3.1. Lift force

3.1.1. General response

Figure 2 shows the lift force, $F_{{L}},$ on a non-rotating spherical intruder in a uniform granular shear flow as a function of the slip velocity, $u_{{s}}$ , for $R=3$ ( ${d}_{i} = 1.5\,\rm cm$ ), $P_0= 1000\,\rm Pa$ and ${\dot {\gamma _0}}=5\, \rm s^-{^1}$ . Here, $F_{{L}}$ is antisymmetric about $u_{{s}}=0$ , i.e. $F_{{L}}(u_{{s}}) = -F_{{L}}(-u_{{s}})$ , as expected. Consequently, for the remainder of this paper, we only consider $u_{{s}} \geqslant 0$ . Of primary interest is the non-monotonic dependence of $F_{{L}}$ on $u_{{s}}$ : for $0 \lt u_{{s}} \lessapprox 0.1\,\rm m\,s^-{^1}$ , $F_{{L}}$ increases until it reaches a maximum, but then decreases monotonically for $u_{{s}} \gt 0.1\,\rm m\,s^-{^1}$ , changing sign at $u_{{s}} \approx 0.3\,\rm m\,s^-{^1}$ . Also shown in the figure are theoretical predictions of the lift force on a non-rotating spherical intruder in a uniform fluid shear flow by Saffman (Saffman Reference Saffman1965) as well as from a subsequent and improved calculation (Shi & Rzehak Reference Shi and Rzehak2019), where the fluid kinematic viscosity is 0.0536 m $^2$ /s in order to match the granular viscosity determined by the $\mu (I)$ rheology with fluid density $\rho_f=1460\,\textrm{kg}/\textrm{m}^3$ , as described in § 3.3. Surprisingly, the direction of $F_{{L}}$ for an intruder in a granular shear flow is opposite that for the Saffman lift force in a fluid at lower slip velocities ( $0 \lt u_{{s}} \lessapprox 0.3\,\rm m\,s^-{^1}$ ), as first noted by Yennemadi & Khakhar (Reference Yennemadi and Khakhar2023). For $u_{{s}} \gt 0.3\,\rm m\,s^-{^1}$ , $F_{{L}}$ acts in the same direction as the fluid lift force. Although the granular and fluid lift forces for large ${u}_{{s}}$ are similar, they have different origins based on their vertical stress distributions, as discussed in § 3.4. The fluid lift force predicted by Shi & Rzehak (Reference Shi and Rzehak2019) becomes less negative above $u_{{s}}\approx 0.7\,\rm m\,s^-{^1}$ , but we do not verify whether the granular lift force also decreases in magnitude at larger $u_{{s}}$ due to the size of the computational domain necessary for simulations at large values of $u_{{s}}$ .

Figure 2. Variation of intruder lift force, $F_{{L}},$ with slip velocity, $u_{{s}},$ (bottom axis) and non-dimensionalized slip velocity, $u^*_{{s}}=u_{{s}}/(\dot \gamma _0({d}_{i}+d)),$ (top axis) in a granular shear flow for size ratio $R=3$ , overburden pressure $P_0= 1000\,\rm Pa$ and shear rate $\dot \gamma _0=5\,\rm s^-{^1}$ from DEM simulations ( $\circ$ ). Error bars indicate standard error of $F_{{L}}$ considering temporal correlations (Zhang Reference Zhang2006). Solid blue and red curves indicate predicted lift forces in comparable fluid shear flows, see text. Results from fluid CFD simulations ( $+$ ) match predictions of Shi & Rzehak (Reference Shi and Rzehak2019), as discussed in § 3.3.

We non-dimensionalize the slip velocity as $u^*_{{s}}=u_{{s}}/(\dot \gamma _0({d}_{i}+d))$ , where the velocity scale $\dot \gamma _0({d}_{i}+d)$ captures the shear-induced velocity difference across the intruder based on the undisturbed velocity of contacting bed particles immediately above and below the intruder. The maximum lift force occurs at $u^*_{{s}} \approx 1$ (as indicated on the horizontal axis at the top of figure 2), which corresponds to relative bed particle velocities of $-\dot \gamma _0({d}_{i}+d)/2$ just above the intruder and $-{3\dot {\gamma }}_0({d}_{i}+d)/2$ just below the intruder. The value of $F_{{L}}$ changes sign at $u^*_{{s}} \approx 3$ , with corresponding relative bed particle velocities of $-5\dot \gamma _0({d}_{i}+d)/2$ above the intruder and $-{7\dot {\gamma }}_0({d}_{i}+d)/2$ below the intruder. The remainder of this paper explores the scaling of $F_{{L}}$ and the conspicuous differences between $F_{{L}}$ in granular and fluid shear flows.

To verify the lift-force results shown in figure 2, we examine the cross-flow ( $z$ -axis) trajectory of an intruder that is unconstrained in the $z$ -direction for different fixed values of the bed velocity at the vertical centreline, $u_0,$ corresponding to different initial values of the non-dimensionalized slip velocity, $u^*_{{s},0}$ . The intruder starts at $z=0$ with an initial slip velocity of $u_{{s}}=-u_0$ , but as time progresses, the intruder should rise or sink due to a non-zero lift force. Results for an intruder with $R=3$ are shown in figure 3(a). For $u_0=0$ , where the lift force is zero, the intruder neither rises nor sinks, corresponding to the case where the bed particle velocities above and below the intruder are equal and opposite. For $0 \lt u^*_{{s},0} \lt 3$ , where the initial lift force is positive, the intruder moves upward, as shown in figure 3(a), but then stops moving upward when it reaches a position where it has the same streamwise velocity as the local bed particles, i.e. $u_{{s}}=0$ at $z_{i}/({d}_{i}+d) = u^*_{{s},0}$ . We do not quantify the upward velocity of the intruder in this regime due to the relatively large collision-induced fluctuations, but it is evident that the upward velocities are similar for different $u_{{s}}$ and tend to slow as the intruder reaches its steady-state position. However, for $u^*_{{s},0} = 3$ , where the initial lift force is approximately zero, the intruder does not rise for the first half-second. Nevertheless, once the intruder has risen slightly, its instantaneous slip velocity decreases, resulting in a positive and increasing lift force that eventually carries the intruder to its equilibrium $z$ -position at $z_{i}/({d}_{i} + d)=3$ .

Figure 3. Cross-flow trajectories of free intruder for various initial non-dimensionalized slip velocities $u^*_{{s},0}$ : (a) $0 \leqslant u_{{s},0}^* \leqslant 3$ and (b) $0 \leqslant u_{{s},0}^* \leqslant 5$ with $R=3$ , $P_0=1000$ Pa and $\dot \gamma _0=5\,\rm s^-{^1}$ . When $F_{{L}}\lt 0$ (for $u_{{s},0}^* = 4$ or $5$ here, see figure 2), the intruder moves downward until it reaches the vicinity of the lower wall.

A very different trajectory occurs when $u^*_{{s},0} \gt 3$ , where the initial lift force is negative, as shown in figure 3(b) which has different vertical scale limits than in figure 3(a) and includes the cases for $u^*_{{s},0} \leqslant 3$ for comparison. When $F_{{L}} \lt 0$ , the intruder moves downward faster than the intruder rises when $F_{{L}} \gt 0$ . As the intruder moves downward, the slip velocity increases and the lift force becomes increasingly negative as $u^*_{{s}}$ increases (recall figure 2). Although not shown in the figure, the intruder in these cases continues moving downward with increasing speed until it nearly reaches the bottom wall. The main point to be taken from figure 3 is that the free intruder trajectories are consistent with the lift-force measurements for a vertically tethered intruder, shown in figure 2, thereby confirming the surprising result of positive and non-monotonic lift at small $u^*_{{s},0}$ , which is opposite to that which occurs in the fluid shear case at small $u^*_{{s},0}$ .

3.1.2. Lift-force scaling

To understand the dependence of the lift force on flow conditions, we vary the shear rate $\dot \gamma _0$ , the overburden pressure $P_0$ , and the particle size ratio $R$ . Simulations are limited to $u_{{s}}^* \leqslant 9$ , above which computations are increasingly more difficult due to the large computational domain that is necessary. Figure 4(a) demonstrates how the relation for $F_{{L}}$ versus $u_{{s}}$ depends on $\dot \gamma _0$ . All three curves exhibit the same non-monotonic trend with increasing $u_{{s}}$ , and their peak values of $F_{{L}}$ are similar. However, with increasing $\dot \gamma _0$ , the maximum and zero-crossing points of $F_{{L}}$ both shift to larger values of $u_{{s}}$ , indicating that $F_{{L}}$ is linked to the intruder slip velocity relative to the bed shear rate rather than to the absolute intruder slip velocity.

Figure 4. Lift-force $F_{{L}}$ vs. slip velocity $u_{{s}}$ for a non-rotating intruder with varying (a) $\dot {\gamma }_0$ with $R= 3$ , $P_0= 1000\,\textrm {Pa}$ , (b) $P_0$ with $R= 3$ , $\dot {\gamma }_0 = 5\,\rm s^-{^1}$ and (c) $R$ with $P_0= 1000\,\textrm {Pa}$ , $\dot {\gamma }_0 = 5\,\rm s^-{^1}$ . Note that two data points at large $u_{{s}}\gt 1.25\,\rm s^-{^1}$ for $\dot {\gamma }_0 = 10\,\rm s$ in (a) are not shown to keep the horizontal axes consistent in all three panels.

Varying the overburden pressure $P_0$ has a different impact on $F_{{L}}(u_{{s}})$ , as shown in figure 4(b). Although the curves follow the same non-monotonic trend, reaching their peaks and decreasing to zero at similar values of $u_{{s}}$ , the magnitude of $F_{{L}}$ increases with increasing $P_0$ . This increase in $F_{{L}}$ with $P_0$ is not a consequence of closer bed particle packing at higher pressures, since packing density $\phi \approx 0.58$ regardless of overburden pressure. Instead, it appears to be due to the proportionality of overburden pressure and stresses in the granular bed.

Figure 4(c) shows $F_{{L}}(u_{{s}})$ for different values of $R$ . The same non-monotonic trend occurs with different $R$ , but the $F_{{L}}$ maximum and the zero-crossing point both shift to larger $u_{{s}}$ values with increasing $R$ , suggesting that the slip velocity scales with the intruder size. In addition, the magnitude of $F_{{L}}$ increases with increasing $R$ , which is reasonable given that both the velocity difference across the intruder and the intruder surface area increase as the intruder diameter is increased.

The qualitatively similar dependence of $F_{{L}}$ on $u_{{s}}$ over a range of parameter values as illustrated in figure 4 suggests that the results may collapse when scaled appropriately. Accordingly, we non-dimensionalize the lift force as $F^*_{{L}}=F_{{L}}/ (P_0({d}_{i}+d)^2/4 )$ , since $P_0({d}_{i}+d)^{2}/4$ is proportional to the intrinsic contact force scale in this dense linear shear flow, and plot it versus the scaled slip velocity $u^*_{{s}}=u_{{s}}/(\dot \gamma _0({d}_{i}+d))$ . This simple scaling is surprisingly effective, as shown in figure 5(a), which includes all the data sets from figure 4. All of the $F^*_{{L}}(u^*_{{s}})$ curves collapse for $u^*_{{s}}\leqslant 1$ . Furthermore, the maximum in $F^*_{{L}}$ occurs at $u^*_{{s}} \approx 1$ , suggesting that the slip velocity scaling is not only dimensionally correct, but that the velocity scale $\dot {\gamma }_0 ({d}_{i}+d)$ reflects the magnitude of the slip velocity for maximal lift. Likewise, the maximum value for the lift force of $F^*_{{L}}\approx 1$ indicates that the intrinsic contact force scale of $P_0({d}_{i}+d)^{2}/4$ is appropriate. The collapse is less satisfying for $u^*_{{s}} \gt 1$ , although the spread is surprisingly small given the range of values of $\dot {\gamma }_0$ , $P_0$ and $R$ included in the plot.

Figure 5. Scaled lift force vs. scaled slip velocity for all data in figure 4. Panel (a) shows $F^*_{{L}}$ vs. $u^*_{{s}}$ , where the symbols are the same as those in figure 4. Pentagram symbols indicate data from Yennemadi & Khakhar (Reference Yennemadi and Khakhar2023) for an intruder in linear shear flow with $R\in \{2, 4, 6\}$ . Inset: low $u^*_{{s}}$ data. Panel (b) shows $F_{{L}}/F_{{L,{max}}}$ vs. $u_{{s}}/u_{{s},F_{{L}}=0}$ , where $F_{{L,{max}}}$ and $u_{{s},F_{{L}}=0}$ indicate the peak force and slip velocity where $F_{L}=0$ , respectively. Inset: $u^{*}_{{s},F_{{L}}=0}$ vs. $I$ .

While the scaled data show good collapse in figure 5(a), the maximum lift force, $F_{{L,max}}$ , and the slip velocity where $F_{{L}}=0$ at finite slip velocity, $u_{{s},F_{{L}}=0}$ , vary slightly between parameter sets. To more directly test if $F_{{L}}(u_{{s}})$ is self-similar and to further validate the proposed scaling, we alternatively scale the slip velocity by $u_{{s}, F_{{L}}=0}$ and the lift force by $F_{{L,max}}$ , where, for each data set, $u_{{s}, F_{{L}}=0}$ is obtained from a linear fit of $F_{{L}}(u_{{s}})$ about the zero crossing and $F_{{L,max}}$ is obtained from a parabolic fit about the maximum. The result, shown in figure 5(b), demonstrates excellent collapse, thereby confirming that $F_{{L}}(u_{{s}})$ is indeed self-similar for the parameter range this work explores. The greater variation of the collapsed data in figure 5(a) compared with figure 5(b) is due to the increase in $u^{*}_{{s},F_{{L}}=0}$ for conditions with the lowest inertial numbers, namely, $\{2.5\,\mathrm {s}^{-1}, 1000\,\mathrm {Pa}, 3\}$ and $\{5\,\mathrm {s}^{-1}, 2000\,\mathrm {Pa}, 3\}$ , for which $I=0.020$ and $I=0.028$ , respectively. The inset of figure 5(b) shows that $u^{*}_{{s},F_{{L}}=0} = u_{{s},F_{{L}}=0}/ (\dot \gamma _0({d}_{i}+d) )$ increases with decreasing inertial number for $I\lt 0.03$ , a range approaching the lower end of the dense granular flow regime, $0.01 \leqslant I \leqslant 0.5$ (Jop Reference Jop2015) where the flow approaches that quasistatic flow regime.

In a previous study, Yennemadi & Khakhar (Reference Yennemadi and Khakhar2023) directly measure the lift force for $R\in \{2, 4, 6\}$ for linear shear flow with $u^*_{{s}} \lt 0.2$ , a much smaller range of scaled slip velocities than explored here. Their data (pentagram symbols in figure 5(a) and its inset) are consistent with our results and scaling. In another study, van der Vaart et al. (Reference van der Vaart, van Schrojenstein Lantman, Weinhart, Luding, Ancey and Thornton2018) indirectly estimate the lift force in gravity-driven chute flow with even smaller slip velocities, $\left |u^*_{{s}}\right |\leqslant 0.01$ . They find a much larger lift force with the opposite sign compared with our results and those of Yennemadi & Khakhar (Reference Yennemadi and Khakhar2023). We believe this can be attributed to their assumptions made to estimate the buoyancy force, as explained in Appendix B. We further note that additional simulations performed with gravity in the cross-flow (depthwise) direction confirm that the lift force is unaffected by the pressure gradient since the total cross-flow force acting on the intruder matches an additive combination of the predicted lift and granular buoyancy (Jing et al. Reference Jing, Ottino, Lueptow and Umbanhowar2020) forces.

Figure 6. Drag coefficient, $C_{{D}},$ vs. intruder Reynolds number, $Re_{{i}},$ for the parameter value combinations in figure 4. The CFD drag results ( $+$ ) are discussed in § 3.3. Symbols are the same as in figures 4 and 5.

3.2. Drag

The intruder lift force is determined by the cross-flow component of the integral of the normal and tangential stresses acting on the intruder surface. Similarly, the drag force equals the component of the same surface integral in the direction opposite to the intruder slip velocity. Given the common dependence of both forces on the surface stresses, we now examine the drag force on the spherical intruder as it slips relative to the flow in the streamwise direction as well as the lift-to-drag ratio, which highlights the differing dependence of the two forces on the slip velocity. The drag force is measured directly as the net streamwise force on the intruder due to contacts with bed particles and is typically averaged over 4 s (corresponding to $10/\dot {\gamma }_0$ to $40/\dot {\gamma }_0$ ).

The results are recast into the form of a drag coefficient, $C_{{D}}=8|F_{{D}}|/({\rho }{\pi }{d}^{2}_{i}\ u^{2}_{{s}})$ , and its dependence on the intruder Reynolds number, $Re_{{i}}={\rho }{d}_{i}u_{{s}}/{\eta }$ . The viscosity, $\eta$ , is based on the granular $\mu (I)$ rheology (Jop et al. Reference Jop, Forterre and Pouliquen2006), where $\mu (I)=\frac {\tau }{P_0}$ is the stress ratio (also called the macroscopic friction coefficient (Dumont et al. Reference Dumont, Bonneau, Salez, Raphael and Damman2023)) in the flow. The viscosity is estimated as $\eta = \mu (I) \frac {P_0}{\dot {\gamma }_0}$ , where $\mu (I) = \mu _{{s}} +\frac {\mu _2-\mu _{{s}}}{I_{{c}}/I+1}$ with $\mu _{{s}} = 0.36$ , $\mu _2 = 0.91$ and $I_c = 0.73$ for simple shear flow using identical bed particles (Jing et al. Reference Jing, Ottino, Umbanhowar and Lueptow2022). Figure 6 shows a general Stokes-like linear relationship between $C_{{D}}$ and $1/Re_{{i}}$ with $8/Re_{{i}} \lessapprox C_{{D}} \lessapprox 24/Re_{{i}}$ over all simulations, indicating that the drag force is proportional to the slip velocity. This result is consistent with previous measurements showing that the drag force on an intruder in a granular shear flow is Stokes-like (Tripathi & Khakhar Reference Tripathi and Khakhar2011; Jing et al. Reference Jing, Ottino, Umbanhowar and Lueptow2022). Just as interesting is that previous drag results apply to an intruder moving normal to the shear flow (across flow streamlines), while our results are for an intruder moving in the streamwise direction of the shear flow (parallel to flow streamlines). That the drag is consistent between these two scenarios suggests universality of the Stokes drag relation for spherical intruders in sheared granular flows.

To further explore similarities and differences between granular and fluid drag, we consider the pre-factor $c$ in the Stokes drag expression, $c\pi \eta {d}_{i}u_{{s}}$ . In a viscous fluid at low Reynolds number, $c=3$ , while $1 \lessapprox c \lessapprox 3$ for $10^{-6} \lessapprox Re_{{i}} \lessapprox 10^1$ for an intruder that moves normal to flow streamlines in a granular shear flow (Jing et al. Reference Jing, Ottino, Umbanhowar and Lueptow2022). We plot $c=|F_{{D}}|/(\pi \eta {d}_{i}u_{{s}})$ for the fluid and granular cases in figure 7(a). It is immediately evident that $c \approx 3$ for the fluid cases, as expected. Furthermore, the magnitude of the streamwise drag force (parallel to flow streamlines) is within the same range as found previously for cross-flow drag (across streamlines), i.e. $1 \lessapprox c \lessapprox 3$ (Jing et al. Reference Jing, Ottino, Umbanhowar and Lueptow2022), which corresponds to the $8/Re_{{i}} \lessapprox C_{{D}} \lessapprox 24/Re_{{i}}$ bounds in figure 6. This suggests that the range of $1 \lessapprox c \lessapprox 3$ for granular Stokes drag is broadly applicable for other types of granular flow and intruder slip directions relative to those flows. However, there are two regimes for each set of flow and particle conditions in the granular cases. At lower $Re_{{i}}$ , $c\approx 2.5$ over approximately an order of magnitude in $Re_{{i}}$ , and at higher $Re_{{i}}$ , $c$ decreases to $\approx 1$ . This behaviour matches the transition from a viscous-like flow regime to an inertial flow regime at $|F_{{D}}|/(P_0{d}_{i}^2) \approx 5$ for an intruder moving across the streamlines (Jing et al. Reference Jing, Ottino, Umbanhowar and Lueptow2022).

Given the strong similarities to the previous drag results for intruder motion across flow streamlines, we consider the dependence of $c$ on size ratio $R$ and inertial number $I$ as well as slip velocity $u^*_{{s}}$ . Using the fit for $c(R,I)$ from Jing et al. (Reference Jing, Ottino, Umbanhowar and Lueptow2022) for drag motion across flow streamlines, we normalize the measured $F_{{D}}$ with $c(R,I)\pi \eta {d}_{i}u_{{s}}$ and plot it versus the slip velocity in figure 7(b). With the exception of $R=1$ where the intruder is the same size as the bed particles, the streamwise drag data collapse well across the entire range of $u^*_{{s}}$ . The streamwise drag data in this study (circles), as well as from Yennemadi & Khakhar (Reference Yennemadi and Khakhar2023) (pentagrams), are well predicted by $c(R,I)\pi \eta {d}_{i}u_{{s}}$ , the cross-flow drag model (Jing et al. Reference Jing, Ottino, Umbanhowar and Lueptow2022) in the viscous-like regime (left of the shaded vertical band where $|F_{{D}}|/(P_0{d}_{i}^2)\leqslant 5$ ), with the exception of the $R=1$ data set. The consistent behaviour between these two scenarios suggests that the Stokes drag relation is independent of slip direction for spherical intruders in sheared granular flows. Moreover, for the parameter value combinations in figure 4(b), the transition point $|F_{{D}}|/(P_0{d}_{i}^2)=5$ corresponds to the shaded vertical band $1.35\leqslant u_{{s}}^*\leqslant 1.57$ in figure 7(b), which is close to $u_{{s}}^*=1$ where the lift force reaches the maximum value. This indicates that the transition between the viscous-like and the inertial drag force regimes may have the same physical origin as the transition between different lift-force regimes discussed in the next subsection.

Third, the drag force measured here not only follows Stokes law, but it is also nearly within the range found previously, $8/Re_{{i}} \leqslant C_{{D}} \leqslant 24/Re_{{i}}$ for $10^{-6} \leqslant Re_{{i}} \leqslant 10^1$ , indicating that the bounds associated with granular Stokes drag are likely broadly applicable to other types of flow and intruder slip directions relative to those flows. Note, however, that the slope of $C_{{D}}$ versus $Re_{{i}}$ is less than $-1$ at higher $Re_{{i}}$ values in each data set and that the scaling of $C_{{D}}$ changes from $C_{{D}}\approx 20/Re_{{i}}$ at lower $Re_{{i}}$ and appears to approach the limiting form drag relation of $C_{{D}}\approx 8/Re_{{i}}$ at larger $Re_{{i}}$ . This change suggests that, at large slip velocities, pressure dominates friction based on the analogy to drag on an intruder in a viscous fluid flow having a value of $8/Re_{{i}}$ for the form (pressure) contribution to the overall drag.

Figure 7. Drag force scaling for all data in figure 6. (a) Pre-factor $c$ in the Stokes drag expression, $c=|F_{{D}}|/(\pi \eta {d}_{i}u_{{s}}),$ vs. $Re_{{i}}$ . (b) Drag force scaled by drag model prediction (Jing et al. Reference Jing, Ottino, Umbanhowar and Lueptow2022), $F_{{D}}/(c(R,I)\pi \eta {d}_{i}u_{{s}})$ , vs. $u_{{s}}^*$ , where shaded region indicates viscous/inertial regime transition near $|F_{{D}}|/(P_0{d}_{i}^2)=5$ , as noted in the same model. Symbols are the same as in figures 4 and 5.

As mentioned above, the lift and drag forces are the cross-flow and streamwise components of the integrated surface stresses acting on the intruder. If these two forces were simply proportional to the slip velocity, the lift-to-drag ratio, $-F_{{L}}/F_{{D}}$ (note that $F_{{D}}$ is negative as shown in figure 1) would be independent of $u^*_{{s}}.$ However, this is not the case, as figure 8 shows. The data for different simulation parameters collapse reasonably well, but $-F_{{L}}/F_{{D}}$ is obviously not constant except, perhaps, for $u^*_{{s}}\lt 0.3$ , where $0.2 \lt -F_{{L}}/F_{{D}} \lt 0.3$ for the different data sets (see inset). These ratios are somewhat higher than previous results with low $u^*_{{s}}$ (Yennemadi & Khakhar Reference Yennemadi and Khakhar2023), particularly for $R=4$ and $6$ , because their drag forces are higher due to higher values of $I=0.23$ and $R$ up to 6, in agreement with Jing et al. (Reference Jing, Ottino, Umbanhowar and Lueptow2022). For increasing $u^*_{{s}}$ , $-F_{{L}}/F_{{D}}$ decreases with decreasing slope magnitude, becomes negative when $F_{{L}}$ reverses to match the Saffman lift direction in a fluid and appears to approach an asymptotic value of $-F_{{L}}/F_{{D}}\approx -0.03$ for large $u^*_{{s}}$ . In all cases, the lift-force magnitude is approximately one order of magnitude smaller than the drag force magnitude over the range of $u^*_{{s}}$ that we consider.

Figure 8. Lift-to-drag ratio, $-F_{{L}}/F_{{D}},$ vs. $u^*_{{s}}$ for varying parameters (symbols and colours) as indicated in figures 4 and 5. Pentagram symbols are from Yennemadi & Khakhar (Reference Yennemadi and Khakhar2023) for an intruder in linear shear flow and $R\in \{2,4,6\}$ (Yennemadi & Khakhar Reference Yennemadi and Khakhar2023). Inset: low $u^*_{{s}}$ data.

3.3. Fluid lift comparison

To gain further insight into the lift force on a spherical intruder in granular shear flow, we simulate the analogous problem in a fluid and compare and contrast how shear and pressure contribute to lift in both cases. Of course, many factors contribute to the lift force in a fluid (see Appendix C). Nevertheless, the point of this fluid comparison is to examine how the constitutive differences between granular flow and incompressible Newtonian flow contribute to the different lift forces evident in figure 2. The incompressible Newtonian fluid simulations are implemented in ANSYS Fluent, a commercial CFD simulation package, for the same flow geometry as that shown in figure 1. The intruder and fluid parameters are matched to the DEM simulations in §. 3.1.1 for $R=3$ : ${d}_{i}=1.5\,\rm cm$ , fluid density $\rho _f=1450\,\rm kg\,m^-{^3}$ , which equals the bed particle bulk density in granular flow simulations for $\phi =0.58$ , and kinematic viscosity equal to $0.0536\,\rm m^2\, s^-{^1}$ , which is based on the $\mu (I)$ rheology (Jop et al. Reference Jop, Forterre and Pouliquen2006) for $I=0.04$ as described in § 3.2. The intruder is modelled as a stationary non-rotating sphere with a no-slip boundary condition at its surface. The fluid velocities on all outer boundaries (see figure 1), i.e. upper and lower moving walls and the periodic inlet and outlet planes are prescribed (Salem & Oesterle Reference Salem and Oesterle1998; Mikulencak & Morris Reference Mikulencak and Morris2004) to produce a linear shear velocity profile in the $z$ -direction with $\dot {\gamma }_0={5}\,\textrm {s}^{-1}$ and a streamwise fluid velocity of $-u_{{s}}$ at $z=z_{i}=0$ sufficiently far from the intruder.

In incompressible Newtonian flow, shear stress is independent of absolute pressure so that an arbitrary background hydrostatic pressure can be superimposed on the pressure field. Here, the average background pressure on the outer boundaries is set to match the granular flow overburden pressure, $P_0= {1000}\,\textrm {Pa}$ , for ease of comparison, and gravity is set to zero. Note that, since we do not model the liquid–gas phase transition (i.e. cavitation) and the background pressure value is arbitrary with respect to the absolute pressure, the local fluid pressure can be negative. In the granular flow, however, the background pressure is physically relevant as it sets the scale of the lift and drag forces on the intruder in dense flows as we demonstrate above. Additionally, the pressure cannot be negative in granular flow, since zero pressure represents a region devoid of bed particles for the non-cohesive particle flows we study here.

Regarding the spatial discretization used in the solver, the gradients of solution variables at cell centres are least-squares cell based (Barth Reference Barth1991). Second-order and second-order upwind integration schemes are used for pressure and momentum interpolations, respectively. At convergence, the momentum and continuity equations have a maximum scaled residual of $10^{-12}$ . As in granular shear flow, the lift force in a fluid shear flow is sensitive to the domain size (Salem & Oesterle Reference Salem and Oesterle1998; Shi & Rzehak Reference Shi and Rzehak2019), necessitating a domain size convergence study to ensure that the system size is sufficiently large to not affect the results. The domain size for which the lift force is unaffected by the bounding walls for the range of parameters and slip velocities investigated is $200{d}_{i}\times 200{d}_{i}\times 200{d}_{i}$ , consistent with results from a recent numerical study (Andersson & Jiang Reference Andersson and Jiang2019), and much larger than the computational domain for the granular case ( $10{d}_{i}\times 10{d}_{i}\times 54{d}_{i}$ ). Lift and drag forces are calculated by integrating the normal and tangential stresses on the intruder in the cross-flow ( $z$ ) and streamwise ( $x$ ) directions, respectively. These forces agree with analytical and empirical results for the lift and drag forces (Stokes Reference Stokes1851; Saffman Reference Saffman1965; Shi & Rzehak Reference Shi and Rzehak2019) over the entire range of slip velocities considered in the DEM simulations, as shown in figures 2 and 6, thereby confirming the validity of the fluid simulation results.

We first compare pathlines for granular and fluid shear flows based on the velocity field in an $xz$ -plane centred on the intruder at various dimensionless slip velocities in figure 9, where $d$ in the expression for $u^*_{{s}}$ is set to $d=5$ mm in the fluid case for consistency with the granular case. For the granular case, the average velocity field in a $d$ -wide slice centred on the intruder is found using a binning procedure, and pathlines are determined by interpolating the average velocity in each bin. The horizontal magenta line indicates the location of zero velocity (matching the intruder velocity) at $z/d=(R+1)u^*_{{s}}$ , where the flow direction above the line is to the right and below the line is to the left. The velocity magnitude is indicated by the pathline dot spacing; dots are spaced at 0.05 s time increments. The background in figure 9 indicates the local pressure, which equals the background pressure of $P_0=1000\,\rm Pa$ throughout most of the domain except near the intruder. Insets in the granular flow panels (top row) indicate the local packing density, $\phi$ , near the intruder. Insets in the fluid flow panels (bottom row) compare the streamwise velocity profile in front of the intruder with uniform shear flow (dotted black line). The orange dashed circles for two cases of granular flow indicate the boundary of the streamwise velocity controller-free region in all granular simulation cases. Note that the full computational domains for both granular and fluid systems extend vertically far beyond the figure limits to avoid wall effects.

Figure 9. Comparison of flow field pathlines for various scaled slip velocities, $u^*_{{s}}$ in granular (top row) and fluid (bottom row) simulations averaged over time and a $d$ -wide spanwise region centred on the intruder. Continuous colour shading in the main panels indicates pressure, while colour in the upper insets indicates the mean local packing density, $\phi$ , around the intruder in granular flow. White in the pressure field indicates $P=0$ , while the white packing fraction contour in the upper insets indicates $\phi =$ 0.3. The lower insets compare the streamwise velocity profiles, $v_x,$ in a region just to the right of the intruder, in the fluid (red) and granular (solid black) flows, where dotted black lines indicate the base shear flow. The orange dashed circles represent the boundary of the controller-free region with radius 3 $d_{i}$ in the granular simulations (shown only in the first and last columns for reference). Flow conditions are $P_{0} =1000\,\rm Pa$ , $\dot \gamma _0=5\,\rm s^-{^1}$ and $R=3$ .

It is immediately evident from figure 9 that the granular and fluid flow fields are quite similar. Fields and pathlines are symmetric above and below the intruder at $u^*_{{s}}=0$ and asymmetric for $u^*_{{s}}\gt 0$ with much larger deviations in the pathlines above the intruder compared with below it and in the pressure in front compared with behind the intruder. One particularly noticeable feature is that, instead of sweeping across the $xz$ -plane horizontally, between one and three pathlines on the left and right sides of the intruder cross the zero-velocity plane at $z/d=(R+1)u^*_{{s}}$ , reverse direction and never reach nor pass the intruder centre ( $x=0$ ). For low slip velocities, the flow reverses just above the intruder and does not pass over the top of the intruder from right to left (evident for $u^*_{{s}}\leqslant 1$ ). At higher $u^*_{{s}}$ (for instance, $u^*_{{s}}=2$ ) pathlines flow from right to left over the top of the intruder. The situation where the flow does not pass over the intruder from right to left at low slip velocities has been noted for fluid flow (Robertson & Acrivos Reference Robertson and Acrivos1970; Poe & Acrivos Reference Poe and Acrivos1975) and is known as the blocking phenomenon (Camassa et al. Reference Camassa, McLaughlin and Zhao2011). Increasing the slip velocity shifts the zero-velocity plane further above the intruder, thereby exposing the intruder to a higher right-to-left velocity in the shear field, thereby eliminating the blocking phenomenon.

Intuitively, the transition from blocking to no blocking should occur at $u^*_{{s}}=0.5$ , when the zero-velocity plane is tangent to the top of the intruder. However, this ignores frictional effects. The frictional tangential boundary condition on a non-rotating intruder interferes with the intrinsic rotating component of the flow field, $\dot \gamma _0/2$ , in linear shear flow. An additional set of fluid simulations with a free-slip boundary condition on the intruder confirms a transition from blocking to no blocking at $u^*_{{s}}\approx 0.5$ in the absence of friction. In contrast, in the fluid simulations with a no-slip boundary condition on the intruder and in DEM granular simulations with friction, the blocking phenomenon continues until $u^*_{{s}}\approx 1$ , as shown in figure 9. As a check, we note that Camassa et al. (Reference Camassa, McLaughlin and Zhao2011) solved the flow field around a sphere in shear flow in the viscous limit and found that the transition from blocking to no blocking occurs at $u_{{s}}\approx 4\dot {\gamma }_0{d}_{i}/3$ . This corresponds to $u^*_{{s}}\approx 1$ for $R=3$ , which is consistent with our results that the transition from blocking to no blocking occurs for $1\leqslant u_{{s}}^*\leqslant 2$ , although we did not attempt to isolate the exact value of the transition.

Despite the pathline reversal just above the intruder, mass is conserved in both fluid and granular flows so that each granular or fluid element on the right side of the intruder that shifts from moving left to moving right is matched by a corresponding element on the left side of the intruder that shifts from moving right to moving left. In an infinitely long streamwise domain, the pathlines that change direction above the intruder would extend to negative infinity on the left and positive infinity on the right. However, with the periodic streamwise boundary conditions used here, they instead form a horizontally elongated circulation roll, evident on the right side for $u_{{s}}^*=5$ in the granular case.

It is tempting to infer that lift forces result from the asymmetry of pathlines above and below the intruder, but this cannot be correct. Even though the pathlines are similar for the granular and fluid cases, the lift force has the opposite sign for the granular and fluid cases for $0\lt u_{{s}}^*\lt 3$ . Nevertheless, there are subtle differences in the granular and fluid pathlines such as the slight shift of the point where the pathlines reverse direction above the intruder to the left in the granular case and to the right in the fluid case for $u_{{s}}^*\geqslant 2$ .

The pressure fields around the intruder are also quite similar between the granular and fluid flow cases. Pressure is higher on the leading side (right side) of the intruder, most evident for large $u_{{s}}^*$ , due to the incoming right-to-left flow with respect to the intruder. The pressure appears nearly symmetric above and below the intruder, although the pressure is slightly higher above the intruder in the granular case for $u_{{s}}^*\geqslant 2$ and slightly higher below the intruder for $u_{{s}}^*=0.5,\,1$ in the fluid case. More importantly, the pressure field behind the intruder in the fluid case is negative for $u_{{s}}^*\geqslant 2$ , whereas the corresponding pressure in the granular case is zero since it cannot go lower.

It is also helpful to examine the interactions of the bed particles with the intruder in the granular case. The packing density, $\phi$ , shown in the insets in the top row of figure 9, is near zero in a region behind the intruder for $u_{{s}}^*\geqslant 2$ . Thus, most of the particle contacts on the intruder occur on the leading side, while a void forms on the trailing side. This corresponds to a constitutive difference between the granular shear flows, which sustains no tension on the trailing side of the intruder, and the fluid shear flows where the fluid remains in contact with the intruder on its trailing side and generates a negative pressure.

Lastly, we compare the streamwise velocity profiles in the region just to the right of the intruder in the bottom row insets of figure 9. At low values of $u^*_{{s}}$ , there is almost no difference in the granular and fluid velocity profiles, indicating that the velocity field is not particularly helpful in understanding the reversal in the sign for the lift between the two cases. At higher values of $u^*_{{s}}$ there are minor differences in the velocity profiles between the granular cases and the fluid cases in that the granular velocity profiles have a sharper peak and the effect of the intruder in the $z$ -direction in the granular case is narrower than that in the fluid case. In all cases, the velocity immediately in front of the fixed intruder is zero, which differs by approximately $4u^*_{{s}}$ from that of the background shear flow at $z/d=0$ , as would be expected for $R=3$ and the definition of $u^*_{{s}}$ . We also note that the relatively narrow width of the region affected by the intruder in the $z$ -direction for the granular case demonstrates that the radius of the controller-free region in the simulations (extending in the $z$ -direction by $\pm 3 d_{i}=\pm 9 d$ for $R=3$ ) is adequate to prevent the controlled regions of the granular flow from affecting the lift or drag measured on the intruder.

A more direct approach to understanding lift-force differences between granular and fluid shear flows than considering the entire pressure field is to examine the pressure on the intruder surface that develops due to flow interactions. Figure 10 compares the pressure distributions on the intruder in granular DEM simulations (a–f) with fluid CFD simulations (g–l) at multiple slip velocities. The pressure distributions at $u^*_{{s}}=0$ are similar in granular flow (a) and fluid flow (g), although the maximum pressure on the intruder in the granular case is higher. As the slip velocity increases, the pressure increases on the leading side of the intruder in both cases. However, there is a large difference in the pressure distribution on the trailing side for the two cases as is clearly evident in figures 10(b–f) and 10(h–l). The pressure on the trailing side in the fluid case becomes negative (blue regions), and the negative pressure zone grows larger with increasing $u_{{s}}^*$ . In the granular case, the minimum pressure at large $u_{{s}}^*$ on the trailing side of the intruder is zero, corresponding to the void evident in the insets in the top row of figure 9.

Figure 10. Intruder pressure distribution (colour map), $P$ , in (a–f) granular DEM simulation and (g–l) fluid CFD simulation with ${d}_{i}=1.5$ cm. See text for fluid simulation parameters.

The qualitative similarity between the pressure distributions on the leading side of the intruder in granular and fluid shear flows suggests that it might be possible to artificially account for the constitutive differences between the fluid and granular media by simply considering only the contributions to the lift force on the leading side of the intruder. This eliminates the differences between a tractionless void in the granular case, shown in a snapshot from a DEM simulation in figure 11(a), and the negative pressure in the fluid case on the trailing side of the intruder. In other words, including only the stresses on the leading (right) hemisphere of the intruder in the fluid case (‘leading-hemisphere’ modification) might be a simple way to modify the fluid case to mimic the granular case. figure 11(b) shows the fluid pressure distribution on the leading hemisphere of the intruder in the fluid case as well as vectors corresponding to the normal (red) and tangential (black) components of the stress for $u_{{s}}^*=5$ . The associated lift force calculated using only the leading hemisphere in the fluid case is shown in figure 11(c) versus $u_{{s}}$ (bottom horizontal axis) and $u^*_{{s}}$ (top horizontal axis). The sign of the lift force for the modified fluid case flips so that $F_{{L}}$ is in the same direction and similar in magnitude to $F_{{L}}$ in the granular case. This suggests that the void on the trailing side of the intruder in granular flow is at least partially responsible for the opposite sign of the granular lift force compared with the fluid case. However, this is clearly only part of the story. First, this approach results in a finite value for $F_{{L}}$ at $u_{{s}}=0$ , which is physically incorrect. Second, $F_{{L}}$ based on the leading-hemisphere-only modification does not become negative as $u_{{s}}$ increases, even at large slip velocities (we simulated values as large as $u_{{s}}^*=45$ , not shown in figure 11(c)). To fully understand the differences between the two cases, a more sophisticated analysis of the $z$ -components of the stresses on the intruder is necessary.

Figure 11. Leading-hemisphere modification of the fluid results for $R=3$ , $P_0= {1000}\,\textrm {Pa}$ and $\dot \gamma _0={5}\,\textrm {s}^{-1}$ . (a) Snapshot of intruder and bed particles in DEM simulation in $y=0$ plane (centred on intruder) showing the void behind the intruder at dimensionless slip velocity $u^*_{{s}}=5$ . (b) Pressure field, $P$ , (colour map) on the leading hemisphere of intruder in a fluid, where arrows are proportional to the normal (red) and tangential (black) stress components for $u^*_{{s}}=5$ in the $y=0$ plane. (c) Lift force, $F_{{L}},$ vs. slip velocity, $u_{{s}},$ for granular flow simulation (black), fluid theory (red) and CFD fluid simulation with leading-hemisphere modification (blue) and $(P\gt 0,\tau =0)$ modification (green).

3.4. Vertical stress

Based on the results in the previous section, it is clear that differences in the stress distributions on the surface of the intruder determine both the sign and magnitude of the lift force. Here we analyse the lift direction component of the combined normal stress and shear stress as well as of each stress component individually to understand their influences on $F_{{L}}$ .

Figure 12. Net vertical stress distribution on the intruder ( $\sigma _{z}$ ) in (a–f) granular flow and (g–l) fluid flow with increasing (left to right) dimensionless slip velocity, $u^*_{{s}}$ . Positive stresses (orange) act upward and negative stresses (purple) act downward. Zero stress contours are omitted in some cases for clarity.

First consider the lift-force component of the combined normal and shear stress fields acting on the intruder surface, $\sigma _z$ , in both granular ( $R=3$ ) and fluid flow, shown in figure 12. The vertical stress distributions differ substantially between the granular and fluid cases over the range of slip velocities. For both flows, when $u^*_{{s}}=0$ the downward stress (lavender) zone on the top left of the intruder and the upward stress (orange) zone on the bottom right of the intruder are approximately symmetric about the intruder centre. This is reasonable because, for $u^*_{{s}}=0$ , the flow above the centre of the intruder impinges on the intruder from the left side, while the flow below the intruder centre impinges on the intruder from the right side, resulting in $F_{{L}}=0$ . With increasing $u^*_{{s}}$ in the granular case, figure 12(a–f), the upward stress zone on the lower right part of the intruder shifts slightly to the right and increases in magnitude, while the downward stress zone on the upper part of the intruder first shifts from the left side of the intruder to the right side before increasing in magnitude. This is explained in terms of the position of the intruder relative to zero velocity in the shear velocity profile: with increasing $u^*_{{s}}$ , bed particles below the centre of the intruder impinge on the intruder from the right at an increasing relative speed, while bed particles above the centre of the intruder shift from impinging from the left for $0 \leqslant u^*_{{s}} \leqslant 1$ to impinging from the right for $u^*_{{s}} \gt 1$ (see figure 9), with the number and relative speed of particles impacting from the right increasing with increasing $u^*_{{s}}$ . The shape of the contours in the granular case are more curved than those in the fluid case. In addition to the region of negative (downward) stress shifting from the trailing side to the leading side of the intruder with increasing $u^*_{{s}}$ , for $0.5\leqslant u^*_{{s}} \leqslant 2$ the positive (upward) stress on the bottom of the intruder is stronger than the negative (downward) stress on the top of the intruder. This leads to $F_{{L}} \gt 0$ in the granular case for $0 \lt u^*_{{s}} \lt 3$ , evident in figure 5. When $u^*_{{s}}=5$ , the downward stress is somewhat larger than the upward stress, resulting in $F_{{L}}\lt 0$ .

In the fluid case, figure 12(g–l), the vertical stress contours are nearly straight (circles in three dimensions) and aligned perpendicular to an axis tilted toward the left, independent of $u^*_{{s}}$ . While the downward stress remains nearly constant with increasing $u^*_{{s}}$ , the upward stress weakens, causing $F_{{L}}$ to become increasingly negative, which is consistent with the lift for the fluid case, shown in figure 2.

To further understand the origin of the differences in $\sigma _z$ for the granular and fluid flow cases, we consider the distribution of vertical components of the shear stress, $\tau _{z}$ , and the normal stress, $P_z$ , separately. From figure 13 showing $\tau _{z}$ on the intruder surface, it is clear that the vertical shear stress in the granular case is as much as five times smaller than that for the fluid case. This likely occurs because the granular shear stress is bounded by $\mu _pP$ and results from intermittent sliding contact of bed particles on the intruder, while the viscous friction in a Newtonian fluid acts continually on the surface of the intruder. There is negligible shear on the trailing side of the intruder in the granular case because of the void that forms behind the intruder. Although the magnitudes are quite different, for $u^*_{{s}}\geqslant 1$ the distributions of vertical shear on the leading side of the intruder share some similarities. In both cases, there is an upward shear component on the upper portion of the leading side as the flow slips upward over the intruder and a downward shear component on the lower portion of the leading side as the flow slips downward below the intruder.

Figure 13. Distributions of intruder vertical shear stress component, $\tau _{z}$ , in (a–f) granular flow and (g–l) fluid flow with increasing dimensionless slip velocity, $u^*_{{s}}$ . Note the different colour bar scales.

Figure 14. Vertical component of normal stress on the intruder, $P_z$ , in (a–f) granular flow and (g–l) fluid flow with increasing dimensionless slip velocity, $u^*_{{s}}$ .

In contrast to $\tau _z$ , the magnitude of $P_z$ is higher for the granular case than the fluid case, as shown in figure 14. Specifically, the absolute value of $P_z$ averaged over the leading hemisphere for the granular case is 1.6–1.9 times higher than that for the fluid case, consistent with the pressure comparison in figure 10. Since $\tau _z$ is so much smaller than $P_z$ for the granular case, $P_z$ nearly matches the distributions of the total stress, $\sigma _{z}$ , for the granular case, in figure 12(a–f) for all $u^*_{{s}}$ . Hence, the lift force in granular flow is primarily a consequence of the vertical component of the normal stress, with a considerably smaller contribution from $\tau _{z}$ . As described in the context of figure 12, the distribution of $P_z$ is readily explained in terms of the position of the intruder relative to zero velocity in the shear velocity profile and the way that bed particles above and below the centre of the intruder impinge on the intruder. Here, $P_z$ in the fluid case, figure 14(g–l), is similar to the granular case at $u^*_{{s}}=0$ , but is increasingly different as $u^*_{{s}}$ is increased. $P_z$ on the leading (right) hemisphere of the intruder remains similar in the two cases with downward $P_z$ on the upper right and upward $P_z$ on the lower right. However, additional normal stress zones emerge on the trailing (left) hemisphere of the intruder in the fluid case for $u^*_{{s}} \geqslant 0.5$ , see figure 14(h–l). In these zones, $P_z$ is opposite in sign to the granular cases and grows to be quite large in magnitude by $u^*_{{s}} =5$ . The signs of $P_z$ on the trailing side of the hemisphere in the fluid case are opposite to those on the leading side: $P_z$ on the top left is upward and on the bottom left is downward. In contrast, the pressure in the void behind the intruder in a granular flow cannot be negative. Although the magnitude of $P_z$ on the bottom of the intruder, no matter upward or downward, is larger than the magnitude of $P_z$ on the top of the intruder in the fluid case, it is the combination of $P_z$ and $\tau _z$ that drives the lift force. In fact, $P_z$ and $\tau _z$ in the fluid case have opposite signs, similar magnitudes and act on the entire intruder surface, as is evident by comparing figure 14(g–l) with figure 13(g–l). This is quite different from the granular case, where $\tau _z$ is an order of magnitude less than $P_z$ and both act primarily on the leading hemisphere of the intruder.

Given that $\tau _z$ is small compared with $P_z$ in the granular case and that the distributions of $P_z$ in the fluid and granular cases are similar on the leading hemisphere of the intruder, we return to the idea of artificially modifying the pressure and shear stress on the intruder in fluid shear flow to mimic that on an intruder in granular shear flow. Consider the distribution of the vertical components of stress, $\sigma _z$ , for the fluid case only where $P \gt 0$ with $\tau$ set to zero, shown in figure 15. The similarity between this figure and the first row of figure 12 suggests that the differences in the lift force between the granular shear flow and the fluid shear flow are related to (i) the relatively low shear stresses exerted by the flow on an intruder in granular shear flow, and (ii) the ability of the fluid to exert a negative gauge pressure at positions on the intruder where there is a void in a granular shear flow.

Figure 15. For the $P\gt 0$ pressure modification of fluid stresses with $\tau =0$ (see text), vertical component of total stresses on the intruder, $\sigma _z$ , with increasing dimensionless slip velocity, $u^*_{{s}}$ .

We return now to figure 11(c) for the lift force to describe how including only $P\gt 0$ regions and setting $\tau = 0$ for the fluid case compares with the granular case. Like the simpler ‘leading-hemisphere’ approach, $F_{{L}}$ changes from negative to positive, matching the sign for $F_{{L}}$ in granular shear flow for small $u_{{s}}$ . Unlike the ‘leading-hemisphere’ approach, the $(P\gt 0,\tau =0)$ modification correctly gives $F_{{L}}=0$ at $u_{{s}}=0$ . However, as $u_{{s}}$ increases, the approach asymptotes to a value that is approximately one half the peak value in the granular case and fails to become negative for large $u_{{s}}$ , as it does in the granular case.

We have implemented more complicated modifications to the fluid shear flow stresses in an attempt to better match the granular shear flow lift force including using shear stresses predicted by the $\mu (I)$ -rheology using the velocity and pressure fields from the fluid flow, accounting for the bed particle diameter by considering the pressure and shear on a spherical shell a distance $d/2$ beyond the intruder surface, and combinations of these approaches. Although $F_{{L}}$ at $u^*_{{s}}=1$ for some approaches reaches a magnitude closer to the peak value of $F_{{L}}$ in the granular case, none are significantly more successful than the two simpler approaches described above because $F_{{L}}$ in the modified cases remains positive as $u^*_{{s}}$ becomes large. This failure to reproduce the granular shear flow lift force by modifying the fluid results demonstrates the sensitivity of $F_{{L}}$ to subtle differences in the interaction of the flow field with the intruder. Despite the general similarity between the modified stress distribution for the fluid case in figure 15 and the stress distribution for the granular case in figure 12(a–f), details of the stress distributions significantly impact the lift force, $F_{{L}}$ .

3.5. Effects of rotation on lift

Up to this point, the intruder has been constrained to not rotate. For completeness, here, we impose angular velocity, $\omega _{0}$ , on the intruder about the $y$ -axis, as well as allowing the intruder to freely rotate with angular velocity $\omega$ about the $y$ -axis in response to the granular shear flow. Figure 16(a) shows the dependence of $F^*_{{L}}$ on $u^*_{{s}}$ for different values of the imposed dimensionless angular velocity $\omega _0^*=\omega _0/\dot \gamma _0$ . Other parameters are the same as in § 3.1.1. For $\omega ^*_0=1$ (clockwise rotation), $F^*_{{L}}$ shifts upward with respect to the non-rotating case, and when $\omega ^*_0=-1$ (counter-clockwise rotation), $F^*_{{L}}$ shifts downward. Near the peak in $F^*_{{L}}$ , the $u^*_{{s}}$ dependent shifts are nearly symmetric with respect to the non-rotating case, while for $u^*_{{s}}\geqslant 3$ , the upward shift for $\omega ^*_0=1$ is 2–3 times larger than the downward shift for $\omega _0=-1$ . For a freely rotating intruder, $F^*_{{L}}$ nearly matches the case where the intruder is not allowed to rotate, except, perhaps, at $u^*_{{s}}=5$ , where $F^*_{{L}}$ is slightly lower.

Figure 16. (a) Dimensionless lift force, $F^{*}_{{L}},$ vs. dimensionless slip velocity, $u^*_{{s}},$ for dimensionless angular velocities, $\omega ^*_0=\pm 1$ , and a freely rotating intruder ( $R=3$ , $P_0= 1000\,\rm Pa$ , $\dot \gamma _0=5\,\rm s^-{^1}$ ). Inset: scaled angular velocity $\omega ^*$ vs. $u^*_{{s}}$ for freely rotating intruder. Vertical component of (b–d) shear stress, $\tau _z$ , and (e–g) normal stress on the intruder, $P_z$ , in granular flow with no rotation and with imposed positive and negative rotations at $u^*_{{s}}=2$ (red box in (a)). Fields in (c) and (f) are the same as figures 13(d) and 14(d), respectively. The vectors (not to scale) compare relative slip direction of contacting bed particles about an intruder for different $\omega ^*_0$ . Red vectors indicate where imposed rotation reverses slip velocity direction, and, consequently, the tangential stress direction.

From the comparison of shear stress, $\tau _z$ , shown in figure 16(b–d), and normal stress on the intruder, $P_z$ , in figure 16(e–g), for different $\omega ^*_0$ at $u^*_{{s}}=2$ , it is apparent that the changes in the lift force with intruder rotation are due to changes in shear stress as the intruder slips relative to the surrounding bed particles, while the normal stress is apparently unaffected by intruder rotation. The vectors on the intruder periphery in figure 16(b–d) illustrate how the slip direction changes with intruder rotation. At $\omega ^*_0=0$ , bed particles flowing over the intruder separate to flow above or below the intruder at approximately the middle of the intruder. When the intruder is rotated clockwise ( $\omega ^*_0=1$ ), the zero-slip separation point shifts downward (red vector indicates change in slip direction), increasing the upward vertical frictional force and shifting $F^*_{{L}}$ higher. The opposite occurs for $\omega ^*_0=-1$ , where the separation point shifts upward, increasing the total downward vertical frictional force and shifting $F^*_{{L}}$ lower.

Returning to the free rotation case, the inset in figure 16(a) shows the dimensionless mean angular velocity $\omega ^*=\omega /\dot \gamma _0$ for a freely rotating intruder. For $u^*_{{s}}=0$ , the intruder rotates with a mean angular velocity of $\dot \gamma _0/2$ corresponding to $\omega ^*=0.5$ , which matches previous results in granular (da Cruz et al. Reference da Cruz, Emam, Prochnow, Roux and Chevoir2005) and laminar fluid shear flows (Kohlman & Mollo-Christensen Reference Kohlman and Mollo-Christensen1965), and is due to the freely rotating intruder’s lack of resistance to the rotational component of the shear flow (Guazzelli & Pouliquen Reference Guazzelli and Pouliquen2018). For $u^*_{{s}}\gt 0$ , the intruder rotates with the unperturbed shear (clockwise) for $u^*_{{s}}\lt 1$ and against the shear (counterclockwise) for $u^*_{{s}}\gt 1$ .