TANG Dexiang, CHEN Feiguo, WANG Xiaowei,. GUO Li & LI Jinghai. Multi-Phase Reaction Laboratory, Institute of Process Engineering, Chi- nese Academy of ...
REVIEW Chinese Science Bulletin 2005 Vol. 50 No. 11 1057—1069
Particle methods for multiscale simulation of complex flows GE Wei, MA Jingsen, ZHANG Jiayuan, TANG Dexiang, CHEN Feiguo, WANG Xiaowei, GUO Li & LI Jinghai Multi-Phase Reaction Laboratory, Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100080, China Correspondence should be addressed to Ge Wei (email: wge@home. ipe.ac.cn)
Abstract The mu lti-scale stru ctu res of comp lex flows have b een great ch allenges to both th eoretical an d engineering r esear ches, and multi-scale modeling is th e natural way in r esponse. P artic le methods (PMs) a re ideal constitutors and pow erful pr obes of multi- scale models, o wing to their phy sical insight and c omputati onal simplic ity. In this pa per, the r ole of differ ent PMs for multi-scale modeling of complex flows is cr itica lly reviewed and possible developme nt of P Ms in this bac kground is pr ospected, with the emphasis on pseudo- par ticle mode ling (PPM ). Th e perfor man ces of some different PMs ar e compar ed in simulations and ne w developme nt in the fundame ntals and applic ations of PP M is also repor ted, demonstr ating P PM as a unique PM for multi-sc ale modeling. Keywords: complex flow, dyna mic simulat ion, multi-scale modeling, pseudo-particle modeling, par ticle method, t r ansport pr ocess, scale-up. DOI: 10.1360/04wb0108
More often than not, flows in industrial or natural processes are complex, which usually involves multi-phases or multi-components, and are typically turbulent or sometimes non-Newtonian. Their kaleidoscopic behavior is very fascinating to researchers on complexity and has provided a favorable test field with ample resources for them. However, this is also the field where engineers are most frustrated, due to the lack of theoretical guidelines and simulation tools in the scaling-up and scaling-down of laboratorial processes to industrial operations. To a great extent, the complexity and hence the scaling problem lie in the multi-scale nature of the dynamic structures in these flows, which displays progressively with increasing scales. It is too expensive, if not impossible, to describe these structures on the micro-scale only, and is obviously insufficient to describe them on the macro-scale only. Therefore, the multi-scale methodology blooms naturally as a practical approach to cope with this complexity. In general, the approach has received increasing attention since the 1990s, as reviewed by ref. [1], but its application is quite limited yet. The single-scale approaches are still prevailing, which not only conceals the Chinese Science Bulletin
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structures below the considered scale, but also smoothes the structures above that scale, causing severe distortion of the simulations results, especially for the transport and reaction processes, since they are more sensitive to the structures. According to ref. [1], there are three mainstreams in multi-scale simulation: descriptive, with different scale descriptions for different spatiotemporal portions of the system integrated into one simulation; correlative, with smaller scale simulations providing constitutive correlations for simulations on the next larger scale; and variational, with equations on different scales correlated by stability conditions to give a lumped description of the system. Also in this order, the emphasis of the three approaches shifts from numerical techniques reproducing the complex phenomena to theoretical insights of the underlying mechanism. The second approach has a relatively long history. Its idea has been well exemplified by the measurement of fluid properties through molecular dynamics simulations which is then fed to numerical calculations of the Navier-Stokes equation to predict the permeability in porous media. Large eddy simulation (LES)[2] or Reynolds stress models for turbulence are also typical examples in this catalogue. However, the first approach is practiced more in recent years, such as the integration of molecular dynamic simulations into finite element calculations to describe fracture developments in material mechanics[3―5] and in micro-flows[6,7]. The third approach is still in its infancy but we believe it to be a more profound solution to the complexity, though its development also relies on the knowledge accumulated in simulations using the other two approaches. For gas-solid flow, the energy minimization multi-scale (EMMS) model[1,8] has been a typical embodiment of this approach. While multi-scale methodology has provided the framework to tackle the complexity, its success also relies on the performance of the constituent models on different scale, for which the so-called “particle methods” (PMs) are attractive candidates for the reasons we will discuss in the rest of the paper, but further improvement are possible as we will demonstrate in pseudo-particle modeling (PPM)[9,10]. 1 Role of particle methods in multi -scale simul ations of compl ex flows PMs are a collection of novel methods in computational mechanics and chemistry, especially complex flows. They reconstruct continuum behavior from the movement and interactions of numerous particles of discrete attributes. Coincidently, PMs are established on different scales, which provide at least structural convenience of “seamless” incorporation into multi-scale simulation. Compared with classic particle-continuum connections between the scales in descriptive approaches, particle-particle connection using different PMs seems to be more flexible and 1057
REVIEW physically more consistent. On the micro-scale, the particles possess mainly molecular features with strong thermal movement and conservative simple interactions. Continuum properties such as pressure and viscosity are expressed as quadratures over many particles. The oldest of all PMs―molecular dynamics (MD)[11,12] is a prototype on this scale. On the macro-scale, the particles are roughly understood as Lagrangian presentation of material elements with stress and energy dissipation between individual particles correlated to their state variables. Typical examples are smoothed particle hydrodynamics (SPH)[13 ― 16] and partially the moving particle semi-implicit (MPS)[17,18]. On the mesoscale in between, we have dissipative particle dynamics (DPD)[19] and its extensions[20,21], where both thermal fluctuation and energy dissipation are considered, so as to describe the collective behavior of a cluster of molecules. The role of these PMs in multi-scale simulation of complex flows is explained in Fig. 1. The most obvious virtue of PMs for simulating complex flows is their numerical simplicity and parallelism in dealing with a variety of challenges for continuum computational fluid dynamics (CFD), such as discontinuities, free surfaces, many-body moving boundaries, large deformation, fractures, non-Newton behavior. Even with the remarkable developments in finite difference or finite element schemes using volume of fluid (VOF)[22], arbitrary Lagrangian-Euler (ALE) [23] , fictitious domain [24] , front-
tracking[25,26], or the level set[27] approaches, the convenience of PMs remains attractive. For the descriptive approach, PMs provide a unified framework for different scale simulations by enabling discrete models for macroscale physics, which facilitate its implementation fundamentally. They also facilitate the incorporation of the models describing the complex physical or chemical processes accompanying the flows, such as phase changes on deforming phase interfaces. In general, PMs are computationally more intensive, but for the description of micro-scale and sometimes meso-scale (such as fine powders and polymers) structures and behavior, they are almost the only choice since no continuum model naturally exists. Besides the traditional examples such as the reaction and transportation processes on particle, droplet or bubble surfaces, micro- and mesoscale features are increasingly concerned in engineering with the wide-spreading use of nano-materials and the advent of micro-manufacturing. It is also important for the correlative approach, since constitutive correlations are provided by a closed statistical description of the smaller scale. So far, theoretical inductions are successful for quite simple systems, say the equation of state for ordinary gas. An essential difficulty is that for equilibrium systems or quasi-equilibrium elements, the description employs only a few variables; but for the non-linear systems typical for complex flows, the large deviation from equilibrium increases dramatically the degrees of freedom in statistical
Fig. 1. Role of particle methods in multi-scale modeling of complex flows.
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REVIEW behavior, and hence the number of variables involved. In this case, simulations score over experiments in that every detail of the results can be studied thoroughly and non-intrusively, which is hardly possible for experiments. Especially, we can digitally experiment with simplified or idealized systems that are impossible or very expensive for physical experiments while still providing clear physical meanings of the results, thereby suggesting a stepwise route to explore the complexity of such systems. For the variational approach, PMs are powerful probes of the consistent mechanism behind multi-scale structures, especially on the micro-scale. Aggregative fluidization in gas-solid systems is a typical example of complex flows displaying strong non-linearity[28]. An apparent difficulty is the lack of clear separation between the multi-scale structures, which further complicates the statistical treatment. That is, a correlative multi-scale approach may prove difficult and the variational approach is in demand. Using PMs, we have got preliminary evidence[10,29] that the hydrodynamic complexity in particle-fluid systems on macro-scales can be traced to micro-scale originations even below the continuum limit, where expression of the stability conditions for the variational approach can be found already[29]. However, as PMs are not originally designed for either probes of complexity or constitutors of multi-scale models, further improvements are expected. PPM and Macro-scale PPM[30,31] we have developed since 1996 present such an attempts by incorporating the virtues of different PMs. 2
Pseudo-parti cle model ing as a pr obe of compl exity
PPM is proposed for microscopic simulation of particle-fluid systems as a combination of MD and direct simulation Monte-Carlo (DSMC[32]; for a review, see ref. [33]). As shown in Fig. 2, each pseudo-particle (PP) has four properties: mass (mf), diameter (df), position (P) and velocity (v), among which mf and df may remain constant in a simulation. All particles move synchronously in uniform time steps. In each time step, all particles move independently, possibly under some external forces. At the end of each step, if the distance between two particles |P1 －P2| is less than the sum of their radius (d1+d2)/2, and the internal product of P1－P2 and v1−v2 is negative, they will collide as two rigid and smooth particles, i.e. v1 = v10 −
(1 + e) ⋅ m2 (v10 − v20 ) ⋅ ( P1 − P2 ) ⋅ ( P1 − P2 ), m1 + m2 | P1 − P2 |2
v2 = v20 −
(1 + e) ⋅ m1 (v10 − v20 ) ⋅ ( P1 − P2 ) ⋅ ( P1 − P2 ), m1 + m2 | P1 − P2 |2
where e is the restitution coefficient. Collisions between PPs are fully elastic, that is, e=1. In the next time step, the particles move to new positions with their new velocities, and so on. Collisions are processed in predefined sequence that can guarantee spatial homogeneity and isotropy. Chinese Science Bulletin
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Fig. 2.
A particle-fluid system as simulated in PPM.
2.1 Physical background and comparison with other PMs PPM is ideal for exploring the mechanism of complex flows ab initio. To final analysis, the complexity of flow behavior is shaped by two ingredients on the micro-scale: the relative displacements and interactions of the numerous molecules. Adding to the generality of the characteristics of complex system identified by ref. [1], we notice that complex structures or behavior are most probably observed when these two ingredients are competitive and hence must compromise, as in the case of emulsions and the so-called soft-matter that includes most bio-systems. When either the displacements or interactions are dominant, as in the case of dilute gas and solid crystals, respectively, complexity is much less spectacular (in terms of geometrical distribution of the particles). Most PMs explicitly consist of these two ingredients, which is operator splitting in a numerical sense, but it is physically more meaningful and concise in PPM. Obviously, the collision dynamics in PPM is identical to that of hard-sphere MD, so that mass, momentum and energy are conserved to machine accuracy. However, the collision detection procedure, which is most time-consuming and difficult to be parallelized in hard-sphere MD, has been greatly simplified to a procedure identical to soft-sphere MD. Actually, the physical model behind such a treatment is essentially different from MD and is more similar to DSMC, but an intrinsic difference is that in DSMC the collisions follow designed statistical rules that are reflection of the real physical processes only in very limited cases such as dilute gas. Although in other cases, it may also reproduce correct hydrodynamic behavior on a macro-scale, the natural way of flows originating from molecular movements is obscured and replaced by an artificial process. The remark also holds for other PMs such as lattice Boltzmann method (LBM, see ref. [34] for reviews) and lattice gas automaton (LGA, see ref. [35] for reviews), where the artifacts are even more pronounced. However, in PPM the PPs can be considered as idealized physical particles of their own, just like other well-known models of real particles such as hard-spheres or Le1059
REVIEW nard-Jones molecules. The collision processes are basically deterministic, reasonably depend on the detailed state of each neighboring particles, and the statistical laws such as the Boltzmann equation (in the dilute limit) and the H-theorem are reproduced from a lower level rather than predefined one. The physical consistency of PPs with real molecules in terms of their behavior relevant to hydrodynamics has been evidenced in different aspects. Qualitatively, we may notice that the model allows fast PPs to get closer in collisions, which agrees well to the shrinking of collisional cross-section of molecules with higher kinetic energy. And in equilibrium simulations of dense gases, PPM has successfully produced the long time tail of the velocity autocorrelation function of the PPs[10], which reflects the correlation between the velocity and position of neighboring particles that are usually considered independent in statistical treatments, and has been a notable fingerprint of hydrodynamic behavior on molecular scales. Other proofs come from the statistical properties of PP fluid that are key issues for the feasibility of the model also. They have been theoretically evaluated earlier1) very preliminarily. We have recently conducted systematic
measurements in simulations with some results reported already[36]. Here Fig. 3 shows the temporal development of plane Poiseuille flow driven by gravities of different intensity (g). The no-slip boundary condition is satisfied by bouncing back of the PPs on walls. As an indication of Newtonian behavior, the final flow profile is parabolic and the dynamic viscosity thus measured, as summarized in Fig. 4, is independent of the mean flow velocity (Vm). Also the compressibility factor (Z) of 2D and 3D PP fluid is shown in Fig. 5, from which the equation of state can be expressed in reduced units of df =2, mf=1 and time step Δt=1 as p=4ηZv02/π, where v0 is the root mean square velocity of the fluctuating PPs. Based on these results, the properties of the simulating PP fluid can be mapped to the simulated physical fluid. Although new attributes should be added to display properties of more complex fluids, we note that most complex flows are actually generated by simple Newtonian fluids. 2.2 Interactions between the pseudo-particles and the solid particles As a PM, PPM can incorporate irregular solid particles simply by discretizing them into collections of constrained
Fig. 3. Development of the flow profile in plane Poiseuille flow as simulated in PPM. 1) Ge, W., Multi-scale simulation of fluidization, Doctor dissertation, Harbin Institute of Technology, Harbin, China, 1998.
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Fig. 4. Kinematic viscosity vt=μf/ρf of the PP fluid as measured in simulations.
can be solved from the momentum, angular momentum and kinetic energy balance. However, to describe the contours with constrained PPs accurately, the solid particles must be large enough compared with the fluid PPs, which may well exceed the optimum (i.e. minimum) value for the required flow description accuracy and hence costs a lot of extra computation. In this case, smooth curves are preferred to describe the contours, and hence the friction or tangential force on the contact point must be considered in order to restore the no-slip boundary condition. Since the fluid PPs are by definition non-rotational, it is simple and also reasonable to treat them as mass points in their collision with the solid particles, so that the relative velocity vf−vs−ω s× (Pf−Ps) before and after the collision should be antisymmetric, which gives the momentum balance. Combined with the angular momentum and kinetic energy balance, a closed second order equation set can be established, and one of its two solutions gives ω s1=−ω s0, which is unreasonable and should be dropped, so the post-collision velocities of the two particles can be determined uniquely1). In fact, the discussion above is closely relevant to a central problem in multi-scale simulation, that is the correlation between different scales. With proper implementation of the no-slip boundary conditions, the micro-scale movement of the PPs is correlated to the macro-scale movement of the solids with physical consistency, and the simulation of particle-fluid systems in PPM is a typical multi-scale simulation in PMs. Three-scale simulations in PMs with colloidal particles, complex polymer molecules and PPs are attempted now. 2.3 Applications of PPM in microscopic simulation of particle-fluid systems
Fig. 5. The compressibility factor of the PP fluid as measured in simulations.
PPs that interact with the free (fluid) PPs just as normal expect that the impacts on the constrained PPs do not take effect immediately, but are first summed over the whole solid particle and then distributed among the PPs according to the dynamics of rigid bodies. MD simulations[37] have suggested that if fixed particles are lined up to produce a certain roughness, the no-slip boundary condition can be satisfied naturally. However, kinetic energy is not conserved precisely in this manner. An alternative is to treat the collision between the fluid PP and the solid particle directly, with its contour presented by the smooth constrained PPs. In this way, the velocity and angular velocity of the solid particle (vs, ω s) and the velocity of the PP (vf)
Simulations in PPM can be carried out in ways very similar to physical experiments. Needless to say, with such a detailed approach, the computational cost is considerable even for the flow around a single particle. Therefore, careful and optimized selection of simulation parameters has been an important factor for the success of PPM. Usually, we have more constraints than the variables available for optimization, so we have to neglect or loose some unimportant constraints. For instance, the Mach number Ma can be enlarged considerably for incompressible flows as long as it is far below unity. The optimization usually results in PP size only one or two orders smaller than that of the solids, so when converted to dimensional values, the size of PPs can be much larger than real molecules, and in this sense, they are not molecules geometrically or computationally. In fact, the physical procedure corresponding to this optimization is to model the real system
1) Li, T., Statistical properties of the pseudo-particle model and its applications. M. Sc. Thesis, Institute of Process Engineering, the Chinese Academy of Sciences, Beijing, 2004.
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REVIEW with a simpler system, i.e. a system with much fewer particles. The price is much higher fluctuation in its flow behavior, a background noise present as the unavoidable computational error. Compared with the numerical errors in continuum approaches, this error has clear physical meaning as reflection of the remaining randomness of molecular movements on micro-scales. As all balances are kept to machine accuracy, the errors are well controlled and the algorithm is very robust. Providing there is enough computing capacity, the size of the PPs and the error of the simulations can be reduced without theoretical limit. A unique advantage of the Lagrangian treatment in PPM is that mass and heat transport processes are automatically integrated into the simulation with the momentum transport, so that no additional computation is needed and all processes can be examined from a molecular level. However, as a physical model, the mass transfer coefficient is a state function of the PP fluid, which adds a constraint in selecting simulation parameters and hence limits the range of applications. A preliminary solution is to take PPs as carriers of each transferred components in addition to their original missions in PPM. The amount of transferred mass in each collision is correlated to the concentrations of each component on the two particles (1,2) involved, say, in the form of δ Ci,1Æ2=ki(Ci,1−Ci,2) with Ci and ki being the relative concentration and collisional mass transfer coefficient of component i. The desired mass transfer rate can be adjusted by ki. In fact, heat transfer and reactive flows can also be simulated in a very similar
way as we have attempted recently. As an example, we simulated the fluidization of relatively large solids by a gas with similar density of ambient air but much higher viscosity. The solids are assumed to have constant concentration (Ca) for the only transferred component “a”. The dynamic distribution of Ca in the flow field is shown in Fig. 6. As the component is emitted from the surfaces of the solids, diffused into the fluid and then convected by the fluid flow, it is more concentrated in the dense phase due to the higher solids concentration and lower slip velocity there. This heterogeneity actually reduces the overall mass transfer rate because of the transfer potential, i.e. the concentration gradient is low in the dense phase and the source supply from the solids is short in the dilute phase. However, we can also find that the forming and dissolution of the clusters (as indicated by the circles and arrows in Fig. 6) cause violet mass exchange between the two-phases, which enhances mass transfer on the other hand. Therefore, it is the competing of these two factors that determines whether a fluidized bed is favorable for the process, and it is crucial for a predictive simulation to capture the dynamic flow structures in detail. This simulation is carried out on an HP DL360 server with one Intel PIII 1.13GHz CPU at a rate of roughly half million steps per week. Because of the strong fluctuation of the PPs, the fluid flow around each particle cannot be seen in satisfactory resolution, for which we have to use parallel computers. However, reasonable local behavior of the solids is reproduced using about half million PPs only.
Fig. 6. Simulated distribution of a gas component “a” emitted from the solids in a fluidized bed. The snapshots begins at t=3.0×106, and the intervals are 20000 steps.
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REVIEW In fact, the U- or inversely U-shaped clusters are observed which are in qualitative agreement with laser sheet meas urements in concurrent-up gas-solid flows[38]. It may be somewhat surprising from a numerical view, but from a physical view, it is naturally considered as a strong indication that the PPs have already started to behave collectively, or hydrodynamically, on a scale much smaller than in statistical predictions based on the molecular chaos assumption. Then the question is whether we can explain macro-scale flow behavior from the molecular level directly, rather than via a continuum description first. To explore this possibility, the molecular details revealed in PPM simulations, which have been considered as undesired by-products resulting in low efficiency from a numerical view, turn to be the most valuable information that is certainly out of the reach of other PMs that have already adopted a statistical treatment, such as DSMC and LBM. Recently, with this powerful tool, we have demonstrated the validity of the stability condition used in the EMMS model, and are encouraged by the discovery of how local and transient dominance of particles or fluid movement tendencies evolves to global and long-term compromise between them by the formation of multi-scale heterogeneity[29], illuminating a promising way to correlate molecular to macro-scale behavior consistently.
⎧(3 − s )5 − 6(2 − s )5 + 15(1 − s )5 , 0 ≤ s < 1, ⎪⎪ W (r ) = A ⎨(1- s )5 − 6(2 − s)5 , 1 ≤ s < 2, ⎪ 5 2 ≤ s < 3, ⎪⎩(1 − s ) , where s=3r/R, and A is the normalization coefficient to satisfy
∫r < R W (r)d v = 1, which is 63/478π for two-dimensional cases. In calculation, however, the integration is over limited number of neighboring points and dv is discretized as mfi/ρi. To correct the numerical errors, the kernel is directly normalized by setting m A = 1 ∑ i Wia fi .
ρi
The schemes thus obtained are listed in Table 1, where mf≡1 and R≡1. Table 1 List of the schemes expressing gradient and Laplacian in macro-scale particle methods Operator Scheme Expression
3 Macr o-scale pseudo-par ticle modeling and r elated particle methods for dir ect simulation While micro-scale PMs are more suitable for exploring the mechanism of complex flows, macro-scale PMs provide the direct interface for engineering applications in most cases and their difficulties are relatively more technological, i.e. we are more concerned about the accuracy, stability, efficiency and generality of their numerical schemes. The following discussion will be restricted to a family of PMs named macro-scale pseudo-particle modeling (MaPPM)[30,31], which can be understood as a generalization of the SPH approach. Mathematically, the basis of MaPPM is to express the operators involved in hydrodynamic equations, such as gradient, divergence and Laplacian, as weighted average of the additive directional difference between neighboring points, i.e. particles. A variety of discretization schemes can be derived systematically as summarized in Table 1. Namely, the weighted function can be fi, fi − fa or fi + fa, denoted as I, N and P, with f and a being the function and the point in consideration, respectively, and i is the neighbor point. The weight function, or kernel can be applied with 0 to 2 orders of derivitives. The form of the kernel is another important factor, and preliminary results[39] show that splines of high order continuity are better choices in general. A comparative study is carried out here on the performance of some typical schemes, for which the quintic spline[40] with a support of r=|rai|
