UNIVERSITI TUNKU ABDUL RAHMAN

Faculty : Engineering and Science Unit Code : UEME3112 Course : Bachelor of Engineering (Hons)

Mechanical/Civil/Chemical Engineering

Unit Title : Fluid Mechanics II

Year/ Semester

:

2/1, 2/2, 3/1 Lecturer

: Dr Lai Soon Onn/Ms. Jaslyn Low Foon Siang

Session : 2010/05

Tutorial No. (1) 1. The three components of velocity in a flow field are given by 222 zyxu ++= ,

2zyzxyv ++= , 42/3 2 += zxzw . a. Determine the volumetric dilatation rate and interpret the result. b. Determine an expression for the rotation vector. Is this an irrotational flow?

2. A combination of a forced and a free vortex is represented by the velocity distribution

( )[ ]2exp11 rr

v = . For r approaches zero, the velocity approaches a rigid body rotation, and

as r becomes large, a free-vortex velocity distribution is approached. Find the amount of rotation (in radians) that a fluid element will experience in completing one circuit around the centers as a function of r. Hint: The rotation rate in a flow with concentric streamlines is given

by ( )rvdrd

rr

v

drdv

12 =+=& . Evaluate the rotation for r = 0.5, 1.0 and 1.5.

3. The velocity components of 2D velocity field are given by the equations ( )xxyu += 12 and ( )12 += xyv . Show that the flow is irrotational and satisfies the continuity equation.

4. The radial velocity component in an incompressible, 2D plane flow field is sin32 2rrvr += . Determine the corresponding tangential velocity component, v , required to satisfy conservation of mass. Determine the corresponding stream function.

5. Consider a 2D velocity potential 22 yxxy += . (a) Is it true that 02 = , and if so, what does it mean? (b) If it exists, find the stream function of this flow. (c) Find the equation of streamline that passes through point )1,2(),( =yx .

6. The stream function for a 2D, inviscid flow field is given by )(6.0 yx = . (a) Is the flow field incompressible? (b) Is the flow field irrotational? If so, find the corresponding velocity potential. (c) Determine the pressure gradient in the horizontal x direction at the point x = 0.6 m, y = 0.6 m.

7. Two sources, one of strength m and the other with strength 3m, are located on the x axis. Determine the location of the stagnation point in the flow produced by these sources.

8. A tornado can be approximated by a free vortex of strength for cRr > , where cR is the radius of the core. Velocity measurements at points A and B (distance = 30 m) indicate that VA = 38 m/s and VB = 18 m/s. (a) Determine the distance from point A to the center of the tornado. (b) Why can the free vortex model not be used to approximate the tornado throughout the flow field ( 0r )?

9. One end of a pond has a shoreline that resembles a half-body. A vertical porous pipe is located near the end of the pond, so that water can be pumped out. When water is pumped at the rate of 0.08 m3/s through a 3-m long pipe, what will be the velocity at point A? Hint: Consider the flow inside a half-body.

10. A Rankine oval is formed by combining a sour-sink pair, each having a strength of 3.3 m2/s and separated by a distance of 3.6 m along the x axis, with a uniform velocity of 3 m/s (in the position x direction). Determine the length and thickness of the oval.

11. For what value of the circulation will the stagnation point be located at: (a) point A, (b) point B?

12. Water flows around a 2-m-diameter bridge pier with a velocity of 4 m/s. Estimate the force (per unit length) that the water exerts on the pier. Assume that the flow can be approximated as an ideal fluid flow around the front half of the cylinder, but due to the flow separation, the average

pressure on the rear half is constant and approximately equal to the pressure at point A.

30 m

U = 4 m/s 2 m