Discontinuity of Gas-dynamic Variables in the Center of the Compression Wave

The purpose of research-the study of the flow in the center of the centered isentropic compression waves. Gas-dynamic discontinuities cover shocks, shockwaves, interfaces and sliding surfaces and also the center of the centered compression wave one-dimensional and two-dimensional. For a long time there has been no analysis of the shockwave structures arising in the center of compression waves. At the same time, the problem of development of supersonic and hypersonic air inlets demands to consider the process of the stream isentropic compression. This problem is connected (three-dimensional case) to the problem of arising inside the streams of hinged shocks as opposite to the usual discontinuities not resulted by interaction of supersonic streams, waves and discontinuities, but like from nowhere. This study sets the problem for study in the terms of the developed theory of the interference of gas-dynamic discontinuities of the area of existing solutions for the structures of possible types. We have obtained the relations describing the parameters in the center of the compression wave. We have considered the neutral polar of neither compression meeting the case when in the center of the compression wave there neither shocks nor depression waves. The analysis of properties of the centered compression wave adds to the theory of stationary gasdynamic discontinuities. We have specified the borders of the shock structure existence area optimal for development of supersonic diffusers.


INTRODUCTION
Objects of study are centered isentropic compression waves, mathematical model of the boundaries of the existence of different shock-wave structures in the center of the wave, as well as application of the theory developed for the design of isentropic air intakes.
If to assign the form of a concave surface according to the equation of the streamline in the Prandtl-Mayer plane wave, when it is covered with a supersonic stream of the compression wave (characteristics) of the Centered Compression Wave (CCW) ω σ cross in the same point (point А in Fig. 1) (Uskov and Bulat, 2012;Bulat and Bulat, 2013;Uskov and Chernishev, 2006a).The shockwave structure forms with the basic shock σ by the limit intensity and with reflected gas-dynamic discontinuity R, which can be a shock, a centered depression wave or a weak discontinuity of the second order (discontinuity characteristics when not the values of gas-dynamic variables but their derivatives of discontinuity characteristic endure a discontinuity).
Until recently, the analysis of Such Shockwave Structures (SWS) has not been completed and classification of discontinuities and SWS arising as the result of their interference (Uskov et al., 1995), did not contain the information on the centered compression waves.This study`s objective is the gap recovery.Stokes (1848) inserted the concept of discontinuity in the field of continuous medium stream and received two conditions for density ρ and gas velocity w on the continuity sides, following the law of mass conservation and the law of momentum.The discontinuities considered by Stokes (1848), according to the modern classification, are called as normal, therefore gas travels through their surface.The normal strong gas-dynamic discontinuities serve as a model for the shockwaves which were named by Riemann (1860).In the Russian literature, stationary waves are often called as shocks and shockwaves mean only running waves.Rankine in 1869-1870 (Rankine, 1869(Rankine, , 1870a, b) , b) received the equation adding the system of the Stokes equations.He specified the link between the parameters on the shockwave sides having considered continuously changing inside medium conditions with equilibrium heat exchange.The total amount of the heat received by the medium he specified as equal to zero.Using the relations of the equilibrium thermodynamics and the formula in the Stokes` work, Rankine received Other types of discontinuity discontinuities.Gas cannot travel through The contact discontinuity concept in 1868 by Von Helmholtz (1868), who, in the works, considered the stationary vortex non viscous medium.Helmholtz laid conditions of the dynamic compatibility (CDC) on the contact discontinuity: the pressure on the sides of its surface and to its surface components of the gas stream The detailed analysis of gas-dynamic (isentropic waves of depression and compression) angle shocks arising in the plane stationary nonviscous low-conductivity perfect gas, in 1908 by Mayer (1908).Starting from work, as the basic parameter characterizing dynamic discontinuity, they consider its relation of static pressure J = P 2 /P 1 on its In an explicit form, the centers of of depression and compression were discontinuities only in the works of the of Uskov and Bulat (2012) and Bulat et M.V. Chernyshev (Uskov and Chernishev 2008) in 2006-2008 analyzed travelling expanded stream around the edge of Laval nozzle, where gas-dynamic parameters discontinuity.Further, P.V. Bulat carried analysis of the centered compression wave Bulat, 2012;Bulat and Bulat, 2013).
Let`s consider the domains of existence SWS appearing in the CCW center.

MATHEMATICAL MODEL O SHOCKWAVE STRUCTURE
It is known that inside the compression stream parameters are described with the solution for the plane centered wave: Then, inserting the concept of compression wave intensity can be written: The angle of the stream turn β following functional dependences: in the center of isentropic compression wave: On the shock: (1 The curves described by these equations we will call as compression polar and correspondingly.Points on the compression polar show the relation of pressure after CCW to the pressure in the undisturbed stream and the stream turn angle in the center of the compression wave.
At the origin (Λ = 0, β = 0, compression polar and the shock polar have the order of contact not less than the second one.This property can be simply expressed as follows: where, ( Then, inserting the concept of the compression wave intensity can be written: β is specified by the following functional dependences: in the center of The curves described by these equations we will and shock polar correspondingly.Points on the compression polar show the relation of pressure after CCW to the pressure in the undisturbed stream and the stream turn angle in the where Λ = lnJ) the n polar and the shock polar have the order of contact not less than the second one.This property can The difference in values of the higher derivatives (i>2) of these curves at J = 1 depends on out elementary computation, one can write for 3.4: where, The difference in values of the higher derivatives depends on ∆ i .Leaving out elementary computation, one can write for ∆ i at i = ( ) ( ) tends to ∞ at M→1 and product roots coincide with the roots The type of reflected discontinuity polars mutual location (Fig. 2).
As it follows from the abovementioned compression polar can go at the the shock polar ( The neutral polar has two branches (Fig. 5).As the isentropic compression wave cannot break the stream up to the velocity lower than the sonic speed, its existence domain is limited with upper sound line J sω .For comparison Fig. 5 shows the shock sound intensity J s .М 2 is a crossing point of the diagram left branchJ н (M) with the sound line of the compression polar J sω .М 3 is a fold point of neutral polar J н (М).Point "s" of crossing J н and J s meets the case when the neutral polar crosses the shock polar in the sound point.
Dependence J н (М, γ) is shown in Fig. 6.Here, the surface J H is in dark color and J s is semi-transparent.The plane γ-M demonstrates diagrams of the characteristic Mach numbers М Нi .One can wee that at γ = 1.67 М Н4 tends to infinity.М Н3 (projected fold line of surface J H (M, γ)) blends at γ = 1.1 with М Н4 .So, at γ = 1.1 there is a feature like a "fold".The resulting picture resembles the domain of existence of characteristic points for regular interaction of follow shocks (Uskov and Starykh, 1990).5 this area is shaded).In Fig. 5 and 6 the special intensity CCW and appropriate Mach number М w is marked with index "w".For any adiabatic index starting number М w , the SWS containing the reflected shock cannot exist (polars have no crossing points, Fig. 7).In the domain of the Mach numbers, large reflected shock is always a depression wave for any values of intensity CCW except for Mach numbers, appropriate intensities CCW and the stream turn angles are given in Table 1.Let`s consider how the mutual location of the shock polar and the compression polar change following increasing the Mach number of undisturbed stream.For the Mach number М<М polar is inside the shock polar in full, consequently, in this range of Mach numbers only SWS reflected discontinuity-depression wave (Fig. 8).The compression polar for the Mach numbers М Н1 near the origin distributes beyond the shock polar crossing it upper.Here we have as a reflected discontinuity a shock or a depression wave, which is specified by intensity CCW (Fig. 9).These two cases is separated with the neutral configurati wave intensity more than J н the reflected discontinuity a compression wave and for the wave intensities less than J н the reflected discontinuity is a shock.es with reflected the depression wave (in Fig. 5 this area is shaded).In Fig. 5 and 6 the special intensity CCW and is marked with index "w".For any adiabatic index starting with the Mach , the SWS containing the reflected shock cannot exist (polars have no crossing points, Fig. 7).In the domain of the Mach numbers, large М w , the reflected shock is always a depression wave for any values of intensity CCW except for J н .The special Mach numbers, appropriate intensities CCW and the stream turn angles are given in Table 1.
Let`s consider how the mutual location of the shock polar and the compression polar change following increasing the Mach number of undisturbed М Н1 , the compression polar is inside the shock polar in full, consequently, in SWS is possible with depression wave (Fig. 8).The compression polar for the Mach numbers М higher than near the origin distributes beyond the shock polar crossing it upper.Here we have as a reflected discontinuity a shock or a depression wave, which is specified by intensity CCW (Fig. 9).These two cases is separated with the neutral configuration, i.e., for the the reflected discontinuity is a compression wave and for the wave intensities less the reflected discontinuity is a shock.Following the increasing of М increases and for М Н2 reaches the sound intensity of the compression wave.For this Mach number the compression polar is beyond the shock polar completely touching it with "upper edge" (point s in Fig. 10).
In the range of Mach numbers compression polar goes beyond the shock polar and does not cross it, correspondingly, characteristic this range cannot arise.For the Mach number to М Н3 the compression polar contacts with the shock polar.In Fig. 11 the contact point is marked with a circle.The circle meets J н and neutral SWS.For all other intensities, the reflected discontinuity is a shock.
Following the decrease of γ, the compression wave intensity, appropriate the contact point, decreases and for γ = 1.1 turns into 1.This γ meets the third order of

Fig. 1 :
Fig. 1: Centered compression wave; a): Common depression wave the expressions for the velocity of normal discontinuity according to medium D and the following stream velocity the known pressures before the discontinuity and specific volume before the discontinuity.The third condition on the normal consequence of the law of conservation consideration of the gas condition shockwave, for the first time was received in 1887-1889 (Hugoniot, 1889).coincides with the earlier condition of its derivation Hugoniot needed no extraOther types of discontinuity discontinuities.Gas cannot travel through The contact discontinuity concept in 1868 by VonHelmholtz (1868), who, in the works, considered the stationary vortex non viscous medium.Helmholtz laid conditions of the dynamic compatibility (CDC) on the contact discontinuity: the pressure on the sides of its surface and to its surface components of the gas streamThe detailed analysis of gas-dynamic (isentropic waves of depression and compression) angle shocks arising in the plane stationary nonviscous low-conductivity perfect gas, in 1908 byMayer (1908).Starting from work, as the basic parameter characterizing dynamic discontinuity, they consider its relation of static pressure J = P 2 /P 1 on itsIn an explicit form, the centers of of depression and compression were discontinuities only in the works of the ofUskov and Bulat (2012) and Bulat et M.V. Chernyshev(Uskov and Chernishev 2008) in 2006-2008 analyzed travelling expanded stream around the edge of Laval nozzle, where gas-dynamic parameters discontinuity.Further, P.V. Bulat carried analysis of the centered compression waveBulat, 2012; Bulat and Bulat, 2013).Let`s consider the domains of existence SWS appearing in the CCW center.
Prandtl-Mayer function ϑ = The angle of velocity vector

Fig. 3 :
Fig. 3: Transformation of polars according transparent polers (of compression and shock) at the origin have third order of contact.Function ∆ 4 (M) has no real Product % " {#{ has extremums at M ∆ : to change of the Mach number the compression polar is toned, ) is nonmonotonic.It has the roots at the Mach number values equal to: 3 1 for any γ.At M = M f1,2 the (of compression and shock) at the origin have the (M) has no real roots.H Ŷ and M =1 and to 0 at M→∞.The product roots coincide with the roots ∆ 3 For some values of M and γ, the compression polar and the shock polar can cross each other.At the crossing point there is equation of the intensities (J ω = J σ ) of the shock σ and CCW, as well as the stream turn angles on these discontinuities.And consequently, the condition of co linearity of velocity vectors on the tangent discontinuity is carried out according to degeneration of the reflected discontinuity R into the characteristic.Let`s call such SWS as neutral, intensity CCW in the polar crossing point we designate J н and relevant curve J н (М,γ) we call as the neutral compression polar.The typical neutral SWS is shown in Fig.4.The compression polar can cross the shock polar distributing at the origin inside it and outside it.

Fig. 5 :
Fig. 5: Neutral polar and domain of CCW existence with reflected discontinuity-shock (shaded) J ω is the intensity of the centered compression wave, J sω is the sound intensity of the centered compression wave, J s is the sound intensity of shock, М 1-4 are the special Mach numbers, М w is the Mach number limiting the domain of existence of shockwave structures with reflected discontinuity-shock

Table 1 :
Special Mach numbers, appropriate intensities CCW and the stream turn angles