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Lane-Kalinske (1941)

The concentration profile in Lane-Kalinske1 is calculated from a constant eddy diffusivity instead of the standard parabolic profile. The parabolic profile reads:

\[ \nu_{T}(y) = \kappa u_{f}y\ \left( 1 - \frac{y}{h} \right) \]

Equation: Parabolic eddy diffusivity profile.

Here one will normally use the total friction (\(u_{f}\)) to determine the friction velocity because the total friction better represents the turbulence level compared to skin friction (\(u_{f}'\)). Lane-Kalinske in MIKE 1D is based solely on using total friction in agreement with the original publications.

The depth-averaged eddy viscosity is:

\[ \epsilon = \frac{\kappa u_{f}}{h}\int_{0}^{h}{y\left( 1 - \frac{y}{h} \right)dy} = \kappa hu_{f}\int_{0}^{1}{\frac{y}{h}\left( 1 - \frac{y}{h} \right)d\left( \frac{y}{h} \right)} = \kappa hu_{f}\int_{0}^{1}{s(1 - s)ds} = \frac{\kappa hu_{f}}{6} = \frac{hu_{f}}{15} \]

Equation: Depth-averaged eddy viscosity from the parabolic profile.

The concentration profile \(c(y)\) is found by solving the vertical advection-diffusion equation using this constant diffusivity:

\[ w_{s}c + \epsilon\frac{dc}{dy} = 0 \]

Equation: Vertical advection-diffusion balance with constant diffusivity.

With the boundary condition \(c(a) = c_{a}\). The non-dimensional form is:

\[ \frac{w_{s}h}{\epsilon}c + \frac{dc}{d\eta} = 0 \]

Equation: Non-dimensional form of the advection-diffusion balance.

Or:

\[ \frac{dc}{d\eta} = -\frac{w_{s}h}{\epsilon}c \]

Equation: Rearranged non-dimensional advection-diffusion balance.

With the solution:

\[ c(\eta) = c_{a}\ \exp\left( -\frac{15w_{s}}{u_{f}}\ (\eta - a) \right) \]

Equation: Exponential concentration profile from constant eddy diffusivity.

The velocity profile is given by:

\[ u(\eta) = 1 + \frac{1}{\kappa}\frac{u_{f}}{V}\ (1 + \ln\eta) \]

Equation: Velocity profile used in the Lane-Kalinske formula.

It is noted that this does not accurately average to unity, and it gives unrealistic profiles for \(V/u_{f}' < 5\).

Lane & Kalinske (1941) do not provide the sediment concentration 5% of the water depth; it is given by Lane & Kalinske (1939)2:

\[ C_{a} = 555\ \left( \frac{1}{2}\ \frac{u_{f}}{w_{s}}\exp\left( -\left( \frac{w_{s}}{u_{f}} \right)^{2} \right) \right)^{1.61} \]

Equation: Near-bed sediment concentration.

Where:

  • \(C_{a}\) — near-bed concentration [ppm]

Note that [ppm] is approximated by [g/m3].

Originally the equation comes with a factor 5.55 and not 555, but the factor 5.55 is multiplied by a percentage (0-100) of the considered fractions. In MIKE 1D we use fraction weights (0-1) in the bed surface, and therefore the near-bed concentration should be multiplied by 100 to convert to the use of fraction weights.

It is noted that the skin friction is better for representing the near-bed concentration due to its entrainment nature. However, the work of Lane & Kalinske (1939) operates with turbulent fluctuations, which means the total friction is better than skin friction. Lane & Kalinske did not even mention skin friction in the 1939 or 1941 papers.

Vanoni (1953) notes that the analysis of Lane & Kalinske (1939) was not carried out to a point where the near-bed concentration was associated with some distance from the bed, and therefore the determined value is used as \(c_{a}\) MIKE 1D3.

The suspended load capacity is calculated by:

\[ q_{s} = \frac{q}{s\ \rho_{w}}\ C_{0}\left( \frac{w_{s}}{u_{f}} \right)P\left( \frac{w_{s}}{u_{f}},\ \frac{V}{u_{f}} \right) \]

Equation: Suspended load capacity in the Lane-Kalinske formula.

Where:

  • \(q\) — water flux
  • \(q_{s}\) — suspended load
  • \(\rho_{w}\) — water density 106 g/m3
  • \(s\) — sediment relative density

The P-function is given by:

\[ P\left( \frac{w_{s}}{u_{f}},\ \frac{V}{u_{f}} \right) = \int_{a}^{1}{\left( 1 + \frac{1}{\kappa}\frac{u_{f}}{V}\ (1 + \ln\eta) \right)\exp\left( -\frac{15w_{s}}{u_{f}}\ (\eta - a) \right)d\eta} \]

Equation: The P-function integrating the velocity and concentration profiles.

The P-function is tabulated internally in the code. In the original paper (Lane & Kalinske, 1941), the integral was done from \(\eta = 0\), but this cannot be done because the logarithmic velocity profile is not defined for \(\eta = 0\). Here we choose to integrate from \(\eta = a\); it is also noted that the adopted logarithmic velocity profile becomes poorly defined for low \(\eta\)-values.

Here we note that the concentration profile is normalized to \(C(a) = C_{a}\).

The suspended load capacity is translated to an equilibrium sediment concentration.

Notes regarding Lane-Kalinske

This is a suspended load formula, does not apply to bed-load.

Lane-Kalinske can be used in graded models, and indeed Vanoni (1953) gives that the near-bed concentration should simply be scaled with the percentage of the considered fraction in the bed surface. This is equivalent to the standard approach in MIKE 21C and MIKE 1D for multi-fraction models; scaling the near-bed concentration with the relative mass of the fraction in the bed surface is the same as scaling the capacity.

  1. Lane, E.W and Kalinske, A.A. (1941) "Engineering calculations of References suspended sediment", Trans.Am.Geophys.Union, vol.22, pp. 603-607. ↩

  2. Lane, E. W., and AAj Kalinske (1939) "The relation of suspended to bed material in rivers." Eos, Transactions American Geophysical Union 20.4 (1939): 637-641. ↩

  3. Vanoni, Vito A. (1953) "A summary of sediment transportation mechanics.", 129-160. ↩