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MAE305_pset01s - M&AE 3050 Problem Set 1 Solution Fa 2009 1...

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M&AE 3050 Problem Set 1 Fa 2009 Solution 1. Redo the dimensional analysis of the force acting on an object of length ` in a uniform steady flow of a gas of density ρ ,v iscos i ty μ and free stream velocity V taking com- pressibility efects into consideration. Assume the force now depends upon ρ , μ , ` and a , the speed of sound. Solution The development is similar to what was done in class, however we now include the speed of sound, a , in the list of variables f ( F, ρ, μ, v, `, a )=0 . In this case m =6 , k =3and n 3 = 3. Selecting ρ , v and ` as the working variables (as before), we obtain the same Π 1 and Π 2 groups. The last group will be Π 3 = α v β ` γ The units are ± L T ²± M L 3 ² α ± L T ² β [ L ] γ Forcing Π 3 to be dimensionless implies M: α =0 , T: 1 β , L: 1 3 α + β + γ . From this we see α , β = 1and γ . WeseethatΠ 3 = a/v = M - 1 is the inverse Mach number. Putting the result in a more historic form we have F ρv 2 ` 2 = f ³ ρv` μ , v a ´ . 2. A model of a harbor is made on the length ratio of 360:1. Storm waves of 2 m am- plitude and 8 m/s velocity occur on the breakwater of the full scale prototype harbor. Signi±cant variables are the length scale, velocity and g , the acceleration of gravity. The scaling of time can be made with the aid of the length scale and velocity scaling factors. (a) Neglecting friction (fluid viscosity), what should be the size and speed of the waves in the model? (b) If the time between tides in the prototype is 12 hrs, what should be the tidal period in the model?
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Solution The variables describing this system are L =s i z eo fh a r b o r , H = height of storm waves
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MAE305_pset01s - M&AE 3050 Problem Set 1 Solution Fa 2009 1...

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