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5. COM and NSL for 1-d unsteady compressible flow read ∂t∂ρ+ρ∂x∂u+u∂x∂ρρ∂t∂u+ρu∂x∂u=−∂x∂p=0 (a) For an isentropic
Posted: Tue Jul 12, 2022 8:48 am
by answerhappygod
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5. COM and NSL for 1-d unsteady compressible flow read ∂t∂ρ+ρ∂x∂u+u∂x∂ρρ∂t∂u+ρu∂x∂u=−∂x∂p=0 (a) For an isentropic flow, show that this pair can be written as ∂t∂ρ+ρ∂x∂u+u∂x∂ρ∂t∂u+u∂x∂u=0=−ρa2∂x∂ρ (b) If the Riemann variable R is defined as dR=aρdρ show that the pair of equations in part (a) becomes ∂t∂R+u∂x∂R+a∂x∂u=0∂t∂u+u∂x∂u+a∂x∂R=0 (c) Add and subtract the pair of equations in part (b) to obtain ∂t∂(u+R)+(u+a)∂x∂(u+R)=0∂t∂(u−R)+(u−a)∂x∂(u−R)=0 (d) from the results of part ( c), determine the slope of the characteristics, i.e. dx/dt as well as the information that is propagated along the characteristics (e) for isentropic flow of a perfect gas, show that if R(0)=a(0)=0, then R=γ−12a