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Stanford CS 237C - Supplementary Reading

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Lecture 10Monday, May 2, 2005Supplementary Reading: Osher and Fedkiw, §18.11 Incompressible FlowRecall the stability condition for compressible flowmaxΩ{|u + c| , |u| , |u − c|} <4x4twhere the quantity on the left of the inequality is the physical wave speed andthe quantity on the right is the numerical wave speed. Then the time step isgiven by4t = α4xmaxΩ{|u + c| , |u| , |u − c|}where α is the CFL number, α < 1.For example, we might have u = 1, c = 300, so that|u + c| = 301,|u| = 1,|u − c| = 299.Observe that the u ± c fields impose a much more severe restriction on thetime step than the u field. If |u|  |c| and we only care about the linearflow phenomena, i.e., the phenomena corresponding to the u field, then we canavoid this difficulty by modeling the flow as incompressible. The assumptionof incompressibility is valid in the limit ascu→ ∞ and is equivalent to thedivergence free condition ∇ ·~V = 0. In fact, the definition of incompressibilityfor a velocity field~V is that ∇ ·~V = 0.Modeling the flow as incompressible allows us to eliminate the severe timestep restriction due to the u ± c fields, and focus on the u field. As a result, welose the nonlinear behavior (e.g., shocks, rarefactions) associated with the u ± cfields.12 EquationsStarting from conservation of mass, momentum and energy, the equations forincompressible flow are derived using the divergence free condition, ∇ ·~V = 0,which implies that there is no compression or expansion in the flow field.2.1 Conservation of MassIn 1D, the equation for conservation of mass isρt+ (ρu)x= 0Applying the chain rule, we getρt+ ρxu + ρux= 0Since the flow is incompressible, ∇ ·~V = 0 which reduces to ux= 0 in 1D, sothat the equation is simplyρt+ uρx= 0In multiple dimension, the equation is given byρt+ ~u · ∇ρ = 0.2.2 Conservation of MomentumStarting with the e quation for conservation of mass,(ρu)t+ρu2+ px= 0we then apply the chain rule to getρtu + ρut+ ρuux+ u (ρu)x+ px= 0.We combine the first and fourth termsu (ρt+ (ρu)x) + ρut+ ρuux+ px= 0.Note that the quantity in parentheses is 0 from conservation of mass, so thatρut+ ρuux+ px= 0.By incompressibility, the second term is 0, so that we are left withρut+ px= 0.2Dividing by ρ, we ge tut+pxρ= 0. (1)In multiple dimension, the equation is given by~ut+ ~u · ∇~u +∇pρ= 0.2.3 Conservation of EnergyThe equation for conservation of energy in 1D isEt+ [(E + p) u]x= 0.Substituting E = ρe +12ρu2, we getρe +12ρu2t+ρe +12ρu2+ pux= 0.Differentiating, we havee +12u2ρt+ρet+ρuut+ρe +12ρu2+ pux+e +12u2uρx+ρuex+ρu2ux+upx= 0Since ux= 0, this b ec omese +12u2ρt+ ρet+ ρuut+e +12u2uρx+ ρuex+ upx= 0Rearranging terms, we havee +12u2(ρt+ uρx) + uρut+pxρ+ ρet+ ρuex= 0By the equations for conservation of mass and conservation of momentum, thisreduces toρet+ ρuex= 0Dividing by ρ, we ge tet+ uex= 0In multiple dimensions the equation for conservation of energy iset+ ~u · ∇e = 0In summary, the equations for incompressible flow (in multiple spatial di-mensions) areρt+ ~u · ∇ρ = 0~ut+ ~u · ∇~u +∇pρ= 0et+ ~u · ∇e = 03Recall that for compressible flow, we had an equation of state p = p(ρ, e).For incompressible flow, we have ∇·~u = 0 and do not have an equation of state.Notice also that the equation for conservation of energy is no longer needed toget a closed system. Instead, we have the closed system∇ · ~u = 0ρt+ ~u · ∇ρ = 0~ut+ ~u · ∇~u +∇pρ= 0In the next lecture, we will see how the pressure is found using an

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