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Unformatted text preview: Problem 7.1 [2] 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 = = = = = ω Substituting into the governing equation * * * 1 * * * 4 4 4 4 2 2 2 2 = ∂ ∂ + ∂ ∂ x y LI L EL t y A L L ω ρ The final dimensionless equation is * * * * * * 4 4 2 2 2 2 = ∂ ∂ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ∂ ∂ x y I L E t y A ω ρ The dimensionless group is ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ 2 2 ω ρ L E Problem 7.4 [2] Problem 7.5 [4] Problem 7.6 [ 2 ] Given: Equations for modeling atmospheric motion Find: Nondimensionalized 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 + ⋅∇ + ∂ ∂ ρ r r r r r The dimensionless groups are V L V p Ω Δ 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. Problem 7.7 [ 2 ] Given: Equations Describing pipe flow Find: Nondimensionalized 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 ν ρ The dimensionless groups are D L V D V p ν ρ 2 Δ Problem 7.8 [ 2 ] Given: Equation for unsteady, 2D compressible, inviscid flow Find: Dimensionless groups Solution: Denoting nondimensional quantities by an asterisk * * * * * * * c L L c t t c c c c v v c u u L y y L x x ψ ψ = = = = = = = Note that the stream function indicates volume flow rate/unit depth!...
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This note was uploaded on 03/09/2010 for the course ME 309 taught by Professor Merkle during the Spring '08 term at Purdue.
 Spring '08
 MERKLE

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