Cospectral budget model describes incipient sediment motion in turbulent flows

Relating incipient motion of sediments to properties of turbulent flows continues to draw significant research attention given its relevance to a plethora of applications in ecology, sedimentary geology, geomorphology, and civil engineering. Upon combining several data sources, an empirical diagram between a densimetric Froude number $F_{dc}=U_c/\sqrt{gh\mathrm{\Delta }}$ and relative roughness $N=d/h$ was recently reported over some six decades of $N$, where $d$ is the grain diameter, $h$ is the overlying boundary-layer depth, $U_c$ is the bulk velocity at which sediment motion is initiated, $g$ is the gravitational acceleration, $\mathrm{\Delta }=s-1$, and $s$ is the specific gravity of sediments. This diagram featured three approximate scaling laws of the form $F_{dc}\sim N^{-\alpha }$ with $\alpha =1/2$ at small $N, \alpha =1/6$ at intermediate $N$, and $\alpha =0$ at large $N$. The individual $\alpha$ values were piecewisely recovered using a combination of (1) scaling arguments linking bulk to local flow variables above the sediment bed and (2) assumed exponents $\sigma$ for the turbulent kinetic energy spectrum $E_{tke}\left(k\right)\sim k^{-\sigma }$, where $k$ is the wave number or inverse eddy size. To explain the $\alpha =1/2$, the aforementioned derivation further assumed the presence of an inverse cascade in $E_{tke}\left(k\right)$ at large wave number (i.e., $\sigma =3$). It is shown here that a single $F_{dc}-N$ curve can be derived using a cospectral budget (CSB) model formulated just above the sediment bed. For any $k$, the proposed CSB model includes two primary mechanisms: (1) a turbulent stress generation formed by the mean velocity gradient and the spectrum of the vertical velocity $E_{ww}\left(k\right)$ and (2) a destruction term formed by pressure-velocity interactions. Hence, a departure from prior work is that the proposed CSB model is driven by a multiscaled $E_{ww}\left(k\right)$ instead of $E_{tke}\left(k\right)$ characterized by a single exponent. Also, the CSB model does not require the presence of an inverse cascade to recover an $\alpha =1/2$. Last, the CSB approach makes it clear that the scaling parameters linking local to bulk flow variables used in prior determinations of $\alpha$ at various $N$ must be revised to account for bed roughness effects.

Figure 1

Sketch of a wide rough channel whose bed is covered by spherical grain particles of uniform diameter $d$. The grains are entrained into the overlying turbulent flow when the surface shear stress ${\tau }_{tb}$ exceeds a threshold. The green arrow depicts turbulent eddies that have multiple sizes, and the grains are represented by brown circles. The cospectral budget (CSB) model is formulated in the roughness sublayer (black dashed line) above the grains but below the region characterized by a logarithmic mean velocity profile (purple dashed line). The forces acting on an individual particle are defined as follows: $F_d$ is the drag force, $F_f$ is the frictional force, $F_G$ is the gravitational force related to particle weight, and $F_L$ is the lift force.

Figure 2

Modified Shields diagram fitted to the original data of Shields [9], where ${\theta }_c\approx 0.06$ is independent of ${\text{Re}}_{*}$ when ${\text{Re}}_{*}>400$ and ${\theta }_c\propto {\text{Re}}_{*}^{-1}$ for ${\text{Re}}_{*}<3$. An intermediate region defined by ${\text{Re}}_{*}\in [3,400]$ exists where $\theta \in [0.02,0.06]$ varies weakly and nonmonotonically with ${\text{Re}}_{*}$.

Figure 3

The $F_{dc}-N$ diagram with data reported in Refs. [3, 26] along with the three scaling laws expressed as $F_{dc}\sim N^{-\alpha }$ with $\alpha =1/2, 1/6$, and 0 (in dashed lines). The original data sources are described in a number of studies, including Refs. [27, 28, 29, 30, 31, 32, 33].

Figure 4

Schematic of the vertical velocity energy spectrum $E_{ww}\left(k\right)$ as a function of wave number $k$ in double-log representation. The right-tail effect is represented with a generic power-law exponent $p$. $K_e, K_f$, and $K_r$ are characteristic wave numbers delineating different energy production and transfer regimes in the vertical velocity. To solve the CSB model without depth integration, $K_e, K_f$, and $K_r$ must be linked to boundary conditions on the flow (i.e., $h$ and $d$).

Figure 5

Fitting the $F_{dc}-N$ derived from the CSB model to the data in Fig. 3. The inset is an enlarged frame associated with the large-roughness regime ($N>0.1$).

Figure 6

Pearson correlation coefficients among $a,c_3$ and $C_p$ corresponding to a given $p$. For example, the Pearson coefficient of two random variables $(X,Y)$ is calculated as the covariance, $\mathrm{cov}(X,Y)$ normalized by the standard deviations of the individual variables (=${\sigma }_X,{\sigma }_Y$), i.e., $\mathrm{cov}(X,Y)/\left({\sigma }_X{\sigma }_Y\right)$. Strong correlations are shown between $p$ and $a$ and between $p$ and $c_3$.