9. Physical Modeling
299
Example 1. Find the expression for tidal prism, P in an estuary where H
is the average tidal range and A is the average surface area of the basin.
Solution
P = A'..4“.K‘
(9.3)
where, K is a non-dimensional constant
Substitute the dimensions on both sides of Eq. (9.3)
M°L3T° =K • (L2)a • (L)6
Equating powers of M, L, T on both sides,
Power of L, 3 = 2a + b
By solving for the above équation, we get a = 1 and b — 1
Substituting values of a and b in Eq. (9.3)
P — K- A1-H1 = AH
9.2.3 Buckingham’s pi theorem
Buckingham’s Pi theorem is another popular method of dimensional analysis to obtain the relationship between the variables involved in a given
physical problem. Buckingham’s Pi theorem States that, “If there are ‘n’
variables in a problem and these variables contain ‘m’primary dimensions
(for example M, L, T) the équation relating ail the variables will hâve (n-m)
dimensionless groups”
The dimensionless groups are represented by % and are called as 7r
groups. According to this theorem, the final expression is given by
7T] = f(7F2, 7T3> • ■ • > ^n—m).
These tt groups formed should be independent of each other and no
groups shall be obtained by multiplying powers of other 7r groups. Compared to the Rayleigh’s method of solving simultaneous équations, this
method is a lot more advantageous and simpler.
Example. Consider a stationary sphere of diameter D is immersed in
a fluid flowing past the sphere in a steady flow where velocity of flow,
v; and the fluid properties, i.e., density, r, and viscosity, p. Therefore, a
functional relationship is expected between the drag force Fd and these
variables,
Fd = (f(D, v, p, g)
(9-4)
299
Example 1. Find the expression for tidal prism, P in an estuary where H
is the average tidal range and A is the average surface area of the basin.
Solution
P = A'..4“.K‘
(9.3)
where, K is a non-dimensional constant
Substitute the dimensions on both sides of Eq. (9.3)
M°L3T° =K • (L2)a • (L)6
Equating powers of M, L, T on both sides,
Power of L, 3 = 2a + b
By solving for the above équation, we get a = 1 and b — 1
Substituting values of a and b in Eq. (9.3)
P — K- A1-H1 = AH
9.2.3 Buckingham’s pi theorem
Buckingham’s Pi theorem is another popular method of dimensional analysis to obtain the relationship between the variables involved in a given
physical problem. Buckingham’s Pi theorem States that, “If there are ‘n’
variables in a problem and these variables contain ‘m’primary dimensions
(for example M, L, T) the équation relating ail the variables will hâve (n-m)
dimensionless groups”
The dimensionless groups are represented by % and are called as 7r
groups. According to this theorem, the final expression is given by
7T] = f(7F2, 7T3> • ■ • > ^n—m).
These tt groups formed should be independent of each other and no
groups shall be obtained by multiplying powers of other 7r groups. Compared to the Rayleigh’s method of solving simultaneous équations, this
method is a lot more advantageous and simpler.
Example. Consider a stationary sphere of diameter D is immersed in
a fluid flowing past the sphere in a steady flow where velocity of flow,
v; and the fluid properties, i.e., density, r, and viscosity, p. Therefore, a
functional relationship is expected between the drag force Fd and these
variables,
Fd = (f(D, v, p, g)
(9-4)