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Energy diffusion in a granular flow of disk shaped solids
Iru. 1. En,pg Sci. Vol. 24, No. 4, pp. 551-555, Printed in Great Britain.
ENERGY
0020-7225186 $3.00 + .CO Q 1986 pcrSamon Pres Ltd.
1986
DIFFUSION IN A GRANULAR OF DISK SHAPED SOLIDS
FLOW
HAYLEY H. SHEN and NORBERT L. ACKBRMANN Clarkson University, Potsdam, NY 13676, U.S.A. (Communicated by A. J. M. SPENCER) Abstract-The kinetic energy of a granular flow consists in part of the energy associated with the random velocity fluctuations u’ of the individual solid particles. When the magnitude of u’ varies throughout the flow field a transfer of kinetic energy pff/2 occurs that must be considered when evaluating the energy balance relationship. In the present investigation this energy transfer or dil’htsion is quantified for the flow of a granular continuum consisting of disk-shaped solids. The theoretical development involves a detailed analysis of the binary collision mechanics which results from the interaction between adjacent disks within the granular mixture. The energy diffusion term developedin this investigation, in conjunction with the stress and energy dissipation functions previously reported by Shen and Ackermann [ 11, provide all of the constitutive relationships that arc necessary to describe the two-dimensional uniform flow of a granular mixture of disk-shaped solids. INTRODUCTION THE
BEHAVIOR
of a granular continuum is described by the solution of the energy, momentum, and mass balance relationships. The balance equations for mass and momentum as presented in eqns (1) and (2) involve only the mean flow velocity u as an explicit kinematic variable
(1) a(PUi) +u.a(pui)=T,
at
’
dXj
Ji.J
.+pb.
1’
(2)
The terms p, T and b refer to the bulk density of the granular mixture, the stresses, and body force, respectively. The energy equation as presented by Ogawa, Umemura and Oshima [2] requires, however, that the random velocity fluctuation u’ of the granular solids also be considered explicitly as a kinematic variable. As first suggested by Jenkins and Cowin [3], the existence of a nonuniform o’ is the means by which energy can be conducted from one part of the granular continuum to another. In eqn (3) the term q describes this energy diffusion process D(Pv’~)/~
Dt
= rijuj,i - qj,j - 73
while the term y represents an energy sink. This sink term equals the rate at which mechanical energy is dissipated through inelastic and frictional collisions between the granular solids. Equations (I) through (3) can be solved for appropriately defined initial and boundary equations if constitutive relationships can be found to relate T, q, and y to p, u, o’ and the material properties of the granular solids. Bagnold [4], Haff [5], Jenkins and Savage [6], Jenkins and Cowin [3], Lun, Savage, Jeffrey, and Chepurniy [7], McTigue [S], Ogawa, Umemura, and Oshima [2], Savage and Jeffrey [9] and Shen and Ackermann [ 1, lo] among others have investigated the formulation of such constitutive relationships. Shen and Ackermann [l] derived the constitutive relationships for the stress r and energy dissipation y in a simple shear flow of disk-shaped solids. The theoretically determined stress state was found to compare favorably with the computer simulated
552
H. H. SHEN and N. L. ACKERMANN
results of Campbell and Brennen [ 1 I]. Such a favorable comparison appears to confirm the validity of the modeling procedure adopted by Shen and Ackermann. This model is now to be extended to describe the diffusion process indicated by the term 4 in eqn (3). In the analyses by Shen and Ackermann [ 1,lo] the fluctuation speed 2)’was considered to be constant through the simple shear field. The results of their analysis showed that u’ is proportional to the gradient Ui,j of the mean velocity. Hence, one may conclude that in a how field where the gradient of the mean velocity is not constant, there would also be a corresponding gradient in the fluctuation velocity energy pu”/2. The existence of such a gradient of the fluctuation energy implies that when two particles, such as A and B in Fig. 1, collide with different fluctuation speeds, energy is not only dissipated, but also transferred. This energy transfer phenomenon can be easily illustrated by considering a system consisting of two identical perfectly elastic spheres. One sphere is initially stationary while the other is moving towards and colliding headon with the stationary one. After the impact has occurred, the originally stationary sphere acquires the speed of the originally moving sphere while the originally moving sphere will come to a complete rest. Energy is thus completely transferred. By quantifying this energy transfer term q within a granular flow, eqns (1)through (3) can then be used to describe complex flows in which the gradient of the mean velocity can vary throughout the flow field.
MODEL
FORMULATION
Shen and Ackermann [ 1, lo] developed the constitutive relationships for T and y following the framework for analysis first used by Bagnold [4] to describe the stress state in a rapidly sheared granular mixture. This same approach will be adopted to determine the diffusion term q in eqn (3). Consider at first the continuum formed from a single layer of moving disks shown in Fig. 1. The stress TV, acting in the Xj direction on the surface normal to xi, was described by Bagnold [4] as TO
where pi is the number of direction, f is the average AMj is the jth component stress generating collisions
=
-pif AMj,
particles per unit area on the surface whose normal is in the Xi frequency of the stress generating collisions per particle and of the average momentum transfer per collision. In Fig. 1 the along the surface where x2 is constant are those collisions that
Isotropic Distribution of fluctuation velocity-,
Fig.
I. Shear flow of disk shaped solids.
553
Energy diffusion in a granular flow of disk shaped solids
occur between the shaded particles such as A and the particles, such as B, which are located on the surface where x2 = constant. The same formulation is used for eqn (5) to describe the energy flux qi across a surface whose normal is in the xi direction -qi = PifET. (5) In eqn (5) ET is the average energy transferred per collision. The transfer occurs between an exterior particle, such as particle A, and one of the particles such as B which temporarily resides on the surface through which energy is being diffused. To quantify first the energy ET transferred per single collision, consider the magnified view of particles A and B which are about to collide as shown in Fig. 2. Let EB be the kinetic energy of particle B as viewed in the coordinate system moving with the mean motion r& and EB be the kinetic energy of particle B after collision. The energy change in B due to such a collision is Eg - Ee: ET = E$ - EB,
(6)
where ET is the energy transferred from A to B. Expressing eqn (6) in terms of the particle mass m and velocity VB of particle B,
ET=:wg*vg- VB’VB],
(7)
L
where the symbol * represents post-collision conditions. It is now considered that the inelastic and frictional effects of particle collision are described solely by the coefficient of restitution 6 and friction coefficient p. The velocity Vg of disk B aBer collision with A therefore becomes
N * (VA- VB )(N + rp),
(8)
where N is the unit vector joining the center of the two disks as shown in Fig. 2. The collision frequency f between two particles is defined as the relative approach velocity of colliding particles divided by their average separation distance S. Hence, the collision frequency for typical particles such as A and B can be expressed as f =
(VA-
VB)*N
s
(9)
.
Disks can only collide, however, if their separation distance decreases. The criterion governing this condition can be expressed quantitatively as VA-N > VB*N.
(10)
By summing over all possible collisions, eqn (5) can then be modified as
9i =
C VA.N>VB-N
Cpiy(Vg.Vg
-VB*VB)I(VATVB)*N~
,
(11)
N
where the summation over N refers to the possible collision contact points in the range described in Fig. 2. from 0 < 8 < ?r.
COMPUTATIONAL
DETAILS
AND
RESULTS
Consider the steady uniform rectilinear shear flow shown in Fig. 1 in which the mean velocity is in the x1 coordinate direction. Using the geometric representation of disks A
H. H. SHEN and N. L. ACKERMANN
554
Fig. 2. Disks at point of collision. Velocity reference frame moving with mean motion us of particle B.
and B shown in Fig. 2, the velocities VA and VB relative to the mean motion of particle B can be expressed as VA =
u’ + 5 2
(D + s) sin 6 (cos aN + sin aP) + 2 I
(D + s) sin 0X1
(12)
2
VB = -u’(cos /3N + sin BP),
(13)
where Xi is a unit vector in the Xi coordinate direction and P is the unit vector normal to N. Shen and Ackermann [I] considered the effect of assuming that o’ % D&i /ax,. They showed that this assumption had a minor effect upon the accuracy of the calculated stresses for a wide range of realistic flow conditions. It is now assumed that both u’ B Dau, /ax2 as well as u’ B Dav’/ax2. Satisfying eqn (10) then reduces to the condition that for
-?r I (YI *.
(14)
Equation (11) can then be rewritten as
The range of integration for 0 is determined by 0r < 19< d1 + ?r where 0, = 0 for diffusion across the surface normal to the x2 direction and 8i = a/2 for diffusion across the surface normal to the xl direction. The number of particles pi are now described using the definition of linear concentration X introduced by Bagnold [4]: X=
(16)
D/s.
Letting C, = 0.907 represent the area concentration packing 2
C’C”& (
1
.
C of the disks at their densest
(17)
Energy diffusion in a granular flow of disk shaped solids
555
Shen and Ackermann [l] have shown that for a randomly arranged pattern of disks, p can be described as the following, with disk thickness being d
(18) By substituting eqns ( 12), ( 13), ( 16) and ( 18) and using the previously stated assumptions that o’ B Ddu, /ax2 and u’ b Dav’/ax2, eqn ( 14) can be integrated to yield q2 =
-[8(1 + c)(l + 4~) + 32p2( 1 + E)‘]
where ps = density of the solid material. Since u’ will be affected by material properties of the granular solids and the local flow conditions, eqn (19) shows that the flux of fluctuation energy does not follow a simple heat conduction law. The identification of the flux term q2 will now enable eqns (I), (2) and (3) to be solved for steady, uniform flows of rapidly sheared granular solids. CONCLUSIONS
The flux of random fluctuation energy in a rapidly sheared granular mixture is formulated through a binary collision model. The local conductivity can be considered to be equivalent to a type of “heat” conductivity since the energy is in the form of the turbulent or random fluctuations of the granular particles. This conductivity is not, however, a constant but a rather complicated function of the size, concentration, material properties, and the local fluctuation energy of the granular solids. This energy diffusion term q, as defined in eqn (19), in conjunction with the energy dissipation functions T and y previously reported by Shen and Ackermann [l], quantify all of the constitutive relationships in the mass, momentum, and energy equations that are needed to describe the uniform flow of a granular continuum of circular disks. Acknowledgement-This
study is partially supported by National science Foundation Grant Number CEE82 16665. REFERENCES
[ 1] H. H. SHEN and N. L. ACKERMANN, Int. J. Engng Sci. 22,829 (1984). [2] S. OGAWA, A. UMEMURA and N. OSHIMA, ZAMP 31,482 (1980). [3] J. T. JENKINS and S. C. COWIN, ASME Symposium on Mechanics Applied to the Transport of Bulk Materials, Bufalo, pp. 19-89 (1979). [4] R. A. BAGNOLD, Proc. R. Sot. London Ser. A 225,49 (1954). [5] P. K. HAFF, J. Fluid Mech. 134,401 (1983). [6] J. T. JENKINS and S. B. SAVAGE, J. Fluid Mech. 130, 187 (1983). [7] C. K. K. LUN, S. B. SAVAGE, D. J. JEFFREY and N. CHAPURNIY, J. Fluid Me&. 140,223 (1984). IS]_ D. F. McTIGUE, Proc. U.S.-Japan Seminar on Continuum Mechanical and StatisticalApproaches in the Mechanics of Granular Materials, p. 266, Sendai, Japan (1978). 191 S. B. SAVAGE and D. J. JEFFREY. J. FluidMech. 110. 255 (1981). iOj H. SHEN and N. L. ACKERMANN, ASCE Engng Me&. Div: 108,748 (1982). I I] C. S. CAMPBELL and C. E. BRENNEN, Proc. U.S.-Japan Seminar on New Models and Constitutive Relations in the Mechanics aGranular Materials, Cornell University, New York (1982). (Received 11 February 1985)
ENERGY
0020-7225186 $3.00 + .CO Q 1986 pcrSamon Pres Ltd.
1986
DIFFUSION IN A GRANULAR OF DISK SHAPED SOLIDS
FLOW
HAYLEY H. SHEN and NORBERT L. ACKBRMANN Clarkson University, Potsdam, NY 13676, U.S.A. (Communicated by A. J. M. SPENCER) Abstract-The kinetic energy of a granular flow consists in part of the energy associated with the random velocity fluctuations u’ of the individual solid particles. When the magnitude of u’ varies throughout the flow field a transfer of kinetic energy pff/2 occurs that must be considered when evaluating the energy balance relationship. In the present investigation this energy transfer or dil’htsion is quantified for the flow of a granular continuum consisting of disk-shaped solids. The theoretical development involves a detailed analysis of the binary collision mechanics which results from the interaction between adjacent disks within the granular mixture. The energy diffusion term developedin this investigation, in conjunction with the stress and energy dissipation functions previously reported by Shen and Ackermann [ 11, provide all of the constitutive relationships that arc necessary to describe the two-dimensional uniform flow of a granular mixture of disk-shaped solids. INTRODUCTION THE
BEHAVIOR
of a granular continuum is described by the solution of the energy, momentum, and mass balance relationships. The balance equations for mass and momentum as presented in eqns (1) and (2) involve only the mean flow velocity u as an explicit kinematic variable
(1) a(PUi) +u.a(pui)=T,
at
’
dXj
Ji.J
.+pb.
1’
(2)
The terms p, T and b refer to the bulk density of the granular mixture, the stresses, and body force, respectively. The energy equation as presented by Ogawa, Umemura and Oshima [2] requires, however, that the random velocity fluctuation u’ of the granular solids also be considered explicitly as a kinematic variable. As first suggested by Jenkins and Cowin [3], the existence of a nonuniform o’ is the means by which energy can be conducted from one part of the granular continuum to another. In eqn (3) the term q describes this energy diffusion process D(Pv’~)/~
Dt
= rijuj,i - qj,j - 73
while the term y represents an energy sink. This sink term equals the rate at which mechanical energy is dissipated through inelastic and frictional collisions between the granular solids. Equations (I) through (3) can be solved for appropriately defined initial and boundary equations if constitutive relationships can be found to relate T, q, and y to p, u, o’ and the material properties of the granular solids. Bagnold [4], Haff [5], Jenkins and Savage [6], Jenkins and Cowin [3], Lun, Savage, Jeffrey, and Chepurniy [7], McTigue [S], Ogawa, Umemura, and Oshima [2], Savage and Jeffrey [9] and Shen and Ackermann [ 1, lo] among others have investigated the formulation of such constitutive relationships. Shen and Ackermann [l] derived the constitutive relationships for the stress r and energy dissipation y in a simple shear flow of disk-shaped solids. The theoretically determined stress state was found to compare favorably with the computer simulated
552
H. H. SHEN and N. L. ACKERMANN
results of Campbell and Brennen [ 1 I]. Such a favorable comparison appears to confirm the validity of the modeling procedure adopted by Shen and Ackermann. This model is now to be extended to describe the diffusion process indicated by the term 4 in eqn (3). In the analyses by Shen and Ackermann [ 1,lo] the fluctuation speed 2)’was considered to be constant through the simple shear field. The results of their analysis showed that u’ is proportional to the gradient Ui,j of the mean velocity. Hence, one may conclude that in a how field where the gradient of the mean velocity is not constant, there would also be a corresponding gradient in the fluctuation velocity energy pu”/2. The existence of such a gradient of the fluctuation energy implies that when two particles, such as A and B in Fig. 1, collide with different fluctuation speeds, energy is not only dissipated, but also transferred. This energy transfer phenomenon can be easily illustrated by considering a system consisting of two identical perfectly elastic spheres. One sphere is initially stationary while the other is moving towards and colliding headon with the stationary one. After the impact has occurred, the originally stationary sphere acquires the speed of the originally moving sphere while the originally moving sphere will come to a complete rest. Energy is thus completely transferred. By quantifying this energy transfer term q within a granular flow, eqns (1)through (3) can then be used to describe complex flows in which the gradient of the mean velocity can vary throughout the flow field.
MODEL
FORMULATION
Shen and Ackermann [ 1, lo] developed the constitutive relationships for T and y following the framework for analysis first used by Bagnold [4] to describe the stress state in a rapidly sheared granular mixture. This same approach will be adopted to determine the diffusion term q in eqn (3). Consider at first the continuum formed from a single layer of moving disks shown in Fig. 1. The stress TV, acting in the Xj direction on the surface normal to xi, was described by Bagnold [4] as TO
where pi is the number of direction, f is the average AMj is the jth component stress generating collisions
=
-pif AMj,
particles per unit area on the surface whose normal is in the Xi frequency of the stress generating collisions per particle and of the average momentum transfer per collision. In Fig. 1 the along the surface where x2 is constant are those collisions that
Isotropic Distribution of fluctuation velocity-,
Fig.
I. Shear flow of disk shaped solids.
553
Energy diffusion in a granular flow of disk shaped solids
occur between the shaded particles such as A and the particles, such as B, which are located on the surface where x2 = constant. The same formulation is used for eqn (5) to describe the energy flux qi across a surface whose normal is in the xi direction -qi = PifET. (5) In eqn (5) ET is the average energy transferred per collision. The transfer occurs between an exterior particle, such as particle A, and one of the particles such as B which temporarily resides on the surface through which energy is being diffused. To quantify first the energy ET transferred per single collision, consider the magnified view of particles A and B which are about to collide as shown in Fig. 2. Let EB be the kinetic energy of particle B as viewed in the coordinate system moving with the mean motion r& and EB be the kinetic energy of particle B after collision. The energy change in B due to such a collision is Eg - Ee: ET = E$ - EB,
(6)
where ET is the energy transferred from A to B. Expressing eqn (6) in terms of the particle mass m and velocity VB of particle B,
ET=:wg*vg- VB’VB],
(7)
L
where the symbol * represents post-collision conditions. It is now considered that the inelastic and frictional effects of particle collision are described solely by the coefficient of restitution 6 and friction coefficient p. The velocity Vg of disk B aBer collision with A therefore becomes
N * (VA- VB )(N + rp),
(8)
where N is the unit vector joining the center of the two disks as shown in Fig. 2. The collision frequency f between two particles is defined as the relative approach velocity of colliding particles divided by their average separation distance S. Hence, the collision frequency for typical particles such as A and B can be expressed as f =
(VA-
VB)*N
s
(9)
.
Disks can only collide, however, if their separation distance decreases. The criterion governing this condition can be expressed quantitatively as VA-N > VB*N.
(10)
By summing over all possible collisions, eqn (5) can then be modified as
9i =
C VA.N>VB-N
Cpiy(Vg.Vg
-VB*VB)I(VATVB)*N~
,
(11)
N
where the summation over N refers to the possible collision contact points in the range described in Fig. 2. from 0 < 8 < ?r.
COMPUTATIONAL
DETAILS
AND
RESULTS
Consider the steady uniform rectilinear shear flow shown in Fig. 1 in which the mean velocity is in the x1 coordinate direction. Using the geometric representation of disks A
H. H. SHEN and N. L. ACKERMANN
554
Fig. 2. Disks at point of collision. Velocity reference frame moving with mean motion us of particle B.
and B shown in Fig. 2, the velocities VA and VB relative to the mean motion of particle B can be expressed as VA =
u’ + 5 2
(D + s) sin 6 (cos aN + sin aP) + 2 I
(D + s) sin 0X1
(12)
2
VB = -u’(cos /3N + sin BP),
(13)
where Xi is a unit vector in the Xi coordinate direction and P is the unit vector normal to N. Shen and Ackermann [I] considered the effect of assuming that o’ % D&i /ax,. They showed that this assumption had a minor effect upon the accuracy of the calculated stresses for a wide range of realistic flow conditions. It is now assumed that both u’ B Dau, /ax2 as well as u’ B Dav’/ax2. Satisfying eqn (10) then reduces to the condition that for
-?r I (YI *.
(14)
Equation (11) can then be rewritten as
The range of integration for 0 is determined by 0r < 19< d1 + ?r where 0, = 0 for diffusion across the surface normal to the x2 direction and 8i = a/2 for diffusion across the surface normal to the xl direction. The number of particles pi are now described using the definition of linear concentration X introduced by Bagnold [4]: X=
(16)
D/s.
Letting C, = 0.907 represent the area concentration packing 2
C’C”& (
1
.
C of the disks at their densest
(17)
Energy diffusion in a granular flow of disk shaped solids
555
Shen and Ackermann [l] have shown that for a randomly arranged pattern of disks, p can be described as the following, with disk thickness being d
(18) By substituting eqns ( 12), ( 13), ( 16) and ( 18) and using the previously stated assumptions that o’ B Ddu, /ax2 and u’ b Dav’/ax2, eqn ( 14) can be integrated to yield q2 =
-[8(1 + c)(l + 4~) + 32p2( 1 + E)‘]
where ps = density of the solid material. Since u’ will be affected by material properties of the granular solids and the local flow conditions, eqn (19) shows that the flux of fluctuation energy does not follow a simple heat conduction law. The identification of the flux term q2 will now enable eqns (I), (2) and (3) to be solved for steady, uniform flows of rapidly sheared granular solids. CONCLUSIONS
The flux of random fluctuation energy in a rapidly sheared granular mixture is formulated through a binary collision model. The local conductivity can be considered to be equivalent to a type of “heat” conductivity since the energy is in the form of the turbulent or random fluctuations of the granular particles. This conductivity is not, however, a constant but a rather complicated function of the size, concentration, material properties, and the local fluctuation energy of the granular solids. This energy diffusion term q, as defined in eqn (19), in conjunction with the energy dissipation functions T and y previously reported by Shen and Ackermann [l], quantify all of the constitutive relationships in the mass, momentum, and energy equations that are needed to describe the uniform flow of a granular continuum of circular disks. Acknowledgement-This
study is partially supported by National science Foundation Grant Number CEE82 16665. REFERENCES
[ 1] H. H. SHEN and N. L. ACKERMANN, Int. J. Engng Sci. 22,829 (1984). [2] S. OGAWA, A. UMEMURA and N. OSHIMA, ZAMP 31,482 (1980). [3] J. T. JENKINS and S. C. COWIN, ASME Symposium on Mechanics Applied to the Transport of Bulk Materials, Bufalo, pp. 19-89 (1979). [4] R. A. BAGNOLD, Proc. R. Sot. London Ser. A 225,49 (1954). [5] P. K. HAFF, J. Fluid Mech. 134,401 (1983). [6] J. T. JENKINS and S. B. SAVAGE, J. Fluid Mech. 130, 187 (1983). [7] C. K. K. LUN, S. B. SAVAGE, D. J. JEFFREY and N. CHAPURNIY, J. Fluid Me&. 140,223 (1984). IS]_ D. F. McTIGUE, Proc. U.S.-Japan Seminar on Continuum Mechanical and StatisticalApproaches in the Mechanics of Granular Materials, p. 266, Sendai, Japan (1978). 191 S. B. SAVAGE and D. J. JEFFREY. J. FluidMech. 110. 255 (1981). iOj H. SHEN and N. L. ACKERMANN, ASCE Engng Me&. Div: 108,748 (1982). I I] C. S. CAMPBELL and C. E. BRENNEN, Proc. U.S.-Japan Seminar on New Models and Constitutive Relations in the Mechanics aGranular Materials, Cornell University, New York (1982). (Received 11 February 1985)