Local speed of sound

Compressible Vlow. The flow of easily compressible fluids, ie, gases, exhibits features not evident in the flow of substantially incompressible fluid, ie, Hquids. These differences arise because of the ease with which gas velocities can be brought to or beyond the speed of sound and the substantial reversible exchange possible between kinetic energy and internal energy. The Mach number, the ratio of the gas velocity to the local speed of sound, plays a central role in describing such flows. 

Most often, the Mach number is calculated using the speed of sound evaluated at the local pressure and temperature. When M = 1, the flow is critical or sonic and the velocity equals the local speed of sound. For subsonic flowM < 1 while supersonic flows have M > 1. Compressibility effects are important when the Mach number exceeds 0.1 to 0.2. A common error is to assume that compressibihty effects are always negligible when the Mach number is small. The proper assessment of whether compressibihty is important should be based on relative density changes, not on Mach number. 

A detonation shock wave is an abrupt gas dynamic discontinuity across which properties such as gas pressure, density, temperature, and local flow velocities change discontinnonsly. Shockwaves are always characterized by the observation that the wave travels with a velocity that is faster than the local speed of sound in the undisturbed mixtnre ahead of the wave front. The ratio of the wave velocity to the speed of sound is called the Mach number. 

Cp and Q are the specific heats, assumed here to be constant for simplicity q is the heat release rate per unit volume X is the thermal conductivity c is the local speed of sound... 

In accordance with the usual convention, we define a detonation as a reaction traveUing faster than the local speed of sound in the unreacted medium, and a deflagration as being a reaction travelling at or slower than the local speed of sound in the unreacted medium. An example of each type of event is given below ... 

A critical characteristic of energetic materials is the Chapman-Jouguet (CJ) state. This describes the chemical equilibrium of the products at the end of the reaction zone of the detonation wave before the isentropic expansion. In the classical ZePdovich-Neumann-Doring (ZND) detonation model, the detonation wave propagates at constant velocity. This velocity is the same as at the CJ point which characterizes the state of reaction products in which the local speed of sound decreases to the detonation velocity as the product gases expand. 

Chapter 5 presented a mathematical model for adiabatic flow through a frictionless nozzle, and we shall use some of that chapter s results. It will be helpful for our present purpose if we derive the Mach number at station 2 , where the Mach number, M, is defined as the ratio of the velocity to the local speed of sound. Recalling equation (5.47), the Mach number at station 2 will be ... 

We will now develop an expression for the sonic speed experienced in the throat/outlet of a convergent-only nozzle and at the throat of a convergent-divergent nozzle when the expansion is frictionally resisted. Sonic conditions will exist in the throat when the velocity calculated by applying equation (14.45) to the convergent section of the nozzle has reached the local speed of sound, i.e. ... 

With this definition all the steady-state drag data on single, smooth spheres moving in infinite , quiescent, newtonian fluids at moderate velocities can be represented by aj single curve on Fig. 6.22. This figure shows also drag coefficients for disks and cylinders, to be discussed later. It is limited to steady velocities of less than about one half the local speed of sound velocities higher than this are discussed elsewhere [18]. 

Someone slows down and is hit by a faster-moving car from behind. The first crash produces a pile of stopped, wrecked cars. This pile then enlarges in the upstream direction jas more and more cars pile into the stopped wreckage.) The rate of propagation of the boundary between stopped and moving fluid (assuming rigid pipe walls) is the local speed of sound. That is not proved here, but may seem clearer after we have discussed the speed of sound in Chap. 8. From Chap. 8 we lean borrow the fact that for water the speed of sound is about c = 5000 ft/s j(1520 m/s), so that the stopped layer of water will reach the reservoir in r =L/c = (3000 m)/(1520 m/s), or about 2 s after the valve is closed. 

But, as shown previously, RkT IM is the square of the speed of sound at state 1, or Cl, so the left side is (1/,/c,). The ratio V/c is called the Mach number M in honor of the Austrian physicist Ernst Mach. This ratio plays a crucial role in the study of high-velocity gas flows (and is widely reported in the press describing the speed of supersonic aircraft). It is the ratio of the local flow velocity to the local speed of sound. For subsonic flows M is less than 1 for sonic flows it equals 1 for supersonic flows it is greater than 1. Making this definition, we can rearrange Eq. 8.15 to... 

Several columns in the second part of App. A.5 apply to normal shock waves, which wej discuss in Sec. 8.5. The table is based on Eqs. 8.16, 8.17, 8.18, and 8.21. In addition, there is a column labeled V/c (the derivation of the equation forjthis ratio is in App. F, Eq. F.12). This ratio is useful in a problem in which we know the velocity and the reservoir conditions and wish to find M. We could solve for the local temperature and local speed of sound, as we did in Example 8.6, but that is tedious. We would like some velocity ratio like V/V toj appear in the table, but that is an impossible choice, because Vfi is zero. The logical choice is VIV = V/c its use is illustrated below. 

Speed of observer s shout = local speed of sound... 

A normal shock wave is a large pressure disturbance which travels faster than the local speed of sound. Normal shock waves are irreversible, causing an increase in -entropy of the fluid flowing through them. 

The product mixture which exists in a particular plane in the reaction zone behind the detonation front obeys the Chapman-Jouguet (C-J) hypothesis. In essence, the C-J or sonic plane differentiates between the part of the reaction zone where the detonative decomposition is completed (and exothermic reactions supply energy with the local speed of sound to the detonation front) and that part in which further energy release due to reactions among the products is not supplied sufficiently rapidly to maintain steady wave propagation. 

In these equations, the q subscript denotes source conditions, M is the Mach number, 7 is the ratio of specific heats (X= 5/3 for a rare gas), a is the local speed of sound, and v is the flow velocity. The definition of a is... 

A dimensionless number defined by the ratio of the local flow velocity to the local speed of sound. A flow with a Mach number exceeding unity is termed supersonic for Mach numbers less than unity the flow is subsonic. ... 

A finite wave may be thought of as a succession of infinitesimal pressure pulses. When the velocity and temperature gradients are not too steep, viscous and heat conduction effects are negligible. Thus, the wave is isentropic, and each elementary part of the wave travels at the local speed of sound with respect to the fluid in which it is propagating. The change in the propagation velocity of a part of the wave with respect to pressure, in a fixed reference frame is given as,... 

The controlling feature of the entire cooldown process is the behaviour of the ambient temperature vapour, as it accelerates down the uncooled portion of pipeline until it reaches the local speed of sound S near the exit. A shockwave develops and all the remaining pressure drop along the pipeline is consumed by increasing the intensity of the shock wave and not by increasing the mass flow. 

This flow, or Fanno flow, is well-known to blow-down wind-tunnel operators, in which the maximum mass flow is determined only by the local speed of sound S at ambient temperature, the exit pressure and the cross-sectional area of the exit (see, e.g. [4]). 

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