f20_sp

# f20_sp - Fluids Lecture 20 Notes 1 Laval Nozzle Flows...

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Fluids – Lecture 20 Notes 1. Laval Nozzle Flows Reading: Anderson 10.3 Laval Nozzle Flows Subsonic ﬂow and choking Consider a duct with a throat, connected at its inlet to a very large still air reservoir with total pressure and enthalpy p r , h r . The duct exit is now subjected to an adjustable exit static pressure p e , sometimes also called the back pressure . As p e is gradually reduced from p r , air will ﬂow from the reservoir to the exit with a mass ﬂow ˙ m . We first note that the stagnation conditions are known from the reservoir values all along the duct. γp o γp r 2 p o = p r , a = ( γ 1) h o = ( γ 1) h r , ρ o = = o ( γ 1) h o ( γ 1) h r If we assume isentropic ﬂow, ˙ m can be computed with the isentropic relations applied at the exit, using the known exit pressure p e and known exit area A e . M γ 1 2 = ( p o /p e ) γ 1 2 e γ 1 γ +1 γp o γ 1 M 2 2( γ 1) ˙ m = ρ e u e A e = M e 1 + 2 e A e (1) ( γ 1) h o The observed relation between p e and ˙ m is shown on the bottom right in the figure. As p e is reduced, ˙ m will first increase, but at some point it will level off and remain constant even if p e is reduced all the way to zero (vacuum). When ˙ m no longer increases with a reduction in p e , the duct is said to be choked . M u ρ throat ρ * * p p r p r 0 m . choked x x p * 1 p e p h e p < p r r large reservoir m . M a throat 1 1 p e r If we examine the various ﬂow properties along the duct, it is evident that the onset of choking cooincides with the throat reaching M = 1 locally. This also corresponds to the 1

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mass ﬂux ρu at the throat reaching
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