Radial Turbine
Theory and Physics
Overview
Is a radial turbine the turbine side of a turbocharger?
Yes. A radial inflow turbine has flow entering from the outer circumference towards the inner diameter and is discharged in the axial direction. It is compact and can achieve a high expansion ratio, making it widely used in automotive turbochargers and small gas turbines.
Total-to-Static Efficiency
How is the efficiency of a radial turbine defined?
Total-to-static efficiency is commonly used.
This definition is appropriate when the exit dynamic pressure is not recovered (free exhaust).
Velocity Ratio
What is the velocity ratio?
It is the ratio of the impeller peripheral speed $U$ to the isentropic expansion velocity $C_s$.
Maximum efficiency is obtained around $U/C_s \approx 0.7$. This corresponds to the condition where the swirl component at the blade inlet becomes optimal.
0.7 is an easy number to remember.
It is the most fundamental indicator for radial turbine design. Plotting the efficiency obtained from CFD against $U/C_s$ and checking if the peak is around 0.7 is the first checkpoint.
Specificities for Turbocharger Use
What are the specific challenges for radial turbines in turbochargers?
Response to exhaust pulses. Engine exhaust is intermittent, and the turbine inlet pressure fluctuates significantly with a cycle of a few milliseconds. Steady-state CFD can only predict time-averaged performance, so for higher accuracy, unsteady calculations with pulsating inlet pressure variations are necessary.
History of Radial Turbines — Establishment of Centripetal Turbine Design Theory (1930s-60s)
The theory and design methods for radial flow turbines (centripetal turbines) were established in the 1930s-40s, with Gehring et al. (1930s) developing high-efficiency centripetal turbine blade profile design theory based on de Laval's impulse steam turbine. The fundamental blade profile design methods for modern turbocharger radial turbines—optimization of backsweep angle, exit blade angle—were completed in the 1960s through joint research by NASA and GE Turbine (Rohlik 1968, etc.). Since then, over 60 years, CFD-based precision design has been layered on top of this foundational theory, leading to modern VGT turbine efficiencies reaching an astonishing 86-90%. Fluid machinery design is not born overnight; it is the product of a complex evolution where CFD refinement is layered upon over 60 years of accumulated experiments and theory.
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 term describes the "state of change." 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"—meaning this term is set to zero. This significantly reduces computational cost, so solving first with steady-state is a basic CFD strategy.
- Convection Term $\nabla \cdot (\rho \mathbf{u} \phi)$: If you drop a leaf into a river, what happens? 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 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" → 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 flow (slow, viscous), diffusion dominates. Conversely, in high Re number flow, 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. When switching to compressible analysis, if results become strange, it might be due to confusion between 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 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 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 number ≥ 0.3, consider compressibility effects
- Boussinesq Approximation (Natural Convection): Consider density changes only in the buoyancy term, using constant density 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 | $Re = \rho u L / \mu$. Indicator for laminar/turbulent transition |
| CFL Number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability |
Numerical Methods and Implementation
Model Configuration
How do you set up a CFD model for a radial turbine?
A typical configuration is as follows.
- Volute/Scroll: Stationary domain, non-axisymmetric
- Nozzle Vane (VGT): Stationary domain, blade row
- Turbine Wheel: Rotating domain
- Diffuser/Outlet Pipe: Stationary domain
For interface treatment, the volute-nozzle interface uses GGI (no pitch difference), and the Nozzle-wheel interface uses Frozen Rotor or Sliding Mesh.
VGT is a variable nozzle, right? Do you also analyze with different opening angles?
Yes. Change the VGT opening angle in 5-10 stages to create a map for each opening. Nozzle angle changes are done parametrically in BladeGen or CAD, and processed sequentially via automatic meshing → CFX batch calculation.
Mesh Generation
What should I be careful about when meshing a radial turbine?
Domain Method Points to Note Volute Unstructured (Tetra+Prism) Fine mesh at the tongue Nozzle Vane TurboGrid or Unstructured Mesh strategy to accommodate variable opening Turbine Wheel TurboGrid (Structured Grid) Curvature of inter-blade passage, splitter blade compatibility Outlet Diffuser Structured or Unstructured Adequately resolve swirl flow decay
Do turbine wheels have splitter blades?
Radial turbines often do not have splitters, but some high-performance designs include splitters to reduce blade loading. TurboGrid also supports topologies with splitters.
Turbulence Model
What turbulence model is suitable for radial turbines?
SST k-omega is standard. If transition on the turbine blade surface is important, add the Gamma-Theta transition model. For unsteady calculations of exhaust pulsations, SAS or SDES are also considered.
Coffee Break Yomoyama Talk
Radial Turbine CFD Numerical Settings — Transonic Flow and Mesh Resolution for Trailing Edge Shock Waves
Radial turbines used in small gas turbines and turbochargers often have exit velocities near sonic speed (Ma≈0.8〜1.0), so proper numerical handling of transonic flow determines accuracy. To accurately capture the oblique shock wave generated from the blade trailing edge, the mesh near the trailing edge should be set to a cell size of at least 1/5 of the trailing edge thickness, and mesh alignment along the shock wave angle is necessary. For numerical schemes, the Roe Flux Differencing Scheme or HLLC (Harten-Lax-van Leer-Contact) scheme is recommended for shock wave capture accuracy, and Preconditioning (low Mach number preconditioning) is required in low Ma number regions (blade pressure surface). Neglecting these settings can lead to CFD overestimation of turbine efficiency by 3-5% compared to experiments, as reported in papers.
Upwind Scheme (Upwind)
1st Order Upwind: Large numerical diffusion but stable. 2nd Order Upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flow.
Central Differencing (Central Differencing)
2nd order accuracy, but numerical oscillations occur for Pe number > 2. Suitable for low Reynolds number diffusion-dominated flow.
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 multi-physics. 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 recommended. Physical meaning: Information should not travel more than one cell per time step.
Residual Monitoring
Convergence is judged when residuals for Continuity Equation, momentum, and 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 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 Scheme
The upwind scheme is a method that "stands in the river flow and prioritizes upstream information." A person in the river cannot tell the source of the water by looking downstream—it's a discretization method reflecting the physics that upstream information determines downstream. Accuracy is first-order, but it is highly stable because it correctly captures flow direction.
What should I be careful about when meshing a radial turbine?
| Domain | Method | Points to Note |
|---|---|---|
| Volute | Unstructured (Tetra+Prism) | Fine mesh at the tongue |
| Nozzle Vane | TurboGrid or Unstructured | Mesh strategy to accommodate variable opening |
| Turbine Wheel | TurboGrid (Structured Grid) | Curvature of inter-blade passage, splitter blade compatibility |
| Outlet Diffuser | Structured or Unstructured | Adequately resolve swirl flow decay |
Do turbine wheels have splitter blades?
Radial turbines often do not have splitters, but some high-performance designs include splitters to reduce blade loading. TurboGrid also supports topologies with splitters.
What turbulence model is suitable for radial turbines?
SST k-omega is standard. If transition on the turbine blade surface is important, add the Gamma-Theta transition model. For unsteady calculations of exhaust pulsations, SAS or SDES are also considered.
Radial Turbine CFD Numerical Settings — Transonic Flow and Mesh Resolution for Trailing Edge Shock Waves
Radial turbines used in small gas turbines and turbochargers often have exit velocities near sonic speed (Ma≈0.8〜1.0), so proper numerical handling of transonic flow determines accuracy. To accurately capture the oblique shock wave generated from the blade trailing edge, the mesh near the trailing edge should be set to a cell size of at least 1/5 of the trailing edge thickness, and mesh alignment along the shock wave angle is necessary. For numerical schemes, the Roe Flux Differencing Scheme or HLLC (Harten-Lax-van Leer-Contact) scheme is recommended for shock wave capture accuracy, and Preconditioning (low Mach number preconditioning) is required in low Ma number regions (blade pressure surface). Neglecting these settings can lead to CFD overestimation of turbine efficiency by 3-5% compared to experiments, as reported in papers.
Upwind Scheme (Upwind)
1st Order Upwind: Large numerical diffusion but stable. 2nd Order Upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flow.
Central Differencing (Central Differencing)
2nd order accuracy, but numerical oscillations occur for Pe number > 2. Suitable for low Reynolds number diffusion-dominated flow.
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 multi-physics. 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 recommended. Physical meaning: Information should not travel more than one cell per time step.
Residual Monitoring
Convergence is judged when residuals for Continuity Equation, momentum, and 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 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 Scheme
The upwind scheme is a method that "stands in the river flow and prioritizes upstream information." A person in the river cannot tell the source of the water by looking downstream—it's a discretization method reflecting the physics that upstream information determines downstream. Accuracy is first-order, but it is highly stable because it correctly captures flow direction.
Practical Guide
Turbine Map Composition
What format is a turbine map?
The horizontal axis is expansion ratio ($p_{01}/p_2$), the vertical axis is corrected mass flow rate $\dot{m}\sqrt{T_{01}}/p_{01}$. Lines for each rotational speed are drawn, with efficiency overlaid as a parameter.
How many calculation points are needed?
5 rotational speed levels × 6-8 expansion ratio points, totaling 30-40 operating points is standard. For VGT, a map is needed for each nozzle opening, so the total can exceed 100 points.
Method for Varying Expansion Ratio
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