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Bottom wall friction coefficients on the dynamic properties of sheared granular flows
Powder Technology 270 (2015) 348–357
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Bottom wall friction coefficients on the dynamic properties of sheared granular flows Chun-Chung Liao, Shu-San Hsiau ⁎, Pei-Sian Chang Department of Mechanical Engineering, National Central University, Jhongli 32001, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 17 February 2014 Received in revised form 23 October 2014 Accepted 26 October 2014 Available online 31 October 2014 Keywords: Sheared granular flows Bottom wall friction coefficient Tangential velocity Granular temperature
a b s t r a c t In this study, we used a two-dimensional annular shear cell to systematically investigate the dynamic properties of granular flow when it is subjected to varying bottom wall friction coefficients. A particle tracking method and image processing technology were employed to measure tangential velocity, slip velocity, local solid fraction, and granular temperature. The results demonstrated that the bottom wall friction coefficient played a crucial role in determining the dynamic properties of sheared granular flows, indicating that slip velocity is larger when a rougher bottom wall is applied. The results also indicated that the tangential velocity and granular temperature were reduced when the roughness of the bottom wall increased because of the strong frictional effect, which caused a larger dissipation of energy. The average granular temperature increased linearly when the solid fraction at each specific bottom wall friction coefficient increased. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Granular materials are found in many industrial processes, such as in coal transportation, pharmaceutical manufacturing, food storage and transport, polymer production, pyrolysis of biomass, and metallurgical engineering, as well as in daily life, such as in sand, salt, sugar, and beans. The handling and processing of granular materials are of economic importance in many industries. Furthermore, to predict and prevent disasters caused by uncontrolled debris flows, avalanches, and landslides, it is crucial to understand the dynamic properties and rheology of these phenomena. However, the present understanding of the behavior of granular flows remains inadequate and a deeper study is necessary. Granular materials do not flow homogeneously like a fluid because the external driven force does not exceed a critical value and energy dissipation resulting from the occurrence of inelastic collisions, as well as from friction between particles and between particles and walls. Therefore, a solid-like region and a liquid-like region (shear band region) can coexist in the same flow system. The thickness of a shear band is approximately four to ten particle diameters, which depends on the external driving conditions, solid fraction, and interstitial fluid viscosity [1–5]. The interactive collisions that occur between particles cause the random motions of these particles, which become the dominant mechanism influencing flow behavior in granular materials [6,7]. Random particle motion in granular flows is analogous to thermal molecular motion. The concept of granular temperature was first proposed by Ogawa [8] to quantify the mean-square value of fluctuation velocities. Granular ⁎ Corresponding author at: No. 300, Jhongda Rd., Jhongli City, Taoyuan County 32001, Taiwan. Tel.: +886 3 426 7341; fax: +886 3 425 4501. E-mail address: [email protected] (S.-S. Hsiau).
http://dx.doi.org/10.1016/j.powtec.2014.10.043 0032-5910/© 2014 Elsevier B.V. All rights reserved.
temperature is defined as the specific fluctuation in the kinetic energy of particles that are present in granular flows, and it plays a role in granular flows that is similar to that of thermodynamic temperature in a gas. Granular materials behave more like a liquid or a gas when they have a higher granular temperature. In the past few decades, shear cells have been widely used for investigating the dynamic properties and rheology of granular materials because they exhibit a relatively simple flow field, which makes them suitable for fundamental research [9–24]. According to the Reynolds shear dilatancy phenomenon, the packing structure of granular matter becomes diluted as a shear force is applied [25]. Mahmood et al. [17] investigated the micromechanics of granular flows by using a twodimensional planar granular Couette flow. They determined that fluctuation velocity and granular temperature are related to the effective shear rate. They also indicated that the distribution of collision angles is anisotropic. Koval et al. [18] studied sheared granular flow by using a discrete element simulation in which an effective wall velocity was used to generate an inertia regime (shear band) near the rotating inner wall, whereas away from the wall a quasi-static regime prevailed, in which the granular material was in a solid or near-solid phase and particle motions were correspondingly slow and weak. The wall friction effect exerts a significant influence on dynamic properties and flow behaviors in granular flows. Hsiau and Yang [15] indicated that the rougher the condition of a wall, the greater the stress that is induced, and the higher the shear rate in sheared granular flows. Jasti and Higgs [21] experimentally studied granular flows in an annular shear cell and observed that particle velocity and granular temperature increased with increasing shearing wall roughness. Marinack et al. [26] used different shearing wall surface materials to produce different coefficients of restitution between the granular and shearing
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Bottom wall Fig. 1. Schematic drawing of (a) the shear cell experimental apparatus; (b) side view of the shear cell.
surface materials with the same shearing wall roughness and found that the velocity and granular temperature increased with the increase of restitution coefficient. Hsiau et al. [19] observed that convection and segregation rates increased with an increasing side wall friction coefficient in a vertical vibration bed. Kose et al. [22] experimentally investigated the rheology of particle–liquid mixtures by using both smooth and rough wall surfaces in sheared granular flows. They observed that the effective mixture viscosity is larger in cases involving rougher wall surfaces than in cases involving smoother wall surfaces. They demonstrated that wall slip substantially affects the apparent viscosity. In the past, the effect of driving wall roughness on dynamic properties had received a lot of attention. The influence of bottom wall roughness on the dynamic properties in sheared granular flows has not been previously examined. Additionally, the bottom surface is usually uneven in most industrial and natural granular systems. Hence, it is important to study the effect of bottom wall friction coefficient on dynamic
Table 1 Parameters used in the current experiments. Bottom wall friction coefficient (μw) 0.384 (#220 (68 0.378 (#240 (61 0.354 (#280 (51 0.316 (#320 (45 0.300 (#400 (38
μm) sandpaper) μm) sandpaper) μm) sandpaper) μm) sandpaper) μm) sandpaper)
0.384 (#220 (68 0.378 (#240 (61 0.354 (#280 (51 0.316 (#320 (45 0.300 (#400 (38
μm) sandpaper) μm) sandpaper) μm) sandpaper) μm) sandpaper) μm) sandpaper)
Inner wall velocity Ui (m/s)
Area solid fraction (v)
0.23 0.34 0.45 0.57 0.68 1.02 1.02
0.82
m/s m/s m/s m/s m/s m/s m/s
0.78 0.80 0.82 0.84
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properties in sheared granular flows. In the present study, the effects of various bottom wall friction coefficients on the dynamic properties of sheared granular materials were investigated. The tangential velocities, slip velocities, local solid fractions, and granular temperatures were determined and discussed. 2. Experimental setup A schematic drawing of a two-dimensional annular shear cell is shown in Fig. 1(a). The inner driving wall has a radius, ri, of 105 mm, and the distance from the center of the inner wheel to the rim of the
outer driving wheel, ro, is 150 mm (Fig. 1(b)). In this study, the radial position was normalized as (r–ri)/(ro–ri) from 0 (inner wall boundary) to 1 (outer wall boundary), with r being the distance from the center of the inner wheel. The inner and outer walls were driven independently by two server motors. The width of the annular trough between the inner wheel and the rim of the outer wheel was 45 mm. The granular materials were placed on the annular trough for testing. The rotational speeds of the inner and outer walls were measured individually by using a tachometer, and the velocities, Ui for the inner wheel and U0 for the outer wheel, were calculated as a product of the rotational speed and radius of the wall. In this study, the inner wall velocity was
Fig. 2. Distributions of tangential velocities with radial position for different wall velocities, (a) μw = 0.384; (b) μw = 0.378; (c) μw = 0.354; (d) μw = 0.316; (e) μw = 0.300 at a specific area solid fraction v = 0.82.
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varied and the outer wall was fixed in a stationary position to investigate the effect of the bottom wall friction coefficient on the dynamic properties of sheared granular flows. The top surface of the apparatus was fabricated using transparent glass to allow visual observation and reduce electrostatic buildup. Five different bottom wall surface conditions were used in this study: #220 sandpaper, #240 sandpaper, #280 sandpaper, #320 sandpaper, and #400 sandpaper were glued to the bottom wall surface. The friction coefficients of 3.0 mm glass beads and the five bottom wall surface conditions were measured using a commercial Jenike shear tester with direct shear model: 0.384 (#220 sandpaper), 0.378 (#240 sandpaper), 0.354 (#280 sandpaper), 0.316 (#320 sandpaper), and 0.300 (#400 sandpaper). By altering the rotational speed of the inner wall and the area solid fractions accordingly, the effect of the bottom wall friction coefficient on the dynamic
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properties of sheared granular flows was investigated. The detailed experimental parameters are listed in Table 1. Mono-sized glass beads with a diameter of 3 mm, a standard deviation of 0.09 mm, and a density of ρp = 2.476 g/cm3 were used as the granular materials in this study. In this study, the internal friction coefficient of 3 mm glass beads was also measured by a commercial Jenike shear tester with direct shear model and the value is 0.54. Additionally, the restitution coefficient of the glass bead used in this study, measured by the drop test is 0.90. The kinetic energy of particles was dissipated due to the inelastic collisions and friction effect. Only one layer of beads was placed in the container to enable the two-dimensional treatment of the granular system. The particles are in continuous contact with the bottom wall in this study. The average area solid fraction, ν, was defined as v = Ap / As, where Ap was the cross-sectional area of
Fig. 3. Distributions of tangential velocities with radial position for different area solid fractions at a specific inner wall velocity, Ui = 1.02 m/s, (a) μw = 0.384; (b) μw = 0.378; (c) μw = 0.354; (d) μw = 0.316; (e) μw = 0.300.
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the particles that occupied the shear cell, as can be seen from the top view, and As was the entire area of the shear cell base [5]. To generate sufficient shear in the flow field, a layer of 3-mm glass beads was glued to the surfaces of the inner and outer walls. A high-speed CCD camera (IDT X-3 plus) was fixed above the shear cell to record the motions of the beads, as shown in Fig. 1(a). A capture speed of 500 FPS and with a resolution of 1200 × 400 pixels was used, and all images were taken after the system had been shearing for at least 1 min to ensure a steady state of the flow field. The particle tracking velocimetry technique was used to calculate the velocity of the beads by locating their centers in the high-speed images and determining their displacements between two consecutive images [5,14,27]. The test section was radially divided into 10 regions. The local area solid fraction in each radial region can be determined as follows vi = Api / Asi, where Api was the cross-sectional area of the particles that occupied the radial region i and Asi was the area of the radial region i [5,21,26]. The ensemble averages of the tangential bVθN and radial bVrN velocities in each radial region were obtained from approximately 1200 particles, as follows: N X
V θ;k
k¼1
bV θ N ¼
N N X
bV r N ¼
;
ð1Þ
;
ð2Þ
V r;k
k¼1
N
where k represents the kth particle, N is the total number of velocities used for averaging the mean values, and Vθ,k and Vr,k are the velocities of the kth particle measured from two consecutive images containing the kth particle. The fluctuation velocities in the two directions were determined as follows:
02
1=2
bV θ N
02
1=2
bV r N
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N 2 uX u V θ; k −bV θ N u t ; ¼ k¼1 N−1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N 2 uX u V r; k −bV r N u t k¼1 : ¼ N−1
ð3Þ
ð4Þ
where Ui was the tangential velocity of the inner wall and Vθ,1 was the average tangential velocity of the bin adjacent to the boundary of the driving inner wall. It is noted that the particles of the bin adjacent to the boundary of the driving inner wall are in contact with the driving inner wall. In this study, each case was repeated at least three times to calculate the average tangential velocity and average granular temperature. 3. Results and discussion Fig. 2(a) shows the distributions of the tangential velocities across the normalized radial position, (r–ri)/(ro–ri), with r being the distance from the center of the inner wheel for different inner wall velocities, Ui, at a specific bottom wall friction coefficient, μw = 0.384, and solid fraction, v = 0.82. The velocity profile illustrates that these velocities decreased gradually from the shearing boundary to the stationary wall. The velocity gradient was larger near the driving wall because of the higher shear rate. Additionally, this velocity profile illustrates that the velocity gradient became larger with the higher wall velocity, which caused stronger particle motions. Interactive inelastic collisions occurred in areas that were closer to the driving wall boundary, and grew weaker the further they were from the driving wall. The particles that were close to the driving wall behaved like a fluid and constituted the region known as the shear band [3–5,10]. The behavior of the granular flow in the shear band was primarily influenced by the interactive inelastic collisions that were occurring when a dynamic state was achieved. When they were further away from the driving wall, the particles received less kinetic energy, and the dominant mechanism influencing granular flow behavior became the sliding and rolling contact that is characteristic of a quasi-static state. Thus, the dense granular packing and solid-like behavior of the granular material were observed close to the stationary wall; this area is known as the solid-like or frictional region [4,5,21]. Fig. 2(b)–(e) displays the distribution of tangential velocities across various radial positions, specifically regarding the various inner wall velocities that were acting at a specific area solid fraction of v = 0.82, with the varying bottom wall friction coefficients of μw = 0.378, 0.354, 0.316, and 0.300. As previously explained, the shear band, with its higher velocity gradient, was observed close to the driving inner wall in each case. From Fig. 2(a)–(e) the shear band thickness is similar with different inner wall velocities and bottom wall friction coefficients at the specific solid fraction. However, the greater bottom wall friction coefficient causes the smaller velocity gradient. Additionally, the velocity gradient was enhanced with an increase
The granular temperature, T, was used to quantify the kinetic energy of the granular flow and was calculated from the average of the mean square of the fluctuation velocities in both directions. The granular temperature in a quasi-two-dimensional system was calculated as T¼
bV θ
02
02
þ Vr N : 2
ð5Þ
In this study, only one layer of particles was used, therefore every particle was always in contact with the solid bottom wall of the container. When the granular materials were sheared to flow, the granular materials interacted continuously with the solid bottom wall of the container. Naturally, friction occurred between the container walls and particles, and this, as well as the inelastic collisions and internal friction between the particles themselves, meant that the kinetic energy of the particles was continuously dissipated. Therefore, external energy had to be constantly introduced into the granular system to maintain the granular temperature. The slip velocity at the driving wall was determined using the following equation based on previous studies [5,21,26]: S ¼ U i −V θ;1 ;
ð6Þ
Fig. 4. Average tangential velocity plotted as a function of the area solid fraction with different bottom wall friction coefficients μw, at the same inner wall velocity, Ui = 1.02 m/s. The error bars correspond to the standard deviation of at least three experimental tests in each case.
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of driving wall velocity in each specific bottom wall friction coefficient condition. The results also show that the tangential particle velocity considerably increased as the bottom wall friction coefficient decreased. The frictional effect is stronger as the larger bottom wall friction coefficient was applied. Higher dissipation of kinetic energy from particles would result in the smaller velocity gradient and tangential particle velocity. Fig. 3(a)–(e) shows the variations in tangential velocity across various radial positions for various area solid fractions, and at wall velocities of Ui = 1.02 m/s, U0 = 0, with varying bottom wall friction coefficients,
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which were (a) μw = 0.384; (b) μ w = 0.378; (c) μ w = 0.354; (d) μw = 0.316; and (e) μw = 0.300. The velocity profile was similar to that shown in Fig. 2. A shear band that exhibited strong particle motions and interactive collisions near the driving wall boundary was observed in each case. This figure also demonstrates that the tangential velocity was greater at a higher solid fraction. The frequency of the inelastic collisions that occurred between the particles and the driving wall boundary were substantially higher in the systems that contained higher solid fractions. As a result of these contacts between the particles and the driving wall, the particles obtained kinetic energy from the
Fig. 5. Distributions of local solid fraction with radial position for different area solid fractions at a specific inner wall velocity, Ui = 1.02 m/s, (a) μw = 0.384; (b) μw = 0.378; (c) μw = 0.354; (d) μw = 0.316; (e) μw = 0.300.
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driving wall, which led to their motions being strengthened and the flow field becoming more active, which led to a greater tangential velocity. Again, Fig. 3(a)–(e) shows that the shear band thickness is similar with different solid fractions and bottom wall friction coefficients at the specific inner wall velocity. The smaller bottom wall friction coefficient causes the greater velocity gradient. Fig. 4 displays the average tangential velocity, which was plotted on a graph as a function of the area solid fraction with various bottom wall friction coefficients at a specific inner wall velocity (Ui = 1.02 m/s, U0 = 0). The average tangential velocity was greater at the smaller bottom wall friction coefficient, and the variation of the average tangential velocity of the various bottom wall frictions increased with the area solid fraction. The frictional effect between the particles and the bottom wall was enhanced when the larger bottom wall friction coefficient was applied. Consequently, a larger amount of the kinetic energy of the particles was dissipated because of the serious frictional sliding contacts that occurred between the particles and the bottom wall, which caused a smaller velocity gradient and average tangential velocity. Regarding the larger solid fraction, the granular packing was denser, which caused a higher collision frequency between the particles and the driving wall. Therefore, the particles had larger kinetic energy and leaded to the serious frictional interactions between the particles and the bottom wall surface. Consequently, the variance of the average tangential velocity at various bottom wall friction coefficients became larger with the increase of the solid fraction. This phenomenon has not been discussed in previous studies. The result indicated that the average tangential velocity increased in conjunction with the area solid fraction at a specific bottom wall friction coefficient. The result is in agreement with the previous study [13]. Accordingly, it could be inferred that a greater solid fraction caused a higher inelastic collision frequency between the particles and the driving wall at the same wall velocity [5]. As mentioned, the particles acquired kinetic energy from the driving wall, which strengthened their motions and enhanced the fluidization of the flow field, and caused a higher average tangential velocity. The results can be explained from the distribution of the local solid fraction (as shown in Fig. 5) at wall velocities of Ui = 1.02 m/s, U0 = 0 with: (a) μw = 0.384; (b) μw = 0.378; (c) μw = 0.354; (d) μw = 0.316; and (e) μw = 0.300. These graphs show the distributions of local solid fractions with the radial position for various area solid fractions at a constant wall velocity. The local solid fraction was small when it was close to the driving wall because of the shear dilation effect. However, it grew larger with increased distance from the driving wall because the lower shear rates caused weaker particle motions. The local solid fraction became smaller as a smaller solid fraction was applied to the system, consistent with the findings of Liao et al. [5] and Jasti and Higgs [21]. Additionally, the local solid fraction in the shear band was larger when the solid fraction was increased in each bottom wall friction coefficient case. Thus, the average tangential velocity was larger when there was an increase of the higher local solid fraction, which caused a faster frequency of interactive collisions, and a larger momentum transfer between the particles and the driving wall boundary, as shown in Fig. 4. Because of the existence of a slip velocity between the driving wall boundary and the particles near the driving wall boundary, the external driving energy was not fully introduced into the granular system. It was crucial to determine the slip velocity to quantify the external energy that was introduced into the granular system. Fig. 6 shows the slip velocity plotted as a function of the inner wall velocity, using varying bottom wall friction coefficients at a specific area solid fraction. The result indicated that the slip velocity increased linearly in conjunction with the inner wall velocity. These results were in good agreement with those obtained in previous studies [5,21]. The slip velocity was shown to be greater at a greater bottom wall friction coefficient at any given inner wall velocity. A larger amount of the kinetic energy of the particles was dissipated because of the stronger frictional effect that occurred between the particles and the bottom wall, which caused a larger slip velocity when there was a greater bottom wall friction coefficient present.
Granular temperature is critical to the study of the flow behaviors and dynamic properties of granular materials [5–8,15,21]. Fig. 7(a)–(e) shows the distributions of the granular temperature across the radial position at various driving inner wall velocities with varying bottom wall friction coefficients at v = 0.82. The local granular temperature could be regarded as the local specific kinetic energy in this study. It was observed that granular temperature was increased when the driving wall velocity was higher at each bottom wall friction coefficient. A larger amount of energy was introduced to the granular system at the higher driving wall velocity, which led to a more active fluid field, stronger particle fluctuations, and increased granular temperature. The granular temperature was also observed to be higher close to the driving inner wall (in the shear band region). Kinetic energy was transmitted mainly through interactive inelastic collisions that occurred between particles that comprised the granular materials. The strength of the motions of the particles and interactive collisions was enhanced, causing additional particle fluctuations caused by the higher velocity gradient in the shear band region. Thus, the granular temperature was greater closer to the driving wall boundary. This was consistent with previous studies that indicated that granular temperature is produced by the velocity gradient [5,7]. As the distance from the driving wall increased, the lower shear rate (smaller velocity gradient) caused weaker fluctuations of the particles (in the solid-like region), leading to a decrease in the granular temperature. The results were in agreement with those of previous studies [5,21]. Marinack et al. [26] reported that the normalized translational kinetic energy increased with the increase of global solid fraction. However, they didn't investigate the effect of bottom wall friction coefficient on granular temperature (kinetic energy). It is important to study the effect of bottom wall friction coefficient on dynamic properties in sheared granular flows because the bottom surface is usually uneven in most industrial and natural granular systems. In this study, we focus on the effect of bottom wall friction coefficients on the dynamic properties in sheared granular flows. To quantify the influence of the bottom wall friction effect on the granular temperature in sheared granular flows, the average granular temperatures, which were averaged from the local granular temperatures in the 10 regions, were plotted as a function of the solid fraction for various bottom wall friction coefficients at a specific inner wall velocity, as shown in Fig. 8. The average granular temperature decreased when the bottom wall friction coefficient increased in each specific area solid fraction. A larger amount of kinetic energy from the particles was dissipated and caused the smaller average granular temperature because of the stronger frictional effect that occurred when there was a larger bottom wall friction coefficient. The results are not similar to most previous studies, where the granular
Fig. 6. Slip velocity plotted as a function of inner wall velocity with different bottom wall friction coefficients. The lines indicate the linear fits to the data.
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temperature was strengthened with the greater shearing wall friction coefficient [15,21]. Based on the experimental results, the influence of wall friction on the dynamic properties of granular flows is not the
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same but depends on the granular system. As shown in Fig. 8, the average granular temperature increased linearly with the solid fraction in each specific bottom wall friction coefficient increased. The slopes
Fig. 7. Distributions of granular temperature with the radial position for different wall velocities, (a) μw = 0.384; (b) μw = 0.378; (c) μw = 0.354; (d) μw = 0.316; (e) μw = 0.300, at a specific area solid fraction v = 0.82.
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4. Conclusion
Fig. 8. Average granular temperature plotted as a function of the area solid fraction with different bottom wall friction coefficients μw, at a specific inner wall velocity, Ui = 1.02 m/s. The lines indicate the linear fits to the data. The error bars correspond to the standard deviation of at least three experimental tests in each case.
were 4.94 × 10− 2, 2.93 × 10− 2, 2.03 × 10− 2, 1.62 × 10− 2, and 1.72 × 10− 2, which correspond to the cases where μw = 0.300, 0.316, 0.354, 0.378, and 0.384, respectively. Liao et al. [5] found that velocity gradient is the predominant parameter influencing the granular temperature. From Fig. 3(a)–(e), the shear band thickness is similar with different solid fractions at the specific inner wall velocity and bottom wall friction coefficient. However, the velocity gradient is larger with the higher solid fraction. The larger velocity gradient leads to the stronger particle fluctuations and the greater average granular temperature. Therefore, the average granular temperature became larger when the higher the solid fraction was applied. Fig. 8 indicates that the variation of the average granular temperature at various bottom wall friction coefficients became larger as a higher solid fraction was applied. The frictional interactions between the particles and the bottom wall were strengthened with the increase of solid fraction. Thus, the variation between average granular temperatures with different bottom wall friction coefficients increased with the increase of the solid fraction, as shown in Fig. 8. Fig. 9 shows average granular temperature plotted as average tangential velocity with the same data as shown in Figs. 4 and 8. The data are collapse and shows that average granular temperature is enhanced with the increasing average tangential velocity.
Average granular temperature (m2/s2)
0.01 0.009 0.008 0.007 0.006
µw = µw = µw = µw = µw =
0.005
0.384 0.378 0.354 0.316 0.300
0.004 0.003 0.005
0.01
0.015
0.02
0.025
Average tangential velocity (m/s)
0.03
Fig. 9. Average granular temperature plotted as average tangential velocity with the same data as shown in Figs. 4 and 8.
This study investigated the effects of bottom wall friction coefficients on the dynamic properties of two-dimensional sheared granular flows. The experiments were conducted at a two-dimensional annular shear cell. The particle tracking velocimetry technique and image processing technology were used to successfully determine the tangential velocity, slip velocity, granular temperature, and local solid fraction. The results revealed that the bottom wall friction coefficient exerted a critical influence on the dynamic properties of sheared granular flows, a phenomenon that has not previously been reported. The dynamic properties were mitigated with the rough bottom wall. Additionally, the results showed that the slip velocity increased when a greater bottom wall friction coefficient was applied. The average granular temperature increased linearly with the solid fraction and decreased when the bottom wall friction coefficient increased because of the strong frictional effect, which caused the smaller velocity gradient and a large dissipation of energy. Furthermore, the variation in the average granular temperature when different bottom wall friction coefficients were applied became larger with the higher solid fraction because of the severe frictional effect between the particles and the bottom wall. This study demonstrated that the bottom wall roughness impacts the flow differently than shearing wall roughness and decreased the dynamic properties within the sheared granular system. A further study with different systems is required to clarify this wall friction effect. Acknowledgments The authors would like to acknowledge the support of the National Science Council Taiwan for this work through grant NSC 100-2221-E008-078-MY3. References [1] H.M. Jaeger, S.R. Nagel, R.P. Behringer, Granular solids, liquids and gases, Rev. Mod. Phys. 68 (1996) 1259–1273. [2] R.R. Hartley, R.P. Behringer, Logarithmic rate dependence of force networks in sheared granular materials, Nature (London) 421 (2003) 928. [3] C.C. Liao, S.S. Hsiau, Influence of interstitial fluid viscosity on transport phenomenon in sheared granular materials, Chem. Eng. Sci. 64 (2009) 2562–2569. [4] C.C. Liao, S.S. Hsiau, J.S. Li, C.H. Tai, The influence of gravity on dynamic properties in sheared granular flows, Chem. Eng. Sci. 65 (2010) 2531–2540. [5] C.C. Liao, S.S. Hsiau, W.J. Yu, The influence of driving conditions on flow behavior in sheared granular flows, Int. J. Multiphase Flow 46 (2012) 22–31. [6] C.S. Campbell, Rapid granular flows, Annu. Rev. Fluid Mech. 22 (1990) 57–92. [7] C.S. Campbell, Granular material flows—an overview, Powder Technol. 162 (2006) 208–229. [8] S. Ogawa, Multi-temperature theory of granular materials, Proceedings of US–Japan Seminar on Continuum-Mechanical and Statistical Approaches in the Mechanics of Granular Materials, Gakujutsu Sunken Fukyu-Kai Publishers, Tokyo, 1978, pp. 208–217. [9] S.B. Savage, M. Sayed, Stresses developed by dry cohesionless granular materials sheared in an annular shear cell, J. Fluid Mech. 142 (1984) 391–430. [10] D. Fenistein, J.W. van de Meent, M. van Hecke, Universal and wide shear zones in granular bulk flow, Phys. Rev. Lett. 92 (2004) 094301. [11] D.S. Grebenkov, M.P. Ciamarra, M. Nicodemi, A. Coniglio, Flow, ordering, and jamming of sheared granular suspensions, Phys. Rev. Lett. 100 (2008) 078001. [12] D. Howell, R.P. Behringer, Stress fluctuations in a 2D granular Couette experiment: a continuous transition, Phys. Rev. Lett. 82 (1998) 5241–5244. [13] C.T. Veje, D.W. Howell, R.P. Behringer, Kinematic of a two-dimensional granular Couette experiment at the transition to shearing, Phys. Rev. E. 59 (1999) 739–745. [14] S.S. Hsiau, Y.H. Shieh, Fluctuations and self-diffusion of sheared granular material flows, J. Rheol. 43 (1999) 1049–1066. [15] S.S. Hsiau, W.L. Yang, Stresses and transport phenomena in sheared granular flows, Phys. Fluids 14 (2002) 612–621. [16] P. Wang, C. Somg, C. Briscoe, H.A. Makse, Particle dynamics and effective temperature of jammed granular matter in a slowly sheared three-dimensional Couette cell, Phys. Rev. E. 77 (2008) 061309. [17] Z. Mahmood, S. Dhakal, K. Iwashita, Measurement of particle dynamics in rapid granular shear flows, J. Eng. Mech. ASCE 135 (2009) 285–294. [18] G. Koval, J. Roux, A. Corfdir, F. Chevoir, Annular shear of cohesionless granular materials: from the inertial to quasistatic regime, Phys. Rev. E. 59 (2009) 0213061–02130616. [19] S.S. Hsiau, P.C. Wang, C.H. Tai, Convection cells and segregation in a vibrated granular bed, AIChE J. 48 (2002) 1430–1438.
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Powder Technology journal homepage: www.elsevier.com/locate/powtec
Bottom wall friction coefficients on the dynamic properties of sheared granular flows Chun-Chung Liao, Shu-San Hsiau ⁎, Pei-Sian Chang Department of Mechanical Engineering, National Central University, Jhongli 32001, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 17 February 2014 Received in revised form 23 October 2014 Accepted 26 October 2014 Available online 31 October 2014 Keywords: Sheared granular flows Bottom wall friction coefficient Tangential velocity Granular temperature
a b s t r a c t In this study, we used a two-dimensional annular shear cell to systematically investigate the dynamic properties of granular flow when it is subjected to varying bottom wall friction coefficients. A particle tracking method and image processing technology were employed to measure tangential velocity, slip velocity, local solid fraction, and granular temperature. The results demonstrated that the bottom wall friction coefficient played a crucial role in determining the dynamic properties of sheared granular flows, indicating that slip velocity is larger when a rougher bottom wall is applied. The results also indicated that the tangential velocity and granular temperature were reduced when the roughness of the bottom wall increased because of the strong frictional effect, which caused a larger dissipation of energy. The average granular temperature increased linearly when the solid fraction at each specific bottom wall friction coefficient increased. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Granular materials are found in many industrial processes, such as in coal transportation, pharmaceutical manufacturing, food storage and transport, polymer production, pyrolysis of biomass, and metallurgical engineering, as well as in daily life, such as in sand, salt, sugar, and beans. The handling and processing of granular materials are of economic importance in many industries. Furthermore, to predict and prevent disasters caused by uncontrolled debris flows, avalanches, and landslides, it is crucial to understand the dynamic properties and rheology of these phenomena. However, the present understanding of the behavior of granular flows remains inadequate and a deeper study is necessary. Granular materials do not flow homogeneously like a fluid because the external driven force does not exceed a critical value and energy dissipation resulting from the occurrence of inelastic collisions, as well as from friction between particles and between particles and walls. Therefore, a solid-like region and a liquid-like region (shear band region) can coexist in the same flow system. The thickness of a shear band is approximately four to ten particle diameters, which depends on the external driving conditions, solid fraction, and interstitial fluid viscosity [1–5]. The interactive collisions that occur between particles cause the random motions of these particles, which become the dominant mechanism influencing flow behavior in granular materials [6,7]. Random particle motion in granular flows is analogous to thermal molecular motion. The concept of granular temperature was first proposed by Ogawa [8] to quantify the mean-square value of fluctuation velocities. Granular ⁎ Corresponding author at: No. 300, Jhongda Rd., Jhongli City, Taoyuan County 32001, Taiwan. Tel.: +886 3 426 7341; fax: +886 3 425 4501. E-mail address: [email protected] (S.-S. Hsiau).
http://dx.doi.org/10.1016/j.powtec.2014.10.043 0032-5910/© 2014 Elsevier B.V. All rights reserved.
temperature is defined as the specific fluctuation in the kinetic energy of particles that are present in granular flows, and it plays a role in granular flows that is similar to that of thermodynamic temperature in a gas. Granular materials behave more like a liquid or a gas when they have a higher granular temperature. In the past few decades, shear cells have been widely used for investigating the dynamic properties and rheology of granular materials because they exhibit a relatively simple flow field, which makes them suitable for fundamental research [9–24]. According to the Reynolds shear dilatancy phenomenon, the packing structure of granular matter becomes diluted as a shear force is applied [25]. Mahmood et al. [17] investigated the micromechanics of granular flows by using a twodimensional planar granular Couette flow. They determined that fluctuation velocity and granular temperature are related to the effective shear rate. They also indicated that the distribution of collision angles is anisotropic. Koval et al. [18] studied sheared granular flow by using a discrete element simulation in which an effective wall velocity was used to generate an inertia regime (shear band) near the rotating inner wall, whereas away from the wall a quasi-static regime prevailed, in which the granular material was in a solid or near-solid phase and particle motions were correspondingly slow and weak. The wall friction effect exerts a significant influence on dynamic properties and flow behaviors in granular flows. Hsiau and Yang [15] indicated that the rougher the condition of a wall, the greater the stress that is induced, and the higher the shear rate in sheared granular flows. Jasti and Higgs [21] experimentally studied granular flows in an annular shear cell and observed that particle velocity and granular temperature increased with increasing shearing wall roughness. Marinack et al. [26] used different shearing wall surface materials to produce different coefficients of restitution between the granular and shearing
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Bottom wall Fig. 1. Schematic drawing of (a) the shear cell experimental apparatus; (b) side view of the shear cell.
surface materials with the same shearing wall roughness and found that the velocity and granular temperature increased with the increase of restitution coefficient. Hsiau et al. [19] observed that convection and segregation rates increased with an increasing side wall friction coefficient in a vertical vibration bed. Kose et al. [22] experimentally investigated the rheology of particle–liquid mixtures by using both smooth and rough wall surfaces in sheared granular flows. They observed that the effective mixture viscosity is larger in cases involving rougher wall surfaces than in cases involving smoother wall surfaces. They demonstrated that wall slip substantially affects the apparent viscosity. In the past, the effect of driving wall roughness on dynamic properties had received a lot of attention. The influence of bottom wall roughness on the dynamic properties in sheared granular flows has not been previously examined. Additionally, the bottom surface is usually uneven in most industrial and natural granular systems. Hence, it is important to study the effect of bottom wall friction coefficient on dynamic
Table 1 Parameters used in the current experiments. Bottom wall friction coefficient (μw) 0.384 (#220 (68 0.378 (#240 (61 0.354 (#280 (51 0.316 (#320 (45 0.300 (#400 (38
μm) sandpaper) μm) sandpaper) μm) sandpaper) μm) sandpaper) μm) sandpaper)
0.384 (#220 (68 0.378 (#240 (61 0.354 (#280 (51 0.316 (#320 (45 0.300 (#400 (38
μm) sandpaper) μm) sandpaper) μm) sandpaper) μm) sandpaper) μm) sandpaper)
Inner wall velocity Ui (m/s)
Area solid fraction (v)
0.23 0.34 0.45 0.57 0.68 1.02 1.02
0.82
m/s m/s m/s m/s m/s m/s m/s
0.78 0.80 0.82 0.84
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properties in sheared granular flows. In the present study, the effects of various bottom wall friction coefficients on the dynamic properties of sheared granular materials were investigated. The tangential velocities, slip velocities, local solid fractions, and granular temperatures were determined and discussed. 2. Experimental setup A schematic drawing of a two-dimensional annular shear cell is shown in Fig. 1(a). The inner driving wall has a radius, ri, of 105 mm, and the distance from the center of the inner wheel to the rim of the
outer driving wheel, ro, is 150 mm (Fig. 1(b)). In this study, the radial position was normalized as (r–ri)/(ro–ri) from 0 (inner wall boundary) to 1 (outer wall boundary), with r being the distance from the center of the inner wheel. The inner and outer walls were driven independently by two server motors. The width of the annular trough between the inner wheel and the rim of the outer wheel was 45 mm. The granular materials were placed on the annular trough for testing. The rotational speeds of the inner and outer walls were measured individually by using a tachometer, and the velocities, Ui for the inner wheel and U0 for the outer wheel, were calculated as a product of the rotational speed and radius of the wall. In this study, the inner wall velocity was
Fig. 2. Distributions of tangential velocities with radial position for different wall velocities, (a) μw = 0.384; (b) μw = 0.378; (c) μw = 0.354; (d) μw = 0.316; (e) μw = 0.300 at a specific area solid fraction v = 0.82.
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varied and the outer wall was fixed in a stationary position to investigate the effect of the bottom wall friction coefficient on the dynamic properties of sheared granular flows. The top surface of the apparatus was fabricated using transparent glass to allow visual observation and reduce electrostatic buildup. Five different bottom wall surface conditions were used in this study: #220 sandpaper, #240 sandpaper, #280 sandpaper, #320 sandpaper, and #400 sandpaper were glued to the bottom wall surface. The friction coefficients of 3.0 mm glass beads and the five bottom wall surface conditions were measured using a commercial Jenike shear tester with direct shear model: 0.384 (#220 sandpaper), 0.378 (#240 sandpaper), 0.354 (#280 sandpaper), 0.316 (#320 sandpaper), and 0.300 (#400 sandpaper). By altering the rotational speed of the inner wall and the area solid fractions accordingly, the effect of the bottom wall friction coefficient on the dynamic
351
properties of sheared granular flows was investigated. The detailed experimental parameters are listed in Table 1. Mono-sized glass beads with a diameter of 3 mm, a standard deviation of 0.09 mm, and a density of ρp = 2.476 g/cm3 were used as the granular materials in this study. In this study, the internal friction coefficient of 3 mm glass beads was also measured by a commercial Jenike shear tester with direct shear model and the value is 0.54. Additionally, the restitution coefficient of the glass bead used in this study, measured by the drop test is 0.90. The kinetic energy of particles was dissipated due to the inelastic collisions and friction effect. Only one layer of beads was placed in the container to enable the two-dimensional treatment of the granular system. The particles are in continuous contact with the bottom wall in this study. The average area solid fraction, ν, was defined as v = Ap / As, where Ap was the cross-sectional area of
Fig. 3. Distributions of tangential velocities with radial position for different area solid fractions at a specific inner wall velocity, Ui = 1.02 m/s, (a) μw = 0.384; (b) μw = 0.378; (c) μw = 0.354; (d) μw = 0.316; (e) μw = 0.300.
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the particles that occupied the shear cell, as can be seen from the top view, and As was the entire area of the shear cell base [5]. To generate sufficient shear in the flow field, a layer of 3-mm glass beads was glued to the surfaces of the inner and outer walls. A high-speed CCD camera (IDT X-3 plus) was fixed above the shear cell to record the motions of the beads, as shown in Fig. 1(a). A capture speed of 500 FPS and with a resolution of 1200 × 400 pixels was used, and all images were taken after the system had been shearing for at least 1 min to ensure a steady state of the flow field. The particle tracking velocimetry technique was used to calculate the velocity of the beads by locating their centers in the high-speed images and determining their displacements between two consecutive images [5,14,27]. The test section was radially divided into 10 regions. The local area solid fraction in each radial region can be determined as follows vi = Api / Asi, where Api was the cross-sectional area of the particles that occupied the radial region i and Asi was the area of the radial region i [5,21,26]. The ensemble averages of the tangential bVθN and radial bVrN velocities in each radial region were obtained from approximately 1200 particles, as follows: N X
V θ;k
k¼1
bV θ N ¼
N N X
bV r N ¼
;
ð1Þ
;
ð2Þ
V r;k
k¼1
N
where k represents the kth particle, N is the total number of velocities used for averaging the mean values, and Vθ,k and Vr,k are the velocities of the kth particle measured from two consecutive images containing the kth particle. The fluctuation velocities in the two directions were determined as follows:
02
1=2
bV θ N
02
1=2
bV r N
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N 2 uX u V θ; k −bV θ N u t ; ¼ k¼1 N−1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N 2 uX u V r; k −bV r N u t k¼1 : ¼ N−1
ð3Þ
ð4Þ
where Ui was the tangential velocity of the inner wall and Vθ,1 was the average tangential velocity of the bin adjacent to the boundary of the driving inner wall. It is noted that the particles of the bin adjacent to the boundary of the driving inner wall are in contact with the driving inner wall. In this study, each case was repeated at least three times to calculate the average tangential velocity and average granular temperature. 3. Results and discussion Fig. 2(a) shows the distributions of the tangential velocities across the normalized radial position, (r–ri)/(ro–ri), with r being the distance from the center of the inner wheel for different inner wall velocities, Ui, at a specific bottom wall friction coefficient, μw = 0.384, and solid fraction, v = 0.82. The velocity profile illustrates that these velocities decreased gradually from the shearing boundary to the stationary wall. The velocity gradient was larger near the driving wall because of the higher shear rate. Additionally, this velocity profile illustrates that the velocity gradient became larger with the higher wall velocity, which caused stronger particle motions. Interactive inelastic collisions occurred in areas that were closer to the driving wall boundary, and grew weaker the further they were from the driving wall. The particles that were close to the driving wall behaved like a fluid and constituted the region known as the shear band [3–5,10]. The behavior of the granular flow in the shear band was primarily influenced by the interactive inelastic collisions that were occurring when a dynamic state was achieved. When they were further away from the driving wall, the particles received less kinetic energy, and the dominant mechanism influencing granular flow behavior became the sliding and rolling contact that is characteristic of a quasi-static state. Thus, the dense granular packing and solid-like behavior of the granular material were observed close to the stationary wall; this area is known as the solid-like or frictional region [4,5,21]. Fig. 2(b)–(e) displays the distribution of tangential velocities across various radial positions, specifically regarding the various inner wall velocities that were acting at a specific area solid fraction of v = 0.82, with the varying bottom wall friction coefficients of μw = 0.378, 0.354, 0.316, and 0.300. As previously explained, the shear band, with its higher velocity gradient, was observed close to the driving inner wall in each case. From Fig. 2(a)–(e) the shear band thickness is similar with different inner wall velocities and bottom wall friction coefficients at the specific solid fraction. However, the greater bottom wall friction coefficient causes the smaller velocity gradient. Additionally, the velocity gradient was enhanced with an increase
The granular temperature, T, was used to quantify the kinetic energy of the granular flow and was calculated from the average of the mean square of the fluctuation velocities in both directions. The granular temperature in a quasi-two-dimensional system was calculated as T¼
bV θ
02
02
þ Vr N : 2
ð5Þ
In this study, only one layer of particles was used, therefore every particle was always in contact with the solid bottom wall of the container. When the granular materials were sheared to flow, the granular materials interacted continuously with the solid bottom wall of the container. Naturally, friction occurred between the container walls and particles, and this, as well as the inelastic collisions and internal friction between the particles themselves, meant that the kinetic energy of the particles was continuously dissipated. Therefore, external energy had to be constantly introduced into the granular system to maintain the granular temperature. The slip velocity at the driving wall was determined using the following equation based on previous studies [5,21,26]: S ¼ U i −V θ;1 ;
ð6Þ
Fig. 4. Average tangential velocity plotted as a function of the area solid fraction with different bottom wall friction coefficients μw, at the same inner wall velocity, Ui = 1.02 m/s. The error bars correspond to the standard deviation of at least three experimental tests in each case.
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of driving wall velocity in each specific bottom wall friction coefficient condition. The results also show that the tangential particle velocity considerably increased as the bottom wall friction coefficient decreased. The frictional effect is stronger as the larger bottom wall friction coefficient was applied. Higher dissipation of kinetic energy from particles would result in the smaller velocity gradient and tangential particle velocity. Fig. 3(a)–(e) shows the variations in tangential velocity across various radial positions for various area solid fractions, and at wall velocities of Ui = 1.02 m/s, U0 = 0, with varying bottom wall friction coefficients,
353
which were (a) μw = 0.384; (b) μ w = 0.378; (c) μ w = 0.354; (d) μw = 0.316; and (e) μw = 0.300. The velocity profile was similar to that shown in Fig. 2. A shear band that exhibited strong particle motions and interactive collisions near the driving wall boundary was observed in each case. This figure also demonstrates that the tangential velocity was greater at a higher solid fraction. The frequency of the inelastic collisions that occurred between the particles and the driving wall boundary were substantially higher in the systems that contained higher solid fractions. As a result of these contacts between the particles and the driving wall, the particles obtained kinetic energy from the
Fig. 5. Distributions of local solid fraction with radial position for different area solid fractions at a specific inner wall velocity, Ui = 1.02 m/s, (a) μw = 0.384; (b) μw = 0.378; (c) μw = 0.354; (d) μw = 0.316; (e) μw = 0.300.
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driving wall, which led to their motions being strengthened and the flow field becoming more active, which led to a greater tangential velocity. Again, Fig. 3(a)–(e) shows that the shear band thickness is similar with different solid fractions and bottom wall friction coefficients at the specific inner wall velocity. The smaller bottom wall friction coefficient causes the greater velocity gradient. Fig. 4 displays the average tangential velocity, which was plotted on a graph as a function of the area solid fraction with various bottom wall friction coefficients at a specific inner wall velocity (Ui = 1.02 m/s, U0 = 0). The average tangential velocity was greater at the smaller bottom wall friction coefficient, and the variation of the average tangential velocity of the various bottom wall frictions increased with the area solid fraction. The frictional effect between the particles and the bottom wall was enhanced when the larger bottom wall friction coefficient was applied. Consequently, a larger amount of the kinetic energy of the particles was dissipated because of the serious frictional sliding contacts that occurred between the particles and the bottom wall, which caused a smaller velocity gradient and average tangential velocity. Regarding the larger solid fraction, the granular packing was denser, which caused a higher collision frequency between the particles and the driving wall. Therefore, the particles had larger kinetic energy and leaded to the serious frictional interactions between the particles and the bottom wall surface. Consequently, the variance of the average tangential velocity at various bottom wall friction coefficients became larger with the increase of the solid fraction. This phenomenon has not been discussed in previous studies. The result indicated that the average tangential velocity increased in conjunction with the area solid fraction at a specific bottom wall friction coefficient. The result is in agreement with the previous study [13]. Accordingly, it could be inferred that a greater solid fraction caused a higher inelastic collision frequency between the particles and the driving wall at the same wall velocity [5]. As mentioned, the particles acquired kinetic energy from the driving wall, which strengthened their motions and enhanced the fluidization of the flow field, and caused a higher average tangential velocity. The results can be explained from the distribution of the local solid fraction (as shown in Fig. 5) at wall velocities of Ui = 1.02 m/s, U0 = 0 with: (a) μw = 0.384; (b) μw = 0.378; (c) μw = 0.354; (d) μw = 0.316; and (e) μw = 0.300. These graphs show the distributions of local solid fractions with the radial position for various area solid fractions at a constant wall velocity. The local solid fraction was small when it was close to the driving wall because of the shear dilation effect. However, it grew larger with increased distance from the driving wall because the lower shear rates caused weaker particle motions. The local solid fraction became smaller as a smaller solid fraction was applied to the system, consistent with the findings of Liao et al. [5] and Jasti and Higgs [21]. Additionally, the local solid fraction in the shear band was larger when the solid fraction was increased in each bottom wall friction coefficient case. Thus, the average tangential velocity was larger when there was an increase of the higher local solid fraction, which caused a faster frequency of interactive collisions, and a larger momentum transfer between the particles and the driving wall boundary, as shown in Fig. 4. Because of the existence of a slip velocity between the driving wall boundary and the particles near the driving wall boundary, the external driving energy was not fully introduced into the granular system. It was crucial to determine the slip velocity to quantify the external energy that was introduced into the granular system. Fig. 6 shows the slip velocity plotted as a function of the inner wall velocity, using varying bottom wall friction coefficients at a specific area solid fraction. The result indicated that the slip velocity increased linearly in conjunction with the inner wall velocity. These results were in good agreement with those obtained in previous studies [5,21]. The slip velocity was shown to be greater at a greater bottom wall friction coefficient at any given inner wall velocity. A larger amount of the kinetic energy of the particles was dissipated because of the stronger frictional effect that occurred between the particles and the bottom wall, which caused a larger slip velocity when there was a greater bottom wall friction coefficient present.
Granular temperature is critical to the study of the flow behaviors and dynamic properties of granular materials [5–8,15,21]. Fig. 7(a)–(e) shows the distributions of the granular temperature across the radial position at various driving inner wall velocities with varying bottom wall friction coefficients at v = 0.82. The local granular temperature could be regarded as the local specific kinetic energy in this study. It was observed that granular temperature was increased when the driving wall velocity was higher at each bottom wall friction coefficient. A larger amount of energy was introduced to the granular system at the higher driving wall velocity, which led to a more active fluid field, stronger particle fluctuations, and increased granular temperature. The granular temperature was also observed to be higher close to the driving inner wall (in the shear band region). Kinetic energy was transmitted mainly through interactive inelastic collisions that occurred between particles that comprised the granular materials. The strength of the motions of the particles and interactive collisions was enhanced, causing additional particle fluctuations caused by the higher velocity gradient in the shear band region. Thus, the granular temperature was greater closer to the driving wall boundary. This was consistent with previous studies that indicated that granular temperature is produced by the velocity gradient [5,7]. As the distance from the driving wall increased, the lower shear rate (smaller velocity gradient) caused weaker fluctuations of the particles (in the solid-like region), leading to a decrease in the granular temperature. The results were in agreement with those of previous studies [5,21]. Marinack et al. [26] reported that the normalized translational kinetic energy increased with the increase of global solid fraction. However, they didn't investigate the effect of bottom wall friction coefficient on granular temperature (kinetic energy). It is important to study the effect of bottom wall friction coefficient on dynamic properties in sheared granular flows because the bottom surface is usually uneven in most industrial and natural granular systems. In this study, we focus on the effect of bottom wall friction coefficients on the dynamic properties in sheared granular flows. To quantify the influence of the bottom wall friction effect on the granular temperature in sheared granular flows, the average granular temperatures, which were averaged from the local granular temperatures in the 10 regions, were plotted as a function of the solid fraction for various bottom wall friction coefficients at a specific inner wall velocity, as shown in Fig. 8. The average granular temperature decreased when the bottom wall friction coefficient increased in each specific area solid fraction. A larger amount of kinetic energy from the particles was dissipated and caused the smaller average granular temperature because of the stronger frictional effect that occurred when there was a larger bottom wall friction coefficient. The results are not similar to most previous studies, where the granular
Fig. 6. Slip velocity plotted as a function of inner wall velocity with different bottom wall friction coefficients. The lines indicate the linear fits to the data.
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temperature was strengthened with the greater shearing wall friction coefficient [15,21]. Based on the experimental results, the influence of wall friction on the dynamic properties of granular flows is not the
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same but depends on the granular system. As shown in Fig. 8, the average granular temperature increased linearly with the solid fraction in each specific bottom wall friction coefficient increased. The slopes
Fig. 7. Distributions of granular temperature with the radial position for different wall velocities, (a) μw = 0.384; (b) μw = 0.378; (c) μw = 0.354; (d) μw = 0.316; (e) μw = 0.300, at a specific area solid fraction v = 0.82.
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4. Conclusion
Fig. 8. Average granular temperature plotted as a function of the area solid fraction with different bottom wall friction coefficients μw, at a specific inner wall velocity, Ui = 1.02 m/s. The lines indicate the linear fits to the data. The error bars correspond to the standard deviation of at least three experimental tests in each case.
were 4.94 × 10− 2, 2.93 × 10− 2, 2.03 × 10− 2, 1.62 × 10− 2, and 1.72 × 10− 2, which correspond to the cases where μw = 0.300, 0.316, 0.354, 0.378, and 0.384, respectively. Liao et al. [5] found that velocity gradient is the predominant parameter influencing the granular temperature. From Fig. 3(a)–(e), the shear band thickness is similar with different solid fractions at the specific inner wall velocity and bottom wall friction coefficient. However, the velocity gradient is larger with the higher solid fraction. The larger velocity gradient leads to the stronger particle fluctuations and the greater average granular temperature. Therefore, the average granular temperature became larger when the higher the solid fraction was applied. Fig. 8 indicates that the variation of the average granular temperature at various bottom wall friction coefficients became larger as a higher solid fraction was applied. The frictional interactions between the particles and the bottom wall were strengthened with the increase of solid fraction. Thus, the variation between average granular temperatures with different bottom wall friction coefficients increased with the increase of the solid fraction, as shown in Fig. 8. Fig. 9 shows average granular temperature plotted as average tangential velocity with the same data as shown in Figs. 4 and 8. The data are collapse and shows that average granular temperature is enhanced with the increasing average tangential velocity.
Average granular temperature (m2/s2)
0.01 0.009 0.008 0.007 0.006
µw = µw = µw = µw = µw =
0.005
0.384 0.378 0.354 0.316 0.300
0.004 0.003 0.005
0.01
0.015
0.02
0.025
Average tangential velocity (m/s)
0.03
Fig. 9. Average granular temperature plotted as average tangential velocity with the same data as shown in Figs. 4 and 8.
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