Bed roughness and friction factors in estuaries
Contents
Introduction
Water movements in an estuary are driven by gravity, through waterlevel gradients induced by tides, wind and river discharge and through density gradients. Momentum is dissipated along the course of the flow because of a number of reasons: friction of the water against the bed and the banks, irregularities of the bed (socalled “bedforms”), channel bends, turbulence, density currents, sediment transport, friction at the free surface, waves, irregularities of the cross section, groins, sills, ... Not all of these phenomena are accounted for individually and explicitly in operational models.
1D models
In 1D models, the equations describing the flow are the continuity equation (mass conservation) and momentum conservation :
[math] \frac{\partial h}{\partial t} + \frac{\partial hu}{\partial x} =0 , [/math]
[math] \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + g \frac{\partial (z_b+h)}{\partial x} = W , [/math]
in which [math]h(x,t)[/math] is the mean water depth in a cross section, [math]u(x,t)[/math] the mean velocity averaged over the cross section, [math]x[/math] is the distance along the estuary, [math]z_b[/math] the bed level relative to a horizontal datum, [math]t[/math] is time, [math]\rho[/math] the water density and [math]g[/math] is the gravity acceleration.
[math]W[/math] is the resistance term or momentum loss per unit weight of water. [math]W[/math] is assumed to be caused mainly by bottom friction and is expressed as a bed shear stress [math]\tau[/math] divided by the hydraulic radius [math]R,[/math]
[math] W = \frac{\tau}{\rho R} . [/math]
The hydraulic radius is given by the ratio crosssection to wetted perimeter; in wide shallow estuaries it is approximately equal to the water depth [math]h.[/math] The bed shear stress is related to the velocity [math]u[/math] by a dimensionless friction factor [math]c_D[/math],
[math] \tau = c_D \rho u^2 .[/math]
The bottom friction factor [math]c_D[/math] incorporates all flow resisting effects and therefore has no welldefined physical meaning. Instead of [math]c_D[/math], sometimes the Chezy coefficient [math]C \; [m^{1/2} s^{1}][/math] is used as friction factor; it is related to [math]c_D[/math] by [math]C=\sqrt{g/c_D}.[/math]
The friction factor is not a constant but (even in the absence of other drag forces than friction) depends for example on the water depth. It can (for a particular cross section or a “homogenous” river stretch) be calculated by an empirical formula e.g. ManningStrickler or by the semiempirical WhiteColebrook formula. For estuaries, typical values of [math]c_D[/math] are in the range 0.0010.004.
ManningStrickler formula
The ManningStrickler formula for the friction coefficient introduces an explicit dependence on the depth [math]h[/math]:
[math]c_D = g n^2 h^{1/3}= g K^{2} h^{1/3} , [/math]
in which [math]n[/math] is the Manning coefficient and [math]K[/math] the Strickler coefficient. Stricklertype equations (quadratic friction laws) are applicable to wideshallow channels (widthdepth ratio greater than 10) where the hydraulic radius can be replaced by the mean depth.
For granular beds and in the absence of bedforms, the Manning (Strickler) coefficient can be estimated by ^{[1]}: [math] n = 1/K = (0.04  0.047) \times d_{90}^{1/6} , [/math] where [math]d_{90}[/math] is the [math]90\%[/math] grain diameter of the bed material ([math]90\%[/math] is finer).
WhiteColebrook (Thysse) equation
For large Reynolds numbers  the case of estuaries  the empirical WhiteColebrook reads ^{[2]}
[math] c_D \approx 0.03 \left[ log_{10} \left( \frac{k_s}{12 h} \right) \right]^{2} . [/math]
Here is [math]k_s[/math] [m] the roughness height, which is of the order of [math]d_{90}[/math] or 23 times the median grainsize [math]d_{50}[/math] for a flat bed.
Since the friction factor includes all other flow resistances, [math]n[/math] or [math]k_s[/math] is not a real physical parameter but rather a calibration parameter. Calibration is done by trial and error, by comparing computed water surface profiles with measured water surface profiles (maregraph stations or limnigraphs along the estuary). A different data set should be used for the validation of the roughness model. The ‘overall’ quality of the calibration/ validation is measured by statistical methods as there are: RMSE, standard deviation, or the percentage of time that the deviation between the measurement and the model remains within a chosen accuracy margin.
In case of a sinusoidal velocity, [math] u = U \cos ( 2 \pi t/T) [/math], the quadratic friction is sometimes replaced by an equivalent linear friction term
[math] W = r \frac{u}{ h} [/math] with [math]r \approx \frac{ 8 c_D}{3 \pi} U .[/math]
2D models
In 2D models, the governing equations are (conservation of momentum and mass)
[math] \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + g \frac{\partial \eta}{\partial x}  fv =  c_D \frac{u (u^2+v^2)^{1/2}}{h} + c_W \frac{W^2 \cos \theta}{h} + N_1 \Delta u , [/math]
[math] \frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + g \frac{\partial \eta}{\partial y} + fu =  c_D \frac{v (u^2+v^2)^{1/2}}{h} + c_W \frac{W^2 \sin \theta}{h} + N_2 \Delta v , [/math]
[math] \frac{\partial \eta}{\partial t} + \frac{\partial hu}{\partial x} + \frac{\partial hv}{\partial y} =0 . [/math]
These are the so called “de St. Venant equations” or “shallow water equations”, in which [math]\eta(x,y,t)[/math] is the water level relative to a horizontal datum, [math]u(x,y,t), v(x,y,t)[/math] are the [math]x,y[/math]components of the current velocity, [math]f[/math] is the Coriolis parameter (at [math]51^0[/math] latitude [math]f= 1.13 10^{4} s^{1}[/math]), [math]c_W[/math] the surface friction factor due to wind, [math]W[/math] the wind velocity, [math]\theta[/math] the wind angle. The kinematic turbulent diffusion coefficients [math]N_1, N_2 [/math] should be calculated by an appropriate turbulence model, for example: constant eddy viscosity, mixing length model (diffusion coefficients depending on the scale of the vortices that can develop) or [math]k\epsilon[/math] model (introducing additional equations for turbulent kinetic energy [math]k[/math] and turbulent energy dissipation [math]\epsilon[/math] ^{[3]}). In 2D and 3D models, the use of [math]k\epsilon[/math] is state of the art. The actual value of [math]c_D[/math] depends on the bed roughness, which may be decomposed into two parts (see also Bedforms and roughness):
 Skin friction or surface drag, which is mainly a function of the bed roughness or the diameter of the grains that compose the bed; [math]c_D[/math] is either assumed to be a constant in a particular section or stretch of the estuary or calculated by an empirical formula e.g. Manning  Strickler or by the WhiteColebrook formula.
 Form drag or form roughness (bar or shape resistance), which is caused by the presence of bedforms on the bottom of the estuary. Bedforms may vary between small ripples (with a height which is a few orders of magnitude smaller than the water depth) and megaripples or dunes (with a height of the same order of magnitude as the water depth). Their formation and appearance is a function of the flow velocity (incl. the effect of waves) and varies from place to place and with time (a.o. as a function of river discharge or the presence of mud).
The flow resistance due to form roughness can be calculated with a WhiteColebrook type equation. For currents only, an empirical formula for the ripple bed roughness ([math]k_r[/math]) was derived by van Rijn (2007^{[4]}) for a sandy bed ([math]d_{50}\lt 0.5 [/math] mm),
[math] k_r \approx d_{50} \left(8565 \tanh[0.015(\psi150)]\right) ,[/math]
where [math]\Psi[/math] is the mobility number [math]\psi = \rho (u^2+v^2)/(g d_{50} \Delta \rho ) [/math] and [math]\Delta \rho[/math] is the sedimentwater density difference.
Calibration takes into account all the other neglected drag forces; [math]c_D[/math] (or [math]n[/math] or [math]K[/math]) then basically becomes a calibration parameter. Calibration of the friction factor is done by comparing calculated water surface profiles with measured ones, for different river discharges and tides (spring, neap). After calibration, validation has to be done using a different data set.
3D models
In 3D models momentum dissipation is represented by Reynolds stresses (time correlations of the turbulent velocity fluctuations [math]u', v', w'[/math]) and by boundary conditions at the bed^{[5]}. In the case of well mixed estuaries, a simplified turbulence model can be chosen for the Reynolds stresses. In the case of partially mixed or very turbid estuaries, turbulence models must incorporate the effect of density stratification induced by vertical gradients in salinity or suspended sediment^{[6]}. Even weak stratification can strongly suppress turbulence and momentum dissipation ^{[7]}. Modelling of the 3D structure of the salinity or suspended sediment distributions (including the occurrence of fluid mud layers) is thus required in such situations. Calibration of the turbulence model is done by comparing model results with measured velocity and salinity profiles or suspended sediment concentration profiles.
Related articles
References
 ↑ Ackers, P. and White, W.R. (1973) Sediment Transport: New Approach and Analysis. Journal of the Hydraulics Division, ASCE, No. HY11
 ↑ ASCE Task Force on Friction Factors in Open Channels (1963) Friction factors in open channels. J. Hydraulics Div. ASCE 89: 97–143
 ↑ Launder, B.E. and Spalding, D.B. (1974) The numerical computation of turbulent flows. Computer Methods in Applied Mechanics and Engineering 3: 269–289
 ↑ Van Rijn, L. (2007) Unified View of Sediment Transport by Currents and Waves. I: Initiation of Motion, Bed Roughness, and BedLoad Transport. Journal of Hydraulic Engineering ASCE 133: 649667
 ↑ Chanson, H. (2004) The hydraulics of open channel flow. Elsevier Butterworth Heinemann
 ↑ Abraham, G. (1988) Turbulence and mixing in stratified tidal flows. In: Physical Processes in Estuaries (Eds. J. Dronkers and W. van Leussen), Springer, pp. 278291
 ↑ Friedrichs, C. T. and Wright L. D. (1997) Sensitivity of bottom stress and bottom roughness estimates to density stratification, Eckernforde Bay, J. Geophys. Res., 102, 5721–5732
Further reading
Tennekes, H. and Lumley, T.L. (1872) A first course in turbulence. M.I.T. Press, Cambridge
Soulsby, R. (1997) Dynamics of marine sands, Thomas Telford, London
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