As winds blow past buildings, complex flow patterns can develop due to various factors such as flow separation and interactions between adjacent buildings. (See Video \(\vee 7.13\).) Assume that the local gage pressure, \(p\), at a particular location on a building is a function of the air density, \(\rho,\) the wind speed, \(V\), some characteristic length, \(\ell,\) and all other pertinent lengths, \(\ell_{i},\) needed to characterize the geometry of the building or building complex. (a) Determine a suitable set of dimensionless parameters that can be used to study the pressure distribution. (b) An eight-story building that is \(100 \mathrm{ft}\) tall is to be modeled in a wind tunnel. If a length scale of 1: 300 is to be used, how tall should the model building be? (c) How will a measured pressure in the model be related to the corresponding prototype pressure? Assume the same air density in model and prototype. Based on the assumed variables, does the model wind speed have to be equal to the prototype wind speed? Explain.

Step by step solution

01

Calculate Non-Dimensional Parameters

The first task calls for determining non-dimensional parameters impacting pressure distribution. The variables given are: gage pressure (p), air density (ρ), wind speed (V), characteristic length (ℓ), and other lengths (ℓi). Applying the Buckingham Pi theorem, \(Pi = p/ρV^2\) can be obtained.

02

Simulation Height Calculation

In this step, the aim is to calculate the height of the model for a wind tunnel simulation with a scale of 1:300. If the actual building is 100 ft tall, the model should be \(100/300 = 0.33 ft\) or approximately 4 inches tall.

03

Relating Pressure in Model and Prototype

With air density the same in the model and prototype, by using the previous derived non-dimensional groups, we find that \(Pi_{model} = Pi_{prototype}\) or \(p_{model} / ρ_{model} V_{model}^2 = p_{prototype} / ρ_{prototype} V_{prototype}^2\). Therefore, \(p_{prototype} = p_{model} (V_{prototype} / V_{model})^2\). The wind speed in the model does not need to be the same as in the prototype. If it was, there wouldn't be any advantage in making a model and the dimensionless groups wouldn't make sense.

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