CO-LOCATED DIFFUSE APPROXIMATION METHOD FOR TWO DIMENSIONAL INCOMPRESSIBLE CHANNEL FLOWS
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1 CO-LOCATED DIFFUSE APPROXIMATION METHOD FOR TWO DIMENSIONAL INCOMPRESSIBLE CHANNEL FLOWS C.PRAX ad H.SADAT Laboratoire d'etudes Thermiques,URA CNRS , Aveue du Recteur Pieau Poitiers Cedex, Frace Itroductio The mai cotributio of this paper is the formulatio of a diffuse approximatio method(dam), for two-dimesioal chael flows. The proposed method is based o the vorticity-streamfuctio formulatio. The DAM which estimates derivates of a scalar field has the remarkable advatage to work o discretizatio poits (thus avoidig mesh geeratio). It has bee show that the DAM is much better tha the fiite elemet method for the computatio of gradiets [-2]. I a previous paper [3], we have show that it ca be used to solve lamiar atural covectio problems. I this work, we discuss the applicability of this method to chael flows with a particular emphasis o the form of the weightig fuctio. The Diffuse Approximatio Cosider a scalar field ϕ(x,y), i a two dimesioal domai, ad a set of poits Mi(x i,yi) i the viciity of a chose poit M(x,y). The diffuse approximatio provides estimates of ϕ ad its derivatives at M from the odal values ϕ i. The startig poit is to estimate the Taylor expasio of ϕ at M by a weighted least squares method which uses oly the values of ϕ at the earest poits Mi. By trocatig the series at order k, oe obtais the correspodig estimates of the derivatives at the same order. Let us the estimate the secod-order Taylor expasio of ϕ i at M as: ϕ i *(x i,y i ) = < p(m i,m)>. < α M > T () where <p(mi,m)> is the lie vector of polyomial basis ad <α M > T is the trasposed vector of the approximatio defied as : <p(m i,m)> = <,(x i -x),(y i -y),(x i -x) 2,(x i -x).(y i -y),(y i -y) 2 > (2)
2 2 <α M > T = < α 0, α, α 2, α 3, α 4, α 5 > T (3) The variables α M are determied by miimizig the quadratic expressio: I(α M ) = {ω(m,m i ). [ϕ i - < p(m i,m)>. < α M > T ] 2 } (4) i= where ω(m,m i ) is a cotiuous weightig fuctio, havig its maximum value at M ad decreasig rapidly to zero. Thus oly the earest poits to M are ivolved i (4). By writig the six coditios : α I(α M ) =0 j, j=0,5 (5) we obtai the (6x6) liear system : where [ A M ]. < α M > T = < B M > T (6) [ A M ]= ω(m,m i ).< p(m i,m)> T < p(m i,m)> (7) i= < B M > T = ω(m,m i ).< p(m i,m)> T.ϕ i (8) i= Oce the system (6) has bee solved, oe fially obtais the desired estimates of the derivatives at M: φ φ φ(x,y)=α 0; =α ; =α 2; x y φ φ φ =α 2 3; =α 4; =α 2 5; x y x y (9) The weightig fuctio ca be chose i may ways. Its radius must be large eough to overlap at least a umber of odes equal to the umber of terms αi. However, the situatio where the selected odes are aliged must be avoided i order to get a o sigular [A M ] matrix.
3 3 Implemetatio of the Diffuse Approximatio We cosider the Navier-Stokes equatios i the vorticitystreamfuctio formulatio : Ψ + ω = 0 (0) uω vω + = υ ω x y () Where ν is the kiematic viscosity. Our method for solvig the equatios (0) ad () by usig the diffuse approximatio has a remote similarity with the fiite differece method. The partial derivatives at a give poit are expressed as fuctios of the eighbourig odal values of Ψ or ω (by ivertig the matrix [A M ]): φ φ φ φ T φ M T = A 2 2 i i i x y x y x y i= φ,,,,,. ω (M,M ). p(m,m ).φ (2) By usig the relatios (2), the goverig equatios (0-) are ow replaced (at every ode M) by algebraic expressios i terms of the eighbourig odal values Ψ i or ω i. Two systems are the obtaied ad solved iteratively after the itroductio of the boudary coditios. I this work a relaxatio factor of 0.2 is used for each variable ad the covergece criteria iclude the relative chages betwee cosecutive iteratios: Ψ Ψ ω ω 0 ; 0 Ψ ω ew old 3 ew old 3 ew max ew max (3) The boudary coditio for the streamfuctio, at the outflow of the chael, is: Ψ x = 0 The vorticity values at the boudary are calculated i terms of the eighbourig streamfuctio values by usig the method of Kettleborough et al. [4].
4 4 Applicatios: Flow betwee Parallel Plates To evaluate the accuracy of the method, the developig lamiar flow betwee two parallel plates was computed. A uiform velocity is imposed at the ilet sectio, ad a parabolic profile is expected to form at about 0.04 Re [5], where Re is the Reyolds umber referred to the width of the chael. Calculatios were made for a chael with legth-to-width ratio L/D=0. We first cosidered a 0* grid ad the followig gaussia widow: ω (M,M i ) = Exp ( -2.( r σ ) 2 ) ω (M,M i ) = 0 if r 2 > σ 2 (4) where r 2 = (x i -x) 2 + (y i -y) 2 We the used a 5*2 grid where the ode desity ratio is equal to four ( x=4 y). I this case, the previous widow selects more poits i the trasverse directio tha i the streamwise directio ad the method fails to coverge. We have cosequetly modified the gaussia widow by settig: r 2 = (x i-x) (y i -y) 2 widows ode calculatio poit,5 U*,4,3,2, CVFEM (0*) DAM (0*) CVFEM (5*2) DAM (5*2) X* 0 FIG. FIG. 2 Weightig fuctios Ceterlie velocity These two widows are schematically represeted o Fig.. The calculated ceterlie velocity as a fuctio of the distace from the ilet for a Reyolds umber of 00 is show o Fig.2. The
5 5 secod mesh gives better results as expected. The umerical value i the fully developed regio agrees well with the aalytical value of.5. I order to compare the DAM with a classical method, we have also reported o FIG.2, the results obtaied by usig a cotrol volume fiite elemet method (CVFEM)[6]. It appears that the results obtaied by the DAM are comparable to those obtaied by the well established CVFEM. Lamiar Flow over a Backward Facig Step L=2 h= H=.5 x FIG. 3 Backward facig step Of cocer here is the lamiar flow over a facig step (FIG.3). A fully developed parabolic lamiar flow is prescribed at the iflow sectio. The goverig equatios are odimesioalized by defiig X = x H-h ; Y = y H-h ; U = u U max ; V = v U max ; Re = U max.(h-h) ν where U max is the maximum velocity at the iflow sectio. Calculatios were performed util X=24 for Re=50 ad Re=50 o two differet grids (62* ad 86*6) by usig a gaussia widow. I Table, the preset calculated reatachmet legth is compared with the results obtaied i [7] with a (62*50) mesh. The results obtaied by usig the CVFEM o a (62*) grid are also reported o Table for compariso. I FIG.4, the axial velocity profiles are preseted. We ca see a rather good agreemet with the referece ad with the CVFEM results.
6 6 TABLE. Reyolds umber SOU (62*50) H 2 2 Ref[7] DAM (62*) DAM(86*6) X=.6 DAM (62*) CVFEM(62*) DAM (86*6) Reattachemet legh,xr/(h-h) 0 X U FIG. 4 Axial velocity profiles X Coclusio The diffuse approximatio method has bee applied to fluid flow i chaels ad compared with a cotrol volume fiite elemet method. Its accuracy has bee show o two test cases. Fially, we have show that it is better to use a elogated weightig fuctio whe the ode desity is greater i oe directio. REFERENCES. B. Nayroles, G. Touzot ad P. Villo : L'approximatio diffuse, C.R.Acad. Sci. Paris, t. 33, Série II, p , Y.Marechal,J.L. Coulomb, G. Meuier ad G. Touzot : Use of the diffuse elemet method for electromagetic field computatio., IEEE Trasactios o magetics, vol 29, 2 March H.Sadat ad C.Prax : Applicatio of the diffuse approximatio method to the umerical solutio of fluid flow ad heat trasfer problems, (to be published i Iteratioal Joural of Heat ad Mass Trasfer). 4. C.F.Kettlebourough, S.R.Hussai ad C.Prakash, Solutio of Fluid Flow Problems with the Vorticity-Streamfuctio Formulatio ad the Cotrol-Volume-Based Fiite-Elemet Method, Numerical Heat Trasfer, Part B,Vol.6, pp.3-58, H.Schlichtig,Boudary-Layer theory, 7th ed., McGraw-Hill, New York, H.Sadat, P.Salagac, Further Results for Lamiar Natural Covectio i a two dimesioal trapezoidal eclosure,numerical Heat Trasfer, Part A, vol.27,pp , M.C.Melaae, Nostaggered calculatio of lamiar ad turbulet flows usig curviliear oorthogoal coordiates, Numerical Heat Trasfer, Part A, vol.24,pp , 993
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