Theory of Breakdown of an Arbitrary Gas-dynamic Discontinuity-2D Flows Interaction

We have considered the theory of breakdown of an arbitrary gas-dynamic discontinuity for the space-time dimension equal to two. The link of this task with the geometrical theory of reconfiguration of shock-waves and wave fronts is shown. We consider the Riemann problem of the breakdown of an arbitrary discontinuity of parameters at angular collision of two flat flows. The problem is solved as accurate stated. We consider the solution region with different types of the shock-wave structure. The Mach number region is discovered and the angles of flows interaction for which there is no solution. We demonstrate the generality of solutions for one-dimensional non-stationary and two-dimensional stationary cases.


INTRODUCTION
The problem considered here is breakdown of an arbitrary gas-dynamic discontinuity in the space-time with the dimension equal to two. Let's remind the basic concepts and terminology. The zero-order gas-dynamic discontinuity is the area of dramatic, discontinuous changes of the gas-dynamic variables. A non-stationary discontinuity, through which the flow goes, is traditionally called a shock-wave, a stationary one is called as a compression shock. There are discontinuities, through which the gas does not flow and the pressure on its sides is the same, the density and other parameters can differ. Such a discontinuity is called tangential (slip line), if its surface is parallel to the velocity vectors on its sides. In the other cases it is called a contact discontinuity. A simple compression wave or depression wave is the area of smooth variation of parameters, restricted on two sides by the first-order discontinuities where the first-order derivatives of gasdynamic discontinuities jump.
If in the space-time there is a discontinuity of parameters occurred due to for some reasons, at variation of one of the parameters, for example, the time or dimensional coordinate, it transforms (a breakdown happens) into the shock-wave structure composed of a few waves and discontinuities ( Fig. 1). At the same time, from the kinematic point of view, the time and space coordinates are absolutely equal and there is no difference between the breakdown of onedimension non-stationary and two-dimensional stationary discontinuity. In studied in general the problem of an arbitrary discontinuity breakdown (Kotchine, 1927) for polytropic gases. In 1946-1953Landau and Lifshits (1953 carried out more total study of the given problem and in the modern terms it was made by Kobzeva and Moiseev (2003).
An arbitrary discontinuity of the gas-dynamic parameters takes place, for example, at breakthrough of the shock tube diaphragm, at two shock-waves interaction with each other or with the contact discontinuity.

Geometrical model of the discontinuity breakdown:
We would like to remind how the geometrical concept of the gas-dynamic discontinuity and its modifications (breakdown into new discontinuities and waves) is introduced when a parameter is changing.
Let`s consider, for example, a one-dimensional medium of particles moving along a straight line at a constant velocity. The particle free motion law looks as x = ϕ(t) = x 0 +ut, where u is the particle velocity. Function ϕ satisfies the Newton equation: On the other hand, the Euler equation describing the field of noninteracting particles has got the same form: Thereby, the motion descriptions with the help of the Euler equation for the field of gas-dynamic variables and with the help of the Newton particles are equal. We know that differential equations in partial derivatives by means of characteristics construction. equation characteristics are equivalent law for a moving particle (Arnold, 1978 way, the problem of wave propagation can construction of the characteristics along particles move. In the example under they are horizontal lines. Let a distribution u(x) be given (Fig. 2а). When you plot horizontal lines fr particles move at their constant velocity line. Then, at some points of time t = velocity distribution form u(x) changes point of time (t = 2 in Fig. 2b) the reflection Res. J. Appl. Sci. Eng. Technol., 11(2): 127-134, 2015 Equivalence of the problems of an arbitrary discontinuity breakdown; (a): One-dimensional problem of two flows collision with formation in time t of two shock-waves D 1 , 2 and a contact discontinuity K, 1 and 4 are gas undisturbed zones, 2 and 3 are the cocurrent flow caused by a wave areas; (b): One-dimensional problem of pressure equalization with formation of a shockwave D2, simple Riemann wave Rc and contact discontinuity K; c) two-dimensional problem of interaction of two inclined supersonic jets with pressure P 1, 2 and velocity v 1, 2 , with formation of two outgoing jumps ; d) two-dimensional problem of interaction of two inclined supersonic jets with formation Meyer wave ω 1 (b) (c) wave formation; (а): Velocity initial distribution; (b): Solution peculiarity formation; Newton equation for that quasilinear derivatives are solved construction. The Euler equivalent to the Newton 1978). In such a can be solved by along which material under consideration, velocity initial from this curve, velocity along every = t 1 , t 2 … t n the changes (Fig. 2b). At a reflection u(x) ceases to be the function graph, i.e., there are met by some values u. In this condition of particle interaction absence motion through each other, which You need to deduce a model of their example, the model of the Universe (Zeldovich, 1970) covers the Universe the gravitational interaction. Addition conditions leads to peculiarities in the areas where the particle (galaxy) maximal. Such areas (set of critical caustics (Fig. 3). Once appeared, the caustic decompose with formation of new cannot disappear. This model well formation of uneven (cellular) structure dimensional problem of two flows , 1 and 4 are gas undisturbed dimensional problem of pressure equalization with dimensional problem of , with formation of two outgoing jumps σ 1, 2 dimensional problem of interaction of two inclined supersonic jets with formation there are values х which this field the physical absence means their which is not physically. their interaction. For Universe creation offered by Universe expansion and Addition of such the solution, i.e., the alaxy) concentration is critical values) are called caustic can transform, new features, but it well describes the structure of the Universe from the initial sudden fluctuations of the energy density. In the sample at hand, it introduce a model of inelastic collision Then, in the point of this collision appears which means discontinuity of particle movement (Fig. 2c). The primitive particles inelastic collision is the Burgers (Karman and Burgers, 1939) which describes dynamic field in the smooth areas of interaction of gas particles inside the shock At low viscosity it approximates the in the areas of parameter smooth change. wave right and left, the flow is described equations, inside the shock-wave discontinuity) -with an equation similar conduction equation. We do not give here geometrical theory of the reflection specified with hyperbolic equations derivatives, we just mark the remarkable both caustics and shock-waves al transformation are completely mapped. be emphasized too strongly. In the classical gas-dynamic discontinuities all solutions conservation laws. There can be a few solutions selection of one of them which is realized is necessary to attract extra considerations. geometrical theory allows to do that and account the direction of change of the parameter influences onto transformation (hysteresis) Model of a discontinuity breakdown interaction of two flat supersonic flows: consider interaction of flat supersonic viscous gas with different gas-dynamic meeting at angle β 0 ( Fig. 1c and d). In interaction, the outgoing from the point waves 1 and 2 appear, which can be isentropic waves (ω), as well discontinuity τ.
The problem of breakdown of stationary discontinuity is set as follows: the given values of gas-dynamic variables discontinuities 1 and 2 to specify the substance and it is necessary to collision of particles. collision a shock-wave of parameters of primitive model of the p before 1 is more or equal to before 2. Introducing the intensity and intensities 1 J and 2 J of discontinuities The conditions can be rewritten as follows: On the isentropic depression wave angle is calculated according to the formula: Here ω(M) is the prandle-meyer function: where, М and М 1 of the Mach number before and after the wave,  (1) pressures following waves turning angles on these consider that static pressure static pressure 2 p intensity of interaction 1 2 J − discontinuities in 1 and 2: (2) follows: (3) compression shock appearance: wave the flow turning formula: function: of the Mach number before and after is the adiabatic index. Relation of the Mach numbers М and М 1 is specified with the help of common for the isentropic and shock (7) Values E and J is connected with the Rankinwaves and with the Poisson adiabats isentropic waves: (5) is account of (7), (8) by means of the wave according to formula: It is accepted to select two points important analysis of the region of the discontinuity solutions. The first is the intensity J 1 , respondent maximal angle of the flow turning compression shock: The flow turning angle on the compression with intensity J 1 is determined form the where, E l = E D (J l ) is expression of the Rankin adiabat (8). Point l divides the cordiform parts, one meets the range [ ] 1 1, J and is branch of the shock polar and its second as its strong branch.
The second special point on the connected with the concept of cordiform possible, due to variables before the parameters after it.

DISCUSSION
solutions for the for flat cases: It is (9) on the plane which are also called its top in the point important for the discontinuity breakdown respondent to the turning on the single 2 (1 2 )( 1) 2 compression shock the relation: Rankin-Hugoniot cordiform curve into two is called a weak second part is called the shock polar is cordiform curve where, E e = E D (J e ) is the expression Hugoniot adiabat (8). One can see that the envelope exists for 2 M ≥ Curves J l -β l (Μ) иJ e -β e (Μ) divide of the plane Λ, β into three subareas where outgoing discontinuity 2 is a jump wave (III) and, dependently on the either a jump or a wave (II).
In such a way, the analysis different shock-wave structures discontinuity breakdown in the point the given angle of two flat supersonic construction on the plane of shock curves Λ l -β l , Λ e -β e and of determination Mach number dividing the region II This problem e similar analysis in the and their proving is given by V. (Uskov et al., 2000).
Let`s consider different solutions polars at the given М. If you construct М 1 ) from the origin of coordinates and М 2 ), corresponding the wave 2, from coordinates { } given curves reflect on the plane of cordiform the problem mathematical solution. Fig. 5 that in the point of discontinuity different shock-wave structures cab arise. that these two cases are divided with where the outgoing discontinuity 2 degenerates discontinuity characteristic. Further, Fig. 6 and 7 show the special intensities J m , J l , J e on the Mach adiabat index γ and also special curves polars J l -β l , J e -β e . The dependence of the One can see in discontinuity breakdown two arise. Obviously, with the structure degenerates into the dependences of Mach number and curves on the plate of the limit angles of the flow turn at the jump -β l , angles the point of contact if the shock polar β e on the Mach number is given in Fig.  Figure 6 the definition area meeting of physical sense is limited with conditions It is also necessary to mark that for lower than 2 M = , the envelope family is absent. Figure 7 shows parametric curves parameter is the Mach number which  , therefore the section below the abscissa has no physical sense. Figure 8 shows dependences of special limit angle of the flow turn at jump -β l , angles of the flow turn in the point of contact of the shock polar with envelope β e on the Mach number. In the same way in Fig. 7, the envelope J e -β e is determined only for the Mach numbers higher tan 2 M = . One can see that βe>β l and this difference first increases, reaches its maximal value at the medium Mach numbers and then decreases. Both special angles of turn asymptotically tend, at M → ∞ , to the limit for this adiabatic index value (Uskov, 1980): β lim (γ ) = arctg 1− ε 2 ε The given dependences completely describe twodimensional interaction of two angle supersonic jets.

CONCLUSION
The development of new computation algorithms for the regions of existence of different solutions is actual. In 2-D case the conditions of dynamic compatibility is not enough for solution selection. The developed by Russian mathematicians geometrical theory of transformation of shock-waves and wave fronts allows both to select physically realized solutions from the set of solutions meeting the dynamic compatibility and to take into account the hysteresis depending on the direction of variation of the problem parameter.
For the whole range of technical applications (flow of the airfoil acute edge, obstacle reflection of a shockwave, shock-wave processes in jet streams) it is necessary to solve the problem of discontinuity breakdown in fine setting, with no simplifications. In the present work the regions of existence of such solutions for a 2-D case is analyzed. In the form of easy-to-use diagrams here are given the basic dependences allowing completely to define the type of outgoing discontinuities and the solution pattern for the problem of an arbitrary discontinuity breakdown.