Supersonic Flow And Shock Waves Courant Pdf Free !!TOP!!
Abstract:This work studies the impact of a shock wave traveling with non-constant velocity over straight surfaces, generating an unsteady and complex reflection process. Two types of shock waves generated by sudden energy released are studied: cylindrical and spherical. Several numerical tests were developed considering different distances between the shock wave origin and the reflecting surface. The Kurganov, Noelle, and Petrova (KNP) scheme implemented in the rhoCentralFoam solver of the OpenFOAMTM software is used to reproduce the different shock wave reflections and their transitions. The numerical simulations of the reflected angle, Mach number of the shock wave, and position of the triple point are compared with pseudo-steady theory numerical and experimental studies. The numerical results show good accuracy for the reflected angle and minor differences for the Mach number. However, the triple point position is more difficult to predict. The KNP scheme in the form used in this work demonstrates the ability to capture the phenomena involved in the unsteady reflections.Keywords: shock wave; reflection; OpenFOAM
supersonic flow and shock waves courant pdf free
ical attack. The first part concerns flow in nozzles and jets, atopicwith increasingly important applications in many fields, e.g.,rocketand jet propulsion. The second part is concerned with flowagainst con-ical obstacles such as projectiles, and gives anintegrated summary ofsome work by Taylor and by Busemann. Theproblem of spherical waves,
c of the quiet gas. If the piston moves into the gas thesituationmay become more complicated through the emergence of asupersonicdiscontinuous shock wave as we shall see in Section C. Atany rate,in Sections A and B we confine ourselves to aconsideration ofcontinuous motions satisfying the differentialequations (at leastnear the piston). Such a continuous motion ofthe gas can becompletely determined by the simple wave theory ofChapter II, Art. 16,
A tempting try would be to differentiate first equation (2.3) to obtain an equation for v, then use the irrotationality to solve u (once v is solved), and finally use (2.4) to solve the density ρ. In order to show the equivalence between these equations and the original potential flow equation (2.5), an additional one-point boundary condition is required for v. However, it is unclear how the boundary condition is to be deduced for v for the problem. Moreover, along the boundaries Γshock and Γ2wedge which meet at the corner, the derivative boundary conditions of the deduced second-order elliptic equation to v are second-kind boundary conditions, i.e. without the viscosity, compared to [26]. This implies that the results from [35,36] could not be directly used. To overcome this, the following directional velocity (w,z) is introduced whose relation with (u,v) is
Given , we obtain (u0,v0) and ρ0 by using the shock polar curve in figure 8 for steady potential flow. In figure 8, θsw is the wedge angle such that line intersects with the shock polar curve at a point on the circle of radius , and θdw is the wedge angle such that line is tangential to the shock polar curve and there is no intersection between line and the shock polar curve when .
For any wedge angle , line and the shock polar curve intersect at a point (u0,v0) with and ; whereas, for any wedge angle , they intersect at a point (u0,v0) with u0>ud and . The intersection state (u0,v0) is the velocity for steady potential flow behind an oblique shock S0 attached to the wedge tip with angle θw. The strength of shock S0 is relatively weak compared with the other shock given by the other intersection point on the shock polar curve. Thus, S0 is called a weak shock and the corresponding state (u0,v0) is a weak state.
The importance of the potential flow equation (1.1) with (1.2) in the time-dependent Euler flows even through weak discontinuities was also observed by Hadamard [51] through a different argument. Moreover, for the solutions containing a weak shock, the potential flow equation (1.1) and (1.2) and the full Euler flow model (5.1) match each other well up to the third order of the shock strength. Also see Bers [52], Glimm & Majda [2] and Morawetz [22].
It is shown experimentally that, in steady flow, transition to Mach reflection occurs at the von Neumann condition in the strong shock range (Mach numbers from 2.8 to 5). This criterion applies with both increasing and decreasing shock angle, so that the hysteresis effect predicted by Hornung, Oertel & Sandeman (1979) could not be observed. However, evidence of the effect is shown to be displayed in an unsteady experiment of Henderson & Lozzi (1979).
Unlike solitons (another kind of nonlinear wave), the energy and speed of a shock wave alone dissipates relatively quickly with distance. When a shock wave passes through matter, energy is preserved but entropy increases. This change in the matter's properties manifests itself as a decrease in the energy which can be extracted as work, and as a drag force on supersonic objects; shock waves are strongly irreversible processes.
The abruptness of change in the features of the medium, that characterize shock waves, can be viewed as a phase transition: the pressure-time diagram of a supersonic object propagating shows how the transition induced by a shock wave is analogous to a dynamic phase transition.
Shock waves are formed when a pressure front moves at supersonic speeds and pushes on the surrounding air.[9] At the region where this occurs, sound waves travelling against the flow reach a point where they cannot travel any further upstream and the pressure progressively builds in that region; a high pressure shock wave rapidly forms.
Shock waves are not conventional sound waves; a shock wave takes the form of a very sharp change in the gas properties. Shock waves in air are heard as a loud "crack" or "snap" noise. Over longer distances, a shock wave can change from a nonlinear wave into a linear wave, degenerating into a conventional sound wave as it heats the air and loses energy. The sound wave is heard as the familiar "thud" or "thump" of a sonic boom, commonly created by the supersonic flight of aircraft.
When analyzing shock waves in a flow field, which are still attached to the body, the shock wave which is deviating at some arbitrary angle from the flow direction is termed oblique shock. These shocks require a component vector analysis of the flow; doing so allows for the treatment of the flow in an orthogonal direction to the oblique shock as a normal shock.
When an oblique shock is likely to form at an angle which cannot remain on the surface, a nonlinear phenomenon arises where the shock wave will form a continuous pattern around the body. These are termed bow shocks. In these cases, the 1d flow model is not valid and further analysis is needed to predict the pressure forces which are exerted on the surface.
Shock waves can form due to steepening of ordinary waves. The best-known example of this phenomenon is ocean waves that form breakers on the shore. In shallow water, the speed of surface waves is dependent on the depth of the water. An incoming ocean wave has a slightly higher wave speed near the crest of each wave than near the troughs between waves, because the wave height is not infinitesimal compared to the depth of the water. The crests overtake the troughs until the leading edge of the wave forms a vertical face and spills over to form a turbulent shock (a breaker) that dissipates the wave's energy as sound and heat.
Similar phenomena affect strong sound waves in gas or plasma, due to the dependence of the sound speed on temperature and pressure. Strong waves heat the medium near each pressure front, due to adiabatic compression of the air itself, so that high pressure fronts outrun the corresponding pressure troughs. There is a theory that the sound pressure levels in brass instruments such as the trombone become high enough for steepening to occur, forming an essential part of the bright timbre of the instruments.[10] While shock formation by this process does not normally happen to unenclosed sound waves in Earth's atmosphere, it is thought to be one mechanism by which the solar chromosphere and corona are heated, via waves that propagate up from the solar interior.
To produce a shock wave, an object in a given medium (such as air or water) must travel faster than the local speed of sound. In the case of an aircraft travelling at high subsonic speed, regions of air around the aircraft may be travelling at exactly the speed of sound, so that the sound waves leaving the aircraft pile up on one another, similar to a traffic jam on a motorway. When a shock wave forms, the local air pressure increases and then spreads out sideways. Because of this amplification effect, a shock wave can be very intense, more like an explosion when heard at a distance (not coincidentally, since explosions create shock waves).
Shock waves can also occur in rapid flows of dense granular materials down inclined channels or slopes. Strong shocks in rapid dense granular flows can be studied theoretically and analyzed to compare with experimental data. Consider a configuration in which the rapidly moving material down the chute impinges on an obstruction wall erected perpendicular at the end of a long and steep channel. Impact leads to a sudden change in the flow regime from a fast moving supercritical thin layer to a stagnant thick heap. This flow configuration is particularly interesting because it is analogous to some hydraulic and aerodynamic situations associated with flow regime changes from supercritical to subcritical flows.
Astrophysical environments feature many different types of shock waves. Some common examples are supernovae shock waves or blast waves travelling through the interstellar medium, the bow shock caused by the Earth's magnetic field colliding with the solar wind and shock waves caused by galaxies colliding with each other. Another interesting type of shock in astrophysics is the quasi-steady reverse shock or termination shock that terminates the ultra relativistic wind from young pulsars.