On the Computation of Steady Hopper Flows

II: von Mises Materials in Various Geometries

Pierre A. Gremaud a,1, John V. Matthews b,2,Meghan O’Malley a

aDepartment of Mathematics and Center for Research in Scientific Computation,North Carolina State University, Raleigh, NC 27695-8205, USA

bDepartment of Mathematics, Duke University, Durham, NC 27708-0320, USA

Abstract

Similarity solutions are constructed for the flow of granular materials through hop-pers. Unlike previous work, the present approach applies to nonaxisymmetric con-tainers. The model involves ten unknowns (stresses, velocity, and plasticity function)determined by nine nonlinear first order partial differential equations together witha quadratic algebraic constraint (yield condition). A pseudospectral discretizationis applied; the resulting problem is solved with a trust region method. The influenceof the geometry of the hopper on the flow is illustrated by several numerical experi-ments of industrial relevance. The corresponding effects are found to be important.

Key words: elliptic, granular, similarity, spectralPACS:

1 Introduction

This is the second in a series of papers on the mechanics of a granular materialunder the influence of gravity flowing through converging hoppers of simple

Email addresses: gremaud@unity.ncsu.edu (Pierre A. Gremaud),jvmatthe@math.duke.edu (John V. Matthews), msomalle@unity.ncsu.edu(Meghan O’Malley).

URLs: http://www.ncsu.edu/math/∼gremaud (Pierre A. Gremaud),http://www.math.duke.edu/∼jvmatthe (John V. Matthews).1 Partially supported by the National Science Foundation (NSF) through grantsDMS-9818900 and DMS-020445782 Partially supported by the National Science Foundation through grant DMS-9983320

geometry. While such devices are common in many industrial processes, thewithdrawal of stored materials from hoppers and bins is well known to beproblematic. Segregation, lack of flow, flooding (uncontrolled flow) and struc-tural failures are often encountered [12]. A better understanding of such flowsshould lead to improved hopper design criteria.

Commonly used models are subject to severe restrictions [7,9–11,14,20,23]:

(i) The granular material is modeled as a continuum, with ad hoc constitutivelaws.

(ii) The flow is assumed to be steady.(iii) Only similarity solutions are considered.(iv) The flow domain is pyramidal, i.e., it is invariant under the transforma-

tion (expressed in spherical coordinates) r 7→ c r, for any c > 0.(v) The flow domain is axisymmetric.

The combination of assumptions (iv) and (v) corresponds to a right circularcone, see Figure 1, upper left

(r, θ, φ) : 0 < r <∞, 0 ≤ θ < θw, (θw = constant). (1)

In that case, it was observed by Jenike [10,11] in the late 1950’s that sim-ilarity solutions, as in (iii), can be constructed (see (8) below for a precisedefinition). To date, this approach remains a central component in silo design.In [6], we removed restrictions (iii) and (iv), see Figure 1, upper right, see also[13,15,17] for more results in that direction. In the present paper, (iii) and(iv) are retained while (v) is removed. More precisely, the hopper is an infinitepyramidal domain

(r, θ, φ) : 0 < r <∞, 0 ≤ θ < C(φ), (2)

where C is a given piecewise smooth 2π-periodic function describing the bound-ary of the cross section of the hopper, see Figure 1, bottom left. The presentstudy was started in [8], where small perturbations of the axisymmetric case,specifically boundaries of equation C(φ) = θw + ε cosmφ, were considered.Equations for first order corrections in ε were derived and solved. Here, crosssections of arbitrary shape can be considered. The authors are aware of noother contribution studying the effects of the geometry on the flow of granularmaterials in this setting. The case of fully three-dimensional hoppers such asthe transitional hopper displayed in Figure 1, bottom right, appears to beopen.

Jenike’s solutions are such that grains follow radial lines (only the r-componentof the velocity is non zero). Experimental evidence confirms the important roleplayed by those radial solutions in practice [13,14] in conical containers (1).

2

Fig. 1. Various geometries and corresponding numerical domains of resolution; topleft: axisymmetric self-similar domain: Jenike’s approach on a cone reduces to solv-ing ODEs on a r constant line; top right: axisymmetric non self-similar, see e.g.part I of this series, [6]: nonlinear conservation laws are solved in the shadoweddomain; bottom left: nonaxisymmetric self-similar: this paper; bottom right: generalthree-dimensional hopper: open problem.

Since general pyramidal hoppers are self similar (see (iv)), one could expectthe radial character of the flow to hold in those cases as well. Our study, whichmakes use of assumptions (i) through (iv), finds this to be far from true. Weshow that a loss of axisymmetry in the hopper geometry induces secondarycirculation as well as strong nonuniformity of the stresses in the azimuthaldirection.

The paper is organized as follows. The model, geometry, and physical assump-tions are discussed in Section 2. Simple scaling arguments reveal the similaritycharacter of the solutions in the r direction. As a result, the solutions are com-puted on spherical caps of various shapes. The numerical approach is pseu-dospectral in nature and is described in Section 3. It uses Fourier collocationin longitude and, to account for the boundary conditions, Chebyshev-Gauss-Radau collocation in latitude. Section 4 is devoted the description of severalnumerical experiments. Conclusions are offered in Section 5.

2 The model

The physical quantities and corresponding equilibrium equations are expressedin spherical polar coordinates, with the origin corresponding to the vertex of

3

the hopper. For a particular pyramidal domain, a coordinate system that isbetter suited to the geometry can be constructed. However, such a coordinatesystem is not typically orthogonal, complicating greatly the structure of thebasic equations of Continuum Mechanics [2]. To simplify the numerics, suchalternate coordinate systems are introduced in Section 4 through a change ofvariables, but the individual components of the stress tensor and velocity arestill measured in terms in the original spherical coordinates.

The unknowns are the 3-component velocity vector v, the 3 × 3 symmetricstress tensor T , and a scalar plasticity coefficient λ. (The density ρ is a con-stant.) In total, there are 3+6+1=10 unknown functions. In writing the equa-tions for these variables, we need the strain rate tensor V = −1/2(∇v+∇vT )and the deviatoric part of the stress tensor dev T = T− 1

3(trT ) I. Note the sign

convention: V measures the compression rate of the material; analogously, pos-itive eigenvalues of T correspond to compressive stresses. This sign conventionreflects the fact that granular materials disintegrate under tensile stresses.

Following [20], we require that these variables satisfy

∇ · T = ρ g, (3)

V =λ dev T, (4)

| dev T |2 = 2s2(trT /3)2, (5)

where g is the (vector) acceleration of gravity, | · | denotes the Frobenius norm

|T |2 =3∑

i,j=1

T 2ij = trT 2

(the latter equality only for symmetric tensors), and s = sin δ, with δ beingthe angle of internal friction of the material under consideration (see [14]).Equation (3) expresses force balance, i.e., Newton’s second law with inertianeglected because the flow is assumed slow; it is equivalent to three scalarequations. Equations (4) and (5) are constitutive laws, the alignment con-dition (or flow rule) and the von Mises yield condition 3 , respectively; theyare equivalent to six and to one scalar equations. Thus (3–5) is a determinedsystem, 10 equations for 10 unknowns. Since (5) contains no derivatives, thissystem has a differential-algebraic character. Taking the trace of (4), we seethat div v = −trV = 0; thus, incompressibility is part of the constitutiveassumptions. Incidentally, for a solution to be physical, the function λ in (4)must satisfy λ ≥ 0 everywhere; otherwise friction would be adding energy tothe system rather than dissipating it. In fact, we want λ to be strictly pos-

3 The name is kept from the corresponding metal plasticity model even though theyield surface is cylindrical there while conical here [20].

4

itive since one of the assumptions underlying the derivation of (3–5) is thatmaterial is actually deforming.

The alignment condition (4) of the eigenvectors of T and V in effect neglectsthe rotation of a material element during deformation, a controversial assump-tion. There is experimental evidence that misalignment may occur under somecircumstances [5]. A few alternative models which allow for the above eigen-vectors to be somewhat out of alignment have been proposed, see e.g. [23,24].While such models have been successfully applied to axisymmetric geometries,their formulation becomes very involved in more general cases and has, to ourknowledge, never been fully described. Further, there does not seem to beenough experimental data to favor one type of model over the other. We referto [9] for a lucid, if somewhat dated, account of the situation. Our contributionto this debate is the first calculation of solutions to any model in relativelygeneral geometries 4 .

We seek solutions of (3–5) in a pyramidal domain (2). On the boundary ∂Ω =(r, C(φ), φ), wall impenetrability imposes one boundary condition on thevelocity: i.e.,

vN = 0, (6)

where vN is the normal velocity. Two additional boundary conditions comefrom Coulomb’s law of sliding friction. The surface traction τ—i.e., the forceexerted by the wall on the material—is given by

τi =3∑

j=1

TijNj,

where N is the unit interior normal to ∂Ω. If the vector τ has normal compo-nent τN and tangential component τT = τ − τNN , then we require that

τT = −µw τN(v/|v|), (7)

where µw is the coefficient of friction between the wall and the material. Notethat: (a) If T is positive definite (i.e., if all stresses are compressive), thenτN > 0. (b) While τN is a scalar, τT is effectively a two-component vector;thus, (7) is equivalent to two scalar equations. (c) Because of (6), the velocityv is tangential to ∂Ω; we are assuming that v 6= 0 at the boundary.

4 Comparison with experimental data is underway; joint work with R.P. Behringer(Duke, Physics), J. Wambaugh (Duke, Physics) and D.G. Schaeffer (Duke, Mathe-matics).

5

Putting together assumptions (iii) and (iv) from the Introduction, we seeksimilarity solutions of the form

Tij = r Tij(θ, φ), vi =1

r2Vi(θ, φ), λ =

1

r4L(θ, φ) (8)

where Tij and vi are any component of the stress tensor and velocity vec-tor respectively. Equations (3,4,5) lead then the following system for the tenunknowns Trr, Trθ, Tθθ, Trφ, Tθφ, Tφφ, Vr, Vθ, Vφ and L.

∂θTrθ + csc(θ) ∂φTrφ + 3 Trr − Tθθ − Tφφ + cot(θ) Trθ = −ρg cos(θ), (9)

∂θTθθ + csc(θ) ∂φTθφ + 4 Trθ + cot(θ) (Tθθ − Tφφ) = ρg sin(θ), (10)

∂θTθφ + csc(θ) ∂φTφφ + 4 Trφ + 2 cot(θ) Tθφ = 0, (11)

2Vr = L (Trr − P), (12)

−1

2(∂θVr − 3Vθ) = LTrθ, (13)

−∂θVθ − Vr = L (Tθθ − P), (14)

−1

2(csc(θ) ∂φVr − 3Vφ) = LTrφ, (15)

−1

2(∂θVφ + csc(θ)∂φVθ − cot(θ)Vφ) = LTθφ, (16)

−(csc(θ)∂φVφ + Vr + cot(θ)Vθ) = L (Tφφ − P), (17)

| devT |2 = 2s2P2, (18)

where T = 1rT and P = 1

3trT = 1

3(Trr +Tθθ +Tφφ). The above equations have

to be satisfied in the spherical domain (θ, φ);−π < φ < π, 0 < θ < C(φ).

The boundary conditions (6, 7) can be expressed in terms of the above un-knowns. The unit interior normal vector N on the outside wall θ = C(φ) takesthe form

N = [Nr, Nθ, Nφ] = [0,− sin θ, C ′(φ)]/√

sin2 θ + C ′(φ)2.

Therefore, when θ = C(φ), 0 < φ < 2π, one has

VθNθ + VφNφ = 0, (19)

Vφ(TrθNθ + TrφNφ)−Vr[−TθθN2θNφ + TθφNθ(1− 2N2

φ) + TφφNφN2θ ] = 0, (20)

(TrθNθ + TrφNφ)2 + (TθθNθNφ − TφφNθNφ + Tθφ(N

2φ −N2

θ ))2

= µ2w(TθθN

2θ + 2TθφNθNφ + TφφN

2φ)2, (21)

where (19, 20, 21) correspond respectively to the conditions (6), τT × v = 0, avector relation yielding two nontrivial scalar relations, and | τT | = µwτN , both

6

expressed in the new variables. Note that the above set of equations definesthe unknowns Vr, Vθ, Vφ and L only up to a multiplicative constant. This is aconsequence of the similarity character of the present approach and is foundwith Jenike’s solution as well. To eliminate this indeterminacy, we fix the valueof one component of the velocity, say Vr, at one point in the computationaldomain. Here, we choose

Vr(Θ0,Φ0) = v?, (22)

where Θ0 and Φ0 are defined in the next section and where v? in effect scalesthe flow rate out of the hopper. For Jenike’s solution, a typical normalizationis to take the maximum (radial) velocity equal to -1. All three one-parameterfamilies of nonaxisymmetric domains considered in Section 4 admit the rightcircular cone as a special case. In all three cases, the boundary value v? istaken as the value of Jenike’s radial velocity at the wall under the abovenormalization for a right circular cone. Note that no conditions are needed atθ = 0 (or more precisely, considering only nonsingular solutions at that pointis a boundary condition). Further, 2π-periodicity is imposed in the φ direction.

Along with the above unknowns, a stream function Ψ is also computed. Underassumption (8), the incompressibility condition div v = 0 reads here

∂θ(sin θ Vθ) + ∂φVφ = 0.

A stream function Ψ can be introduced through

∂φΨ = sin θ Vθ, and ∂θΨ = −Vφ.

In fact, Ψ can be characterized as the solution to a Poisson problem on thespherical cap under consideration

∆Ψ =−ξr, for 0 < θ < C(φ),−π < φ < π, (23)

Ψ = 0, on θ = C(φ),−π < φ < π, (24)

where ∆ = 1sin θ

∂θ(sin θ∂θ·) + 1sin2 θ

∂φφ· is the Laplace operator on the sphereand ξr is r3 times the r component of the vorticity ξ = ∇ × v, i.e., ξr =cot θ Vφ +∂θVφ− 1

sin θ∂φVθ. As above, periodicity is assumed in the φ direction.

3 Numerical analysis

In order to simplify the numerics, the problem is mapped onto a rectangularcomputational domain. We define the new coordinates

7

Θ = θwθ

C(φ)and Φ = φ.

Note that Θ,Φ is not an orthogonal coordinate system. However, the com-putational domain is now simply (0, θw) × (−φw, φw), where φw correspondsto the smallest interval of periodicity of the solution in the φ direction. Moreprecisely, for a pyramidal hopper of general cross section, φw = π; for a squarecross section for instance, one can take φw = π/4 and construct the solutionfrom −π to π by duplicating the solution obtained between −π/4 and π/4 inthe obvious way.

The equations (9-21) are written in the new coordinate system. The reader isspared the expression of the corresponding equations. The resulting problemcan be discretized by collocation; Chebyshev collocation at the Chebyshev-Gauss-Radau points is used in Θ, while Fourier-cosine collocation at theFourier collocation points is used in Φ. More precisely, we set

UNM(Θ,Φ) =N−1∑n=0

M/2−1∑m=−M/2

Unmψn(Θ)eim π( Φφw

+1), (25)

where ψnN−1n=0 are the Lagrange interpolation polynomials at the Chebyshev-

Gauss-Radau nodes on [0, θw], i.e.

Θj =θw

2

(1 + cos

(2πj

2N − 1

)), j = 0, . . . , N − 1. (26)

The above choice, as opposed to the more standard Chebyshev-Gauss-Lobattocollocation, see e.g. [3], §2.4, results from the nature of the boundary condi-tion along Θ = θw. For completeness, we derive below the expression of thecollocation derivative for Chebyshev-Gauss-Radau nodes (which we have notbeen able to find in the literature).

Lemma 1 The Lagrange interpolation polynomials on the Chebyshev-Gauss-Radau nodes (26) are given by

ψj(Θ) =1

cj

θw −Θ

Θ−Θj

(1

NT ′

N

(2Θ

θw

− 1)

+1

N − 1T ′

N−1

(2Θ

θw

− 1))

, j = 0, . . . , N−1,

where

c0 = 1− 2N,

cj =− θw

2Θj

(N cos

2πNj

2N − 1+ (N − 1) cos

2π(N − 1)j

2N − 1

), j = 1, . . . , N − 1,

8

and TN(x) = cos(N arccos x), |x| ≤ 1, is the Chebyshev polynomial of degreeN .

The above result can easily be verified through the use of L’Hospital’s ruleand elementary properties of the Chebyshev polynomials. Interpolation at thenodes (26) of a function u of Θ defined in [0, θw] simply takes the form

INu(Θ) =N−1∑j=0

u(Θj)ψj(Θ).

By definition, the Chebyshev collocation derivative of u at those nodes is then

(INu)′(Θl) =

N−1∑j=0

u(Θj)ψ′j(Θl) =

N−1∑j=0

Dlju(Θj),

with Dlj = ψ′j(Θl). The collocation derivative at the nodes can then be ob-tained through matrix multiplication. Elementary albeit tedious calculationslead to the following expressions

Dlj =

23θw

N(N−1), if l = j = 0,

2c0θw sin 2 2πl

2N−1(N cos 2Nπl

2N−1+(N−1) cos

2(N−1)πl2N−1 ) if j = 0,

l = 1, . . . , N − 1,

cl

cj

1Θl−Θj

if j = 1, . . . , N − 1,

j 6= l, l = 0, . . . , N − 1,

− 14Θj

3θw−2Θj

θw−Θj− θw

4cjΘj

1√Θj(θw−Θj)

·(N2 sin(N arccos(

2Θjθw

−1))

+(N−1)2 sin((N−1) arccos(2Θjθw

−1))

) if j = l = 1, . . . , N − 1.

To minimize round-off errors, trigonometric identities are used to express thequantities Θl −Θj [18], namely

Θl−Θj = θw sin(

π

2N − 1(j + l)

)sin

(π

2N − 1(j − l)

), j, l = 0, . . . , N−1.

Further, as suggested in [1], the differentiation matrix D is made to representexactly the derivative of a constant by numerically satisfying the identity∑N−1

j=0 Dlj = 0, l = 0, . . . , N−1. More precisely, the off-diagonal terms only are

9

computed using the above formula while the diagonal entries are determinedby

Dll = −N−1∑

j=0,j 6=l

Dlj, l = 0, . . . , N − 1.

In the Φ direction, the collocation points are taken as the usual Fourier collo-cation nodes, i.e.,

Φl = φw

(2l

M− 1

), l = 0, . . . ,M − 1. (27)

Let U be the N × M matrix of coefficients Unm, n = 0, . . . , N − 1, m =0, . . . ,M/2 − 1,−M/2, . . . ,−1 for one of the variables under considerationand let W be the M ×M Fourier matrix

W =

1 1 1 . . . 1

1 ωM ω2M . . . ωM−1

M

1 ω2M ω4

M . . . ω2(M−1)M

. . . . . . . . . . . . . . .

1 ωM−1M ω

2(M−1)M . . . ω

(M−1)2

M

,

where ωM = e2πi/M is the primitive M -th root of unity. Further, if Λ is theM ×M diagonal matrix with diagonal [0, . . . ,M/2− 1,−M/2, . . . ,−1], thenfor any pair j and l, j = 0, . . . , N − 1, l = 0, . . . ,M − 1, the nodal values ofUNM and its derivatives can be expressed as follows

UNM(Θj,Φl) = (UW )jl,

∂ΘUNM(Θj,Φl) = (DUW )jl,

∂ΦUNM(Θj,Φl) = 4i (UΛW )jl.

Each of the ten unknowns Trr, Trθ, Tθθ, Trφ, Tθφ, Tφφ, Vr, Vθ, Vφ and L is nowexpanded according to (25). Their coefficient matrices are denoted Urr, Urθ,Uθθ, Urφ, Uθφ, Uφφ, Ur, Uθ, Uφ and U , respectively. The discretized equations 5

are obtained as follows

5 From here on down and unless stated otherwise, we refer to the equations writtenin the new variables (Θ,Φ). As those can easily be obtained from (9-21), we do notrewrite them explicitly, for the sake brevity.

10

0 0.2 0.4 0.6 0.8 10

100

200

300

400

500

600

eccentricity

stre

ss d

iffer

ence

per

cent

age

2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

120

140

160

p

stre

ss d

iffer

ence

per

cent

age

Fig. 2. Effect of the geometry on the normal wall stress. Left: effect of the eccentric-ity in vertical hoppers with elliptical cross-sections given by (28) for E between 0and 0.866. Right: effect of the parameter p on the normal wall stress in vertical hop-pers with square like cross-sections given by (29) for p between 2 and 20. Materialparameters correspond to corn in a steel hopper: angle of internal friction δ = 32.1,angle of wall friction = 11.7.

• equations (9-18), written in terms of (Θ,Φ), are satisfied at the collocationpoints (Θj,Φl), j = 1, . . . , N − 1, l = 0, . . . ,M − 1,

• at the boundary nodes Θ0,Φl), l = 0, . . . ,M − 1, equations (9-11) arereplaced by the boundary conditions (19-21), i.e., equations (12-18) and(19-21), all written in terms of (Θ,Φ), are satisfied there,

• at (Θ0,Φ0), the boundary condition (19) is replaced by (22).

This results in a system of nonlinear equations

F (X) = 0,

where the unknown X = [Urr, Urθ, Uθθ, Urφ, Uθφ, Uφφ, Ur, Uθ, Uφ, U ]T is a 10N×M matrix and F : R10N×M → R10N×M is a quadratic function with variablecoefficients. The current nonlinear solver uses a trust region dogleg method[16] with finite difference Jacobians.

The problem (23, 24) for the stream function is discretized using the sameapproach as above, but is solved independently in a post-processing step.

4 Numerical results

Several experiments are presented in order to display the importance of the lossof axisymmetry on the stress and velocity fields. A detailed material parameterstudy will be presented elsewhere. Note that for ease of interpretation allthe numerical results below correspond to values obtained after projection

11

onto a horizontal cross section and not a spherical cap as would be directlyobtained from the model. The symmetry of both domain and solution is takeninto account to reduce computational costs. The discretization is taken asN = M = 20; solutions were checked to have converged as expected forspectral methods of this type (i.e., further refinement does alter the results inany significant way).

4.1 Vertical elliptical hoppers

A family of hoppers described by the following equation are considered

C(φ) =π/6√

1− E2 cos2 φ, 0 ≤ E < 1, (28)

where E stands for the eccentricity. For various eccentricities, the percentageof difference between the largest and smallest values of the normal stress ona horizontal cross section of the wall was computed, see Figure 2, left. Theeccentricity varies from E = 0 (circular cross section) to E = 0.866 whichroughly corresponds to a 2 to 1 ratio between major and minor axes. Therelative difference (as a percentage) between the largest and smallest valuesσM and σm taken by the normal stress on a horizontal cross section of the wall,i.e., 100σM−σm

σmvaries correspondingly from 0 (1.81×10−11 to be precise) in the

axisymmetric case to about 600% for E = 0.866. In other words, the normalstress varies by more than a factor 6 for this range of hopper geometries.

Figure 3 clearly illustrates the effects of the loss of axisymmetry. This is fur-ther supported by Figure 4, which depicts the behavior of the normal stressalong the boundary. In the present elliptical case, the normal stress reachesits maximum value at the points of largest curvature. This type of nonunifor-mity, while ignored in standard models in this field, may be at the root of veryimportant effects such as shell buckling and other structural failure modes rou-tinely observed in silos [19]. One observes from the graph of Vr that motionmainly takes place in a central circular core of the domain. Further, secondarycirculation, while totally absent from Jenike’s theory, is clearly present here asshown by the graph of the stream function: four vortices (for the full domain)are formed. However, the angular components of the velocity, Vθ and Vφ, areabout two orders of magnitude smaller than Vr.

12

0 0.1 0.2 0.3 0.4 0.5 0.6

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

average stress

0.5

1

1.5

2

2.5

3

0 0.1 0.2 0.3 0.4 0.5 0.6

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

λ

5

10

15

20

25

0 0.1 0.2 0.3 0.4 0.5 0.6

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

stream function

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1x 10

−3

0 0.1 0.2 0.3 0.4 0.5 0.6

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Vr

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0 0.1 0.2 0.3 0.4 0.5 0.6

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Vθ

−8

−6

−4

−2

0

2

4

6

8x 10

−3

0 0.1 0.2 0.3 0.4 0.5 0.6

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Vφ

−6

−4

−2

0

2

4

x 10−3

Fig. 3. Flow in a vertical elliptical hopper (28) with E = 0.866 (major axis/minoraxis ≈2); material parameters correspond to corn in a steel hopper: angle of inter-nal friction δ = 32.1, angle of wall friction = 11.7. Only half of the domain isrepresented.

4.2 Vertical square hoppers

In spite of being well-known to have suboptimal flowing properties, hopperswith square cross sections are still widely used in many applications.

A family of hoppers described by the following equation are considered

C(φ) =π/6

(cosp φ+ sinp φ)1/p, 2 ≤ p <∞. (29)

13

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

−π −3π/4 −π/2 −π/4 0 π/4 π/2 3π/4 π

Fig. 4. Behavior of the normal stress along the wall; dash-dot line: vertical ellipti-cal hopper (28) with E = 0.866; solid line: vertical “square-like” hopper (29) withp = 20; dashed line: tilted right circular hopper (30) with θw = 30 and a tilt angleα = 14; material parameters correspond to corn in a steel hopper: angle of internalfriction δ = 32.1, angle of wall friction = 11.7. For comparison purposes, themaximal values are normalized to 1.

The above domain corresponds to the unit disk with respect to the p-norm;for p 2, the domain approximates a square.

In Figure 2, right, the parameter p varies from 2 (circular cross section) to20. The relative difference (as a percentage) between the largest and smallestvalues σM and σm taken by the normal stress on a horizontal cross sectionof the wall, i.e., 100σM−σm

σmvaries correspondingly from 0 in the axisymmetric

case p = 2 to about 150%.

As can be seen from Figure 4, the nonuniformity of the stresses is less promi-nent for (almost) square hoppers than it is for elliptical ones. It is worth notingthat the normal stress is largest at the corners, a feature of possible importanceto structural engineers. The profile of the radial velocity Vr is much steeperthan it was in Figure 3, a feature reminiscent of funnel flows. Additionally, theangular components of the velocity are here about three orders of magnitudesmaller than the radial one; compared to the elliptical hopper, circulation isless pronounced. In the full domain, eight circulation cells are present, as canbe seen from the stream function graph in Figure 5. The structure of the flowdoes not appear to change significantly for larger values of the parameter p in(29).

14

0 0.2 0.4 0.6−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5average stress

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.2 0.4 0.6−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5λ

100

200

300

400

500

600

700

0 0.2 0.4 0.6−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5stream function

−1.5

−1

−0.5

0

0.5

1

1.5

x 10−3

0 0.2 0.4 0.6−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Vr

−35

−30

−25

−20

−15

−10

−5

0 0.2 0.4 0.6−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Vθ

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0 0.2 0.4 0.6−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Vφ

−8

−6

−4

−2

0

2

4

6

8x 10

−3

Fig. 5. Flow in a vertical “square-like” hopper (29) with p = 20; material parameterscorrespond to corn in a steel hopper: angle of internal friction δ = 32.1, angle ofwall friction = 11.7. Only a quarter of the domain is represented.

4.3 Tilted circular hoppers

Industrial applications sometimes call for the use of an off-center outlet atthe bottom of a large silo. We consider here the case of tilted right circularcones. More precisely, a right circular cone of (half) opening angle θw is tiltedin the yz-plane (φ ≡ 0) by an angle α. Elementary trigonometry shows thefunction θ = C(φ) describing the boundary of the computational domain tobe implicitly defined by

cos θw = sinα sinφ sin θ + cosα cos θ. (30)

15

−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

average stress

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44

−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

λ

3

4

5

6

7

8

−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

stream function

−5

−4

−3

−2

−1

0

1

2

3

4

x 10−5

−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Vr

−0.6

−0.55

−0.5

−0.45

−0.4

−0.35

−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Vθ

−5

−4

−3

−2

−1

0

1

2

3

4

5

x 10−4

−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Vφ

−4

−3

−2

−1

0

1

2

3

x 10−4

Fig. 6. Flow in a tilted right circular hopper (30) with θw = 30 and a tilt angleα = 14; material parameters correspond to corn in a steel hopper: angle of internalfriction δ = 32.1, angle of wall friction = 11.7.

Even for relatively small angles of tilt, our numerical experiments show thepresence of very steep gradients in some of the unknowns. While this createscomputational difficulties, this phenomenon in fact results from the choiceof measuring the components of the stress tensor with respect to sphericalcoordinates. We have verified that, when measured with respect to Cartesiancoordinates, the components of the stress tensor are all smooth.

16

Figure 6 illustrates the same fields as displayed for the previous two experi-ments. Only two circulatory cells are observed. The main component of thevelocity Vr is found to be more uniform than in the previous two experiments.The same is true of the average stress and the normal stress on the wall.Figure 4 shows that the largest value of the normal stress is reached in thedirection toward which the hopper is tilted.

5 Conclusions

We have shown the effects of the geometry of the containers on granular flowsare to be important. Secondary circulation is brought to the fore; this phe-nomenon, while predicted in an earlier study of a linearized problem [8], isabsent from other existing studies. More importantly, significant stress nonuni-formities are observed. Both results may have important practical applicationsin hopper and silo design and in the prediction of shell buckling. In a forth-coming paper, comparison between the present work, results obtained withother constitutive laws (such as the Matsuoka-Nakai yield condition [4]), andexperimental results will be considered.

Acknowledgments

The authors thank Bob Behringer, Tim Kelley, Michael Rotter, Tony Royaland John Wambaugh for many helpful discussions. They are also grateful toDavid Schaeffer for many insightful suggestions and remarks without whichthis paper would not have been possible.

References

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