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NASA/TM-- 1998-208844
The Space-Time Conservation Element andSolution Element Method--A New
High-Resolution and Genuinely
Multidimensional Paradigm for SolvingConservation LawsII. Numerical Simulation of Shock Waves and Contact
Discontinuities
Xiao-Yen Wang
University of Minnesota, Minneapolis, Minnesota
Chuen-Yen Chow
University of Colorado at Boulder, Boulder, Colorado
Sin-Chung Chang
Lewis Research Center, Cleveland, Ohio
December 1998
https://ntrs.nasa.gov/search.jsp?R=19990018834 2018-06-18T14:09:36+00:00Z
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NASA/TM-- 1998-208844
The Space-Time Conservation Element andSolution Element Method--A New
High-Resolution and Genuinely
Multidimensional Paradigm for SolvingConservation LawsII. Numerical Simulation of Shock Waves and Contact
Discontinuities
Xiao-Yen Wang
University of Minnesota, Minneapolis, Minnesota
Chuen-Yen Chow
University of Colorado at Boulder, Boulder, Colorado
Sin-Chung Chang
Lewis Research Center, Cleveland, Ohio
National Aeronautics and
Space Administration
Lewis Research Center
December 1998
Trade names or manufacturers' names are used in this report for
identification only. This usage does not constitute an official
endorsement, either expressed or implied, by the National
Aeronautics and Space Administration.
NASA Center for Aerospace Information7121 Standard Drive
Hanover, MD 21076Price Code: A04
Available from
National Technical Information Service
5285 Port Royal Road
Springfield, VA 22100Price Code: A04
The Space-Time Conservation Element and Solution Element Method
mA New High-Resolution and Genuinely Multidimensional Paradigm
for Solving Conservation Laws
II. Numerical Simulation of Shock Waves and Contact Discontinuities
Xiao-Yen Wang
Department of Aerospace Engineering and Mechanics
University of Minnesota
Minneapolis, Minnesota 55455
e-mail: [email protected]
Chuen-Yen Chow
Department of Aerospace Engineering Sciences
University of Colorado at Boulder
Boulder, Colorado 80309-0429
e-mail: chow@ rastro.colorado.edu
and
Sin-Chung Chang
NASA Lewis Research Center
Cleveland, Ohio 44135
e-mail: [email protected]
web site: http://www.lerc.nasa.gov/www/microbus
Abstract
Without resorting to special treatments for each individual test case, the 1D and 2D CE/SE
shock-capturing schemes described in Part I [ 1] are used to simulate flows involving phenomena
such as shock waves, contact discontinuities, expansion waves and their interactions. Five 1D
and six 2D problems are considered to examine the capability and robustness of these schemes.
Despite their simple logical structures and low computational cost (for the 2D CE/SE shock-
capturing scheme, the CPU time is about 2/,tsecs per mesh point per marching step on a Cray
C90 machine), the numerical results, when compared with experimental data, exact solutions or
numerical solutions by other methods, indicate that these schemes can accurately resolve shock
and contact discontinuities consistently.
NASA/TM-- 1998-208844 1
i. Introduction
NASA/TM-- 1998-208844 2
2. Boundary CondiLion_
9.. L "t
_'_b,-".,. _'_b,-".,9..9. "i
Tx'_ (1) P> -- L..,"2,_..,"2,.%..,"2.... ifj_ {g _ I,_lf {ng_,g_,t': _n_L Iiil P> -- O, L,2 .... ifj'_ ig
I"2.2"l ar_
<-",J.l_([
T,_. j'6
NASA/TM-- 1998-208844 3
:_--2.9, _,--0., #-- L.O, p-- L.O..."L.."J 2.,5 "1
I>'1 Tn t.l,e _ .hP,,PT3,
:_- 2.6L9_, _,----0..506_2, #-- L.TO00, }>-- L..52._2 2.6 _
(e'l Tn t.l,e _ P,,_P,
:_--2.40L.5, _,--0., #--2.6._T2, }>--2.9340 2._'1
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.._.__l,e tLl>l>et" I)otLn([&t'y .h Jr.), lot" &ll , -- L,.,"2,L,_,.,"2,2 ..... ({) :_ &t_ e',_ltL&t.e([ tLP.{l"l_
Eq. (2.()), &n<[ ({{) :_¢ &n<[ :_ &t_ e',_ltL&t.e<[ tL_.{n_, Eq. (2.._).
NASA/TM-- 1998-208844 4
_.. LOI
2. L2I
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• " 11 " " 11
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2. L.'_"l
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6m_" Ic.vc.I .. T,_'_ I";.'l,S'_ - 5' if 5' _._c.vc.n: _n<L _n<L I";.';.'l,S'_ - 5' - L if 5' _._o<kL. Tl,_'n I";.'l
._ -- .'1,.% L2..... 2,S'_ if e_ -- L,.,"2,_,.,"2..... an,L I'{{'1 ._ -- L,.5,_ ..... 2,S'_ L if e_ -- L,2 .....
F'ttvt, l,_,r_-__cw_, I_y 1.L_{l"lg Gq. ((".._'1 ¢_fPav_ T[L] will. 6 -- 0, i_ can I_" _l.¢_'vn _l.a_ Eq_. 2. L_I
an,L f2. L."JIarv _,qu{_,al_,nt, t,¢_
z.:,:,-- L.,2., 2.L,5l
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":l I_"") !':l ")11-- __ " . " 112. LGI
rv_l>_,c_iv_,ly. Gqttat,{_, (2. L2), f2. L.5I an,L f2. LGI a_ _I,_ I)r'_ttl_([a_ r ¢¢_,Li_i¢_ a_ _I,_ Icw¢_v
wall f a _,:-:,1";.,:L-wall'l. In _1 ,_r- ',:v_ L_,_1,_ ma_l ,'h',g v_.al _1_'__,:-:,¢';.a,:._,:[ ',:v';.,:.l,_1,_ ,'_',_1, I>Nn _
I:,d,:-:w ,:1,_ _,:-:,li,:L-wall will I:,_ ,:Lo:_m':_,h,o:LLL_h,g ':1,_ _qLLa':i_,_.
.",,': ':1,_ ,:':,LL':fl,:':,WI;,C:,LLn,;L_r C-"_, f,:':,L'_ny _':,-- L,'9,3 ..... _r,,:L r. -- L,2,3 ..... _, we. _LLr':_,_
• 11 " " 11 -- L/----.
2. L,_I
NASA/TM-- 1998-208844 5
all(L
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o1" Pat=t. T ILl w{t.l, b -- O, _.t. can I_e _l,_wn t.l,at. Gcj_. 1:2. L._'l ancL f2. Lgi
• • 11 • • 11-- l ,.'__.
(:.,_,.._ _ _L ( -- :.,_.,.,,) ,, P:,:,- L,,2,,_,,.I f2.20i
wall _._, i.e..,
(=-' - (=-'L.+ P_- L,,_,,.'_,,an,L (';.,__.')_.__._.,-- -- (';.,__')""__._t'?..?.?.'t
a._t (=,_+,)_.__.,- (=,_+,)_.__._,,.- t._.._ ,?..?._,
• " 11 " " 11
a._L (:_-%.)_._.._.,- -(:_-%.L._.._ ,?..?.4,• " 11
an([
_an not. I:,e tt_e,L ,:L';._t.ly _ t.l,e _olkt-wall I:,ot,r,,:La_r cvm,Lit.i_.
Two al>l>ro_'_'l,e_ can I:,e Lt_e(L t.O ,:-:,",,,erc,:-_"=_'_et.l,e alcove (Li_,:'-Ltlt.y. T-n t.l,e fi_t. al>l>ro_'_'l,,
t.l,e f,:-:,llowir,g al>l>roxh:_at.e fr-:,ta'__gof Eqg. I'?..?.?.'1-1"?..?.._'1aL"C.Ltge,L_ t.l,e g,:-:,li,:L-wall I)oLtn*La_'e,:-m,:LM_', g"
• " - " + " " - " " - :"---),-.__s , 2.2.5 ,(=,_.,).,,__._.. (=,_.,).,,__._, m L,_,."I, an,L (=,__.).,,__._.. -( ""
NASA/TM-- 1998-208844 6
• , ._. - , _-'-')_._s ' P:,:,-- L,_.,.I 2.2,3"1- - "-'
_1"1 ,:L
• • 11 - • 11- L,,.'_ - • 11 - • 11-- L,,"-
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-r':_,e.+l,i>,:-:,{-nt. ( r', 2,G',I':, -- L,.,"2) wit.l, t.l,e, a.i,:Lof t.l,e. G+t.-c_r-,:Le.r-T<"_.yl¢:,r"+ e...+l>a-n+i,:m,.
ln t.l,e, e_e.ev:,ncLa.ncLge.ne.r-a.lly _-_+_¢.<',_'-e-ur-a.t.e.al>l>-rv:,<-",Mi,t.lle, i>-t-'-e.+e.-nt. -r:',a-rc-lrh',g I>rV:,ev,:Lttt"e.
ie_a.l>l>lie.cLo.r.,e.r-a. ,L-ua.Ime.e4,. It. wa+--:,e...+l>la.ine.cLin SJe.c..+,of Part. T t.l,a.t.a. ,L-ua.I 1-_+e.e_l,ca.n I+e.
ev:,ne;kLe.r-e.cLa+--:,t.l,e, ev:,mI+ina.t.ic:,nof t.wo e_t.<',,+.,_.+.,e.-t'-e.cLeq><-',_x'.-t.h-_w.1-_+e.e_l,e.e_,i.e.., 1-_+e.e_l,L (Pig. 41a.))
a.ncL 1-_+e.e_l,2 IiT;'ig. 411+))...'+.e_ e4,c:,,:vnin Plg. 4(1+), a. 1-_+e.e_l,i>,:-:,i-nt,of 1-_+e.e_l,2 ie_<-',.le_okLe.nt.iGe.cL
I:,y ,:.we:,+l:,a,:.ial i,,,:Li_+ t.'.a,,,:L .._, a,,,:L _I,_, ,:.ir:,e.-le.ve.I -r,-tt-r':',l:,_t'- t:,. "F'-ttt'-t.l,_t'--r':',¢_v.,a m_,e_l, i>c:,in,:.
(t.'., .%P:,l of-r':,e.+ll 2 i+ -r':,a-L-'-ke.,:LI:,y (1) a +oli,:L ,:.rla-r,gle. if P:,- L..,"2,+..,"2,.3..,"2..... a-r,,:LIiil a I1_11¢_:',:"
,:.rla-r,gle. if P:, - O, L, 2 .....
T.,c.t..({) t" a r,,:L .._ I:,e. a r,y" I:,aiv ¢:,i"wl,¢:,le, i r,,:.e.ge._ a r,,:L P:, I:,e. a r,y" wl,¢:,le. _ I,ali"ir,,:.e.ge.v roLeIn
_l,a_ (1%._, i'>) rvl>rv+_,n_+ a m_'+l, l>Nn_ c_fm_,+l, L, ancL I{{I i.>+ -- i'> . L.."2. Tl,_'n, <'_'cc_Ling _o
T;'i_.. fJl'_'l _I_¢[ fJ(l>), (r,._, I'>_) rvl>rv+_'n_+ a m_'+l, l>C_h_c_i'm_,+l, 2 at. _I,_" P>+_l,_h;w I_'-,,_'Iwi_l,
i,:.+ +l:,a,:.ial Ic:,¢-a,:.i¢:,nI:,e.i,,g kte.,,,:.ical ,:.c:,,:.l,a,:.c:,i"(t",,._,, P:"I. _Ltt"t:.l,_L"-l":_'n_r,_'.,, ({) i"r'W"I':, -- t"),, L.,2 .....
(r, 2.S' L, t>'i i+ a m_'+l, l>Nn_ c_f m_,+l, L wl,il_" (r, 2.S', t+'i i+ a m_'+l, l>Nn_ c_f m_,+l, 2, I'{{'I fr-yr
_i'1;_'_.ll L, _n¢[ l'iii'l i'r-_r I'> -- t'], L,.,"2, L,_,.,"2 ..... t.li_" I;_'_.II l>_int._. (r, 2,_' L, I'>'I _n¢[ (r, 2,_', I'>'I
lle. at. t.l,e. _ame. t.h:w, le.ve.I anaL, friar.lye, t._ 0'.,"9, arv mimw h:_-'_,e._ _f e._l, _t.l,e.v. ..%_ a
L"_mLlt.., _.Iie. +,:-:,li,:L-wa.lll+¢':,LLncLa._'_:,,,,:LM¢:,,,_ at. CD ca,, l:,e. i-ml>,:-:,+e.,:L"LL_{'I'Ig "I'_c'J_. (2.22"I-1"2.2._),
wk.Ii , - O, L,.,"2,L,_,.,"2..... _-','_.e. _.Ii<-',+._._.Iie. ma.-rc-lrh+g _m-L+L<-',+.l+le.e_+e_+_.<-',+._.e.cLwk.Ii me.e41e.e_ L
a r,,:L2 are. r,¢_:',:"CC:,LLI>IC',:Lt.lrL"C:,LL_II_l,c.+c. I),:':,LLr',,:Lat"y_:,r,,:LM¢:,r,+...".v.l+,:-:,-n,:-:+e._l,a_ _l,c. "[:'..,q+.(2.2_),
(2.2.'J), 1"2.21_'1_r',,:L 1"2.2']"1c_r', c._'s.--_.{ly"I:,c.cc:,-,,-,._-L-'-,:.c.,:L,:.,:-:,,:.l,c.vc._ic:,-,,_ _,:-:,cia,:.c.,:L ,:vi,:.l, ,:.l,c.(_,,_)
,_",C:.:'yL"_Lift <-",J._,_._I)y LL_if't _ "F".,c'j. (_" .._ ) ,:-:,l""_<-",J.L"_T r L] ,:vi,:.l,& - O.
Tr":_,i>,:-:,_{-r,_ t.Iic.cc:,-r,,:L{t.{c:,-r,_ l:'..,q_. ( 2.22 1-12.2.'J 1,:Li-L-'C.c-,:.ly t"C.c'JLL{t"C._t.IIc.LL_,_.,:':,l"<-".J.,:LLL_Ir":_',c._I i...".v.
<-"+.L"_+LLIP..,,it. I,_ t.l,e. ,Li_a,L_nt._-"_,e. of ,:L¢:,LLI:,I{n__:,r':_'_l>LLt.<-'+.t.{,:':,r,<-'+.l¢,:-:,_t...T"Jr"_%_t"..t.l,e, e._t.ra _x-_t.
..",._ will I:,e.e._l>l_i-,,e.,:Li-,, i'-Lt,:.Ltt"e.i>al>e._, ,:.l,i_i-,',,:Le.e.,:Li_ ,:.i,e.¢-_e. i'br--,:',_-,',yi>r-a¢,:.i¢-alal>l>li¢-a,:.ic:,-r,_
i-n¢-I-Lt,:Li-,,:_,:.l,,:-:,_e.-r-e.q-Ltlri-,,:_,:.i,e.Lt_e. ,:':,fLt,',_,:.r-Lt¢,:.Ltt"e.,:L,:.rla-,,g-Ltlar--,:,e._l,e._.
NASA/TM-- 1998-208844 7
_. Numerical Results
_.i. On,c-Dimenslonal Problems
..%p,<-",+.i>re.lh"=4na_,, <-",+.mLl::,t.let.y re.lat.e.,:Lt.c:,t.l,e. P,l>e.cifi,:at.{_l ,:-:,1"n'LL'r"='_e'Ld.,:'al{nit.ial ec:,n,:LM_lp,
will I:,e. ,:L{P,,:-LLP,P,c.,:LIw.t'v....%p,<-%-ne."+ar'=,i>le,we. ec:,np,i,:Le.r-<-",+.P,ItC:,ck-t.'LLI:,C"i>eol:,le.r':_,,:Le.6ne.,:LI:,y t.l,e.
ec:,np,e._t.i_, law E,"j. (2. Lgt {-n Part.. T ILl an,:L t.l,e, e.."+<',_"-t,h,Mal ec:,n,:LM_,p,- <-",+.t.,' - O,
:__ - :__ (_, O),L___ (_. tt
i., _.110
[ r+j,
if j -- j'd L,.,"2,j'd _,.,"2..... j'b -- L,.,"2
if j -- j'd -- L,.,"2,j'd -- +,.,"2..... --j'+ L/2
._ 1
NASA/TM-- 1998-208844 8
T+_t.c,r - .P.:,_,,:,:'it.I, j'_ ..- {. L,.,"2, . _,.,"2...... (j_,-- L,.,"2)}. Tl,_'ll (j',+,0) t_l>t_llt._ _ 1___1,
l>_.lIP, ,.- _. T:'¢L_P,li_t'-n_'u% <-'w_[_.ll_ P,_ t:'.+_'J.(_.L), :_ _II<[ O:_,."O.P _ ll_P, <[_G11_<[ _ _I,_
I_C_P,I._'_I_ _i" P,li{_. I;_'_.II I>_{I_P,...%_. _ t'_.tLIP,, P,li_" 1.1_I.P,I._I C¢)I_[I.P,I._'_I_. F'_Q_.. I'.'{..'{'I _I_<L l'.%fJ'l _t'_ I_
will. (j, 0) ..- _, w_, I._,_,_,(1)
_ _ :__ (_:, 0), {f j - j_(
t (_'_ _':, ),.,"2, {f j - +ie
1_.,5 l
NASA/TM-- 1998-208844 9
L.O,0.4, --2.0), if _, • • 0...3
L.O,0.4, 2.0), if _, • • 0...3
_.i._. $}m-Osher Problem
if _ • • -4
if _ • -4
_.,qlf _.,q,..37,LO._, 2.629),
L 0.2 _in ..3_,,L.O, 0.0),
1_.91
if _ • • -2.0
if --2.0 • _, • • L...3
if L...3• _"
( 7. L,q2_, 2,.3.40 LS, _.,_.q28,.3),(p, t_, :_) -- f L.._82 L, 2..q..3._, 0.,_2L8),
t L.O, L.O, 0.0),
NASA/TM-- 1998-208844 10
(t.O, LOOO,OZI, {f_ I "OIL
L.0, 0.0 L, 0), {fO. L I I _ I
U.O,UO0,0), {fO.9 I I _
I 0 "_ I" "/_ " LOI
(20.0, 20.0, 0.0), {/" ._ • " 0.2.5
L.0, L.0, 0.0), {/" _ • [').2.5
I_.L LII
NASA/TM--1998-208844 11
8._. Two-l_imensional Problems
NASA/TM-- 1998-208844 12
NASA/TM-- 1998-208844 13
NASA/TM-- 1998-208844 14
8._.-,4. ___'racL_on o_I'5;}1ock IA'ave down a 5;Lep
NASA/TM-- 1998-208844 15
NASA/TM-- 1998-208844 16
NASA/TM-- 1998-208844 17
4. Conclus{ons and D{scuss{ons
l_y ___ I>_n_ P..I,e.C'LL'L"_I.1I:.n'LLl%'_e._l "L"e.mLIP.._vc{_l,e.Yl>e._l%'_e.nP.._l,:L&t.&,e.Y<-',_'t._,:':,ILLP..'[_I
r, tLr':,e.-r{,:&l _oltL_';.,_',_ge.l.1e.t'-&_e.,:LI:,y o_l,e.t+ l':',e._l,o,:L_,";._I,_ I:,e.e.l.1_1,_,_"1.1_1,<-",+.__l,e. LD &l.1,:L2D
_'"F".,,.,"£'F".,_l,o,:k-,:&l>ULr'h.1g _,:l,e.l':',e._ ,:&1.1&,:,e-tLt'-&_e.ly_ol-,.'e. _l,o,:k &l.1,:Le,:m.1_&,:_,:L'[_e,:m.1_'[1.1tL'[_'[e._
c,:-:,1.1;.'[;.t.e.1.1t.ly."[;"LLt"t.lle.t'-i%',OPe,"it. li<'m.-'..&h-'.,:-:,l:,e.e.i', ;.Im,,;v1.1t..li<-",,.P..P..lle.l>t"e.;.e.1.1':.;.,:l,e.,;',e.;. &t"e._e.i',LL'[i',e.l',.."
t_l)tL_.P.,{.e'., li+._+_e. +]'ca+._ym+f-+:e'_-,_+:m+e_-e++f_+.l,_-_++._",_e+:e.+]'ce.,_f.+:e._+_-aeeli_-a_" ACe. ae+:_+e.%'e.d rr+'_+f.+:-
01ii I. _-e.,ex_'_-iI.J';'._12_il.O ,_,l':'e.,gJ'_'_.i' il._-e.ail.;']'_e.;'._il._ _'_- e.a,9_: J';'._,:.l'J"_'J',:.l'lia.i' ,9+'_,':_C'.
"L_e.,C'_'LLP.e.Of t.l,e."i.t";."h;',l>le.l,o,_,:&l P.I:.L"-LL,CI:.LLL"_P._1.1,:Lt.,:-:,t.&lly e...+l>li.,:i.t.I"I_I:.'LL"L"_,, t.l,e, l>t'-e.;.e.1.1t.
+,:l,e.l':,e.+ <-",+._<-",+.1+oI,{gl,ly _:':,1%'_l>'Ltt-&t-{,:':,'i.1&le._.e.'i.1t.. A+ &1.1e..'.'.X&l':',l>le.,e_+e;[<Le.t"& ve.+_.+_4.e+e.<Lm<Le.
•h":'_l>le.l':'_e.l.1t.'h.1gt.l,e. +'h.1gle.-l':'_e.+l,2D _'"F".,,.,"£'F".,+l,o,:k-,:&l>t.U_l.1g +,:l,e.l':'_e.._,:-_"& .++00:>::L2t"]l':'_e.+l,,
t.l,e. ('T"U t.'[l':'_e.,_.1 & ("r&y ('90 t'e.qu'[t"e.,:L+..oe._e.eut.e. L_t"]l':'_&_l,'h.1g +t.e.l>+(T -- L_t"]:>::(",,t,.,"2))
•[+_+ly L.+'J+e.e+'m_[+,"i.e..,_l+¢':,-tL_.2. L6 +'++e.e+i>e.t'-+%w.+l,I>O'[n_.I>e.t"+%+_-r..:l,'[n_+_'e.l>-
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':."h;',e';.'I_I,_ ,:-:,i"& ,:.yl>[,:&l ;.h.1gle.-;.,:.e.l>L"e_,tLl<-",,.t"-i%',e.;.--.It;.'[1%',LLl&t.'[,:':,i.1{f e.<+,w-i,;.'[1%',LLl&t.'[,:-:,i.1LLP.e.P._ ,G'X ,_me._I,_n<L I,o_I,I,_:_"_I,e._'__e.",,.,"alt,e.;-'.oi" ,5', .F'+,-',_'_n<L _o_I ;.'h;',t,l&t.'i.,:-:,1.1t."i.-,;',e.T. TT_",:'e.ve.t'-,
_1,';.+,:L'[+&,:Lv_l.1_&ge.,:&1.1I:,e. e,:m":',l>e.l.1+&_e.,:Li"_t"-I:,y o_l,e.t+ <-",+.,:L',."__',_<-',,+_,e.+_l,e. I>re.+e.n_.++i,e.+%+e.i,<-',+-_+,
+-tt,:l, _ I,'[gl,e.t+-<',+e-e-tt_," &l.1,:LI_,_"e.-t+-e,:m":',l>tt_&_'[,_.1&le,_,+_i>e.-t+-l':',e.+l, I>_'[1.1_I>e.t+ l':',&_l,'h.1g +_e.l>.
..%;-'.&1.1e..'.'.X&l':',l>le.,e,:m.1+'[,:Le.t+-_l,e. _e.+_I>r,_l:,le.l':',,:L'[+e-tt++e.,:L{1", £e.,:..++.2.6. TI,e. T'+,,-'D_+-ttl_+ _Ve.l.1
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_.{I"_ULI&P.{ot.1.TI,e. _l{gl,_ (L{_&(Lm_n_<'_;,e.o/" _l,e. i>t_e.n P._-I,e.1-__e.e&n I,e. 6Lt"t.l_e.t"e+m-__l>e.n_&_e.(L/'or
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e.&Ply _'_;,e. +il"_ULI&P.e.([I,y _l,e. TVD 1;_e.t.l,o(L.
NASA/TM-- 1998-208844 18
R_f_r_nc_z
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.,:.:;_,_",_o_'.'.<',:::._,',_,,".,_,f.af.#_,.'.a,i'7:'f_,h] f")_'_'.'.a_",_iP.,:.:,I>- L_+ L,,Lake Tal.oe, _Ne,,.,"a,:La,,L995.
NASA/TM-- 1998-208844 19
NASA/TM-- 1998-208844 20
J = -Jb
j =-3/2 j =-1/2 j = 1/2 j = 3/2
j=-I j=O j=l J =Jb
m.._ X
x -.-ih x -3/2 x -1 x -1/2 xo x1/2 Xl x3/2 _Xib
t
mn=2
-- n = 3/2
mn=l
-- n=1/2
-- n = 0 (t = O)
Figure 1: A staggered space-time mesh with the spatial boundaries at j = + jb (jb = 2).
A D
0 _ i _ vl $ C_ xB 1 E2 3 4
Figure 2: The computation domain and the shock
locations of a steady-state oblique shock problem.
NASA/TM-- 1998-208844 21
Inflow
boundary
_A s _-- Upper/ boundary
T " f---e-------o-r' i (1, 2) (1, 3)
(1 ,£11)
I •I (2, 2)
(2 ,_11)
--I_1 •
(3!1)(3, 2)
• Mesh points with n = 1/2, 3/2, 5/2, - - -.
o Mesh points with n = 0, 1, 2, - - -.
D
(1, 6) (1, 2S-1)
• o " ,j S+l)_h°2-_-(1, 4) (1, 5) (1, 2S) (1o • o ._
(2, 3) (2, 6) (2, 2S-1)• 0 •
(2, 4) (2, 5) (2, 2S) (2,12S + 1)o • o Outflow
(3, 3) (3, 6) (3, 2S-1) /l_--b°undary• 0 • 0
(3, 4) (3, 5) (3, 2S) (3,12S + 1)0 • 0
(R, 2) (R, 3) (R, 6) (R, 2S-1) i I(R,__)___ -o (Re' 4) (R° 5) (Re,2S) (R, 2S+ 1)
-- -e- -- -q--- o- --ii_x
B (R + 1, 2) (R + 1, 3) (R + 1, 6)', (R, 2S-1) C0 • 0 1 • 0
(R+1,1) (R+1,4) (R+1,5) _ (R+1,2S)(R+1,2S+1)
_-- Solid wall
Figure 3: The spatial locations and the mesh indices (r, s) of mesh points used in
a steady-state oblique shock problem (R = S = 4).
NASA/TM-- 1998-208844 22
s oA,_ (0,1)_m 4 .....
r
(a)
(1, 0)
(2, 0)
(3, 0)
(R,0)
(0, 4).ore
B
(R+1,1)
Mesh points with n = 1/2, 3/2, 5/2, - - -.
Mesh points with n = 0, 1, 2, - - -.
o •
(0, 5) (0, 2S) D
(1, 2)o
(1, 1)
(2, 2)o
(2, 1)
(3, 2)o
(3, 1)
(R, 2)o
(R, 1)
(R + 1, 2)o
(1, 3)
(1, 4)
o(2, 3)
(2, 4)
o(3, 3)
(3, 4)o
(R, 3)
(R, 4)
(1, 6) (1, 2S-1) Io • I o
(1, 5) (1, 2S) I(1, 2S + 1)• o I
(2, 6) (2, 2S-1) Io • I o
(2, 5) (2, 2S) I(2, 2S + 1)
• o I(3, 6) (3, 2S-1) I
o • I o(3, 5) (3, 2S) 1(3, 2S + 1)
• o I
(R, 6) (R, 2S-1) I Ay
o . II o T(R, 5) (R, 2S) (R, 2_ + 1)
(R + 1, 6) (R + 1, 2S-1) CO •
(R + 1, 5) (R + 1, 2S)
-(3,
(R + 1, 3)
(R + 1, 4)
_[--_s A ....
rA
(1, 0)
A(2, 0)
A
(3, 0)
A(R, 0)
(b)
B
(R+1,1)
(0,1)
(1, 2)
(1, 1)A
(2, 2)
(2, 1)A
(3, 2)
(3, 1)A
(R, 2)
(R, 1)
(R + 1, 2)
• Mesh points with n = 1/2, 3/2, 5/2, - - -.
A Mesh points with n = 0, 1, 2, - - -.
A • A
(o,,) (o,5)(1, 3) (1, 6) (1, 2S-1) I
Z_ • Z_ I •(1, 4) (1, 5) (1, 2S) I(1, 2S + 1)
• Z_ • I
(2, 3) (2, 6) (2, 2S-1) IZ_ • Z_ I •
(2, 4) (2, 5) (2, 2S) I(2, 2S + 1)• Z_ • I
(3, 3) (3, 6) (3, 2S-1) IZ_ • Z_ I •
(3, 4) (3, 5) (3, 2S) 1(3, 2S + 1)• Z_ • I
(R, 3) (R, 6) (R, 2S-1) IZ_ • Z_ I •
(R, 4) (R, 5) (R, 2S) i(R, 2S+ 1).... ._. _- .... _ -- __ ..=
(R + 1, 3) (R + 1, 6) (R + 1, 2S-1) C
(R + 1, 4) (R + 1, 5) (R + 1, 2S)
Figure 4: The spatial locations and the mesh indices (r, s) of mesh points used in a problem
with both horizontal and vertical walls. (a) Mesh 1. (b) Mesh 2.
NASA/TM-- 1998-208844 23
2.5
1.5
0.5
_ -0.5
-1.5
-2.5
0.0
• . . i . . . i . . . i . . . i . . .
0.2 0.4 0.6 0.8 1.0X
0.50
0.40
0.30L.
GO® 0.20L
a. 0.10
0.00
-0.10
0.0
• . . i . . . i . . . i . . . i . . .
0.2 0.4 0.6 0.8 1.0X
2.5
1.5
0.5
o_ -0.5
-1.5
-2.5
0.0
• . . i . . . i . . . i . . . i . . .
0.2 0.4 0.6 0.8 1.0X
Figure 5: The CE/SE solution and the exact solution of the Sj6green problem.
NASA/TM-- 1998-208844 24
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5 .........................
-, -3 -2 -1 0 1 2 3X
12.510.5
8.5
A
6.5L
o_ 4.5
2.5
0.5 • . . i . . . i . . . i . . . i . . . i . . . i . . .
-3 -2 -1 0 1 2
X
0
.2°
3.3
2.8
2.3
1.8
1.3
0.8
0.3
-0.2
A
• . . i . . . i . . . i . . . i . . . i . . . i . . .
-3 -2 -1 0 1 2
X
Figure 6: The CE/SE solution of the Shu-Osher problem.
NASA/TM-- 1998-208844 25
"i
8.'''''''''
? -..........-.__----_._
6
5
4
3
!0 ....................................... :
-3 -2 -1 0 1 2 3 4 5 6 7X
0
>
2
1
0
-!
-3
. . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . .
-2 -1 0 1 2 3 4 5 6 7X
27
24
21
18
15
12
9
6
3o-3 -2 -1 0 1 2 3 4 5 6 7
X
Figure 7: The CE/SE solution and the exact solution of the shock-wave merging
problem (t = 0.675, o_ = 2).
NASA/TM-- 1998-208844 26
e_
8
7
6
5
4
3
2
1
0
-3 -2 -1 0 1 2 3 4 5 6 7X
0
o
5
4
3
2
1
0
-1
-3
. . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . .
..,...,...,...,...,...,...,...,...,...
-2 -1 0 1 2 3 4 5 6 7X
27
21
18 i
15 i
12
9
6
o-3 -2 -1 0 1 2 3 4 5 6 7
X
Figure 8: The CE/SE solution and the exact solution of the shock-wave merging
problem (t = 1.1205, o_= 2).
NASA/TM-- 1998-208844 27
8
7
6
5
4
3
2
1
0
-3
. . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . .
. . . 1 . . . 1 . . . 1 . . . 1 . . . 1 . . . 1 . . . 1 . . . 1 . . . 1 . . .
-2 -1 0 1 2 3 4 5 6 7X
0
o
5
4
3
2
I
0
-1
-3
. . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . .
. . . 1 . . . 1 . . . 1 . . . 1 . . . 1 . . . 1 . . . 1 . . . 1 . . . 1 . . .
-2 -1 0 1 2 3 4 5 6 7X
27 E.:.-.._2_:__._-_:_'_.:.:._.__ .........24 ........
21 I %'_=_*==
IX,
01- ....................................
-3 -2 -1 0 1 2 3 4 5 6 7X
Figure 9: The CE/SE solution and the exact solution of the shock-wave merging
problem (t = 1.62, o_= 2).
NASA/TM-- 1998-208844 28
8
7
6
5
4
3
2
1
. . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . .
A
0 - - - ' - - - ' - - - ' - - - ' - - - ' - - - ' - - - ' - - - ' - - - ' - - -
-3 -2 -1 0 1 2 3 4 5 6 7X
0
5
4
3
2
1
0
-1
-3
. . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . .
L
. . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . .
-2 -1 0 1 2 3 4 5 6 7X
27 .......... - .......
01- ....................................
-3 -2 -1 0 1 2 3 4 5 6X
M
7
Figure 10: The CE/SE solution and the exact solution of the shock-wave merging
problem (t = 1.62, o_ = 3).
NASA/TM-- 1998-208844 29
81 #2f f0.0 0.2 0.4 0.6
X
A
A
A
A
A
A
A
m
0.8 .0
14
12
10
•8 80
6
4
2
0
-2
16
0.0 0.2 0.4 0.6 0.8 1.0X
42O
36O
3OO
24o18o
t.
120
60
0
0.0
A
i• . . , . . . , . . . , . . . , . . .
0.2 0.4 0.6 0.8 1.0
X
Figure 11: The CE/SE solution of the Woodward-Colella problem.
NASA/TM-- 1998-208844 30
3
o 4 8x
?
6
5
e_ 43
2
AUSM ÷
,,] , , , , io0 4 8
X
"L.
7
6
5
4
3
2
1
00
AUSMD V
4 8X
7
6
5
4el.
3
2
' O'
o
Roe
i
4 81(
7 V=_ Lee'r/H_..eL6
5
2
o t _
X
Figure 12: Upwind solutions of the Woodward-Colella problem.
NASA/TM--1998-208844 31
__O 0.25 1.0
- t_L4
t_L.1
t_L09
C_. (3o_taetSurra_ I,S. l.neldcnts]to¢_l_
lg.F. Igx'im_slmFa.n I_. I_fJoet_dSho¢_
T_. Tra.nsmltl_ S]toek
Figure 13: Waves in a shock-tube with closed ends.
NASA/TM-- 1998-208844 32
w
21
18
15
12
9
6
3
0 - - - ' - - - ' - - - ' - - - ' - - - ' - - - ' - - - ' - - - ' - - - ' - - -
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.60 0.90 1.00X
o
1.4
1.1
0.8
0.5
0.2
-0.1
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
X
w
n
21
18
15
12
9
6
3
0
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
X
Figure 14: The CE/SE solution and the exact solution of the waves
in a shock-tube with closed ends problem (t = 0.09).
NASA/TM--1998-208844 33
21
18
15
,._ 9
6
3
0 --'---'---'---'---'---'---'---'---'---
0.00 O.lO 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
X
0
>
1.4
1.1
0.8
0.5
0.2
-0.1
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00X
21 . . . i . . . i . . . i . . . i . . . i . . . i . . . i . . . i . . . i . . .
18
15
= 12
0 r .................. 7.7:7i7:7.7:7.7Y7.7:7;
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
X
Figure 15: The CE/SE solution and the exact solution of the waves in a
shock-tube with closed ends problem (t = 0.3).
NASA/TM-- 1998-208844 34
r_
21
18
15
12
9
6
3
0 - - ' - - - ' - - - ' - - - ' - - - ' - - - ' - - - ' - - - ' - - - ' - - -
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.60 0.90 1.00X
_o
1.4
1.1
0.8
0.5
0.2
-0.1
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.60 0.90 1.00X
L
n
21
18
15
12
9
6
3
0 - - ' - - - ' - - - ' - - - ' - - - ' - - - ' - - - ' - - - ' - - - ' - - -
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.60 0.90 1.00X
Figure 16: The CE/SE solution and the exact solution of the waves in a
shock-tube with closed ends problem (t = 0.4).
NASA/TM--1998-208844 35
e_
21
18
15
12
9
6
3
0 ---,---,---,---,---,---,---,---,---,---
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.60 0.90 1.00
X
00
1.4
1.1
0.8
0.5
0.2
-0.1
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.60 0.90 1.00
X
L
Yl
k
e_2112151869 ." ...... " " ' " " " ' " " " ' " " " ' " " " ' " " "
3 _4 ....................................................
0 . . . i . . . i . . . i . . . i . . . i . . . i . . . i . . . i . . . i . . .
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00X
Figure 17: The CE/SE solution and the exact solution of the waves in a
shock-tube with closed ends problem (t = 0.585).
NASA/TM--1998-208844 36
1.0
;_ 0.5
0.0
0 1 2 3 4X
0.75
0.50
0.25
0.00
-0.250
_ x J..L _ _..i
1 2 3 4X
Figure 18: Pressure contours and pressure coefficient at y = 0.5 of the
oblique shock problem (60x20 mesh).
0.5
0.0
0.5
0 1 2 3 4X
0.75
0.50
0.25
0.00
-0.25
0
............................ At A .............
:::::_::::_._ .............................
1 2 3 4
X
Figure 19: Pressure contours and pressure coefficient at y = 0.5 of the
oblique shock problem (120x40 mesh).
NASA/TM-- 1998-208844 37
loI 10.8 Ma = 3p= 1
0.6 p = 1.4
0.4
0.2
O0
0.0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.0
X
o oo oo oo eo
o oo oo oo oo
oo oo oo oo ooo oo oo oo oo
eo oo oo oo oo
o oo oo oo oo
eo oo oo oo oo
o oo oo oo oo
i
o olo oo oo ooIe_
0 o oo
• o
• o
• o
• o
• o
• o
X
Figure 20: Geometry and Grid distribution of the 2D supersonic flow past a
step problem.
1.0
0.8
0.6
0.4
0.2
0.0
0.0 0.6 1.2 1.8 2.4 3.0
X (60x20)
1.0 " " _
0"8 I
0.6
0.4
0.2
0.0
0.0 0.6 2.41.2 1.8 3.0
X (120x40)
loI0.8
0.6
0.4
0.2
0.0
0.0 0.6 1.2 1.8 2.4 3.0
X (_Ox80)
Figure 21: Density contours of the 2D supersonic flow past a step problem
generated using 60x20, 120x40, and 240x80 meshes.
NASA/TM-- 1998-208844 38
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
M.= 1.75
Pl-- 1.0135x 10 s N/m e
p,= 1.2014 kg/m s
u,=v,=O.O m/s
p_/pl=3.42, p_/p,=2.28
D= 152 mm
x/D
Figure 22: The initial conditions and geometry (cross section) of a cylindrical shock tube
for the blast wave problem.
OO
O0 O0 O0 O0 O0 O0 O0 O0
°°
O 00 I0
O0 O0 •
0 00 IO
O0 O0 •
O QO IO
O0 O0 •
0 00 I0
O0 O0 •
0 00 IO
O0 O0 •
0 O0 O0
O0 O0 •
0 O0 IO
©
©
• 0 •0 IO
• 0 00 •0 00 00 00 00 00
0
• 0 00 00 00 00 00
0 •0 •0 •0 •0 •0
• 0 00 00 00 00 00
0 •0 •0 •0 •0 •0
• 0 00 00 00 00 00
0 •0 •0 •0 •0 •0
• 0 00 00 00 00 00
0 •0 •0 •0 •0 •0
• 0 00 00 00 00 00
0 •0 •0 •0 •0 •0
• 0 00 00 00 00 00
0 •0 •0 •0 •0 •0
• 0 00 00 00 00 00
0 •0 •0 •0 •0 •0
• 0 00 00 00 00 00
0 •0 •0 •0 •0 •0
• 0 00 00 00 00 00
0 •0 •0 •0 •0 •0
• 0 00 00 00 00 00
• 0 • 0 • 0 • 0 • 0 • 0 • 0 • 0
0 • 0 • 0 • 0 • 0 • 0 • 0 • 0 • 0
•.io • o • o • o • o • o • o • o ,3o
rdl)
Figure 23: The computational domain and mesh-point distribution of the blast wave
problem (planar-flow version).
NASA/TM--1998-208844 39
3.0 ...,...,...,...,...,...,...,...
2.5(a)t=0.1996 msec
2.0
1.00.5
0.0 .................
-1.0-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
X/D
r_"_ 1.5
3.0 ...,...,...,...,...,...,...,...
2.5(b)t=0.4937 msec
2.0
1.00.5
0.0 ........
-1.0-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
X/D
r_1.5
3.0
2.5
2.0
1.0
0.5
0.0
-1.0-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
_D
1.5
3.0
2.5
2.0
1.0
0.5
0.0
-1.0-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
X/D
r_"_ 1.5
3.0
2.5
2.0
1.0
0.5
0.0
-1.0-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
_D
_. 1.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-1.0-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
X/D
3.0
2.5
2.0
1.0
0.5
0.0
-I. -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
X/D
1.5
Figure 24: Pressure contours of the blast wave problem at eight different time levels.
NASA/TM-- 1998-208844 40
3.0f" "', "'" ,'" "," "', "'', "'" ,'" "," ""
2.5 (a)t=0.1996 msec
2.0
1.5
1.0_
0.5
0.0 .................
-I.0-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
X/D
3.0 ...,...,...,...,...,...,...,...
2.5
(b)t=0.4937 msec
2.0
1.00.5
0.0 ......
-I.0-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
X/D
_- 1.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-I.0-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
X/D
ao I'\2.5
2.0
1.0 I0.5
0.0 ,.
-1.0-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
X/D
_- 1.5
3.0
2.5
• e
2.0
1.0
0.5
0.0
-1.0-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
X/D
1.5
a.o ...................... I"]
2.5
-1.0-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
X/D
3.0
2.5
2.0
"_ 1.5
1.0
0.5
0.0
-I.0-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
X/D
Figure 25: Density contours of the blast wave problem at eight different time levels.
NASA/TM-- 1998-208844 41
3.5_ ,
3.0
2.5
2.0
>-
1.5
1.0
0.5
(a)t=O.420.0 ............
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5X
3.5
3.0
2.5
2.0
>-
1.5
1.0
0.5
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5X
3.5-- _ " , " " , " , " , "
3.0
2.5
2.0
>-
1.5
1.0
0.5
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
X
Figure 26: The computational domain and density contours at three different
time levels of the diffraction of shock wave down a step problem (first case).
NASA/TM--1998-208844 42
: iiiiiii_;;:i!i:i:iiiii:i!i!iliiii:i;ii:iiiiill: ,ii:i_i:i_ii_i_:i_iiiiiiiiiiiiiii:iiiiiiii:_ii/iiii:i::_ii!ii:_:_!ii::i:, ::i_ ?i?::: i:::i:i_::_i';:i:!iil;::_:_iii:_:i:;i::
Figure 27: Experimental results of the diffraction of shock wave down a
step problem (first case--three different time levels).
NASA/TM-- 1998-208844 43
>-
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
(a)t=0.275
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5X
3.5
3.0
2.5
2.0
>-
1.5
1.0
0.5
0.0
(b)t=O.875
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5X
>-
3.5 , , • , • '_ _.__
3.0
2.5
2.0
1.5
1.0
0.5 (c)t=l •3750.0 ...........
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5X
Figure 28: The computational domain and density contours at three different time
levels of the diffraction of shock wave down a step problem (second case).
NASA/TM-- 1998-208844 44
': :: +:: _:::i:!_:::i:i_!::i:i_:i;: :;i:i i:::: :: !::ii::i_i; ::::::i!::ii:_:i/i%iiiiiiiii
_ :: :_ _i _,_ _ _ __, :_ : _:_ _ _ _: , _ _ _::: ::_ :: ::,: :_ _ ,:i:;:ilf: U_:_I_I:III_
Figure 29: Experimental results of the diffraction of shock wave down a
step problem (second case--three different time levels).
NASA/TM-- 1998-208844 45
shock y,
Y
o
X t
X
Figure 30: Shock moving past a wedge with a dust layer.
i i i i
3
ent shock wave
1
o-0.50.0 0.5 1.O 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
x'/L
2
Figure 31" The computational domain of the dust layer problem.
NASA/TM-- 1998-208844 46
0 . ae_. ..........................
-1 0 1 2 3 4 5 6x/L
4
3
2
1
• • • i • • • i • • • i • • • i • • • i • • • i • • • i • • •
-1 0 1 2 3 4 5 6 7x/L
Figure 32: Density contours for the dust layer problem (Ow = 30 °) at four
different time levels. (a) t = 0.5, (b) t = 1.75, (c) t = 3, (d) t = 4.
NASA/TM-- 1998-208844 47
5
4
3
2
1
0
-1
• • • I • • • I • • • I • • • I ...... I • • • I • • •
0 1 2 3 4 5 6x/L
7
5.5
4.4
3.3
2.2
1.1
0.0
-1
• • • I • • • I • • • I • • • I • • • I • • • I • • • I • • •
0 1 2 3 4 5 6x/L
Figure 32: (continued)
7
NASA/TM-- 1998-208844 48
5.5
4.4
3.3
2.2
1.1
0.0
-1
• • • I • • • I • • • I • • • I • • • I ...... I • • •
0 1 2 3 4 5 6 7x/L
Figure 33: Density contours at t = 3.8 for the dust layer problem ( Ow = 20°).
5.5
4.4
3.3
2.2
1.1
0.0
-1
• • • I • • • I • • • I • • • I ...........
0 1 2 3 4 5 6 7x/L
Figure 34: Density contours at t = 3.0 for the dust layer problem ( Ow = 40°).
NASA/TM-- 1998-208844 49
Figure 35: A schlieren photography for (0w = 20°).
Figure 36: A schlieren photography for (0w = 30°).
Figure 37: A schlieren photography for (0w = 40°).
NASA/TM-- 1998-208844 50
2
ii ................(a)
-2 -1 0 1 --22 -1 0 1 --22 -1 0 1 2
t=0.203 t=0.387 t=0.428
2
-2
z
. m . m . m . -_ . m
-1 0 1 --22 -1
t=0.392
I I0
t=0.504
m . --9 . m
I "-22 --I
|
0
t=0.639
| .
1 2
2
(c)-2
. | . | . |
-1 0 1
t=0.39
0
1'
--22 -1 0 1 -22 -1 0 1 2
t=0.484 t=0.694
Figure 38: Pressure contours for implosion/explosion in a square box with
different initial shock wave configurations.
(a) an equilateral triangle. (b) a square.
(c) a regular pentagon.
NASA/TM--1998-208844 51
(a)-2 -1 0 1 -22 -1 0 1 _ -1 0 1 2
t=0.203 t=0.387 t=0.428
2 - ' - ' - ' - Z - ' - - ' - Z - ' ' ' -
ii ................_c_ -2 -_ o _ -_ -_ o _ -_ -_ o _ 2
t=0.39 t=0.484 t=0.694
Figure 39: Density contours for implosion/explosion in a square box with
different initial shock wave configurations.
(a) an equilateral triangle. (b) a square.
(c) a regular pentagon.
NASA/TM-- 1998-208844 52
2 -1 0 1
t=0.394
-22 -1 0 1 -22 -1 0 1 _"
t=0.488 t=0.761
2 -1 0 1 _ -1 0 1 --22 -1 0 1 ;
t= 1. 172 t= 1.965 t=2.465
- m - m - m - _ m - m - m - _ m m - m
2 -1 0 1 --22 -1 0 1 --22 -1 0 1 ;
t=3.166 t =3.452 t=4.17,5
Figure 40: Pressure contours for implosion/explosion of a hexagonal shock in a
square box.
NASA/TM--1998-208844 53
2 -1 0 1 -22 -1 0 1 -22 -1 0 1 ;t=0.394 t=0.488 t=0.761
..........
2 -1 0 1 -22 -1 0 1 -22 -1 0 1 ;
t=3.166 t=3.457 t=4.175
Figure 41" Density contours for implosion/explosion of a hexagonal shock in a
square box.
NASA/TM-- 1998-208844 54
Form ApprovedREPORT DOCUMENTATION PAGEOMB No. 0704-0188
Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources,
gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this
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Davis Highway, Suite 1204, Arlington, VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC 20503.
1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVEREDDecember 1998 Technical Memorandum
4. TITLE AND SUBTITLE 5. FUNDING NUMBERS
The Space Time Conservation Element and Solution Element Method_ New Higl_
Resolution and Genuinely Multidimensional Paradigm for Solving Consmwation LawsII. Numerical Sinmlation of Shock Waves and Contact Discontinuities
6. AUTHOR(S)
Xiao-Yen Wang, Chuen-Yen Chow, and Sin-Chung Chang
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)
National Aeronautics and Space Administration
Lewis Research Center
Cleveland, Ohio 44135- 3191
9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)
National Aeronautics and Space Administration
Washington, DC 20546- 0001
WU-538-03-11-00
8. PERFORMING ORGANIZATION
REPORT NUMBER
E- 11457
10. SPONSORING/MONITORING
AGENCY REPORT NUMBER
NASA TM--1998-208844
11. SUPPLEMENTARY NOTES
Xiao-Yen Wang, Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, Minnesota
55455; Chuen-Yen Chow, Department of Aerospace Engineering and Science, University of Colorado at Boulder, Boulder,
Colorado 80309-0429; Sin-Chung Chang, NASA Lewis Research Center. Responsible person, Sin-Chung Chang, organi-
zation code 5880, (216) 433-5874.
12a. DISTRIBUTION/AVAILABILITY STATEMENT
Unclassified - Unlimited
Subject Categories: 34, 59, 61 Distribution: Nonstandard
This publication is available from the NASA Center for AeroSpace Information, (301) 6214)390.
12b. DISTRIBUTION CODE
13. ABSTRACT (Maximum 200 words)
Without resorting to special treatment for each individual test case, the 1D and 2D CE/SE shock-capturing schemes
described in Part I are used to simulate flows involving phenomena such as shock waves, contact discontinuities, expan-
sion waves and their interactions. Five 1D and six 2D problems are considered to examine the capability and robustness of
these schemes. Despite their simple logical structures and low computational cost (for the 2D CE/SE shock-capturing
scheme, the CPU time is about 2 psecs per mesh point per marching step on a Cray C90 machine), the numerical results,
when compared with experimental data, exact solutions or numerical solutions by other methods, indicate that these
schemes can accurately resolve shock and contact discontinuities consistently.
14. SUBJECT TERMS
Space-Time; Flux conservation; Conservation element; Solution element; Shocks;
Contact discontinuities
17. SECURITY CLASSIFICATIONOF REPORT
Unclassified
NSN 7540-01-280-5500
15. NUMBER OF PAGES
6016. PRICE CODE
A0418. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT
OF THIS PAGE OF ABSTRACT
Unclassified Unclassified
Standard Form 298 (Flev. 2-89)Prescribed by ANSI Std. Z39-1B298-102
