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).
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.
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.
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.
To solve these problems, we need to consider boundary layer theory and apply it to the given wind tunnel conditions. Let's go through each question step by step:
1) Estimating centerline outlet velocity:
(i) For laminar boundary layers, we assume H = 2.59. The first step is to estimate the boundary layer development using the simple inviscid model. According to the model, the velocity distribution along the tunnel can be approximated as ue(x) = Vin * (x/ℓ)^H, where ue(x) is the velocity at a distance x along the tunnel, and Vin is the inlet velocity.
Using this equation, we can determine the outlet velocity Vout. We substitute Vin = 5 m/s and ℓ = 1 m into the equation:
Vout (laminar) = ue(ℓ) = Vin * (ℓ/ℓ)^H = Vin * (1/1)^H = Vin * 1^H = Vin = 5 m/s
So, for laminar boundary layers, the centerline outlet velocity is Vout = 5 m/s.
(ii) For fully turbulent boundary layers, we assume H = 1.5. Using the same equation as before, we substitute the values:
Vout (turbulent) = ue(ℓ) = Vin * (ℓ/ℓ)^H = Vin * (1/1)^H = Vin * 1^H = Vin = 5 m/s
So, for fully turbulent boundary layers, the centerline outlet velocity is also Vout = 5 m/s.
However, the inviscid model is not accurate for determining outlet velocity in reality. A more accurate model, such as solving the boundary layer equations, would be necessary to obtain a more precise value for Vout.
2) To design a test section with a constant ue(x) = Vin = 5 m/s along the entire length, we need to determine the height function h(x) required to achieve this. We know h(0) = 0.1 m and ℓ = 1 m.
The velocity distribution along the tunnel can be expressed as ue(x) = Vin * (h(x)/h(0))^(2/3).
Since we need ue(x) = Vin = 5 m/s along the entire length, we can set up the equation:
5 = 5 * (h(x)/h(0))^(2/3)
Canceling out the Vin and solving for h(x), we have:
(h(x)/h(0))^(2/3) = 1
Taking both sides to the power of 3/2, we get:
(h(x)/h(0)) = 1^(3/2) = 1
Therefore, h(x) = h(0) = 0.1 m along the entire length of 1 m.
So, the height function h(x) required to achieve a constant ue(x) = Vin = 5 m/s is h(x) = 0.1 m (constant).
3) If the tunnel inlet speed is reduced to Vin = 3 m/s on the new test section while maintaining laminar flow, we can use the same height function h(x) = 0.1 m (constant) as in the previous question.
As the flow is laminar, we can estimate the centerline outlet velocity Vout by using the inviscid model:
Vout (laminar) = ue(ℓ) = Vin * (ℓ/ℓ)^H = Vin * (1/1)^H = Vin * 1^H = Vin = 3 m/s
So, when the tunnel inlet speed is reduced to Vin = 3 m/s on the new test section, the estimated centerline outlet velocity Vout is 3 m/s.