Vol.44 No. 4SCIENCE IN CHINA( Series E)August 2001Explicit analytical wave solutions of unsteady 1D ideal gasflow with friction and heat transferCAI Ruixian(蔡睿贤)& ZHANGNa(张娜)Institute of Engineering Thermophysics , Chinese Academy of Sciences , Beiing 100080 , ChinaCorrespondence should be addressed to Cai Ruixian( email crx@ mail. etp. ac. cn)Received December 26 , 2000AbstractSeveral families of algebraically explicit analytical wave solutions are derived for the un-steady 1D ideal gas flow with friction and heat-transfer , which include one family of travelling wave S0-lutions , three families of standing wave solutions and one standing wave solution. Among them , theformer four solution families contain arbitrary functions , so actually there are infinite analytical wave So-lutions having been derived. Besides their very important theoretical meaning , such analytical wave S0-lutions can guide the development of some new equipment , and can be the benchmark solutions topromote the development of computational fluid dynamics. For example , we can use them to check theaccuracy , convergence and effectiveness of various numerical computational methods and to improvethe numerical computation kills such as differential schemes , grid generation ways and so on.Keywords : explicit analytical solution , travlling wave , standing wave , friction , heat transfer. .Analytical solutions of various fundamental governing equations are very important for thescientific problem described by the equations , both theoretically and practically. For example ,many analytical solutions of incompressible non- viscous flow made a great contribution to the earlydevelopment of fluid dynamics. Even in recent years , the computational fluid dynamics and com-puters have made marvelous progress and the computational solutions of many scientific and tech-nical problems can be obtained by the numerical method , analytical solutions still have their ownirreplaceable role. For example , as benchmark solutions they can be used to check the accuracy ,convergence and effectiveness of various numerical methods , and to improve the numerical com-putational skills such as diferential schemes , grid generation ways and so on. Of course , analyti-cal solutions keep their theoretical meaning also. They are the strictly accurate and typical an-swers to the scientific and technical problems described by the governing equations ; in addition ,sometimes they can even be directly applied to practical engineering. Therefore , to derive analyt-ical solutions is one of the topics much concerned by many disciplines .However , derivation of analytical solutions becomes more difficult when the fundamentalgoverning equations describing the practical phenomena become more, accurate and complicated.For example , the goverming equations of incomp中国煤化工linear ; they have manyanalytical solutions. But as the governing equaticMHC N M H Gl flow are strongly non-linear , it is very difficult to derive analytical solutions with common methods. There had been on-ly three kinds of analytical solutions of steady 2D compressible flow before the 1990s which werefound 50 years ago ; they are the vortex flow , source and sink flow and Ringleb flow1. In theearly 1990s , magy algebraically explicit analytical solutions of unsteady geometrical 1D , 2D方方数据’No.4UNSTEADY 1D IDEAL GAS FLOW415and 3D compressible flows were first derived by Cai with uncommon methods 231. However , theydo not include solutions with at least two wave dependent variables. Actually , wave solutions notonly have their own theoretical meaning , but also can guide the development of some equipment .For example , they are useful for the thermal-acoustic heat engines and refrigerators 41. There-fore , based on the previous experience of the authors , some algebraically explicit analytical solu-tions of unsteady 1D compressible flow with travelling wave or standing wave are derived to enrichthe unsteady compressible flow theory.1 Governing equation setThe goverming equation set of unsteady 1D ideal gas compressible flow with friction and heattransfer can be expressed asaρ duaρ+ pu.1dA= 0,( 1a)+pxxA dxJu.. du. 1 p+ux+( lb)ρ 0xU(h- 1))0(9+/采利;=-+u卫_kp(?ρ( 1c)JtJxρ \dtwhere x and l are the geometric and time coordinates respectively ,u ,ρ φ and h are the veloci-ty , density , pressure and specific heat ratio of the working substance respectively ,A is the flowarea , f is the friction coefficient and q is the heat transfer per unit flow rate.To physically solve equation set( 1 ) , commonly one should first give the variationsoff( x ,t ),( x ,t ),A( x ) and sufficient initial and boundary conditions , and then derive p( x ,t ),dx ,t )and u( x ,t )with equation set( 1 ). However , the main purpose of this paper is to findout some possible algebraically explicit analytical solutions ( without infinite series and specialfunctions ) as benchmark solutions but not to derive the special solutions for given initial andboundary conditions . Therefore , in the following derivation , we first deduce the unknown func-tions which satisfy equation set( 1 ) , and then determine the corresponding initial and boundaryconditions. For the same reason , in order to make it easy to derive possible analytical solutions,arbitrary three physical parameters are chosen from f ,q A Ip rρ and u as given functions andthen other three parameters are solved .2 An explicit analytical travelling wave solution family for constant friction coefficientand flow areaWhen f= Const. and A = Const. ,ifu=Const.=土uo,(2)then it follows from continuity eq. ( 1a ) thatρ=Po土中国煤化工(3 )is the solution of density ρ , where g is an arbitYHC N M H Gant. Substituting P andu into momentum equation( 1b ) , we have the expression ot pressure as follows :p = Po干~u[Pox士)g(x干uot )x].(4)If the integral in eq.(4 ) can be explicitly integrated , then eq. ( 4 ) represents the explicitanalytical' soH剂描of pressure. Now , in energy equation( lc ) , all terms except heat transfer per416SCIENCE IN CHINA( Series E)Vol. 44unit flow q are known ,so q can be derivedq=+uldt. atJx=干fu8土P0(5)(h- 1)pJSince there are infinite integrables g( x干uot ),eqs.(2) ( 4 )actually represent infiniteexplicit analytical solutions. For example ,if g= ρisir( x干uot ), then a simple typical travel-ling analytical solution can be derivedA = Const. f = Const. ，q =干人fui{1(k- 1Iρ土p1sir( x干uol )]u = Const. =士uo ,ρ = ρo士ρ1sin(x干uot ),(6 )(p = po干λjfuaρox土ρ1co(x干uol )]To ensure ρ>0 and p> 0 , the following conditions have to be satisfied : ρo> ρ1| andpo>V π/AfuK.ρoxmax+ p1 ),where xXnax is the maximum value of x in the region discussed.The initial and boundary conditions of this solution can be deduced by substituting the initial andboundary values into equation set( 6). For example , when 1=0 ,u=土uo ,ρ= Po土ρ1sinx.ππPop=po干〈4fu[ pρox干p1cosx],q=干. gfuB1+The case is(h-1 ρo土 ρisinx )」the same with the following solutions and we will not address it in detail.The typical variations in p. ,p and q are shown in fig. 1 ,where u= uo.Equation set( 6 ) shows that for the constant ve-locity flow in constant flow area tube with a constantfriction coeffcient the density travelling wave and heattransfer travelling wave are in phase , but the pressureptravelling wave is π/2 out of phase with density travel-ling wave. This situation can be understood also fromeq.( lb): since ap/ax~ - ρ when u= Const. ,p~co( x + lot ),when ρ~ sir( x + uot ). Note thatthe case of constant friction coefficient , constant flow0.0area and constant velocity flow is the commonest ,so it .is believed that this solution is not only a benchmarksolution of computational fluid dynamics , but will be4.08.012.0of other typical significances. In addition , the above-x-uJmentioned analytical solution derivation procedure ofstron中国煤化工ential equation set( 1 )Fig. 1. Typical variation curves of solution( 6).is veMYHC N M H Gwledge , there have notyet been such procedure and solution being published.Another simple and meaningful solution of constant friction cofficient , constant flow areaand constant velocity can be obtained withu= uo ,ρo=0 andg=ρ/[1 +( x- uot尸], whereρ1 is the maximwe density in the process , occuring at x = uot. Mathematically ρ is a witchNo.4UNSTEADY 1D IDEAL GAS FLOWcurve with independent variable x一u0t. The solution is as follows :u=u0,(7a)ρ = ρ1[1 +(x- uol }],(7b)p= Po-NAfuEρptan~'(x- uot ),(7c)q =-N(7d)where po has to be larger than入人fupρ1 to ensure that p >0. Mathematically μx- uol )is .an arctangent curve .It is valuable to mention that the flow cooling rate of this solution is a constant.Shown in fig. 2 are the p and p curves of this solution at t =0. As time goes on , all curvespass to right with fixed curve shape and constant velocity uo.By the way , the first explicit analytical solutionfamily given by ref.[ 2]is actually a special case ofsolution( 6 ).3 Explicit analytical standing wave solution fam-ilies with ρ(x )Let f= Const. ,( dA/dx )A = C( throughoutthis paper each C; is A = c2e*1* constant ),a ρ =d x),u= T( t )X( x ). Then solving continuityequation( la ) with ordinary method of separation 0.0一variables , we have0+ XTp'+ pX'T+ C1ρXT= 0，i.e. -10.0-5.00.05.010.0(8 )Fig.2. Typical variation curves of solution( 7).It follows from eq.( 8 )that T( t )can be an ar-Xbitrary function T= h( t ) andρ=-了-C1.Since ρ and X are functions of x , we have= g'(x)=---C1，(9 )(where g( x ) is an arbitrary function of x. It is easy to solve eq. ( 9 ) and obtainρ =Poe'( 10)u = uoe-C1(11 )中国煤化工Substituting the above expressions of ρ u0HCNMHGGation( 1b )( when f=0，lul/u can be unity-11 ; otherwise u mustap=- pouoe-Cx. h'(t)+ poud C1+ g'(x)}-2C,x-(x). [h(t)FJxπ人C pufe-sC,s.26-). [h(l)].(12)418SCIENCE IN CHINA( Series E)Vol. 44When C1=0 and f=0 ,or g( x )is a linear function of x ,eq.( 12 ) can be integrated and theexplicit analytical solution of pressure p can be deduced .The first case is equivalent to non-viscous constant area flow. The final expressions of its .analytical solutions areA = Const. ,f=0,q =h- 1“0,{- xe-6(x). h"(t)+ e-2d(x)(tI kuoxg'( x)h'(t)- 3uoh'(l )- kpog"( x )ρo] .( 13)+(h: + 1 )uje-3(x)g'(x) [h(t)]},u = uoe"(x). h(t)，ρ = poebx),p = po- pouoxh'(l)- poue-s(x). [h( l)}.Since there are two arbitrary functions g( x ) and h( l ) and their derivatives in this solu-tion , a family of infinite explicit analytical solutions has been obtained. In addition , the specificflow rate pu of this solution family is only a function of time.For example ,if g( x )= Cssin( nx+β),h( t )=sin( wt +a )( where a β ,w and n arearbitrary constants ) , the following standing wave solutions can be deduced :A = Const. ,f= 0,q = uo{xe-Csir(nx+B). w2sin( wt + a)- 3uoe-2Csin nx+B)wsi[2Xwt + a)]2- kpe-2c.idnx+B)Csnco( nx + β);i( wt + a )po+ kuoxe-2C.sin nmx+B)Csncos( nx + β )usir[ x ot + a )]2(14)+(h: + 1 )u6e-3CSsidmx+B)Csnco( nx + β)sin( wt + a)}( k- 1 ),u = upe-csinnx+B). sin(wt + a),p = peSsin(nx+B) ,lp = po- poluoxwco( wt + a)- pou5e- Csindmx+B){1 - co[x wt + a )}/2.po has to be larger than PouoXmaxW + Pouδe's/2 to ensure p > 0 , where Xmax is the maximumvalue of x in the region discussed. In addition , the different phases of various parameters areworth noticing .The typical variation curves of solution( 14 ) are shown in fig. 3. When Cs= 0 , solution( 14 ) degenerates into a solution of incompressible flow.The second integrable case of eq.( 12)g( x )= C3x+ C4 is a flow with friction and vari-able area. The final expressions of its analytical solutions are[A = C2eC.x ,f= Const. ,u = upe-(C+C3)x+Ci1. h(t),ρ = poeC:*+Cg ,p =po+Pluoe-Cr*h(1)- pouKCi + C3)+(2C.+C)2+C.[h(l)}(2Ci+ C3)C(15)+/ C,foune(2.5C.+C,)+C.[ h(心2中国煤化工The analytical expression of q can be dedud.TYHC N M H Glytical expressionofu，ρ and p into eq.( 1c ). Since f≠0 , u ,h( t ) themselves should have the same sign in the re-gion discussed. In addition , po has to be large enough to ensure p > 0.There is an arbitrary function h( t ) and its derivative in solution( 15 ). Therefore , anotherfamily of infinite explicit analytical solutions has also been obtained. By the way ,( 15 ) degener-ates into丽方熬据pressible solution when C3= 0.No.4UNSTEADY 1D IDEAL GAS FLOW41910π/712x/ 78π/7t=06π/72π/ 74π/ 7| (a)
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