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 design levels, $\eta_{is}=90\sim92\%$ for HP stages and $88\sim90\%$ for LP stages in gas turbines.
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 aerospace engine manufacturers. NUMECA FINE/Turbo is efficient for setting up multi-stage turbines and is used by companies like Rolls-Royce. STAR-CCM+ has strengths in CHT (Conjugate Heat Transfer) analysis for turbine blade cooling.
The Foundation of Turbine Efficiency Theory—Rankine Cycle and Thermal Efficiency (1859)
The theory of the Rankine Cycle, which quantifies the thermal efficiency of steam turbines, was established by Scottish engineer William Rankine (1820-1872). Compared to Carnot efficiency, the method of determining the efficiency achievable by actual steam turbine systems through Rankine Cycle calculations remains the foundation of power plant design today. From 1859, when Rankine compiled his thermodynamic analysis of steam engines into a paper, 165 years later, modern gas turbine combined cycle (GTCC) power generation has advanced to achieve thermal efficiencies of up to 64%. Approximately one-third of this improvement is contributed by CFD-driven airfoil optimization (including cooling technology for high-temperature regions), demonstrating how the theoretical framework from 160 years ago continues to function in combination with modern CFD.
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 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 looks only at "after sufficient time has passed and the flow has settled down"—meaning this term is set 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 due to air, the "carrier," transporting 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'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 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 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 creates 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 go wrong immediately after switching to compressible analysis, it might be due to confusing 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 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 source terms. 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 room with the heater on 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, 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 (shock wave capturing required), free surface flows (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 $\nu = \mu/\rho$ [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
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 engineering effort |
| L4 | CHT (fluid + solid coupling) | Highest | Predicts internal blade temperature distribution |
Are L3 and L4 practical?
L3/L4 for a single blade 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 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 is calculated by outputting the adiabatic wall temperature on the blade surface.
Numerical Settings for Gas Turbine Blade CFD—Selection of High-Temperature Combustion Gas Properties and Heat Transfer Models
In CFD analysis of gas turbine blades, accurate setting of the working gas (high-temperature combustion gas) properties is crucial for predicting heat transfer accuracy. Post-combustion gas is a mixture of CO2, H2O, N2, and O2, requiring temperature-dependent specific heat Cp(T), viscosity mu(T), and thermal conductivity k(T) to be set via polynomial approximation or the WSGG model. The commonly used approximation of "using constant air property values at 1500K" underestimates viscosity by 10–15% on the high-temperature side, causing prediction errors in boundary layer thickness and HTC (Heat Transfer Coefficient) on the blade surface. NASA's CHT (Conjugate Heat Transfer) benchmark experiments showed that CFD with appropriate polynomial approximation for properties kept the error in blade surface HTC distribution within ±8% compared to experiments, demonstrating that property setting accuracy is fundamental to high-temperature blade CFD.
Upwind Scheme
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 Scheme (MUSCL, QUICK, etc.)
Maintains 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 method: CFL ≤ 1 is the stability condition. Implicit method: 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 each equation—Continuity, momentum, energy—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.
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 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 the Upwind Scheme
The upwind scheme 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 discretization method reflects the physics that upstream information determines downstream conditions. Although first-order accurate, it is highly stable because it correctly captures 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 considerations.
- 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: Resolution of supersonic patches on the suction side and trailing edge shock waves.
If the trailing edge is 0.3mm, the mesh must be quite fine, right?
The O-grid around the trailing edge should have at least 10 cells in the radial direction, and the wake region immediately behind the trailing edge should also have a fine mesh. TurboGrid's trailing edge cutoff function can control the trailing edge shape.
Transonic Turbine Blade Row
Does the flow in turbines 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?
A cell size orthogonal to the shock wave direction of 0.5% of the chord or less, with at least 10 cells before and after the shock, is recommended. Adaptive Mesh Refinement (AMR) to concentrate mesh at shock wave locations 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 | ±10–2 |
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