MASS+MOMENTUM u a. v x. u x. v a. u y. v y. v x ORGANIZE SO AS TO SOLVE FOR THE VELOCITY

Transcription

1 The Method of Chrcteristics

2 D Isentropic Compressible Flow IRROT. v x u y MASS+MOMENTUM u u x v v y uv ( 1 ) (1 ) v x ORGANIZE SO AS TO SOLVE FOR THE VELOCITY DERIVATIVES

3 Prtil Differentil Eqns.? TO MATRIX SOLVE FOR VELOCITY DERIVATIVES u u v v uv v (1 ) (1 ) x y x v v dy dx y x dv u v dx dy x x du y, v x, u U Mch Nozzle Flow

4 Mtrix Solution (Crmer s Rule) Mtrix Solution (Crmer s Rule) dv y v x u dx dy uv v u ) (1 ) (1 du dv x v y v dy dx dx dy dv dy v u ) (1 ) (1 SOLVING FOR vx D N uv v u du dx dv dy x v ) (1 ) (1 vx dy dx dx dy uv v u ) (1 ) (1

5 y, v U x, u CHARACTERISTIC EQUATIONS Denomintor (1 u dx ) (1 v dy ) uv dx dy D

6 dy dx dy dx tn( ) tn( ) Chrcteristic Lines M>1

7 y, v U Numertor x, u COMPATABILITY RELATIONS (1 u (1 u ) (1 v (1 v dy dv dx du ) ) uv dy dx dx dy ) N D

8 Chrcteristics Summry C+ (left running) = const. Stremline C (right running) + = const.

9 Reflected Expnsion Fn C+ C M 1 1 d e f g h i 3 l b c k j

10 Reflected Expnsion Fn C+ SOLVING FOR FLOW PROPERTIES IN THE UNIFORM AND SIMPLE REGIONS C Given M 1 = nd so 1 = 6.38 o nd 1 = 3 o nd geometry d e f g h i 3 l b c d e b c k f 55 o 1 o j g h i

11 Reflected Expnsion Fn C+ SOLVING FOR FLOW PROPERTIES IN THE COMPLEX REGION C Given M 1 = nd so 1 = 6.38 o nd 1 = 3 o nd geometry d e f g h i 3 l b c d e b c k f 55 o 1 o j g h i

12 Reflected Expnsion Fn C+ SOLVING FOR THE WAVE GEOMETRY C 1 Given M 1 = nd so 1 = 6.38 o nd 1 = 3 o nd geometry d e PowerPoint P tsolution b c k 55 o f 1 o g h i 3 l b c d e f g j h i

13 Vrition C+ 1. Wve origintes t sudden turn C 1 Given M 1 = nd so 1 = 6.38 o nd 1 = 3 o nd geometry d g i e h f 3 l O j k

14 Vrition C+. Wve reflects from jet edge C 1 Given M 1 = nd so 1 = 6.38 o nd 1 = 3 o nd geometry d e g f h i 3 O 1 o PowerPoint Solution

15 Vrition C+ 3. Wve cnceltion t wll C 1 Given M 1 = nd so 1 = 6.38 o nd 1 = 3 o e g f h b i c d O = 1 o

16 Design Applictions 1. Constnt Mch number turn. Wind Tunnel Type Nozzle 3. Rocket Type Nozzle

17 1. Constnt Mch number turn. Choose initil turn geometry bc of ngle b. Compute reflection of wve s though from jet jtedge of constnt tmch hmm 1. Shpe of jet edge gives shpe of upper wll. c. Choose lower wll shpe to cncel reflected wve t jkl Given M 1 1 d e g f h i b c 3 j k l

18 . Wind Tunnel Type Nozzle Strightening Section Expnsion Section Subsonic M e

19 3. Rocket Engine Type Nozzle i M e PowerPoint Exmple

20 Mtlb Codes nd Scripts for the Method of Chrcterisics Computing nd Plotting Simple wves Computing nd Plotting Complex Regions Oh Other functions Appliction Exmples

21 Bsic Simple Wve function [,n,x,y]=simple(i,ni,xi,yi,le,g) 3 Terminting chrcteristic Flow le N 1 3 N Incoming wve boundry

22 Other Simple Wve Functions function [,n,x,y]= simplecncel(i,ni,xi,yi,c,x,y,,g) function simpleplot(,n,x,y,g,cl,ch) Outgoing wve boundry 1 3 N (x,y) Flow 1 3 N Incoming wve boundry

23 Bsic Complex Wve function [,n,x,y]=complex3(i,ni,xi,yi,bc,g) (1,1) (,) (N,N) bc (1,)

24 Other Complex Wve Functions function [,n,x,y]=complex3curve(i,ni,xi,yi,bc,g) function [,n,x,y]=complex3free(i,ni,xi,yi,bc,g) function [,n,x,y]=complex4(p,np,xp,yp,n,nn,xn,yn,g) function complex3plot(,n,x,y,g,cl,ch) function complex4plot(,n,x,y,g,cl,ch)

25 Extrs function uniformplot(,n,x,y,g,cl,ch) function n=nu(m,g) function m=m_nu(n,g) function [x,y]=intercept(x1,y1,t1,x,y,t) function [x,y,]=interceptcurve(x1,y1,t1,cf),,

26 Exmple: Rocket Engine Nozzle ( M e) i i M e function [,n,x,y]=simple(i,ni,xi,yi,le,g) function [,n,x,y]=complex3(i,ni,xi,yi,bc,g) function [,n,x,y]= simplecncel(i,ni,xi,yi,c,x,y,,g) ArocketEngineNozzle.m Mtlb Demo

27 Exmple: Rocket i Engine Nozzle M e cler ll %Mch minimum length nozzle, initil turn ngle i=.3 rdins g=1.4; i=[ :.1:.3]; %8 wves xi=zeros(size(i));yi=ones(size(i));ni=i; le=-1; figure(1);clf(1);cl=1;ch=; [1,n1,x1,y1]=simple(i,ni,xi,yi,le,g); simpleplot(1,n1,x1,y1,g,cl,ch); [,n,x,y]=complex3(1(end,:),n1(end,:),x1(end,:),y1(end,:),,g); %complex wve reflection complex3plot(,n,x,y,g,cl,ch); x cl ch); [3,n3,x3,y3]=simpleCncel((:,end),n(:,end),x(:,end),y(:,end),1,xi(end),yi(end),i(end),g); simpleplot(3,n3,x3,y3,g,cl,ch); uniformplot([1(1,1) ( 1(,1) ],[n1(1,1), n1(,1) ],[x1(1,1), x1(,1) x1(1,1)], uniformplot([1(1,end) 1(,end) 3(,1)],[n1(1,end) n1(,end) n3(,1)], uniformplot([3(1,end) 3(,end) 3(1,end)],[n3(1,end) n3(,end) n3(1,end)], hold off;cxis([cl ch]);colorbr;xis imge

28 Pre Pckged Pckged Scripts Constnt Mch Number Turn Minimum length nozzle Wve interction Underexpnded jet Wind tunnel nozzle