Question 6.9: 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 \rho , coefficient of dynamic viscosity of liquid \mu , 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, and n . The number of fundamental dimensions in which the variables can be expressed = 3. Therefore, the number of independentp terms is (6–3) = 3. We use the Buckingham’s \pi theorem to find the \pi 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
\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}=\left(\mathrm{ML}^{-3}\right)^{a}\left(\mathrm{LT}^{-1}\right)^{b}(\mathrm{~L})^{c}\left(\mathrm{ML}^{-1} \mathrm{~T}^{-1}\right) (6.42)
\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}=\left(\mathrm{ML}^{-3}\right)^{a}\left(\mathrm{LT}^{-1}\right)^{b}(\mathrm{~L})^{c} \mathrm{~L} (6.43)
\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}=\left(\mathrm{ML}^{-3}\right)^{a}\left(\mathrm{LT}^{-1}\right)^{b}(\mathrm{~L})^{c} \mathrm{~T}^{-1} (6.44)
Equating the exponents of M, L and T in the above equations we get, From Eq. (6.42),
a+1 =0
-3 a+b+c-1 =0,-b-1=0
which give a = –1, b = –1 and c = –1
Hence, p_{1}=m / r V D_{h}
From Eq. (6.43),
a=0
-3 a+b+c+1 &=0
-b =0
which give a = b = 0, c = –1
Hence, p_{2}=B / D_{h}
From Eq. (6.44),
a=0
-3 a+b+c=0
-b-1=0
Which give a = 0, b = –1, c = 1
Hence, p_{3}=\left(n D_{h}\right) / V
Therefore, the governing dimensionless parameters are
\left(\frac{\rho V D_{h}}{\mu}\right)\left(=1 / p_{1}\right) the Reynolds number
\frac{B}{D_{h}} \quad\left(=p_{2}\right) ratio of the width of the body to hydraulic diameter of the duct
\frac{n D_{h}}{V} \quad\left(=p_{3}\right) the Strouhal number