Materials Science Forum 649 (2010) pp 205-210 © (2010) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.649.205
Meshless Approach to Solving Freezing with Natural Convection Kosec Gregor1,a, Božidar Šarler1,b 1
University of Nova Gorica Vipavska 13, SI-5000 Nova Gorica, Slovenia a
[email protected],
[email protected]
Keywords: Newtonian fluid flow, Stefan problem, freezing, primitive variables, natural convection, pressure-velocity coupling, mesh-less methods, local radial basis function collocation method, metals, rectangular cavity.
Abstract. This paper for the first time explores the application of the meshless approach, structured on the Local Radial Basis Function Collocation Method (LRBFCM), for solving the freezing process with convection in the liquid phase for a metals-like material in a closed rectangular cavity. The enthalpy one-domain formulation is used to avoid inclusion of additional boundary conditions at the fluid-solid interface. To avoid numerical instabilities, the freezing of a pure substance is modeled by a narrow phase change interval. The fluid flow is solved by a local pressure-velocity coupling, based on the mass continuity violation [1-3], and the explicit time stepping is used to drive the system to the free boundary solution. The results are presented through temperature and streamfunction contours and the liquid-solid interface position at the steady state, as well as the time development of the average Nusselt number and the time development of the cavity average liquid fraction. Results are validated with already benchmarked melting example [3]. The paper represents first steps in solution of the Hebdich and Hunt experiment by an alternative numerical technique, different from the classical finite volume or finite element methods [4]. Introduction Liquid-solid phase changes are central to many oldest as well as very modern processes for producing useful implements of the mankind. The conditions under which melt solidifies is crucial for the final microstructure and properties of the commercial products. Since 1980’s, there is an exceedingly growing interest in physical understanding and numerical modeling of liquid-solid phase changes in multiconstituent materials of the metallic, ceramics and polymer types. The modeling of complex metallic alloys solidification might be, in the very basic approach, collapsed into the situation of freezing of a pure substance driven by natural convection in a rectangular cavity. Analytical solutions for melting or freezing of a pure substance for special cases, without convection (Stefan’s problem) are well known [5], but the inclusion of the natural convection, phase-change and surface driven convection, and related complex flow structures, makes the problem unsolvable in a closed form. The discrete approximate approach is thus required to obtain the solution. Respectively, the properties of numerical methods for solving these problems emerged as one of the central issues in description of liquid-solid systems. From the physical point of view, relatively simple problem of melting, driven by natural convection, has been proposed by Gobin and Le Quéré in 1998 as a first systematic benchmark for numerical methods in liquid-solid systems [6]. It has been solved by several laboratories with quantitatively quite different results [6-11]. In addition, the Hebditch and Hunt solidification experiment (with solution of Sn-5wt pct Pb alloy and Pb-48 wt pct Sn alloy) has been modeled and solved by the finite element and finite volume methods [4] with several quantitative discrepancies. Respectively, there is a substantial need in further development of robust, accurate, and simple numerical methods for phase change problems. This paper belongs to this category. This paper deals with the freezing of the metals like material (Prandtl number approximately 0.01) in the differentially heated closed rectangular cavity. The completely local numerical approach is used to solve the governing equations where LRBFCM is used to calculate all required spatial derivatives and local pressure velocity coupling is adopted to enforce the divergent free velocity field. The method has been already successfully applied to
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diffusion problem [12], convection-diffusion problems [13], classical De Vahl Davis problem [1], natural convection in porous media [2], and melting of the anisotropic metals [3]. Governing equations
r Consider a connected fixed domain Ω with boundary Γ and its normal n , exposed to gravity r acceleration g and occupied by a pure substance, defined by constant density ρ0 , specific heat c p , latent heat L , melting temperature Tm , lower and upper bounds TS and TL of the computationally smoothened melting interval, thermal conductivity λ , viscosity µ , thermal expansion coefficient β , and reference temperature Tref . It is the purpose of this paper to determine the position of the r interphase boundary, temperature T , pressure P , and liquid velocity field v in the system at point r p as a function of time t . The system is observed in the 2 dimensional Cartesian coordinate system r r r r r (with coordinates px , p y and base vectors ix , iy ), i.e. p = px ix + p y iy . The governing equations for mass, energy and momentum transport in the system are assumed to take the following form: r ∇⋅v = 0 , (1) ∂ r (2) ( ρ h ) + ∇ ⋅ ( ρ vh ) = ∇ ⋅ ( λ∇T ) , ∂t ∂ r rr r r (3) ( ρ v ) + ∇ ⋅ ( ρ vv ) = −∇P + ∇ ⋅ ( µ∇v ) + f ∂t With the following constitutive equations for the enthalpy h , liquid fraction fl , and Boussinesq r body force f : (4) h = c pT + f l L ,
0 ; T ≤ Ts f l = (T − Ts ) / ( Tl − Ts ) ; Ts < T < Tl , 1 ; T ≥ Tl r r f = ρ g 1 − β (T − Tref ) .
(5)
(6)
r The initial temperature at time t0 is set to T0 , the initial velocity to v0 , and the initial pressure to P0 . We seek the solution of the temperature, velocity and pressure fields at the time t0 + ∆t , where ∆t represents a positive time increment. The boundary conditions for the velocity are assumed of the non-slip type, and the boundary conditions for the temperature are assumed of the Neumann and of the Dirichlet type on the parts of the boundary Γ D and Γ ∂ r r r r r v = 0; p ∈ Γ T = T D ; p ∈ ΓD (7) r T = T ; p∈Γ , ∂n r respectively. The calculated velocity v at time t0 + ∆t is multiplied by the liquid fraction at the same time level in order to fix the zero velocity in the solid phase.
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Solution procedure The meshless numerical method is used where the solution is represented on the arbitrarily distributed set of nodes without any additional topological relations between them. The meshless methods represent a promising technique to avoid the meshing problems [14-18]. This paper employs the simplest class of such methods in development today, the Radial Basis Function [19] Collocation Methods [20]. The pressure, velocity and enthalpy fields are interpolated on the coincident grid points by Hardy’s multiquadrics RBF. The arbitrary function θ is represented on each of the local sub-domains as
θ (p ) ≈ ∑α nΛn (p ) ,
(8)
n =1
with p, Λ n , α n and standing for the position vector, the basis function, the collocation coefficient and the number of the collocation points, respectively. Hardy’s multiquadrics basis functions are defined as Λ n ( p ) = rn2 ( p ) + c 2 r02 ;
rn2 = ( p − p n ) ⋅ ( p − p n ) ,
(9)
Through the collocation criteria the coefficients α n are obtained and with known collocation coefficients all required derivatives are calculated through the derivatives of the equation (8). The explicit time scheme is adopted to cope with the transience terms in the momentum and energy equations. The Navier-Stokes equations (1) and (3) are solved iteratively by completely local pressure-velocity coupling based on the pressure correction predicted from the velocity divergence[1]. After the divergent free velocity has been calculated the energy equation (2) can be directly solved.
Benchmark test and discussion A freezing problem driven by natural convection, with differentially heated vertical walls and adiabatic horizontal walls, is solved. The initial temperature is set to value T0 = 1 , the melting temperature is set to value TM = 0.0 , approximated by TS = −0.005 , TL = +0.005 . The whole domain is initially in the superheated solid phase. The east wall temperature is set to the value TE = T0 . The west wall temperature is set to TW = Tm − ∆T . The problem is characterised by the dimensionless Prandtl, Rayleigh, and Stefan numbers, and the dimensionless time:
Pr =
µcp kl
,
Ra =
g β∆T l3 ρ02c p kl µ
,
Ste=
∆Tc p L
,
Fo=
kl t. ρc pl2
(10)
The case with Pr = 0.02 , Ra=2.5 ⋅10 4 and Ste=0.01 is assumed in this paper. A complementary case for melting (the initial temperature is set below melting temperature and the hot side cavity wall is set above the melting temperature: TM = 1.0 , T0 = 0 and TW = Tm + ∆T ) with the same dimensionless numbers is calculated as well. Both cases were computed on the uniform discretization 101x101 points, five points overlapping sub-domain strategy, and with the multiquadrics shape parameter c = 30 . The streamline plots with step 0.2 and temperature contour plots with step 0.1 are superimposed together with the liquid-solid final free boundary, and presented in the Figure 1. The average cavity liquid fraction and the average Nusselt number (for freezing on the cold side and for melting on the hot side of the cavity) as a function of dimensionless time are presented in the Figure 2. Both cases are compared.
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Figure 1: Temperature contour plots (solid lines), liquid-solid interface (bold solid line) and streamline plots (dotted lines) for freezing (left) and melting (right).
Figure 2: Cavity average liquid fraction (left) and average Nusselt number (right) as a function of a Fourier time: comparison of freezing and melting. The steady states of both (freezing and melting) are identical (taking account the mirror transformation) as expected, however, the system dynamics is a bit more intense in the early stages of freezing compared to the dynamics of melting. All calculation have been done with dimensionless time step 10e-5 . Verification of the new approach The represented novel numerical approach has been thoroughly verified by performing the Gobin and Le Quéré benchmark test [11], where four different melting cases have been simulated. Two with Ste=0.01, Pr=0.02, differing in Ra=2.5e4 and 2.5e5, and two with Ste=0.1, Pr=50, differing in Ra=10e7 and Ra=10e8. In all four cases, good agreement has been achieved and our results are within the dispersion of other nine laboratories which attempted the benchmark. Furthermore, since the most difficult to cope with mechanism in the simulation is fluid flow, additional tests have been done to confirm the reliability of the represented new method. The natural convection in the tall cavity filled with low Pr-fluid has been computed and results compared with the fine grid FVM and
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spectral FEM methods [21]. The hot side Nusselt number evolution perfectly converges towards reference results with time-step and spatial discretisation refinement. A detailed report on verification of the method will be given in one of our future publications.
Summary In this paper, the suitability of the proposed new meshless method, for performing simulations of phase change phenomena coupled with fluid flow is demonstrated. The LRBFCM is simple to implement, accurate, it has low numerical diffusion and it does not require any kind of polygonisation or integration. The important feature of the method is straightforward node redistribution capability which might play a key role in more complex solidification processes due to the intense system dynamics near the solid-liquid interface. The results have been validated through performing melting and freezing simulations with the same dimensionless numbers where the steady-state is mirror-identical, despite the different time evolution of the transient. The influence of the natural convection on the freezing reflects in a smaller frozen domain; instead of 50% only 40% of material is frozen. The algorithm has been implemented in C++ computational language with OpenMP library used to parallelize the calculation. Future work will be focused to the N-adaptive time dependent node distribution, the adaptive time step implementation, and the inclusion of more complex physics. This work represents a preliminary study for simulation of solidification [4] by RBFCM.
Acknowledgements The authors would like to thank the Slovenian Research Agency for support in the framework of the projects J2-0099 Multiscale Modeling and Simulation of Liquid-Solid Processes (BŠ) and Young Researcher Ph.D. Programme (GK).
References [1] G. Kosec and B. Šarler: International Journal of Numerical Methods for Heat and Fluid Flow Vol. 18 (2007), p. 868-882. [2] G. Kosec and B. Šarler: CMES Vol. 25 (2008), p. 197-208. [3] G. Kosec and B. Šarler: Cast Metals Research, in press. [4] N. Ahmad, H. Combeau, J. L. Desbiolles, T. Jalanti, G. Lesoult, J. Rappaz, M. Rappaz and C. Stomp: Metallurgical and Materials Transactions A Vol. 29A (1998), p. 617. [5] B. Šarler: Engineering Analysis with Boundary Elements Vol. 16 (1995), p. 83-92. [6] O. Bertrand; B. Binet; H. Combeau; S. Couturier; Y. Delannoy; D. Gobin; M. Lacroix; P. Le Quéré; M. Médale; J. Mencinger; H. Sadat; G. Vieira: Int. J. Therm. Sci. Vol. 38 (1998), p. 5-26. [7] P. Le Quéré and D. Gobin: Int. J. Therm. Sci. Vol. 38 (1999), p. 595-600. [8] F. Stella and M. Giangi: Numerical Heat Transfer, Part A Vol. 38 (2000), p. 193-208. [9] J. Mencinger: Journal of Computational Physics Vol. 198 (2003), p. 243-264. [10] N. Hannoun, V. Alexiades and T. Z. Mai: Numerical Heat Transfer, Part B Vol. 44 (2003), p. 253-276. [11] D. Gobin and P. Le Quéré: Comp. Assist. Mech. Eng. Sc. Vol. 198 (2000), p. 289-306.
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[12] R. Vertnik and B. Šarler: International Journal of Numerical Methods for Heat and Fluid Flow Vol. 16 (2006), p. 617-640. [13] B. Šarler and R. Vertnik: Computers and Mathematics with Applications Vol. 51 (2006), p. 1269-1282. [14] E. J. Kansa: Computers and Mathematics with Applications Vol. 19 (1990), p. 127-145. [15] S. N. Atluri and S. Shen: Computer Modelling in Engineering & Sciences Vol. 3 (2002), p. 11-52. [16] G.R. Liu, Mesh Free Methods, CRC Press, Boca Raton, 2003. [17] G.R. Liu and Y.T. Gu, An Introduction to Meshfree Methods and Their Programming, Springer, Dordrecht, 2005. [18] S .N. Atluri and S. Shen, The Meshless Method, Tech Science Press, Encino, 2002. [19] M.D. Buhmann, Radial Basis Functions, Cambridge University Press, Cambridge, 2000. [20] B. Šarler: From global to local radial basis function collocation method for transport
phenomena, Advances in Meshfree Techniques, Springer Verlag, Berlin, 2007, p. 257-282. [21] M. Založnik, S. Xin and B. Šarler: International Journal of Numerical Methods for Heat & Fluid Flow Vol. 18 (2005), p. 308-324.
