Abstract

Cavitation is possible when the local pressure approaches the effective inception pressure. For a pure liquid with an ideal surface, this pressure could be below the equilibrium vapor pressure. Real process liquids contain microscopic nuclei, dissolved gas, particles, rough surfaces, and surfactants, so inception often occurs before the ideal tensile limit is reached. The vapor pressure $p_v(T)$ is therefore necessary but not sufficient for predicting inception. Temperature increases $p_v$, while higher dissolved gas content generally increases the number of available nuclei.

2. Physical Basis and Hydrodynamic Generation

2.1 Pressure, Velocity, and Cavitation Inception

For steady, incompressible flow through a device, the volumetric flow rate is:

$$\begin{array}{ccc}& Q=AU& \text{(2.1)}\end{array}$$

where $Q$ is volumetric flow rate, $A$ is local cross sectional area, and $U$ is the area averaged liquid velocity. A decrease in area increases velocity. Along a streamline, a useful engineering form of the mechanical energy balance is:

$$\begin{array}{ccc}& \frac{p_1}{\rho g}+{\alpha }_1\frac{U_1^2}{2g}+z_1=\frac{p_2}{\rho g}+{\alpha }_2\frac{U_2^2}{2g}+z_2+h_L& \text{(2.2)}\end{array}$$

where $p$ is static pressure, $\rho$ is liquid density, $g$ is gravitational acceleration, $z$ is elevation, $\alpha$ is the kinetic energy correction factor, and $h_L$ is the irreversible head loss. In a short horizontal restriction, elevation is usually negligible. The acceleration term converts static pressure into kinetic energy, while separation, turbulence, two phase flow, and mixing consume part of the recoverable pressure.

Cavitation is possible when the local pressure approaches the effective inception pressure. For a pure liquid with an ideal surface, this pressure could be below the equilibrium vapor pressure. Real process liquids contain microscopic nuclei, dissolved gas, particles, rough surfaces, and surfactants, so inception often occurs before the ideal tensile limit is reached. The vapor pressure $p_v(T)$ is therefore necessary but not sufficient for predicting inception. Temperature increases $p_v$, while higher dissolved gas content generally increases the number of available nuclei.

The most common dimensionless index is the cavitation number:

$$\begin{array}{ccc}& \sigma =\frac{p_{ref}-p_v}{\frac{1}{2}\rho U_{ref}^2}& \text{(2.3)}\end{array}$$

where $p_{ref}$ and $U_{ref}$ are defined reference pressure and velocity. Smaller $\sigma$ generally indicates stronger susceptibility to cavitation. The definition is not universal. Some studies use downstream pressure, some use upstream pressure, and some use throat velocity or an inferred velocity. Consequently, a numerical value of $\sigma$ is meaningful only when the measurement locations and velocity definition are stated [1–3].

For geometrically similar restrictions, the following groups are also useful:

$$\begin{array}{ccc}& Re=\frac{\rho UD}{\mu },We=\frac{\rho U^{2}D}{\gamma },Eu=\frac{\mathrm{\Delta }p}{\frac{1}{2}\rho U^2}& \text{(2.4)}\end{array}$$

where $D$ is a characteristic diameter, $\mu$ is dynamic viscosity, $\gamma$ is interfacial or surface tension, and $\mathrm{\Delta }p$ is pressure drop. Reynolds number characterizes the ratio of inertial to viscous effects. Weber number characterizes inertial stress relative to surface tension. Euler number relates pressure drop to dynamic pressure. These groups do not uniquely define the cavity field, but they help organize comparisons among devices and scales.

2.2 Pressure Recovery and Cavitation Regime

The collapse intensity is governed not only by the minimum pressure but also by the pressure to which the cavities are subsequently exposed. A device with low downstream pressure may generate a long attached cavity that collapses weakly or far from the desired treatment zone. A device with stronger pressure recovery may generate shorter cavities and more frequent collapse events, but excessive recovery can cause severe erosion if collapse occurs at a wall. For this reason, both the cavitation number and pressure recovery measure should be reported. A simple ratio is:

$$\begin{array}{ccc}& {\mathrm{\Pi }}_r=\frac{p_2}{p_1}& \text{(2.5)}\end{array}$$

where $p_1$ is the upstream pressure and $p_2$ is the downstream pressure. An alternative recovery coefficient can be defined using the minimum pressure $p_{\min }$:

$$\begin{array}{ccc}& C_{pr}=\frac{p_2-p_{\min }}{p_1-p_{\min }}& \text{(2.6)}\end{array}$$

The first ratio is easy to measure, while the second is more physically descriptive but usually requires local pressure measurement or a validated flow model. The qualitative regime map in Figure 2.1 illustrates why a single index is insufficient. Incipient cavitation, developed cloud cavitation, transient cavity shedding, and supercavitation may all occur at similar nominal pressure drops in different geometries.

Figure 2.1. Qualitative cavitation regime map using cavitation number and pressure recovery ratio. Boundaries are device and liquid dependent and are shown only to organize the discussion. [1–3].

2.3 Device Families

Static and dynamic cavitators can be grouped by the mechanism that produces the low pressure zone. Figure 2.2 summarizes common families. Orifice plates produce jets and separated shear layers. Venturis accelerate the liquid more gradually and can reduce irreversible losses relative to a sharp orifice. Multihole plates divide the flow and increase the number of cavities. Vortex devices use tangential momentum to create a low pressure core. Rotor stator devices combine rotational shear with pressure fluctuations. Multistage baffle devices impose repeated acceleration and recovery.

Figure 2.2. Common hydrodynamic cavitation device families.

Table 2.1. Principal device characteristics and typical engineering implications.

Device	Cavitation zone	Strengths	Main limitations
Single orifice	Jet vena contracta and downstream shear layer	Simple, compact, inexpensive, easy to characterize	High pressure loss; clogging risk; wall collapse and erosion if poorly designed
Multihole plate	Multiple jets and interacting cavity clouds	High event density; distributes flow; geometry is easily varied	Hole fouling; unequal flow distribution; manufacturing tolerances matter
Venturi	Throat and diffuser	Smoother acceleration; potentially lower irreversible loss	Longer device; cavity position depends strongly on diffuser and backpressure
Vortex chamber	Low pressure vortex core	Cavitation can occur away from walls; open flow passage; scale out is practical	Swirl stability and outlet geometry are critical; pressure measurements may not represent core pressure
Rotor or rotor stator	Rotor tip, gap, and wake regions	High throughput and strong mixing; tunable rotational speed	Moving parts, seals, mechanical wear, and more complex scale up
Multistage baffles	Repeated local restrictions	Multiple treatment zones in one body; adjustable designs are possible	Cumulative pressure loss and potential solids deposition

2.4 Nucleation, Gas Content, and Liquid Properties

The same hydraulic condition can produce different cavitation behavior in filtered water, a gas saturated liquid, an emulsion, and a high solids slurry. Nuclei may be trapped in surface crevices, carried on particles, stabilized by surfactants, or introduced intentionally. Dissolved gas can enter a growing cavity by diffusion, making collapse less violent because the noncondensable gas cushions the final compression. Conversely, a greater number of nuclei can increase the number of collapse events and the spatial coverage of treatment.

Viscosity damps radial bubble motion and suppresses small scale turbulence. Surface tension resists cavity growth and droplet or particle interface deformation. Vapor pressure increases strongly with temperature, which favors inception but can reduce collapse severity because more vapor must condense. Temperature also changes reaction kinetics, gas solubility, viscosity, and the stability of sensitive products. A valid process description should therefore include at least temperature, viscosity, density, surface tension when relevant, gas content or degassing condition, solids concentration, and surfactant content.

Table 2.2. Dimensionless groups and engineering quantities used in cavitation processing.

Quantity	Definition	Primary use	Important qualification
Cavitation number $\sigma$	$(p_{ref}-p_v)/\left(\rho U_{ref}^2/2\right)$	Inception and relative cavitation intensity	State pressure and velocity reference locations
Reynolds number $Re$	$\rho UD/\mu$	Dynamic similarity and turbulence	Does not capture nuclei or two phase effects
Weber number $We$	$\rho U^{2}D/\gamma$	Droplet and interface deformation	Use the appropriate interfacial tension
Euler number $Eu$	$\mathrm{\Delta }p/\left(\rho U^2/2\right)$	Pressure loss and geometric comparison	Sensitive to velocity definition
Pressure recovery ratio ${\mathrm{\Pi }}_r$	$p_2/p_1$	Collapse driving condition	Does not provide minimum pressure
Specific energy consumption	$P_{el}/\dot{m}$ or $\mathrm{\Delta }pQ/\dot{m}$	Energy normalized process comparison	State whether electrical or hydraulic power is used
Cavitational yield	Product change divided by input energy	Useful outcome per energy input	The product metric must be explicitly defined

2.5 Forms and Regimes

Table 2.3. Common cavitation forms and regimes. [7]

Form or regime	Typical geometry or behavior	Primary concern
Bubble cavitation	Discrete cavities grow in a low pressure region and collapse after convection.	Noise, local pitting, and inception diagnostics.
Sheet cavitation	An attached vapor layer forms on a blade or hydrofoil suction surface.	Performance change, reentrant jet instability, and cloud shedding.
Cloud cavitation	A group of cavities is periodically shed and collapses collectively.	Strong pressure pulses, vibration, and high erosion potential.
Vortex cavitation	A vapor core forms in a concentrated vortex such as a propeller tip or leakage vortex.	Noise, long range interaction, and remote collapse damage.
Traveling cavitation	Cavities nucleate in the free stream and pass through a pressure field.	Test tunnel nuclei sensitivity and isolated impact events.
Partial cavitation	Vapor occupies only part of a passage or surface.	Often controllable but may be unstable or erosive.
Supercavitation	A cavity extends over most of the body or blade.	Drag and load control, cavity closure, ventilation, and stability.
Choked two phase flow	Vapor volume limits mass flow or pressure recovery.	Severe performance loss and inability to gain flow with added pressure drop.

2.6 Collective Effects and the Limits of Single Index Descriptions

Cavitation may appear as isolated bubbles, attached sheet cavities, detached clouds, vortex cores, or an extended supercavity. The regime depends on geometry, nuclei, pressure level, residence time, and pressure recovery. Isolated bubbles can often be interpreted with single bubble models, whereas dense clouds behave collectively: neighboring bubbles alter the local pressure field, shield one another from the driving disturbance, and can collapse as a coordinated structure that produces stronger pressure pulses than an isolated event [8,9].

A lower minimum pressure does not necessarily produce a more useful cavitation field. If vapor occupies most of the flow passage, pressure recovery may occur outside the intended active zone or may be cushioned by a persistent vapor cloud. Effective generation therefore requires control of inception, growth time, void fraction, cloud transport, and collapse location. These considerations apply to both acoustic bubble clouds and hydrodynamic cavity structures [2,10].

Nomenclature

The following key defines the symbols and abbreviations used in this chapter. Unless otherwise stated, quantities are expressed in SI units.

Symbol	Definition	SI unit or status
A	Flow cross sectional area	m²
$C_{pr}$	Pressure recovery coefficient	dimensionless
D	Characteristic hydraulic diameter	m
$E_{in}$	Input energy	J
Eu	Euler number	dimensionless
g	Gravitational acceleration	m s⁻²
$h_L$	Irreversible head loss	m
ṁ	Mass flow rate	kg s⁻¹
p	Absolute static pressure	Pa
$p_{ref}$	Reference pressure used in a cavitation number	Pa
$p_v$	Saturation vapor pressure	Pa
$P_{el}$	Electrical input power	W
Q	Volumetric flow rate	m³ s⁻¹
Re	Reynolds number	dimensionless
U	Mean liquid velocity	m s⁻¹
$U_{ref}$	Reference velocity	m s⁻¹
We	Weber number	dimensionless
Y	Energy normalized treatment yield	application dependent
z	Elevation relative to a datum	m

Greek Letters

Symbol	Letter name	Meaning in this chapter
α	Alpha	Kinetic energy correction factor
γ	Gamma	Surface or interfacial tension
μ	Mu	Dynamic viscosity
ρ	Rho	Liquid density
σ	Sigma	Cavitation number
${\mathrm{\Pi }}_r$	Pi	Pressure recovery ratio
Δ	Delta	Finite change, such as pressure drop

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