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equal. However, if the flow is isentropic, total pressure is constant along streamlines. Density Analogous to the total pressure, the total density in a compressible flow is given by (12.28) where is the local or static density. If the flow is isentropic, then t is a constant along streamlines and Eq. (12.28) can be used to determine the variation of gas density with the Mach number. In literature dealing with compressible flows, one often finds reference to stagnation conditions that is, stagnation temperature and stagnation pressure. By definition, stagnation refers to the conditions that exist at a point in the flow where the velocity is zero, regardless of whether or not the zero velocity has been achieved by an adiabatic, or reversible, process. For example, if one were to insert a Pitot-static tube into a compressible flow, strictly speaking one would measure stagnation pressure, not total pressure, since the deceleration of the flow would not be reversible. In practice, however, the difference between stagnation and total pressure is insignificant. Kinetic pressure The kinetic pressure, q = V2/2, is often used, as seen in Chapter 11, to calculate aerodynamic forces with the use of appropriate coefficients. It can also be related to the Mach number. Using the ideal gas law to replace gives (12.29) Then using the equation for the speed of sound, Eq. (12.11), results in (12.30) where p must always be an absolute pressure since it derives from the ideal gas law. The use of the equation for kinetic pressure to evaluate the drag force is shown in Example 12.4. The Bernoulli equation is not valid for compressible flows. Consider what would happen if one decided to measure the Mach number of a high-speed air flow with a Pitot-static tube, assuming that the Bernoulli equation was valid. Assume a total pressure of 180 kPa and a static pressure of 100 kPa were measured. By the Bernoulli equation, the kinetic pressure is equal to the difference between the total and static pressures, so Solving for the Mach number, and substituting in the measured values, one obtains 4 of 6 1/15/2009 1:14 AM Mach Number Relationships http://edugen.wiley.com/edugen/courses/crs2436/crowe9771/crowe9771... EXAMPLE 12.4 DRAG FORCE O A SPHERE The drag coefficient for a sphere at a Mach number of 0.7 is 0.95. Determine the drag force on a sphere 10 mm in diameter in air if p = 101 kPa. Problem Definition Situation: A sphere is moving at a Mach number of 0.7 in air. Find: The drag force (in newtons) on the sphere. Properties: From Table A.2, kair = 1.4. Plan The drag force on a sphere is FD = q CD A. 1. Calculate the kinetic pressure q from Eq. (12.30). 2. Calculate the drag force. Solution 1. Kinetic pressure 2. Drag force: The correct approach is to relate the total and static pressures in a compressible flow using Eq. (12.26). Solving that equation for the Mach number gives (12.31) and substituting in the measured values yields Thus applying the Bernoulli equation would have led one to say that the flow was supersonic, whereas the flow was actually subsonic. In the limit of low velocities (pt/p ! 1), Eq. (12.31) reduces to the expression derived using the Bernoulli equation, which is indeed valid for very low (M 1) Mach numbers. It is instructive to see how the pressure coefficient at the stagnation (total pressure) condition varies with Mach 5 of 6 1/15/2009 1:14 AM Mach Number Relationships http://edugen.wiley.com/edugen/courses/crs2436/crowe9771/crowe9771... number. The pressure coefficient is defined by Using Eq. (12.30) for the kinetic pressure enables one to express Cp as a function of the Mach number and the ratio of specific heats. The variation of Cp with Mach number is shown in Fig. 12.7. At a Mach number of zero, the pressure coefficient is unity, which corresponds to incompressible flow. The pressure coefficient begins to depart significantly from unity at a number of about 0.3. From this observation it is inferred that compressibility effects in the flow field are unimportant for Mach numbers less than 0.3. Figure 12.7 Variation of the pressure coefficient with Mach number. Copyright 2009 John Wiley & Sons, Inc. All rights reserved. 6 of 6 1/15/2009 1:14 AM [ Pobierz całość w formacie PDF ] |