V. FLOW IN THREE DIMENSIONS

Transcription

1 V. FLOW IN THREE DIMENSIONS 78. Introdtion 33 A. Flow in Nozzles and Jets 79. Nozzle flow Flow throgh ones De Laal's nozzle Varios types of nozzle flow Shok patterns in nozzles and jets Thrst Perfet nozzles 53 B. Conial Flow 86. Qalitatie desription. The seond type of problem treated in this hapter that of "onial flow" permits a rather far-reahing analysis on the basis of the differential eqations. It onerns steady isentropi irrotational flow with symmetry abot the x-axis and nder a frther assmption that the flow is onial i.e. that the qantities ρ p q retain onstant ales on ones (onsidered infinite) with a ommon ertex the origin. Flow satisfying this ondition may or for instane at the onial tip of a projetile opposed to a spersoni stream of air. The flow against a one is analogos to the flow against a wedge and as in the ase of a wedge two ases mst be distingished. If the one angle is not too large the defletion of the flow is ahieed by a shok front whih begins at the tip of the one and has the shape of a straight one (Fig. 14a). If howeer the one angle exeeds a ertain extreme ale (Fig. 14b) no sh onial shok front is possible. 1

2 Instead a red shok front stands ahead of the one. Only the first ase an be handled on the basis of the assmption that the flow is onial. We therefore onfine orseles to this ase. In reality projetiles are not represented by infinite ones; they hae a onial tip and then taper off e.g. into a ylindrial shape of finite length. Ths farther bak the wae from the onial tip interats with other waes sh as expansion waes oming from the bend of the projetile. It is worth while noting that in the ase of a shok wae standing ahead of the projetile the distane nder otherwise eqal onditions is the greater the farther the one extends before tapering off. Retrning now to the idealized ase of a stritly onial flow we may desribe the sitation qalitatiely as follows. Ahead of the shok front the air is in a onstant state flowing in the diretion of the axis with onstant eloity. Sine the shok front is a straight one making eerywhere the same angle with the inident flow the state behind it is also onstant and it is therefore lear that the flow is isentropi behind the shok front. Moreoer it an be ontined so as to satisfy the basi assmption that the flow is onial. The state of the air beyond the shok one will therefore be onstant on o-axial ones. The angle between sh a one and the flow diretion approahes zero when this one approahes the obstale one. 87. The differential eqations. For a mathematial treatment let x be the absissa along the axis r be the distane from the axis and be the omponents of the flow eloity q in the diretion of the axis and in the diretion perpendilarly away from the axis respetiely. The differential eqations for isentropi flow are then (19) x = r (0) r( ρ ) ( rρ) = 0 x + r where ρ is gien by the relation (1) ρ ρ * = and Bernolli s law () * /( γ 1) * 1 1 µ ( q / q = µ Inserting (1) and () into (0) one has * ) q = +.

3 (3) = 0 x + r x. r The basi assmption of onial flow now implies that and hene depend only on the ratio (4) x t =. r Eqation (19) then beomes (5) t = 0 t + t while (3) is reded to (6) = 0 t tt t. Eqations (5) and (6) are a pair of differential eqations of the first order for the two fntions and of t. Clearly this pair is eqialent to one eqation of the seond order for one fntion. This eqation of seond order assmes a form whih is partilarly amenable to treatment when is introded as fntion of. From (5) we hae (7) t =. Differentiation of this relation with respet to t yields (8) t 1 =. This relation together with (7) and (5) gies (9) t =. Insertion of eqations (7) (8) and (9) into (6) gies 1 or + 1 = 0 (30) ( + ) = 1+. Eery setion of a soltion of eqation (30) 1 gies a flow proided that ondition (3) 0 is satisfied bease then x and r an be introded as independent ariables by = t = x / r. Ths the ray to whih ales of and are to be attahed is determined. The diretion of this ray in the xy-plane is eidently normal to the re = () at the point () in 1 Bsemann gies an elegant geometri interpretation of eqation (30): See Figre 16 and eqation (31). Here R is the radis of ratre. 3 N R= U 1

4 the hodograph plane. The flows so obtained are in a ertain way analogos to entered simple waes for two-dimensional flows. Howeer while in the ase of two-dimensional steady flow the simple waes are represented in the hodograph plane by two families of fixed harateristis (epiyloids) the images of the speial flows onsidered here in the hodograph plane orrespond to a greater ariety of res namely a whole family throgh eah point. 88. Conial shoks. The relations goerning the transition throgh a onial shok are the same as for the plane obliqe shok; the ratre of the shok one does not enter. When the shok one is a straight one as is assmed the jmps of p and of the entropy are onstant along eah ray when the assmption of onial flow is satisfied on one side; onseqently this assmption remains satisfied on the other side. The flow may ontine as a onial flow with onstant entropy after rossing the shok. In other words the assmption of proper onial shoks is ompatible with the basi assmption. Sppose a flow haraterized by p0 ρ rosses sh a onial shok. (It is to be noted that this an or only if the speed q = + is spersoni i.e. if q 0 > ). The eloity q = ) immediately past the shok front is loated on the loop 1 ( 1 1 of the strophoid in the -plane. The inlination of the ray whih generates the shok one is perpendilar to the straight onnetion between ) and ) ( 0 0 ( 1 1. The positions of the ones orresponding to the ases (a): 1 > 0 and (b): 1 < 0 are indiated in Fig. 17. When the flow on either the front or the bak side of the shok is to be ontined aording to differential eqation (30) the slope of the -re is to be so determined that the ray gien by (7) oinides with the shok. Sine this ray is to be normal to the -re on the one hand and perpendilar to the straight segment onneting 4 with on the other hand the -re shold begin or enter in the diretion of this segment. The slope of the -re is ths gien by (33) 1 0 =. 1 0

5 The disssion of onial shok fronts by Bsemann and by Taylor and Maoll is restrited to ase (a) with = q 0 and 0 0 > 0 = 0. This ase (Fig. 18a) ors when a onstant axial flow is defleted by a onial projetile. We shall indiate briefly how Bsemann treats this problem. Throgh the shok transition relations the flow eloity ) past the shok is gien (obsere that the ( 1 1 third transition relation garantees that the Bernolli onstant ˆ ( 1/ ) q is the same before and after the shok) A soltion of eqation (30) is to be fond whose graph passes throgh the point ( 1 1 ). The slope of this re is gien by (33). The soltion is now to be so ontined that t = x/r inreases i.e. in iew of (7) dereases p to a point at whih the flow and the ray hae the same diretion i.e. where / = x/r or where the normal passes throgh the origin; sh a point may be alled an end point. This end point depends on the hoie of the point ) on the strophoid. The ( 1 1 manifold of endpoints that an be reahed from ( q 0 0) forms a re whih Bsemann alls the "apple re" in iew of its peliar shape see Fig. 19. In this proedre the shok is presribed and the end diretion is fond. If the end diretion is presribed one may find the orresponding point on the apple re by interseting it with the appropriate ray throgh the origin. In general there will be two intersetions of whih the one orresponding to the weaker shok is likely to or in reality. The ales of pressres and angles allated on the basis of the preeding onsiderations agree exeedingly well with experimental ales (see Taylor and Maoll [57] [58]). It may be mentioned that in the proedre of Taylor and Maoll one begins with the end diretion the shok then being fond by following the soltion of (6) bakwards. This proedre has adantages when single ases are to be inestigated. 5

6 C. Spherial Waes 89. General remarks. Spherial wae motion is obiosly a sbjet of basi importane for the stdy of explosion waes in water air and other media. In spherial motion the eloity is radial and its magnitde as well as that of density pressre temperatre and entropy depends only on the distane r from the origin and on the time t. Sh motion might be onsidered in a ertain sense as somewhat analogos to one-dimensional motion in a tbe nder the inflene of a piston. In the three-dimensional spae the piston is replaed by an expanding (or ontrating) sphere whih impresses a motion on the medim inside or otside. The simplest model wold be that of a "spherial piston" pshing at onstant eloity into an infinite srronding medim. Sh a model orresponds to the niform "piston motion" in one dimension as stdied in Chapter III in partilar in Art. 41. One shold bear in mind howeer that in three-dimensional spae an energy spply at an inreasing rate is reqired to maintain onstant speed of the piston. In better agreement with atal sitations is the assmption that the total energy aailable for the motion is gien. This is the ase for spherial blast waes ased by the explosion of a gien mass of explosie. While in the first of these two models the shok wae raing ahead of the piston has onstant speed so that the shok onditions are ompatible with the assmption of isentropi flow on both sides of the disontinity this is no longer tre of blast waes. In the latter the strength of the shok and hene the hange of entropy rapidly dereases so that behind the shok front the flow is no longer isentropi. Moreoer in blast waes the air or water after haing rossed the shok front and haing thereby ndergone ompression will rapidly expand again to a pressre in general een below that in front of the shok wae This stion phase is an important featre of motion ased by explosions. A phenomenon of major importane is that of refletion of spherial shok fronts; a ontrating spherial wae preeded by a shok front may be "refleted" at the enter with the reslt of 6

7 enormos pressre inrease behind the refleted shok front. At the present state of knowledge all that an be done along the lines of mathematial analysis is to find and to disss some partilar soltions of the differential eqations of spherial waes whih are approximately in agreement with the additional onditions of the problems. One may hope that these soltions display at least qalitatiely important featres of reality. It is remarkable that sh an nambitios approah seems to be sffiient to lead to a ertain degree of nderstanding and ontrol of atal phenomena. 90. Analytial formlations. Assming that the eloity is radially direted and that the radial omponent of eloity the pressre p and the density ρ depend only on the distane r from the enter at the time t the differential eqations are (see II (F) Art. 8) 1 (34) t = r + pr = 0 ρ (35) ρ t + ρ r + ( r + ) = 0 r γ t γ r (36) ( pρ ) + ( pρ ) = 0 assming that the medim is polytropi with the adiabati exponent γ. The third eqation expresses the fat that the entropy is onstant along the path of a partile. It is not assmed that the entropy is onstant throghot sine as stated before the entropy does not in general remain onstant behind a shok front. If the head of the wae is gien as a fntion (37) r = R(t) the total energy arried by the wae is expressed as R 1 1 (38) E = + p 4π ρ r dr. 0 γ 1 E is learly a fntion of the time t. Another important qantity the implse I per nit area reeied by a setion of the srfae of the sphere at distane r is gien by (39) I = pdt T where T = T(r) is the time at whih the wae front arries at the plae r. T(r) is onneted with R(t) throgh r = R(T(r)). Clearly I is a fntion of r. 7

8 91. Speial soltions. Aording to lassial proedre one may obtain partilar soltions of the differential eqations by assming a speifi form for the soltion to rede the problem to one inoling ordinary differential eqations. Ths soltions are obtained whih hae been alled progressing waes. These are soltions oneniently assmed in the form (40) = t β ξu (ξ ) ρ = t δ P(ξ ) p = ε t ξ T ( ξ ) ρ where ξ is the ombination (41) α ξ = rt. In other words a progressing motion is a speial motion for whih the qantities β t ρt δ who moes on a path gien by ε δ pt appear onstant for an obserer α rt = onst. The exponents α β δ ε shold be so adjsted that pon insertion of (40) and (41) in (34) (35) (36) eqations reslt whih inole only the ariable ξ and no longer the ariables r and t expliitly. One immediately erifies that to this end one mst set (4) β =α 1 ε = β. The eqations for U P T are then ξp' (34 ) ( U α )( ξu ' + U ) + βu + ξt ' + T + T = 0 P (35 ) α ) ξp' + δp+ ( ξu ' + 3U ) P = 0 ξp' α. P (36 ) ( U ) ξt ' ( γ 1)( U α ) + T + { β ( γ 1) δ} T = 0 It is interesting that after elimination of ξp' P by (35') eqations (34') and (36') an be reded to one eqation of first order for T as a fntion of U. The eqations (34') (35') (36') are of orse amenable to a nmerial soltion. When the head of the wae is gien by (37 ) r = R( t) = Ξt or ξ = Ξ Ξ being a onstant we obtain for the energy 5α+ δ 1 1 (38 ) E = 4πt U + T Pξ dξ γ 1 and for the implse per nit area 8 Ξ 0

9 1 ( γ 1) / α+ P( ξ ) T ( ξ ) (39 ) I = r. ( γ 1) / α+ 1 ξ Ξ α 0 9. Disssion of Speial Cases. We shall disss the simplifiations reslting from seeral speial assmptions. (a) If the flow is isentropi implying onstant strength of the shok ahead of it then relations (40) and (4) reqire that (43) δ = β. γ 1 If in partilar δ =β = 0 α = 1 the head of the wae 9 r = Ξt moes with onstant eloity Ξ and ρ and p are onstant behind it. Sh wae motion is therefore ompatible with a onstant shok front. As mentioned before a wae of this type will reslt if a sphere is sddenly expanded with onstant eloity. After rossing the shok front eery air partile aqires the same pressre density entropy and eloity. Thereafter as an be shown the air partiles are frther ompressed and aelerated and their eloity approahes asymptotially that of the expanding sphere. (b) Exept for the ase jst mentioned a shok at the head of the wae is not exatly ompatible with a progressing wae. Strong shoks howeer are ompatible to a good approximation if the exponents β and δ are properly related. Denoting by ρ 0 the density ahead of the wae and setting the pressre ahead of the wae eqal to zero [This is the simplifying approximation orresponding to the assmption of a strong shok.] the shok transition onditions rede to (44) ρ = µ ρ0 p ( 1 µ ) R& = ρ0 = (1 µ ) R& as an be inferred by setting p 0 = 0 ρ 0 = 0 0 = 0 and ξ = R& in IV (i )(ii N) (iii N). Insertion shows that these relations are ompatible with (40) (4) and (37') only if (45) δ = 0 i.e. if the density remains onstant on the paths α r = ξt. For the wae motion behind the shok front one then obtains the bondary ales (46) P ( Ξ ) = µ ρ0 U ( Ξ ) = (1 µ )( β + 1). T ( Ξ ) = µ (1 µ )( β + 1)

10 A sitation of partilar interest arises if the shok wae ontrats toward the origin and is eentally refleted by another progressing wae preeded by a strong shok. Sh an orrene an be expressed in terms of progressing waes of the type onsidered here only if α = (or α = for ylindrial waes). It is ery signifiant (see [64] Fig.4) that the pressre past the refleted shok front is abot 6 times the pressre behind the inident shok front (for air γ = 1. 4 ) as ompared with a 17-fold inrease for ylindrial motion and an 8-fold inrease for one-dimensional motion. () The ondition that the energy remains onstant leads by (38') to the ondition (47) δ = 5 α + If in addition the wae is to possess a strong shok at its head so that δ = 0 we hae (48) 3 β = 5 α = 5 6 ε =. 5 The motion of the shok front is then gien by (49) / 5 r = Ξt. The pressre behind the shok (50) approahes zero as 4 6 / (1 ) ρ0ξ t = (1 µ ) ρ0ξ p= µ R 5 5 t. Conseqently the assmption that the shok is strong will eentally be iolated. As long as this assmption is alid howeer the soltion represents a progressing blast wae. G. I. Taylor who first reognized its existene has arried ot the soltion nmerially and has been able to draw important onlsions from the reslts (see [63] ) althogh atal blast waes are in general not of this simple "progressing" type. The diffilties of determining non-progressing spherial waes are ery great and for that reason inferenes from arios approximate treatments hae been attempted. The "inompressible approximation" arises when one lets γ and aordingly beome infinite while ρ remains onstant. For water with γ = 7 this appears to be aeptable. The "soni approximation" reslts when the deiation from the state at rest is small so that only linear terms in these deiations need be onsidered. Finally it may be mentioned that ertain onlsions an be drawn from the differential eqations (l) () (3) by a prely dimensional 10

11 analysis. Any soltion ~ ( r t) ~ ρ ( r t) ~ p( r t) leads to a ariety of other soltions (51) = ~ r t 0 r0 t0 = ~ r t ρ ρ0 ρ r0 t0 = ~ r t p p0 p r0 t0 when r0 t0 0 ρ 0 p0 are any fixed qantities (of obios dimensions) satisfying (5) r 0 = 0t0 p 0 ρ00 =. One may hoose p 0 as the pressre 0 as the sond speed ahead of the shok wae. In the ase of a blast wae with the energy E 0 one may set (53) 1/ 3 E 0 0 = p 0 r. Then one finds for the implse per nit area (see (39) Art. 91) (54) 1/ 3 ~ r ( E 0 p0 ) ~ r I = p0t0i = I. r0 0 r0 In onlsion it might be emphasized again that the theory of flow in three dimensions is still in a state where one an proeed only by groping for sh les as may ome from typial examples whih an be handled by some speial deie. There is mh need for and some prospet of progress in this field. 11