Bed roughness and friction factors in estuaries

Introduction

Water movements in an estuary are driven by gravity, through water-level gradients induced by tides, wind and river discharge and through density gradients. The momentum dissipation along the course of the flow is influenced by multiple processes: friction of the water against the bed and the banks, irregularities of the bed (so-called “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 processes are accounted for individually and explicitly in operational models.

1D models

In 1D models for rectangular channels of constant width, the equations describing the flow are the continuity equation (mass conservation) and momentum conservation :

[math] \dfrac{\partial h}{\partial t} + \dfrac{\partial hu}{\partial x} =0 \, , \qquad (1)[/math]

[math] \dfrac{\partial u}{\partial t} + u \dfrac{\partial u}{\partial x} + g \dfrac{\partial (z_b+h)}{\partial x} = -F \, , \qquad (2) [/math]

in which [math]h=h(x,t)[/math] is the mean water depth in a cross section, [math]u=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 and [math]g[/math] is the gravity acceleration.

[math]F[/math] is the resistance term that incorporates momentum loss by all the processes not explicitly accounted for in the 1D description. In practice, [math]F[/math] is often assumed to be dominated by boundary friction and is expressed in terms of an effective boundary shear stress [math]\tau[/math] and the hydraulic radius [math]R[/math],

[math] F = \dfrac{\tau}{\rho R} \, . \qquad (3)[/math]

The hydraulic radius is given by the ratio cross-section to wetted perimeter; in wide shallow estuaries it is approximately equal to the water depth [math]h[/math]. The effective boundary shear stress [math]\tau[/math] is related to the velocity [math]u[/math] by a dimensionless friction factor [math]c_D[/math],

[math] \tau = c_D \rho |u| u \, , \qquad (4)[/math]

where [math]\rho[/math] the water density. In a simplified estuary model, the calibrated friction coefficient [math]c_D[/math] generally represents the combined effect of bed and bank friction and unresolved flow resistance. It should therefore be regarded as an effective model parameter rather than a direct measure of the physical roughness of the sediment bed. 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. Manning-Strickler or by the semi-empirical White-Colebrook formula. For estuaries, typical values of [math]c_D[/math] are in the range 0.001-0.004.

Manning-Strickler formula

The Manning-Strickler formula for the friction coefficient introduces an explicit dependence on the depth [math]h[/math]:

[math]c_D = g n^2 R^{-1/3}= g K^{-2} R^{-1/3} \, , \qquad (5)[/math]

in which [math]n[/math] is the Manning coefficient and [math]K[/math] the Strickler coefficient. The Manning-Strickler formula applies to channels of different cross-sectional geometries when expressed in terms of the hydraulic radius [math]R[/math]. For wide, shallow channels (width-depth ratio greater than 10), the hydraulic radius can be approximated by the mean water depth [math]h[/math].

For granular beds and in the absence of bedforms, the Manning (Strickler) coefficient can be estimated by

[math] n = 1/K = (0.038 - 0.047) \times d_{90}^{1/6} \, , \qquad (6)[/math]

where [math]d_{90}[/math] is the [math]90\%[/math] grain diameter of the bed material in meters ([math]90\%[/math] is finer).

White-Colebrook (Thijsse) equation

For hydraulically rough channels - the case of estuaries - the semi-empirical White-Colebrook reads

[math] c_D \approx 0.03 \Big[ log_{10} \big( \dfrac{k_s}{12 h} \big) \Big]^{-2} \, . \qquad (7)[/math]

Here is [math]k_s[/math] [m] the roughness height, which is of the order of [math]d_{90}[/math] or 2-3 times the median grainsize [math]d_{50}[/math] for a flat bed.

For a flat granular bed, the equivalent roughness height [math]k_s[/math] can be related to the sediment grain size. In estuary models, however, calibrated values of [math]k_s[/math] or [math]n[/math] generally incorporate additional flow resistance, including form drag and unresolved effects of channel geometry. They therefore represent effective hydraulic roughness parameters. Calibration is done by trial and error, by comparing computed water surface profiles with measured water surface profiles (tide gauges / water-level gauges along the estuary). A different data set should be used for the validation of the roughness model. Discharge measurements are particularly valuable for distinguishing errors in flow resistance from errors associated with the representation of channel geometry. The 'overall' quality of the calibration/ validation is measured by statistical methods such as: 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 term can be approximated by an equivalent linear term retaining its fundamental harmonic:

[math] F = r \dfrac{u}{ h} [/math] with [math]r \approx \dfrac{ 8 c_D}{3 \pi} U \, . \qquad (8)[/math]

2D models

In 2D models, the governing equations are (conservation of momentum and mass)

[math] \dfrac{\partial u}{\partial t} + u \dfrac{\partial u}{\partial x} + v \dfrac{\partial u}{\partial y} + g \dfrac{\partial \eta}{\partial x} - fv = - c_D \dfrac{u (u^2+v^2)^{1/2}}{h} + c_W \dfrac{\rho_{air}}{\rho} \dfrac{W^2 \cos \theta}{h} + N_1 \Delta u \, , \qquad (9)[/math]

[math] \dfrac{\partial v}{\partial t} + u \dfrac{\partial v}{\partial x} + v \dfrac{\partial v}{\partial y} + g \dfrac{\partial \eta}{\partial y} + fu = - c_D \dfrac{v (u^2+v^2)^{1/2}}{h} + c_W \dfrac{\rho_{air}}{\rho}\dfrac{W^2 \sin \theta}{h} + N_2 \Delta v \, , \qquad (10)[/math]

[math] \dfrac{\partial \eta}{\partial t} + \dfrac{\partial hu}{\partial x} + \dfrac{\partial hv}{\partial y} =0 \, . \qquad (11)[/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]h(x,y,t)[/math] is the total depth, [math]\rho_{air}[/math] the air density, [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 \sim 1.3 \times 10^{-3}[/math] the surface friction factor due to wind, [math]W[/math] the wind velocity, [math]\theta[/math] the wind angle.

The terms involving [math]N_1, N_2 [/math] represent horizontal turbulent momentum diffusion in simplified form. More general formulations account for spatially varying eddy viscosity and water depth. Turbulent momentum diffusion can be represented by empirical eddy-viscosity formulations or by turbulence closure models. In 2D depth-averaged models, simple horizontal eddy-viscosity formulations are widely used (constant eddy viscosity or diffusion coefficients depending on the scale of the vortices that can develop), whereas 3D models commonly employ more elaborate turbulence closures.

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 is the shear resistance exerted directly by the sediment surface, 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 White-Colebrook formula.
• Form drag or form roughness (bar or shape resistance) results from pressure differences and flow separation around bedforms. Bedforms may vary between small ripples (with a height which is a few orders of magnitude smaller than the water depth) and mega-ripples 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, see River tides and Estuarine turbidity maximum).

Empirical formulas for the flow resistance due to form roughness are given in the article Bedforms and roughness. The total drag can be estimated by calibrating the value of [math]c_D[/math] (or [math]n[/math] and [math]K[/math]). Calibration is usually done by comparing calculated water surface profiles with measured ones, for different river discharges and tides (spring, neap). Spatial current-velocity measurements provide additional constraints on the calibration of 2D models. After calibration, validation has to be done using a different data set.

3D models

For a fully rough turbulent bed boundary layer the velocity profile is logarithmic,

[math]u(z) = \dfrac{u_*}{\kappa} \ln \big( \dfrac{z}{z_0} \big) \; ; \quad z_0 \approx \dfrac{k_s}{30} \; ; \quad \kappa \approx 0.4 \, . \qquad (12)[/math]

Here [math]z[/math] is the distance above the bed and [math]u_*[/math] is the friction velocity.

The expression for [math]c_D[/math] follows by integrating the logarithmic profile over the water depth, assuming that the profile is applicable over most of the water column. Integrating over the depth gives the depth-averaged friction coefficient

[math]c_D = \Big[ \dfrac{\kappa}{\ln (h /e \, z_0) }\Big]^2 \, , \qquad (13)[/math]

which links the roughness height [math]k_s[/math] to the 2D friction factor and to the bed boundary condition used in 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. In the case of well mixed estuaries, a simplified turbulence model can be chosen for the Reynolds stresses. Many 3D models use turbulence models such as [math]k-\epsilon[/math], introducing additional equations for turbulent kinetic energy [math]k[/math] and turbulent energy dissipation [math]\epsilon[/math] .

In the case of partially mixed or very turbid estuaries, these turbulence models must incorporate the effect of density stratification induced by vertical gradients in salinity or suspended sediment. Stable density stratification can strongly suppress vertical turbulent mixing and modify the velocity profile and bottom stress, see Estuarine circulation. Consequently, the relationship between depth-averaged velocity and bottom friction may differ substantially from that in a well-mixed estuary. 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. Fluid mud layers can additionally modify the near-bed flow through their high density and non-Newtonian rheological properties, see Fluid mud. Calibration of the turbulence model is done by comparing model results with measured velocity and salinity profiles or suspended sediment concentration profiles.

The physical interpretation of the friction coefficient depends on the dimensionality and spatial resolution of the model. In 1D models, a calibrated friction coefficient represents a broad range of unresolved flow-resistance effects. In 2D models, part of the horizontal flow structure is resolved, but vertical momentum transfer remains parameterized. In 3D models, vertical velocity gradients are resolved and turbulent momentum transport is represented by turbulence closure models, allowing the bottom boundary condition to be related more directly to physical bed roughness. Nevertheless, calibrated roughness parameters may still incorporate unresolved bedforms and other small-scale processes.

Related articles

Bedforms and roughness

Coastal and marine sediments

River tides

Estuarine turbidity maximum

Tidal asymmetry and tidal inlet morphodynamics

Further reading

Tennekes, H. and Lumley, J.L. (1972) A first course in turbulence. M.I.T. Press, Cambridge

Soulsby, R. (1997) Dynamics of marine sands, Thomas Telford, London

The main author of this article is Jean Berlamont Please note that others may also have edited the contents of this article. Citation: Jean Berlamont (2026): Bed roughness and friction factors in estuaries. Available from http://www.coastalwiki.org/wiki/Bed_roughness_and_friction_factors_in_estuaries [accessed on 26-10-2026] For other articles by this author see Category:Articles by Jean Berlamont For an overview of contributions by this author see Special:Contributions/U0006958

Article reviewed by

Job DronkersSee the discussion page