Pump Cavitation
Theory and Physics
Overview
Pump cavitation is the phenomenon where bubbles form and pop, causing damage, right?
That's roughly correct. When the static pressure of the liquid falls below the saturated vapor pressure, vapor bubbles form and then collapse in the downstream high-pressure region. This collapse generates a localized shock pressure of several GPa, eroding the impeller surface.
Several GPa!? That would definitely cause damage...
It's also serious for performance. It causes head drop, increased vibration, and increased noise. That's why avoiding cavitation is one of the top priorities in pump design.
Definition of NPSH
I often hear about NPSH, but what is its precise definition?
NPSH (Net Positive Suction Head) represents how much margin the liquid has above the vapor pressure on the suction side.
$p_{atm}$: Atmospheric pressure, $p_v$: Saturated vapor pressure, $z_s$: Height from liquid surface to pump center, $h_f$: Friction head loss in the suction pipe. This is the system-side NPSH (Available).
There's also an NPSH on the pump side, right?
NPSH_r (Required) is the minimum NPSH required by the pump itself, defined as the point where the head drops by 3%. For safety, this condition must be met:
It can also be expressed by the cavitation coefficient (Thoma number).
Rayleigh-Plesset Equation
Is there an equation that describes bubble growth and collapse?
The Rayleigh-Plesset equation is fundamental.
$R$: Bubble radius, $p_B$: Pressure inside the bubble, $p_\infty$: Surrounding pressure, $S$: Surface Tension. CFD cavitation models simplify this into a mass transport equation.
The History of Cavitation Troubles for Submarines
The bubble dynamics of pump cavitation (Rayleigh-Plesset equation) began to be treated as a serious practical problem starting with submarine propellers during World War I. Bubbles violently formed and collapsed near the propeller blade tips, causing the dual problems of reduced propulsion efficiency and metal erosion. Post-war research systematized the concept of NPSH (Net Positive Suction Head), which has been carried over into modern pump design.
Physical Meaning of Each Term
- Temporal Term $\partial(\rho\phi)/\partial t$: Think of the moment you turn on a faucet. At first, water comes out in an unstable, spluttering manner, but after a while, it becomes a steady flow, right? This term describes that "period 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 down"—in other words, it sets this term to zero. Since this significantly reduces computational cost, solving first in steady-state 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 contains "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, 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 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: The "pressure" in CFD is often gauge pressure, not absolute pressure. If results become strange immediately after switching to compressible analysis, it might be due to mixing up absolute/gauge pressure.
- Source Term $S_\phi$: Heated air rises—why? Because it becomes lighter (lower density) than its surroundings, so it's pushed upward 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's electromagnetic pump... All these are effects 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, if you forget to include buoyancy, the fluid won't move at all—you get 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 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
Homogeneous Mixture Model
How do you calculate cavitation in CFD?
The most widely used is the Homogeneous Mixture model. It treats the liquid and vapor phases as a single fluid and solves a transport equation for the vapor volume fraction $\alpha_v$.
$\dot{m}^+$ is the evaporation (bubble formation) source term, and $\dot{m}^-$ is the condensation (bubble collapse) source term.
What kind of models are there for the source terms?
Let's compare three representative ones.
| Model | Features | Used in Solvers |
|---|---|---|
| Zwart-Gerber-Belamri | Based on nucleation site density, easy parameter adjustment | CFX (default), STAR-CCM+ |
| Schnerr-Sauer | Based on Rayleigh-Plesset, requires bubble number density specification | OpenFOAM (interPhaseChangeFoam), Fluent |
| Singhal (Full Cavitation) | Considers non-condensable gases, practical | Fluent |
So the Zwart model is standard in CFX, huh.
Yes. Default values are evaporation coefficient $F_{vap}=50$, condensation coefficient $F_{cond}=0.01$. Typical values are nucleation site volume fraction $\alpha_{nuc}=5 \times 10^{-4}$, initial bubble radius $R_B=10^{-6}$ m.
Key Points for Numerical Settings
Are there any tips for getting cavitation calculations to converge?
It's a difficult calculation, so there are several key points.
- Time Step: Unsteady is mandatory. Aim for 1/20 to 1/50 of the time for one blade passage.
- Convergence Criteria: Target RMS residuals of $10^{-5}$ or better (sometimes they only drop to $10^{-4}$ due to cavitation oscillations).
- Initial Conditions: First, get a steady-state solution without cavitation, then turn the cavitation model ON from there.
- Compressibility: Due to the large vapor-liquid density ratio, settings that numerically consider compressibility are necessary.
Zwart, Merkle, Singhal—The Battle of Three Models
A common question heard on the CFD cavitation analysis front lines: "Which model should I use?" The Zwart model explicitly handles nucleation site density, the Merkle model is an empirical form that reacts directly to pressure difference, and the Singhal (Full Cavitation) model is the most complex configuration that even considers dissolved gases. Benchmark studies sometimes show the ranking changing depending on the case, so it's hard to definitively say "this one is the best." Since results are heavily influenced by initial values, mesh, and empirical constant settings, validation with in-house experiments is essential.
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 Schemes (MUSCL, QUICK, etc.)
Maintain 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 methods: CFL ≤ 1 is the stability condition. Implicit methods: 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 continuity, momentum, and energy each 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, reconsider 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 then velocity is revised using the corrected pressure—this back-and-forth is repeated to approach the correct solution. It's similar to 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 can't tell where the water comes from by looking downstream—this discretization method reflects the physics that upstream information determines downstream conditions. It's first-order accurate but highly stable because it correctly captures flow direction.
Practical Guide
Procedure for Obtaining NPSH Characteristic Curve
How do I plot an NPSH curve using CFD?
The general method is to gradually lower the inlet total pressure.
1. Baseline Calculation: Obtain a steady-state solution at a sufficiently high NPSHa (no cavitation)
2. Lower Inlet Pressure: Decrease inlet total pressure in steps of 0.1~0.2 atm
3. Unsteady Calculation at Each Point: Perform unsteady calculation for several rotations with the cavitation model ON
4. Record Time-Averaged Head: The point where the head drops by 3% from the baseline is NPSH_r
Isn't a 0.1 atm step size rather coarse?
Since the head drops sharply near NPSH_r, it's efficient to first get a rough overall picture with coarse steps, then refine near the 3% drop point with 0.02~0.05 atm steps.
Visualization and Evaluation
How should I look at cavitation results?
The isosurface of vapor volume fraction $\alpha_v = 0.5$ represents the cavity shape. Related Topics
なった
詳しく
報告