MEEN344_Practice_Problems_Exam2B.pdfProblem 7.1.pdfProblem 7.2.pdfProblem 7.3.pdfProblem 7.4.pdfProblem 7.5.pdfProblem 7.6.pdfProblem 7.7.pdfProblem 7.8.pdfProblem 7.9.pdfProblem 7.10.pdfProblem 7.11.pdfProblem 7.12.pdfProblem 7.13.pdfProblem 7.14.pdfProblem 7.15.pdfProblem 7.16.pdfProblem 7.17.pdfProblem 7.18.pdfProblem 7.19.pdfProblem 7.20.pdfProblem 7.21.pdfProblem 7.22.pdfProblem 7.23.pdfProblem 7.24.pdfProblem 7.25.pdfProblem 7.26.pdfProblem 7.27.pdfProblem 7.28.pdfProblem 7.29.pdfProblem 7.30.pdfProblem 7.31.pdfProblem 7.32.pdfProblem 7.33.pdfProblem 7.34.pdfProblem 7.35.pdfProblem 7.36.pdfProblem 7.37.pdfProblem 7.38.pdfProblem 7.39.pdfProblem 7.40.pdfProblem 7.41.pdfProblem 7.42.pdfProblem 7.43.pdfProblem 7.44.pdfProblem 7.45.pdfProblem 7.46.pdfProblem 7.47.pdfProblem 7.48.pdfProblem 7.49.pdfProblem 7.50.pdfProblem 7.51.pdfProblem 7.52.pdfProblem 7.53.pdfProblem 7.54.pdfProblem 7.55.pdfProblem 7.56.pdfProblem 7.57.pdfProblem 7.58.pdfProblem 7.59.pdfProblem 7.60.pdfProblem 7.61.pdfProblem 7.62.pdfProblem 7.63.pdfProblem 7.64.pdfProblem 7.65.pdfProblem 7.66.pdfProblem 7.67.pdfProblem 7.68.pdfProblem 7.69.pdfProblem 7.70.pdfProblem 7.71.pdfProblem 7.72.pdfProblem 7.73.pdfProblem 7.74.pdfProblem 7.75.pdfProblem 7.76.pdfProblem 7.77.pdfProblem 7.78.pdfProblem 7.79.pdfProblem 7.80.pdfProblem 7.81.pdfProblem 7.82.pdfProblem 7.83.pdfProblem 7.84.pdfProblem 7.85.pdfProblem 7.86.pdfProblem 7.87.pdfProblem 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 LxxLIIttLyyLAA ===== *****42ω Hence *****42xLxILIttyLyALA =====ω Substituting into the governing equation 0***1***44442222=∂∂+∂∂xyLILELtyALLωρ The final dimensionless equation is 0******442222=∂∂⎟⎟⎠⎞⎜⎜⎝⎛+∂∂xyILEtyAωρ The dimensionless group is ⎟⎟⎠⎞⎜⎜⎝⎛22ωρLEProblem 7.5 [4]Problem 7.6 [2] Given: Equations for modeling atmospheric motion Find: Non-dimensionalized equation; Dimensionless groups Solution: Recall that the total acceleration is VVtVDtVDrrrr∇⋅+∂∂= Nondimensionalizing the velocity vector, pressure, angular velocity, spatial measure, and time, (using a typical velocity magnitude V and angular velocity magnitude Ω): LVttLxxpppVVV ==ΩΩ=ΩΔ== *****rrrr Hence ***** tVLtxLxpppVVV ==ΩΩ=ΩΔ==rrrr Substituting into the governing equation *1**2*****pLpVVVVLVVtVLVV ∇Δ−=×ΩΩ+⋅∇+∂∂ρrrrrr The final dimensionless equation is **2*****2pVpVVLVVtV∇Δ−=×Ω⎟⎠⎞⎜⎝⎛Ω+⋅∇+∂∂ρrrrrr The dimensionless groups are VLVpΩΔ2ρ 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: LVttLrrLxxpppVuu ===Δ== ***** Hence ***** tVLtrDrxLxpppuVu ===Δ== Substituting into the governing equation ⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+∂∂Δ−=∂∂=∂∂***1**1**11**222rurruDVxpLptuLVVtuνρ The final dimensionless equation is ⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂⎟⎠⎞⎜⎝⎛⎟⎠⎞⎜⎝⎛+∂∂Δ−=∂∂***1******222rurruDLVDxpVptuνρ The dimensionless groups are
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