6.6.2: Sediment continuity
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In order to obtain the sediment concentration, a mass balance equation for the sediment needs to be solved. The general conservation statement for the sediment reads:
\[\underbrace{\dfrac{\partial c}{\partial t}}_{\begin{array} {c} {\text{change in}} \\ {\text{sediment}} \\ {\text{concentration}}\end{array}} \underbrace{+\dfrac{\partial uc}{\partial x} + \dfrac{\partial vc}{\partial y}}_{\begin{array} {c} {\text{net import of sediment}} \\ {\text{by the horizontal}} \\ {\text{fluid velocity}}\end{array}} \underbrace{+\dfrac{\partial wc}{\partial z}}_{\begin{array} {c} {\text{net upward transport}} \\ {\text{of sediment by the}} \\ {\text{vertical fluid velocity}}\end{array}} \underbrace{-\dfrac{\partial w_s c}{\partial z}}_{\begin{array} {c} {\text{net downward transport}} \\ {\text{with the fall velocity}} \end{array}} = 0\label{eq6.6.2.1}\]
or using the continuity equation for the fluid:
\[\dfrac{\partial c}{\partial t} + u \dfrac{\partial c}{\partial x} + v \dfrac{\partial c}{\partial y} + w \dfrac{\partial c}{\partial z} - \dfrac{\partial w_s c}{\partial z} = 0\label{eq6.6.2.2}\]
Please note that in Eqs. \(\ref{eq6.6.2.1}\) and \(\ref{eq6.6.2.2}\) the velocity and the concentration are the total signals consisting of a mean, an oscillatory and a turbulent part.
The horizontal advective terms are often (but not always) smaller than the vertical advective terms (see Intermezzo 6.6). Let us for simplicity neglect the horizontal advective terms of Eq. \(\ref{eq6.6.2.1}\) so that we are left with:
\[\dfrac{\partial c}{\partial t} + \dfrac{\partial wc}{\partial z} - \dfrac{\partial w_s c}{\partial z} = 0\label{eq6.6.2.3}\]
Similarly to the drift of water (Stokes’ drift, Sect. 5.5.1), there is also a drift of suspended sediment. Under the wave crest the suspended sediment concentration is stretched out, whereas under the trough it is compressed. Stokes’ drift was explained from the higher water levels during the forward orbital motion than during the backward motion. Analogously, a suspended sediment drift occurs. This drift is only taken into account when the horizontal advective terms are included in the advection-diffusion equation.
As said previously, the velocity and the concentration consist of a mean, an oscillatory and a turbulent part. Hence, \(w = W + \tilde{w} + w'\) and \(c = C + \tilde{c} + c'\). We are certainly not going to resolve the turbulent motion and would like to average over that motion. This is called Reynolds’ averaging. If we average Eq. \(\ref{eq6.6.2.3}\) over the turbulence scale, most terms with turbulent fluctuations average out, except for one term, viz. \(\partial \langle c', w' \rangle / \partial z\). Here the brackets denote averaging over the turbulence timescale and \(\langle c', w' \rangle\) is the (upward) sediment flux by turbulence. Further, the Reynolds averaged vertical water velocity can be assumed to be negligible compared to the fall velocity of the sediment. With the simplification of a constant fall velocity (in reality the fall velocity depends on the concentration, see Sect. 6.2.3), we end up with the following often used form of the advection-diffusion equation:
\[\dfrac{\partial c}{\partial t} \underbrace{- w_s \dfrac{\partial c}{\partial z}}_{\begin{array} {c} {\text{sediment net going downward}} \\ {\text{with its fall velocity}} \end{array}} \underbrace{+ \dfrac{\partial \langle c', w' \rangle}{\partial z}}_{\begin{array} {c} {\text{sediment net going upward}} \\ {\text{with fluid turbulence}} \end{array}} = 0\]
Please pay attention that in this equation the concentration \(c\) now denotes the turbulence averaged concentration: \(c = C + \tilde{c}\).
In order to model the sediment flux due to turbulence, we make a similar assumption of upward transport due to turbulent diffusion as for the fluid (see for instance Eqs. 5.5.5.14 and 5.5.5.17):
\[-\langle c', w' \rangle = v_{t, s} \dfrac{\delta c}{\delta z}\]
in which \(v_{t, s}\) is the turbulent diffusivity of sediment mass in \(m^2/s\) and \(c\) is now defined as volume concentration (see Sect. 6.2.2). This upward transport by turbulent diffusion can be understood as follows. Turbulent exchange makes sure that a sediment laden fluid parcel goes upward to a level with a lower sediment concentration. A sediment parcel going downward contains less sediment than the average parcel at the level where it arrives. Since more sediment particles are carried upward than downward the net effect is an upward transport.
Sometimes the turbulent diffusivity \(v_{t, s}\) of sediment mass is taken equal to the turbulent viscosity \(v_t\) of the water. However, it can also be argued that the mixing of water and sediment are two different things. Whatever approach is taken, normally the damping of turbulence due to high sediment concentrations is taken into account. This refers to the influence that sediment particles have on the turbulence structure of the fluid. This effect becomes increasingly important for high sediment concentrations that result in stratification and hence damping of turbulence. This affects both the water motion and the sediment distribution. Empirical formulations are sometimes used, which reduce the eddy viscosity dependent on the sediment concentration.
The non-steady advection-diffusion equation now reads:
\[\dfrac{\partial c}{\partial t} - w_s \dfrac{\partial c}{\partial z} - \dfrac{\partial }{\partial z} v_{t, s} \dfrac{\delta c}{\delta z} = 0\label{eq6.6.2.6}\]
As a bottom boundary condition often a sediment concentration at a certain level near the bed is prescribed, the so-called reference concentration. This concentration is prescribed at a certain reference level, for instance \(z_a = 2D_{50}\), also depending on the bed load formulation. The reference concentration is often assumed to be a function of the bed shear stress (much like the bed load transport formulations). Hence, at the bed the response is quasi-steady, whereas higher in the vertical the sediment concentration lags behind the shear stress at the bed. An alternative boundary condition is a pick-up function that prescribes the vertical concentration gradient instead of the concentration.
Detailed models that also resolve the wave boundary layer (Sect. 5.4.3) resolve the time-dependent concentration in the vertical in order to model sheet-flow transport and suspended load transport. The time-dependent suspended sediment concentration can be determined with an unsteady advection-diffusion equation such as Eq. \(\ref{eq6.6.2.6}\). This approach involves the modelling of the turbulence, which increases and decreases during a wave period, and is difficult and time-consuming (see also Intermezzo 6.1).