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CE 3305 Engineering Fluid Mechanics
Summer 2022 Exercise Set 12

LAST NAME, FIRST NAME

R00000000

Purpose :

Application of Dimensional Analysis and Similitude.

Assessment Criteria :

Completion, plausible solutions, use JupyterLab as a calculator.

Problem 1.

A smooth pipe prototype designed to carry crude oil:

  • $D = 47$ inches
  • $\rho= 1.75$ slugs/$ft^3$
  • $\mu= 4 \times 10^{−4}~ lbf − s/ft^2$

is to be modeled with a smooth pipe 4 inches in diameter carrying water (T =60$^o$F ).

If the mean velocity in the prototype is to be 2 ft/s, what should be the mean velocity of the water in the model to ensure dynamically similar conditions?

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# solution (step-by-step)
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# discussion

Problem 2.

Figure 1: Standing wave on leading edge of a bridge pier

Flow around a bridge pier is studied using a model at $\frac{1}{12}$ scale. When the velocity in the model is 0.9 m/s, the standing wave at the pier nose is observed to be 2.5 cm in height.

What are the corresponding values of velocity and wave height in the prototype?

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# list known quantities
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# list unknown quantities
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# governing principles
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# solution (step-by-step)
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# discussion

Problem 3. (Application of ENGR-1330 Computational Thinking)

Experiments are performed to measure the pressure drop in a pipe with water at 20$^oC$ and crude oil at the same temperature. Data was gathered with pipes of two diameters, 5 cm and 10 cm for pressure drop per unit length of pipe, and are tabulated below:

Liquid Pipe Diameter (cm) Velocity (m/s) Pressure Drop (N/m$^3$)
Water 5 1 210
Oil 5 1 310
Water 5 2 730
Oil 5 2 1040
Water 5 5 3750
Oil 5 5 5300
Water 10 1 86
Oil 10 1 130
Water 10 2 320
Oil 10 2 450
Water 10 5 1650
Oil 10 5 2210

The presure drop per unit length is postulated to be a function of pipe diameter, liquid density, liquid viscosity, and flow velocity as:

$$\frac{\Delta p}{L}=f(\rho,\mu,V,D)$$

Perform a dimensional analysis to identify suitable $\pi-$groups and create a Jupyter Notebook to reduce the data and plot the results using the dimensionless values. Select an appropriate data model based on the plot.

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