# Problems of Chapter 7

Problem 7.1

Solution: a) 6 = 0.005 m, = 0.00172 m, S2 = 0.00066 m, Hn = 2.591.

b) 6 = 0.0233 m, 61 = 0.00292 m, 62 = 0.00227 m, H12 = 1.286, 6vs = 0.000116 m, 6sc = 0.000535 m.

Problem 7.2

Solution: a) 6 = 0.0158 m, 61 = 0.00544 m, 62 = 0.00209 m, H12 = 2.591.

b) 6 = 0.147 m, 61 = 0.0184 m, 62 = 0.0143 m, H12 = 1.286, 6vs = 0.000146 m, 6sc = 0.000848 m.

Problem 7.3

Table 7.3 gives besides others the dependence of the thicknesses on the running length x. The numerical results show that indeed in the turbulent cases a stronger increase is given than in the laminar cases. The thickness of the viscous sub-layer, respectively the scaling thickness, grows only very weakly. These thicknesses govern the skin friction and the heat flux in the gas at the wall in turbulent boundary layers.

Problem 7.4

Integrate the general expression for the skin friction, eq. (7.146), along the flat plate and find with Reref = pref vref Lref /^ref, and the exponent of the power-law relation for the viscosity, eq. (4.15), u^2 = 0.65 the skin-friction drag with reference-temperature extension of the flat plate wetted on one side

The skin-friction drag coefficients for the flat plate in compressible flow— wetted on both sides—are the familiar ones, but now with reference – temperature extension (note that Aref = bL):

and

Remember that eq. (7.146), and hence also these equations, hold only in a certain Reynolds number range. See in this regard the discussion in the summary of Sub-Section 7.2.1.

Problem 7.5

Re^,L = 1.691 108, Яж = 30,165.63 Pa. a) Df, c,lam = 8,963 N, b) Df, c,lam = 8,308 N.

Problem 7.6

Df, turb = a) 72480 N, b) 54,437 N.

Problem 7.7

The turbulent skin-friction drag is much higher than the laminar one. The ratio ‘turbulent drag’ to ‘laminar drag’ is a) 8.1 and b) 6.5.

A higher wall temperature reduces the skin-friction drag, much more for turbulent than for laminar flow. For laminar flow the drag is reduced by 7.3 per cent for the higher wall temperature, for turbulent flow by 25 per cent.

Problem 7.8

We take as reference temperatures the boundary-layer edge temperatures each. The dynamic pressure, the skin-friction coefficient, and the skin-friction drag at the windward side are qe, w = 51,433 Pa, CD, f,w = 8.022-10-4 and Df, w = 76,742 N, and at the leeward side qe, l = 15,692 Pa, CD, f,l = 7.343-10~4 and Df, l = 21,699 N. The total skin-friction force is Df, t = 98,441 N.

The lift component is Lf = —sin a Df, t = —10,290 N and the drag component Df = cos a Df, t = 97,902 N.

Problem 7.9

We add the inviscid and the skin-friction parts and find L = 1,21 -106 N and D = 0.226-106 N. Compared to the purely inviscid case the lift-to-drag ratio is reduced to L/D = 5.35. This is a realistic order of magnitude for a CV at

Mж = 6, see, e. g., [1].

The viscous forces are of large importance for CAV’s. This is in contrast to RV’s, where the viscous drag is only a small part of the total drag. Note that for this problem only the skin-friction drag was taken into account, not also the form drag. This is allowed, because of the large wetted surface of the CAV.

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