Secondary Flow
Secondary Flow: Theoretical Foundations
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.
Computational Methods for Secondary Flow
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."
Secondary Flow in Practice
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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