Solution Found!
Based on your results of Prob. 4-120. discuss the
Chapter 4, Problem 117P(choose chapter or problem)
Problem 117P
Based on your results of Prob. 4-120. discuss the compressibility (or incompressibility) of this flow.
PROBLEM 4-120: Consider the flow field of Prob. 4‒117 (flow over a circular cylinder). Calculate the two linear strain rates in the rθ-plane; i.e., calculate Ɛrr andƐθθ· Discuss whether fluid line segments stretch (or shrink) in this flow field.(Hint: The strain rate tensor in cylindrical coordinates is given in Prob. 4‒109.)
PROBLEM 4-177:There are numerous occasions in which a fairly uniform free-stream flow encounters a long circular cylinder aligned normal to the flow (Fig. P11–58). Examples include air flowing around a car antenna, wind blowing against a flag pole or telephone pole, wind hitting electrical wires, and ocean currents impinging on the submerged round beams to support oil platforms. In all these cases, the flow at the rear of the cylinder is separated and unsteady, and usually turbulent. However, the flow in the front half of the cylinder is much more steady and predictable. In fact, except for a very thin boundary layer near the cylinder surface, the flow field may be approximated by the following steady, two-dimensional velocity components in the xy-or rθ-plane: Is this flow field rotational or irrotational? Explain.
FIGURE P11–58
The strain rate tensor in cylindrical coordinates:
Questions & Answers
QUESTION:
Problem 117P
Based on your results of Prob. 4-120. discuss the compressibility (or incompressibility) of this flow.
PROBLEM 4-120: Consider the flow field of Prob. 4‒117 (flow over a circular cylinder). Calculate the two linear strain rates in the rθ-plane; i.e., calculate Ɛrr andƐθθ· Discuss whether fluid line segments stretch (or shrink) in this flow field.(Hint: The strain rate tensor in cylindrical coordinates is given in Prob. 4‒109.)
PROBLEM 4-177:There are numerous occasions in which a fairly uniform free-stream flow encounters a long circular cylinder aligned normal to the flow (Fig. P11–58). Examples include air flowing around a car antenna, wind blowing against a flag pole or telephone pole, wind hitting electrical wires, and ocean currents impinging on the submerged round beams to support oil platforms. In all these cases, the flow at the rear of the cylinder is separated and unsteady, and usually turbulent. However, the flow in the front half of the cylinder is much more steady and predictable. In fact, except for a very thin boundary layer near the cylinder surface, the flow field may be approximated by the following steady, two-dimensional velocity components in the xy-or rθ-plane: Is this flow field rotational or irrotational? Explain.
FIGURE P11–58
The strain rate tensor in cylindrical coordinates:
ANSWER:
Solution 117P
We need to determine whether the given flow is compressible or incompressible.
A flow is said to be incompressible if the volumetric strain rate is zero everywhere.
We need to calculate the linear strain rates in the plane.
Step 1
The linear strain rate in the r-direction is calculated as