P0264 - 2.64 CHAPTER 2 PROBLEM 64 235 2.64 Chapter 2 Problem 64 Problem Experimental studies are planned for a rotational axisymmetric flowfield

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2.64. CHAPTER 2, PROBLEM 64 235 2.64 Chapter 2, Problem 64 Problem: Experimental studies are planned for a rotational axisymmetric flowfield. The vorticity of the flow, ω , has dimensions 1 /T and depends on initial circulation, Γ o (dimensions L 2 /T ), radius, r ,t ime , t , and kinematic viscosity of the fluid, ν .How many dimensionless groupings are there? What are the dimensionless groupings? Solution: The dimensional quantities and their dimensions are [ ω ]= 1 T , [ Γ o L 2 T , [ r L, [ t T, [ ν L 2 T There are 5 dimensional quantities and 2 independent dimensions ( L, T ) ,sot h a tth e number of dimensionless groupings is 3. The appropriate dimensional equation is [ ω ]=[ Γ o ] a 1 [ r ] a 2 [ t ] a 3 [ ν ] a 4 Substituting the dimensions for each quantity yields T 1 = L 2 a 1 T a 1 L a 2 T a 3 L 2 a 4 T a 4 = L 2 a 1 + a 2 +2 a 4 T a 1 + a 3 a 4 Thus, equating exponents, we arrive at the following two equations. 0= 2 a 1 + a 2 +2 a 4 1= a 1 + a 3 a 4 We can solve immediately for a 1 from the second equation, viz., a 1 =1+ a 3 a 4 Substituting into the first equation yields
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This note was uploaded on 02/09/2012 for the course AME 309 taught by Professor Phares during the Spring '06 term at USC.

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P0264 - 2.64 CHAPTER 2 PROBLEM 64 235 2.64 Chapter 2 Problem 64 Problem Experimental studies are planned for a rotational axisymmetric flowfield

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