05 book 2007/5/15 page 58 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 58 Chapter 7. Advanced particulate flow models d c Figure 7.3. Introduction of a cutoff function. Thus, the preceding analysis indicates that, for the three-dimensional case, an interaction “cutoff” distance (d c ) should be introduced (Figure 7.3), ||r i − r j || = d c ≤ d (+) , (7.14) to avoid long-range (central-force) instabilities. Remark. By introducing a cutoff distance, one can circumvent a loss-of-convexity instability. However, introducing such acutoff can induce another type of instability. Specif- ically, if the particles are in static equilibrium, or are not approaching one another, and if the cutoff distance, d c , is much smaller than the static equilibrium separation distance, d e , then the particles will not interact at all. Thus, we have the following two-sided bounds on the cutoff for near-field forces to play a physically realistic role:  α 2 α 1  1 −β 1 +β 2 = d (−) ≤ d c ≤ d (+) =  β 2 α 2 β 1 α 1  1 −β 1 +β 2 . (7.15) Clearly, since β 2 >β 1 , d (−) is a lower bound(dictatedbythe minimum interaction distance), while d (+) is an upper bound (dictated by the (convexity-type) stability). 7.4 A simple model for thermochemical coupling As indicated earlier, in certain applications, in addition to the near-field and contact effects introduced thus far, thermal behavior is of interest. For example, applications arise in the study of interstellar particulate dust flows in the presence of dilute hydrogen-rich gas. In many cases, the source of heat generated during impact in such flows can be traced to the reactivity of the particles. This affects the mechanics of impact, for example, due to thermal softening. For instance, the presence of a reactive substance (gas) adsorbed onto the surface of interplanetary dust can be a source of intense heat generation, through thermochemical reactions activated by impact forces, which thermally softens the material, thus reducing the coefficient of restitution, which in turn strongly affects the mechanical impact event itself (Figure 7.4). To illustrate how one can incorporate thermal effects, a somewhat ad hoc approach, building on the relation in Equation (2.50), is to construct a thermally dependent coefficient 05 book 2007/5/15 page 59 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 7.4. A simple model for thermochemical coupling 59 REACTIVE FILM TWO IMPACTING PARTICLES ZOOM OF CONTACT AREA Figure 7.4. Presence of dilute (smaller-scale) reactive gas particlesadsorbed onto the surface of two impacting particles (Zohdi [217]). of restitution as follows (multiplicative decomposition): e def =  max  e o  1 − v n v ∗  ,e −  max  1 − θ θ ∗  , 0  , (7.16) where θ ∗ can be considered as a thermal softening temperature. In order to determine the thermal state of the particles, we shall decompose the heat generation and heat transfer processes into two stages. Stage I describes the extremely short time interval when impact occurs, δt  t, and accounts for the effects of chemical reactions, which are relevant in certain applications, and energy release due to mechanical straining. Stage II accounts for the postimpact behavior involving convective and radiative effects. 7.4.1 Stage I: An energy balance during impact Throughout the analysis, we shall use extremely simple, basic, models. Consistent with the particle-based philosophy, it is assumed that the temperature fields are uniform in the particles. 30 We consider an energy balance, governing the interconversions of mechanical, thermal, and chemical energy in a system, dictated by the first law of thermodynamics. Accordingly, we require the time rate of change of the sum of the kinetic energy (K) and the stored energy (S) to be equal to the sum of the work rate (power, P) and the net heat supplied (H): d dt (K + S) = P +H, (7.17) where the stored energy comprises a thermal part, S = mCθ, (7.18) 30 Thus, the gradient of the temperature within the particle is zero, i.e., ∇θ = 0. Thus, a Fourier-type law for the heat flux will register a zero value, q =−K ·∇θ = 0. 05 book 2007/5/15 page 60 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 60 Chapter 7. Advanced particulate flow models where C is the heat capacity per unit mass and, consistent with our assumptions that the particles deform negligibly during impact, we assume that there is an insignificant amount of mechanically stored energy. The kinetic energy is K = 1 2 mv ·v. (7.19) The mechanical power term is due to the forces acting on a particle, namely P = dW dt =  tot · v, (7.20) and, because dK dt = m ˙ v ·v, (7.21) and we have a balance of momentum m ˙ v ·v =  tot · v, (7.22) we have dK dt = dW dt = P, (7.23) leading to dS dt = H. (7.24) For example, in certain applications of interest, such as the ones mentioned, we consider that the primary source of heat is due to chemical reactions, where the reactive layer generates heat upon impact. The chemical reaction energy is defined as δH def =  t+δt t H dt. (7.25) Equation (7.24) can be rewritten for the temperature at time = t +δt as θ(t + δt) = θ(t) + δH mC . (7.26) The energy released from the reactions is assumed to be proportional to the amount of the gaseous substance available to be compressed in the contact area between the particles. A typical ad hoc approximation in combustion processes is to write, for example, a linear relation δH ≈ κ min  | I n | I ∗ n , 1  πb 2 , (7.27) where I n is the normal impact force; κ is a reaction (saturation) constant, energy per unit area; I ∗ n is a normalization parameter; and b is the particle radius. For details, see Schmidt [172], for example. For the grain sizes and material properties of interest, the term in Equation (7.26), δH mC , indicates that values of approximately κ ≈ 10 6 J/m 2 will generate 05 book 2007/5/15 page 61 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 7.4. A simple model for thermochemical coupling 61 significant amounts of heat. 31 Clearly, these equations are coupled to those of impact through the coefficient of restitution and the velocity-dependent impulse. Additionally, the postcollision velocities are computed from the momentum relations coupled to the temperature. Later in the analysis, this equation is incorporated into an overall staggered fixed-point iteration scheme, whereby the temperature is predicted for a given velocity field, and then the velocities are recomputed with the new temperature field, etc. The process is repeated until the fields change negligibly between successive iterations. The entire set of equations are embedded within a larger overall set of equations later in the analysis and are solved in a recursively staggered manner. 7.4.2 Stage II: Postcollision thermal behavior After impact, it is assumed that a process of convection, for example, governed by Newton’s law of cooling, and radiation, according to a simple Stefan–Boltzmann law, occurs. As be- fore, it is assumed that the temperature fields are uniform within the particles, so conduction within the particles is negligible. We remark that the validity of using a lumped thermal model, i.e., ignoring temperature gradients and assuming a uniform temperature within a particle, is dictated by the magnitude of the Biot number. A small Biot number indicates that such an approximation is reasonable. The Biot number for spheres scales with the ratio of the particle volume (V ) to the particle surface area (a s ), V a s = b 3 , which indicates that a uniform temperature distribution is appropriate, since the particles, by definition, are small. We also assume that the dynamics of the (dilute) gas does not affect the motion of the (much heavier) particles. The gas only supplies a reactive thin film on the particles’ surfaces. The first law reads d(K +U) dt = m ˙ v ·v + mC ˙ θ =  tot · v    mechanical power − h c a s (θ − θ o )    convective heating −Ba s (θ 4 − θ 4 s )    far-field radiation , (7.28) where θ o is the temperature of the ambient gas; θ s is the temperature of the far-field surface (for example, a container surrounding the flow) with which radiative exchange is made; B = 5.67 × 10 −8 W m 2 −K is the Stefan–Boltzmann constant; 0 ≤  ≤ 1 is the emissivity, which indicates how efficiently the surface radiates energy compared to a black-body (an ideal emitter); 0 ≤ h c is the heating due to convection (Newton’s law of cooling) into the dilute gas; and a s is the surface area of a particle. It is assumed that the radiation exchange between the particles (emission exchange between particles) is negligible. 32 For the applications considered here, typically, h c is quite small and plays a small role in the heat transfer processes. 33 From a balance of momentum, we have m ˙ v · v =  tot · v, and Equation (7.28) becomes mC ˙ θ =−h c a s (θ − θ o ) − Ba s (θ 4 − θ 4 s ). (7.29) 31 By construction, this model has increased heat production, via δH, for increasing κ. 32 Various arguments for such an assumption can be found in the classical text of Bohren and Huffman [33]. 33 The Reynolds number, which measures the ratio of the inertial forces to viscous forces in the surrounding gas and dictates the magnitude of these parameters, is extremely small in the regimes considered. 05 book 2007/5/15 page 62 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 62 Chapter 7. Advanced particulate flow models Therefore, after temporal integration with the previously used finite difference time step of t  δt, we have 34 θ(t + t) = mC mC + h c a s t ¯ θ(t) − tBa s  mC + h c a s t  θ 4 (t +t) −θ 4 s  + h c a s tθ o mC + h c a s t , (7.30) where ¯ θ(t) def = θ(t+δt) is computed from Equation (7.26). This implicit nonlinear equation for θ(t + t), for each particle, is solved simultaneously with the equations for the dy- namics of the particles by employing a multifield staggering scheme, which we shall discuss momentarily. Remark. Convection heat transfer comprises two primary mechanisms, one due to primarily random molecular motion (diffusion) and the other due to bulk motion of a fluid, in our case a gas, surrounding the particles. As we have indicated, in the applications of interest, the gas is dilute and the Reynolds number is small, so convection plays a very small role in the heat transfer process for dry particulate flows in the presence of a dilute gas. The nondilute surrounding fluid case will be considered in Chapter 8. Also, we recall that a black-body is an ideal radiating surface with the following properties: • A black-body absorbs all incident radiation, regardless of wavelength and direction. • For a prescribed temperature and wavelength, no surface can emit more energy than a black-body. • Although the radiation emitted by a black-body is a function of wavelength and temperature, it is independent of direction. Since a black-body is a perfect emitter, it serves as a standard against which the radia- tive properties of actual surfaces may be compared. The Stefan–Boltzmann law, which is computed by integrating the Planck representation of the emissive power distribution of a black-body over all wavelengths, 35 allows the calculation of the amount of radiation emitted in all directions and over all wavelengths simply from the knowledge of the temperature of the black-body. We note that Equation (7.30) is of the form θ(t + t) = G(θ (t + t)) + R, (7.31) where R = R(θ(t +t)) and G’s behavior is controlled by tBa s  mC + h c a s t , (7.32) which is quite small. Thus, a fixed-point iterative scheme such as θ K (t +t) = G(θ K−1 (t +t)) +R (7.33) would converge rapidly. 34 For this stage, since δt  t, we assign θ(t) = θ(t + δt) = θ(t) + δH mC and replace θ(t) with it in Equation (7.30). 35 Radiation is idealized as requiring no medium to transmit energy. 05 book 2007/5/15 page 63 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 7.5. Staggering schemes 63 7.5 Staggering schemes Broadly speaking, staggering schemes proceed by solving each field equation individually, allowing only the primary field variable to be active. After the solution of each field equation, the primary field variable is updated, and the next field equation is addressed in a similar manner. Such approaches have a long history in the computational mechanics community. For example, see Park and Felippa [161], Zienkiewicz [206], Schrefler [173], Lewis et al. [133], Doltsinis [52], [53], Piperno [162], Lewis and Schrefler [132], Armero and Simo [7]–[9], Armero [10], Le Tallec and Mouro [131], Zohdi [208], [209], and the extensive works of Farhat and coworkers (Piperno et al. [163], Farhat et al. [65], Lesoinne and Farhat [130], Farhat and Lesoinne [66], Piperno and Farhat [163], andFarhat et al.[67]). Generally speaking, if a recursive staggering process is not employed (an explicit scheme), the staggering error can accumulate rapidly. However, an overkill approach involving very small time steps, smaller than needed to control the discretization error, simply to suppress a nonrecursive staggering process error, is computationally inefficient. Therefore, the objective ofthenextsectionistodevelopa strategy to adaptively adjust, in fact maximize, the choice of the time step size to control the staggering error, while simultaneously staying below the critical time step size needed to control the discretization error. An important related issue is to simultaneously minimize the computational effort involved. The number of times the multifield system is solved, as opposed to time steps, is taken as the measure of computational effort, since within a time step, many multifield system re-solves can take place. We now develop a staggering scheme by following an approach found in Zohdi [208]–[210]. 7.5.1 A general iterative framework We consider an abstract setting, whereby one solves for the particle positions, assuming the thermal fields fixed, A 1 (r L+1,K ,θ L+1,K−1 ) = F 1 (r L+1,K−1 ,θ L+1,K−1 ), (7.34) and then one solves for the thermal fields, assuming the particle positions fixed, A 2 (r L+1,K ,θ L+1,K ) = F 2 (r L+1,K ,θ L+1,K−1 ), (7.35) where only the underlined variable is “active,” L indicates the time step, and K indicates the iteration counter. Within the staggering scheme, implicit time-stepping methods, with time step size adaptivity, will be used throughout the upcoming analysis. Continuing where Equation (3.28) left off, we define the normalized errors within each time step, for the two fields, as  rK def = ||r L+1,K − r L+1,K−1 || ||r L+1,K − r L || and  θK def = ||θ L+1,K − θ L+1,K−1 || ||θ L+1,K − θ L || . (7.36) We define the maximum “violation ratio,” i.e., the larger of the ratios of each field variable’s error to its corresponding tolerance, by Z K def = max(z rK ,z θK ), where z rK def =  rK TOL r and z θK def =  θK TOL θ , (7.37) 05 book 2007/5/15 page 64 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 64 Chapter 7. Advanced particulate flow models with the minimum scaling factor defined as  K def = min(φ rK ,φ θK ), where φ rK def =    TOL r  r0  1 pK d   rK  r0  1 pK   ,φ θK def =    TOL θ  θ0  1 pK d   θK  θ0  1 pK   . (7.38) SeeAlgorithm 7.1. The overall goal is to deliver solutions where the staggering (incomplete coupling) error is controlled and the temporal discretization accuracy dictates the upper limits on the time step size (t lim ). Remark. As in the single-field multiple-particle discussion earlier, an alternative approach is to attempt to solve the entire multifield system simultaneously (monolithically). This would involve the use of a Newton-type scheme, which can also be considered as a type of fixed-point iteration. Newton’s method is covered as a special case of this general analysis. To see this, let w = (r,θ), (7.39) and consider the residual defined by  def = A(w) − F . (7.40) Linearization leads to (w K ) = (w K−1 ) +∇ w | w K−1 (w K − w K−1 ) + O(||w|| 2 ), (7.41) and thus the Newton updating scheme can be developed by enforcing (w K ) ≈ 0, (7.42) leading to w K = w K−1 − (A TAN ,K−1 ) −1 (w K−1 ), (7.43) where A TAN ,K = ( ∇ w A(w) ) | w K = ( ∇ w (w) ) | w K (7.44) is the tangent. Therefore, in the fixed-point form, one has the operator G(w) = w − (A TAN ) −1 (w). (7.45) One immediately sees a fundamental difficulty due to the possibility of a zero or near-zero tangent when employing a Newton’s method on a nonconvex system, whichcan lose positive definiteness and which in turn will lead to an indefinite system of algebraic equations. 36 Therefore, while Newton’s method usually converges at a faster rate than a direct fixed- point iteration, quadratically as opposed to superlinearly, its convergence criteria are less robust than the presented fixed-point algorithm, due to its dependence on the gradients of the solution. Furthermore, for the problems considered, the solutions are nonsmooth and nonconvex, primarily due to the impact events, and thus we opted for the more robust “gradientless” staggering scheme. 36 Furthermore, the tangent may not exist in some (nonsmooth) cases. 05 book 2007/5/15 page 65 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 7.5. Staggering schemes 65 (1) GLOBAL FIXED-POINT ITERATION (SET i = 1 AND K = 0): (2) IF i>N p , THEN GO TO (4); (3) IF i ≤ N p , THEN (FOR PARTICLE i) (a) COMPUTE POSITION: r L+1,K i ≈ t 2 m   tot (r L+1,K−1 )  + r L i + t ˙ r L i ; (b) COMPUTE TEMPERATURE (FOR PARTICLE i): θ L+1,K i = θ L i + δH L+1,K−1 mC ; θ L+1,K i = mC mC + h c a s t θ L+1,K−1 i − tBa s  mC + h c a s t  (θ L+1,K−1 i ) 4 − θ 4 s  + h c a s tθ o mC + h c a s t ; (c) GO TO (2) AND NEXT PARTICLE (i = i + 1); (4) ERROR MEASURES (normalized): (a)  rK def =  N p i=1 ||r L+1,K i − r L+1,K−1 i ||  N p i=1 ||r L+1,K i − r L i || , θK def =  N p i=1 ||θ L+1,K i − θ L+1,K−1 i ||  N p i=1 ||θ L+1,K i − θ L i || ; (b) Z K def = max(z rK ,z θK ) where z rK def =  rK TOL r ,z θK def =  θK TOL θ ; (c)  K def = min(φ rK ,φ θK ) where φ rK def =     TOL r  r0  1 pK d   rK  r0  1 pK    , φ θK def =     TOL θ  θ0  1 pK d   θK  θ0  1 pK    ; (5) IF TOLERANCE MET (Z K ≤ 1) AND K<K d , THEN (a) INCREMENT TIME: t = t + t; (b) CONSTRUCT NEW TIME STEP: t =  K t; (c) SELECT MINIMUM: t = min(t lim ,t)AND GO TO (1); (6) IF TOLERANCE NOT MET (Z K > 1) AND K = K d , THEN: (a) CONSTRUCT NEW TIME STEP: t =  K t; (b) RESTART AT TIME = t AND GO TO (1). Algorithm 7.1 05 book 2007/5/15 page 66 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 66 Chapter 7. Advanced particulate flow models 7.5.2 Semi-analytical examples For the class of coupled systems considered in this work, the coupled operator’s spectral radius is directly dependent on the timestepdiscretizationt. Weconsiderasimpleexample that illustrates the essential concepts. Consider the coupling of two first-order equations and one second-order equation a ˙w 1 + w 2 = 0, b ˙w 2 + w 3 = 0, c ¨w 3 + w 1 = 0. (7.46) When this is discretized in time, for example, with a backward Euler scheme, we obtain ˙w 1 L+1 = w L+1 1 − w L 1 t , ˙w 2 L+1 = w L+1 2 − w L 2 t , ¨w 3 L+1 = w L+1 3 − 2w L 3 + w L−1 3 (t) 2 , (7.47) which leads to the following coupled system:    1 t a 0 01 t b (t) 2 c 01         w L+1 1 w L+1 2 w L+1 3      =      w L 1 w L 2 2w L 3 − w L−1 3      . (7.48) For a recursive staggering scheme of Jacobi type, where the updates are made only after one complete iteration, considered here only for algebraic simplicity, we have 37    100 010 001         w L+1,K 1 w L+1,K 2 w L+1,K 3      =      w L 1 w L 2 2w L 3 − w L−1 3      −        t a w L+1,K−1 1 t b w L+1,K−1 2 (t) 2 c w L+1,K−1 3        . (7.49) Rewriting this in terms of the standard fixed-point form, G(w L+1,K−1 ) + R = w L+1,K , yields    0 t a 0 00 t b (t) 2 c 00       G      w L+1,K−1 1 w L+1,K−1 2 w L+1,K−1 3         w L+1,K−1 +      w L 1 w L 2 2w L 3 − w L−1 3         R =      w L+1,K 1 w L+1,K 2 w L+1,K 3         w L+1,K . (7.50) 37 A Gauss–Seidel approach would involve using the most current iterate. Typically, under very general con- ditions, if the Jacobi method converges, the Gauss–Seidel method converges at a faster rate, while if the Jacobi method diverges, the Gauss–Seidel method diverges at a faster rate. For example, see Ames [5] for details. The Jacobi method is easier to address theoretically, so it is used for proof of convergence, and the Gauss–Seidel method is used at the implementation level. 05 book 2007/5/15 page 67 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 7.5. Staggering schemes 67 The eigenvalues of G are given by λ 3 = (t) 4 abc and, hence, for convergence we must have |max λ|=     (t) 4 abc     1 3 < 1. (7.51) We see that the spectral radius of the staggering operator grows quasi-linearly with the time step size, specifically superlinearly as (t) 4 3 . Following Zohdi [208], a somewhat less algebraically complicated example illustrates a further characteristic of such solution processes. Consider the following example of reduced dimensionality, namely, a coupled first-order system a ˙w 1 + w 2 = 0, b ˙w 2 + w 1 = 0. (7.52) When discretized in time with a backward Euler scheme and repeating the preceding pro- cedure, we obtain the G-form  0 t a t b 0     G  w L+1,K−1 1 w L+1,K−1 2     w L+1,K−1 +  w L 1 w L 2     R =  w L+1,K 1 w L+1,K 2     w L+1,K . (7.53) The eigenvalues of G are λ 1,2 =±  (t) 2 ab . (7.54) We see that the convergence of the staggering scheme is directly related (linearly in this case) to the size of the time step. The solution to the example is w L+1 1 = abw L 1 + btw L 2 ab −(t) 2 = w L 1 − w L 2 a t    first staggered iteration + w L 1 ab (t) 2    second staggered iteration +··· (7.55) and w L+1 2 = abw L 2 + atw L 1 ab −(t) 2 = w L 2 − w L 1 b t    first staggered iteration + w L 2 ab (t) 2    second staggered iteration +···. (7.56) As pointed out in Zohdi [208], the time step induced restriction for convergence matches the radius of analyticity of a Taylor series expansion of the solution around time t, which converges in a ball of radius from the point of expansion to the nearest singularity in Equations (7.55) and (7.56). In other words, the limiting step size is given by setting the denominator to zero, ab −(t) 2 = 0, (7.57) which is in agreement with the condition derived from the analysis of the eigenvalues of G. [...]... 0. 75 TOTAL KINETIC ENERGY TOTAL KINETIC ENERGY 0.7 0. 65 0. 65 ENERGY (N-m) 0.7 ENERGY (N-m) 05 book 2007 /5/ 15 page 71 ✐ 0.6 0 .55 0.6 0 .55 0 .5 0 .5 0. 45 0. 45 0.4 0.4 0 0 .5 1 1 .5 2 2 .5 TIME 3 3 .5 4 4 .5 5 0 0 .5 1 1 .5 2 2 .5 3 3 .5 4 4 .5 5 TIME Figure 7.7 Top to bottom and left to right, with clustering forces: the total kinetic energy in the system per unit mass with eo = 0 .5, µs = 0.2, µd = 0.1, α 1 = 0 .5, ... 3 35 316 330 314 TEMPERATURE TEMPERATURE 312 310 308 306 304 3 25 320 3 15 310 302 3 05 300 298 300 0 0 .5 1 1 .5 2 2 .5 3 3 .5 4 4 .5 5 0 0 .5 1 1 .5 2 TIME 2 .5 3 3 .5 4 4 .5 5 TIME 1000 1800 TEMPERATURE TEMPERATURE 1600 900 1400 TEMPERATURE TEMPERATURE 800 700 600 1200 1000 800 50 0 600 400 400 300 200 0 0 .5 1 1 .5 2 2 .5 3 3 .5 4 4 .5 5 0 0 .5 1 1 .5 TIME 2 2 .5 3 3 .5 4 4 .5 5 TIME Figure 7.9 Top to bottom and left to. .. 0. 65 0.64 0. 65 0.64 0.63 0.63 0.62 0.62 0.61 0.61 0.6 0.6 0 0 .5 1 1 .5 2 2 .5 3 3 .5 4 4 .5 0 5 0 .5 1 1 .5 2 TIME 2 .5 3 3 .5 4 4 .5 5 TIME 0.7 0.7 TOTAL KINETIC ENERGY TOTAL KINETIC ENERGY 0.69 0.68 0.68 0.67 0.67 0.66 0.66 ENERGY (N-m) 0.69 ENERGY (N-m) 05 book 2007 /5/ 15 page 72 ✐ 0. 65 0.64 0. 65 0.64 0.63 0.63 0.62 0.62 0.61 0.61 0.6 0.6 0 0 .5 1 1 .5 2 2 .5 3 3 .5 4 4 .5 5 0 0 .5 1 1 .5 TIME 2 2 .5 3 3 .5 4 4 .5 5... is to allow the system to adjust to the physics of the problem Some further remarks elaborating on this issue can be found in Zohdi [208]–[210] ✐ ✐ ✐ ✐ ✐ ✐ ✐ 7 .5 Staggering schemes 71 0. 75 0. 75 TOTAL KINETIC ENERGY TOTAL KINETIC ENERGY 0. 65 ENERGY (N-m) 0.7 0. 65 ENERGY (N-m) 0.7 0.6 0 .55 0.6 0 .55 0 .5 0 .5 0. 45 0. 45 0.4 0.4 0 0 .5 1 1 .5 2 2 .5 3 3 .5 4 4 .5 0 5 0 .5 1 1 .5 2 TIME 2 .5 3 3 .5 4 4 .5 5 TIME 0. 75. .. 300 300 0 0 .5 1 1 .5 2 2 .5 3 3 .5 4 4 .5 5 0 0 .5 1 1 .5 2 TIME 2 .5 3 3 .5 4 4 .5 5 TIME Figure 7.10 Top to bottom and left to right, without clustering forces: the average particle temperature with eo = 0 .5, µs = 0.2, µd = 0.1, α 1 = 0 .5, and α 2 = 0. 25: (1) κ = 106 J/m2 , (2) κ = 2 × 106 J/m2 , (3) κ = 4 × 106 J/m2 , and (4) κ = 8 × 106 J/m2 (Zohdi [217]) Table 7.1 The number of time steps and fixed-point... Figures 7.14 and 7. 15 ✐ ✐ ✐ ✐ ✐ ✐ ✐ 74 Chapter 7 Advanced particulate flow models 307 312 TEMPERATURE TEMPERATURE 310 3 05 308 TEMPERATURE 306 TEMPERATURE 05 book 2007 /5/ 15 page 74 ✐ 304 303 306 304 302 302 301 300 300 298 0 0 .5 1 1 .5 2 2 .5 3 3 .5 4 4 .5 0 5 0 .5 1 1 .5 2 TIME 2 .5 3 3 .5 4 4 .5 5 TIME 330 400 TEMPERATURE TEMPERATURE 390 3 25 380 370 TEMPERATURE TEMPERATURE 320 3 15 360 350 340 310 330 320 3 05 310 300... the 5- s simulation, if the time steps stayed at the starting value 39 The computation time scales were, approximately, no worse than the number of particles squared For example, a thousand particles took approximately 10 min ✐ ✐ ✐ ✐ ✐ ✐ ✐ 70 05 book 2007 /5/ 15 page 70 ✐ Chapter 7 Advanced particulate flow models Z X Z Y X Z X Y Z Y X Y Figure 7.6 Top to bottom and left to right, the dynamics of the particulate. .. threshold velocity of v ∗ = 10 m/s in Equation (7.16) The simulation duration was set to 5 s, with an upper bound on the time step size of t lim = 10−2 s and a starting time step size of 10−3 s The tolerances of both fields (TOLr and TOLθ ) for the fixed-point iterations were set to 10−6 and the upper limit on the number of fixed-point iterations was set to K d = 102 Two main types of computational tests... and the total mass could be written as M = i=1 mi = ρν, while that of ρν an individual particle, assuming that all are the same size, was mi = Np = ρ 4 π bi3 In order 3 to visualize the flow clearly, we used Np = 100 particles The length scale of the particles 3 was L = 0. 25, which resulted in a corresponding volume fraction of vf = 4π3L = 0.0 655 and particulate radii of b = 0. 053 9 A mass density of. .. = (1, 2) ✐ ✐ ✐ ✐ ✐ ✐ ✐ 7 .5 Staggering schemes 05 book 2007 /5/ 15 page 69 ✐ 69 Z X Z Y X Z X Y Z Y X Y Figure 7 .5 Top to bottom and left to right, the dynamics of the particulate flow with clustering forces: An initially fine cloud of particles that clusters to form structures within the flow Blue indicates a temperature of approximately 300◦ K, while red indicates a temperature of approximately 400◦ K (Zohdi . 71 0.4 0. 45 0 .5 0 .55 0.6 0. 65 0.7 0. 75 0 0 .5 1 1 .5 2 2 .5 3 3 .5 4 4 .5 5 ENERGY (N-m) TIME TOTAL KINETIC ENERGY 0.4 0. 45 0 .5 0 .55 0.6 0. 65 0.7 0. 75 0 0 .5 1 1 .5 2 2 .5 3 3 .5 4 4 .5 5 ENERGY (N-m) TIME TOTAL. ENERGY 0.4 0. 45 0 .5 0 .55 0.6 0. 65 0.7 0. 75 0 0 .5 1 1 .5 2 2 .5 3 3 .5 4 4 .5 5 ENERGY (N-m) TIME TOTAL KINETIC ENERGY 0.4 0. 45 0 .5 0 .55 0.6 0. 65 0.7 0. 75 0 0 .5 1 1 .5 2 2 .5 3 3 .5 4 4 .5 5 ENERGY (N-m) TIME TOTAL. 73 298 300 302 304 306 308 310 312 314 316 318 320 0 0 .5 1 1 .5 2 2 .5 3 3 .5 4 4 .5 5 TEMPERATURE TIME TEMPERATURE 300 3 05 310 3 15 320 3 25 330 3 35 340 0 0 .5 1 1 .5 2 2 .5 3 3 .5 4 4 .5 5 TEMPERATURE TIME TEMPERATURE 300 400 50 0 600 700 800 900 1000 0