Research Journal of Applied Sciences, Engineering and Technology 8(22): 2248-2254,... ISSN: 2040-7459; e-ISSN: 2040-7467

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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.,
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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.,
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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.,
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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 35 
CD
 3  ˆ34   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.,
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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  vv / 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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CONCLUSION
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