Question

6a. The aerodynamic drag acting on a cylinder at a very low Reynolds number (Stokes flow or creeping motion) is a function of dynamic viscosity of fluid (u), diameter of cylinder (D) and freestream velocity (U). Find a non-dimensional parameter using dimensional analysis. 6b. A model test is to be conducted in a water tunnel using a 1: 20 model of a submarine, which is to travel at a speed of 12 km/h deep under sea surface. The water temperature in the tunnel is maintained, so that is kinematic viscosity is half that of sea water. At what speed is the model test to be conducted to produce useful data for the prototype?

Answer 1

619) diameter of cylinder (0) Drag force (F) = [MLT-2 dynamic viscosity (u) = [ML" + ] [L] freestream Velouty (0) = (LT-' ] by Buckingham r- theorem no. of variable (n) = 4 fixed variable (m) - 3 n-m= 4-3 = 1 mo. q r terus laking , M ,D&U variables a -b U? fixed r = Mo lopo] - [MLT-² (MLT [M° °T°) - (mitaj [l'ab+c] (1-2-4-2] by comparing 7 a + 1 = 0 a=-1

o li l-a tbt c = 0 1-(-1) + 6+2 = 0 Tb + c = -2 T: -2-a-ca -2-(-1) -C=O -|- C = 0 KE - b -1 = -2 1b =-1 J. to Mo ADU 6 (6) for water tunnel Re VD Given : De Vm= 12 km/hr 20 Dm &p= Um ² p = / um 2 (Re) prototype (Re model model Um Dm Vp Dp ve Vm

Up um Dm X/Up Dp Vm up 12x18e Um Dm Dp 12 X (%) х Vp = lv, Re km Ihr
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