f19_sp - Fluids – Lecture 19 Notes 1 Compressible Channel...

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Unformatted text preview: Fluids – Lecture 19 Notes 1. Compressible Channel Flow Reading: Anderson 10.1, 10.2 Compressible Channel Flow Quasi-1-D Flow A quasi-one-dimensional flow is one in which all variables vary primarily along one direction, say x . A flow in a duct with slowly-varying area A ( x ) is the case of interest here. In practice this means that the slope of the duct walls is small. Also, the x-velocity component u dominates the y and z-components v and w . A x dr dx 1 r y z A(x) u v,w 1−D Flow Quasi−1−D Flow Governing equations Application of the integral mass continuity equation to a segment of the duct bounded by any two x locations gives V · ˆ circlecopyrt ρ vector n dA = 0 x − ρ 1 u 1 dA + ρ 2 u 2 dA = 0 1 2 − ρ 1 u 1 A 1 + ρ 2 u 2 A 2 = 0 1 2 The quasi-1-D approximation is invoked in the second line, with u and ρ assumed constant on each cross-sectional area, so they can be taken out of the area integral. Since stations 1 or 2 can be placed at any arbitrary location x , we can define the duct mass flow which is constant all along the duct, and relates the density, velocity, and area. ρ ( x ) u ( x ) A ( x ) ≡ m = constant (1) ˙ If we assume that the flow in the duct is isentropic, at least piecewise-isentropic between shocks, the stagnation density...
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f19_sp - Fluids – Lecture 19 Notes 1 Compressible Channel...

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