Research Journal of Applied Sciences, Engineering and Technology 8(22): 2248-2254, 2014 ISSN: 2040-7459; e-ISSN: 2040-7467 © Maxwell Scientific Organization, 2014 Submitted: September 13, 2014 Accepted: October 17, 2014 Published: December 15, 2014 Classification of Gas-dynamic Discontinuities and their Interference Problems 1 Vladimir Nikolaevich Uskov, 1Pavel Viktorovich Bulat and 2Lyubov Pavlovna Arkhipova 1 University ITMO, Kronverksky Pr., 49, Saint-Petersburg 197101, 2 Saint Petersburg State University, Universitetsky Prospekt, 28, Peterhof, St. Petersburg, 198504, Russia Abstract: The aim of the study is to give a common classification of gasdynamic discontinuities, shock-wave structures and shock-wave processes. We have considered the classification of gas-dynamic discontinuities, shockwave processes, shock-wave structures, discontinuity interaction problems. We have considered different classification criteria: thermodynamic, cinematic, transiency, discontinuity direction, arriving and outgoing discontinuities. A comprehensive list of discontinuity interference problems is given, as well as classification of possible transformations and wave front reorganization up to the case of the problem two-dimensional transient definition. Keywords: Gas-dynamic discontinuity interference, riеmann waves, shock, shockwave, simple waves INTRODUCTION Objects of study are gas-dynamic discontinuities, shock-wave structure and shock-wave processes, as well as their classification. The Shock-Wave Process (SWP) should be understood as the process of transformation of gasdynamic variables in waves and discontinuities: f→f0 (1) Variables f are a great number of cinematic (uspeed, w-acceleration), thermodynamic ( p-pressure, pdensity, t-temperature), fo-parameters of deceleration, entropy change ΔS = Cvlnϑ/ϑ, where ϑ = p/pγ-LaplacePoisson invariant and h and ho-enthalpy, as well as thermal and physical parameters (thermal capacity cp and cv, γ = cp/cv-adiabatic index, viscosity index etc.), which can change during the SWP. We have set the problem to classify all possible types of interaction of all types of waves and gasdynamic discontinuities. Let us remind briefing information about the Gas-Dynamic Discontinuities (GDD). As we know, supersonic streamline can include areas where parameters change leap, abruptly. In such case, within the perfect gas model they say about existence of gas-dynamic discontinuities. Gas-dynamic discontinuities in supersonic streamline can be zero-order Ф0: the depression/compression wave center, shock wave and sliding surface where the streamline gas-dynamic parameters endure discontinuity (pressure P, total pressure Р0, speed u, velocity vector angle ϑ) and of the first order called as weak discontinuities (discontinuity characteristics, weak tangential discontinuities) Ф1, where first-order derivatives of gas-dynamic variables endure discontinuity. It is possible to specify features (discontinuities) Фi of the space of any order gasdynamic variables. Dynamic compatibility conditions (DCC)at GDD Ф0 (Uskov and Mostovykh, 2010a), connecting the streamline parameters before discontinuity and after it, are deduced from the conservation laws for streamline of matter, power streamline, momentum flux component written before discontinuity and after it. This ratio`s parameter is discontinuity intensity J (more often it is specified as the ratio of pressure after discontinuity and pressure before it). Differential conditions of dynamic compatibility (DCDC) Ф0 connect the streamline irregularity before discontinuity and after it (Uskov and Mostovykh, 2012): 5 N i ci Aij N j (2) j 1 Coefficients Аij, сi are published in the works by Uskov and Mostovykh (2012) and Uskov et al. (1995). For generality, equation included N4 = δ/y (δ = 0 in two-dimensional streamline) and N5 = Kσ (compression shock curvature). DCDC for known streamline field before discontinuity, discontinuity intensity and curvature allow to calculate derivatives from gasdynamic variables after the discontinuity (Uskov and Mostovykh, 2010b). If one of irregularities is known, you can find the discontinuity curvature in the given point. It allows, in some cases, to calculate the streamline field with clear selection of gas-dynamic discontinuities and to calculate their geometry by means of DCDC. For example, at the supersonic jet Corresponding Author: Pavel Viktorovich Bulat, University ITMO, Kronverksky Pr., 49, Saint-Petersburg 197101, Russia 2248 Res. J. Appp. Sci. Eng. Technol., T 8(22)): 2248-2254, 2014 2 (arriving into the atmossphere from thhe Laval nozzzle) boundary, N1 = 0, this alllows to calcullate the curvatuure of the sttreamline bou undary at thhe nozzle eddge (Bulat et al., 1993; Uskov v and Chernyshev, 2006). DCC and DCDC allow to make the total list of possible configurations c of interactingg GDD and to investigatee their exisstence domaiin. The basic monographh (Uskov et al., 1995) is devoted to the t problem annd, for the firrst time, it deffines the modeern full-blown theory of the stationary s GDD D interference.. MATERIALS S AND METH HODS uity classificattion: Discontinu Classificattion accordin ng to the thermodynam mic principle: The importan nt thermodynam mic difference of simple waves and discontinuities is the entroopy behavior in the streamliines going thrrough them. The T basic param meter of such h waves is stattic pressure raatio (discontinuuity intensity) J pˆ p afterr and before the t wave. Densitty ratio J pˆ p is connecteed with the waave intensity by Laplace-Poissson is entropyy (invariant) (ϑ ϑ= const), thatt is: JE 1 ( (3) Or by meanns of Rankine--Hugoniot shocck adiabat: E 1 J J Table 1: Possible waves annd discontinuities h0 0 h0 0 S 0 S 0 ------------------------------J<1 J=1 J 1 r v c ------------------------------J<1 J=1 J >1 Rr - Rr - ,K D d thhese derivativees can change. Weak weak discontinuity disconttinuities (discoontinuity charracteristics) arre the front and a trailing edges of isenntropic waves and degeneerating into thenn intense discoontinuities (J→ →1). At weak discontinuities, d not only first--order derivativves (f′) can chaange, but highher-order derivvatives. All poossible waves and discontiinuities are tabulated t (Tablle 1). Discon ntinuity classiffication: Classiffication according to the cinematic prin nciple: Under the cinematicc principle, waves w and GD DD are dividedd into stationaary and transiient ones. Thee first cover waves w generateed in supersonnic streamliness (Fig. 1а), likke Prandtl-Meyyer ( ) waves and comprression shocks (σ) (standing shock waves, Fig. F 1b). Wavee front and tailling edges ) and surface off discontinuityy in the gas suppersonic stream mlines going thrrough them. Thhe Riemann prrogressive wavve edges and shock wave edges e move in space and timee. For stationarry and transiennt discontinuitties ratio of total pressuree J0 = p02/p01 and total enthhalpy H0 = h02/h / 01 looks diffeerently (Kochinn et al., 1963)). In steady strreamlines H0 = 1the ratio gooes over: 1 J 0 JE 1 ( (4) fi case defin nes is entropicc acoustic wavves The first like Riemaann waves (R Meyer ( ) . The T R) or Prandtl-M second casse defines shock non is entroppic waves ( D ) . Depression and com mpression waaves. Waves are a classed intto depression waves (J≤1) and a compressiion waves (JJ>1). The last l ones cover isentroppic compressioon waves ( c and a Rc ) and shoock waves ( D ) . Intense an nd weak wavees: If the intenssity values J ≠ 1, the wave is intense. In degenerate waave (J = 1), the t gas-dynam mic variables do d not changee, but at suchh a (5) Foormula (5) desccribes the totall pressure loss factor in supeersonic steady waves (comppression shockks ( ) and in isentropic i Pranndtl-Meyer wavves where JEγ = 1). In transient streaamlines, the tottal enthalpy chhanges (H0 ≠ 1), 1 so, in Riem mann waves annd progressive shock waves changes c J0 anddH0 are related by formula: 1 H 1 J 0 0 JE And thee most easy in simple R wavves: Fig. 1: Prandtl-Meyer wave (а) and stationaary shock 2249 (6) Res. J. Appp. Sci. Eng. Technol., T 8(22)): 2248-2254, 2014 2 J 0 H 0 1 (77) uity classificattion: Discontinu Classificattion by the wave edge trravel directioon: One-dimennsional wave trravel directionns for the originnal gas stream mline is possiible to characcterize by factor ( 1) of the edge traavel direction. For χ = +1 the ed dges travel with w the originnal streamline travel and th he wives are called as waake waves. Thee edge speeds of isentropic waves w relative to the stream mline particles are sonic onees and have the t propagatioon speed u+a (а-sound speeed). The norm mal shock wavve edge has speed D>a annd overruns the t original strreamline particcles. Values 1 meeet the oncomiing waves whiich he original strreamline. As the t fronts travvel towards th front edgees of simple waves w relative to gas particles travel at the t sound speed, in the suppersonic originnal streamline (u>a) they meeet the streamlline particles, but b drift downstream. We calll such waves as a drift waves. Therebby, in the diirection of traavel, the wavves relative too the streamlin ne are possiblle to class innto: wake wavves, streamlin ne co-directioonal waves and a oncoming waves. Oncom ming simple waaves spreadingg to the supersoonic streamlinee are drift wavees. If the shock wave speed s propagaation is less thhan the speed of incoming supersonic strreamline, suchh a b also a drift one. o wave will be Particuularly, a samplle of drift wavee is normal shoock wave whicch is called as standing s shock wave. Fig. 2: Optimal O SWC inn the air inlet Fig. 3: Shock-wave S triple configuration w is especiially importantt for investigattion of ones, which causes of random diiscontinuity deecay. The reasson of the struucture formation is waves and discontinnuities arrivingg in the same point. Outgoing waves (disconntinuities) aree the result of interactioon of incominng waves (discontinuities). Becausee of interacttion of isentroppic waves, the outgoing wavves can be onlyy centered deppression wavess ( r or Rw ), which centers coincide with the cross point of incoming waves. The compresssion wave cennters are formeed due to crosssing characteriistics of one faamily with form mation of a shoock-wave struccture in this point. In these structurres, the currennt lines go thhrough differennt systems of incoming andd outgoing wavves. A typical sample is Tripple Configurations (TC) com mposed n of one incoming σ1 and two outggoing σ2, σс normal disconttinuities (Fig. 3) dividedd with tanggential disconttinuity (τ). Figgure 3 σ1 show ws compressionn shocks, from which σ1 is inncoming shockk branched intoo shocks (2) annd (c), of the following streeamlines are suupersonic. In shocks s somee current liness serially go thhrough wave structures the waave system (1) and (2) andd another parrt-only throughh wave (c). Thhe structures arre usually calcculated based on the dynaamic compatibbility conditioons at tangenttial (or contact)) discontinuityy. uity classificattion: Discontinu Shock-Waave structure Classification n (SWC): In the t shock-wavve process not n only singgle waves and a discontinuiities can take part, p but also their t systems and a Shock-Wave Structures (SWC). Shocck-wave system ms s waves and a (structures) are considerred a set of some w current lines or particcle discontinuiities though which trajectoriess sequentially y go in stabble or transieent streams. SWC S is ofteen used for control of gas g streamline gas-dynamic parameters andd with their heelp s differen nt aero gas dyynamics appliied you can solve problems. For this, in original streaamlines wave or mal properties are a compressioon shock systeems with optim specially created. Atyp pical sample is compressiion i plane`s supeersonic air inlets shock systtems (Fig. 2) in (Ovsyannikkov, 2003). Shockk-wave structu ures occur as a a result of RESULT TS AND DISCU USSION interactionn (crossing, interference) of waves or discontinuiities among themselves, with tangentiial, ntinuity interaaction Classiffication of waave and discon contact, freee or solid surffaces. problems: We can distin nguish folloow, oncomiing Classiffication by the gas-dyn namic indiccation: discontinuiities, disconttinuity decay or branchiing Accordding to the gas--dynamic indiccation, waves can c be (Uskov andd Mostovykh, 2010a). dividedd into two types: isenttropic wavess and Relativve to the point wheree SWC form ms, disconttinuities. Thenn you can separate three typpes of discontinuiities is classed onto incominng and outgoiing wave innteraction (Uskkov, 2000): 2250 Res. J. Appp. Sci. Eng. Technol., T 8(22)): 2248-2254, 2014 2 Crossiing gas-dynam mic discontinuuities (includiing shock waves and contact c disconntinuities) amoong themseelves Interacction of Riemaann isentropic waves w Interacction of isentrropic waves with w gas-dynam mic disconntinuities Besidees, waves and d discontinuitties can interaact with solid surfaces. ontact disconttinuities) cannnot Entroppy waves (co cross, therrefore, the fiirst type coveers shock-wavves interactionn or shock--wave-contact discontinuitties interactionn. Interaction process p of D or o R wave with w contact disscontinuity is called c refractioon and its reasson is possiblee to indicate as a a sum of сумму с incomiing waves: D K or R K . This process is accompaniied with wavee refraction an nd wave refleection at contaact discontinuiity. Refraction D K coveers gas-dynam mic discontinuiities interactio on and refracttion R K is of mixed type. The rest caases of wave sinter action are a called as innterference ( D D, R R, D R ). Fig. 4: G Generic shockk-wave structuure; 1-2 inccoming c compression shhocks of the same directioon; 3: R Reflected discoontinuity (jump or shock); 4: Main o outgoing discontinuity; 5: Incoming counter c c compression shoock. - - - - tangenntial discontinuitties depresssion wave. This classificationn is in completee, as it includees no simplle waves. Also, A the ceentered compreession wave ceenter, which is also a discontiinuity, is misseed. Neevertheless, for an inddividual casee of dimenssionality 2, the integratted SWC allows a introduuction of coomprehensive classificatioon of interferrence problemss. Relative to thhe interferencee point Т (Fig. 4), gas-dynam mic discontinuiities in the gennerated SWC are a divided into incoming (R ( a) and и outtgoing (Rp). For F incoming discontinuitiess, a velocity vector componnent for discontinuity directtion goes to pooint Т and forr outgoing discontinuities-bacck from it. Taaking in to consideration the directioon of interactting waves (w wake (W) andd contra direcctional Problem dimensionalitty and generic shock-waave structure: From the point of view off dimensionaliity, time t is a coordinate like l space cooordinates. In thhis context, one-dimensionaal transient ceentered Riemaann wave is totally equiv valent to Praandtl-Meyerplaane stationary wave (Fig. 1аа). Oblique shoock is equivaleent to progresssive one-dimen nsional D-wavve. Curved shoock wave is equivalent to D-wave travelling with w acceleratioon. There are differen nt approaches to shock-waave structure classification c and a their interaaction. In offerred classificatiion based on the integratedd SWC (Uskoov, 1979). Inntegrated SWC consists of o all possibble discontinuiities and wavees: three incom ming, one maain, one tangenntial and one reeflected disconntinuities (Fig. 4). The last onne may be both h a compressiion shock andd a w relative to the originnal streamlinee), we (W) waves obtain two classes of o interaction problems: p incoming waves of the same ( W W or W W ) or diffferent B classes off problems occuur for ( W W ) directions. Both Table 2: Classification of the in nteractions Incooming discontinuitty characteristics ------------------------------------------------------------------------------------ No 1 2 3 2 5 ------------------- ------------------ ------------------ J1 1 ------------------ Desirred intensities 3 1 W RD D 4 ˆ3 4 3 0 I 0 J3 J1 I J5 I I J I J2 I I 5 J1 0 I 0 J3 , J 4 ˆ 1 4 3 4 3 D SD 4 ˆ3 4 3 J1 I 0 J - J3 , J4 J1 0 J - J3 , J4 4 ˆ2 4 1 2 4 ˆ3 4 3 FD ˆ 3 4 3 4 0 0 J2, J4 0 J3 , J4 0 J3 , J4 ˆ 1 5 3 35 CD 3 ˆ34 4 Interrference formula 4 1 2 0 0 0 I I I J1 I - I J1 I J2 I J1 I J2 J1 I J1 I 0 I I 0 I I I I 0 I I I I 0 - I I 0 J3 , J5 I 0 J5 I I 0 J3 , J4 2251 N Notes T TC-2 T TC-3 T TC-1 Res. J. Appp. Sci. Eng. Technol., T 8(22)): 2248-2254, 2014 2 wave inteerference and for wave refraction. r It is obviously that differeent directionn waves crooss constantly and possibilitty of the same direction wavves interactionn needs supporttive analysis. The interfeerence formulaa can be generaally written as: R (kk) R (k) a p ( (8) For exxample, interaaction of follow compressiion shocks (thhe same direection disconttinuity) may be presents ass: 1 2 r3 ˆ 4 ( (9) Tangeential discontin nuity separatess two streamlinnes going throuugh discontinu uities 1 and 2 (follow ( incomiing compressioon shocks) and d reflected disccontinuity 3. For F contra dirrectional com mpression shhock we haave (Table 2): 1 5 3 ˆ 4 (110) Topological classificatio on of transforrmations shocckwave structures: Articlee by Bogaevskky (2002) coveers c in the discoontinuity reorrganization classification potential solutions s of Bu urgers equationn and in viscoous solutions of Hamilton-Jacob equatioon with convvex ut that all thesse classificatioons Hamiltoniaan. It turns ou are the sam me. A sym mptotics of po otential solutioons ( v x S ) of Burgers equation e t v / t vv / x v when the viscosity ε tends to zero o expresses in function, wheere F(λ) = minn fλ(y), where λ = (t, х) is a space-time s poiint, f-smooth function f family y. Even the f family f is smoooth; function F has its own features. fe At eveery point of tim me these featuures represent the shock-waave system in хspace. At change of tim me t, the shoock-wave systeem c can work as t. reorganizes. Any space coordinate a We are interesteed in only features and reorganization stable reelative to anyy small enouugh mooth initial conditions. c Othher disturbancees of the sm Fig. 5: T Typical shockk-wave reorgganization d dimensional (d = 1) case in one- features and reorganization are struucturally instabble and t cannot be used in prractice. It turnns out that at typical points of o time genericc shock wave has h features froom the finite list l (Arnold et e al., 1991) and are subject to reorgannization at indiividual points of time. To deescribe them, we w consider soo called world shock wave lyying in space-ttime. Instantaneous shock waaves are combiination of worlld shock-wave crossing by isoochrones t = coonst. Features of thee functions of o general poosition families minimum deepending on a few parameterrs, are examinned in Bryzgallova (1977, 19978). First, Gurrbatov and Saaichev (1984) and Gurbatovv et al. (1984) paid their attention a to existence off reorganizatioon of minimuum functions unrealizable by shock waves. w Baryshhnikov (1990) that at anyy initial condditions homotoopy types of innstantaneous shock s wave adddition sat the reorganizationn moment and just j after it coiincide. Or, what is the same, around the pooint of reorganiization in space-time, instaantaneous shock waves at close points of o time, directlly following thhe reorganizatioon, are the poinnt homotopic. Leet us connsider one-dimensional case (dimennsionality d = 1). Instantaaneous shock wave consists of isolated points p and worrld shock wavve is a curve on o the surface with regular, triple t and end points (Fig. 5)). r in two-dimensioonal (d = 2) case Fig. 6: Shocck-wave typical reorganizations 2252 Res. J. App. Sci. Eng. Technol., 8(22): 2248-2254, 2014 The first line covers the designation of the world shock wave features. The wave can arise at some moment (endpoint), propagate in space (regular points) and decay with formation of three waves (triple points). At typical moments of time, an instantaneous shock wave cane be reorganized according to black arrows in Fig. 5. Particularly, any triple point of the world shock wave gives a couple of instantaneous shock wave points (shown in the second line) and the world shock wave end point generates a new point of instantaneous shock wave. The set wore organizations totally include all reorganizations of the general position (Bogaevsky, 2002). Figure 6 illustrates a flat case (d = 2). World shock wave is a surface with features; all these features are shown in the second line. Instantaneous shock wave is a curve which can have triple points and end points (the same features which world shock waves have in case of dimensionality d = 1. Instantaneous shock wave can be reorganized according to black arrows in Fig. 6. All reorganizations of wave fronts and shock-wave given in Fig. 5 and 6 totally cover possible types of interference of one-dimensional transient waves and two-dimensional stationary and transient waves and discontinuities. In the same wave, the classification of triple stationary and transient waves may be introduced. In this case, the world wave will be a hyper surface in four-dimensional space-time. And its isochront crosssections will be instantaneous triple SWCs. This case is not covered by the work. discontinuities, the total enthalpy changes, but at stationary compression shocks (standing wave) it remains constant. ACKNOWLEDGMENT This study was prepared as part of the "1000 laboratories" program with the support of SaintPetersburg National Research University of Information Technologies, Mechanics and Optics (University ITMO) and with the financial support of the Ministry of Education and Science of the Russian Federation (the Agreement №14.575.21.0057). 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