The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

A method is presented which enables the computation of the bed-load transport as the product of the saltation height, the particle velocity and the bed-load concentration. The equations of motions for a solitary particle are solved numerically to determine the saltation height and particle velocity. Experiments with gravel particles (transported as bed load) are selected to calibrate the mathematical model using the lift coefficient as a free parameter. The model is used to compute the saltation heights and lengths for a range of flow conditions. The computational results are used to determine simple relationships for the saltation characteristics. Measured transport rates of the bed load are used to compute the sediment concentration in the bed-load layer. A simple expression specifying the bed-load concentration as a function of the flow and sediment conditions is proposed. A verification analysis using about 600 (alternative) data shows that about 77% of the predicted bed-load-transport rates are within 0.5 and 2 times the observed values.

Sediment transport; Part I, Bed load transport

Development and validation of a three-dimensional morphological model

A method is presented which enables the computation of the suspended load as the depth-integration of the product of the local concentration and flow velocity. The method is based on the computation of the reference concentration from the bed-load transport. Measured concentration profiles have been used for calibration. New relationships are proposed to represent the size gradation of the bed material and the damping of the turbulence by the sediment particles. A verification analysis using about 800 data shows that about 76% of the predicted values are within 0.5 and 2 times the measured values.

Sediment Transport, Part II: Suspended Load Transport

A critical review of the intrinsic nature of vortex-induced vibrations

Scour around pipelines in combined waves and current

This paper summarizes the results of an experimental and theoretical investigation of: (1) pipeline flotation in a soil (liquefied under waves); and (2) density of the liquefied soil. In the experiments, the soil was silt with d50 =0.078 mm . Pipeline models of 2 cm diameter were used. They were buried in the soil at different depths in the range 3–15.5 cm . The total depth of the silt layer was 17.5 cm . Waves (with 17 cm wave height and 1.6 s wave period, the water depth being 42 cm ) were used to liquefy the soil. The pipes with specific gravity smaller than 1.85–2.0 floated when the soil was liquefied, the critical specific gravity for pipe flotation. The lower bound of the above range corresponds to the initial pipe position near the surface of the bed, and the upper bound to that near the impermeable base. Furthermore, the pipe floated (or sank) to a depth where the pipe specific gravity was equal to the previously mentioned critical specific gravity for flotation. The density of liquefied soil was ...

Critical Flotation Density of Pipelines in Soils Liquefied by Waves and Density of Liquefied Soils

Numerical modeling of shoreline undulations part 1: Constant wave climate

[1] A numerical model solving incompressible Reynolds-averaged Navier-Stokes equations, combined with a two-equation k-ω turbulence closure, is used to study converging-diverging effects from a sloping bed on turbulent (oscillatory) wave boundary layers. Bed shear stresses from the numerical model are first validated against hydraulically smooth and rough data from flat bed experiments in the form of wave friction factor diagrams. The results show that the model provides acceptable accuracy for wave orbital amplitude to roughness length ratios a/kN ≥ 20. Validation against experimental measurements for steady streaming induced by a skewed free stream velocity signal is also provided. We then simulate a series of experiments involving oscillatory flow in a convergent-divergent smooth tunnel, and a good match with respect to bed shear stresses and streaming velocities is achieved. The streaming is conceptually explained using analogies from steady converging and diffuser flows. A parametric study is undertaken to assess both the peak and time-averaged bed shear stresses in converging and diverging half periods under rough-turbulent conditions. The results are presented as friction factor diagrams. A local similarity condition is derived for relating oscillatory flow in a convergent-divergent tunnel, as considered herein, to shoaling shallow-water waves by matching spatial gradients in the free stream orbital length.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2008JC005045

Bed slope effects on turbulent wave boundary layers: 1. Model validation and quantification of rough‐turbulent results

This paper considers the lift forces acting on a pipe with or without a small gap between the pipe and the seabed. From simple physical considerations, it is explained why the potential theory that predicts a downward lift force fails. A modified description of the potential flow in front of the pipe is developed, from which the lift force is evaluated if the undisturbed flow has no velocity gradients (shear‐free flow). The paper also discusses the effect of shear on the lift coefficient. The theoretical findings are supported by experiments.

Jørgen Fredsøe

Papers

Scour around pipelines in combined waves and current

Critical Flotation Density of Pipelines in Soils Liquefied by Waves and Density of Liquefied Soils

Numerical modeling of shoreline undulations part 1: Constant wave climate

Bed slope effects on turbulent wave boundary layers: 1. Model validation and quantification of rough‐turbulent results

Lift Forces on Pipelines in Steady Flow