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Chapter 7

# Chapter 7 - Problem 7.4 Problem 7.5 Nondimensionalizing the...

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Problem 7.5 Nondimensionalizing the velocity, pressure, spatial measures, and time: L V t t L r r L x x p p p V u u = = = = = * * * * * Hence * * * * * t V L t r D r x L x p p p u V u = = = = = Substituting into the governing equation + + = = * * * 1 * * 1 * * 1 1 * * 2 2 2 r u r r u D V x p L p t u L V V t u ν ρ The final dimensionless equation is + + = * * * 1 * * * * * * 2 2 2 r u r r u D L V D x p V p t u ν ρ The dimensionless groups are D L V D V p ν ρ 2

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Problem 7.6 Recall that the total acceleration is V V t V Dt V D r r r r + = Nondimensionalizing the velocity vector, pressure, angular velocity, spatial measure, and time, (using a typical velocity magnitude V and angular velocity magnitude ): L V t t L x x p p p V V V = = = = = * * * * * r r r r Hence * * * * * t V L t x L x p p p V V V = = = = = r r r r Substituting into the governing equation * 1 * * 2 * * * * * p L p V V V V L V V t V L V V = × + ⋅∇ + ρ r r r r r
The final dimensionless equation is * * 2 * * * * * 2 p V p V V L V V t V = ×  Ω + ⋅∇ + ρ r r r r r The dimensionless groups are V L V p 2 ρ The second term on the left of the governing equation is the Coriolis force due to a rotating coordinate system. This is a very significant term in atmospheric studies, leading to such phenomena as geostrophic flow.

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Problem 7.8 Given: That drag depends on speed, air density and frontal area Find: How drag force depend on speed Apply the Buckingham Π procedure c F V ρ A n = 4 parameters d Select primary dimensions M , L , t e 2 3 2 L L M t L t ML A V F ρ r = 3 primary dimensions f V ρ A m = r = 3 repeat parameters g Then n m = 1 dimensionless groups will result. Setting up a dimensional equation, ( ) 0 0 0 2 2 3 1 t L M t ML L L M t L F A V c b a c b a = = = Π ρ

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Summing exponents, 2 0 2 : 1 0 1 2 3 : 1 0 1 : = = = = + + = = + a a t c c b a L b b M Hence A V F 2 1 ρ = Π h Check using F , L , t as primary dimensions [ ] 1 2 2 2 4 2 1 = = Π L t L L Ft F The relation between drag force F and speed V must then be 2 2 V A V F ∝ ρ The drag is proportional to the square of the speed.

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Problem 7.17 (In Excel) Given: That drain time depends on fluid viscosity and density, orifice diameter, and gravity Find: Functional dependence of t on other variables Solution We will use the workbook of Example Problem 7.1, modified for the current problem The number of parameters is: n = 5 The number of primary dimensions is: r = 3 The number of repeat parameters is: m = r = 3 The number of Π groups is: n - m = 2 Enter the dimensions ( M , L , t ) of the repeating parameters, and of up to four other parameters (for up to four Π groups).

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Chapter 7 - Problem 7.4 Problem 7.5 Nondimensionalizing the...

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