Compressible Fluid

Despite the existence of large pressure, fluids undergo very little change in density. Water needs a pressure change of 20,000kPa to have 1 percent change in its density. Fluids of constant density are termed incompressible fluids and assumed during computation the density is constant. A flowing fluid is said to be compressible when appreciable density changes are brought about the motion. The variation of density is usually accompanied by temperature changes as well as heat transfer. The variation of density means that a group of fluid elements can spread out into a larger region of space without requiring a simultaneous shift to be made of all fluid elements in the fluid. In compressible media a small shift of fluid element will induce similar small movements in adjacent elements and by this a disturbance called an acoustic wave propagates at a relatively high speed through the medium. In the incompressible flows, these propagation have infinite speed which adjustment (disturbance) took place simultaneously through the entire flow and there are now wave to be considered. Thus, compressibility means admission of elastic waves having finite velocity.

Up to now, we needed four scalar equations (the equation of continuity and three components of the equation of linear momentum) to describe fully a flow field. For compressible flow, the density and pressure changes are also accompanied by temperature changes. Thus we need equation of energy and equation of state(p = p(p,T)). The general theory of compressible flow is very complicated, not only because of large number of the equations involved, but also because of the wave propagation phenomena that are predominant at flow speed higher than the speed of sound.

The speed of sound is defined as the rate of propagation of an infinitesimal pressure disturbance(wave) through a continuous medium. Sound is the propagation of compressible an expansion wave of finite but small amplitude such that the ear can detect them. Frequencies range from 20 to 20, 000 Hertz while the magnitude is typically less than 10 Pa.

Consider a long tube filled with motionless fluid and having a piston at one end, as shown in figure 2.7. By tapping the piston we may cause a pressure increase dp on the right of A — A. Now two things will happen. Due to molecular action, the pressure will increase to the left of A — A and this increase in pressure will move in the tube at high speed C. We thus have a pressure wave of speed V moving to the left due to microscopic action. The second effect is on the macroscopic level. According to Newton's second law, the fluid just to the left of wavefront described above must accelerate as a result of the pressure difference dp to a velocity dv. Once the pressure rise dp has been established in the fluid, there is no further change in velocity, so it remains at dv. Behind the wavefront, the fluid is thus moving to the left at speed dv. During an interval of dt the wave has progressed a distance Cdt and is shown at position B in figure 2.7. Meanwhile, fluid particle at A move a distance dvdt to position A1. At an intermediate position, such as halfway between A and B , shown in diagram as D, the fluid velocity dv has persisted for a time interval d . Consequently, fluid initially at D has moved a distance to position Di.

DI D

Al A

dvdt/2

cMett dV

Figure 2.7: Wave front movement and fluid movement

By tapping, as infinitesimal pressure disturbance which will move down the tube at a constant speed, the fluid behind the wave is slightly compressed, while the fluid ahead of the wave remains undisturbed. This is an unsteady state problem. However, if we assume an infinitesimal control volume around the wave, travelling with the wave velocity C, we can apply a steady state analysis. The wave is thus stationary while the fluid flows with an approach velocity C, as shown in figure 2.8. Neglecting friction effects, the velocity profile can be assumed flat. The continuity equation may be written as:

pAC = (p + dp)(C - dv)A Neglecting infinitesimal quantities of higher order (dpdv) gives:

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