Einstein decomposition

The total shear stress is decomposed into skin and form for the sediment transport calculations with the following general principles:

• The Rouse number is always calculated from the total friction, being a better reflection of the turbulence levels.
• Interaction with the bed (entrainment, bed load) is reflected by the skin friction unless specifically stated that it is done using the total friction, as in some sediment transport formulas.

Using the Einstein decomposition, the total shear stress is divided into skin and form contributions¹:

\[ \tau = \tau' + \tau'' \]

Equation: Decomposition of the total shear stress into a skin and a form contribution.

Where:

• \(\tau\) — total shear stress
• \(\tau'\) — skin friction shear stress
• \(\tau''\) — form friction shear stress

The same can be used for friction factors:

\[ f = f' + f'' \]

Equation: Decomposition of the total friction factor into a skin and a form contribution.

Where:

• \(f\) — friction factor for the total resistance
• \(f'\) — friction factor for the skin resistance
• \(f''\) — friction factor for the form resistance

The relation between shear stress and friction factor is:

\[ \tau = \rho f V^{2} \]

Equation: Relation between shear stress and friction factor.

Where:

• \(\rho\) — water density
• \(V\) — flow speed

Here we use the term friction factor, while some publications will call \(f/2\) the friction factor. The form friction coefficient is also sometimes referred to as a form drag coefficient (e.g. Parker, 2004²).

Currently in the MIKE 1D model there are three methods for calculating the skin friction shear stress:

1. Total friction
2. Law of the wall (full depth)
3. Law of the wall (skin friction depth)

If using the first method, the skin friction simply equals the total friction used in the HD model.

Law of the wall (full depth)

The easiest approach for calculating the skin friction is to use the law of the wall, which can be calculated by various means, e.g. from the relation (e.g. van Rijn, 1984³):

\[ C' = 18 \log_{10}\left( \frac{4h}{d_{90}} \right) \]

Equation: Chezy number representing skin friction, from the law of the wall over the full depth.

Where:

• \(C'\) — Chezy number representing skin friction
• \(h\) — water depth
• \(d_{90}\) — 90% fractile of the local particle size distribution curve

The friction factor for skin resistance can then be calculated from the Chezy number representing skin friction using:

\[ f' = \frac{g}{(C')^{2}} \]

Equation: Friction factor for skin resistance derived from the skin friction Chezy number.

Where:

• \(g\) — gravitational acceleration

Law of the wall (skin friction depth)

The division of the shear stress into skin and form can be extended to the water depths via the static relation for shear stress:

\[ \tau = \rho g h S \]

Equation: Static relation for shear stress.

Where:

• \(S\) — water surface slope

Resulting in:

\[ h = h' + h'' \]

Equation: Decomposition of the total water depth into a skin and a form contribution.

Where:

• \(h\) — total water depth
• \(h'\) — water depth carried by skin friction
• \(h''\) — water depth carried by form friction

Alternative to the simple law of the wall (full depth), the following implicit formulation can be used instead:

\[ (f')^{-1/2} = 6 + 2.5 \ln\left( \frac{h'}{3 d_{90}} \right) \]

Equation: Implicit formulation of the skin friction factor based on the depth carried by skin friction.

We note here that – for legacy reasons – the formulation is slightly different than what is used when the depth is not partitioned. When using this formulation, the expression for \(f'\) become implicit with the depth carried by skin friction (\(h'\)) relating to the total depth by:

\[ \frac{h'}{h} = \frac{\tau'}{\tau} = \frac{f'}{f} \]

Equation: Relation between the depth ratio, the shear stress ratio and the friction factor ratio.

Yielding the implicit equation solved for \(f'\):

\[ (f')^{-1/2} = 6 + 2.5 \ln\left( \frac{f'}{f} \frac{h}{3 d_{90}} \right) \]

Equation: Implicit equation solved for the skin friction factor.

This equation is solved iteratively where the total resistance is known from the hydrodynamic model. Method 3 with friction-carried depth yields larger skin friction shear stress because the roughness length to depth ratio is seen as larger in the formulation. The difference between Method 2 and Method 3 depends on how much of the total resistance is due to form.

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1. Fredsøe, J. & Deigaard, R. (1992) "Mechanics of Coastal Sediment Transport", World Scientific. ↩

2. Parker, G. (2004) "CHAPTER 9: RELATIONS FOR HYDRAULIC RESISTANCE IN RIVERS", downloadable from hydrolab.illinois.edu. ↩

3. van Rijn, Leo C. (1984a) "Sediment transport, part I: bed load transport." Journal of Hydraulic Engineering 110.10 (1984): 1431-1456. ↩