Assignment 1

1. Explain dimensionally homogeneous equation. Give any 2 examples.(10)

2. Explain three types of similarities in model analysis.(8)

3. Derive an expression for kinematic and dynamic similarities.(4)

4. Define buoyancy, metacentre and meta-centric height.(5)

5. State and prove the Buckingham Π - theorem. Also explain its advantages over Rayleigh's method of dimensional analysis.(6)

6. Explain the Froude's model law. Derive the different scale ratio for Froude model law.(10)

7. Using Buckingham's Π - theorem, show that the velocity through a circular orifice is given by,

where H is head causing flow, μ is coefficient viscosity, ρ is mass density and g is gravitational acceleration.(10)

8. The frictional torque, T of a disc of diameter, D rotating at a speed, N in a fluid of viscosity, μ and density, ρ in a turbulent flow using dimensional analysis. Prove

9. In the model test of a spillway, the discharge and velocity of flow over the model were 2.5m³/s and 2m/s respectively. Calculate the velocity and discharge over the prototype which is 40 times the model size.(8)

10. A ship 300m long moves in a sea water, whose density is 1030kg/m³, a 1:100 model of this ship is to be tested in a wind tunnel. The velocity of air in the wind tunnel around the model is 30m/s and the resistance of the model is 60N. Determine the velocity of ship in sea water and also the resistance of the ship in sea water. The density of air is 1.24kg/m³. Take the kinematic viscosity of sea water and air as 0.012 stokes and 0.018 stokes respectively.(12)

11. Derive on the basis of dimensional analysis suitable parameters to present the thrust developed by a propeller. Assume that the thrust P depends upon the angular velocity W, speed of advance V, diameter D, dynamic viscosity μ, mass density ρ, elasticity of the fluid medium which can be demoted by the speed of sound in the medium C.(10)

12. A solid cylinder 4m in diameter and 4m high is floating in water with its axis vertical. If the specific gravity of the material of cylinder is 0.60, find its meta centric height. State also whether the equilibrium is stable or unstable.(8)