SIAM J. APPL. MATH.Vol. 52, No. 4, pp. 1076-1088, August 1992

1992 Society for Industrial and Applied Mathematics

012

STEADY-STATE COMPUTATIONS OF GRANULAR FLOW IN ANAXISYMMETRIC HOPPER*

FENG WANGt, CARL L. GARDNER:I:, AND DAVID G. SCHAEFFER

Abstract. Axisymmetric steady-state computations of plastic incompressible granular flow in a converg-ing hopper are presented. The granular flow equations are solved as a first-order system of conservationlaws plus a set of constitutive relations, using a conservative finite difference method. Newton’s method isused to solve the nonlinear partial differential equations (PDEs).

The steady-state granular flow equations may change type from elliptic to hyperbolic. In this investiga-tion, flow parameters and boundary conditions are chosen to ensure that the PDE system is elliptic. Thetransition from a "radial solution" similarity flow at the top of the hopper to a more general flow involvinga boundary layer near the exit of the hopper is analyzed.

Key words, granular flow, nonlinear partial differential equations (PDEs), conservation laws, finitedifference methods, Newton’s method

AMS(MOS) subject classifications. 76A10, 76M20

1. Introduction. Many challenging scientific problems are posed by the flow ofgranular materials in a hopper. Regardless of the geometry, the governing partialdifferential equations (PDEs) (in a continuum formulation) are vulnerable to changeof type [5], often leading to ill-posed initial value problems. In the silo/hoppergeometry, the flow is often complicated by the presence of free boundaries separatingregions undergoing plastic deformation from stagnant, or at least nondeforming,regions. Not only is it difficult to include such boundaries in a computation, but thegoverning equations become singular as the deformation rate tends to zero. Moreover,the modeler’s task is complicated by the bewildering variety of constitutive laws thathave been proposed in the literature.

As an initial investigation in this daunting environment, we make the followingsimplifying assumptions. (i) We use the conceptually simplest constitutive law dueessentially to Coulomb [3]. (ii) We choose the geometry to eliminate the risk ofequations that change type. (iii) We solve a problem on a fixed domain, disregardinga free boundary at the hopper exit (and the free-boundary condition). (iv) We solvesteady-state equations, neglecting the very important possibility of time-dependentflow. (v) We consider only axisymmetric flow, not fully three-dimensional flow. Evenwith all these simplifications, the unusual structure of the granular flow equationspresents challenging problems, and these are the subjects of this paper.

2. Granular flow equations. The plastic flow of an incompressible granular materialis described by conservation laws for mass and momentum, with a constitutive lawrelating stress and strain. The constitutive law (3), the simplest of many proposed,

* Received by the editors August 12, 1991; accepted for publication August 28, 1991.t Department of Applied Mathematics and Statistics, University at Stony Brook, Stony Brook, New

York 11794.$ Department of Computer Science, Duke University, Durham, North Carolina 27706. This author’s

research was supported in part by National Science Foundation grant DMS-8804592, which includes fundsfrom the Air Force Office of Scientific Research.

Department of Mathematics, Duke University, Durham, North Carolina 27706. This author’s researchwas supported in part by National Science Foundation grant DMS-8804592, which includes funds from theAir Force Office of Scientific Research.

1076

COMPUTATIONS OF GRANULAR FLOW IN A HOPPER 1077

assumes incompressibility and rigid perfectly plastic behavior. (See [5] for a mathemati-cal derivation.) The granular flow equations are

(1) V "v=O,

(3) To G60 + ko"

where v is the velocity, p is the density, T is the stress tensor (with compressive stresspositive), g=-g is the gravitational acceleration, cr Trace { T}/3 is the average ornormal stress, k (0, xf2] is a dimensionless material constant specifying the angleof internal friction, V is the strain rate or deformation rate tensor

l(Ovi(4) Vj - \Oxj+ -x/

and [VI is the Euclidean norm of V. Repeated spatial indices i, j 1, 2, 3 are summedover. Equation (1) expresses conservation of mass, (2) expresses conservation ofmomentum, and (3) is the granular analogue of the von Mises constitutive law forplastic deformation of metals. In (2) we have dropped the advective term pv. V vi,

since, for the slow flows in typical industrial hoppers, this term is much smaller thanthe other terms in (2).

We numerically solve for the steady-state solution of (1)-(3) in an axisymmetricconverging hopper in spherical coordinates

(5) {(r, O, qg)lRinner<--rNRouter, O<--O<--Ow, ONqg<27r},

where Rinne and Route are, respectively, the inner and outer computational boundariesof the hopper, and Ow is the angle between the axis and the conical wall of the hopper(see Fig. 1). The steady-state flow is specified by the geometrical parameters Router/Rinneand Ow, and the material parameters k and wall friction 6w (set equal to zero for thepresent study). We measure stresses in terms of/)gRinne Note that the velocity canbe scaled by an overall factor.

/outer

\/

Ow\\\11

FIG. 1. Schematic of an axisymmetric industrial hopper.

1078 F. WANG, C. L. GARDNER, AND D. G. SCHAEFFER

Imposing axial symmetry implies that

0(6) -0, v =0,

and that T has the form

To 0 t(7) T= Tro Too 0

0 0 T,,In steady state, (1)-(3) simplify to the following seven equations in the seven

unknowns Vr, VO, O’, T,.,., Too, T,,,, and Tro:

(8) f,, Or(r2 sin Ov)+10o (r sin Ovo O,

1(9) f=O(r sin OTrr)+-Oo(r2 sin OTo)-rsin O(Too+ T,)+(r sin O)pg cos 0=0,

1fo =0(r2 sin OTro)+-Oo(r sin OToo)+ r sin O(To- T, cot 0)

r(r2 sin 0)pg sin 0 0,

(11 fr [V[(T ) + kgOrv O,kg

(12) Too Vl(T00 )+ (OoVo + ) O,

kg(13) L =[Vl(T,-g)+(Vo cot 0+ v) 0,

(14) fr=lVlT+ OrVO+--(OOVr--DO)r =0.

Equation (8) is the mass conservation law, (9) and (10) are the momentum conservationlaws, and (11)-(14) are the constitutive relations. We multiplied (1) and (2) by theJacobian r sin 0 of the transformation from Caesian to spherical coordinates to putthe continuity and momentum equations in conservation form in spherical coordinates.

The condition =Trace{T}/3 is enforced by the sum of (11)-(13) plus thecontinuity equation (8).

For axisymmetric problems, the strain rate tensor V has the following componentsin spherical coordinates:

1 1Vr --OVr, VO0 (00VO + Vr), V VO cot 0 + V),

(5)

where Igl is given by

(16) vl= Vr+ Vo + V@ + 2 VoNote that, for axisymmetric flow, V does not contain any velocity derivatives.

For an infinite hopper, there is a similarity solution of the steady-state equations

u(o)(7) v=

rvo =0, . ro.(o),

with u(0)> 0 and Too =T. This "radial solution" was discovered by Jenike [4]. u(0)

(10)

COMPUTATIONS OF GRANULAR FLOW IN A HOPPER 1079

is the solution of an ordinary differential equation. We analyze the transition from aradial solution flow at the top of the hopper to a more general flow near the exit ofthe hopper.

The principal part of the granular flow equations (1)-(3) is sixth-order and canhave both elliptic and hyperbolic modes in steady state. For a sufficiently steep smoothhopper, the principal part of the steady-state granular flow equations is elliptic [5]. Inthis investigation we focus on elliptic flow in the hopper. Therefore, we must specifythree boundary conditions at the top and bottom computational boundaries and alongthe hopper wall. Since v 0, we must specify two additional boundary conditions.

At the axis of the hopper, we impose the symmetry conditions: Vo and Tro areodd functions of 0, while Vr, tr, Trr, Too, and T are even functions of 0.

At the top "inflow" computational boundary r Router, we specify vr and Vo fromthe radial solution. At the bottom "outflow" computational boundary r Rinner, wespecify that Vro 0 and vr v(0). Along the hopper wall, Vo--0. Assuming the wallis frictionless, we also specify that Tro 0 along the wall.

The current investigation may be viewed as a prelude to computing the free outflowsurface that marks the boundary between converging flow in the hopper and rigid"plug" flow with no deformation below the hopper. With a free outflow surface, wewould prescribe Vo in addition to v and Vro. Rigid plug flow below the hopper maybe achieved experimentally by connecting a cylindrical tube with a piston to the exitof the hopper and withdrawing the piston with a uniform velocity. The experimentsof [2] implemented this type of boundary condition in a different configuration.

We now show that if we specify vertical flow at the outflow boundary, then theproblem becomes hyperbolic near the outflow boundary. In this investigation, werestrict our attention to elliptic flows.

To prove that the problem becomes hyperbolic near the outflow boundary for

(18) D v cos 0, Vo v sin 0, v 0,we first combine (15) and (18) to conclude that V 0. Since the material is incompress-ible, contraction along one direction must be accompanied by expansion along anotherdirection, i.e., Trace { V} V v 0.

The strain rate tensor has the following form for axisymmetric problems:

vo o I(19) V= Vo Voo 0

0 0 VThe trace of the upper left 2 x 2 block is zero since Trace { V} 0 and V 0. SinceV 0, the two eigenvalues of the upper left 2 x 2 block must be equal in magnitudeand opposite in sign. The middle eigenvalue of V is zero. We have proved the followinglemma.

LEMMA 1. At a point where the velocity is vertical, the middle eigenvalue of thestrain rate tensor V is zero.

Using the stability analysis results in [5], we prove the following proposition.PROPOSITION 1. The steady-state granular flow equations (8)-(14) are hyperbolic

if the flow is vertical.Proof. Section 3 of [5] shows that the type of the granular flow equations depends

on the middle eigenvalue a2 of Vl- V. Specifically, a2 0 implies that the steady-state

If "far-field" boundary conditions for all seven variables are specified at Route from the radial

solution, then the outer radius Route must be > 10R to prevent small oscillations from occurring in voand Tro near Route

1080 F. WANG, C. L. GARDNER, AND D. G. SCHAEFFER

equations are of hyperbolic type. a2 0 follows if the velocity is vertical since Lemma1 remains true for IVl-lV as long as v # o. D

3. Numerical method. The granular flow equations (8)-(14) are discretized usingthe box method (flux method) [1] in spherical coordinates with central differences.

Since the Jacobian of the transformation of coordinates r2 sin 0 vanishes at theaxis of the hopper, we impose the conservation laws (8)-(10) at 0 A0/2, instead ofat 0 0. We use one-sided first-order accurate derivatives at the wall and the outflowboundary.

We use Newton’s method to solve the following discrete granular equations(8)-(14)"

(20) J 8TrrTooT,,,,_BTro

fr=-f,foof

where J is the Jacobian and f is the 7 x 1 residual. Note that f,,, fr, and fo are linear.The new solution is obtained by setting- r"

9r Dro Vo o

(21) :Prr Trr + Trro Too

Tj 6T

The discrctizcd Jacobian has the following block structure:

f f0 0 0 0 05Vr Vo

fr fr fr fr0 0 0Trr Too T Tro

afo afo fo0 0 0 0

T T Trofr fr f, f,

() 0 0 0Vr Vo Tr

afoo afoo afo afoo0 0 0v vo Tooaf af af o o f o

Vr VO Tafro af afro o o o afro

Each block is diagonal, tridiagonal, or tridiagonal with fringes. Newton’s equation(20) may be solved directly by banded or sparse matrix techniques with paial pivoting.We use a LINPACK banded solver.

COMPUTATIONS OF GRANULAR FLOW IN A HOPPER 1081

As an initial guess for the solution, we take the radial solution (17). The Newtonmethod converges quadratically.

4. Computations. We performed numerical simulations of granular flow in ahopper for the following two different outflow boundary conditions:

(23a) v -tl(O)/R2inner

where u(O) is determined from the radial solution, and

(23b) Vr --const (e COS (rO/Ow) + 1),

where the constant is determined by requiring the outflowing mass at the bottom ofthe hopper to be equal to the mass inflowing at the top.

For both boundary conditions (23a) and (23b), we have that OoV 0 at the cornerof the outflow boundary with the wall and with the axis of symmetry since Vro- 0 atthe outflow boundary and Vo =0 along the wall and the axis of symmetry (see (15)).This condition ensures that there is no singularity in Vr at the corners.

We present computational results for Ow 30, 6 20, where k x/ sin 6,Router/Rinner--3, and boundary conditions (23a) and (23b) at the outflow boundary.All computations were done on an 80At x 20A0 grid, except for the mesh refinementresults in Fig. 11.

In the computations, the overall velocity scale is set (at the inflow boundary) bychoosing u(0 0)= 1 in the radial solution (17). The units of stress are chosen so thatpgRinner 1.

Figures 2-11 illustrate the flow for a hopper with boundary condition (23a). Notethat the flow is very close to the radial solution flow. The radial component of thevelocity field plotted in Fig. 5 along the outflow surface is just the radial solution v,from boundary condition (23a). The theta component of the velocity field plotted inFigs. 6 and 7 differs from the radial solution in that Vo O. The average stress tr inFig. 8 is basically linear in r for fixed 0, as in the radial solution (17). Figure 10 showsthat To, however, deviates substantially from the radial solution. Both Vo and Todisplay an exponentially damped oscillation as functions of r about the radial solution.This type of behavior is also exhibited by solutions of the linearized granular flowequations. Figure 11 indicates convergence ofthe computations under mesh refinement.

tiIIII //I,

iiiI,iI/::/:/./!!l/i’/:/iiiii/:

FiG. 2. The velocity field in a hopperfor Ow 30 and boundary condition (23a). 80x20 grid.

1082 F. WANG, C. L. GARDNER, AND D. G. SCHAEFFER

-------.----- ,

FIG. 3. e eenvalues and eendirections the stress tensor (Ow 30, bounda condition (23a)).

13.5Exit

21.0

28.5

FIG. 4. v,. (0w =30, boundary condition (23a)).

2.5

ii2.0

-0.98

-0. 985

-0.99-

-0. 995-

/

6 12 18 24 30

FIG. 5. Vr along the outflow surface (Ow 30, boundary condition (23a)).

COMPUTATIONS OF GRANULAR FLOW IN A HOPPER 1083

28.5

1.5Axis

2.0

FIG. 6. vo (0w 30, boundary condition (23a)).

0.006

0.004

0.002

..--.. 2 2.5

FIG. 7. v along the straight line 0 Ow/2 (Ow 30, boundary condition (23a)).

5

4

3

2 2.5

ii

13.5Exit 1.5

21.0

28.5FIG. 8. Average stress tr (Ow 30, boundary condition (23a)).

1084 F. WANG, C. L. GARDNER, AND D. G. SCHAEFFER

1.66"

1.64"

1.62

1.58’

24 30

FIG. 9. o along the outflow surface (Ow 30, boundary condition (23a)).

-0. 005

-0.01-

-0 015-

-0 02-

-0. 025-

-0.03-

8.5

FIG. 10. Tro (Ow 30, boundary condition (23a)).

1.5 2 2.5 3

OoooooooooO.

FIG. 11. Convergence under mesh refinement for Tro along the straight line 0 Ow/2 Ow 30, boundarycondition (23a)). 20x20 (solid) vs. 80x20 grid (dots).

COMPUTATIONS OF GRANULAR FLOW IN A HOPPER 1085

Figures 12-17 present a more complicated flow for a hopper with boundarycondition (23b). Note the visibly nonradial flow near the corner ofthe outflow boundarywith the wall of the hopper in Figs. 12-14. The average stress r in Fig. 15 is stillbasically linear in r for fixed 0, except near the corner of the outflow boundary withthe wall of the hopper. Again, Vo # 0 (see Fig. 16).

The interesting structure in the solutions near the outflow boundary in Figs. 6 and16 for vo and in Figs. 10 and 17 for Tro may be understood in the following way. Att" Rinner, Vro 0. From (15),

1(24) Vo --- Oovr + terms in vo.zr

Since the first term OaOV l 0 from both boundary conditions (23a) and (23b), thesecond term in (24) is forced to be nonzero at r Rinne Asymptotically, as r Router,vo O. However, just inside r Rinner, v0 remains nonzero, and large gradients develop

lll ll!Ii.

III /l

lllll////z//III ///

FIG. 12. The velocity field for Ow 30 and boundary condition (23b) with e 0.4. 80 x 20 grid.

FIG. 13. The eigenvalues and eigendirections of the stress tensor Ow 30, boundary condition (23b) withe 0.4).

1086 F. WANG, C. L. GARDNER, AND D. G. SCHAEFFER

13.5Exit

21.0

2.5

ii2.0

28.5

FIG. 14. vr (0w =30, boundary condition (23b) with e =0.4).

28.Z

6.0

FIG. 15. Average stress tr (Ow 30, boundary condition (23b) with e =0.4).

in Vo and Vro. Since Tro is proportional to Vo (see (3)), large gradients develop in Toas well. In this way, a boundary layer in the solutions near r Rinne is created.

5. Conclusion. The present computations are a "ground-breaking" effort, prepar-ing the way for more difficult problems to follow. These computations study steadyflow in a regime where the flow, at least qualitatively, resembles Jenike’s radial solution,and are somewhat incomplete--the outflow surface in the full problem should be afree boundary on which three boundary conditions are imposed (rather than two, asin this paper). We anticipate that, when a code incorporating the free boundary isdeveloped, flows deviating more substantially from the radial solution will be calculable.

Comparison with experiment is a long-term goal for simulations of the typereported here. Several extensions of this work are needed for such comparisons to be

COMPUTATIONS OF GRANULAR FLOW IN A HOPPER 1087

xit

.0

%0-0.025-0.05}-0.075-o.

2.5

FIG. 16. vo (0w =30, boundary condition (23b) with e =0.4).

28.5 0

.i

.2

Ps. 0.35

1.5

2 0Axis

.o

FIG. 17. Tro (Ow 30, boundary condition (23b) with e =0.4).

meaningful. Most significantly, (i) equations that change from elliptic to hyperbolicmust be handled to model the free boundaries between flowing and stagnant regions(shear bands), and (ii) time-dependent computations are needed to compute oscillatoryphenomena.

REFERENCES

[1] R. E. BANK, D. J. ROSE, AND W. FICHTNER, Numerical methods for semi-conductor device simulation,SIAM J. Sci. Statist. Comput., 4 (1983), pp. 416-435.

[2] G. BAXTER, R. BEHRINGER, T. FAGERT, AND G. JOHNSON, Pattern formation in flowing sand, Phys.Rev. Lett., 62 (1989), pp. 2825-2826.

1088 F. WANG, C. L. GARDNER, AND D. G. SCHAEFFER

[3] C. COULOMB, Essai sur une application des rbgles de maximis et minimis gt quelques problgmes de statique,

relatifs fi l’architecture, Memoirs Mathematiques de l’Acadamie Royale des Sciences (Paris), 7(1776), pp. 343-382.

[4] A. JENIKE, Gravity flow of bulk solids, Bulletin No. 108, Utah Engineering Experimental Station,University of Utah, Salt Lake City, UT, 1961.

[5] D. G. SCHAEFFER, Instability in the evolution equations describing incompressible granular flow, J.Differential Equations, 66 (1987), pp. 19-50.