FEEG2003W2 FLUID MECHANICS SEMESTER 2 EXAMINATION 2022-2023
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FEEG2003W2
SEMESTER 2 EXAMINATION 2022-2023
TITLE: FLUID MECHANICS
DURATION: Two Hours (Online and Open Book)
Question 1
(a) A wing section with a chord of c and a span of b is mounted at zero angle of attack in a wind tunnel. A pitot probe is used to measure the velocity profile in the viscous region downstream of the wing section as shown in the figure. The measured velocity profile is u(z) = U∞ – (U∞ /2) cos[πz/(2 w)] for – w ≤ z ≤ w . Here, w = 0.02c. Assuming a constant pressure p = p∞ along the streamlines
(dashed lines in the figure) and across the wake where the velocity was measured, calculate the friction drag coefficient CD f of the wing section.
[5 marks]
(b) Consider a thin flat plate at zero angle of attack in an airflow at ρ∞ = 1.225 kg/m3 , T∞ = 288 K and μ∞ = 1.7894 × 10–5 kg/m/s. The length of the plate is 2 m and the span is 0.5 m. Assume the boundary layers on the plate are laminar throughout (on the upper and lower surfaces both) where θLBL (x)/x = 0.664/√ Rex applies. The freestream velocity is 100 m/s. Calculate the friction drag (Df) of the first half of the plate (0 ≤ x ≤ 1 m); and then, that of the second half (1 ≤ x ≤ 2 m). [5 marks]
(c) Repeat the question (b) when the freestream velocity is 1000 m/s and there is a normal shock formed in front of the plate. Use Sutherland’s law for the viscosity, i.e. μ/μ∞ = (T/T∞ )3/2 (T∞ +110)/(T +110). [10 marks]
Question 2
(a) For a turbulent BL (TBL) on a flat plate at zero PG, the momentum thickness is approximately estimated by
θ TBL (x) = .
This equation is valid only if the turbulent BL starts right at the leading edge (LE) of the plate (x = 0). However, normally, a laminar BL develops from the LE and it becomes turbulent at some point downstream via a transition process. We denote the transition point xT (see the figure below). In this case, the TBL cannot be assumed to have originated from the LE, and we imagine that it came from a “virtual” origin denoted by x0 which is somewhere between the LE and xT. This means that the above equation should be modified to
θ TBL (x) = for x ≥ x0 .
The relationship between x0 and xT is given by:
x0 = xT(1 – 38.22ReX(–)T(3/)8 ).
In this question, xT = L/2 and ReL = 3.8164 × 105 . Calculate the ratio of drag produced by the TBL and that by the LBL, i.e. Df , TBL /Df ,LBL . Use θLBL (x) given in Q1(b) if needed. Here, con- sider the upper surface of the plate only.
[10 marks]
(b) Using the incompressible Navier-Stokes equations derived in the class, find a solution of the steady plane Poiseuille flow (described in the figure below), i.e. u1(x2). The solution may be written in terms of dp/dx1 . Then, obtain the mass flow rate inside the channel. Here, u1 is a function of x2 only and u2 = u3 = 0 at all times.
[10 marks]
(c) The Poiseuille flow described in (b) is a fully-developed flow inside the channel. In fact, a uniform flow enters the channel first and, after undergoing a developing stage, the Poiseuille flow is achieved. This scenario is depicted in the figure below. Firstly, using a control volume and mass conservation, determine dp/dx1 in terms of the inlet velocity (U0). Here, dp/dx1 is assumed constant regardless of position. Secondly, re-write the Poiseuille solution in terms of U0and find the ratio between the maximum velocity of the fully-developed flow and the inlet velocity. Lastly, using momentum conservation, determine the drag of the plates in the developing zone.
[10 marks]
Question 3
This question concerns the superposition of a uniform flow U m/s in the x direction with two irrotational line vortices of strength K > 0 (in m2/s) placed at (0, a) and (0, 一a) respectively as shown in figure 3.1.
[Note that the line vortex in the lower half-plane is NOT a simple image source as in this question it is rotating in the SAME direction as the source in the upper half plane.]
Line Vortex
at (0, a)
Line Vortex
at (0, 一a)
y
x
Figure3.1.Schematicdiagramshowing mean flow and line vortices.
(a) What types of flow can be modelled by superposing velocity potentials which are solutions of Laplace’s equation? [2 marks]
(b) Write down the velocity potential for the problem described above as a function of position (x, y) in Cartesian coordinates. [2 marks]
(c) Using the substitutions tan θ1 = and tan θ2 = show that the velocity in the x direction is given by
ux = u 一
Page 6 of 9
(y 一 a)K (y + a)K |
x2 + (y 一 a)2 x2 + (y + a)2 |
[6 marks]
(d) What is the value of ux on the x axis? Give a physical reason why this should be the case. [2 marks]
(e) Derive a formula for the velocity in the y direction (uy). [3 marks]
(f) Are there any stagnation points? If so calculate their positions. [5 marks]
If the x and y coordinates are normalised by a and velocities normalised by U, then the flow field is entirely determined by the non-dimensional parameter K / (ua).
(g) Copy the graph below and sketch the streamlines for the case with K,(ua) = 2, marking the position of any stagnation points. [5 marks]
y/a
x/a
Figure3.2.Outlinegraphto copy for part(g).
Question 4
In this question all flows can be assumed to be one-dimensional inviscid flows of a perfect gas with gas constant R = 287m2s一2k一1 and constant ratio of specific heats Y = 1.4. Gravity can be neglected.
An engineer proposes to reduce air pressure using a double convergent- divergent nozzle with two shocks as shown schematically in figure 4.1 below.
The flow comes from a large vessel in which the air is stationary at
pressure Po and temperature To . The minimum duct areas are A1(*) and
A2(*) as shown in the figure. The flow exhausts to a large vessel at
pressure pex .
Figure4.1. Schematic(nottoscale)of double convergent-divergent nozzle.
(a) In which regions of the flow can the isentropic flow relations be applied? [2 marks]
(b) Write down the equation for mass flow through a choked nozzle in terms of stagnation density and stagnation temperature. [1 mark]
(c) Based on the above equation, explain why the engineer has to set the second minimum area larger than the first: A2(∗) > A1(∗) . [3 marks]
(d) Sketch (qualitatively) the static pressure profile along the duct. [5 marks]
(e) This part refers to how the shock positions change due to
increasing the pressure at either end of the duct. The changes can be assumed sufficiently small that the flow regime (i.e. two shocks in the duct) remains the same.
Copy & complete the following table using the words
‘downstream’, ‘upstream’ or ‘same’ to describe the change in each shock position.
Change in pressure |
Shock position xs1 |
Shock position xs2 |
Increase Po |
|
|
Increase pex |
|
|
[5 marks]
For the remaining parts, assume that the duct is designed for inlet total pressure Po = 200kPa, inlet total temperature To = 288k. At the design point the target mass flow is 100kg/s and the pre-shock Mach number M1s for the first shock is 2.0.
(f) Calculate the areas A1(∗) and A2(∗), showing your working. [6 marks]
(g) The exit pressure pex is raised to exactly the point where the second shock disappears. Sketch (qualitatively) the static
pressure profile along the duct in this case. [3 marks]
2023-08-05