R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 54

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This simplified theory does not howeveravoid the basic non-linearity of the problem and therefore may play, incertain respects, the role of a mathematical model. I n this sense, it is ofinterest that here most steps can actually be carried out explicitly andwithout too great difficulty, in contrast to the general situation where moredifficult and delicate methods are required (see A r t . 21). Finally, an exactexistence proof for subsonic flow past a body has been given in this case, in acomparatively simple way.24* This limit behaviour holds not only for an ellipse in the 2-plane but for fairlygeneral profiles, as seen by (49").

I t holds true even for analogous constructions inthe polytropic case, as we shall see in A r t . 21.18.1 DEFINITION AND BASIC PROPERTIES287Article 18Simple Waves1. Definition and basic propertiesA s in the preceding article we shall consider here plane potential flow(see Sec. 16.1). Simple wave or simple wave solution (see also A r t . 13 regard­ing the (x,t) -problem) is the name of an important type of solution of thebasic equations (16.7).25Simple waves can be derived and introduced inseveral essentially equivalent ways. Here we take the following startingpoint.

W e ask for a flow in which the lines q = constant and Θ =constantcoincide; we call such lines <£-lines. Each <£-line is thus mapped, by defini­tion, onto one point of the hodograph. W e assume that these points do notall coincide, i.e., that the solution does not merely represent a region of con­stant state, but that the whole set of <£-lines, or the whole region R in the£,i/-plane covered by «£-lines, is mapped onto one line Λ of the hodograph.This fact, namely, that a two-dimensional flow region has a one-dimensionalhodograph, was the starting point in Sec. 13.1 (where the speedgraphserved the same purpose as does the hodograph in the present problem).T h e existence of the line A in the hodograph plane implies the existenceof a relation between q and 0, and, as a consequence, the vanishing through­out R of the Jacobian(i)i= *MN o w using Eqs.

(16.7) and d(x,2/)/d(s,r&) =«')The1, we obtainS S - J [(I)' <«•-»-(?„)']•rightsideof(1')cannotvanishinR,forΜ<dq/ds = dq/dn = 0 and, by (16.7), either q = 0 or θθ/dn =1,unlessθθ/ds =0which means constant flow. Therefore, the flow under consideration cannotbe subsonic, and real characteristics exist in R.A m o n g the lines crossing the «£-lines there must be at least one set ofcharacteristics, and the image of each of these must lie on A. ( I n fact sucha characteristic C intersects each £-line at a point. Each of these pointsmust map onto A since each <£-line is mapped as a whole onto a point of A;hence, each point of C is mapped onto a point of A.) I t follows that Λ is aΓ+or a Γ~, and that each M a c h line of the second set—eachif A is aΓ — i s mapped onto a single point of A. Thus the <£-lines form this secondτset of characteristics.

Since on each of them both 0 and q, and therefore a288IV. P L A N E S T E A D Y P O T E N T I A LFLOWas a function of q, are constant, it follows that θ =F a, that is, the slope ofeach <£-line, is constant. Hence the <£-lines are straight.Since the whole region R covered by the <£-lines is mapped onto one char­acteristic, Γ or Γ~, the equation of this characteristic, Q =F θ = constant,[see (16.41) and (16.42)] is valid throughout R. On the other hand, it willnow be proved that this supersonic pattern represents a flow, i.e., that qand θ satisfy Eqs.

(16.7). Consider an arbitrary point in the x,?/-plane;through it passes an £-line, say a C , which makes the angle a with thestream direction. On account of the constancy of q and θ along this line,we have dq/dl = 0, dQ/dl = 0 along it, or++dq. dq .— cos a + — sin a = 0,dsdn'Λand from Q(q) +d0. dd .— cos a + — sin a = 0,dsdn'Λθ = constant everywhere in the wave regiondq - dOQ -ξ +,Λ— = 0,^whereQcot adQ= ~p =.dsdsdqqThen, from the first and third of these equations,dq^ dq1 dddd-~ = -cot a= cot a— = q —,dndsQ dsdsAand from the second and third equations,ddιdddqcot a dq— = - c o t a — = cotaQ=dndsdsgdsand these are the equations (16.7).W e now state the definition: A plane, steady, irrotational flow in a regionR is called a simple wave if one set of Mach lines consists of straight lines oneach of which q = constant.

The image of R in the hodograph is an arc of aΤ-characteristic. If it is a T ("forward" wave), then Q — θ has a constantvalue throughout R; if it is a T~ ("backward" wave), the same will hold forQ + Θ*Since pressure, density, and absolute temperature are functions of q, thestraight M a c h lines are also the isobars, isotherms, etc.

For a forward wave,the straight characteristics are the C~ and the others, the cross-characteristics,are the C . In a backward wave the straight M a c h lines are the C and theC~ are the cross-characteristics.T h e flow pattern introduced here forms the intermediate case betweenthe general case, where a region of the physical plane is mapped onto anarea of the hodograph, and the completely degenerate case, where q =2+++26* T h e terms " f o r w a r d " and " b a c k w a r d " , which have here no particular physicalmeaning, are used in analogy to A r t . 13.18.1 D E F I N I T I O N A N D B A S I CPROPERTIES289constant in a region of the x,y-plane and this whole region is mappedonto a single point.

W e have seen that throughout the simple-wave regioni = d(q,d)/d(x,y)= 0. Therefore, the exchange of the variables x,y andq,6, essential for the hodograph method, is not possible here, and simplewaves are indeed " l o s t solutions", which cannot be obtained as solutionsof linear equations in the hodograph plane.T h e following is a basic property of simple waves: Adjacent to a regionof constant state is either another region of constant state or a simple wave. Inother words, a region of constant state which maps onto a single hodographpoint cannot be directly adjacent to a region of general flow which corre­sponds to an area in the hodograph. A simple wave must form the linkbetween them. T o prove this statement, consider a region Ri of constantstate and some region R of nonconstant state adjacent to it. T h e lineseparating these two flow regions of different types must be an arc S of acharacteristic, say a C , since, as was shown in A r t .

9, it is only acrosssuch a curve that the derivatives of the flow variables can change discontinuously. Since S belongs to Ri, it carries constant values of q, and there­fore its image in the hodograph is a point P. Through each point of R ,passes a characteristic C~. Consider the subregion R of R whose (^-char­acteristics intersect the segment S. T h e image of each C~ in R must lie on acharacteristic Γ , passing through the point P.

Since there is, however, onlyone T~ passing through a given point, the images of all these C~ lie on oneand the same T~ = Γ Ό~. Hence the flow in β is a simple wave.2+22-A simple wave can connect any uniform supersonic state q = qi, 0 = θιwith another uniform supersonic state q = q , 0 = 0 provided either Q + θor Q — 0 has the same value in both states. B y combining a forward w a v eand a backward wave, and inserting a uniform state between the two, agiven final state q , 0 can be reached, in general, and in many cases int w o ways.2222A n individual wave may be specified in several ways. W e may, e.g., desig­nate a certain characteristic To~ as the image of the whole backward wave, andin addition give in the rr,?/-plane a family of straight lines to represent thestraight C . If these C all have a point in common, that is, if their envelopedegenerates to a point, we speak of a centered wave.

T h e velocity distribu­tion for the wave, centered or not, follows easily from the above definitions.Call φ the angle which a straight Mach line, a C in a backward wave, aC~ in a forward wave, makes with the positive ^-direction. Then the tworelations hold:++27+(2)Q(q) =F 0 = constant,0 =F a = φ,where, as in the remainder of this article, the upper (lower) sign holds for aforward (backward) wave. T h e constant is known since the Γο" ( Γ ^ ) is290IV. P L A N E S T E A D Y P O T E N T I A LFLOWassumed given and φ, the angle made b y the straight C~ ( C ) with the+x-axis, is known.

Therefore, along each straight M a c h line the velocity q isdetermined b y ( 2 ) .W e know that the C~ (C )through a point Ρ is perpendicular to the+Γ( Γ ~ ) through the corresponding point P'+in the hodograph. Hence astraight C~ in a forward wave is perpendicular to the tangent to the fixedΓο" at the point P' which corresponds to that whole line C~. T o sum u p :denote by 6 the angle between the g -axis and the "initial direction of the,,xtTo in the hodograph (see Fig.

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