Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
86 Dynamics and mechanics<br />
Characteristic numbers<br />
Reynolds<br />
number<br />
Froude<br />
number a<br />
Strouhal<br />
number b<br />
Prandtl<br />
number<br />
Mach<br />
number<br />
Rossby<br />
number<br />
Re = ρUL<br />
η<br />
=<br />
inertial force<br />
viscous force<br />
F = U2<br />
Lg = inertial force<br />
gravitational force<br />
S = Uτ<br />
L<br />
P = ηc p<br />
λ<br />
M = U c =<br />
Ro = U ΩL<br />
=<br />
evolution scale<br />
physical scale<br />
=<br />
momentum transport<br />
heat transport<br />
speed<br />
sound speed<br />
=<br />
inertial force<br />
Coriolis force<br />
(3.311)<br />
(3.312)<br />
(3.313)<br />
(3.314)<br />
(3.315)<br />
(3.316)<br />
a Sometimes the square root <strong>of</strong> this expression. L is usually the fluid depth.<br />
b Sometimes the reciprocal <strong>of</strong> this expression.<br />
Re<br />
ρ<br />
U<br />
L<br />
η<br />
F<br />
g<br />
S<br />
τ<br />
P<br />
c p<br />
λ<br />
M<br />
c<br />
Ro<br />
Ω<br />
Reynolds number<br />
density<br />
characteristic velocity<br />
characteristic scale-length<br />
shear viscosity<br />
Froude number<br />
gravitational acceleration<br />
Strouhal number<br />
characteristic timescale<br />
Prandtl number<br />
Specific heat capacity at<br />
constant pressure<br />
thermal conductivity<br />
Mach number<br />
sound speed<br />
Rossby number<br />
angular velocity<br />
Fluid waves<br />
( ) 1/2 ( ) 1/2<br />
Sound waves K dp<br />
v p = =<br />
(3.317)<br />
ρ dρ<br />
In an ideal gas ( ) 1/2 ( ) 1/2<br />
(adiabatic<br />
γRT γp<br />
v p =<br />
=<br />
(3.318)<br />
conditions) a µ ρ<br />
ω 2 = gktanhkh (3.319)<br />
Gravity waves on<br />
⎧<br />
a liquid surface b ⎨1<br />
( g<br />
) 1/2<br />
(h ≫ λ)<br />
v g ≃ 2 k<br />
(3.320)<br />
⎩<br />
(gh) 1/2 (h ≪ λ)<br />
Capillary waves<br />
(ripples) c<br />
Capillary–gravity<br />
waves (h ≫ λ)<br />
ω 2 = σk3<br />
ρ<br />
ω 2 = gk+ σk3<br />
ρ<br />
a If the waves are isothermal rather than adiabatic then v p =(p/ρ) 1/2 .<br />
b Amplitude ≪ wavelength.<br />
c In the limit k 2 ≫ gρ/σ.<br />
v p<br />
K<br />
p<br />
ρ<br />
wave (phase) speed<br />
bulk modulus<br />
pressure<br />
density<br />
γ ratio <strong>of</strong> heat capacities<br />
R molar gas constant<br />
T (absolute) temperature<br />
µ mean molecular mass<br />
v g<br />
h<br />
λ<br />
k<br />
g<br />
ω<br />
group speed <strong>of</strong> wave<br />
liquid depth<br />
wavelength<br />
wavenumber<br />
gravitational acceleration<br />
angular frequency<br />
(3.321) σ surface tension<br />
(3.322)
Change language