To determine The set of dimensional parameters. The drag for the prototype. Explanation Write the functions for the drag. d = f ( D , V , ρ ) (I) Here, the drag is d , the density of the fluid is ρ , the velocity of the fluid is V and the parachute diameter is D . Write the dimension of the diameter in F L T unit system. D = [ L ] (II) Here, the dimensions for diameter is L . Write the dimension of the drag in F L T unit system. d = [ F ] (III) Here, the dimensions for drag is F . Write the dimension of the velocity in F L T unit system. V = [ L T − 1 ] (IV) Here, the dimension for distance is L and the time is T . Write the expression for density. ρ = m v (V) Here, the mass is m , the volume is v and the density is ρ . Substitute [ F L − 1 T 2 ] for m and [ L 3 ] for v in Equation (IV). ρ = [ F L − 1 T 2 ] [ L 3 ] = [ F L − 4 T 2 ] (VI) Here, the dimensions for force is F and the dimension for diameter is L . Write the expression for the number of pi-terms. n = k − r (VII) Here, the number of variable is k , the number of dimensions is r and the number of pi terms is n . The number of variable is 4 and the number of dimensions is 3 . Substitute 4 for k and 3 for r in Equation (VII). n = 4 − 3 = 1 Write the expression for pi term. Π = d D a V b ρ c (VIII) Here, the constant are a , c and b . Write the dimension for pi term. Π = L 0 T 0 (IX) Write the expression for drag. d = ( ρ ρ m ) ( V V m ) 2 ( D D m ) 2 d m (X) Here, the drag for prototype is d , the density for the prototype is ρ , the velocity for the prototype is V and the parachute diameter for prototype is D , the drag for the model is d m , the density for the model is ρ m , the velocity for the model is V m and the parachute diameter for the model is D m . Conclusion: Substitute [ F ] for d , [ F 0 L 0 T 0 ] for Π , [ L T − 1 ] for V , [ L ] for D and [ F L − 4 T 2 ] for ρ in Equation (VIII). [ F 0 L 0 T 0 ] = [ F ] [ L ] a [ L T − 1 ] b [ F L − 4 T 2 ] c [ F 0 L 0 T 0 ] = [ F ] [ L ] [ L b T − b ] [ F c L − 4 c T 2 c ] [ F 0 L 0 T 0 ] = [ F 1 + c L a + b − 4 c T − b + 2 c ] (XI) Compare the exponents of F in Equation (XI)

Munson, Young and Okiishi's Fundamentals of Fluid Mechanics, Binder Ready Version

8th Edition

ISBN: 9781119080701

Author: Philip M. Gerhart, Andrew L. Gerhart, John I. Hochstein

Publisher: WILEY

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Chapter 7.8, Problem 53P

To determine

The set of dimensional parameters.

The drag for the prototype.

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Chapter 7.8, Problem 52P

Chapter 7.8, Problem 54P

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Munson, Young and Okiishi's Fundamentals of Fluid Mechanics, Binder Ready Version

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