2.4 Nucleation in flowing liquids and cavitation inception

The nucleation theory developed so far has been static — given a sample of liquid at uniform ambient pressure $p_\infty$, when does it cavitate? In engineering practice the more common question is dynamic: given a flow with spatially-varying pressure (a ship propeller, a hydrofoil, a valve, a vascular constriction in the body), at what flow speed does cavitation first appear, and where? This lesson develops the standard engineering parameterisation — the cavitation inception number $\sigma_i$ — and identifies the physical reasons it differs from the static thresholds of the previous lessons.

The inception number

For a flow of incompressible liquid past a body at far-field speed $U_\infty$ and far-field pressure $p_\infty^*$, Bernoulli’s equation places the local pressure at any point on the body surface at

$p = p_\infty^* + \tfrac{1}{2} \rho U_\infty^2 (1 - C_p \cdot \text{geometry factor}),$

with $C_p = (p - p_\infty^*) / (\tfrac{1}{2} \rho U_\infty^2)$ the local pressure coefficient. $C_p$ is a function of position on the body and is determined by the flow geometry alone (independent of $U_\infty$ in incompressible inviscid flow). The minimum value $C_{p,\text{min}}$ on the body sets where the cavitation will occur first.

Cavitation inception in the simplest model occurs when the local pressure drops to the vapour pressure $p_v$:

$p_{\text{min}} = p_\infty^* + \tfrac{1}{2} \rho U_\infty^2 C_{p,\text{min}} = p_v.$

Solving for the conditions at inception:

$\boxed{\sigma_i \;=\; \frac{p_\infty^* - p_v}{\tfrac{1}{2} \rho U_\infty^2} \;=\; -C_{p,\text{min}}.}$

The cavitation inception number $\sigma_i$ is the dimensionless ratio of the ambient-vs-vapour pressure gap to the dynamic pressure. It is a property of the geometry — set by where the streamlines diverge most strongly on the body. For a 2-D hydrofoil at zero angle of attack, $\sigma_i$ is typically 0.3 to 1.5 depending on the shape; for a sphere it is $\sim 0.6$; for sharp-edged obstacles it can be 10 or more.

When the operating $\sigma = (p_\infty^* - p_v) / \tfrac{1}{2} \rho U_\infty^2$ falls below $\sigma_i$ — that is, the flow is too fast for the ambient pressure to keep the minimum body-surface pressure above vapour — cavitation begins. As $\sigma$ drops further, cavitation spreads over an increasing fraction of the body surface, eventually becoming fully developed or supercavitating.

Why measured $\sigma_i$ differs from the inviscid prediction

The Bernoulli-based prediction $\sigma_i = -C_{p,\text{min}}$ is an upper bound on the cavitation inception number. Real measurements are typically lower (cavitation appears at smaller $\sigma$ — faster flow — than Bernoulli alone would suggest), because three additional effects intervene:

1. Insufficient residence time

A nucleus passing through the low-pressure region of the body surface experiences the tension only for the time $\tau_\text{res} \sim L / U_\infty$ that it takes to traverse the region. For a 10 cm body at 10 m/s, $\tau_\text{res} \sim 10$ ms. If the nucleus’s equilibration time — the time required for the gas pocket to expand to a free bubble after the tension is applied — is comparable to or longer than $\tau_\text{res}$, the nucleus does not have time to fully nucleate before being swept back into a higher-pressure region downstream.

Equilibration times for typical crevice gas pockets are sub-millisecond, so for slow flows this effect is small. For very fast flows (high $U_\infty$, small body) it can substantially shift $\sigma_i$ below the static prediction.

2. Need for a nucleus to be present at the right place

A flow might have a localised low-pressure region that would cavitate if a nucleus were present there — but if the local $N(R)$ is low and no nucleus happens to occupy that fluid parcel during the brief residence time, cavitation does not occur. The probability of inception in a given fluid parcel scales with $\tau_\text{res} \cdot \dot{N}_E$ where $\dot{N}_E$ is the cavitation event rate per unit volume per unit time at the local pressure. For dilute samples (well-degassed water in a clean facility) the event rate can be low enough that cavitation inception is intermittent — occurring only when a particularly nucleation-active fluid parcel happens to pass.

This is why some cavitation tunnel measurements show $\sigma_i$ that depends on the air content of the water: gas-rich water has more nuclei per unit volume, so inception is reliably triggered as soon as the Bernoulli condition is met; gas-poor water has sparse nuclei and shows $\sigma_i$ values that drift over time and across runs.

3. Viscous effects and boundary-layer separation

The Bernoulli prediction is inviscid; real flows have boundary layers, and at the trailing edge or at a point of geometric separation the boundary layer can produce locally-different pressures than the inviscid prediction. Boundary-layer separation in particular produces a region of recirculation behind the body where the local pressure can be substantially below the Bernoulli prediction. Cavitation in this region — called base cavitation or wake cavitation — produces $\sigma_i$ values that depend on the Reynolds number (because the separation point shifts with Re) in addition to the geometry alone.

For the canonical case of a 2-D hydrofoil, the experimentally measured $\sigma_i$ at $\text{Re} = 10^6$ is typically 5–20% below the inviscid Bernoulli prediction. The discrepancy is large enough that engineering design of cavitation-sensitive systems must use experimentally-calibrated $\sigma_i$ values rather than inviscid calculations alone.

Scaling and Reynolds-number effects

When does cavitation appear at the same $\sigma_i$ for a scaled-up version of a body? Approximately, when the Reynolds number is the same — so the boundary-layer behaviour is similar. In practice, the inception number changes by a few percent as Re changes by an order of magnitude, with the largest effects in flows where boundary-layer separation is critical.

Brennen’s chapter on cavitation inception (his §1.17) collects measurements of $\sigma_i$ for many canonical bodies (spheres, cylinders, hydrofoils at various angles of attack, NACA sections) at Reynolds numbers from $10^5$ to $10^8$. The data is essential reference material for ship-propeller, valve, and pump design — fields in which cavitation-inception margins are designed-for explicitly.

What the inception number does and does not tell us

$\sigma_i$ tells us the threshold at which cavitation first appears in a given flow. It does not tell us:

• The acoustic-noise spectrum produced by the resulting cavitation.
• The surface erosion produced by collapsing bubbles.
• The lift-and-drag changes that fully-developed cavitation imposes on the body.

These are downstream consequences that engineers also need to understand. But all of them begin with the inception number — without a nucleated bubble there is no cavitation event to develop, and the inception number tells us the minimum operating condition that produces that event.

Closing the chapter

That closes Chapter 2. The arc: a perfectly homogeneous liquid would withstand $\sim -1000$ atm of tension before thermal fluctuations alone produced a vapour bubble — but no real sample is perfectly homogeneous. Heterogeneous nucleation at preexisting sites — overwhelmingly gas pockets trapped in hydrophobic surface crevices — sets the actual tensile strength of any normal sample at $-0.1$ to $-10$ atm, depending on cleanliness. The population of these sites is distributed as $N(R) \sim R^{-4}$ across several decades of nucleus radius, with the cavitation threshold set by whichever nucleus is largest. In flowing systems, this all gets parameterised by the cavitation inception number $\sigma_i$, a dimensionless ratio of ambient pressure to dynamic pressure that captures the geometry of the flow.

We have now built up enough machinery to study what happens after a bubble appears. The next chapter develops the Rayleigh–Plesset equation — the momentum balance on the liquid surrounding a spherical bubble — which describes how the bubble grows and collapses once nucleation has produced it.

Next chapter: Ch 3 — The Rayleigh–Plesset equation.