A small wind tunnel is being built for studying insect flight. The first test section built has a uniform height h=0.1m and length ℓ=1m. During operation its walls are found to have boundary layers which effectively start at the inlet at x=0 (the BL growth is negligible in the rapid contraction).
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CONSTANT CROSS-SECTION (2 points possible)
1) During operation it is found that there's a slight but undesirable centerline velocity increase from the inlet to the outlet of the test section. For an inlet velocity of Vin=5m/s at sea level where ν=1.45×10−5m2/s, estimate the centerline outlet velocity Vout for the case of
(i) laminar boundary layers, assuming H=2.59
(ii) fully turbulent boundary layers, assuming H=1.5.
For both cases, for the purpose of estimating the boundary layer development you may assume a ue(x) distribution obtained via the simple inviscid model of Section 3.1. Then determine Vout using a more accurate model.
Hint: Use the average skin friction coefficient, C¯f, following section 4.13 in the book.
For case 1(i), give Vout in m/s with 3 significant digits.
laminar Vout=- unanswered
For case 1(ii), give Vout in m/s with 3 significant digits.
turbulent Vout=- unanswered
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VARIABLE CROSS-SECTION (2 points possible)
2) A new test section is designed to obtain a constant ue(x)=Vin=5m/s along the entire test section with laminar boundary layers. Determine the height function h(x), in meters, required to achieve this, with h(0)=0.1m and ℓ=1m. Enter numerical values with 3 significant digits.
h(x)=- unanswered
3) What Vout will result when the tunnel inlet speed is reduced to Vin=3m/s on the new test section, still with laminar flow? Round your answer to three significant digits.
Vout=- unanswered
To solve these problems, we need to apply the principles of boundary layer theory and use equations that relate to the behavior of laminar and turbulent boundary layers.
1) For case 1(i) with laminar boundary layers and an inlet velocity of Vin = 5 m/s, we need to estimate the centerline outlet velocity Vout. We are given that the boundary layer starts at the inlet (x = 0) and that the growth of the boundary layer is negligible in the rapid contraction.
To estimate the boundary layer development, we can use the inviscid model and assume a ue(x) distribution. However, this inviscid model will not give us an accurate outlet velocity. So, we need to use a more accurate model.
To determine Vout using a more accurate model, we need the average skin friction coefficient, Cf. The formula for Cf is given in section 4.13 of the book. We can use this formula to relate Cf to Vout and Vin.
To use the formula, we need a constant called H. For laminar boundary layers, we can assume H = 2.59. Now, we can calculate Vout.
2) For case 1(ii) with fully turbulent boundary layers, we need to estimate the centerline outlet velocity Vout. Similarly to case 1(i), we can use the inviscid model to estimate the boundary layer development and then use the more accurate model with the average skin friction coefficient, Cf. In this case, we assume H = 1.5.
3) For the variable cross-section case, we are designing a test section with a constant ue(x) of Vin = 5 m/s along the entire section. We need to determine the height function h(x) that will achieve this.
Given that h(0) = 0.1 m and ℓ = 1 m, we can use the boundary layer theory equations to calculate h(x).
4) Finally, for the reduced inlet speed case, where Vin = 3 m/s, we need to determine the resulting Vout. We can use the same methods as in case 1(i) to calculate Vout.
Make sure to follow the steps outlined above and use the provided formulas and constants to obtain the answers to the questions accurately.