MESA Vol 4 No 3, 2013, Paper 7 Pages 265-271

8/10/2019 MESA Vol 4 No 3, 2013, Paper 7 Pages 265-271

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MATHEMATICS IN ENGINEERING, SCIENCE AND AEROSPACE

MESA - www.journalmesa.com

Vol. 4, No. 3, pp. 265-271, 2013

c CSP - Cambridge, UK; I&S - Florida, USA, 2013

Gasdynamic wave-wave interactions. A short comparative

review of some significant results

Liviu Florin DINU1,

1

Institute of Mathematics of the Romanian Academy.

Corresponding AuthorE-mail: [email protected]

Dedicated to the memory of late Professor Mihai Popescu.

Abstract. Two distinct contexts [isentropic; strictly anisentropic] are considered for a hyperbolic

quasilinear system of a gasdynamic type [Euler]. For each of these two contexts two genuinely non-

linear, geometrical, analytical and constructive approaches are considered [of a Burnat type and, re-

spectively, of a Martin type]. Each of these two approaches leads particularly to a pair of classes of

solutions [wave solutions; wave-wave regular interaction solutions]. We finally parallel the mentioned

approaches by using various comparisons between the mentioned pairs of classes. This parallel leadsus to consider an ad hoc relation of continuation between the two mentioned approaches.

1 Introduction

Two distinct contexts [isentropic; strictly anisentropic] are considered for a hyperbolic quasilinear

system of a gasdynamic type [Euler]. For each of these two contexts two genuinely nonlinear, geo-

metrical and analyticalconstructive approaches are considered: a Burnat type approach [centered on

a duality connection between the hodograph character and the physical character] and, respectively, a

Martin type approach [centered on a MongeAmpere type representation]. Each of these approachesleads particularly to a pair of classes of solutions [wave solutions; wave-wave interaction solutions]

via a corresponding significant and specific construction. In an analytical constructionthe wave-wave interaction solutions are associated with a

character. Our analysis suggeststhat a character reflects amultidimensionaland skewinteraction construction generally.

The two mentioned constructions show some distinct, complementary, valences. They are as-sociated with some distinct dimensional characterizations: the Burnat type approach allows mul-

tidimensional objects, while the Martin type approach is restricted to two independent variables.

For the Martin type approach we consider two significant versions a [hyperbolic] unsteady one-dimensional version, and an [elliptic-hyperbolic] steady two-dimensional and rotational one [which

2010 Mathematics Subject Classification35A30, 35L60, 35L65, 35Q35, 76N10, 76N15

Keywords: two geometrical approaches (Burnat type, Martin type): details of a parallel, two-dimensional isentropic

solutions, quantifiable amount of genuine nonlinearity, multidimensional and skew wave-wave regular interaction.


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266 Liviu Florin DINU

appears again to show a hyperbolic character in presence of a supersonic description]. In theisen-tropic context the two approaches arecoincident: in the sense that the two mentioned pairs of classes

are coincident. In the anisentropic context the two approaches appear to be distinct: we make evi-dence of some consonances [a certain complementarity] and, concurrently, of some nontrivial sig-

nificant contrasts. We are finally led to consider an ad hoc relation of continuation between the two

mentioned approaches.

2 Burnat type approach. Genuine nonlinearity restrictions

2.1 Introduction

For the multidimensional first order hyperbolic system of a gasdynamic typen

j=1

m

k=0ai jk(u)

uj

xk=0, 1 i n (2.1)

the Burnat type approach ([1]) starts with identifying dual pairs of directions, [we write

]

connecting [via their duality relation] the hodograph [= in the hodograph space Hof the entities u]

and the physical [= in the physical space E of the independent variables] characteristic details. The

duality relation atuHhas the form:n

j=1

m

k=0

ai jk(u)kj= 0, 1 i n. (2.2)

Here isan exceptional direction[= normal characteristic direction(orthogonalinthephysicalspace E

to a characteristic character)]. A direction dual to an exceptional direction

is said to be

ahodograph characteristicdirection. The reality of exceptional / hodograph characteristic directionsimplied in (2.2) is concurrent with the hyperbolicity of (2.1).

Example 2.1(Lax [7]). For theone-dimensionalstrictly hyperbolic version of the system (2.1) afinite

numbern of dual pairsi

i consisting in

i =

R iand

i =i(u)[i(u),1], where

R iis a right

eigenvector of the n n matrix a and i is an eigenvalue ofa, are available (i= 1,...,n). Each dualpair associates in this case, at each uR[for a suitable region RH], to a vector

asingledual

vector.

Example 2.2(Peradzynski [10]). For thetwo-dimensionalisentropic version of (2.1) an infinitenum-

ber of dual pairs are available at each uH. Each dual pair associates, at the mentioned u, to a vector asingledual vector

.

Example 2.3 (Peradzynski [11]). For the three-dimensional isentropic version of (2.1) an infinite

number of dual pairs are available at eachuH. Each dual pair associates, at the mentionedu, to avector

afinite[constant, =1] number ofkindependent exceptional dual vectors

j, 1 jk; and

therefore has the structure(

1 ,...,

k).

Definition 2.4(Burnat [1]). A curve CHis said to becharacteristicif it is tangent at each point ofit to a characteristic direction

. A hypersurfaceSHis said to be characteristicif it possesses at

least a characteristic system of coordinates.

2.2 Genuine nonlinearity. Simple waves solutions

Remark 2.5. In case of an one-dimensional strictly hyperbolic version of (2.1) a hodograph charac-

teristic curve CRH, of index i, is said to be genuinely nonlinear (gnl) if the dual constructive


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A gasdynamic parallel between two geometrical approaches 267

pairi

i is restricted [the restriction is on the pair!] by

i (u)

i (u)

R i(u)gradui(u) = 0

in R; see Example 2.1. This condition transcribes the requirement d

d = 0 along each hodograph

characteristic curveC.Definition 2.6. We naturally extend the gnl character of a hodograph characteristic curve C to the

cases corresponding to Examples 2.2 and 2.3, by requiring along C:

dd=0 and, respectively,

k

=1

d d=0.See [2].

Definition 2.7a. A solution of (2.1) whose hodograph is laid along a gnlcharacteristic curve is said to

be a simple waves solution(here below also calledwave). Thegnl character implies anondegeneracy

[in the sense of a funning out] of such a solution.

Here are three types of simple waves solutions, presented in an implicit form respectively asso-ciated, in presence of agnlcharacter, to the Examples 2.12.3 above [(x, t)results from the implicitfunction theorem; the solution is structured by (2.2)]

u =U[(x, t)]; =(), =xi()t,

u =U[(x, t)]; =(), =m=0[U()]x =m=0 {U[()]}x,

u =U[(x, t)]; =(1, . . . ,k), j=m=0j[U()]x; 1 jk

dU

d= alongC.

2.3 Genuine nonlinearity: a constructive extension. RiemannBurnat invariants.A subclass of wave-wave regular interaction solutions

Remark 2.8.LetR1, . . . ,Rp be gnl characteristic coordinates on a given p-dimensional characteristic

regionRof a hodograph hypersurfaceSwith the normal n. Solutions of the

system

ulxs

=p

k=1

kkl (u)ks(u), u R; 1 ln,0sm; k

n , 1kp (2.3)

appear to concurrently satisfy the system (2.1) [we carry (2.3) into (2.1) and take into account (2.2)].

This indicates a key importance of the algebraic concept of dual pair.

Definition 2.7b. A solution of (2.1) whose hodograph is laid on a characteristic hypersurface is said to

correspond to awave-wave regular interactionif its hodograph possesses agnl system of coordinates

andthere exists a set of

R(x), structuring the dependence on x of thesolutionu by aregularinteraction representation

ul=ul [R1(x0,...,xm),...,Rp(x0,...,xm)], 1 ln. (2.4)

Remark 2.9. We consider next a of the wave-wave solutions of (2.1). This subclass results

whether (2.1) is replaced by (2.3) in Definition 2.7b [because, the solutions of (2.3) concurrentlysatisfy (2.1); cf. Remark 2.8]. To construct this subclass we have to put together (2.3) and (2.4). We

compute from (2.4)ulxs

=p

k=1

ulRk

Rkxs

=p

k=1

kl (u)Rkxs

, 1 l n; 0 s m (2.5)

and compare (2.5) with (2.3), taking into account the independence of the characteristic directionsk. It results that for a wave-wave regular interaction solution in the mentioned subclass, Ri(x) in

(2.4) must fulfil areasonable(overdetermined and Pfaff) system

Rkxs

=kks[u(R)], 1 k p, 0 s m. (2.6)

Sufficient restrictions for solving (2.6) are proposed, for example, in [5], [6], [10], [11].


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2.4 Wave-wave regular interaction solutions. Some remarks.

Quantifiable amount of genuine nonlinearity

Four circumstances appear to be significant for solutions with a characteristic hodograph: thecase of a characteristic hodograph surface for which all the coordinate systems are gnl, thecase of a characteristic hodograph surface for which only a part of the coordinate systems are

gnl, the case of a characteristic hodograph surface for which all the coordinate systems arelinearly degenerate (ldg) (= 0 is replaced by = 0 in Definition 2.6), the case of a hodo-graph surface which is not Burnat characteristic (Definition 2.4); for such a circumstance a char-

acteristic character of the hodograph surface may persist in an alternative sense (ex. in a Mar-

tin type sense; cf. section 3 bellow). The first two cases above can be exemplified by two sig-nificant analytical and exactself-similar gasdynamic isentropic solutions. These two exemplifying

solutions are taken from an exhaustive list of solutions with a suggestive suitable structure ([2]).

The hodograph of the first of these solutions is structured by three characteristic gnl fields it

appears to be a characteristic surface with three gnl characteristic systems of coordinates. The firstmentioned solution appears to be associated locally to a set ofthreeconcurrent and distinct structures

[wave-wave regular interaction structures; RiemannBurnat representations]. The hodograph of thesecond of these solutions is structured by threecharacteristic fields too but only two of these fields are

gnl the third appears to be ldg. In this case we dispose of a single gnl hodograph system of char-acteristic coordinates. The mentioned solution appears to be associated locally to a singlewave-wave

regular interaction structure [RiemannBurnat representation].

3 Martin type approach. Martin linearization

3.1 Two significant examples of anisentropic flows

An anisentropic flow is present in the region behind a shock discontinuity which propagates with a

non-uniform velocity [unsteady one-dimensional; curved shock path; cf. the theory of the compressive

piston] or in the region behind a curved shock [steady two-dimensional; rotational]. Because, the jump

of the entropy through the shock depends on the velocity of the shock [one-dimensional] or on the

inclination of the shock line [two-dimensional; rotational].

The Burnat type approach 2.1-2.3, leading to a wave-wave regular interaction structure, is still

valid for an anisentropic form of (2.1). In particular for the case of two independent variables. This

approach provides a pair of anisentropic classes of solutions of this mentioned system [Burnat type

wave and wave-wave regular interaction solutions]. This Burnat type pair of classes could be com-pleted by a Martin type pair of classes whether we construct solutions of the anisentropic system (2.1)

with a Martin type approach.

3.2 Martin type approach

In a Martin type approach we interchange the pairs of independent variables x, t or x,y with

,p [usual gasdynamic notations] on the [natural] assumptions ptx

px

t= 0 [or, respectively,

px

y

py

x = 0] and are led to compute the solution of the system (2.1) in two independent vari-

ables through an entity (unsteady one-dimensional [8]) or through a couple of entities,[steady

two-dimensional; rotational] fulfilling each a MongeAmpere type equation in the independent vari-ables,p. The steady two-dimensional and rotational version corresponds to a natural adaptationof the Martins unsteady one-dimensional details ([4]). In the unsteady one-dimensional case the hy-

perbolicity of (2.1) results in the hyperbolicity of the MongeAmpere type equation. In the steady


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two-dimensional rotational case we have to add for hyperbolicity the restriction of a supersonicchar-

acter of the flow. We also remark a pseudo isentropic character of the Martin type anisentropic

description. Precisely ([4]): in each of the two mentioned cases [unsteady one-dimensional; steadytwo-dimensional, rotational] the degenerate fields [particle lines, streamlines] are not characteristic.

There exist, for each mentioned case, two genuinely nonlinear characteristic fields which are incorrespondence with the two characteristic fields of the MongeAmpere type equation associated.

3.3 Martin type linearization. Unsteady one-dimensional gas dynamics

A Martin type construction of a solution of (2.1) needs a pair [F+,F] of intermediate integrals

of the MongeAmpere equation [letCits characteristic fields] associated to (2.1)

F+=R+= constant+ (alongC+), F=R= constant(alongC). (3.1)

The essential aspect of a Martin type construction is that a is induced in

presence of such a pair of linear intermediate integrals. The nature of such a geometrical linearizationis shortly presented in sections 3.4, 3.5 here below. In the unsteady one-dimensionalisentropicgas

dynamics a pair of intermediate integrals F

(p)dpcan be identified directly by using de-

tails of the MongeAmpere representation [vx =] to transcribe the well known Riemann invariance

relations: vx (p)dp= R. The two approaches [Burnat type, Martin type] appear to be coinci-

dentin an isentropic context. Finally, in order to construct an anisentropicextension of the Martinlinearization approach we have to identify some anisentropic pairs of linear intermediate integrals

of the MongeAmpere equation associated to (2.1). There are cases of an anisentropic Martintype linearization (a finite number; [9]). A strictly anisentropic linearized Martin type constructionappears to correspond to a extension [continuation] of a Burnat type construction.

3.4 Unsteady one-dimensional pseudo interaction solution

If in (3.1) R+,R depend on the characteristic [C+,C] we can construct a pseudo interactionsolution by considering R+,Ras new independent variables to show ([8]) that in these independent

variables the entities of the flow fulfil some linearEulerPoissonDarboux equations [for which thesolutions are well known]. We present these representations by

p =p(R+,R), =(R+,R), vx=vx(R+,R), t=t(R+,R), x =x(R+,R) (3.2)

wherex(R+,R)results by quadratures. Reversing (3.2)4,5intoR=R(x, t)will induce a form of so-lution (3.2), parallel to (2.4) [asRhave a characteristic nature]. We callR(x, t) . In the isentropic case these invariants appear to be concurrently RiemannBurnat invariants.

Remark 3.1.We prove ([4]) that the pseudo interaction hodograph (3.2) is nota Burnat characteristic

surface. Still, incidentally and essentially for the linearized approach, this hodograph appears to be

associated with an example of surface for which a characteristic character persists in a Martinsense. For a pseudo interaction solution this results from the following relation between the Burnat

type () and the Martin type () hodograph characteristic directions ([3], [4])

=

p

R,vxR

,S

R

t=

+0 with S(R+,R)F[(R+,R)], = 1

vxR

,= SR

.

We notice that in an anisentropic context the Burnat type approach and the Martin type approach are

distinct. These two approaches are stillcoincidentin an isentropic context.


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3.5 Unsteady one-dimensional pseudo simple waves solution

If in (3.1) R+ or R are overall constants then a solution of the linearequation F+=R+ orF=Rappears to be automatically a solution of the mentioned linearized MongeAmpere type equation.We use this solution to get a solution of (2.1) and call the computed solution a pseudo simple waves

solution. We can prove ([4]) that, in contrast with the Burnat type construction, a pseudo simplewaves solution has a two-dimensional hodograph and for it none of the characteristic fields C in

the physical plane x, tis made of straightlines generally. The paper [4] also includes a detailed and

significant example of unsteady one-dimensional pseudo simple waves solution.

3.6 Unsteady one-dimensional final remarks

The hodograph of a formal regular interaction of pseudo simple waves solutions will be then struc-tured by glueing, along suitable M- characteristics [induced on the hodograph surface by the char-

acteristics of the MongeAmpere type equation considered], a hodograph of a pseudo interactionsolution with some suitable hodographs of pseudo simple waves solutions.

We notice that in an anisentropic context the Burnat type approach and the Martin type approacharedistinct. These two approaches are stillcoincidentin an isentropic context.

The two approaches [Burnat type, Martin type] have both a genuinely nonlinear nature [imposed(Burnat), natural (Martin); cf. Definition 2.7band subsection 3.2]

The geometrical linearization is an element of the Martin type construction. In a parallelwith the Burnat type approach it corresponds to the intermediate element (2.3).

A linearized Martin type construction appears to correspond to a extension [continuation] ofa Burnat type construction [Remark 3.3]. We have to notice in this respect that there are

cases of

an anisentropic Martin type linearization [(a finite number; [9]); in contrast with the availability of

a Burnat type construction]. Concurrently, we have to make mention that, in contrast with a Bur-nat type construction, a Martin type construction requires

[two independent

variables] and it is initiated from a of the system of equations. The structureof a [Martin type] pseudo simple waves solution is in contrast with the structure of a [Burnat type]

simple waves solution [two-dimensional hodograph (subsection 3.5) vs. one-dimensional hodograph

(Definition 2.7a)].

3.7 Steady two-dimensional (rotational and supersonic) Martin type linearization

In this case a Martin type linearization is not possible [cf. a paper in final progress of this author].

Particulary, even an isentropic linearization is not available in this case (in absence of a Riemann

invariance; see comparatively subsection 3.3).


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Math. J., 11 (1970), pp.279309.

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Institute for Math. Sci., Cambridge UK, No. 29, 2006. This work has been essentially considered in the classifying

paper J. LI A N D Y. ZHENG, Interaction of four rarefaction waves in the bi-symmetric class of the two-dimensional

Euler equations, Commun. Math. Phys., 296 (2010), pp.303321.

[3] DIN U L.F. AND DIN U M.I., Nondegeneracy, from the prospect of wave-wave regular interactions of a gasdynamic

type, Proceedings of the International Conference on Mathematical and Numerical Aspects of Waves, University of

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[6] FERAPONTOVE.V. A ND K HUSNUTDINOVAK.R.,The Haahtjes tensor and double waves in multidimensional systemsof hydrodynamic type: a necessary condition for integrability, Proc. Roy. Soc., A 462 (2006), pp.1197-1219.

[7] LAX P.D.,Hyperbolic systems of conservation laws (II), Comm. Pure and Appl. Math., 10 (1957), pp.537566.

[8] MARTIN M.H.,A new approach to problems in two dimensional flow, Quarterly Appl. Math., 8 (1950), pp.137150.

[9] MARTIN M.H., The MongeAmpere partial differential equation rt s2 +2 = 0, Pacific J. Math., 3 (1953),

pp.165187.

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[11] PERADZYNSKIZ .,Riemann invariants for the nonplanar k-waves, Bull. Acad. Polon. Sci., Ser. Sci. Tech., 19 (1971),

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MESA Vol 4 No 3, 2013, Paper 7 Pages 265-271