SOLUTION The acceleration of air through the round test section of a wind tunnel is to be calculated, and a redesign of the test section is to be recommended.
Assumptions 1 The flow is steady and incompressible. 2 The walls are smooth, and disturbances and vibrations are kept to a minimum. 3 The boundary layer is laminar.
Properties The kinematic viscosity of air at 19°C is \nu=1.507 \times 10^{-5} m ^{2} / s.
Analysis (a) The Reynolds number at the end of the test section is approximately
\operatorname{Re}_{x}=\frac{V x}{v}=\frac{(4.0 m / s )(0.30 m )}{1.507 \times 10^{-5} m ^{2} / s }=7.96 \times 10^{4}
Since \operatorname{Re}_{x} is lower than the engineering critical Reynolds number, \operatorname{Re}_{x, cr }= 5 \times 10^{5}, and is even lower than \operatorname{Re}_{x, \text { critical }}=1 \times 10^{5}, and since the walls are smooth and the flow is clean, we may assume that the boundary layer on the wall remains laminar throughout the length of the test section. As the boundary layer grows along the wall of the wind tunnel test section, air in the region of irrotational flow in the central portion of the test section accelerates as in Fig. 10–105 in order to satisfy conservation of mass. We use Eq. 10–73 to estimate the displacement thickness at the end of the test section,
Displacement thickness, laminar flat plate: \frac{\delta^{*}}{x}=\frac{1.72}{\sqrt{\operatorname{Re}_{x}}} (Eq. 10-73)
\delta^{*} \cong \frac{1.72 x}{\sqrt{\operatorname{Re}_{x}}}=\frac{1.72(0.30 m )}{\sqrt{7.96 \times 10^{4}}}=1.83 \times 10^{-3} m =1.83 mm (1)
Two cross-sectional views of the test section are sketched in Fig. 10–107, one at the beginning and one at the end of the test section. The effective radius at the end of the test section is reduced by 𝛿* as calculated by Eq. 1. We apply conservation of mass to calculate the average air speed at the end of the test section,
V_{\text {end }} A_{\text {end }}=V_{\text {beginning }} A_{\text {beginning }} \quad \rightarrow \quad V_{\text {end }}=V_{\text {beginning }} \frac{\pi R^{2}}{\pi\left(R-\delta^{*}\right)^{2}} (2)
which yields
V_{\text {end }}=(4.0 m / s ) \frac{(0.15 m )^{2}}{\left(0.15 m -1.83 \times 10^{-3} m \right)^{2}}=4.10 m / s (3)
Thus the air speed increases by approximately 2.5 percent through the test section, due to the effect of displacement thickness.
(b) What recommendation can we make for a better design? One possibility is to design the test section as a slowly diverging duct, rather than as a straight-walled cylinder (Fig. 10–108). If the radius were designed so as to increase like 𝛿*(x) along the length of the test section, the displacement effect of the boundary layer would be eliminated, and the test section air speed would remain fairly constant. Note that there is still a boundary layer growing on the wall, as illustrated in Fig. 10–108. However, the core flow speed outside the boundary layer remains constant, unlike the situation of Fig. 10–105. The diverging wall recommendation would work well at the design operating condition of 4.0 m/s and would help somewhat at other flow speeds. Another option is to apply suction along the wall of the test section in order to remove some of the air along the wall. The advantage of this design is that the suction can be carefully adjusted as wind tunnel speed is varied so as to ensure constant air speed through the test section at any operating condition. This recommendation is the more complicated, and probably more expensive, option.
Discussion Wind tunnels have been constructed that use either the diverging wall option or the wall suction option to carefully control the uniformity of the air speed through the wind tunnel test section. The same displacement thickness technique is applied to larger wind tunnels, where the boundary layer is turbulent; however, a different equation for 𝛿*(x) is required.
