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# ch07 - Problem 7.1[2 Problem 7.2[2 Problem 7.3 Given...

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Problem 7.1 [2]

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Problem 7.2 [2]
Problem 7.3 [ 2 ] Given: Equation for beam Find: Dimensionless groups Solution: Denoting nondimensional quantities by an asterisk L x x L I I t t L y y L A A = = = = = * * * * * 4 2 ω Hence * * * * * 4 2 x L x I L I t t y L y A L A = = = = = Substituting into the governing equation 0 * * * 1 * * * 4 4 4 4 2 2 2 2 = + x y LI L EL t y A L L ωρ The final dimensionless equation is 0 * * * * * * 4 4 2 2 2 2 = + x y I L E t y A The dimensionless group is 2 2 L E

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Problem 7.4 [2]
Problem 7.5 [4]

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Problem 7.6 [ 2 ] Given: Equations for modeling atmospheric motion Find: Non-dimensionalized equation; Dimensionless groups Solution: 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 Ω ): L V t t L x x p p p V V V = = Ω Ω = Ω Δ = = * * * * * r r r r Hence * * * * * t V L t x L x p p p V V V = = Ω Ω = Ω Δ = = 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 Δ = × Ω Ω + ⋅∇ + ρ r r r r r The final dimensionless equation is * * 2 * * * * * 2 p V p V V L V V t V Δ = × Ω ⎛ Ω + ⋅∇ + r r r r r The dimensionless groups are V L V p Ω Δ 2 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.
Problem 7.7 [ 2 ] Given: Equations Describing pipe flow Find: Non-dimensionalized equation; Dimensionless groups Solution: Nondimensionalizing the velocity, pressure, spatial measures, and time: L V t t L r r L x x p p p V u u = = = Δ = = * * * * * Hence * * * * * t V L t r D r x L x p p p u V u = = = Δ = = Substituting into the governing equation + + Δ = = * * * 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 + + Δ = * * * 1 * * * * * * 2 2 2 r u r r u D L V D x p V p t u The dimensionless groups are D L V D V p 2 Δ

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Problem 7.8 [ 2 ] Given: Equation for unsteady, 2D compressible, inviscid flow Find: Dimensionless groups Solution: Denoting nondimensional quantities by an asterisk 0 0 0 0 0 * * * * * * * c L L c t t c c c c v v c u u L y y L x x ψ = = = = = = = Note that the stream function indicates volume flow rate/unit depth! Hence * * * * * * * 0 0 0 0 0 ψψ c L c t L t c c c v c v u c u y L y x L x = = = = = = = Substituting into the governing equation () 0 * * * * * 2 * * * * * * * * * * * * 2 3 0 2 2 2 2 3 0 2 2 2 2 3 0 2 2 3 0 2 2 3 0 = + + + + + y x v u L c y c v L c x c u L c t v u L c t L c The final dimensionless equation is ( ) 0 * * * * * 2 * * * * * * * * * * * * 2 2 2 2 2 2 2 2 2 2 2 2 2 = + + + + + y x v u y c v x c u t v u t No dimensionless group is needed for this equation!
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ch07 - Problem 7.1[2 Problem 7.2[2 Problem 7.3 Given...

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