Page 76 - Mechanical Engineers' Handbook (Volume 4)

8 Dimensionless Numbers and Dynamic Similarity 65

2
Inertia force LV 2 V 2 V
or , the Weber number We
Surface tension force L / L / L
2
Inertia force LV 2 V 2 V
or , the Mach number M
Compressibility force KL 2 K/ K/
If a system includes n quantities with m dimensions, there will be at least n m
independent dimensionless groups, each containing m repeating variables. Repeating varia-
bles (1) must include all the m dimensions, (2) should include a geometrical characteristic,
a fluid property, and a flow characteristic and (3) should not include the dependent variable.
Thus, if the pressure gradient
p/L for flow in a pipe is judged to depend on the pipe
diameter D and roughness k, the average flow velocity V, and the fluid density , the fluid
viscosity , and compressibility K (for gas flow), then
p/L ƒ(D, k, V, , , K)orin
4
2
2
3
2
dimensions, F/L ƒ(L, L, L/T, FT /L , FT/L , F/L ), where n 7 and m 3. Then
there are n m 4 independent groups to be sought. If D, , and V are the repeating
variables, the results are
ƒ

p DV k V
,
2
V /2 , D K/
or that the friction factor will depend on the Reynolds number of the flow, the relative
roughness, and the Mach number. The actual relationship between them is determined ex-
perimentally. Results may be determined analytically for laminar flow. The seven original
variables are thus expressed as four dimensionless variables, and the Moody diagram of Fig.
32 shows the result of analysis and experiment. Experiments show that the pressure gradient
does depend on the Mach number, but the friction factor does not.
The Navier–Stokes equations are made dimensionless by dividing each length by a
characteristic length L and each velocity by a characteristic velocity U. For a body force X
due to gravity, X g g( z/ x). Then x x/L, etc., t t(LU), u u/U, etc., and p
x
2
p/ U . Then the Navier–Stokes equation (x component) is
u u u u
u v w
x y z t

gL p u u
2
u
2
2

U 2 x UL x 2 y 2 z 2
2
2
2
u
1 p u u
1

Fr 2 x Re x 2 y 2 z 2
Thus for incompressible flow, similarity of flow in similar situations exists when the Reyn-
olds and the Froude numbers are the same.
For compressible flow, normalizing the differential energy equation in terms of temper-
atures, pressure, and velocities gives the Reynolds, Mach, and Prandtl numbers as the gov-
erning parameters.
8.2 Dynamic Similitude
Flow systems are considered to be dynamically similar if the appropriate dimensionless
numbers are the same. Model tests of aircraft, missiles, rivers, harbors, breakwaters, pumps,