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 adiabatic efficiency is the ratio of work between an isentropic process and the actual process.
So it's calculated from temperature. In CFD, can we get it 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 tip speed of a centrifugal compressor impeller 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 a shock wave within 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 in TurboGrid from the meridional shape of a centrifugal impeller. 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 meshes + automatic prisms, but the mesh quality in inter-blade passages often does not match TurboGrid.
Turbocharger Thermodynamics—Why Exhaust Gas Can Compress Air
The turbocharger is a very clever mechanism from an energy perspective, using "waste exhaust energy" to drive the compressor. In a typical passenger car turbo, the compressor side compresses air to a pressure ratio of 2-3 times, while the turbine side converts the expansion energy of the exhaust gas into mechanical work to spin the compressor. In CFD analysis, the target design adiabatic efficiency is typically around 70-80% for the compressor and 70-75% for the turbine. A 1% drop in this efficiency worsens fuel consumption by about 0.5%, so turbocharger manufacturers optimize blade profiles down to 0.1mm increments using numerical analysis.
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 in an unstable, spluttering manner, 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/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. This significantly reduces computational cost, so 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, it naturally mixes after a while. That's molecular diffusion. Next 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 in a "thick" manner. 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 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. "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 when switching to compressible analysis, it might be due to mixing up absolute/gauge pressure.
- Source Term $S_\phi$: Heated air rises—why? Because it becomes lighter (lower density) than its surroundings, so it's 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 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 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 number ≥ 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 (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 $\nu = \mu/\rho$ [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
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, physical surge is a dynamic instability of the entire system, so 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. Rotating stall cells propagate circumferentially, so the full circumference (360 degrees) must be calculated unsteadily. The cell count is the number of blades per pitch times, so for 20 blades, the computational cost is also 20 times.
Harmonic Balance Method
Is there a lighter method?
There are methods like Harmonic Balance and Non-Linear Harmonic. They capture unsteady fluctuations in the frequency domain, greatly reducing computational cost in the time direction. In CFX, it's implemented as the Time Transformation method; in FINE/Turbo, as the Nonlinear Harmonic method.
How much cost reduction can be achieved?
It often suffices with 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?
In centrifugal compressors, stall in the diffuser often triggers surge. Especially with vaned diffusers (VD), when the VD incidence angle 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.
High-Speed Rotating Body Meshing—Spatial Challenges in Turbocharger CFD
Mesh generation is the most troublesome aspect of turbocharger CFD analysis. For compressor impellers with backswept blades (blades curved backward), the passage height varies significantly between the hub and shroud sides, requiring specialized skills to create meshes with uniform quality. On the turbine side, thermal expansion due to high-temperature exhaust (over 900°C) can change the actual clearance from the design value, sometimes necessitating a procedure to remesh with the post-thermal-deformation geometry. An industry rule of thumb is "at least 5-8 elements in the clearance region" for accuracy assurance, and it's not uncommon for the total mesh count to exceed 10 million elements.
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 (Central Differencing)
2nd order accuracy, 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 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 time step until a steady solution converges. Inner 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 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 looking downstream cannot tell where the water comes from—it's a discretization method reflecting the physics that upstream information determines downstream. It's first-order accurate but highly stable because it correctly captures 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 with 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 with BladeGen or AxSTREAM.
3. 3D Blade Shape Definition: Output full 3D shape including splitter blades with BladeGen.
4. Mesh Generation: Generate H/J/L type structured grids with TurboGrid.
5. CFD: Steady MRF analysis with CFX (design point) →
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