A Versatile Multiphysics Modeling Tool 385 M ∇Q(R, V)= J Q(R , V )∇W(|R − R |, H) (2) Ρ(R ) J J J J=I J
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Sunden CH010.tex 10/9/2010 15: 22 Page 383 10 SPH–aversatile multiphysics modeling tool Fangming Jiang1 and Antonio C.M. Sousa2, 3 1The Pennsylvania State University, USA 2Universidade de Aveiro, Portugal 3University of New Brunswick, Canada Abstract In this chapter, the current state-of-the-art and recent advances of a novel numerical method – the Smoothed Particle Hydrodynamics (SPHs) will be reviewed through case studies with particular emphasis to fluid flow and heat transport. To provide sufficient background and to assess its engineering/scientific relevance, three par- ticular case studies will be used to exemplify macro- and nanoscale applications of this methodology. The first application deals with magnetohydrodynamic (MHD) turbulence control. Effective control of the transition to turbulence of an electri- cally conductive fluid flow can be achieved by applying a stationary magnetic field, which is not simply aligned along the streamwise or transverse flow direction, but along a direction that forms an angle with the main fluid flow in the range of 0◦ (streamwise) to 90◦ (transverse). The SPH numerical technique is used to interpret this concept and to analyze the magnetic conditions. The second application deals with non-Fourier ballistic-diffusive heat transfer, which plays a crucial role in the development of nanotechnology and the operation of submicron- and nanodevices. The ballistic-diffusive equation to heat transport in a thin film is solved numerically via the SPH methodology. The third application deals with mesoscopic pore-scale model for fluid flow in porous media. SPH simulations enable microscopic visu- alization of fluid flow in porous media as well as the prediction of an important macroscopic parameter – the permeability. Keywords: CFD, Numerical methods, Smoothed particle hydrodynamics 10.1 Introduction A novel numerical method – the smoothed particle hydrodynamic (SPH) offers a relatively flexible tool for heat and fluid flow computations, as it can cope with a wide range of space scales and of physical phenomena. SPH is a meshless particle- based Lagrangian fluid dynamics simulation technique, in which the fluid flow is WIT Transactions on State of the Art in Science and Engineering, Vol 41, © 2010 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) doi:10.2495/978-1-84564-144-3/10 Sunden CH010.tex 10/9/2010 15: 22 Page 384 384 Computational Fluid Dynamics and Heat Transfer represented by a collection of discrete elements or pseudo-particles. These par- ticles are initially distributed with a specified density distribution and evolve in time according to the constitutive conservation equations (e.g., mass, momentum, energy). Flow properties are determined by an interpolation or smoothing of the nearby particle distribution using a special weighting function – the smoothing kernel. SPH was first proposed by Gingold and Monaghan [1] and by Lucy [2] in the context of astrophysical modeling. The method has been successful in a broad spectrum of problems, among others, heat conduction [3, 4], forced and natural convective flow [5, 6], low Reynolds number flow [7, 8], and interfacial flow [9–11]. Many other applications are given by Monaghan [12] and Randle and Libersky [13], which offer comprehensive reviews on the historical background and some advances of SPH. In comparison to the Eulerian-based computational fluid dynamics (CFD) methods, SPH is advantageous in what concerns the follow- ing aspects: (a) particular suitability to tackle problems dealing with multiphysics; (b) ease of handling complex free surface and material interface; and (c) relatively simple computer codes and ease of machine parallelization. These advantages make it particularly well suited to deal with transient fluid flow and heat transport. This chapter is organized in six main sections, namely: 1. Introduction; 2. SPH theory, formulation, and benchmarking; 3. Control of the onset of turbulence in MHD fluid flow; 4. SPH numerical modeling for ballistic-diffusive heat conduc- tion; 5. Mesoscopic pore-scale SPH model for fluid flow in porous media, and 6. Concluding remarks. 10.2 SPH Theory, Formulation, and Benchmarking SPH, despite its promise, is far from being at its perfected or optimized stage. Problems with the SPH, such as handling of solid boundaries [14], choice of smooth- ing kernel and specification of the smoothing length [15], treatment of heat/mass conducting flow [5, 6], and modeling of low Reynolds number flow, where the surface/interface effects act as a dominating role relative to the inertial effect [10], have prevented its application to some practical problems. This section reports an effort to enhance the SPH methodology and develop appropriate computer programmable formulations. 10.2.1 SPH theory and formulation In the SPH formulation, the continuous flow at time t is represented by a collec- tion of N particles each located at position ri(t) and moving with velocity vi(t), i = 1, 2, ......, N. The “smoothed” value of any field quantity q(r, v) at a space point r (bold text designates vector or tensor) is a weighted sum of all contributions from neighboring particles, namely: m q(r, v)= j q(r , v ) w (|r − r |, h) (1) ρ(r ) j j j j j WIT Transactions on State of the Art in Science and Engineering, Vol 41, © 2010 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Sunden CH010.tex 10/9/2010 15: 22 Page 385 SPH – a versatile multiphysics modeling tool 385 m ∇q(r, v)= j q(r , v )∇w(|r − r |, h) (2) ρ(r ) j j j j=i j where, mj and ρ(rj), respectively, denote the mass and density of particle j. w(|r|, h) is the weight or smoothing function with h being the smoothing length. Density update Based on equation (1), the local density at position ri(t) is: ρi = mjwij (3) j The summation is over all neighboring particles j including particle i itself. Equa- tion (1) conserves mass accurately if the total SPH particles are kept constant. In terms of equation (2), another formulation for density calculation can be derived from the continuity equation, namely dρ i = m v ·∇w (4) dt j ij i ij j=i The summation is over all neighboring particles j with exception of particle i itself. Here, vij = vi − vj, rij = ri − rj, and wij = w(|ri − rj|, h); the subscript in the oper- ator ∇i indicates the gradient derivatives are taken with respect to the coordinates at particle i r ∂w ∇ = ij ij iwij (5) rij ∂ri Equation (4) is the Galilean invariant, which is more appropriate for interface reconstruction in free surface flow simulations. The use of this equation has the computational advantage over equation (3) of calculating all rates of change in one pass over the particles; whereas the use of equation (3) requires one pass to calculate the density, then another one to calculate the rates of change for velocity, and different scalars, such as temperature. Equation (4) is employed to perform the density update in the case studies to be presented. Incompressibility A quasi-compressible method is implemented to determine the dynamic pressure p [8], in which an artificial equation of state is used ρ γ p = p0 − 1.0 (6) ρ0 where, p0 is the magnitude of the pressure for material state corresponding to the reference density ρ0. The ratio of the specific heats γ is artificially a large value WIT Transactions on State of the Art in Science and Engineering, Vol 41, © 2010 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Sunden CH010.tex 10/9/2010 15: 22 Page 386 386 Computational Fluid Dynamics and Heat Transfer (say, 7.0) to guarantee the incompressibility of the fluid. The speed of sound, cs, can be formulated as: 2 = γp0 = 2 cs αU (7) ρ0 where, U is the characteristic or maximum fluid velocity. The choice of α is a compromise: it should be an adequate value to avoid making p0 (or cs) too large, which will force the time advancement of the simulation to become prohibitively slow, and also to avoid yielding Mach numbers that can violate the incompressibility condition of the fluid; α takes 100 in the case studies to be presented, which ensures the density variation be less than 1.0%. Velocity rate of change The SPH formulation of viscous effects in fluid flow requires careful modeling to satisfy its adaptivity and robustness. In this respect, Morris et al. [8] presented the following expression, equation (8), to simulate viscous diffusion. $ % 1 m (µ + µ )v 1 ∂w ∇·µ∇ v = j i j ij ij (8) ρ ρ ρ |r | ∂r i j=i i j ij i This expression has proven to be highly accurate in particular for the simulations of low Reynolds number planar shear flows [7, 8]. The variable µ denotes the dynamic viscosity of the fluid. In what concerns equation (2), the pressure gradient terms in the Navier– Stokes (N–S) equations can be formulated using the following symmetrized [6] SPH version: 1 pi pj − ∇p =− mj + ∇iwij (9) ρ ρ2 ρ2 i j=i i j Therefore, the SPH momentum equation yields ⎡ ⎤ p p 1.0 ⎢ j + i ∇ w − ⎥ dv ⎢ 2 2 i ij ⎥ i =− ρj ρi ρiρj + mj⎢ ⎥ Fi (10) dt ⎣ ∇iwij · rij ⎦ j=i µ + µ v i j 2 + 2 ij rij 0.01h where, F is the body force per unit mass (m/s2). The quantity 0.01 h2 is used to prevent singularities for rij ≈ 0. Position update To assure the SPH particles move with a velocity consistent with the average velocity of its neighboring particles it used the variant proposed by Morris [10] dr m v i = v + ε j ij w (11) dt i ρ¯ ij j ij Here, ρ¯ij = (ρi + ρj)/2.0 and the factor ε is chosen as ε = 0.5.