Vintage Loss
Theory and Physics
Overview
What is windage loss? It's a term I'm not very familiar with.
It's the power loss due to friction between the air (or fluid) and a rotating body and a stationary wall. It becomes significant enough to be non-negligible in high-speed rotating machinery. It's particularly important in turbine generators, flywheels, high-speed motors, etc.
Friction Loss of a Rotating Disk
Specifically, how much loss are we talking about?
The friction loss of a rotating disk in a closed space is expressed as follows.
$C_M$ is the moment coefficient, which is a function of the rotational Reynolds number $Re_\theta = \rho \omega r^2 / \mu$ and the gap ratio $s/r$.
It's proportional to $\omega^3$? The effect of rotational speed is significant.
Yes. Doubling the rotational speed makes the windage loss eight times greater. That's why it can become a dominant loss source in high-speed rotating machinery.
Flow Regime
Are there laminar and turbulent flows in the gap flow as well?
Yes. The classification by Daily & Nece (1960) is widely used.
| Regime | $Re_\theta$ | Gap Ratio | Characteristics |
|---|---|---|---|
| I: Laminar・Merged BL | Low | Small | Boundary layers from both walls merge |
| II: Laminar・Separated BL | Low | Large | BLs are independent, core rotation |
| III: Turbulent・Merged BL | High | Small | Most common industrial condition |
| IV: Turbulent・Separated BL | High | Large | Core rotation present |
Most industrial turbo machinery falls under Regime III or IV.
Empirical Formula for $C_M$
Are there empirical formulas for the moment coefficient?
For Regime III, Daily's formula is representative.
By comparing CFD results with this empirical formula, the validity of the calculation can be conveniently verified.
Fluid Mechanics of Rotating Disks—von Karman's Disk Analysis (1921) and Stokes' Pioneering Work
The flow around a disk rotating in a stationary fluid was analyzed by Theodore von Karman (1921). He not only studied Karman vortices behind rotors but also derived an exact analytical solution for the boundary layer on a rotating disk (von Karman boundary layer), formulating the relationship between disk torque and eddy viscosity. Cochran (1934) computed this solution with higher accuracy, and Daily & Nece (1960) expanded the model through experiments, laying the foundation for today's correlation formulas. Modern turbo machinery disk cavity CFD is an evolved form that starts from von Karman's analysis a century ago, numerically adding complex three-dimensional effects such as multi-stage, multi-body rotation, and hot gas ingestion. It is noteworthy that the work of a scientist who left revolutionary achievements through a mathematical approach is directly applied to jet engine cooling design 100 years later.
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 due to heartbeats, and the flow fluctuation each time an engine valve opens and closes are all unsteady phenomena. So what is steady-state analysis? It looks only at "after sufficient time has passed and the flow has settled down"—meaning setting this term to zero. Since computational cost is significantly reduced, trying a steady-state solution first 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 end 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 things" → They are completely different! Convection is carried by flow, conduction is transmitted by molecules. There is 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, next question—honey and water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. When viscosity is high, the diffusion term becomes strong, and the fluid moves in a "thick" manner. In low Reynolds number flows (slow, viscous), diffusion is dominant. Conversely, in high Re number flows, 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 piston side is high pressure, and the needle tip is low pressure—this pressure difference becomes 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. 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 is 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 source terms. 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 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: Linear relationship between shear stress and strain rate (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 variation only in the buoyancy term, using constant density in other terms
- Non-applicable Cases: Rarefied gas (Kn > 0.1), supersonic/hypersonic flow (shock capturing required), 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
CFD Model Configuration
How do you model windage loss calculation in CFD?
Model the gap space between the rotating disk and stationary wall as the fluid domain.
- Rotating Wall: Disk surface (No-Slip, specify rotational speed)
- Stationary Wall: Casing inner surface (No-Slip, zero velocity)
- Inlet/Outlet: Set if there is leakage flow into the gap
If axisymmetric, a 2D axisymmetric model is sufficient. If there are 3D effects (bolt heads, cooling holes, etc.), use a sector model.
Turbulence Model Selection
Which turbulence model is suitable?
SST k-omega is recommended. Since the BL near the rotating wall is important, ensure y+ < 1 with a Low-Re approach. k-epsilon is not recommended as it struggles to accurately capture rotational effects near walls.
| Turbulence Model | Windage Prediction Accuracy | Notes |
|---|---|---|
| SST k-omega (Low-Re) | ±5～10% | Recommended |
| k-epsilon + Wall Function | ±15～25% | Insufficient accuracy near wall |
| RSM (Reynolds Stress) | ±3～8% | Captures swirling effects, high cost |
| LES | ±2～5% | Highest accuracy but high cost |
Is RSM good?
In rotating disk flow, the anisotropy of Reynolds stress is strong between the circumferential and radial directions. RSM (e.g., BSL-RSM) directly models this anisotropy, which can sometimes improve accuracy over SST. However, computational cost increases by 1.5～2 times.
Mesh Requirements
How much mesh is needed inside the gap?
A guideline is 40～60 cells in the gap direction (axial) and 100～200 cells in the radial direction. Place prism layers on both walls and ensure y+ < 1. For 2D axisymmetric, tens of thousands of cells are sufficient.
Windage Loss Calculation—Daily-Nece Correlation and CFD Accuracy Comparison
The windage loss torque M of a rotating disk can be calculated from the experimental correlation formula by Daily & Nece (1960). Different coefficient formulas apply to each of the four flow regimes (combinations of G and Reomega). In the turbulent regimes (III, IV), M is proportional to the square of rotor speed and the fifth power of radius, and inversely proportional to Re to the 1/5 power. This correlation is still widely used for initial estimates in engine/turbine design, but errors can exceed 30% when conditions deviate from the applicable range (G=0.02~0.2, Reomega=10^4~10^7). Comparison studies with CFD report that for disk gaps G<0.02 (narrow gaps) or actual shapes with recesses/holes in the disk, the Daily-Nece formula tends to predict 10~20% lower loss than CFD, according to multiple papers. For complex engine disk shapes, the standard procedure is to perform an initial estimate with the correlation formula, then always verify precisely with CFD.
Upwind Differencing (Upwind)
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 (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 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 timestep.
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 especially 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 timestep until a steady solution converges. Internal iteration count: 5~20 times is a guideline. If residuals fluctuate between timesteps, review the timestep 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 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 technique 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 reflects the physics that upstream information determines downstream. Although it's first-order accurate, it is highly stable because it correctly captures the flow direction.
Practical Guide
Rotor System Temperature Rise
Does temperature increase due to windage loss?
All loss becomes heat. In a sealed space, if cooling is insufficient, temperature continues to rise. This becomes a problem in steam turbine chambers or motor air gaps.
If the cooling flow rate $\dot{m}_{cooling}$ is low, the temperature rise becomes large.
Can CFD also predict temperature distribution?
By enabling the energy equation and solving with CHT, the ro...
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