Fundamentals of Fluid Mechanics A Assignment 7 — Dimensional Analysis
 Instructions
Write your solutions in single column format, with one statement following another vertically. Write your solutions neatly so that they are easy to read and verify. Don't write one line with two equal signs. Highlight your answers using a box. Failure to do this will result in a lower score and fewer comments on my part.
 Question #1
The model of a wind turbine (windmill) is to be tested in a low speed wind tunnel. The starting torque, and the power output at various angular velocities, are the dependent variables of specific interest. Determine suitable non-dimensional forms for these two parameters and the criteria of similarity on which they depend. Discuss what difficulties are likely to arise if dynamically-similar tests are made in a wind tunnel. A 1/20$^{\rm th}$ scale model is tested in a tunnel and it is decided that the effects of viscosity will be unimportant. The model produces 50 W in a wind speed of 5 m/s when rotating at $240$ rpm. Determine the power output of the prototype when the angular velocity is 30 rpm under dynamically-similar conditions. What is the corresponding wind speed? The air density is the same for the model and the prototype.
 08.01.19
 Question #2
You are assigned the task to design a prototype of a new airship. The prototype has a length $L$ of 100 m and its maximum diameter $D$ is of 20 m:
It is wished to estimate the drag force that would act on the prototype when it is at an altitude of 10 km and when the air speed $q_\infty$ is of 5 m/s (at an altitude of 10 km, the air pressure and temperature can be taken equal to 0.26 atm and 223 K respectively). To do so, you decide to build a small scale model and measure its drag in a wind tunnel under dynamically similar conditions, and further use non-dimensional analysis to relate the drag force measured in the model to the one acting on the prototype. For this purpose, do the following:
 (a) Find the non-dimensional numbers associated with this problem, and show how they are related to the Reynolds number and the drag coefficient (the criteria of similarity). (b) If the length of the model used in the experiment is fixed to 2 m and the air density in the wind tunnel is of $1.2$ kg/m$^3$, determine the diameter of the model airship and determine the wind tunnel air flow speed (c) Knowing that the drag force measured on the model is of 150 N, estimate the drag force acting on the prototype.
Notes: the air viscosity can be taken as $10^{-5}$ kg/ms for both the prototype and the model, and the gas constant for air is $286$ J/kgK.
 Question #3
Small droplets of liquid are formed when a liquid jet breaks up in spray and fuel injection processes. The resulting droplet diameter, $d$, is thought to depend on liquid density, liquid viscosity, and liquid surface tension, as well as jet speed $V$, and jet diameter $D$. How many dimensionless ratios are required to characterize this process? Determine these ratios. Hint: the surface tension has dimensions of force per unit length or of energy per unit area.
 $\pi$