by     Christopher Earls Brennen     © Oxford University Press 1995



a Amplitude of wave-like disturbance
A Cross-sectional area or cloud radius
b Body half-width
B Tunnel half-width
c Concentration of dissolved gas in liquid, speed of sound, chord
ck Phase velocity for wavenumber k
cP Specific heat at constant pressure
CD Drag coefficient
CL Lift coefficient
, Unsteady lift coefficients
CM Moment coefficient
, Unsteady moment coefficients
Cij Lift/drag coefficient matrix
Cp Coefficient of pressure
Cpmin Minimum coefficient of pressure
d Cavity half-width, blade thickness to spacing ratio
D Mass diffusivity
f Frequency in Hz.
f Complex velocity potential, φ+iψ
fN A thermodynamic property of the phase or component, N
Fr Froude number
g Acceleration due to gravity
gx Component of the gravitational acceleration in direction, x
gN A thermodynamic property of the phase or component, N
(f) Spectral density function of sound
h Specific enthalpy, wetted surface elevation, blade tip spacing
H Henry's law constant
Hm Haberman-Morton number, normally gμ4/ρS3
i,j,k Indices
i Square root of -1 in free streamline analysis
I Acoustic impulse
I* Dimensionless acoustic impulse, 4πI {\cal R} / ρL U RH2
IKi Kelvin impulse vector
j Square root of -1
k Boltzmann's constant, polytropic constant or wavenumber
kN Thermal conductivity or thermodynamic property of N
KG Gas constant
Kij Added mass coefficient matrix, 3Mij/4ρπR3
Kc Keulegan-Carpenter number
Kn Knudsen number, λ/2R
Typical dimension in the flow, cavity half-length
L Latent heat of vaporization
m Mass
mG Mass of gas in bubble
mp Mass of particle
Mij Added mass matrix
n Index used for harmonics or number of sites per unit area
N(R) Number density distribution function of R
Cavitation event rate
Nu Nusselt number
p Pressure
pa Radiated acoustic pressure
ps Root mean square sound pressure
pS A sound pressure level
pG Partial pressure of gas
P Pseudo-pressure
Pe Peclet number, usually WR/αL
q Magnitude of velocity vector
qc Free surface velocity
Q Source strength
r Radial coordinate
R Bubble radius
RB Equivalent volumetric radius, [3τ/4π]1/3
RH Headform radius
RM Maximum bubble radius
RN Cavitation nucleus radius
RP Nucleation site radius
Distance to measurement point
Re Reynolds number, usually 2WR/νL
s Coordinate measured along a streamline or surface
s Specific entropy
S Surface tension
St Strouhal number, 2fR/W
t Time
tR Relaxation time for relative motion
t* Dimensionless time, t/tR
T Temperature
u,v,w Velocity components in cartesian coordinates
ui Velocity vector
ur,uθ Velocity components in polar coordinates
u′ Perturbation velocity in x direction, u-U
U, Ui Fluid velocity and velocity vector in absence of particle
V, Vi Absolute velocity and velocity vector of particle
U Velocity of upstream uniform flow
w Complex conjugate velocity, u-iv
w Dimensionless relative velocity, W/W
W Relative velocity of particle
W Terminal velocity of particle
We Weber number, 2ρW2R/S
z Complex position vector, x+iy


α Thermal diffusivity, volume fraction, angle of incidence
β Cascade stagger angle, other local variables
γ Ratio of specific heats of gas
Γ Circulation, other local parameters
δ Boundary layer thickness or increment of frequency
δD Dissipation coefficient
δT Thermal boundary layer thickness
ε Fractional volume
ζ Complex variable, ξ+iη
η Bubble population per unit liquid volume
η Coordinate in ζ-plane
θ Angular coordinate or direction of velocity vector
κ Bulk modulus of compressibility
λ Mean free path of molecules or particles
Λ Accommodation coefficient
μ Dynamic viscosity
ν Kinematic viscosity
ξ Coordinate in ζ-plane
Logarithmic hodograph variable, χ+iθ
ρ Density
σ Cavitation number
σc Choked cavitation number
σijStress tensor
Σ Thermal parameter in bubble growth
τ Volume of particle or bubble
ø Velocity potential
ø′ Acceleration potential
φ Fractional perturbation in bubble radius
Φ Potential energy
χ log(qc/|w|)
ψ Stream function
ω Radian frequency
ω* Reduced frequency, ωc/U


On any variable, Q:

Qo Initial value, upstream value or reservoir value
Q1,Q2,Q3Components of Q in three Cartesian directions
Q1,Q2 Values upstream and downstream of a shock
Q Value far from the bubble or in the upstream flow
QB Value in the bubble
QC Critical values and values at the critical point
QE Equilibrium value or value on the saturated liquid/vapor line
QG Value for the gas
Qi Components of vector Q
Qij Components of tensor Q
QL Saturated liquid value
Qn Harmonic of order n
QP Peak value
QS Value on the interface or at constant entropy
QV Saturated vapor value
Q* Value at the throat


On any variable, Q:

Mean value of Q or complex conjugate of Q
Complex amplitude of oscillating Q
Laplace transform of Q(t)
Coordinate with origin at image point
Rate of change of Q with time
Second derivative of Q with time
Q+,Q-Values of Q on either side of a cut in a complex plane
δQ Small change in Q
Re{Q} Real part of Q
Im{Q} Imaginary part of Q


In 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 m2/s2) will occasionally be used.

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Last updated 12/1/00.
Christopher E. Brennen