Final Comments

This chapter and the previous two have dealt with boundary layers, especially those on a flat plate. We end with the presentation of a photograph in Figure 19.2 showing the development of velocity profiles in the boundary layer over a flat plate. The fluid is water, which flows from left to right. The profiles are made visible by the hydrogen bubble technique, the same used for Figure 16.13. The Reynolds number is low (the freestream velocity is only 0.6 m/s); hence, the boundary-layer thickness is large. However, the thickness of the plate is only 0.5 mm, which means that the boundary layer shown here is on the order of 1 mm thick—still small on an absolute scale. In any event, if you need any further proof of the existence of boundary layers, Figure 19.2 is it.

19.5 Summary

Approximations for the turbulent, incompressible flow over a flat plate are

0.37 x

[19.1]

Re[./5

0.074

[19.2]

Cf = —– Г77

Re,1/5

To account for compressibility effects, the data shown in Figure 19.1 can be used, temperature method can be employed.

or alternatively the reference

Problems

Note: The standard sea level value of viscosity coefficient for air is д = 1.7894 x 10-5 kg/(m • s) = 3.7373 x 1СГ7 slug/(ft • s).

1. The wing on a Piper Cherokee general aviation aircraft is rectangular, with a span of 9.75 m and a chord of 1.6 m. The aircraft is flying at cruising speed (141 mi/h) at sea level. Assume that the skin friction drag on the wing can be approximated by the drag on a flat plate of the same dimensions. Calculate the skin friction drag:

(a) If the flow were completely laminar (which is not the case in real life)

(b) If the flow were completely turbulent (which is more realistic)

Compare the two results.

2. For the case in Problem 19.1, calculate the boundary-layer thickness at the trailing edge for

(a) Completely laminar flow

(b) Completely turbulent flow

3. For the case in Problem 19.1, calculate the skin friction drag accounting for transition. Assume the transition Reynolds number = 5 x 10s.

4. Consider Mach 4 flow at standard sea level conditions over a flat plate of chord 5 in. Assuming all laminar flow and adiabatic wall conditions, calculate the skin friction drag on the plate per unit span.

5. Repeat Problem 19.4 for the case of all turbulent flow.

6. Consider a compressible, laminar boundary layer over a flat plate. Assuming Pr = 1 and a calorically perfect gas, show that the profile of total temperature through the boundary layer is a function of the velocity profile via

To = Tw + (7oe — Tw) — ue

where Tw = wall temperature and To,,, and ue are the total temperature and velocity, respectively, at the outer edge of the boundary layer. [Hint: Compare Equations (18.32) and (18.41).]

7. Consider a high-speed vehicle flying at a standard altitude of 35 km, where the ambient pressure and temperature are 583.59 N/m2 and 246.1 K, respectively. The radius of the spherical nose of the vehicle is 2.54 cm. Assume the Prandtl number for air at these conditions is 0.72, that cp is 1008 joules/(kg K), and that the viscosity coefficient is given by Sutherland’s law. The wall temperature at the nose is 400 K. Assume the recovery factor at the nose is 1.0. Calculate the aero­dynamic heat transfer to the stagnation point for flight velocities of (a) 1500 m/s, and (b) 4500 m/s. From these results, make a comment about how the heat transfer varies with flight velocity.

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