Question 6.8: In the study of vortex shedding phenomenon due to the presen...
In the study of vortex shedding phenomenon due to the presence of a bluff body in a flow through a closed duct, the following parameters are found to be important: velocity of flow V, density of liquid ρ, coefficient of dynamic viscosity of liquid μ, hydraulic diameter of the duct D_{h}, the width of the body B and the frequency of vortex shedding n. Obtain the dimensionless parameters governing the phenomenon.
The problem is described by 6 variables V, \rho, \mu, D_{h}, B, n. The number of fundamental dimensions in which the variables can be expressed = 3. Therefore, the number of independent π terms is (63) = 3. We use the Buckinghams π theorem to find the π terms and choose \rho, V, D_{h} as the repeating variables.
Hence, \pi_{1}=\rho^{a} V^{b} D_{h}^{c} \mu
\pi_{2}=\rho^{a} V^{b} D_{h}^{c} B
\pi_{3}=\rho^{a} V^{b} D_{h}^{c} n
Expressing the equations in terms of the fundamental dimensions of the variables we have
M ^{0} L ^{0} T ^{0}=\left( ML ^{-3}\right)^{a}\left( LT ^{-1}\right)^{b}( L )^{c}\left( ML ^{-1} T ^{-1}\right) (6.49)
M ^{0} L ^{0} T ^{0}=\left( ML ^{-3}\right)^{a}\left( LT ^{-1}\right)^{b}( L )^{c} L (6.50)
M ^{0} L ^{0} T ^{0}=\left( ML ^{-3}\right)^{a}\left( LT ^{-1}\right)^{b}( L )^{c} T ^{-1} (6.51)
Equating the exponents of M, L and T in the above equations we get, From Eq. (6.49),
a + 1 = 0
3a + b + c 1 = 0, b 1 = 0
which give a = 1, b = 1 and c = 1
Hence, \pi_{1}=\mu / \rho V D_{h}
From Eq. (6.50),
a = 0
3a + b + c + 1 = 0
b = 0
which give a = b = 0, c = 1
Hence, \pi_{2}=B / D_{h}
From Eq. (6.51),
a = 0
3a + b + c = 0
b 1 = 0
which give a = 0, b = 1, c = 1
Hence \pi_{3}=\left(n D_{h}\right) / V
Therefore the governing dimensionless parameters are
\left(\frac{\rho V D_{h}}{\mu}\right)\left(=1 / \pi_{1}\right) the Reynolds number
\frac{B}{D_{h}}\left(=\pi_{2}\right) ratio of the width of the body to hydraulic diameter of the duct.