Predictive model for local scour downstream of hydrokinetic turbines in erodible channels

A modeling framework is derived to predict the scour induced by marine hydrokinetic turbines installed on fluvial or tidal erodible bed surfaces. Following recent advances in bridge scour formulation, the phenomenological theory of turbulence is applied to describe the flow structures that dictate the equilibrium scour depth condition at the turbine base. Using scaling arguments, we link the turbine operating conditions to the flow structures and scour depth through the drag force exerted by the device on the flow. The resulting theoretical model predicts scour depth using dimensionless parameters and considers two potential scenarios depending on the proximity of the turbine rotor to the erodible bed. The model is validated at the laboratory scale with experimental data comprising the two sediment mobility regimes (clear water and live bed), different turbine configurations, hydraulic settings, bed material compositions, and migrating bedform types. The present work provides future developers of flow energy conversion technologies with a physics-based predictive formula for local scour depth beneficial to feasibility studies and anchoring system design. A potential prototype-scale deployment in a large sandy river is also considered with our model to quantify how the expected scour depth varies as a function of the flow discharge and rotor diameter.

Figure 1

Schematic of the two theoretical scenarios. Model case 1 (left) considers the effect of the turbine rotor drag on the scour. Model case 2 (right) considers the effect of the support tower drag under accelerated flow. The inset shows the characteristic velocity scales within the scour region.

Figure 2

Experimental values of the scour depth function $S_e$ compared to the model curves for (a) the rotor drag, model case 1 [Eq. (16)], with dashed lines indicating coefficient range $K_1=0.075$ to 0.21 and the solid line indicating the midpoint $K_1=0.15$; (b) tower drag, model case 2 [Eq. (17)], with dashed lines indicating coefficient range $K_2=0.17$ to 0.40 and the solid line indicating the midpoint $K_2=0.29$. The model curves follow the form $S_e=K{(U/U_c)}^{\theta }$ where ${\theta }_1=-1.1$ and ${\theta }_2=-1.89$. $U/U_c=1$ indicates clear water (CW) conditions.

Figure 3

Functional dependency of the rotor drag (model case 1) on the (a) Froude number $\text{Fr}$ where ${\left(\widetilde{y_s/y}\right)}_1={\left(\frac{y_s}{y}\right)}_1{\left[{\left(\frac{U}{U_c}\right)}^{\theta }{(s-1)}^{-3/5}{C}_T^{2/5}{\left(\frac{D}{d}\right)}^{2/5}{\left(\frac{D}{y}\right)}^{2/5}\right]}^{-1}$ is the left-hand side in Eq. (18), and (b) rotor diameter normalized by the sediment size $D/d$. The solid lines represent the best fit of the experimental data for the theoretically derived power laws: (a) ${\text{Fr}}^{6/5}$; (b) ${(D/d)}^{2/5}$. The dashed lines mark the bounds of the coefficient range $K_1=0.075$–0.21. Refer to Fig. 2 for data marker definitions.

Figure 4

Functional dependency of the tower drag (model case 2) on the (a) Froude number $\text{Fr}$ where ${\left(\widetilde{y_s/y}\right)}_2={\left(\frac{y_s}{y_1}\right)}_2{\left[{\left(\frac{U}{U_c}\right)}^{\theta }{(s-1)}^{-1}{C}_d^{2/3}{\left(\frac{c}{d}\right)}^{2/3}{\left(1+\frac{a}{2k_t}\right)}^2\right]}^{-1}$ is the left-hand side in Eq. (19), and (b) rotor submergence represented by $1+a/2k_t$. The solid lines represent the best fit of the experimental data for the derived power laws: (a) ${\text{Fr}}^2$; (b) ${(1+a/2k_t)}^2$. In (a) the dashed lines indicate the bounds of the coefficient range $K_2=0.17$–0.40, while in (b) the dashed lines indicate the model case 1 scour prediction range with $K_1=0.075$–0.21 for comparison. Refer to Fig. 2 for data marker definitions.

Figure 5

Time-resolved instantaneous bed elevation measurements (gray lines) downstream of the MHK turbine, located at $(x-x_T)/D=0$, for (a) experiment 5 (ripples in the Main Channel); (b) experiment 3 (dunes in the Tilting Bed Flume). The instantaneous gray curves in panels (a) and (b) are not shown for all times. Average scour (solid black line) and minimum and maximum (dashed lines) bed elevation envelope curves are included for reference. Probability density function of instantaneous scour depth immediately downstream of the turbine for (c) experiment 5; (d) experiment 3. Average scour depth $\overline{y_s}$ (solid line) and bedform amplitude $A_{\text{bf}}$ (dashed line) are indicated.

Figure 6

(a) Probability density function of model coefficients $K_1$ (bottom axis) and $K_2$ (top axis) corresponding to the instantaneously measured scour depths reported in Figs. 5 and 5 immediately downstream of the turbine, for experiments 3 and 5. (b) Cumulative density function for the same quantities. Prescribed ranges for $K_1$ and $K_2$ included are for reference.

Figure 7

Predicted scour depth in the Mississippi River at Audubon Park, New Orleans, LA, as a function of the flow discharge and rotor diameter for (a) model case 1 using $K_1=0.15$ and (b) model case 2 using $K_2=0.29$. This estimation assumes optimal turbine performance, fixed local depth $y=25$ m, river width $b=600$ m, median grain size $d=0.0002$ m, critical velocity $U_c=0.42{\mathrm{ms}}^{-1}$, support tower diameter $c=1$ m, and drag coefficient $C_d=1$. For model case 2 the turbine top tip elevation is fixed 3 m below the water surface. The red dotted line indicates the discharge $Q_c$ corresponding to the critical flow velocity for the sediment incipient motion and thus to the transition between clear water equations (11) and (15), and live bed equations (16) and (17). The critical flow velocity for the median grain size employed was estimated following [20, 21].