## CAVITATION AND BUBBLE DYNAMICS

byChristopher Earls Brennen© Oxford University Press 1995

Nomenclature

ROMAN LETTERS

aAmplitude of wave-like disturbance ACross-sectional area or cloud radius bBody half-width BTunnel half-width cConcentration of dissolved gas in liquid, speed of sound, chord c_{k}Phase velocity for wavenumber k c_{P}Specific heat at constant pressure C_{D}Drag coefficient C_{L}Lift coefficient ,Unsteady lift coefficients C_{M}Moment coefficient ,Unsteady moment coefficients C_{ij}Lift/drag coefficient matrix C_{p}Coefficient of pressure C_{pmin}Minimum coefficient of pressure dCavity half-width, blade thickness to spacing ratio DMass diffusivity fFrequency in Hz. fComplex velocity potential, φ+iψ f_{N}A thermodynamic property of the phase or component, N FrFroude number gAcceleration due to gravity g_{x}Component of the gravitational acceleration in direction, x g_{N}A thermodynamic property of the phase or component, N (f)Spectral density function of sound hSpecific enthalpy, wetted surface elevation, blade tip spacing HHenry's law constant HmHaberman-Morton number, normally gμ ^{4}/ρS^{3}i,j,kIndices iSquare root of -1 in free streamline analysis IAcoustic impulse I^{*}Dimensionless acoustic impulse, 4πI {\cal R} / ρ _{L}U_{∞}R_{H}^{2}I_{Ki}Kelvin impulse vector jSquare root of -1 kBoltzmann's constant, polytropic constant or wavenumber k_{N}Thermal conductivity or thermodynamic property of N K_{G}Gas constant K_{ij}Added mass coefficient matrix, 3M _{ij}/4ρπR^{3}KcKeulegan-Carpenter number KnKnudsen number, λ/2R ℓTypical dimension in the flow, cavity half-length LLatent heat of vaporization mMass m_{G}Mass of gas in bubble m_{p}Mass of particle M_{ij}Added mass matrix nIndex used for harmonics or number of sites per unit area N(R)Number density distribution function of R Cavitation event rate NuNusselt number pPressure p_{a}Radiated acoustic pressure p_{s}Root mean square sound pressure p_{S}A sound pressure level p_{G}Partial pressure of gas PPseudo-pressure PePeclet number, usually WR/α _{L}qMagnitude of velocity vector q_{c}Free surface velocity QSource strength rRadial coordinate RBubble radius R_{B}Equivalent volumetric radius, [3τ/4π] ^{1/3}R_{H}Headform radius R_{M}Maximum bubble radius R_{N}Cavitation nucleus radius R_{P}Nucleation site radius Distance to measurement point ReReynolds number, usually 2WR/ν _{L}sCoordinate measured along a streamline or surface sSpecific entropy SSurface tension StStrouhal number, 2fR/W tTime t_{R}Relaxation time for relative motion t_{*}Dimensionless time, t/t _{R}TTemperature u,v,wVelocity components in cartesian coordinates u_{i}Velocity vector u_{r},u_{θ}Velocity components in polar coordinates u′Perturbation velocity in x direction, u-U _{∞}U, U_{i}Fluid velocity and velocity vector in absence of particle V, V_{i}Absolute velocity and velocity vector of particle U_{∞}Velocity of upstream uniform flow wComplex conjugate velocity, u-iv wDimensionless relative velocity, W/W _{∞}WRelative velocity of particle W_{∞}Terminal velocity of particle WeWeber number, 2ρW ^{2}R/SzComplex position vector, x+iy

GREEK LETTERS

αThermal diffusivity, volume fraction, angle of incidence βCascade stagger angle, other local variables γRatio of specific heats of gas ΓCirculation, other local parameters δBoundary layer thickness or increment of frequency δ_{D}Dissipation coefficient δ_{T}Thermal boundary layer thickness εFractional volume ζComplex variable, ξ+iη ηBubble population per unit liquid volume ηCoordinate in ζ-plane θAngular coordinate or direction of velocity vector κBulk modulus of compressibility λMean free path of molecules or particles ΛAccommodation coefficient μDynamic viscosity νKinematic viscosity ξCoordinate in ζ-plane Logarithmic hodograph variable, χ+iθ ρDensity σCavitation number σ_{c}Choked cavitation number σ_{ij}Stress tensor ΣThermal parameter in bubble growth τVolume of particle or bubble øVelocity potential ø′Acceleration potential φFractional perturbation in bubble radius ΦPotential energy χlog(q _{c}/|w|)ψStream function ωRadian frequency ω^{*}Reduced frequency, ωc/U _{∞}

SUBSCRIPTSOn any variable,

Q:

Q_{o}Initial value, upstream value or reservoir value Q_{1},Q_{2},Q_{3}Components of Q in three Cartesian directions Q_{1},Q_{2}Values upstream and downstream of a shock Q_{∞}Value far from the bubble or in the upstream flow Q_{B}Value in the bubble Q_{C}Critical values and values at the critical point Q_{E}Equilibrium value or value on the saturated liquid/vapor line Q_{G}Value for the gas Q_{i}Components of vector Q Q_{ij}Components of tensor Q Q_{L}Saturated liquid value Q_{n}Harmonic of order n Q_{P}Peak value Q_{S}Value on the interface or at constant entropy Q_{V}Saturated vapor value Q_{*}Value at the throat

SUPERSCRIPTS AND OTHER QUALIFIERSOn any variable,

Q:

Mean value of Qor complex conjugate ofQComplex amplitude of oscillating QLaplace transform of Q(t)Coordinate with origin at image point Rate of change of Qwith timeSecond derivative of Qwith timeQ^{+},Q^{-}Values of Qon either side of a cut in a complex planeδQSmall change in QRe{Q}Real part of QIm{Q}Imaginary part of Q

UNITSIn most of this book, the emphasis is placed on the nondimensional parameters that govern the phenomenon being discussed. However, there are also circumstances in which we shall utilize dimensional thermodynamic and transport properties. In such cases the International System of Units will be employed using the basic units of mass (

kg), length (m), time (s), and absolute temperature (K); where it is particularly convenient units such as a joule (kg m) will occasionally be used.^{2}/s^{2}

Last updated 12/1/00.

Christopher E. Brennen