Introduction: Example: The velocity of propagation of a pressure wave through a liquid can be expected to depend on the elasticity of the liquid represented by the bulk modulus K, and its mass density ρ. Establish by D. A. the form of the possible relationship.
Rayleigh's method of dimensional analysis is a conceptual tool used in physics, chemistry, and engineering. This form of dimensional analysis expresses a functional relationship of some variables in the form of an exponential equation. It was named after Lord Rayleigh.
The method involves the following steps:
- Gather all the independent variables that are likely to influence the dependent variable.
- If R is a variable that depends upon independent variables R1, R2, R3, ..., Rn, then the functional equation can be written as R = F(R1, R2, R3, ..., Rn).
- Write the above equation in the form where C is a dimensionless constant and a, b, c... m are arbitrary exponents.
- Express each of the quantities in the equation in some fundamental units in which the solution is required.
- By using dimensional homogeneity, obtain a set of simultaneous equations involving the exponents a, b, c... m.
- Solve these equations to obtain the value of exponents a, b, c... m.
- Substitute the values of exponents in the main equation, and form the non-dimensional parameters by grouping the variables with like exponents.
Assume: u = C Kaρ b
U = velocity = L T-1, ρ = M L-3, K = M L-1T-2
LT-1= MaL-aT-2ax Mb L-3b
M: 0 = a b
L: 1 = -a – 3b
T: -1 = - 2a
Therefore: a = ½, b = -½, and a possible equation is u = C
Rayleigh’s method is not always so straightforward. Consider the situation of flow over a U-notched weir.
Q = f (ρ, µ, H, g), [Q] = [C ρaµbHc gd] [ ] => dimensions of
Using the M, L, T system,
[L3T-1] = [ML-3]a [M L-1T-1] b [L]c [L T-2]d, M: 0 = a b (1), L: 3 = -3a –b c d (2),T: -1 = - b – 2d (3)
We have only 3 equations, but there are 4 unknowns. Need to express a, b, c, in terms of d.
b = 1 – 2d
a = -b = 2d -1
c = 3 3a b – d = 1 3d
Q = C ρ (2d-1) µ (1-2d) H (1 3d) g (d)
2 dimensionless groups. Please check. This method can be very tedious if there are more variables.