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Compressible Flow

الكلية كلية الهندسة     القسم  الهندسة البيئية     المرحلة 3
أستاذ المادة عدي عدنان جهاد الخيكاني       05/10/2012 11:54:55
Introduction
Motivation. All eight of our previous chapters have been concerned with “low-speed’’ or “incompressible’’ flow, i.e., where the fluid velocity is much less than its speed of sound. In fact, we did not even develop an expression for the speed of sound of a fluid. That is done in this chapter.
When a fluid moves at speeds comparable to its speed of sound, density changes become significant and the flow is termed compressible. Such flows are difficult to obtain in liquids, since high pressures of order 1000 atm are needed to generate sonic velocities. In gases, however, a pressure ratio of only 2_1 will likely cause sonic flow. Thus compressible gas flow is quite common, and this subject is often called gas dynamics. Probably the two most important and distinctive effects of compressibility on flow are
(1) choking, wherein the duct flow rate is sharply limited by the sonic condition,
(2) shock waves, which are nearly discontinuous property changes in a supersonic flow.
The purpose of this chapter is to explain such striking phenomena and to familiarize the reader with engineering calculations of compressible flow.
Speaking of calculations, the present chapter is made to order for the Engineering Equation Solver (EES) in App. E. Compressible-flow analysis is filled with scores of complicated algebraic equations, most of which are very difficult to manipulate or invert. Consequently, for nearly a century, compressible-flow textbooks have relied upon extensive tables of Mach number relations (see App. B) for numerical work. With EES, however, any set of equations in this chapter can be typed out and solved for any variable— see part (b) of Example 9.13 for an especially intricate example. With such a tool, App. B serves only as a backup and indeed may soon vanish from textbooks. to see when we might safely neglect the compressibility inherent in every real fluid. We found that the proper criterion for a nearly incompressible flow was a small Mach number

Ma = V /a <1
where V is the flow velocity and a is the speed of sound of the fluid. Under small-Machnumber conditions, changes in fluid density are everywhere small in the flow field. The energy equation becomes uncoupled, and temperature effects can be either ignored or put aside for later study. The equation of state degenerates into the simple statement that
density is nearly constant. This means that an incompressible flow requires only a momentum and continuity analysis, as we showed with many examples later . This part treats compressible flows, which have Mach numbers greater than about 0.3 and thus exhibit no negligible density changes. If the density change is significant, it follows from the equation of state that the temperature and pressure changes are also substantial. Large temperature changes imply that the energy equation can no longer be neglected. Therefore the work is doubled from two basic equations to four
1. Continuity equation
2. Momentum equation
3. Energy equation
4. Equation of state

to be solved simultaneously for four unknowns: pressure, density, temperature, and flow velocity (p, _, T, V). Thus the general theory of compressible flow is quite complicated, and we try here to make further simplifications, especially by assuming a reversible adiabatic or isentropic flow.

The Mach Number
The Mach number is the dominant parameter in compressible-flow analysis, with different effects depending upon its magnitude. Aerodynamicists especially make a distinction between the various ranges of Mach number, and the following rough classifications are commonly used:
Ma < 0.3: incompressible flow, where density effects are negligible.
0.3 < Ma <0.8: subsonic flow, where density effects are important but no
shock waves appear


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