軸流圧縮機段
Theory and Physics
Overview
Professor, axial compressors have many rows of blades lined up, right? How much does the pressure increase per stage?
Good question. For subsonic axial compressors, the pressure ratio per stage is roughly around 1.15 to 1.4. Achieving an overall pressure ratio of 10:1 or more by stacking multiple stages is typical for gas turbine compressors.
Only that much per stage? Why not compress it all at once?
Because the flow decelerates during compression, and rapid deceleration can cause boundary layer separation. Keeping the Diffusion Factor (DF) per blade row low is key to stable operation. According to Lieblein's criterion, DF < 0.6 is a guideline.
Governing Equations ― Euler Work Equation
What equation represents the energy transfer per stage?
It's Euler's turbomachinery equation, which is fundamental to turbomachinery.
Here, $U$ is the blade speed, and $C_{\theta}$ is the swirl component of the absolute flow. Subscript 1 denotes inlet, 2 denotes outlet. This equation is derived rigorously from energy conservation and holds true for both compressors and turbines.
So, a larger change in swirl velocity means more work is done, right?
Exactly. However, if the swirl increases too much, the blade loading becomes excessive. So, drawing velocity triangles to check the relative Mach number and flow angles is the ABCs of design. The degree of reaction $R$ is also used to manage the work distribution.
A 50% reaction stage with $R = 0.5$ results in symmetric velocity triangles for the rotor and stator, making it a widely used design that tends to minimize losses.
de Haller Criterion and Diffusion Limit
How much deceleration is allowed in a blade row?
The de Haller number is a simple, practical index often used.
$W$ is the relative velocity. If this ratio falls below 0.72, the risk of boundary layer separation increases sharply. For a more precise evaluation, Lieblein's Diffusion Factor is used.
$\sigma$ is the solidity (chord/pitch). DF < 0.6 is the guideline design limit.
Are these two criteria used in the preliminary 1D design stage, before CFD?
Exactly. The standard workflow is to determine the velocity triangles through 1D Mean-Line Analysis first, then proceed to CFD.
Commercial Tools and Blade Row Design
What software is used for axial compressor blade row design?
Let me list some representative ones.
| Tool | Purpose | Features |
|---|---|---|
| Ansys CFX / TurboGrid | 3D CFD + Dedicated Blade Row Mesher | High-quality structured meshing of inter-blade passages |
| NUMECA FINE/Turbo | 3D CFD (AutoGrid5) | Specialized for multi-stage turbomachinery, supports Non-Matching Interface |
| Concepts NREC (AXIAL) | 1D/2D Preliminary Design | Integrated Mean-Line + Throughflow analysis |
| AxSTREAM (SoftInWay) | 1D to 3D Integrated Design | Seamless linkage from preliminary design to CFD |
I've heard of TurboGrid. What's good about it?
Its ATM optimization (Automatic Topology and Meshing) automatically generates H/J/C/L type topologies suited to the blade shape. O-grids around the leading and trailing edges are also placed automatically, making boundary layer resolution significantly easier.
The Engineers of the Past Who Couldn't Draw Velocity Triangles
The "velocity triangle"—that vector diagram representing the relationship between absolute velocity, relative velocity, and blade speed, essential for axial compressor design—was solved by engineers in the late 19th century through hand calculations and trial and error. Although Euler's equation itself was established in the 1750s, systematic methods for applying it to blade row design (Mean-Line method) weren't fully developed until the 1940s. The rapid demand from jet engine development historically accelerated the "practical application of velocity triangles."
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 "period of change" is described by the temporal term. The pulsation of blood flow from 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's looking only at "after sufficient time has passed and the flow has settled down"—meaning setting this term to zero. Since this drastically reduces computational cost, solving first in 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 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" → Not at all! 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 it naturally mixes, right? 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 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 water release 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 go wrong immediately after switching to compressible analysis, it might be due to confusing absolute/gauge pressure.
- Source Term $S_\phi$: Warmed 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 generated by 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 source terms. What happens if you forget the source term? In a natural convection analysis, forgetting to include buoyancy means the fluid doesn't move at all—a physically impossible result, like turning on a heater in a winter room but the warm air doesn't rise.
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 variation is considered only in the buoyancy term; constant density is used in other terms.
- Non-applicable Cases: Rarefied gases (Kn > 0.1), supersonic/hypersonic flows (requires shock capturing), free surface flows (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
Rotating Frame Formulation
When solving for compressor rotor blades in CFD, how is the rotation handled?
The most common method is the MRF (Multiple Reference Frame) method, i.e., steady-state analysis in a rotating coordinate system. Coriolis and centrifugal body force terms are added to the momentum equation.
$\mathbf{v}_r$ is the relative velocity in the rotating frame, and $\boldsymbol{\omega}$ is the angular velocity vector.
The Coriolis force part is the addition. For steady-state calculation, the time derivative is zero, right?
Yes, for steady MRF, the first term on the left-hand side is zero. A Mixing Plane (circumferential averaging) is placed at the rotor-stator interface to connect blade rows with different pitches in a steady-state manner.
Turbulence Model Selection
Which turbulence model is standard for compressor CFD?
The practical standard is the SST (Shear Stress Transport) k-omega model. It can capture separation due to adverse pressure gradients on blade surfaces more accurately than k-epsilon models. In CFX, simply selecting "SST" enables automatic near-wall switching.
| Turbulence Model | Strengths | Weaknesses | Recommended Use |
|---|---|---|---|
| SST k-omega | Strong against adverse pressure gradients, good near-wall accuracy | Does not capture transition | Steady-state analysis at design point |
| Gamma-Theta Transition Model | Can predict laminar-turbulent transition | Requires calibration parameters | Performance evaluation of low-Re airfoils |
| SAS / DES | Resolves unsteady structures of large-scale separation | High computational cost | Analysis near stall / surge |
In what situations is a transition model necessary?
For low-speed fans or small compressors where the chord-based Reynolds number is below $5 \times 10^5$, the laminar region on the blade surface can be quite long. Since the transition location directly affects losses, the Gamma-Theta (Langtry-Menter) model is effective.
Boundary Condition Settings
What kind of conditions are set at the inlet and outlet?
Typical boundary conditions for a compressor stage are as follows.
- Inlet: Total temperature $T_0$, total pressure $p_0$, flow angle (swirl component), Turbulence Intensity (usually around 5%)
- Outlet: Static pressure or mass flow rate specification
- Blade Surface: No-slip, adiabatic wall is typical
- Hub/Shroud: No-slip rotating wall (rotor side) or stationary wall (stator side)
- Periodic Surface: Rotational periodic boundary condition for one pitch
So if the outlet is set to static pressure, the mass flow rate comes out as a result, right?
Yes. To draw a performance map, you repeatedly calculate by gradually increasing the back pressure at the outlet towards operating points near surge. The point where mass flow rate drops sharply is near the stall limit. However, steady-state calculations cannot fully capture true surge, so unsteady analysis becomes necessary near the limit.
How Boundary Conditions Change the Compressor Map
In axial compressor CFD, "which boundary condition to use" actually significantly affects the results. Whether you specify static pressure or mass flow rate at the outlet completely changes the behavior near the surge limit. In practice, it's often said that "back pressure specification converges more easily," but in return, the flow rate becomes a result of the calculation. The difficulty of turbomachinery CAE lies in these choices of "what to fix and what to solve for," which is where the engineer's skill comes into play.
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 (C
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