Ch13 INTRODUCTION TO NUMERICAL TECHNIQUES FOR NONLINEAR SUPERSONIC FLOW

Ch13 INTRODUCTION TO NUMERICAL TECHNIQUES FOR NONLINEAR SUPERSONIC FLOW

Goerning Eqtions for Unste Iniscid Compressible Flow Eler's eqtion Stte eqtions finite-difference nmericl techniqes

Goerning Eqtions for Ste Iniscid Compressible Flow Eler's eqtion Stte eqtions finite-difference nmericl techniqes

Assme D spersonic, ste, iniscid, irrottionl, isentropic flow: irrottionl isentropic Velocity potentil Velocity Potentil Eqtion for Compressible Flow

Velocity Potentil Eqtion for Compressible Flow nonliner prtil differentil eqtion THE LINEARIZED VELOCITY POTENTIAL EQUATION for 0 M or 1. M 0.8; 5 Method of Chrcteristics finite-difference nmericl techniqes

Assme D spersonic, ste, iniscid, irrottionl, isentropic flow:

d x y x d x y y

0 1 y y x x d y y x x 0 d y y x x 0 D N d d y x 0 0 0 0 0 0 0 D d d d d N 0 0 0

indeterminte

0 0 D

0 D 0 1 chr chr 4 chr chr 1 4 chr

sin V ; cos V 1 chr cos 1 sin cos V V V chr / 1/sin M V cos 1 sin cos V V V chr sin cos 1 sin 1 sin sin cos chr tn chr ] tn tn ]/[1 tn [tn tn

chr tn

0 1 d d N chr d d 1 chr 1 1 ] [ d d 1 d d

1 d d sin 1 sin cos sin 1 sin cos M M M V V V d d sin 1 sin cos cos sin M M M V d V d

Ch9 expnsion we

Prndtl-Meyer Fnction The differentil eqtion cn be integrted if we first express V in terms of M. Eqtion then becomes which cn now be integrted from ny point 1 to ny point in the Prndtl- Meyer we. 3 4 Ch9 expnsion we

3 Eqtion 3 cn be pplied to ny two points within n expnsion fn, bt the most common se is to relte the two flow conditions before nd fter the fn. Reerting bck to or preios nottion where is the totl trning of the corner, the reltion between nd the pstrem nd downstrem Mch nmber is 5 Ch9 expnsion we This cn be considered n implicit definition of M M 1,, which cn be elted grphiclly sing the M fnction plot, s shown in the figre.

The dntge of the chrcteristic lines nd their ssocited comptibility eqtions is: to sole the nonliner spersonic flow, we need del only with ordinry differentil eqtions or in the present cse, lgebric eqtionsinstedof the originl prtil differentil eqtions. Finding the soltion of sch ordinry differentil eqtions is slly mch simpler thn deling with prtil differentil eqtions.

1 1 1 3 3 K 3 3 K ] [ 1 1 1 3 ] [ 1 1 1 3

=>M 3 from App. C. =>P 3,T 3 from App. A. =>V 3

SUPERSONIC NOZZLE DESIGN

A line clled the limiting chrcteristic is sketched jst downstrem of the sonic line. Let s ssme tht by independent clcltion of the sbsonic-trnsonic flow in the throt region, we know the flow properties t ll points on the limiting chrcteristic. Tht is, we se the limiting chrcteristic s or initil dt line. For exmple, we know the flow properties t points 1 nd on the limiting chrcteristic in Fig. 13.7.

Consider the nozzle contor jst downstrem of the throt. Letting denote the ngle between tngent to the wll nd the horizontl, the section of the diergent nozzle where is incresing is clled the expnsion section. The shpe of the expnsion section is somewht rbitrry; typiclly, circlr rc of lrge rdis is sed for the expnsion section of mny windtnnel nozzles. Conseqently, in ddition to knowing the flow properties long the limiting chrcteristic, we lso he n expnsion section of specified shpe; i.e., we know 1, 5, nd 8 in Fig.

The end of the expnsion section occrs where = mx point 8 in Fig.. Downstrem of this point, decreses ntil it eqls zero t the nozzle exit. The portion of the contor where decreses is clled the strightening section. The prpose of or ppliction of the method of chrcteristics now becomes the proper design of the contor of the strightening section from points 8 to 13 in Fig.