Experimental comparison of initiation of motion for submerged objects resting on fixed permeable and impermeable beds

Systematic experiments were conducted to study differences in the threshold flow conditions for initiation of motion of submerged objects (spheres and cylinders) resting on permeable and impermeable beds. The threshold flow conditions were characterized with the Shields parameter, which was estimated with upstream flow velocity measurements obtained with the particle image velocimetry (PIV) technique under conditions of unidirectional, open-channel flow in a laboratory flume. Flow-bed interface geometries were maintained across both permeable and impermeable bed experiments. Plotting the Shields parameter against the ratio of the particle size to bed roughness scale revealed clear differences between permeable and impermeable beds, including a larger Shields parameter value requirement to initiate motion on permeable beds than on impermeable ones, which is connected with the lower lift force on a permeable bed. Neglecting the presence of bed permeability can result in up to half an order of magnitude of variation in critical Shields parameter values. Experimental results also show that greater Shields parameter values are required to initiate the motion of a sphere than for a cylinder with identical diameter for large $L/D$ ratio $(L$ and $D$ are the length and diameter of the cylinder). As the $L/D$ ratio decreases, the cylinder approaches the sphere geometry, and the critical Shields parameter value is similar.

Figure 1

The forces acting on a submerged object, where $U\left(y\right)$ is the streamwise flow velocity profile along the $y$ direction, $F_b$ is the Archimedean buoyant force, $F_d$ is the fluid drag, $W_i$ is the weight of the test object, $D$ is the diameter of the object, $\varphi$ is the contact angle between the object and the neighboring bed particles, and $k$ is the length scale of the bed particles.

Figure 2

Illustration of experimental setup for the impermeable bed condition. (Left image) Profile view of the facility. The PIV measurement window is denoted as the shaded box. (Middle image) Cross section view of the flume. The test object on the bed is a cylinder in the illustration. (Right image) A close top view of the test section, showing the positioning of the laser sheet.

Figure 3

(a) The mean velocity field measured at the top of the bed particles, (b) the Reynolds stress field measured at the top of the bed particles, and (c) viscous stress, ${\tau }_{\nu }$, Reynolds stress ${\tau }_R$, form-induced shear ${\tau }_f$, and total stress ${\tau }_t$ profiles (defined as the origin of $y$ axis at the tangent surface of the bed) for the case of the sphere with a diameter of 6.3 cm and specific density of 1.40 on a permeable bed.

Figure 4

For (a) spheres, (b) cylinder set 1 (constant length), and (c) cylinder set 2 (constant $L/D$ ratio), the threshold Shields parameter was plotted against the particle-size ratio, $D/k$, value, compared with James [23]. Open and filled symbols stand for permeable and impermeable condition, respectively. The solid and dashed line are the entrainment equation [Eq. (9)] corresponds to impermeable and permeable condition, respectively. The dashed light gray line with a minus one slope is superimposed for comparison. Refer to Table 1 for the specifics of the objects.

Figure 5

The $\frac{U_T^2-U_B^2}{u_{*}^2}$ ratio was plotted against the bulk Reynolds number, $\text{Re}=\frac{D{U}_{\infty }}{\nu }$ to show that the lift is smaller for objects resting on a permeable bed than on an impermeable one. Filled and open symbols stand for the impermeable and permeable conditions, respectively.

Figure 6

For (a) permeable and (b) impermeable cases, the threshold Shields parameter for cylinders and spheres were plotted for different values of the ratio of the particle size to bed roughness scale, $D/k$. See Table 1 for the specifics of cylinder sets 1 and 2.

Figure 7

For (a) impermeable and (b) permeable cases, the Shields parameter for cylinder are plotted against the length over diameter ratio, $L/D$, for different ratio of the particle size to bed roughness scale, $D/k$. See Table 1 for the specifics of cylinder sets 1 and 2.

Figure 8

The relative variation, $\mathrm{\Delta }$, was plotted against the $L/D$ ratio. The square and triangle symbol denote cylinder set 1 and 2, respectively. The relative variation for spheres is denoted with circle symbols. Although the $L/D$ ratio of spheres is not defined in Table 1, a value of $L/D\sim 1$ is assigned to the spheres to illustrate the trend despite of the shape effect. The open symbol does not denote the permeable condition in this plot. See Table 1 for the specifics of the test objects.

Figure 9

Comparison of experimental results and the theoretical Shields curve for a single sphere on a bed of uniform spheres [23]. The square and triangular symbols denote cylinder sets 1 and 2, respectively. The spheres are denoted with circle symbols. Open and filled symbols stand for permeable and impermeable conditions, respectively.