Turbine CFD Analysis
Theory and Physics
Overview
What's the difference between turbine CFD and compressor CFD?
Turbines are on the side that extracts energy from the fluid. Because the flow accelerates, large-scale separation like in compressors is less likely to occur. Instead, blade cooling, secondary flow losses, and transonic shock waves become the main challenges.
Stage Work and Isentropic Efficiency
How is turbine work expressed?
Output and efficiency are defined from the Euler equation.
$h_{02s}$ is the enthalpy after isentropic expansion. For modern gas turbine HP stages, $\eta_{is}=90\sim92\%$, and for LP stages, $88\sim90\%$ is the current design standard.
Blade Loading Coefficient
How do you evaluate the magnitude of blade loading?
The Zweifel blade loading coefficient is the standard.
$s$: pitch, $c_x$: axial chord. $Z_w \approx 0.8$ has been considered the traditional optimum, but in recent high-load designs, $Z_w > 1.0$ is also being researched.
Software Selection
What software is used for turbine CFD?
Ansys CFX + TurboGrid is the most widely used among aero-engine manufacturers. NUMECA FINE/Turbo is efficient for multi-stage turbine setup and is used by companies like Rolls-Royce. STAR-CCM+ has strengths in CHT (Conjugate Heat Transfer) analysis for turbine blade cooling.
Beyond Bernoulli — 100 Years of Hydraulic Turbine Theory
The fundamental equations for hydraulic turbines date back to Euler's turbomachinery equation (1754). This is a simple principle: "The change in angular momentum of the fluid determines the turbine output." However, in actual design, it's necessary to consider the viscous boundary layer around the blades, the centrifugal field of the runner, and the diffusion efficiency of the draft tube. Design cannot converge with 1D theory alone. Before CFD became widespread, designers combined potential flow theory with empirical loss correction factors (loss maps), repeatedly performing hand calculations and model tests to find the optimal airfoil. The process of replacing that empirical knowledge with CFD continues today.
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 "during the 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? 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 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 air, as a "carrier," transports heat via convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, when 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 things" → They're 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 left milk in coffee without stirring? Even without mixing, after a while, they naturally blend, 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. When viscosity is large, the diffusion term becomes strong, and the fluid moves in a "thick" manner. In low Reynolds number flow (slow, viscous), diffusion dominates. Conversely, in high Re number flow, convection overwhelmingly dominates, 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 plunger 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's 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 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, if you forget to include buoyancy, the fluid doesn't move at all—you get 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)
- Incompressible 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 gas (Kn > 0.1), supersonic/hypersonic flow (shock wave 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 | $Re = \rho u L / \mu$. A criterion for laminar/turbulent transition |
| CFL Number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability |
Numerical Methods and Implementation
Importance of Blade Cooling
How is turbine blade cooling handled in CFD?
HP turbine inlet gas temperatures reach 1500–1800°C, far exceeding the heat resistance limit of blade materials (about 1000°C for Ni-based superalloys). Internal cooling passages and film cooling are used to lower the blade surface temperature.
Cooling Model Hierarchy
How do you incorporate cooling into CFD?
There are multiple levels depending on the trade-off between accuracy and cost.
| Level | Model | Computational Cost | Accuracy |
|---|---|---|---|
| L0 | No cooling flow (adiabatic wall) | Lowest | Baseline evaluation without cooling |
| L1 | Source Term (mass/energy injection) | Low | Rough estimate for film cooling |
| L2 | Discrete Hole (individual cooling hole BC) | Medium | Quantitative evaluation of film effectiveness |
| L3 | Resolved Cooling Holes (holes meshed) | High | Highest accuracy but high effort |
| L4 | CHT (fluid + solid coupling) | Highest | Predicts internal blade temperature distribution |
Are L3 and L4 practical?
L3/L4 for a single blade row is practically used as a Singleton calculation. STAR-CCM+'s CHT is highly rated for this application. L3/L4 for multi-stage is currently at the research level.
Film Cooling Effectiveness
How do you evaluate the effectiveness of film cooling?
It is defined by the adiabatic film cooling effectiveness.
$T_g$: Mainstream gas temperature, $T_{aw}$: Adiabatic wall temperature, $T_c$: Cooling air temperature. $\eta_f = 0$ means no cooling, $\eta_f = 1$ means perfect cooling. In CFD, it's calculated by outputting the adiabatic wall temperature on the blade surface.
Numerical Methods for Rotor-Stator Interaction — The "Interface" Problem in Hydraulic Turbines
One of the most difficult problems in hydraulic turbine CFD analysis is handling the hydrodynamic interaction between the rotating runner and the fixed guide vanes. The Frozen Rotor method is fast but ignores unsteady effects of rotor-stator interaction. The Sliding Mesh (moving mesh) method is accurate but computational cost can reach 5–10 times that of Frozen Rotor. A practical decision criterion is "the unsteadiness of the physical quantity being evaluated"—for predicting efficiency maps, Frozen Rotor is often sufficient, but for pressure pulsation or fatigue evaluation, Sliding Mesh is essential.
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 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 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 equations drop by 3–4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factors
Typical initial values: Pressure: 0.2–0.3, Velocity: 0.5–0.7. 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 then velocity is revised with 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 information from upstream." A person in the river looking downstream cannot tell where the water comes from—it's a discretization method that reflects the physics that upstream information determines downstream. Although it's 1st-order accurate, it is highly stable because it correctly captures the flow direction.
Practical Guide
Turbine Blade Row Mesh
Is the mesh for a turbine blade row the same as for a compressor?
The basic structure is the same, but there are turbine-specific points to note.
- Trailing Edge Thickness: Turbine blades have very thin trailing edges (0.3–0.8mm). Sufficient cells are needed around the trailing edge in the O-grid.
- Cooling Holes: Local refinement around cooling holes is necessary for L2/L3 models.
- Transonic Regions on Blade Surface: Resolving the supersonic patch on the suction side and the trailing edge shock wave.
If the trailing edge is 0.3mm, the mesh must be quite fine, right?
For the trailing edge O-grid, at least 10 cells in the radial direction, and also refine the mesh in the wake region immediately behind the trailing edge. TurboGrid's trailing edge cutoff function can control the trailing edge shape.
Transonic Turbine Blade Row
Does turbine flow become supersonic?
In HP turbines, the blade-to-blade Mach number reaches 1.1–1.3. After accelerating to supersonic speed on the suction side, oblique shock waves are emitted from the trailing edge. Accurate prediction of this Trailing Edge Shock System, where the shock wave impinges on the adjacent blade, is key to CFD accuracy.
How much mesh is needed to resolve shock waves?
Cell size orthogonal to the shock wave direction should be less than 0.5% of the chord, with at least 10 cells before and after the shock wave is recommended. Adaptive Mesh Refinement (AMR) to concentrate mesh at the shock wave location is also effective. AMR functions in Fluent or STAR-CCM+ can be used.
Performance Prediction Accuracy
What is the accuracy of turbine CFD?
| Metric | Accuracy |
|---|---|
| Stage Efficiency (multi-stage) | ±0.5–1.5 points |
| Blade Surface Pressure Distribution | Good (qualitatively matches experiment) |
| Blade Surface Heat Transfer Coefficient | ±10–20% (depends on turbulence model) |
| Trailing Edge Shock Wave Location | ±2% of chord |