Tutorial-1: Dimensional Analysis & Similitude
1. The drag force F on a body in supersonic flow is considered to depend on the length
(L), velocity (V), diameter (D) and fluid properties like density (ρ), dynamic viscosity
(μ) and bulk modulus of elasticity (E). Derive an expression for F.
2. The discharge (Q) over a small rectangular weir is known to depend upon the head
(H) over the weir, the weir height (P), gravity (g), width of the weir (L) , and fluid
properties density (ρ), dynamic viscosity (μ) and surface tension (σ). Express the
relationship between the variables in dimensionless form.
3. A small sphere of density (ρs) and diameter (D) settles at a terminal velocity (V) in a
liquid of density (ρf) and dynamic viscosity (μ). Gravity (g) is known to be a
parameter. Express the functional relationships between these variables in a
dimensionless form.
4. Show that frictional torque (T) of a disc of diameter (D) rotating at a speed of N in a
fluid of viscosity (μ) and density (ρ) in a turbulent flow condition is related as –
(
)
5. Using Buckingham’s pie theorem, show that the velocity (V) of flow of fluid of mass
density (ρ) and dynamic viscosity (μ), through a circular orifice of diameter (D) under
a head (H) is given by√
[
]
In this expression g is the acceleration due to gravity.
6. A fluid flow phenomenon is to be studied in a model which is to be constructed by
using Reynolds model law. Find the expressions for model to prototype ratios of
velocity, discharge, pressure, work and power.
7. For Froude model law, find the ratios of velocity, discharge, force, work and power in
terms of the length scale.
8. A 1: 6 scale model of a passenger car is tested in a wind tunnel. The prototype
velocity is 60 km/h. If the model drag is 250 N what is the drag and the power
required to overcome the drag in the prototype? The air in the model and prototype
can be assumed to have the same properties.
9. A model boat, 1/100 size of its prototype has 0.12 N of resistance when simulating a
speed of 5 m/s of the prototype. Water is the fluid in both the cases. What is the
corresponding resistance in the prototype? [The frictional forces can be neglected].
10. A pipe of diameter 1.5 m is required to transport an oil of relative density 0.9 and
kinematic viscosity = 3 × 10– 2 stoke at a rate of 3.0 m3/s. If a 15 cm diameter pipe
with water at 20°C (v = 0.01 stoke) is used to model the above flow, find the velocity
and discharge in the model.
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