Secondary Flow
Theory and Physics
Overview
Secondary flow is the flow that goes in a different direction than the main flow inside the blade row, right?
Yes. Flow within the blade passage that has a velocity component perpendicular to the main flow direction is called secondary flow. It is generated when the boundary layer on the endwalls (hub/shroud) is bent sideways by the pressure difference across the blade row.
Main Vortex Structures
What kind of vortex structures are there?
Let me list the typical vortex structures.
| Vortex Name | Generation Mechanism | Impact |
|---|---|---|
| Passage Vortex | Endwall BL rolls up due to inter-blade pressure difference | Major source of secondary flow loss |
| Horseshoe Vortex | Endwall BL splits at the blade leading edge | SS side merges with passage vortex |
| Corner Vortex | Occurs at blade-endwall intersection | Induces separation |
| Tip Leakage Vortex | Leakage flow from tip clearance | Main cause of efficiency drop (rotating blades) |
| Scraper Vortex | Relative motion of shroud wall surface | Prominent in transonic stages |
Please explain the relationship between the passage vortex and the horseshoe vortex.
The horseshoe vortex splits into two at the blade leading edge. The pressure side (PS leg) heads towards the adjacent blade, while the suction side (SS leg) is directly entrained into and strengthens the passage vortex. This merged vortex forms the main body of the passage vortex.
Quantification of Secondary Flow Loss
How significant is secondary flow loss?
It is said that 30-50% of the total loss in a turbine blade row is attributed to secondary flow. In CFD, visualizing using entropy generation rate is effective.
By calculating and volume-integrating this quantity in CFD-Post, you can separately evaluate blade profile loss, endwall loss, and tip leakage loss.
Turbo Machinery Secondary Flow Theory—Hawthorne (1955) and the Systematization of the Horseshoe Vortex
The person who theoretically organized "Secondary Flow" in turbo machinery blade rows was the British W.R. Hawthorne (1955). Hawthorne described the process where the incident boundary layer vorticity splits and stretches into a horseshoe shape at the blade leading edge, forming the "Horseshoe Vortex," using the vorticity transport equation. This theory was the first to quantitatively explain the mechanism of endwall loss in blade rows and became a pioneering work showing the importance of endwall treatment in turbo machinery design. Hawthorne himself, as an engineering professor at Cambridge University, educated multiple generations of aerospace engineers and produced many researchers who would later form the foundation of modern turbo CFD. His secondary flow theory has been numerically verified with modern CFD, and the correspondence between the shape/strength of the horseshoe vortex predicted by CFD and Hawthorne's classical theory remains a research topic today.
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, 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/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"—meaning 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 "carrier," air, 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" → 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. 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 overwhelms, and diffusion becomes a minor player.
- 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 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. "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 go wrong immediately after switching to compressible analysis, mixing up absolute/gauge pressure might be the cause.
- Source Term $S_\phi$: Warmed 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 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 (molecular 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 and above, 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 coefficient $\nu = \mu/\rho$ [m²/s] |
| Reynolds Number $Re$ | Dimensionless | $Re = \rho u L / \mu$. Criterion for laminar/turbulent transition |
| CFL Number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability |
Numerical Methods and Implementation
Vortex Identification Methods
How do you extract vortex structures from CFD results?
There are multiple vortex identification methods.
| Method | Definition | Features |
|---|---|---|
| Q-Criterion | Magnitude of vorticity tensor > magnitude of strain rate tensor | Most widely used, standard in CFD-Post |
| λ2 Criterion | Second eigenvalue of pressure Hessian is negative | Removes shear effects, more accurate |
| Helicity | $H = \mathbf{v} \cdot \boldsymbol{\omega}$ | Can determine vortex rotation direction |
| Wall Limiting Streamlines | Direction of wall shear stress | Identifies separation/attachment lines |
Is displaying the Q-criterion isosurface the easiest way?
Yes. In CFD-Post, coloring isosurfaces of Q=positive value with total pressure loss coefficient or vorticity makes the 3D structure of passage vortices and tip leakage vortices immediately clear.
Mesh Requirements
How much mesh is needed to accurately predict secondary flow?
Mesh density near the endwalls is key.
- Endwall y+: < 1 (when using Low-Re SST model)
- Endwall Prism Layers: 15–20 layers
- Blade-Endwall Intersection: Mesh refinement (to capture corner vortex)
- Spanwise Direction at Passage Center: 40 cells or more
Mesh quality on the endwalls, not just the blade surfaces, determines secondary flow prediction accuracy.
How do you make the endwall mesh finer in TurboGrid?
Use the Boundary Layer Refinement in TurboGrid to set dedicated prism layers for the endwalls (Hub/Shroud). Independent y+ control is possible for both blade surfaces and endwalls.
Impact of Turbulence Model
Does the choice of turbulence model make a big difference in predicting secondary flow?
SST k-omega and k-epsilon show significant differences in passage vortex position and size. SST more accurately captures adverse pressure gradients near endwalls, so the strength and position of the passage vortex are closer to experiments. LES can resolve even unsteady vortex structures, but computational cost increases by two orders of magnitude or more.
CFD Numerical Methods for Turbo Secondary Flow—Corner Vortex Prediction and SST Model Accuracy
The prediction accuracy of the "Corner Vortex" formed near the blade endwall in turbo machinery strongly depends on the turbulence model used. The standard k-ε model assumes isotropy of shear stress, so it significantly underestimates vortex strength (40–60% underestimation compared to experiments) in the endwall region where strong curvature and pressure gradients act simultaneously. The SST model improves accuracy near endwalls by using a blending function to switch between k-ε and k-ω, but it can still shift the position of secondary flow vortices by ±5–10%. The highest accuracy requires Differential Reynolds Stress Models (DRSM) or LES, but computational cost in the design cycle is problematic. In practice, a staged refinement approach is often adopted: "Understand secondary flow trends with SST, and verify with LES only for the final design."
Upwind Scheme
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
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 shocks 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 timestep.
Residual Monitoring
Convergence is judged when residuals for Continuity, momentum, and energy drop by 3–4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factor
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 timestep until a steady solution converges. Internal iteration count: 5–20 iterations is a guideline. If residuals fluctuate between timesteps, review the timestep 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—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
Endwall Contouring
Are there ways to reduce secondary flow loss?
Non-Axisymmetric Endwall Contouring is one of the most effective techniques. By making the endwall shape convex/concave between blades, the pressure distribution near the endwall is altered, suppressing secondary flow.
How much loss reduction is possible?
Reports indicate a 10–30% reduction in blade row secondary flow loss, improving stage efficiency by 0.5–1.5 points.
CFD-Based Endwall Optimization
Can we optimize endwall contouring with CFD?
Yes. Parameterize the endwall shape using Fourier series or spline surfaces and perform CFD-based optimization.
1. Discretize the endwall with a grid in the circumferential × axial direction between blades (5×5 to 10×10 control points)
2. Use the radial displacement of each point as design variables
3. Objective Function: Minimization of total pressure loss coefficient OR maximization of stage efficiency
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