Proceedings of FEDSM’99 3rd ASME/JSME Joint Fluids Engineering Conference July 18-23, 1999, San-Francisco, California, USA
FEDSM99-7898
DEVELOPMENT AND VALIDATION OF CONTINUUM PARTICLE WALL BOUNDARY CONDITIONS USING LAGRANGIAN SIMULATION OF A VERTICAL GAS/SOLID CHANNEL FLOW Olivier Simonin1;2; de M e´ canique des Fluides de Toulouse Institut National Polytechnique de Toulouse 31400 Toulouse France Email:
[email protected]
Marc Sakiz1 National d’Hydraulique ´ Electricit e´ de France 78400 Chatou France Email:
[email protected]
2 Institut
1 Laboratoire
f,
ABSTRACT This paper concerns numerical simulations of particles suspended in a turbulent gas vertical channel flow, focusing on the near-wall region and the particle/wall interactions. A theoretical approach of the derivation of wall boundary conditions for inelastic frictionnal particle bouncing is developed, in the frame of the kinetic theory of granular medium. The rebound law provides a direct relation between the separate velocities of incident and reflected particles, from which practical boundary conditions for the first order moments can be derived. The theoretical approach is first presented in this paper, leading to the the boundary conditions. Then, discrete particle simulation results are presented, used to test basic relations at the wall and compared to continuum model predictions. Eventually, probability density functions derived from the theory using certain assumptions are compared to those calculated during the simulations.
velocity distribution of incident particles at the wall (m,6 :s3 ) n2 particle number density at the wall (m,3 ) n, incident particle number density at the wall (m,3 ) 2 , T isotropic agitation of incident particles, with respect to 2 ,2 ) < u2 > (m :s u2 instantaneous particle velocity vector (m:s,1) u, v, w streamwise, wall-normal and spanwise velocity components (m:s,1 ) U streamwise mean velocity (m:s,1 ) U , streamwise mean velocity of incident particles (m:s,1 ) u00 , v00 , w00 fluctuations of velocity components, with respect to ,1) < u2 > (m:s α2 particle volumetric fraction (dimensionless) µw friction coefficient for wall-particle interaction (dimensionless) ν1 fluid kinematic molecular viscosity (m2 :s,1 ) ρ1 fluid density (kg:m,3 ) ρ2 particle density (kg:m,3 ) global average , average on incident particles
NOMENCLATURE d p particle diameter (m) ew normal restitution coefficient for wall-particle interaction (dimensionless) f particle velocity distribution function at the wall (m,6 :s3 )
INTRODUCTION Gas-particle turbulent flows are found in a very wide range of applications in power and process industry. In case of a channel flow, the presence of the wall can noticeably influence the
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1
Copyright 1999 by ASME
behaviour of the particles on large distances, especially if the particles have a strong inertia. Two different approaches are used to predict such flows. In the Euler/Lagrange approach, numerous discrete particles are tracked through the fluid flow with various methods to account for particle-particle and fluid-particle interactions (see for example Sommerfeld, 1990). In the Euler/Euler approach (or continuum approach), the two phases are described by few first moments of the instantaneous properties, computed from modelled transport equations. Such modelling approaches have been developed, in parallel, by several authors (Andresen, 1990; Simonin, 1991; Zaichik and Vinberg, 1991), based on the calculation of separate transport equations for the first three moments of the particle velocity distribution function (number density, mean velocity, particle kinetic stress), and a Boussinesq approximation for the third order moments (triple particle velocity correlations). This method allows to account simultaneously for the main mechanisms occurring in the flow (fluid-particle, particle-particle and wall-particle interactions). Complex wall-particle interaction models are easily taken into account in the Euler/Lagrange approach, but such an approach is difficult to handle in dense industrial flows, mainly due to computer limitations. In the Euler/Euler approach, most generally used for such industrial cases, wall-particle interaction mechanism appears through wall boundary conditions which have to be developed. The theoretical approach is first presented in this paper, leading to the basic relations mentioned earlier and to the boundary conditions. Then, discrete particle simulations and their main results are presented, and compared to continuum model predictions. The basic relations at the wall are tested using the simulation results. Eventually, probability density functions derived from the theory using certain assumptions are compared to those calculated during the simulations.
40 mm
A
mean flow
240 mm
B
g
n
u
u v
m m
w
32
x z
Figure 1.
y
FLOW CONFIGURATION
statistics. The physical characteristics of the fluid are: ρ1 = 1:205 kg:m,3 ν1 = 1:515 10,5 m2 :s,1 As for the dispersed phase, two kinds of particles are studied: d p = 1:5 mm and ρ2 = 1032 kg:m,3 , or d p = 0:406 mm and ρ2 = 1038 kg:m,3 , for different volumetric fractions varying from 10,3 to 4 10,2 . The rotation of the particles is neglected in the results presented here. The streamwise, wall-normal and spanwise directions are respectively referenced x, y and z. n is the unit vector normal to the wall and oriented towards the inside on the channel (cf. fig. 1). Lagrangian Simulation The main Lagrangian algorithm for particle tracking is divided in two successive steps: * The first step takes the effects of the fluid and gravity into account. The particles are moved and their velocity is changed through a second-order Runge-Kutta algorithm. Particles that leave the calculation domain are repositioned using the periodicity and rebound conditions. The only fluid-particle interaction considered here is that of the drag, although it is true that other contributions, such as the lift, may also be of some importance and will be evaluated in further studies. * The second step takes the inter-particle collisions into account. The algorithm used is derived from that of Hopkins and Louge (1991). The domain is scanned for overlapping particles. Each pair of overlapping particles is treated: their relative velocities and their relative positions at the time of the impact are calculated; the velocities are then updated, but their position stays untouched. The time step must be small enough in order to limit
GENERAL DESCRIPTION Flow Configuration The proposed test case is a gas-solid vertical fully developed channel flow, very close to the vertical pipe flow studied by Tanaka and Tsuji (1991) and He and Simonin (1993). The computational domain for the Lagrangian simulations is a rectangular box, 240 mm long and 40 mm wide, with periodic boundary conditions in the streamwise and spanwise directions. According to the previous pipe flow studies, the given low solid concentrations and the large particle inertia allow to neglect all interaction between the fluid turbulence and the particles as well as the influence of the particles on the mean fluid flow (no two-way coupling). So, a preliminary single-phase k , ε computation was performed in order to determine the mean fluid velocity profile used for both Lagrangian and Eulerian predictions of the particle 2
Copyright 1999 by ASME
the number of collisions that are missed by this algorithm. The time step used in our simulations is about 10,4 s. This algorithm has been validated and used in gas-solid turbulent flows, for example, by Lavi´eville et al. (1997). Concerning the computation of averaged properties, we use the homogeneity of streamwise and spanwise directions. Therefore, mean variables depend only on the wall-normal coordinate y. The channel is divided in about 30 slices, parallel to the walls and thinner near the walls. At each time step, the properties of each particle are associated to the slices their centers have crossed, ponderated by the time they have spent in them. A statistical routine, called just before and just after the collision step, allows us to calculate, by subtraction, the collisional rate of change, directly. The stationary state is usually reached after about 10 times the fluid/particle interaction characteristic time τF12 , and 100 times the collisional characteristic time τc2 . Afterwards, statistics are calculated over 10000 to 100000 time steps, depending on the number of particles in the channel, which represents about 3 106 particles taken into account for the thinner slices.
For the kinetic stress tensor components (velocity second-order moments):
∂ ∂t
2 j 2i ,R2 im ∂U , R2 jm ∂U ∂x ∂x ;
;
;
;
m
m
WALL BOUNDARY CONDITION DERIVATION Kinetic Theory Formalism Let us consider the velocity distribution at the wall, f . Because of the assumptions of stationarity and homogeneity in the streamwise and spanwise directions, f is only a function of the velocity u2 . f (u2 )du2 is the average number of particles with a velocity u2 in the range du2 at the wall. Considering any function ψ of the velocity, the mean value of ψ writes :
=
1 n2
Z
ψ(u2 ) f (u2 )du2
Z with n2 =
f (u2 )du2 (4)
In order to focus on the effect of the rebound on the wall, we will distinguish f + and f ,, restrictions of f respectively on D+ = fu2 = u2 :n > 0g and D, = fu2 = u2 :n < 0g (Grad, 1949; Cercignani, 1975). The average on incident particles is defined by:
(1)
∂ 1 ∂ Fi + U2; j U2;i = , α2 R2;i j + gi + < > ∂x j α2 ∂x j m +C l (u2;i )
=
Fi is the drag force, and Cl the collisional operator. The different models used for drag, collisional and dispersion terms are given in (Sakiz and Simonin, 1998). As for the boundary conditions, the method we are about to describe yields Dirichlet conditions at the wall for the velocity moments that have an odd dependance on the wall normal velocity. But these conditions can also be seen as flux conditions for the moments with an even dependance on v. In our secondorder system, each of the seven resolved variables α2 , U, V , 00 00 00 00 00 00 00 00 < u u >, < v v >, < w w > and < u v >, will be computed using a Dirichlet conditions, respectively on α2V , < u00 v00 >, V , 00 00 00 00 00 00 00 00 00 00 00 < u u v >, < v v v >, < v w w >, and < u v >. Due to the particle size, these boundary conditions are applied at a distance 1 2 d p from the wall. In the rest of the text, we will focus on boundary conditions derivation for wall A, for which incident particles have a negative wall-normal velocity, with respect to the reference frame (see fig. 1).
For mean velocity components:
∂ + U2; j R2;i j ∂x j
Fj Fi 00 u > + < u002;i > +< m 2; j m 1 ∂ , α ∂x α2S2;i jm + Cl (u002;i u002; j) (3) 2 m
Continuum Model The mean balance equations describing the dispersed phase may be derived by applying kinetic theory formalism to the particle ensemble; see for instance Morioka and Nakajima (1987) or Simonin (1996). A probability density function (pdf) is introduced, which obeys a Boltzmann type equation, that accounts for exchange with the fluid, influence of external fields (gravity), and particle-particle collisions. Transport equations for the velocity moments may be derived by averaging from the pdf equation. The averaging operator will be written < : >, the mean velocity of the particles in the i-th direction U2;i , the velocity fluctuations u002;i = u2;i , U2;i . More generally, the subscript 1 refers to the continuous phase and 2 to the dispersed phase. The following moments will also be considered: R2;i j =< u002;i u002; j > and S2;i jm =< u002;i u002; j u002;m >. Eventually, for simplification purposes, the particle velocity components (u2;x ; u2;y; u2;z) will also be noted (u; v; w), the mean velocity components (U2;x ; U2;y; U2;z) will be noted (U ; V; W ), and the fluctuation components (u002;x ; u002;y; u002;z) will be noted (u00 ; v00 ; w00 ). So, the equations of the first order moments of the particle pdf are derived under the following very general forms. For the volumetric fraction: ∂ ∂ α2 + α2U2;i = 0 ∂t ∂xi
∂ ∂t
, = , n2
Z
ψ(u2 ) f (u2 )du2 with n, 2 =
D,
Z D,
f (u2 )du2 (5)
Copyright 1999 by ASME
Similarly, an average on reflected particles, fined. They are linked by:
< : >
+ , can be de-
, + + n2 < ψ >= n, 2 < ψ > +n2 < ψ >
order to be consistent with our no-rotation assumption in the particle trajectory computation. We consider the impulse J applied to the particle by the wall and the sliding velocity vs = u2 , (u2 :n)n. 1 We also consider the vector t = jvs j vs , if vs 6= 0, t = 0 otherwise.
(6)
We also introduce the probability for an incident particle with a velocity u2 to be reflected with a velocity u˜2 in the range d u˜2 , R(u2 ; u˜2 )d u˜2 . Comparing the incident and reflected mass fluxes, and considering that they must be equal, yields a relation between f + and f , as (cf. Cercignani, 1975): f + (u˜2 )ju˜2 :njd u˜2 =
Z u2 2D,
R(u2 ; u˜2 )d u˜2 f , (u2 )ju2 :njdu2
8 > > m(u˜2 u2 ) = J > > < J = Jn n + Jt t
,
Jn = ,m(1 + ew )(u2 :n) (11) ,m(1 + β0)jvs j if jJt j µwjJn j (no sliding) Jt = ,µw Jn otherwise (sliding)
> > Jt = > > :
(7)
Lagrangian calculations are performed using the system (11). But for the Eulerian averaging, the system is simplified by noticing that the mean streamwise velocity ( 10 , 14 m=s) is much larger than the velocity fluctuations in any direction ( 0:3 m=s). Therefore, the probability of a no-sliding collision is extremely small and we shall consider that there are only sliding rebounds. Also, for the same reason, the system (11) can be linearised. It yields:
In our simulation, the reflected velocity of a bouncing particle depends on the incident velocity in a deterministic manner, by means of a rebound law u˜2 = Φ(u2 ). This leads to R(u2 ; u˜2 ) = δ(u˜2 , Φ(u2 ))
(8)
8 < u˜ = u + µw (1 + ew )w
If we make the further assumption that the bouncing is linear in the wall-normal direction, i.e. Φ(u2 ):n = ,ew u2 :n, then equation (7) leads to a direct relation between f + and f , : f +(u˜2 ) =
1 f , (Φ,1 (u˜2 )) ew JΦ
:
, +
1 ew
,
(12)
Using the system (12), the relation between f + and f , can be explicited (cf. eq. (9)), as well as the relation between < : > and < : >, (cf. eq. (10)). The following relations can then be established:
(9)
(JΦ is the jacobian of Φ). Given this result, the relation between the global averages , can be ob< : > and the averages on incident particles < : > tained: n2 < ψ(u2 ) >= n, 2
v˜ = ,ew v w˜ = w
1 + ew , n ew 2 , , < u >=< u > +µw < v > < v >= 0 00 00 00 00 , < u v >= ,µw ew < v v > 00 00 00 00 , < v v >= ew < v v > 00 00 00 00 00 00 , < u u v >= ,2µw ew < u v v > ,ew (1 + ew)µ2w < v00 v00 v00 00 00 00 00 00 00 , < u v v >= ew < u v v > 2 00 00 00 , +ew µw < v v v > 00 00 00 00 00 00 , < v v v >= ew (1 , ew ) < v v v > 00 00 00 < v w w >= 0 n2 =
(10)
Relation (9) is of great interest if we need to use velocity distribution models, as we shall see below. Indeed, rather than modelling f directly, it allows us to make assumptions only on f , and calculate f + . By doing this, we ensure that our velocity distribution models fully take into account the presence of the wall.
(13)
,
>
The system (13) is almost closed, but we need to evaluate u00 v00 v00 >, and < v00 v00 v00 >, . The former will be computed using the relation on < u00 v00 v00 > in system (13), and a Boussinesq approximation for < u00 v00 v00 >. The latter will be evaluated by making an assumption on f ,.
Rebound Laws and Basic Relations The rebound laws used in our simulations correspond to a classical frictional flat wall (see for instance Jenkins, 1992), except that the inertia of the particles is supposed to be infinite, in
= ,2µw < u00 v00 v00 > 2 4 1 , ew +µ w p pe < v00 v00 2π w 4 1 , ew 00 00 00 00 00 = , p p 2π ew
= 0
=
,µw
3=2
>
>
BOUNDARY CONDITIONS AT THE WALL
We consider f y, (v) =
Z
Error (%)
< u00 v00 >= ,µwew < v00 v00 >, < v00 v00 >= ew < v00v00 >, < u00 u00 v00 >= ,2µwew < u00 v00 v00 >, ,ew (1 + ew)µ2w < v00 v00 v00 >, < u00 v00v00 >= ew < u00 v00 v00 >, +e2w µw < v00 v00 v00 >, < v00 v00v00 >= ew(1 , ew) < v00 v00 v00 >, < v00 v00v00 >, = , p42π < v00 v00 >, 3=2 < v00 v00v00 >= , p42π 1p,ee < v00v00 >3=2
=
f , (u2 )dudw. f y, has the following
w w
A
B
C
0.00
0.00
0.00
0.10
1.71
0.07
0.00
0.00
0.00
0.03
0.45
0.33
0.03
0.34
0.25
0.00
0.00
0.00
11.63
0.53
0.83
11.63
0.53
0.83
Table 2. A PRIORI TESTS
properties:
Z
0
,∞
Z
0
,∞
before the rebound, ψ+ i just after. Reasoning on a fictious averaging cell, of a certain width tending towards zero, we can establish the following practical formulas:
f y, (v)dv = n, 2
00 00 v2 f y, (v)dv = n, 2
Supposing that f y, is Gaussian yields: f y,(v) =
p
2n, 2
2π < v00 v00 >,
,2
exp
(v00 )2
,
(14)
2π
00 00 , 3
2
=
ψ >, =
1 1 , 1 ψi with N , = ∑ , , N , ∑ i jvi j i jvi j 1 1 , 1 + , + , j ψi + jv+ j ψi with N = N + N (16) N∑ j v i i i 2 3
=
1 1 1 1 5 f (u0 )∆u0 = 4 ∑ + ∑ , n2 N jvi j iju+i 2∆u0 jv+i j iju, i 2∆u0
And consequently: 4 00 00 00 , < v v v > = ,p
and < : >, ) at the wall. To avoid any perturbation of the results due to the fluid-particle interaction, we have to reason on the particles as they hit the wall to rebound, rather than all the particles in the first statistical cell. Let us consider the ith particle that hits the wall during the averaging process. For any property ψ, ψ, i will be its value just 5
Copyright 1999 by ASME
0.8
15
0.6
−2
1.2
ew=1 µw=0 (Lagrangian) (Lagrangian) (Lagrangian) (Lagrangian) Eulerian results
ew=0.94 µw=0.325 (Lagrangian) (Lagrangian) (Lagrangian) (Lagrangian) Eulerian results
kinetic stress (m .s )
U2 Lagrangian U2 Eulerian
−1
α2 Lagrangian α2 Eulerian
U2 (m.s )
α2/α2
mean
1.3
2
14
0.4
1.1
0.2
13 1.0 0.0
0.9 0.00
0.01
0.02
0.03
−0.2 0.00
12 0.04
0.01
wall−normal coordinate (m) Figure 2. LAGRANGIAN AND EULERIAN RESULTS FOR PARTICLE CONCENTRATION AND MEAN STREAMWISE VELOCITY, FOR d p = 0:406 mm,
0.02 0.03 wall−normal coordinate (m)
0.04
Figure 3. LAGRANGIAN AND EULERIAN RESULTS FOR THE KINETIC STRESS TENSOR COMPONENTS, FOR d p = 0:406 mm AND α2 = 10,2
α2 = 10,2 , ew = 0:94 AND µw = 0:325
formed in the same configuration but with ew = 0:6 and µw = 0, show, as expected, a decrease in the agitation, with respect to the elastic case. As for the comparisons with the Eulerian results, they are in very good agreement for both cases. Other tests with more dilute configuration (α2 = 10,3) show a noticeable over-evaluation of the diagonal components of the kinetic stress tensor with the Eulerian code. But the error has been found to originate in the modelisation of the dispersion term, and not in the boundary conditions (cf. Sakiz and Simonin, 1998).
librium. Still, even in the dilute case A, the results are rather acceptable. A posteriori tests The continuum model presented earlier and the boundary conditions (table 1) have been implemented in a one-dimensional Eulerian code. Inter-particle collisions are purely elastic for the results presented here, but other tests with inelastic collisions have been performed and yield similar results. Figure 2 shows the particle concentration and mean streamwise velocity for d p = 0:406 mm, α2 = 10,2 , ew = 0:94 and µw = 0:325. Except for a slight underestimation of the velocity at the wall, the Eulerian computations are very close to the Lagrangian results. Kinetic stress components for the same case are presented on figure 3, as well as results obtained for elastic rebounds (ew = 1, µw = 0). The huge difference between the two cases show the strong sensitivity of the system to boundary conditions. In the inelastic case, agitation is stronger and the shear stress < u00 v00 > at the wall is not zero. Inelastic restitution and friction have in fact two opposite effects. Coefficient ew induces dissipation at the wall: < v00 v00 v00 > is always negative, and in our case < u00 u00 v00 > is also negative. But the friction effect (coefficient µw ) increases the shear stress absolute value, which induces a larger production of < u00 u00 > (eq. 3) and, via collisionnal redistribution, an increase in < v00 v00 > and < w00 w00 >. In our case, the frictionnal effect is stronger than the restitution effect. But other tests, per-
WALL-NORMAL VELOCITY DISTRIBUTIONS In the former paragraph, we used a particular assumption on f y, to derive the boundary conditions. In the same way we can define other pdf models, that can be compared to the exact pdf computed during the Lagrangian simulations. As said before, we will make our assumptions on f y, , and calculate f y+ using eq. (9). In this approach, the assumptions on f y, must logically be based on the incident average properties, and not the global average properties. Three models can be defined:
Maxwellian model: f y, (v) = where T , 6
1 = 3
Copyright 1999 by ASME
(with respect to < u2 >). It is the model that would be used instead of eq. (14) if we were using energy-viscosity model.
2.0
Elliptic Gaussian model: It is the one we used earlier for the boundary conditions. It is given by eq. (14).
,
Z,∞
0 > 00 00 > v2 f y, (v)dv = n, > 2 > , ∞ > Z > 0 > > 3 , , 00 00 :
,∞
v f y (v)dv = n2
> f y, (v)dv = n, > 2 > > , ∞ > >Z 0 > , > < v f y, (v)dv = n, 2
α2=10
1.5
Gaussian development model: Based on a Grad-like development, this comes to assuming: f y, (v) = αy + βy v + γy v2 + δy v3 f 0, (v)
simulation Maxwellian model Elliptic Gaussian model Gaussian development model
(19)
0.8
>
v v v00 >,
−1
0 −1 wall−normal velocity fluctuation (m.s )
simulation Maxwellian model Elliptic Gaussian model Gaussian development model
1
α2=10
−2
dp=0.406 mm
0.6
It must be noted that this last assumption cannot be used directly to derive the boundary conditions, because it requires the knowledge of < v >, and < v00 v00 v00 >, , which are not computed in our second-order Eulerian calculations. Still this form is useful to evaluate the degree of anisotropy of the system. Figure (4) shows the wall-normal velocity distribution profiles, calculated from the Lagrangian simulations and modelled, for the small particles, and for α2 = 10,3 and α2 = 10,2 . The less dilute case gives very satisfying results. The elliptic Gaussian model is very accurate, which explains for the very good results for case B in table 2. The more dilute case, shows larger differences between the models, because of the stronger anisotropy of the system. The Maxwellian form is far from the Lagrangian results, and even the elliptic Gaussian form, although undeniably better, is still not very accurate for the low velocities. Hence the less accurate results for case A in table 2. Figure (5) shows the same cases, but with elastic rebounds at the wall. The velocity distributions are symmetrical, but the same discrepancies arise in the more dilute case.
0.4
0.2
0.0
−1 0 1−1 wall−normal velocity fluctuation (m.s )
2
Figure 4. WALL-NORMAL VELOCITY DISTRIBUTION FOR ew = 0:94 AND µw = 0:325
channel flows, with dilute suspension of coarse particles. A theoretical approach, based on kinetic theory, has been presented in order to obtain basic relations between the average properties of the whole particulate phase and that of incident particles. A priori tests on these relations, using results from the Lagrangian simulations, yield excellent results. A further assumption has then been made on the wall-normal velocity distribution, in order to obtain the full boundary conditions, used for the Eulerian computations. A priori tests on the assumption give very close results, especially for the less dilute cases. Complete Eulerian computations also compare favourably
CONCLUSION Lagrangian simulation results and continuum model predictions have been presented for gas-solid fully developed turbulent 7
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2.0
simulation Maxwellian model Elliptic Gaussian model Gaussian development model
α2=10
Turbulent Gas-Particle Flows, PHD thesis, Department of Fluid Mechanics, Technical University of Denmark, Lyngby, Denmark, 1990. Cercignani, C., Theory and Application of the Boltzmann Equation, Elsevier, New-York, 1975. Grad, H., On the Kinetic Theory of Rarefied Gases, Communication on Pure and Applied Mathematics,Vol. 2, nb 4, pp.331407, 1949. He, J., Simonin, O., Non-Equilibrium Prediction of the Particle-Phase Stress Tensor in Vertical Pneumatic Conveying, 5th Int. Symp. on Gas-Solid Flows, ASME FED, Vol. 166, pp. 253-263, 1993. Hopkins, M.A., Louge, M.Y., Inelastic Macrostructure in Rapid Granular Flows for Smooth Disks, Phys. Fluids, Vol. 3, nb 1, pp. 47-57, 1991. Jenkins, T., Boundary Conditions for Rapid Granular Flows: Flat, Frictionnal Walls,Journal of Applied Mechanics, Vol. 59, pp. 120-135, 1992. Lavi´eville, J., Simonin, O., Berlemont, A., Chang, Z., Validation of Inter-Particle Collision Models Based on Large-Eddy Simulation in Gas-Solid Turbulent Homogeneous Shear Flow, Proc. 7th Int. Symp. on Gas-Particle Flows, ASME Fluids Engineering Division Summer Meeting, FEDSM97-3623, 1997. Morioka, S., Nakajima, T., Modelling of Gas and Solid Particles Two-Phase Flow and Application to Fluidized Bed, Journal de M´ecanique Th´eorique et Appliqu´ee/Journal of Theoretical and Applied Mechanics, Vol. 6, pp. 77-88, 1987. Sakiz M., Simonin O., Continuum Modelling and Lagrangian Simulation of the Turbulent Transport of Particle Kinetic Stresses in a Vertical Gas-Solid Channel Flow, 3rd International Conference On Multiphase Flows, Lyon, France, 1998. Simonin, O., Prediction of the Dispersed Phase Turbulence in Particle-Laden Jets, 4th Int. Symp. on Gas-Solid Flows, ASME FED, Vol. 121, pp. 197-206, 1991. Simonin, O., Continuum Modelling of Dispersed Two-Phase Flows, in Combustion and Turbulence in Two-Phase Flows, 1995-1996 Lectures Series Programme, von Karman Institute, Belgium, 1996. Sommerfeld, M., Numerical Simulation of the Particle Dispersion in Turbulent Flow including Particle LiftForces and Different Particle/Wall Collision Models, Numerical Methods for Multiphase Flows, ASME FED-Vol. 91, pp. 11-18, 1990. Tanaka, T., Tsuji, Y., Numerical Simulation of Gas-Solid Two-Phase Flow in a Vertical Pipe: On the Effect of InterParticle Collision, 4th Int. Symp. on Gas-Solid Flows, ASME FED, Vol. 121, pp. 123-128, 1991. Zaichik, L.I., Vinberg, A.A., Modelling of Particle Dynamics and Heat Transfer in Turbulent Flows Using Equations for First and Second Moments of Velocity and Temperature Fluctuations, Proc. 8th Int. Symp. on Turbulent Shear Flows, Munich, Germany, Vol. 1, pp. 1021-1026, 1991.
−3
dp=0.406 mm
1.5
1.0
0.5
0.0
−1.0
0.0 1.0 −1 wall−normal velocity fluctuation (m.s )
simulation Maxwellian model Elliptic Gaussian model Gaussian development model
α2=10
−2
dp=0.406 mm
1.0
0.5
0.0
−1.0
0.0 1.0 −1 wall−normal velocity fluctuation (m.s )
Figure 5. WALL-NORMAL VELOCITY DISTRIBUTION FOR ELASTIC REBOUNDS
with the Lagrangian simulation results. Eventually, wall-normal and streamwise velocity profiles taken from Lagrangian simulations have been presented and compared to different models. Isotropic models are not very accurate, but elliptic Gaussian models show a good approximation of the pdf, especially for the less dilute cases. At present, other studies are being carried out using rough walls and taking into account the particle rotation and lift effect.
REFERENCES Andresen, E., Statistical Approach to Continuum Models for 8
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