. If results become strange immediately after switching to compressible analysis, mixing up absolute/gauge pressure might be the cause.
- Source Term $S_\phi$: Warmed air rises—why? Because it becomes lighter (lower density) than its surroundings, so it's pushed upward 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," expressed by the source term. What happens if you forget the source term? In natural convection analysis, forgetting to include buoyancy means the fluid doesn't move at all—a physically impossible result where warm air doesn't rise in a heated room in winter.
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, 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 (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: 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 | $Re = \rho u L / \mu$. Criterion for laminar/turbulent transition |
| CFL Number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability |
Numerical Methods and Implementation
Surge Prediction Approach
How can I predict the surge line with CFD?
The most practical method is to gradually increase the outlet back pressure in a steady-state calculation and treat the point where convergence fails as the approximate surge limit. However, since physical surge is a dynamic instability of the entire system, accurate prediction requires unsteady Full-Annulus calculations.
Full-Annulus means calculating the full circumference? That sounds tough.
Surge cannot be captured with a single-pitch periodic calculation. Since rotating stall cells propagate circumferentially, the full circumference (360 degrees) must be calculated unsteadily. The cell count is the number of blades per pitch times the number of pitches, so for 20 blades, the computational cost is also 20 times higher.
Harmonic Balance Method
Are there any lighter methods?
There are methods like Harmonic Balance and Non-Linear Harmonic. They capture unsteady fluctuations in the frequency domain, significantly reducing computational cost in the time direction. They are implemented as Time Transformation in CFX and Nonlinear Harmonic in FINE/Turbo.
How much cost reduction can be achieved?
It often requires only 1/5 to 1/20 of the cost of time integration methods. However, there are still limitations for strong unsteadiness where multiple frequency components interfere.
Centrifugal Compressor Surge
Is surge in centrifugal compressors different from axial compressors?
In centrifugal compressors, stall in the diffuser often triggers surge. Especially for vaned diffusers (VD), when the incidence angle of the VD becomes large, stall occurs abruptly. Vaneless diffusers (VLD) have a wider surge margin but lower efficiency.
| Diffuser Type | Surge Margin | Peak Efficiency | Application |
|---|---|---|---|
| Vaneless (VLD) | Wide | Slightly Low | Automotive Turbo, Variable Operation |
| Vaned (VD) | Narrow | High | Industrial, Aircraft Engines |
| Pipe Diffuser | Medium | High | High Pressure Ratio Applications |
So the reason automotive turbochargers use vaneless diffusers is for their wide operating range.
Yes. Since they are used over a wide range of engine speeds, ensuring surge margin is the top priority.
Why Vaneless Diffusers are Common in Automotive Turbos
Vaneless diffusers are widely adopted in centrifugal compressors for automotive turbochargers. The reason is simple: "wide operating range." Vaned diffusers offer high efficiency at the design point, but the surge margin narrows sharply off-design. Since gasoline engines are used from idle to 6000 rpm over a wide range, vaneless diffusers with a broad flow range are chosen. In CFD, verifying this design trade-off across the entire compressor map has become a standard process in engine development.
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 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 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 judged when the residuals for the continuity equation, momentum, and energy each drop by 3-4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factor
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.
Unsteady Calculation Inner Iterations
Iterate within each timestep until a steady solution converges. Inner 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 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 Workflow
Please tell me the typical analysis flow for a centrifugal compressor.
The following steps are standard.
1. 1D Design: Mean-Line design using Concepts NREC's COMPAL or AxSTREAM. Determine basic dimensions from pressure ratio, flow rate, and rotational speed.
2. Meridional Design: Define hub/shroud curves and blade angle distribution using BladeGen or AxSTREAM.
3. 3D Blade Shape Definition: Output full 3D shape including splitter blades using BladeGen.
4. Mesh Generation: Generate H/J/L type structured grids using TurboGrid.
5. CFD: Steady MRF analysis with CFX (design point) → Obtain characteristic curves by varying back pressure.
6. Optimization: Automatic exploration of blade angle and meridional shape using optiSLang or FINE/Design.
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