Local sound speed

The radius and velocity of the bubble waU are given by R and U respectively. The values for H, the enthalpy at the bubble waU, and C, the local sound speed, maybe expressed as foUows, using the Tait equation of state for the Hquid. 

For a shock wave in a solid, the analogous picture is shown schematically in Fig. 2.6(a). Consider a compression wave on which there are two small compressional disturbances, one ahead of the other. The first wavelet moves with respect to its surroundings at the local sound speed of Aj, which depends on the pressure at that point. Since the medium through which it is propagating is moving with respect to stationary coordinates at a particle velocity Uj, the actual speed of the disturbance in the laboratory reference frame is Aj - -Ui- Similarly, the second disturbance advances at fl2 + 2- Thus the second wavelet overtakes the first, since both sound speed and particle velocity increase with pressure. Just as a shallow water wave steepens, so does the shock. Unlike the surf, a shock wave is not subject to gravitational instabilities, so there is no way for it to overturn. 

In materials that support shock waves, the sound speed increases with pressure. It is this same property that causes rarefactions to spread out as they progress. In Fig 2.6(b), an unloading wave is shown propagating into a stationary material with some initial pressure Pq. This time, we consider the evolution of two small decompressional disturbances. The first disturbance moves at the local sound speed of a, into its surroundings, which have begun... 

When a premixed gas-oxidant cloud is ignited, the flame can propagate in two different modes through the gas mixture—deflagration and detonation. Deflagrations propagate at subsonic speeds relative to the unburned mixture and the heat and mass are transported by conduction, diffusion, and convection. Gas mixture detonations propagate at speeds faster than the local sound speed of the unburned gas. In a gas mixture detonation, a shock wave is sustained... 

Conditions for explosive burning can only occur if the core collapses at a speed vs, the local sound speed. In a star with an iron core, vs 107ms 1, which implies a collapse timescale < 3 s - essentially a dynamical timescale. Any slower and the star can reorganize itself without an explosion, so a lot of energy ( 1014 Jkg-1) must be removed rapidly. 

Prior knowledge of the local sound speeds is not required when beginning a simulation. If compression occurs at state A, then > c. If compression stops at state B, then , +c, > v. Note that as a consequence of the instability at point A of Eq. (12), runaway expansion on the tensile strain side of state A is also a valid solution of the steady state Euler equations. Such an expansion solution may have physical significance if there exists a larger volume where... 

Another practical issue associated with the use of this simulation technique is biasing the instability of the starting point. As discussed in the section on stability, as long as the shock speed exceeds the local sound speed, the volume equation of motion Eq. (16) can either force compression or expansion of the volume. While both of these steady solutions can potentially have physical significance, the solutions we focus on in this chapter are the compressive, shock-like solutions. Therefore some technique is required for biasing the initial instability so that only compression occurs. Note that this is simply a selection of the particular type of steady solution to be simulated (compressive shock versus expansion shock) and does not represent nor require an empirical parameter or extra degree of freedom. 

Choked flow also called critical flow is defined in single-phase flow as the flow when the fluid Mach number which is the ratio between the local fluid velocity and the local sound speed in the fluid approaches unity. For compressible single-phase flow or for gas-liquid two-phase flow when Mach number equal to one, the pressure gradient asymptotically... 

The following typical combustion velocities have been observed the turbulent deflagration mode with a velocity of some dozens of meters per second in lean mixtures the sound deflagration mode, high-speed deflagration with 800-1,000 m/ s velocities, when the combustion front moves with the local sound speed relative to the reaction products the quasi-detonation mode when the velocity spectra exceeds 1,100 m/s, but is 200-500 m/s less than the CJ detonation velocity. The quasi-detonation mode velocity deficit, in comparison with thermodynamic defined values, is explained by impulse losses due to interactions with walls and obstacles. 

Sound point/surface A flow zone, where the flow velocity is equal to a local sound speed. In case of a detonation -it is a surface where the sum of a local sound speed and combustion products velocities is equal to a leading wave velocity. In the case of an ideal detonation it is the end of the reaction zone. 

Now the assumption is dropped that the chemical reaction is a rate-controlled conversion to an invariant product composition, and the composition is permitted to vary with local thermodynamic state. Zel dovich, Brinkley Si Richardson, and Kirkwood Wood pointed out that since in a chemically reactive wave, pressure is a function not only of density and entropy but also of chemical composition, the sound speed for a reacting material should be defined as the frozen sound speed... 

If we combine the above-mentioned information then we are able to calculate the effective speed of sound in a bubbly liquid and the damping coefficient as a function of the local sound pressure. These pieces of information can directly be coupled to the sound-field equation and may be calculated by any three-dimensional code capable of solving the wave equation for the sound pressure with variable coefficients. 

Figure 8.1.14 Effective speed of sound c = cu/Re in a bubbly liquid as a function of local sound pressure for water with air bubbles at ambient temperature and ambient static pressure.

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. 

There are certain limitations on the range of usefulness of pitot tubes. With gases, the differential is very small at low velocities e.g., at 4.6 m/s (15.1 ft/s) the differential is only about 1.30 mm (0.051 in) of water (20°C) for air at 1 atm (20°C), which represents a lower hmit for 1 percent error even when one uses a micromanometer with a precision of 0.0254 mm (0.001 in) of water. Equation does not apply for Mach numbers greater than 0.7 because of the interference of shock waves. For supersonic flow, local Mac-h numbers can be calculated from a knowledge of the dynamic and true static pressures. The free stream Mach number (MJ) is defined as the ratio of the speed of the stream (V ) to the speed of sound in the free stream ... 

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... 

As will be outlined below, the computation of compressible flow is significantly more challenging than the corresponding problem for incompressible flow. In order to reduce the computational effort, within a CED model a fluid medium should be treated as incompressible whenever possible. A rule of thumb often found in the literature and used as a criterion for the incompressibility assumption to be valid is based on the Mach number of the flow. The Mach number is defined as the ratio of the local flow velocity and the speed of sound. The rule states that if the Mach number is below 0.3 in the whole flow domain, the flow may be treated as incompressible [84], In practice, this rule has to be supplemented by a few additional criteria [3], Especially for micro flows it is important to consider also the total pressure drop as a criterion for incompressibility. In a long micro channel the Mach number may be well below 0.3, but owing to the small hydraulic diameter of the channel a large pressure drop may be obtained. A pressure drop of a few atmospheres for a gas flow clearly indicates that compressibility effects should be taken into account. 

For N2 molecules in the air at room temperature cr(d is of the order of the speed of sound, 370 ms-1, a is 0.43 nm2 and Z = 5 x 1028 cm-3 s 1. This is a very large number, which means that collisions between molecules occur very frequently and the energy can be averaged between them, ensuring the concept of local thermal equilibrium. Each molecule collides ZAK/NA times per second, which is about 5 x 109 s x once every 0.2 ns. However, in the diffuse ISM where the molecule density is of order 102 cm3 the collision frequency is 5 x 10-8 s-1 or a collision every 1.5 years. 

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