Prototype of a dam spillway ( a structure used for controlled release ...
To determine the minimum length-scale ratio between the prototype and the model for dynamic similarity, we need to consider the Froude number and Reynolds number. Let's analyze these parameters step by step:
Froude Number:
The Froude number (Fr) represents the ratio of inertial forces to gravitational forces and is given by the equation:
Fr = V / sqrt(gL)
where V is the characteristic velocity, g is the acceleration due to gravity, and L is the characteristic length.
Given:
Characteristic length of the prototype (Lp) = 20m
Characteristic velocity of the prototype (Vp) = 2m/s
Density of water (ρ) = 1000 kg/m^3
Acceleration due to gravity (g) = 9.81 m/s^2
Using the values given, we can calculate the Froude number for the prototype:
Frp = Vp / sqrt(gLp) = 2 / sqrt(9.81*20) ≈ 0.201
Now, for dynamic similarity, the Froude number needs to be the same for the model as well. Therefore, Frm = Frp = 0.201.
Reynolds Number:
The Reynolds number (Re) represents the ratio of inertial forces to viscous forces and is given by the equation:
Re = (VL) / μ
where V is the characteristic velocity, L is the characteristic length, and μ is the viscosity.
Given:
Viscosity (μ) = 10^-3 Pa-s
Minimum Reynolds number for the model (Rem) = 100
We can rearrange the Reynolds number equation to solve for the characteristic length of the model (Lm):
Lm = (Rem * μ) / Vm
To find the minimum length-scale ratio, we need to determine the characteristic velocity of the model (Vm) when the Froude number is the same as the prototype.
Solving the Froude number equation for Vm:
Frm = Vm / sqrt(gLm)
Vm = Frm * sqrt(gLm)
Substituting the values:
Vm = 0.201 * sqrt(9.81*20) ≈ 0.902 m/s
Now, substituting the values of Rem, μ, and Vm into the rearranged Reynolds number equation:
Lm = (Rem * μ) / Vm = (100 * 10^-3) / 0.902 ≈ 0.1108 m
Finally, to determine the minimum length-scale ratio, we divide the characteristic length of the prototype by the characteristic length of the model:
Length-scale ratio = Lp / Lm = 20 / 0.1108 ≈ 180.4
Hence, the minimum length-scale ratio between the prototype and the model is approximately 180.4, which corresponds to option 'A' - 1.8 x 10^-4.