Fluid supersonic flow

More recently, shock waves and other details associated with flows at speeds greater than that of sound in the fluid (supersonic flows) have been introduced. Still more recently, the interaction of magnetic fields with the flow of charged particles has been extensively studied and formulated into a body of theory known as magnetohydrodynamics. In connection with the space effort and the performance of vacuum systems, it became necessary to study the flow of fluids at such low pressures that the mean free path of molecular motion plays an important role. At the other extreme, the flow of fluids must be studied under such high pressures that densification sufficient to cause solid-like behavior results. This latter area is still to be explored. 

It will now be shown from purely thermodynamic considerations that for, adiabatic conditions, supersonic flow cannot develop in a pipe of constant cross-sectional area because the fluid is in a condition of maximum entropy when flowing at the sonic velocity. The condition of the gas at any point in the pipe where the pressure is P is given by the equations ... 

For the delivery of atomization gas, different types of nozzles have been employed, such as straight, converging, and converging-diverging nozzles. Two major types of atomizers, i.e., free-fall and close-coupled atomizers, have been used, in which gas flows may be subsonic, sonic, or supersonic, depending on process parameters and gas nozzle designs. In sonic or supersonic flows, the mass flow rate of atomization gas can be calculated with the following equation based on the compressible fluid dynamics ... 

During recent years experimental work continued actively upon the macroscopic aspects of thermal transfer. Much work has been done with fluidized beds. Jakob (D5, J2) made some progress in an attempt to correlate the thermal transport to fluidized beds with transfer to plane surfaces. This contribution supplements work by Bartholomew (B3) and Wamsley (Wl) upon fluidized beds and by Schuler (S10) upon transport in fixed-bed reactors. The influence of thermal convection upon laminar boundary layers and their transition to turbulent boundary layers was considered by Merk and Prins (M5). Monaghan (M7) made available a useful approach to the estimation of thermal transport associated with the supersonic flow of a compressible fluid. Monaghan s approximation of Crocco s more general solution (C9) of the momentum and thermal transport in laminar compressible boundary flow permits a rather satisfactory evaluation of the transport from supersonic compressible flow without the need for a detailed iterative solution of the boundary transport for each specific situation. None of these references bears directly on the problem of turbulence in thermal transport and for that reason they have not been treated in detail. 

For low-speed flow of gases viscous dissipation is rarely important. However, in highspeed flows, where the velocities increase toward the sound speed, and in supersonic flows, viscous dissipation is important. Also for the flow of high viscosity fluids, like oils in a journal bearing, viscous dissipation must be considered. 

Chang, S. S.-H. (1975). Nonequilibrium Phenomena in Dusty Supersonic Flow Past Blunt Bodies of Revolution. Phys. Fluids, 18,446. 

Equations (7.14), (7.15), and (7.20), combined with the relations between the thermodynamic properties at constant entropy, determine how the velocity varies with cross-sectional area of the nozzle. The variety of results for compressible fluids (e.g., gases), depends in part on whether the velocity is below or above the speed of sound in the fluid. For subsonic flow in a converging nozzle, the velocity increases and pressure decreases as the cross-sectional area diminishes. In a diverging nozzle with supersonic flow, the area increases, but still the velocity increases and the pressure decreases. The various cases are summarized elsewhere.t We limit the rest of this treatment of nozzles to application of the equations to a few specific cases. 

Porter, D.H., A. Pouquet, and P. R. Woodward. 1994. Kolmogorov-like spectra in decaying three-dimensional supersonic flows. J. Physics Fluids 6 2133-42. 

Ludwig Prandtl (1875—1953) was Professor for Applied Mechanics at the University of Gottingen from 1904 until his death. He was also Director of the Kaiser-Wilhelm-Institut for Fluid Mechanics from 1925. His boundary layer theory, and work on turbulent flow, wing theory and supersonic flow are fundamental contributions to modem fluid mechanics. 

V and T decrease but P and p will increase with distance down the pipe. That is, a flow that is initially subsonic will approach (as a limit) sonic flow as L increases, while an initially supersonic flow will also approach sonic flow. Thus all flows, regardless of their starting conditions, will tend toward the speed of sound as the gas progresses down a uniform pipe. The only way that a subsonic flow can be transformed into a supersonic flow is through a converging-diverging nozzle, where the speed of sound is reached at the nozzle throat. Supersonic flow relations are found in many fluid mechanics books (e.g.. Hall, 1951 Shapiro, 1953). 

Drag coefficients for compressible fluids increase with increase in the Mach number when the latter becomes more than about 0.6. Coefficients in supersonic flow are generally greater than in subsonic flow. 

Shock waves occur in nature in the air surrounding explosions (the shock wave causes much of the destruction of buildings, etc., in an atomic bomb blast) and in the sudden closing of a valve in a duct with high-velocity flow. Sonic booms are shock waves. Shock waves can be produced in the laboratory in a duct or nozzle with supersonic flow. In such cases the shock wave will stand still in one place while the fluid flows through it. The latter is the easier to analyze mathematically, so we use it as a basis for calculations. The nomenclature for a shock wave is shown in Fig 8.13. 

The theory of detonation is applied to the liquid-vapor phase transition in superheated fluids. It is shown that such detonations are always we detonations, characterized by supersonic flow of the shocked region. The detonation state is therefore determined by the transport properties— the viscosity and reaction rate— rather than by the boundary conditions. A numerical example is presented using the Van der Waals equation of state with parameters api opriate for water superheated approximately 100 degrees at a pressure of 5 bars. Detonation pressures of the order of 100 bars and explosion energies of the order of 10 J/Kg are predicted for this example. 

Supersonic flow Gas-liquid mass transfer coefficient Fluid-bed reactor capacity... 

Computational fluid dynamics (CFD) is the numerical simulation of fluid motion. While the motion of fluids in mixing is an obvious application of CFD, there are hundreds of others, ranging from blood flow through arteries, to supersonic flow over an airfoil, to the extrusion of rubber in the manufacture of automotive parts. Numerous models and solution techniques have been developed over the years to help describe a wide variety of fluid motion. In this section, the fundamental equations for fluid flow are presented. 

Witze Am. In.st. Aeronaut. Astronaut. J., 12, 417-418 [1974]) gives equations for the centerline velocity decay of different types of subsonic and supersonic circular free jets. Entrainment of surrounding fluid in the region of flow establishment is lower than in the region of estabhshed flow (see Hill, J. Fluid Mech., 51, 773-779 [1972]). Data of Donald and Singer (T/V7/1.S. In.st. Chem. Eng. [London], 37, 255-267... 

The temperature or enthalpy of the gas may then be plotted to a base of entropy to give a Fanno line.iA This line shows the condition of the fluid as it flows along the pipe. If the velocity at entrance is subsonic (the normal condition), then the enthalpy will decrease along the pipe and the velocity will increase until sonic velocity is reached. If the flow is supersonic at the entrance, the velocity will decrease along the duct until it becomes sonic. The entropy has a maximum value corresponding to sonic velocity as shown in Figure 4.11. (Mach number Ma < 1 represents sub-sonic conditions Ma > 1 supersonic.)... 

S2 — S is positive when Ma 1 > 1. Thus a normal shock wave can occur only when the flow is supersonic. From equation 4.96, if Mai > 1, then Ma2 < 1, and therefore the flow necessarily changes from supersonic to subsonic. If Mai = 1> Mai — 1 also, from equation 4.96, and no change therefore takes place. It should be noted that there is no change in the energy of the fluid as it passes through a shock wave, though the entropy increases and therefore the change is irreversible. 

The energy of fast fluid flow can be utihzed to intensify processes in chemical reactors and there are two basic ways of doing it by purposefully creating the cavitation conditions in the reacting liquid or by using a supersonic shockwave for fine phase dispersion. 

To understand the difference in stagnation pressure losses between subsonic and supersonic combustion one must consider sonic conditions in isoergic and isentropic flows that is, one must deal with, as is done in fluid mechanics, the Fanno and Rayleigh lines. Following an early NACA report for these conditions, since the mass flow rate (puA) must remain constant, then for a constant area duct the momentum equation takes the form... 

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