《复变函数论》课程教学资源（书籍文献）A Complex Variable Boundary Element Method for the Flow around Obstacles

Proceedings of the World Congress on Engineering 2009Vol IIWCE2009, July1-3,2009, London, U.KA Complex Variable Boundary Element Methodfor the Flow around ObstaclesLuminitaGrecu,ValentinGrecu,MihaiDemian,GabrielaDemianThepaperpresents anapplicationof thecomplexvariableAbstractThepaperpresentsanapplication oftheComplexboundary element method(CVBEM)for solving a boundaryVariable Boundary Element Method (CVBEM)to solve avalue problemover two-dimensional triply connectedboundary value problem over two-dimensional multiplyregions, in fact for the problem of a potential fluid flowconnected regions, in fact for the problem of a potential fluidaround two objects.flow around objects.The CVBEM is a powerful numerical toolThe application of CVBEM for solving problems overfor solvinggenerallytwo-dimensional boundaryvalueproblemstwo-dimensional multiply connected regions has a greatin which appear complex functions, and it represents apractical importancein computational fluid dynamicsnumerical application of CauchyIntegral Theorem.because,for example,there can be developed streamlinesFor solving the boundary integral the problem is reduced atthere can be used different kinds of boundary elements. In thiswithin a riverwith flows past bridge piers, and so it can bepaper there are used linear boundary elements, so theusedtodesign thebridge pier alignment so astominimize thegeometries involved are approximated by polygonal lines anddisturbance.This method can be also successfullyapplied infor the approximation of the unknowns there are used linearother kind of problems of continuum mechanics as heatbasis functions. The CVBEM's advantage overotherconduction[3], cracks, etctechniques, pointed out by the present paper, is the fact thatwhen this method is applied the approximation exactly solvesthe equation, so using this method good approximations can beI. THE CVBEM NUMERICAL STATEMENTfound. A computer code based on this method is developed andnumerical results are obtained for some particular cases.Let us consider a uniform steady potential bi-dimensionalriver flowofan inviscidfluid past some arbitrary obstacles,IndexTerms-complexboundary element method,fluid flow,firstweconsideronlytwo,ofboundary2Wewantlinear boundary element, multiply connected domaindetermine theperturbation induced by the presence of theobstaclesandtheaction exerted bythefluidonthem applyingI.INTRODUCTIONthe CVBEM.Using dimensionless variables, we have:Ap(x,y)=0 on 2,(1)By use of the Cauchy integral equation for complexvariable analytic functions it is obtained an advancedwhere p(x,y)is the perturbation potential, is he fluidmathematicalapproachforsolvingtwo-dimensionaldomain,amultiplyconnecteddomainenclosedbypotential problems as those that arise when we study a fluidboundaries r',Fi,F2 (T=I*Ur,UT2),flowaround one or more objects.The theoretical bases ofthismethodwhereputaround1983byHromadkaanditsand theboundary conditions:gradp·n =0 across theflowcollaborators [1], [2]boundaries on * and on I UT2, where n(nx,n,)is theThe advantage of this method over the othermethods thatoutward unit normal at thecorresponded boundary,andbycanbeusedtosolvethesameproblemscomesfromthefactthat the numerical application in this case is analytic and sodefining an arbitrarily chosen potential drop between thethe approximation exactly solves theequation, whiletheupstream and downstream boundaries, noted and 2other numericaltechniquesdeveloponlyinexactUsing the complex variable z=x+iy,the perturbationapproximationsfortheequation.potential f(=)=p(-)+iy(-), where y() is the streamfunction, and y being related by the Cauchy-RiemannoooManuscript received Mars 24, 2009equations,real-valued functions thataxdyoyaxLuminita Grecu is with the University of Craiova, Faculty ofare harmonic functions for ::Ap=0,=0,we get aEngineering and Management of Technological Systems Dr. Tr. Severin,holomorphicfunctionf+40252333431;(phone:fax:+40252-317219e-mail:lumigrecu@hotmail.com)Weconsideran approximation of theproblemboundaryValentin Grecu is Phd. Student at University of CraiovaMihai Demian is with the University of Craiova,Faculty of EngineeringF as a polygonal line - Urk, where Fr is a straight lineand Management of Technological Systems Dr. Tr. Severin,(phone:+40252333431;fax:+40252-317219;e-mail:mihaidemian@yaho0.com)k=lGabriela Demian is with the University of Craiova, Faculty ofsegment with nodal points at the end-points, noted zk,+1Engineering and Management of Technological Systems Dr. Tr. Severinsituated on the real boundary.+40252333431+40252-317219;e-mail:(phone:fax:gabrielademian@yahoo.com).ISBN:978-988-18210-1-0WCE2009

Abstract— The paper presents an application of the Complex Variable Boundary Element Method (CVBEM) to solve a boundary value problem over two-dimensional multiply connected regions, in fact for the problem of a potential fluid flow around objects. The CVBEM is a powerful numerical tool for solving generally two-dimensional boundary value problems in which appear complex functions, and it represents a numerical application of Cauchy Integral Theorem. For solving the boundary integral the problem is reduced at there can be used different kinds of boundary elements. In this paper there are used linear boundary elements, so the geometries involved are approximated by polygonal lines and for the approximation of the unknowns there are used linear basis functions. The CVBEM’s advantage over other techniques, pointed out by the present paper, is the fact that when this method is applied the approximation exactly solves the equation, so using this method good approximations can be found. A computer code based on this method is developed and numerical results are obtained for some particular cases. Index Terms—complex boundary element method, fluid flow, linear boundary element, multiply connected domain I. INTRODUCTION By use of the Cauchy integral equation for complex variable analytic functions it is obtained an advanced mathematical approach for solving two-dimensional potential problems as those that arise when we study a fluid flow around one or more objects. The theoretical bases of this method where put around 1983 by Hromadka and its collaborators [1], [2]. The advantage of this method over the other methods that can be used to solve the same problems comes from the fact that the numerical application in this case is analytic and so the approximation exactly solves the equation, while the other numerical techniques develop only inexact approximations for the equation. Manuscript received Mars 24, 2009. Luminita Grecu is with the University of Craiova , Faculty of Engineering and Management of Technological Systems Dr. Tr. Severin, (phone: +40252333431; fax: +40252-317219; e-mail: lumigrecu@hotmail.com). Valentin Grecu is Phd. Student at University of Craiova Mihai Demian is with the University of Craiova , Faculty of Engineering and Management of Technological Systems Dr. Tr. Severin, (phone: +40252333431; fax: +40252-317219; e-mail: mihaidemian@yahoo.com). Gabriela Demian is with the University of Craiova , Faculty of Engineering and Management of Technological Systems Dr. Tr. Severin, (phone: +40252333431; fax: +40252-317219; e-mail: gabrielademian@yahoo.com). The paper presents an application of the complex variable boundary element method (CVBEM) for solving a boundary value problem over two-dimensional triply connected regions, in fact for the problem of a potential fluid flow around two objects. The application of CVBEM for solving problems over two-dimensional multiply connected regions has a great practical importance in computational fluid dynamics because, for example, there can be developed streamlines within a river with flows past bridge piers, and so it can be used to design the bridge pier alignment so as to minimize the disturbance. This method can be also successfully applied in other kind of problems of continuum mechanics as heat conduction [3], cracks, etc II. THE CVBEM NUMERICAL STATEMENT Let us consider a uniform steady potential bi-dimensional river flow of an inviscid fluid past some arbitrary obstacles, first we consider only two, of boundary 1 2 Γ , Γ . We want to determine the perturbation induced by the presence of the obstacles and the action exerted by the fluid on them applying the CVBEM. Using dimensionless variables, we have: Δϕ(x, y) = 0 on Ω , (1) where ϕ(x, y)is the perturbation potential, Ω is he fluid domain, a multiply connected domain enclosed by boundaries 1 2 * Γ ,Γ ,Γ ( 1 2 * Γ = Γ ∪Γ ∪Γ ), and the boundary conditions: gradϕ ⋅ n = 0 across the flow boundaries on * Γ and on Γ1 ∪ Γ2 , where ( ) n nx ny , is the outward unit normal at the corresponded boundary, and by defining an arbitrarily chosen potential drop between the upstream and downstream boundaries, noted ϕ1 and ϕ2 . Using the complex variable z = x + iy , the perturbation potential f (z) = ϕ(z) + iψ (z) , where ψ ( )z is the stream function, ϕ and ψ being related by the Cauchy-Riemann equations x ∂y ∂ = ∂ ∂ϕ ψ y ∂x ∂ = − ∂ ∂ϕ ψ , real-valued functions that are harmonic functions for z : Δϕ = 0, Δψ = 0 , we get a holomorphic function f We consider an approximation of the problem boundary Γ as a polygonal line ∪ N k k =1 Γ = Γ , where Γk is a straight line segment with nodal points at the end-points, noted 1 , k k + z z , situated on the real boundary. A Complex Variable Boundary Element Method for the Flow around Obstacles Luminita Grecu, Valentin Grecu, Mihai Demian, Gabriela Demian Proceedings of the World Congress on Engineering 2009 Vol II WCE 2009, July 1 - 3, 2009, London, U.K. ISBN:978-988-18210-1-0 WCE 2009

Proceedings of the World Congress on Engineering 2009 Vol IIWCE2009, July1-3,2009, London, U.KWe choose m nodal points zj, j=1,m+1, on the outerWe further get:curve ",=m+1=zi, numbered in a counterclockwiseF(≤)dr:direction, n nodes =j,j=m+2,m+n+2, on the inner.curve=m+n+2=zm+2,located in a clockwise dirctionzjFj-z,FjdgF-FCaeand p nodes zj, j=m+n+3,m+n+p+3, on the other1,5-202 j1 -Z,Zj+1 -Tinner curve 2,=m+n+3=zm+n+p+3ocated inaclockwieThe two integrals from the right side of the above relationdirection too.can be analytically evaluated:m+n+p+2m+n+1d5_=In(5-0)ESo we have: I"- Urk, Ii=-"Urk, I2 -Urk5-20k=lk=m+2k=m+n+3SThe next step in using the CVBEM is to develop a2j1-20continuous approximation of the unknown f(-) on F by=In州-20+ie(j+1,j)=In,(=0)Inm+n+p+2z, -202,- 20the global trial function F(-)=N(-)Fk, zerk=lk+m+1aedzk*m+n+21- z.)+ 20where Nk(=) is a continuous function representing the202influence of over elements that have zk as nodal point, so=(5j+1 -2,)+ z0ln(5 -20)l =(5j1 -2,)+Over Fk-1 and Fk.The appoximation weconstruct is()(Z j+1 - 20+i0(j+1, J)+zln2元iJZ,-Z0e,the integral been taken in the counterclockwisewhere (j+1,j) is the central angle between straight linedirection.segment joining points zj and zj+1 to central pointBecause F()is continuous on T,f(-) is analytic in Q20Qas an extension of the following theorem given in [1] tomultiply connected regions, and so its real and imaginarySo we deduce:parts satisfy Lapalce equation overQTheorem 1.F()20-211(c0)-LetFbe a simple closed contour with finitelengthL and= Fj F, +Fj-dlsimply connected interior .Let h()be a continuousZo2j+1-2function on T, Then w(-) is analytic in Q, where w(=) is20--2(201(6)dsdefined by the contour integral w()Zi+12元Finally we get:III. LINEAR BASIS FUNCTIONWegetthefollowing discretized form2元if(=)=m*-F)F()dc(F17(=)=2元/el24-j+m+1uritm+n+=m+p+2[Ff(20 -2)-F,(0 -1*m++220In this paper we consider on each boundary element aZj+1-2j=linearapproximationforF(-).Aftersomecalculuswegetttfor the nodal pointjthefollowing linear basisfunctionBecause the first term cancels wededuce:1ZET-Im++p+2[Fj(50 --)-F,(50 -=j+12, 2j-12f(=0)=>z0= j+I 2j+1-2jj=1N,(=)=ZETj+m+1ZJ+1 - ZJj+m+n+20, z@T,, Ur,and furtherISBN:978-988-18210-1-0WCE2009

We choose m nodal points z , j = 1,m + 1 j , on the outer curve * Γ , 1 1 zm = z + , numbered in a counterclockwise direction, n nodes z , j = m + 2,m + n + 2 j , on the inner curve Γ1 , m+n+2 = m+2 z z , located in a clockwise direction and p nodes z , j = m + n + 3,m + n + p + 3 j , on the other inner curve Γ2 , m+n+3 = m+n+ p+3 z z , located in a clockwise direction too. So we have: , 1 * ∪ m k k = Γ = Γ , 1 2 1 ∪ + + = + Γ = Γ m n k m k ∪ 2 3 2 + + + = + + Γ = Γ m n p k m n k . The next step in using the CVBEM is to develop a continuous approximation of the unknown f ( )z on Γ by the global trial function () () ∑ + + + ≠ + + ≠ + = = 2 2 1 1 m n p k m n k m k k Fk F z N z , z ∈Γ , where N ( )z k is a continuous function representing the influence of over elements that have kz as nodal point, so over Γk −1 and Γk . The approximation we construct is ( ) ( ) ζ ζ ζ π d z F i f z ∫ Γ − = 2 ~ 1 , z ∈Ω , the integral been taken in the counterclockwise direction. Because F( )z is continuous on Γ , f ( )z ~ is analytic in Ω as an extension of the following theorem given in [1] to multiply connected regions, and so its real and imaginary parts satisfy Lapalce equation over Ω . Theorem 1. Let Γ be a simple closed contour with finite length L and simply connected interior Ω . Let h( ) ζ be a continuous function on Γ . Then w( )z ~ is analytic in Ω , where w(z) ~ is defined by the contour integral ( ) ( ) ζ ζ ζ π d z h i w z ∫ Γ − = 2 ~ 1 . III. LINEAR BASIS FUNCTION We get the following discretized form: ( ) ( ) ζ ζ ζ π d z F i f z m n p k m n k m k k ∫ + + + ≠ + + ≠ + = Γ − = ∪ 2 2 1 1 2 ~ 1 In this paper we consider on each boundary element a linear approximation for F( )z . After some calculus we get for the nodal point j the following linear basis function: ( ) ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ ∉Γ Γ ∈Γ − − ∈Γ − − = − + + − − − j j j j j j j j j j j z z z z z z z z z z z N z 1 ∪ 1 1 1 1 1 0, , , We further get: ( ) ∫ ∫ ∫ Γ + Γ + + + + Γ − − − + − − − = = − k j k z d z z F F z d z z z F z F d z F j j f j j j j j j j 1 0 1 1 0 1 1 0 ζ ζ ζ ζ ζ ζ ζ ζ . The two integrals from the right side of the above relation can be analytically evaluated: ( ) ( ) () 0 0 1 0 0 1 0 0 0 ln ln 1, ln 1 i j j z z z z z z z z z ln z z d j j j j j z z j j j + + = − − = − − = = − = − + + Γ + ∫ θ ζ ζ ζ ( ) ( ) ( ) ( ) i ( ) j j z z z z z z z z ln z z z z d z z z z d j j j j z z j j j j j j k j ln 1, 0 1 0 0 1 0 0 1 0 1 0 0 1 + + − − + = − + − = − + = − = − + − + + + Γ + Γ + ∫ ∫ θ ζ ζ ζ ζ ζ ζ , where θ ( j + 1, j) is the central angle between straight line segment joining points j z and j+1 z to central point z0 ∈Ω . So we deduce: ( ) ( ) ( ) ( ) 0 1 0 1 0 1 0 1 1 0 l z z z z z F l z z z z z d F F F z F j j j j j j j j j j j j k − − − − − − = − + − + + + + + Γ ∫ ζ ζ ζ Finally we get: ( ) ( ) [ ] ( )( ) ( ) 0 2 2 1 1 1 1 0 0 1 2 2 1 1 0 1 ~ 2 l z z z F z z F z z if z F F j m n p j m n j m j j j j j j j m n p j m n j m j j j ∑ ∑ + + + ≠ + + ≠ + = + + + + + + ≠ + + ≠ + = + − − − − + π = − + Because the first term cancels we deduce: ( ) [ ( ) ( )] ( ) 0 2 2 1 1 1 1 0 0 1 0 ~ 2 l z z z F z z F z z if z j m n p j m n j m j j j j j j j ∑ + + + ≠ + + ≠ + = + + + − − − − π = and further Proceedings of the World Congress on Engineering 2009 Vol II WCE 2009, July 1 - 3, 2009, London, U.K. ISBN:978-988-18210-1-0 WCE 2009

Proceedings of the World Congress on Engineering 2009 Vol IIWCE2009,July1-3,2009,London,U.K+n+p+previousexpressions of A,.Except theelements and2mf(=0)=ZA,(=0)Fj,T,which become singular,this implies a simple replacementi=lj+m+1of zowith z,.With regard to the coefficients coming fromj+m+n+2the singular integral, we do as in [4], we shall use the(F0-=(F0 - = j-1+1with A,(=o)=evaluation of a principal value (in the Cauchy sense)of a（=0）3j -2 j-1"j+1-2jsingular integral of the type ds and the equalitywhen j+1,m+2,m+n+3(5-2)For j=1,lim (=-=p)in(= -=p)=0(see [5]),4(0)(二m) (60)-(2)(0),So weget21-=m22 21for j=m+2,(3i -2j-1))(si-zj+1)An=A;(G)=.(A()= ()/()-2/-zi-"j+1 -.12-二m+nIn(s)=n In-1(6)=n(E0 -=m+3) m2(20)=j-1 -2i2,-zi=m+32m+2for j+1,m+2,m+n+3,andi+j-1,i+j,i+j+1for j=m+n+3,Aj=_lim 4)(=0)=In=j+I-)(=0 - 2m+n+p+2=0→=izj-1 -2jAm++3(=)=(=.)-m++3-2m+#+p+2Aij-1=_ lim ,4(eo)=-3-1--/n-+-"j-1 j+1-2j0→2i3j 2 j-1++P+2m+n+3(=0)Zm+n+p+2-2m+n+3Aj+=_ lim,A4;(co)=--in-j+1Fromtheaboverelation wecan write the complexfunctionf(=o) in terms of nodal values of F ,in fact in terms ofFj,=0→= j+13j-2j-13 j-1 -= j+1so:Similarly we get the other coefficients:=.,m, ..,·..f(=0)=d4n=4(6)=(6)-G2)(),=1 - =m22 21(...+iVfori+m,i+1,i+2(+,.. . ., +1., (*)A11= lim 4(=o)=In2-2)where zo is in =m 2]=0→21Aswe can seetheglobal function is continuous onI,andwe also have: F(-,)=F, =Φ, +ij, the nodal values for(m =2)in 22 -2mAm=theapproximationfunction.Wealsohavethe nodal value of22 - 21z1-2mthethesolutionfunctionforcomplexpotential,fj=,+ij,wherefj=f(-,),and j,jare42 = (52-m)in3-2the values ofthe state and the stream functions.=1=2=m-22For given values of F, =, +i, at each z, the aboverelation gives f an analytic function in Q, and Re() andAm+2/ = A+2(5)= (-=ml),+++ (2.)-Im() both satisfy the Laplace equation in Q If7m+2-7m+n+1()= f() on I, then F()= f() in Q2, and so F(-) is(g-=m) 1m2()the solution to the original boundary value problem.Zm+32mWe need to evaluate, using a limit process the value off(z0) for zo EFfori+m+n+l,i+m+2,i+m+3IVTHELIMITPROCESSANDTHEEXPRESSIONS OF THEAm+2,m+2= ln=m+3-=m+2COEFFICIENTS"m+n+1-2m+2Concerning the calculation of the coefficients, it isperformed by imposing effectively =o -→z,eT in theISBN:978-988-18210-1-0WCE2009

() () ∑ + + + ≠ + + ≠ + = = 2 2 1 1 0 0 , ~ 2 m n p j m n j m j j Fj πif z A z with ( ) ( ) ( ) ( ) ( ) 0 1 0 1 1 0 1 0 1 0 l z z z z z l z z z z z A z j j j j j j j j j − − − − − = + + − − − , when j ≠1,m + 2,m + n + 3 . For j =1, ( ) ( ) ( ) ( ) ( ) 1 0 2 1 0 2 0 1 0 1 0 l z z z z z l z z z z z A z m m m − − − − − = , for j = m + 2, ( ) ( ) ( ) ( ) ( ) 2 0 3 2 0 3 1 0 2 1 0 1 2 0 l z z z z z l z z z z z A z m m m m m n m m n m n m + + + + + + + + + + + + − − − − − − = for j = m + n + 3, ( ) ( ) ( ) ( ) ( ) 3 0 2 3 0 2 2 0 3 2 0 2 3 0 l z z z z z l z z z z z A z m n m n p m n m n p m n p m n m n p m n p m n + + + + + + + + + + + + + + + + + + + + + + + − − − − − − = , From the above relation we can write the complex function ( ) 0 ~ f z in terms of nodal values of F , in fact in terms of Fj , so: ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Φ Ψ Ψ Ψ Ψ Ψ Φ Φ Φ Φ + +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Φ Ψ Ψ Ψ Ψ Ψ Φ Φ Φ Φ = + + + + + + + + + + + + + + + + + + + + + + + + 2 1 2 1 2 0 1 2 1 2 1 2 1 2 0 1 2 1 0 , ,., , ,., ,., , ,., , ,., ,., ~ , ,., , ,., ,., , ,., , ,., ,., ~ ~ m n p m m m n m n p m m m n m n p m m m n m n p m m n z i z m f z ψ ϕ (*) where 0z is in Ω . As we can see the global function is continuous on Γ , and we also have: ( )j j j j F z = F = Φ + iΨ , the nodal values for the approximation function. We also have the nodal value of the solution function for the complex potential, j j j f = ϕ + iψ , where ( ) j j f = f z , and ϕ j ψ j , are the values of the state and the stream functions. For given values of j j j F = Φ + iΨ at each j z the above relation gives f ~ an analytic function in Ω , and (f ) ~ Re and (f ) ~ Im both satisfy the Laplace equation in Ω . If f () () z = f z ~ on Γ , then f () () z = f z ~ in Ω , and so f (z) ~ is the solution to the original boundary value problem. We need to evaluate, using a limit process the value of ( ) 0 ~ f z for z0 ∈Γ . IV. THE LIMIT PROCESS AND THE EXPRESSIONS OF THE COEFFICIENTS Concerning the calculation of the coefficients, it is performed by imposing effectively z0 → zi ∈Γ in the previous expressions of Aj . Except the elements Γi−1 and Γi which become singular, this implies a simple replacement of 0z with i z . With regard to the coefficients coming from the singular integral, we do as in [4], we shall use the evaluation of a principal value (in the Cauchy sense) of a singular integral of the type ( ) ( ) ∫ Γ − ξ ξ ξ d z f and the equality lim ( − )ln( − ) = 0 → p p z z z z z z p (see [5]). So we get: ( ) ( ) ( ) ( ) ( ) j i j j i j j i j j i j ji j i l z z z z z l z z z z z A A z − − − − − = = + + − − − 1 1 1 1 1 ( ) j i j i j i z z z z z − − = +1 ln ln , ( ) j i j i j i z z z z z − − = − − 1 1 ln ln , for j ≠1,m + 2,m + n + 3 , and i ≠ j −1, i ≠ j, i ≠ j +1 ( ) j j j j j z z jj z z z z A A z j − − = = − + → 1 1 0 lim ln 0 , ( ) 1 1 1 1 1 1 1 0 lim ln 0 1 − + − + − + → − − − − − = = − − j j j j j j j j j z z jj z z z z z z z z A A z j ( ) 1 1 1 1 1 1 1 0 lim ln 0 1 − + + − + − → + − − − − = = + j j j j j j j j j z z jj z z z z z z z z A A z j Similarly we get the other coefficients: ( ) ( ) ( ) ( ) ( )i i m i m i m i i l z z z z z l z z z z z A A z 1 2 1 2 1 1 1 − − − − − = = , for i ≠ m, i ≠1, i ≠ 2 ( ) 1 2 1 11 1 0 lim ln 0 1 z z z z A A z z z m − − = = → ( ) m m m m z z z z z z z z A − − − − = − 1 2 2 1 2 1 ln , ( ) 2 1 2 1 2 2 12 ln z z z z z z z z A m m − − − − = ( ) ( ) ( ) ( ) ( ) m i m m i m m n i m m n i m n m i m i l z z z z z l z z z z z A A z 2 3 2 3 1 2 1 1 2, 2 + + + + + + + + + + + + + − − − − − − = = , for i ≠ m + n +1, i ≠ m + 2, i ≠ m + 3 1 2 3 2 2, 2 ln + + + + + + + − − = m n m m m m m z z z z A Proceedings of the World Congress on Engineering 2009 Vol II WCE 2009, July 1 - 3, 2009, London, U.K. ISBN:978-988-18210-1-0 WCE 2009

Proceedings of the World Congress on Engineering 2009 Vol IIWCE2009, July1-3,2009, London, U.Kmatrix involved.The evaluated nodal values enclosed to the(=m+n+1 -=m+3)1n=m+3 -=m+n+l..1Am+2,m+n+1original set ofknown nodal values completely definef()=m+3-=m+2=m+2-=m+n+1onQur(=m+3-2m+n+l)inm+2-m+3TakingintothatAm+2,m+3=accountAj=aji+ibji2m+n+1 -2m+3=m+2-2m+3(aji = Re(Aji),bji = Im(Aj)) and isolating the real andA+3/ = As(,)=(-m)D+2(-)-the imaginary parts in system (**) we obtain the following+n2m+n+3-2m+n+p+2linear system of equations, in terms of real unknowns andcoefficients:(s; - 2 m+n+p+2)++3()[-2元;=aji9j-bj]2m+n+p+2-2m+n+3[2元p,=bjig, +ajiWjfori+m+n+p+2,i+m+n+3,i+m+n+4i,j=1,m+n+1,i+m+1,j+m+1Am++3,m+n+3=ln_=m+n+4=m+n+32m+n+p+2-2m+n+3Imposing that:P;=pforthe N, nodes where thepotential is known and y,=; for the nodes where theAm++3,m++p+2 =stream function is known, and after solving the system wem+n+p+22m+mm+n+4—=m+n+p+2obtainedtheotherunknownvaluesforbothfunctionsinSo all thenodal values arethen known.By replacing them2m+#+42m+#+3≥m+++32m+#+p+2in relation (*)weget the analytic function in Q,f,whichAm+n+3,m+n+4 =satisfies relation f()= f() on T and therefore the relationm+n+4-2m+n+p+2m+n+3-2m+n+4f()= f(=) in Q2. So () is the solution to the originalr7m+n+3-2m+n+4=m+n+p+2-2m+n+4boundaryvalue problem.All the coefficients are so evaluated and theydepend onlyon the nodal points.Weconsider in the aboverelationsV.NUMERICALRESULTSzo=zi,i=l,m+n+p+2,i+m+1,i+m+n+2.Theproblem of the evaluation of the system coefficients.and also that of finding its solution can be easily solved withAsitakes all thesevaluesweobtainasystemofm+n+pacomputercodemadeinMATHCADrelations,intermsof complexnumbers of thefollowingNumerical results canbe obtained for any shape fortheform:two obstacles,but in order to make a checking and to validatem+n+p+2m+n+p+22元(-)=thecomputercodeweconsideraparticularcase,theproblemZA,(-)F)=ZAjFjofapotentialflowbetweentwoplaneparallel wallsaroundai=1i=ljm+1j+m+1circle,becauseitisaproblemwithaknownsolution.Ithasaj+m+n+2j+m+n+2great importancebecause itoffersusthepossibilitytomakeacomparison between the exact solution and the numericalUsing the complex expression of Fj and j,=f()one.AcomputercodeinMATHCADismadeinordertofindF,-Φ,+i, and f,=p,+iyj,we deduce:thenumerical solutions fordifferentpositions oftheobstacle,and they are represented in the graphics below.m+n+p+2ZA(@,+,2元（@，+i）=(**)Aj=l75j+m+11,5j+m+n+2YIf p(x,y) and y(x,y) are known continuously onand F,=Φ,+i,=p,+iy, for all the nodes thanf()=f(=) on QUr . Generally (x,y) and y(x,y)areknown only on portions of T.If there are N nodes letsupposethat thereareN,nodeswhereweknowp(x,y) and N2 nodes where we know y(x,y)N=Ni+N2.The next step is to impose in the aboverelations theboundary conditions:;=,for all the nodeswhere the potential is known and i=; for the nodeswhere the stream function isknown.Doing so we generateimplicit expressions of theunknown nodal values asfunctions of all the unknown variables, so m equations ofmunknowns which can be solved using the computer.Thecomputer is also used for getting the coefficients of theISBN:978-988-18210-1-0WCE2009

( ) 2 1 3 1 3 2 1 3 2, 1 ln + + + + + + + + + + + + + + − − − − = − m m n m m n m m m n m m m n z z z z z z z z A ( ) 1 3 2 3 2 3 3 1 2, 3 ln + + + + + + + + + + + + − − − − = m n m m m m m m m n m m z z z z z z z z A ( ) ( ) ( ) ( ) ( ) m n i m n p m n i m n p m n p i m n m n p i m n p m n i m n i l z z z z z l z z z z z A A z 3 2 3 2 2 3 2 2 3, 3 + + + + + + + + + + + + + + + + + + + + + + + + + − − − − − − = = for i ≠ m + n + p + 2, i ≠ m + n + 3, i ≠ m + n + 4 , 2 3 4 3 3, 3 ln + + + + + + + + + + + + + − − = m n p m n m n m n m n m n z z z z A ( ) 3 2 4 2 4 3 2 4 3, 2 ln + + + + + + + + + + + + + + + + + + + + + + + + − − − − = − = m n m n p m n m n p m n m n m n p m n m n m n p z z z z z z z z A ( ) 2 4 3 4 3 4 4 2 3, 4 ln + + + + + + + + + + + + + + + + + + + + + + − − − − = − = m n p m n m n m n m n m n m n m n p m n m n z z z z z z z z A . All the coefficients are so evaluated and they depend only on the nodal points. We consider in the above relations i z = z 0 , i =1,m + n + p + 2, i ≠ m +1, i ≠ m + n + 2 . As i takes all these values we obtain a system of m+n+p relations, in terms of complex numbers of the following form: () () ∑ ∑ + + + ≠ + + ≠ + = + + + ≠ + + ≠ + = = = 2 2 1 1 2 2 1 1 , ~ 2 m n p j m n j m j ji j m n p j m n j m j πif zi Aj zi Fj A F Using the complex expression of Fj and ( ) j j f f z ~ ~ = : j j j F = Φ + iΨ and j j j f ϕ iψ ~ ~ ~ = + , we deduce: ( ) ∑ ( ) + + + ≠ + + ≠ + = + = Φ + Ψ 2 2 1 1 ~ ~ 2 m n p j m n j m j i i ji j j πi ϕ iψ A i (**) If ϕ( ) x, y and ψ ( ) x, y are known continuously on Γ , and j j j j j F = Φ + iΨ = ϕ + iψ for all the nodes than f () () z f z ~ = on Ω ∪Γ . Generally ϕ( ) x, y and ψ (x, y) are known only on portions of Γ . If there are N nodes let suppose that there are N1 nodes where we know ϕ( ) x, y and N2 nodes where we know ψ (x, y) , N = N1 + N2 . The next step is to impose in the above relations the boundary conditions: ϕi ϕi ~ = for all the nodes where the potential is known and ψ i ψ i ~ = for the nodes where the stream function is known. Doing so we generate implicit expressions of the unknown nodal values as functions of all the unknown variables, so m equations of m unknowns which can be solved using the computer. The computer is also used for getting the coefficients of the matrix involved. The evaluated nodal values enclosed to the original set of known nodal values completely define f (z) ~ on Ω ∪ Γ . Taking into account that ji ji ji A = a + ib ( ( ) ji Aji a = Re , Im( ) ji Aji b = ) and isolating the real and the imaginary parts in system (**) we obtain the following linear system of equations, in terms of real unknowns and coefficients: ⎪⎩ ⎪ ⎨ ⎧ = + − = − i ji j ji j i ji j ji j b a a b πϕ ϕ ψ πψ ϕ ψ ~ 2 ~ 2 i, j =1,m + n +1, i ≠ m +1, j ≠ m +1 Imposing that: ϕi ϕi ~ = for the N1 nodes where the potential is known and ψ i ψ i ~ = for the nodes where the stream function is known, and after solving the system we obtained the other unknown values for both functions. So all the nodal values are then known. By replacing them in relation (*) we get the analytic function in Ω , f ~ , which satisfies relation f (z) = f (z) ~ on Γ and therefore the relation f (z) = f (z) ~ in Ω . So f (z) ~ is the solution to the original boundary value problem. V. NUMERICAL RESULTS The problem of the evaluation of the system coefficients, and also that of finding its solution can be easily solved with a computer code made in MATHCAD. Numerical results can be obtained for any shape for the two obstacles, but in order to make a checking and to validate the computer code we consider a particular case, the problem of a potential flow between two plane parallel walls around a circle, because it is a problem with a known solution. It has a great importance because it offers us the possibility to make a comparison between the exact solution and the numerical one. A computer code in MATHCAD is made in order to find the numerical solutions for different positions of the obstacle, and they are represented in the graphics below. F 2 1.5 1 0.5 0 0.5 1 1.5 2 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 F 2 1.5 1 0.5 0 0.5 1 1.5 2 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 Proceedings of the World Congress on Engineering 2009 Vol II WCE 2009, July 1 - 3, 2009, London, U.K. ISBN:978-988-18210-1-0 WCE 2009

ProceedingsoftheWorldCongressonEngineering2009VolIIWCE 2009, July1-3, 2009,London, U.KReferences[I] Hromadka II T.V..Lai C.,TheComplex Variable5125-Boundary Element Method in Engineering Analysis,Springer-Verlag,New-York,1987475[2] Hromadka II T.V., Whitley R.J., Advances in theacomplex variable boundary element method, Springer-Verlag, 1997.[3] Aliabadi M.H., Brebbia C.A., Advanced formulations in1.7Boundary Element Methods, Computational Mechanics5Publications, Elsevier Applied Science, 1993[4] Grecu L, Petrila T., A Complex Variable BoundaryElement Methodforthe Problem of the Free-surfaceHeavyInviscid flow overanobstacle,General025-Mathematics nr. 2/2008, pag 3-17.-i.a[5] Petrila T., Trif D., Basics of fluid mechanics andintroduction to computational fluid dynamics, SpringerScience New-York,2005In the following figure there are represented the numericalresults obtained for different position of two circularobstacles situated between the walls.Both have the sameradius, and their centers are situated at the same distancesfromthewalls,butdifferentdistancesbetweentheircentershavebeenconsidered.ISBN:978-988-18210-1-0WCE2009

F 2 1.5 1 0.5 0 0.5 1 1.5 2 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 F 2 1.5 1 0.5 0 0.5 1 1.5 2 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 In the following figure there are represented the numerical results obtained for different position of two circular obstacles situated between the walls. Both have the same radius, and their centers are situated at the same distances from the walls, but different distances between their centers have been considered. F 2 1.5 1 0.5 0 0.5 1 1.5 2 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 F 2 1.5 1 0.5 0 0.5 1 1.5 2 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 F 2 1.5 1 0.5 0 0.5 1 1.5 2 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 References [1] Hromadka II T.V., Lai C., The Complex Variable Boundary Element Method in Engineering Analysis, Springer- Verlag, New-York, 1987 [2] Hromadka II T.V., Whitley R.J., Advances in the complex variable boundary element method, Springer￾Verlag, 1997. [3] Aliabadi M.H., Brebbia C.A., Advanced formulations in Boundary Element Methods, Computational Mechanics Publications, Elsevier Applied Science, 1993 [4] Grecu L, Petrila T., A Complex Variable Boundary Element Method for the Problem of the Free-surface Heavy Inviscid flow over an obstacle, General Mathematics nr. 2/2008, pag 3-17. [5] Petrila T., Trif D., Basics of fluid mechanics and introduction to computational fluid dynamics, Springer Science New-York, 2005. Proceedings of the World Congress on Engineering 2009 Vol II WCE 2009, July 1 - 3, 2009, London, U.K. ISBN:978-988-18210-1-0 WCE 2009