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 Fdc=Uc/ghΔ 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, Uc is the bulk velocity at which sediment motion is initiated, g is the gravitational acceleration, Δ=s-1, and s is the specific gravity of sediments. This diagram featured three approximate scaling laws of the form Fdc∼N-α with α=1/2 at small N, α=1/6 at intermediate N, and α=0 at large N. The individual α 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 σ for the turbulent kinetic energy spectrum Etke(k)∼k-σ, where k is the wave number or inverse eddy size. To explain the α=1/2, the aforementioned derivation further assumed the presence of an inverse cascade in Etke(k) at large wave number (i.e., σ=3). It is shown here that a single Fdc-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 Eww(k) 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 Eww(k) instead of Etke(k) characterized by a single exponent. Also, the CSB model does not require the presence of an inverse cascade to recover an α=1/2. Last, the CSB approach makes it clear that the scaling parameters linking local to bulk flow variables used in prior determinations of α at various N must be revised to account for bed roughness effects.

Duke Scholars

Author Gabriel G. Katul Civil and Environmental Engineering