' ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Pg December 15,1948 ANALYSIS OF 'oEE DIMENSIONAL FLOW FROM A JET UP TO A CONICAL FLAME FRONT W1fEN ME NORMAL VELOCISTY AT THE FLAME FRONT IS A CONSTANT. Prepared by i r * Keeve M. Siegel Approved by e L D. M. Brcon ARONAIJTICAL RESEARCH CPTER W:LIdW RUN AIRPORT YPSILAITIe,MICHIGAN'

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF NCICHIIGAN. Analysis of three dimensional flow from a Jet; up to a conical flame front when the ncn1lt velocity at the flse front is a constant. It will be shown that the problem carmot be solved by present hydrodynamic and aerodynamic methods. Jet, u Cone x u = u t cos e - v1 sin 0' (1) V = u' sin o + v' coss ' (2) where u and v are tvelocities parallel to the x-and r-axes respectively and u' and v' are velocities tangent to and nomnal to the flame front reapectively. At the flame front r x tne' (3) From the new boundary condition v' =fr (ge) (4) A question may be raised as to why the axially synmetric cone problem cannot 'be handled as we did the wedge (see EMV3). Both problems are essentially two dimensional. In the wedge there is cross sectional sym8metry, That is,if you can handle the flow for one cross section, you are handling the flow for all cross sections and thus. the total solution in tie plane of the cross-section is he t soluti4on for all streamlines. In the axially symmAetric cone case, if you can handle the solution- in the plane containing the axis of the cone, because of rotational symmetry one would expect that you would be handling the total solution for all such p1mn. of similar cross-section. The answer to thse question is not obvious, because the estadard answer) that the differential equations are different) is not

MV"4 ENGINEERING RESEARCH INSTITUTE Pge UNIVERSITY OF MICHIGAN P,, necessarily the correct one. This implies that the classical methods of compnlex variable cannot be applied to the spatial exially symmetrical problem. There are cases in the literature where the use of the complex variable has been successfully applied to the axially symmetric supersonic differential equations. (Ref. 1) Since the technique employed in the wedge-like flame problem was to add the solution due to the solid wedge to the solution for flow if the wedge were not there, we will try to duplicate that techtnique. Let us then start out by trying to find the solution for flow about a finite solid cone. We will not use comrplex variable methods. However, if we do find the solution it would be wise to then convert to a complex potential function. In the axially asmyetric case, r 8r (7) or = -f uardr (' SintCe our plan is to find the soblution for the solid cone first, we ktaovr that v' is equal to 0 at the btLourdary of the cone, which is the zero ltreanmline. From (7) and (1) fu' cos e r dr = 0 (9) since (as cone would no longer exist) then I u' r dr = 0 (10) Since this equation holds for all limits of the nLtegral finite or infinite. (See appendix I) u' O (11) Thus -we have shown that. in the axially symmetric case if we assume the normal velocity to the flame front is equal to zero we find that we are forcing the tangential velocity also to be zero. Thus the axially symmetric solution for subsonic flow about cones with e> 0 canmot be foun. by this method. It mlght be well to add that for cones of very small solid angle, this method will yield a result.l

ENGINEERING RESEARCH INSTITUTE Pae -4,~l~iQ...... UNIVERSITY OF MICHIGAN 3 For ife _~0O Equations (1) and (2) becomc ~~U-=~~~ U^t~~~ (1)' v U't' (2)' and thus equation (10) becomes fu r dar= o (10o) This equation holds for all u, thus (10)J mast have the solution r 0 But this s sthe X axis which is the streamfline of a thin cone (0e ' 0). Thus this method does yie6l approximate solutions for thin cones. Returning to the.. ca.e in which e8 > 0, we mst try to find a solution which satisfies the boundary conditions and also LsPla es equation, LaPlace's equations for axial symmetry are, 62 41 a2 - 1 = o (12) bX2 + br2 +.r ~r + 0 aX2 8r2 r 3r Let us pick the most genval solution satisgfyiag (15) whLcS. doesn't contain the -Bessel solution*:~f EiV2. ~ = C1X + C2r2 + C3 X r2 + C4 (14) u5-_ 2 C2- 2 CX (15) r JX + Csr (~6) r 6r bX — fu dx=+2 C2X+C. X2+f(r ) (1'f) v =_ b_ _ (l.) ~ Ruled out in 4r1Y-5

2 * =- C 1n dr = _ C~1 I r- 32P + g(X) (21) r 2 + 2 C 2 X +C3 X + Cr5 but (0,O) =O (Boundary conditIon 1) (22) thusa C = 0 C = (23) bfut, (O0,0) 0 (boundary condI imon 2) (24) c o (25) if e a 0 u'= U (boundary condition 3) (26) This +nosn that if the cone does not exist the flow, w;ll re.aln unifomi rectill-near flo-w. thus at 0' = O, u is a constant and thus independent of X but u = -2C2 -2C3 X (15) Chisa w-ou:Ld only be t'ue -If atO = 0 that C,Z = 0 C3 1 (e) whbere h(O). 0 (27) thue C3 h (e O) (28) thus 2 C i U c= (29),cow to regroup our knowledge U + 2, 0) -2 4_ '! + h ( e -) X r' l (so) *= -: e e) -- U X ~ ~_. '*) x-() u = - h( 3 ') x (32) v h (o *). (33)

ENGINEERING RESEARCH INSTITUTIE UNIVERSITXY OF MICHIGAN P5 nsw rewriting (1) and (2) 'uder condition (4) we have ut =- ' coD ~ - f (0 ') sin (34). vfn u + ~ (e') cos) a (35) now eu3bs tttir. g (32) in (54) we obtain t= - 2 h( t (X 3 )?- a- s Ei nc subsatitui'nXg (33) 'in (35) we o-'bt In (e ~') r -' (8 ')oo e' ir \8 Io COS -B (57) sin O' no< cmbinirng (56) and (437) Jand applying (5) +f(e U) cat e + U + ' tane X =.... -, —..... (38) This of course ordy holds at the flame front because of the use of instead of O and becaxse we used conditions (3) ard (4). However, equation (38) tells us that X is a constant at the flame front and this of course is not correct as the flame frant has length and is not ust apoint. This must mean that equation (38) is really an indeterminate form. *. h (e ') S O (39) f(e. ') cote' + e f (e ') tanO t 0 (40' ( e ) ( C-08 - (9. -) + + U sin e' cos-e' cos-e f(e ') +u sine ' 0 f(e ') =- Usine ' (41) now apply.9 ) and Q1) to (57) u = U cose8 (42) from (34)

ENGINEERIN RESEAOF CH INSTITUTE Pae,,,,....UNI..ERSITY OF..ICHIGAN. u= U cos e + U sine (45) v = U cos a' e in O - U sin e' cod O' 0 (44) Thus at the front we Just have uniform rectilinear flow, This obviously means that the cone does t exist as it has had rno efYect spon the flcw. This just means that the conic l flame problem canot be dol.ed by these hydrodynaniLc and aerodynardc m.ethodAs. 'We expected this resultv as soon as we slowed that the solId cone problem was unsolvable sb these methods.

ENGINEERING RESEARCH INSTITUTE Page EIV-'4 UNIVERSITY OF MICHIGAN 7 R =i~EECE; 1. An Investigation of the Exact Solution of the Linearized Equations for the Flow Past Conical Bodies. By q. Laporte and R. C. F. Bartels Bumblebee Series Report No. 75 February 1948

ENGINEERING RESEARCH I-iNSTITUT.FI _UNIVERSITY OF MC.H(:AN APPENDIX I The potential equation is a2F 2.~ 1 )X2 r 0t~I~0~111o (4.5) At the flane front r = X tan ' (3) thuss ax2 =tan2O'- a~~(46) ~2.~ ~ ~ ~ r tan20 I Equation (45)becomes (~nn2 88 f 3 2 4:$d c0 (tn2e'9 + 1) +2 i sea O' 2~ 2. 3 ar2 r T beD sec 2 9' ~+ I ) 3re r -sec2 0 1 ' -r Jr sec2 e +v_ o )-v + C) O ~~(48) 3r r but applying (2) v = UR sin eI + v' oss (2) and for the solid cone v' 0 uC' sin e' (49) = __ (.9) SirB " (5o> ' 3or thus by substittrting (50) end (49) into (48) seco ' Lain O u us r r see 2 E) )U* 'ULz sec~~'sin + 0 ~ -- Q (o rJ~rr sec2 O~ ~ U't but at the front u' is only a functfon of X and r and from (15) r =X tan ' thus ut i.s a function of r only and the fI-X.-::l derivative is really the ordin,.ary derivative. e2 0 ' 88@~~~i B O( dr

fV-4| ENGINEERING RESEARCH INSTITI.'TE Page 9 UNIVERSITY OF MICHIGAN du' _ dr co u' r u' = CrO (53) butfu' r dr = 0 (10) now substitute (53) into (10) 11-C 082 E) 1 Sin'0 f C rC cs e 0 drf C r dr =0 now integrating from 0 tCL sin,+1 0 in0 +1 C sin ' + 1 0 (9 CL =0 (55) where '> andL > 0 thus equation (55) becomes a contradiction unless C = (5,6) but U' = C r (53) thus u' = 0 (54) &,. m