8/3/2019 CUDBOCMZ_4802

1/3

Assignment1- 1 of 3

FLUID DYAMICS AND HYDRAULIC MACHINES

ASSIGNMENT SHEET #1 SUBMIT ON/BEFORE 27th

January 2012

PROBLEMS ON DIMENSIONAL ANALYSIS

1. List the seven fundamental variables.

2. The boundary layer thickness in flows with pressure gradient depends on the

distance x, velocity U, density , viscosity , andx

p

. Perform dimensional

analysis. [midsem 06]

3. The resisting force R of a supersonic aircraft during flight can be considered as

dependent on the length of the aircraft l, velocity v, air viscosity , air density andbulk modulus of air K. Express the functional relationship between these variables

and the resisting force. [endsem 06]

4. The pressure drop p along the bend in a pipe of diameterD depends on velocity of

flow U, pipe roughness (absolute) k, bend radius r, bend angle , and fluid properties

and . Obtain the complete set of dimensionless parameters. [suppl 07]

5. Show that the frictional torque T required to rotate a disc of diameter D at an

angular velocity in a liquid of density and viscosity is given by T/2D5 =

f(D2/). [mid-sem 08]

6. The efficiency of geometrically similar fans depends on mass density and

viscosity of air, angular velocity and diameterD of the fan, and the discharge Q.

Perform dimensional analysis for . [endsem 08]

7. The discharge Q over a triangular notch depends on the head Hover the notch,

notch angle , , , g and . Perform dimensional analysis for Q.

8. Assume that bacterium propels itself by rotating its tail (known as flagellum) in the

form of a spiral (see figure below). If the thrust F developed by the bacterium depends

on the rotational speed N of flagellum, mass density and the viscosity of the fluid, and

the diameter D of the coiled flagellum, perform dimensional analysis.

8/3/2019 CUDBOCMZ_4802

2/3

Assignment1- 2 of 3

PROBLEMS ON PHYSICAL SIMILARITY

9. A quarter scale turbine model is tested under a head of 10.8m. The full-scale

turbine is required to work under a head of 30m and to run at 7.14 rps. At what speed

must the model be run? If it develops 100 kW and uses 1.085 m3/s of water at this

speed, what power will be obtained from the full-scale turbine? What is thedimensionless specific speed of the full-scale turbine? [suppl 08]

10. A ship 300 m long moves in sea water having a density of 1025 kg/m3. A 1:100

model of the ship is to be tested in a wind tunnel. The velocity of air in the wind

tunnel is 30 m/s and the resistance on the model is measured to be 60 N. Determine

the velocity of the ship in sea-water. The density of air in the wind tunnel is 1.24

kg/m3. Take kinematic viscosity of sea-water and air as 1.2 10

-6m

2/s and 1.8 10

-6

m2 /s.

11. An aircraft is to fly at a height of 9 km (where the temperature is 45 C and thepressure is 30.2 kPa) at 400 m/s. A 1:20-scale model is tested in a pressurized wind-

tunnel in which air is at 15 C. For complete dynamic similarity what pressure and

velocity should be used in the model? For air take T3/2/(T+ 117), Es = p,p =

RT, Tis in Kelvin and is the ratio of specific heats. [midsem 06]

12. The torque required to operate the rudder of a deeply submerged submarine

traveling at 3 m/s is studied with a 1/20th

scale model in a fresh-water tunnel. In a

suitable test the torque on the model rudder is 8.29 N-m. What torque may be

expected for the full-size submarine? (Density of sea water = 1025 kg/m3, assume

equal kinematic viscosities for sea and fresh water). [midsem 04]

13. A 1:25 scale model of a ship with surface area of 4.0 m2

and length 6.0 m, whentested in the laboratory at 2.0 m/s velocity gave a total drag force of 40 N. Calculate

the drag force and power of the prototype when it is cruising at the corresponding

speed. Assume 20 C as the temperature of water and calculate skin friction by usingthe formula Cf= 0.074/ReL1/5. [endsem 06]

14. A model of an airplane is to be tested in a wind tunnel operating under a pressure

of 15 atmospheres. If the prototype is to fly at 500 km/hr under atmospheric

conditions, what would be the corresponding speed of the model which is built to a

scale of 1:12? If the model experiences a force of 600 N, what would be the

corresponding force on the prototype. Also find the power required by the prototype.

Assume viscosity of air to be same in both cases. [midsem 11]

15. An airplane of model scale ratio of 1:30 is tested in water, which is 50 times more

viscous and 800 times denser than air. If the pressure drop in the model during the test

is 225.63 N/m2, find the corresponding pressure drop in the prototype.

16. Water flows at the rate of 3500 m3/s in a river, 2.25 m deep and 1500 m long. A

model is prepared to a horizontal scale of 1:400 to accommodate the same in the

available space in the laboratory. For unity slope scale, suggest a suitable depth scale.

8/3/2019 CUDBOCMZ_4802

3/3

Assignment1- 3 of 3

What is the discharge in the model for this scale? Take = 1 10-6

m2 /s. Ans.

0.00626 m3/s.

17. A ship 125 m long at the water-line and having a wetted surface of 3500 m2

is to

be driven at 10 m/s in sea water. A 1:25 model ship is to be tested in fresh water to

determine its resistance. If the total resistance in the model is measured to be 54.2 N,

calculate that for the prototype.

Use2

)2(log

075.0

=

E

fR

C for model and an additional 0.0004 for prototype,

6

rfresh wate 10235.1= m

2/s, 6watersea 10188.1

= m2/s.

The magnitude of skin friction is not easy to determine for the prototype accurately

because the condition of surface after use is seldom known exactly. The surface of a

new hull is quite smooth, but after some time in service it becomes encrusted with

barnacles and coated with slime. Thus the roughness increases and is, however,

indeterminate. Hence, it is necessary to make some allowance for this and the value of

Cfmay be increased by, say about, 45%.