13.Prandtl-Meyer Expansion Flow

Transcription

1 3.Prandtl-eyer Expansion Flow This hapter will treat flow over a expansive orner, i.e., one that turns the flow outward. But before we onsider expansion flow, we will return to onsider the details of the ompressive orner ase that results in oblique shok waves (the topi of Chapter ). If we zoom in to examine a orner that is not a perfetly sharp edge, but rather a orner with a finite radius of urvature: oblique shok oblique shok with a finite jump in pressure, veloity, et., is formed. Thus, even if a orner is loally gradual with a finite radius, an oblique shok will eventually form. The shok angle and other properties of this oblique shok will be the same as if the original orner was sharp and disontinuous. This is the two-dimensional, steady analog of the generation of a normal shok by the gradually aelerating piston disussed in Setion 7.; note the similarity between the drawing here and the x-t diagram of the aelerating piston generation of a shok in Chapter 7. Examining the analogous flow over an expansive orner, we see that the suessive ah lines tend to have a smaller slope, due to the hanging angle of the flow and the inreasing ah number as the flow expands (resulting in a dereasing ah angle). Thus, these disturbanes an never oalese together to form a shok, and the flow remains isentropi. Similar to how a piston that is rapidly extrated from a ylinder generates a rarefation wave, the flow over ah lines we see that the oblique shok is not formed instantaneously. Rather, the first disturbane reated at the orner propagates outward from that point as a ah line at angle sin. This is not a shok wave, but rather a weak ---- disturbane that ommuniates the presene of the orner to the flow. This is followed by other disturbanes whih also propagate at the loal ah angle. Beause the prior disturbanes have defleted the flow upward and ompressed the flow (dereasing the ah number and, thereby, inreasing the ah angle), the later disturbanes will be at a greater angle with respeted to the horizontal than the original disturbane. Thus, the later disturbanes tend to overtake the earlier ones. When these disturbanes merge together, an ah lines 43

2 an expansive orner generates an expansion fan. If the orner is sharp, the resulting flow is alled a entered Prandtl-eyer expansion fan. The analysis developed in this hapter applies to any uniform supersoni flow that enounters a gradual area hange, as long as the disturbanes do not oalese into a shok. Thus, the analysis is valid for the initial stages of flow over a gradual ompressive orner, and is valid for the entire flow over an expansive orner. 3. Working Relations for Prandtl-eyer Flow We begin by onsidering a single disturbane wave generated by a slight defletion of the wall: os tan - - These will be useful in our development. Superimposing the veloity vetors for the flow oming into and out of this wave: + d T + d Note that the tangential omponent for the veloity vetor upstream and downstream of the wave must be equal, as ditated by the momentum equation (review Setion. if this point is unlear). The tangential omponent of veloity is related to the resultant veloity by: T os The flow is defleted downward by an angle. By using our previous expression for os: The disturbane wave is at an angle sin ---- with respet to the approahing flow. We an interpret this relation for the ah angle via the triangle shown here. From this triangle, we an find relations for os and tan: T os - T - Likewise, for the veloity downstream of the wave: T ---- os( + d + ) 44

3 Expanding os( + ) : os( + ) osos sinsin Sine the defletion angle is very small ( << ): Or: d - os, sin dv d Thus: T ---- os + d sin This differential relates the defletion angle to the hange in veloity d as a funtion of ah number. Note that a positive results in a positive d, verifying our initial assumptions in the sign of and d. This agrees with our experiene from one-dimensional isentropi supersoni flow: a diverging area should result in an inrease in veloity. Using our expressions for T -, sin, and os: d We would like to integrate this relation for finite hanges in angle, but we need d to express the term in terms of ah number for this expression to be integrable. Starting with the definition of ah number ( ) and using logarithmi differentiation: Expanding ---- by long division: + d d d - d d d d + d 3 + H.O.T. d Sine <<, we an neglet the higher order terms. Thus: d -- Simplifies to: Sine the flow is adiabati, we an relate the sound speed to the temperature: ---- o Differentiating: T o γ T γ o d γ γ d 45

4 d Thus, we an express entirely in terms of ah number: γ ---- d d γ So, our expression for : an be integrated for a finite defletion of the flow: ν ν Integration yields: ---- d γ d γ d γ Prandtl eyer Funtion ν max funtion is plotted here; tabulated values of ν as a funtion of an be found in Table A-6. Note that in the limit as, the Prandtl-eyer funtion ν reahes an asymptote: π -- γ γ ν [ ν] γ + ν γ tan γ ( ) tan γ + π lim ν -- γ γ (for γ.4 ) Sine this funtion is somewhat umbersome, it is onvenient to hoose a referene angle for ν. Defining ν 0 at, we obtain the Prandtl-eyer Funtion: ν γ γ tan γ ( ) tan γ + Thus, flow that starts out soni annot be turned more than At this angle, the flow is fully expanded and the pressure and temperature derease to zero. What happens if we try to fore flow to turn though a larger angle by using a defletion angle greater than 30.45? The flow an no longer follow the wall and, in priniple, a region of vauum exists between the maximum turning angle and the wall. * This funtion represents the angle ν that an initially soni flow (i.e., flow starting at ah ) has been turned through to reah ah number. This * In a real experiment with real gases, the ideal gas law and alorially perfet gas assumptions break down as the temperature and pressure approah zero. In pratie, a flow annot reah the ideal maximum turning angle predited from theory. 46

5 Thus, the flow approahing the orner in our problem has a value of ν 6.38 assoiated with it. After it has been defleted an additional 0 by the orner, the new value of ν must be: ν ν + θ vauum ν max ν Knowing the new value of the Prandtl-eyer funtion, we an find the ah number of the flow downstream of the orner: How do we use this funtion when we are onsidering a flow that does not originate at soni onditions? This is best illustrated by a numerial example. 3.. Numerial Example: Prandtl-eyer Expansion Problem: A supersoni flow at ah and atm stati pressure enounters a orner with a 0 expansion angle. Find the ah number and pressure of the flow downstream of the orner. p atm θ 0 ν Sine the flow is isentropi, we an use isentropi relations to find the pressure downstream of the orner: p p ---- o ( ) p p p o Thus, p atm. p Solution: The solution tehnique is to onsider that the flow originated at ah where the Prandtl-eyer angle was 0. The flow must then have been turned through ν 6.38 in order to reah ah. ref ν ref 0 p atm ν 6.38 θ 0 47