a polygon-free numerical solution of steady natural …...a polygon-free numerical solution of...

A polygon-free numerical solution of steady natural convection in solid-liquid systems

J. ~ e r k o ' , C.S. Chen2, B. ~a r l e r ' I L a b o r a t o ~ f o r Fluid Dynamics and Tlzermodynamics, Faculty of Mechanical Engineering, University of Ljubljana, Slovenia ' ~ e p r t r n e n t of Mathematical Sciences, Universitj of Nevada, USA

Abstract

This paper describes a primitive variable formulated and Kansa's method solved free boundary problem arising in the steady natural convection of the incompressible Newtonian solid-liquid phase change material. Solution of the coupled mass, momentum, and energy equations in two-dimensions is solved by using global multiquadrics approximation of the fields and global cubic spline approximation of the boundary geometry.

1 Introduction

Various aspects of science and technology are nowadays related to modelling of the solid-liquid phase change systems. X broad spectra of discrete ap- proximative methods have been used in the past decades to evaluate them. In recent years, there is a strong development in meshless methods [l]. Among them, the collocation method using radial basis functions (RBFs). has emerged as a promissing meshless technique. The R.BFs have been orig- inally used for scattered data approximat,ion [ 2 ] and afterwards for solution of the PDEs [3]. The latter technique was named as Kansa's method by Golberg & Chen [4]. The main advantage of the I<ansa's method is in it,s simplicity, applicability to different PDEs. and effectiveness in dealing with arbitrary dimension and complicated domains. The method has recently been successfully applied in many scientific and engineering disciplines [S. G]. The Kansa's method can be modified into Hermite interpolation scheme [7] for which the resulting collocation matrix is guaranteed to be nonsingular.

Transactions on Modelling and Simulation vol 29, © 2001 WIT Press, www.witpress.com, ISSN 1743-355X

Bot,h approaches give similar accuracy. This paper presents the solution of the free boundary problem arising in steady solid-liquid phase-change systems using Kansa's method. The physically equivalent problem has been solved by the Dual Reciprocity Boundary Element Method (DRBEM) in our previous MBP conference paper [8] where the polvgonization and integration over the boundary had to be employed. In the present method the integration as well poloygonization over the boundary has been avoidcd. This fact might be particularly advantageous when solving solidification or melting problems, because thev usually appear in complicated geometric ar- rangements. This paper shows basic implementation of the Kansa's method for solving steady natural convection in single-phase fluid and phase change material. For this purpose the one-phase continuum formulation [g] has been used as a physical framework, and the Poisson formulation [l01 of the general transport equation as the mathematical framework.

2 Governing equations

This paper deals wit,h incompressible Newtonian solid-liquid phase change material confined to the domain R with the boundary T. Consider a connected fixed domain 0 with boundary F occupied by a phase change material described with the temperature dependent desity @ p of the phase P, temperature dependent specific heat at constant pressure c,, effective thermal conductivity k ; viscosity p and the specific latent heat of the solid-liquid phase change h M . The one-phase continuum formulation of the transport equations for the assumed system is

with constant mixture density Q, mixture enthalpy defined as h = fs hs + fL he. mixture velocitv defined as v = fs vs + fL vc and mixture enthalpy defined as h = fs hs + fL he. P stands for mixture pressure. The consti- tutive mixture temperature T - mixture enthalpy relationships are

Thermal conductivity and viscosity can arbitrarily depend on temperature. The variation of the density with temperature is included in the body forre term by using the Boussinesq approximation


with a standing for the acceleration vector, B for the ~olumetr ic thermal expansion coefficient. and T,,f for the reference temperature. The second term in the body force is modelling the phase change, simply constituted through the phase change coefficient c? (large number) and temperature dependent liquid fraction fc which is assurned to vary as

where Ts stands for t,he solidus and TL for the liquidus temperature. We seek for steady mixt,ure temperature, velocity and pressure. The steady state solution is obtained as a limiting case of the transient solution with the timestep At. The solution at time to + At is obtained by assuming known temperature and velocity fields at time to and boundary conditions of the Dirichlet, Neumann and Robin type, respectively.

3 Solution procedure

3.1 General formulation of the transport phenomena

For the purposes of the demonstration of the solution procedure, a transport phenomena problem can be briefly described in a general manner as thtl numerical solution of Eulerian transport equation, defined on a fixed domain fl with boundary F, of the kind

with Q. a. t , v ? and S standing for density. transport variable, time, velocity. source, and diffusion matrix

The dependent variable stands, for instance for the velocity component in each coordinat,e direction. or temperataure. or the mass fraction of a chemical species. The function ;C denotes the relation between the transported and the diffused variable such as for example relation between the enthalpy and the temperature,

The solution of the governing equation for the dependent variable at final time t = to i- At is sought, where to represent,^ the initial time and At the positive time increment. The solution is constructed by the initial and boundary conditions that. follow. The initial value of transport variable @(p, to) a t a point with position vector p and time to is defined through t,lle known fi~nction

@(p, t ) = $ 0 ; p E R U r (8)


Thc boundary r is divided into connected or disjoint parts rD, rN and r R

with Dirichlet, Neumann and Robin type boundary conditions respectively. These boundary conditions are at the boundary point p with normal nr and time to < t 5 to + At defined through known functions @p, @g, @m and @rRref

@ = G D ; p e r D , - a @ = @'d. , p E P

The involved parameters of the governing equat,ion and boundary conditions are assumed to depend on the transport variable, space and time. The solution procedure thus inherently involves iterations. The governing equation is transformed as follows. The diffusion matrix

is split into constant isotropic part D I, with I denoting identity matrix, and the remaining nonlinear anisotxopic part D'

The transport equation is subsequently cast into Poisson form

with

The inhomogcnous terms are Taylor expanded as

with 'bar' denoting known value a t prcvious iteratlion. The final form of t,he transformed equation becomes

~ ~ @ = e + v . 0 = 8 + 8 , ~ ( @ - G ) + V . O + V . G , < ~ ( ~ 6 ) (16)

3.2 Time discretization

The time discretization is made (for simplicity of presentation only) in a

fully implicit (backward Eulcr) m a n n u where subsrrlpt 0 represents th(' value a t the initial time.


3.3 Kansa's method

The unknown field 9 is approximated by the !Y global approximation functions 44,(p) and their coefficients C,

The global radial basis function approximation is based on the multiquadrics wit,h the free parameter 7-0

The coefficients can he calculated from the 12; collocation equations of which 1% are equally distributed over boundary Xr and Na over the donlam R . Let us define the boundary condition indicators in order to be able to represent the boundary collocation equations in a compact form. The Dirichlet, Neumann, and Robin type of boundary condition indicators X D 3 X ~ ~ ' , and XR are

1 PEP . = 1 p E I--\' , XYP) = { 1

0 PW' o p q s " o p e r R (20)

The boundary collocation equations can be respectively written in the fol-

The domain collocation equations can be written in the following form

The coefficients C,, are calculated from the following system of N X A' linear equations

$ C = b (23)


3.4 Meshless representation of the boundary

The length Ci of the contour between the boundary points pi and pi-l can be approximated by the Euclidean distance

with the cyclic condition pi-l = p ~ ~ ; i = 1. The total length of the boundary contour er equals to

The position of the boundary points can he represented by the meshless approximation with the parameter e

p<@) = $:(e)<L;n = 1 ,2 , . . . , 1Vr+~

4; (e) = le - ln 1 3

r r v'N,+I = l $fVr+2 (P) = $Lrr+3 (e) = P2 (28)

The following compatibility conditions are needed for determination of the coefficients c; n = Nr+i, Nr+2,

The coefficients C;, are calculated from the following Nr+3 X N P + ~ systelri of linear equations

,p f b1- < - < (30) with

r 9:TL =$I,,; i = 1,2;..i%. n = P.2,.. .Nr+3

= + K ( & ) - di(0) : i = Nr+l, r L = 1,2;.. , Nr+3

b;, = pc; j = l ) 2 , . . . , % T r

OF, = O ; i = iVr+', (32)

The components of the normal on the boundary can be explicitly calculat~ccl


3.5 Numerical implementation

After each solution of the scalar @ a relaxation with coefficient c,,! is made = 6 +cre, ( - 4 ) and the timestep it,erations are st,opped when the critc-

rion (/@,,,I - 16,,,/)/~,,, < eitr is reached. where subscript avg represents the average gridpoint value. The criterion of reaching the steady state is (l@avg 1 - /@avg ~ I ) / / @ a v g ~ 1 < csts The three governing equations (pressure correction Poisson equation replaces the mass conservation equation) are in each tiniestep solved in an iterative bundle. Their representation within general transport equation context is shown in .Appendix. First, new pressure field is solved: based on the old velocity> pressure and temperature fields. I\/lomentum e q ~ a t ~ i o n is solved afterwards, based on the old tempcr- ature field and new pressure field. Subsequently, pressure correction field is solved based on the new velocity field. The new velocity field is cor- rected through the pressure correction field. Finaly new tempareture field is calculated, based on the new velocity field.

4 Numerical examples

Two ilumerical examples are shown in the present text. They are reprc- sented in dimensional form because of the space constraints. Geometry is a square with dimension l [m]. Upper and lower boundaries are insulated, the left boundary is subject to temperature - 0.5 [C] and the right boundary is subject to temperature + 0.5 [C]. Phase change coefficient is set to O [kgmP3 S-'] in case of single-phase fluid arid to 10' [kg mP3 S-'] in case of phase change material. The solidus and the liquidus temperatures are Ts = - 0.01 [C]. Ts = + 0.01 [C]. Multiquadrics constant ro is set to 0.01. All other material propert,ies except = 104 [ l / K ] are set to unit values. The square is discretized into 120boundary nodes and 900domain nodes tha t give 1020 meshpoints. Other paranleters are At = 0.005 [S], c,,, = 1. eitr = 0.001, eSts = 0.01. No flow has been assumed at the initial time and the temperature is distributed linearly from +O.5 [C] to -0.5 [C] from the left to the right boundary. Figures 1 and 2 show solution of the natural convection with the single-phase fluid and the phase change material, respectively. S~lut~iorl in Figures 1 and 2 reach steady state after 214 and 351 iterat>ions. .A proper single-phasr fluid numerical implementation is confirmed by matching wit,h the classical [l11 benchmarli. A proper phaw- change nunlerical implementation is confirmed by matching with the very recent results obta.ined by the finite volume method [12]. A systematic com- parative study of these two methods will appear elsewhere. The approxirna- tion of the boundary of the square with the cubic splines introduces errors in the position of the normals as seen in Figures 1 and 2.


Figure 1: Isotherms and velocity vectors of the natural convection in a square cavity. Example with t,he single-phase fluid. Maximum velocity is 19.8 [m/s].

Note the meshless representation of the square geometry. The calculated normals to the boundary are denoted by the dashed line.

5 Conclusions

This preliminary study shows basic elements of the Kansa's method based computational modelling of the natural convection in phase change ma- terials as required in numerous natural and technological systems. To our knowledge it represents the first attempt to solve a solid-liquid phase change system by such type of meshless approach. The use of the RBF approximation of the boundary geometry appears to be new as well. The main advantage of the method represent,^ truly polygon-free discretization and simple numerical implementation. The dissadvantage represents resulting full asymmetric algebraic systems of equations that might become ill-conditioned and are difficult to solve in an efficient way in large-scale problems. However, t,here are ways to overcome ~nentioned difficulties, such as the use of compactly supported radial basis functions, tackled in our on- going research. The example represented in Figure 2 mimics (narrow phase change interval) the phase change of a pure substance, a problem that is usually difficult to solve with a fixed-grid technique. The Kansas's method however demonstrates stability even for much coarser discretization and flow


intensity as used in the present paper. Due to the nature of polygon-free discretization, the current development is particularly attractive in solving geometrically complex systems. The successful application of the Kansa's method to present highly non-linear and coupled problem promises a new era of "meshless computing t,echnology" as experienced decades ago with other main-stream numerical methods.

Figure 2: Isotherms and velocity vectors of the natural convection in a square cavity. Example with the phase change material. Maximum velocity is 11.3 [m/s].

Acknowledgement

Present paper represents a part of the projects (1) C O S T - P S : Simulation of Phys- ical Phenomena zn Technologzcal Applicatzons that forms a part of the Slov~nian national research programme Multiphase system,^ supported by the Slovenian X n - istry of Education, Science and Sport, and (2) C O P E R N I K U S U L T R A W A T sup-

ported by the European Union. The second author thanks University of Kevatia. Las Vegas for the support of Faculty Development Leave at the Laboratory for

Fluid Dynamics and Thermodynamics, Faculty of Mechanical Engineering, Uni- versity of Ljubljana durig spring 2001.


References

[l] Golberg, M.,4. & Chen, C S . , Discrete projection methods for integral equations, CMP, Southampton, 1997.

[2] Kansa, E.J . , Muliquadrics - a scattered data approximation scheme with application to computational fluid dynamics - I . , Comp.Math.Appl., 19 , pp. 127-145, 1990.

[3] Kansa, E.J. , Muliquadrics - a scattered data approximation scheme with application to computational fluid dynamics - II., Comp.Math.Appl., 19 , pp. 147-161, 1990.

[4] Golberg, M.A. & Chen, C.S., A bibliography on radial basis function approximation, B.E.Comm., 7, pp. 155-163, 1996.

[5] Hon, Y.C., Lu, M.W., Xue, W M . & Zhu; Y.M., Muliquadric method for numerical solution of a biphasic mixture model, Appl.Math. Comput., 88, pp. 153-175, 1991.

[6] Zerroukat, M., Power, H. & Chen, C.S., A numerical method tbr heat transfer problems using collocation and radial basis functions, Fnt. J. Num,er. Meth. Eng., 42, pp. 1263-1278, 1998.

[7] Fasshauer, G.E., Solving partial differential equations by collocation with radial basis functions, Proceedin,gs of Chamonix 1996, eds. -4.Le Mkhauti., C. Rabut & L.L. Schumaker, iTanderbilt University Press, Nashwille, pp. 1-8, 1997.

[8] Sarler, B., Boundary integral equation solution of steady natural convection in solid-liquid systems, eds. B. ~ a r l e r , C.A. Brcbbia & H. Power.. Com,putational Modelling of Free and Moving Boundary Problems V, WIT Press, Southampton, pp. 149-158, 1999.

[g] Bennon, W.D. & Incropera, F.P., A continuum model for momentum, heat and spccies transport in binary solid-liquid phase change syst*ems, - I. Formulation, 1nt.J.Heat Mass Tra~nsfer, 30, pp. 2161-2170, 1987.

[l01 ~ a r l e r , B.. Solution of the transport phenomena by the dual reciprocity boundary elernent method, European Congress on Computa- tional Methods in Applied Sciences and Engineering, Barcelona, 11-14 September 2000, CD-ROM Proceedings, CIMNE, Barcelona, 2000.

[Ill Dc Vahl Davis, G. Natural convection of air in a square cavit,y: a bench mark solution: Int. J.Numer.Meth,ods Fluids, 3, pp. 249-264, 1983.

[l21 Gobin, D & Le Querit, P., hlelting from an isothermal vertical wall. Syn- t,hesis of a numerical comparison exercise, Comp.Ass.Mech.Eng.Sci., 7, pp. 289-306: 2000.


Appendix: Definition of coefficients

N x N systems of equations

X-Momentum transport

variable 9 C ( @ ) D

Y-Momentum transport

definition

vv

U ,

P 0 - P . y + g a y ( l - p ( T - T , e r ) ) - c ~ ( l - f c ) @

e + / ( A t ~ ) - e @ o / ( l t ~ ) - S I P e l ( A t ~ ) + C P (1 - f c ) l ~ e @ / P 0 * - / P Q U L I P 2 n @ / / - L


Nr xNr systems of equations

Pressure Poisson equation

variable definition P

C(@) p D 1

Pressure correction Poisson equation

variable definition @ P' C ( @ ) P' D C

D;, S 0 B 0 Q,* 0 0, - @ %

0, -@vy @L ,a 0 0 0

Pressure correction Poisson equation Neurnann boundary conditions


a polygon-free numerical solution of steady natural …...a polygon-free numerical solution of...

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