Suspended Load

Suspended load refers to fine particle (\(\lesssim\) 1-2 mm) displacement in the water column. The TELEMAC software suite uses the hydrodynamic Telemac2d/3d models to simulate Suspended load by solving the Advection-Diffusion equations with tracer concentrations. This is why suspended load modeling requires an open boundary LICBOR type for tracers (e.g., 4 or 5) as described in the setup of the boundaries-gaia.cli file.

To activate the simulation of suspended load, add the following to the Gaia steering file:

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/ SUSPENDED LOAD
SUSPENSION FOR ALL SANDS : YES

Additional Sediment Parameters

Fine sediment mixtures involving very fine cohesive particles (less than 0.06-0.1 mm) are referred to as mud in Gaia and so do the keywords in the following paragraphs. More information about mud-related keywords can be found in section 4.2 in the Gaia manual.

Deposition Parameters

For suspended load, the definition of additional sediment properties for every sediment class is required (or enabled).

Particle settling velocities \(w_{s}\) can be defined with the CLASSES SETTLING VELOCITIES keyword to calculate the deposition flux \(q_{s,dep}\):

(14)\[ q_{s,dep} = w_{s} \cdot c_{mud} \cdot \left[1 - \left(\frac{\sqrt{\tau / \rho_{w}}}{u_{cr}}\right)^{2} \right] \]

where \(c_{mud}\) is the concentration of mud in g/l (i.e., g m\(^{-3}\)), \(\tau\) is dimensional bed shear stress (N\(\cdot\)m\(^{-2}\)), and \(u_{cr}\) is the critical shear velocity for mud deposition (m s\(^{-1}\)).

The settling velocity is computed as [Dey14]:

(15)\[ w_{s} = \sqrt{\frac{4}{3}\cdot \frac{(s-1)\cdot g\cdot h}{C_{D}}} \]

where \(C_{D}\) is the dimensionless drag coefficient that is a function of the Reynolds number \(Re\). The fundamental equation for calculating the drag coefficient stems from Stokes [Sto50]:

(16)\[ C_{D} = \frac{24}{Re} \]

Experimental data have shown that Stokes [Sto50]’ original equation requires adaptations for high Reynolds numbers (\(Re >\) 10-100) [Rou39]. Gaia comes with integrated algorithms for calculating the settling velocity \(w_{s}\) and the drag coefficient \(C_{D}\) as a function of the grain size. To take advantage of Gaia’s integrated routines for calculating \(w_{s}\), either do not use the CLASSES SETTLING VELOCITIES keyword in the Gaia steering file, or set its per-class values to -9. Detailed information on the calculation of settling velocities for particular cases (e.g., suspended load calculation for other suspended material than mineral sediment) can be found, for example, in Dey [Dey14] (book section 1.7). Gaia’s settling velocity algorithm is located in the file settling_vel.f.

The critical shear velocity \(u_{cr}\) for mud deposition can be defined with the CLASSES CRITICAL SHEAR VELOCITY FOR MUD DEPOSITION keyword (default is 1000. m s\(^{-1}\)). The fundamental definition of shear-related sediment mobility stems from the Shields parameter, which refers to sediment erosion (i.e., the mobilization of an immobile grain), against the critical shear velocity for mud referring to deposition. Since an onset (initialization) energy must be overcome in the erosion process, the shear threshold for erosion is higher than for deposition. Vice versa, the critical shear stress for deposition is smaller than the shear stress for erosion.

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CLASSES SETTLING VELOCITIES : -9;-9;-9
CLASSES CRITICAL SHEAR VELOCITY FOR MUD DEPOSITION : 1000;1000;1000 / N per m2

Erosion Parameters

Gaia calculates erosion fluxes \(q_{s,e}\) as the ratio of bed shear stress \(\tau\) and its critical value for mud erosion \(\tau_{cr}\):

(17)\[\begin{split} q_{s,e} = \begin{cases} M\cdot \left(\frac{\tau}{\tau_{cr}} - 1\right) & \mbox{ if } \tau > \tau_{cr} \\ 0 & \mbox{ if } \tau \leq \tau_{cr}\end{cases} \end{split}\]

where \(M\) denotes the Krone [Kro62]Partheniades [Par65] erosion constant (in kg m\(^{-2}\) s\(^{-1}\)), which can be defined in Gaia with the LAYERS PARTHENIADES CONSTANT keyword (default value: 1.E-03). Moreover, \(\tau_{cr}\) for mud erosion can be defined with the LAYERS CRITICAL EROSION SHEAR STRESS OF THE MUD keyword (default is 0.01;0.02;0.03;...) in N\(\cdot\)m\(^{-2}\) (not used here).

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LAYERS PARTHENIADES CONSTANT : 1.E-03 / in kg per m2 per s
/ LAYERS CRITICAL EROSION SHEAR STRESS OF THE MUD : 0.01;0.1;0.1 / in N per m2

Suspended Load Formulae

The sediment transport formulae for suspended load modeling can be defined with the SUSPENSION TRANSPORT FORMULA FOR ALL SANDS keyword, which accepts an integer number defining a formula for calculating the equilibrium near-bed concentration \(c_{eq}\) in g/l (i.e., gram per 10\(^{-3}\) m\(^3\)). The calculated \(c_{eq}\) values align with the later definition of initial and boundary conditions for suspended load. The following integer numbers can be used for calculating \(c_{eq}\) with the SUSPENSION TRANSPORT FORMULA FOR ALL SANDS keyword:

Initial and Boundary Conditions

Requires TELEMAC v8p2 or more recent

The setup of initial and boundary conditions described in this eBook requires TELEMAC v8p2 or later. Earlier versions, such as v8p1, will not recognize the keywords for initial conditions.

Gaia enables a class-wise definition of initial concentrations for suspended load following the order of sediment class definitions. The following list definition sets the initial concentration for the 0.5-mm sediment class (recall its definition) to 0.6 g/l (i.e., 0.0006 gram per m\(^3\)) and 0.0 g/l for the 0.02-m and 0.1-m sediment size classes. The definition of initial suspended sediment concentrations can be overridden in 2d at boundary nodes by setting the EQUILIBRIUM INFLOW CONCENTRATION keyword to YES (requires that the tracer boundary is set to 5).

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INITIAL SUSPENDED SEDIMENTS CONCENTRATION VALUES : 6.E-04;0.;0.

Read more about the definition of initial conditions in section 2.1.1 in the Gaia manual.

Boundary Prescriptions

The per-sediment class suspended load concentrations can be prescribed similar to the initial concentrations with the PRESCRIBED SUSPENDED SEDIMENTS CONCENTRATION VALUES keyword. Alternatively, the EQUILIBRIUM INFLOW CONCENTRATION keyword may be used, though none of these keywords is used in this tutorial.

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/ PRESCRIBED SUSPENDED SEDIMENTS CONCENTRATION VALUES : 6.E-04;0.;0.
/ EQUILIBRIUM INFLOW CONCENTRATION : YES / not used in  this tutorial

Gaia can be run with liquid boundary files for assigning time-dependent suspended load fluxes (the outflow should be kept in equilibrium). Solid flux time series can be implemented using the already applied 455-5 upstream boundary type, analogous to the descriptions of the Telemac2d unsteady boundary setup. More information about suspended load boundary conditions can be found in section 2.1.2 in the Gaia manual.

Numerical Parameters

Most numerical parameters for suspended load modeling depend on hydrodynamic Telemac2d/3d steering file definitions. Additional keywords directly affecting the simulation of suspended load should be declared in the Gaia steering file.

For instance, the SCHEME FOR ADVECTION … keywords for velocities, tracers, and turbulence modeling are defined with the hydrodynamics (Telemac2d/3d) steering file’s general numerical parameters for finite elements. In addition, the advection scheme for suspended load can be defined in the Gaia steering file with the SCHEME FOR ADVECTION OF SUSPENDED SEDIMENTS keyword that accepts one of the following integer keywords (for 2d only):

  • 1 for the unconditionally stable, non-conservative, but diffusive (for small timesteps) Characteristics scheme.

  • 2 for the non-conservative Streamline Upwind Petrov Galerkin (SUPG) scheme that uses the CFL condition and is less diffusive than the Characteristics (1) scheme.

  • 3 or 4 for the conservative form of the Continuity equation (Conservative N-scheme) and with timestep reduction based on the CFL condition. This option should not be used in the presence of tidal flats (use 13 or 14).

  • 5 for the mass-conservative PSI distributive scheme (default), which corrects fluxes according to tracer concentrations and is less diffusive than 4 or 14; thus, the computation time with 5 is longer than with 4 or 14. This option should not be used in the presence of tidal flats.

  • 13 and 14 for the Edge-based N-scheme (NERD), which is similar to 3 and 4, but adapted to tidal flats. Option 14 is used in this tutorial according to the recommendation in the Gaia manual.

  • 15 for the mass-conservative ERIA scheme that works with tidal flats.

The options 4 and 14 can be defined along with the keyword definition CONTINUITY CORRECTION : YES that enables a correction of vertical convection velocities. This setting avoids overestimating suspended load, especially in deep waters, but it is not used in this tutorial.

The SCHEME OPTION FOR ADVECTION OF SUSPENDED SEDIMENTS can be additionally defined to either use a strong (default of 1) or a weak (2) form for advection. A weak form decreases Diffusion, is more conservative, and increases computation time (read more in the Telemac2d steady section).

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SCHEME FOR ADVECTION OF SUSPENDED SEDIMENTS : 14
/ CONTINUITY CORRECTION : YES / use when SCHEME is 4 or 14

Read more about the definition of initial conditions in section 2.1.5 in the Gaia manual.

Example Applications

Examples for the implementation of suspended load come along with the TELEMAC installation (in the /telemac/v8p2/examples/gaia/ directory). The following examples in the gaia/ folder feature (pure) suspended load calculations:

  • 2d model of combined cohesive and non-cohesive suspended transport: hippodrome-t2d/

  • 2d model of cohesive mud: mud_conservation-t2d/

  • 3d model of combined cohesive and non-cohesive suspended transport: hippodrome-t2d/

  • 3d model of non-cohesive suspended transport with skin friction correction: lyn-t3d/

  • 3d model of cohesive suspended transport with rouse vertical profile (cf. Gaia manual, section 2.1.2): rouse-t3d/

  • 3d model of a tidal flume with cohesive sediment: tidal_flats-t3d/

  • Coupling with waves: sandpit-t2d/