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

This paper presents the results of an experimental investigation on scour below pipelines exposed to waves. The effect of leewake of the pipe is the key element in the scour process, and it is demo...

Scour Below Pipelines in Waves

This study concerns the flow around the base of a vertical, wall-mounted cylinder - a pile - exposed to waves. The study comprises (i) flow visualization of horseshoe-vortex flow in front of and the lee-wake-vortex flow behind the pile and (ii) bed shear stress measurements around the pile conducted in a wave flume, plus supplementary bed shear stress measurements carried out in an oscillatory-flow water tunnel. The Reynolds number range of the flume experiments is ReD = (2-9) x 103 and that of the tunnel experiments is ReD = 103—5 x 104, in which ReD is based on the pile size. Steadycurrent tests were also carried out for reference. The horseshoe-vortex flow (like leewake-vortex flow) is governed primarily by the Keulegan-Carpenter number, KC. The range of KC was from 0 to about 25 in the flume experiments, and from 4 to 120 in the tunnel experiments. The experiments were conducted mainly with circular piles. The results indicate that no horseshoe vortex exists for KC < 6. The size and lifespan of the horseshoe vortex increase with KC. The influence of the cross-sectional shape of the pile on the horseshoe vortex was investigated. The results show that a square pile with 90° orientation produces the largest horseshoe vortex while that with 45° orientation produces the smallest one, the circular-pile result being between the two. The influence of a superimposed current on the horseshoe vortex was also investigated. The range of the current-to-wave-induced-velocity ratio, Uc/Um, was from 0 to about 0.8. The overall effect of the superimposed current is to increase the size and lifespan of the horseshoe vortex. This effect increases with increasing Uc/Um. Regarding the near-bed lee-wake flow, the flow regimes observed for the two-dimensional free-cylinder case exist for the present case, too, but with one exception: in the present case, no transverse vortex street was observed in the so-called single-pair regime. The results show that the bed shear stress beneath the horseshoe vortex and in the lee-wake area is heavily influenced by KC. The amplification of the bed shear stress with respect to its undisturbed value is maximum (O(4)) at the side edges of the pile, in contrast to what occurs in steady currents where the maximum occurs at an angle of about 45° from the upstream edge of the pile with an amplification of O(10).

The horseshoe vortex and vortex shedding around a vertical wall-mounted cylinder exposed to waves

This work concerns the combined oscillatory flow and current in a circular, smooth pipe. The study comprises wall shear stress measurements, and laser-Doppler-anemometer velocity and turbulence measurements. Three kinds of pipes were used, with diameters D=19 cm, 9 cm, and 1.1 cm, enabling the influence of the parameter R/δ to be studied in the investigation (R/δ ranging from about 3 to 53), where R is the radius of the pipe, and δ is the Stokes layer thickness. The ranges of the two other parameters of the combined flow processes, namely the current Reynolds number, Rec, and the oscillatory-flow boundary-layer (i.e. the wave–boundary layer) Reynolds number, Rew, are: Rec=0−1.6×105, and Rew=0−7×106. The transition to turbulence in the combined flow case occurs at a current Reynolds number larger than the conventional value, ca. 2×103, depending on Rew, and R/δ. A turbulent current can be laminarized by superimposing an oscillatory flow. The overall average value of the wall shear stress (the mean wall shear stress) may retain its steady-current value, it may decrease, or it may increase, depending on the flow regime. The increase (which can be as much as a factor of 4) occurs when the combined flow is in the wave-dominated regime, while the oscillatory-flow component of the flow is in the turbulent regime. The component of the wall shear stress oscillating around the mean wall shear stress can also increase with respect to its oscillatory-flow-alone value. For this to occur, the originally laminar oscillatory boundary layer needs to become a fully developed turbulent boundary layer, when a turbulent current is superimposed. This increase can be as much as O(3–4). The velocity profiles across the cross-section of the pipe change near the wall when an oscillatory flow is superimposed on a current, in agreement with the results of the wall shear stress measurements. The period-averaged turbulence profiles across the cross-section of the pipe behave differently for different flow regimes. When the two components of the flow are equally significant, the turbulence profile appears to be different from those corresponding to the fundamental cases; the level of turbulence increases (only slightly) with respect to those experienced in the fundamental cases.

Turbulent combined oscillatory flow and current in a pipe

The distribution of suspended sediment in the combined wave‐current motion is theoretically predicted in the case of a plane bed. This will be the case if the wave‐induced motion close to the bed is sufficiently strong. The theory is able to predict the average value of the concentration as well as the instantaneous values at a given distance from the bed. The theory is compared with laboratory and field measurements.

Distribution of suspended sediment in large waves

This paper presents the results of an experimental investigation of the complete sequence of sediment behaviour beneath progressive waves. The sediment was silty with d50 = 0.060 mm. Two kinds of measurements were carried out: pore-water pressure measurements (across the sediment depth), and water-surface elevation measurements. The process of liquefaction/compaction was videotaped from the side simultaneously with the pressure and water-surface elevation measurements. The video records were then analysed to measure: (i) the time development of the mudline, (ii) the time development of liquefaction and compaction fronts in the sediment and (iii) the characteristics of the orbital motion of the liquefied sediment including the motion of the interface between the water body and the sediment. The ranges of the various quantities in the tests were: wave height, H = 9–17 cm, wave period, T = 1.6 sec, water depth = 42 cm, and the Shields parameter = 0.34–0.59. The experiments reveal that, with the introduction of waves, excess pore pressure builds up, which is followed by liquefaction during which internal waves are experienced at the interface of the water body and the liquefied sediment, the sequence of processes known from a previous investigation. This sequence of processes is followed by dissipation of the accumulated excess pore pressure and compaction of the sediment which is followed by the formation of bed ripples. The present results regarding the dissipation and compaction appear to be in agreement with recent centrifuge wave-tank experiments. As for the final stage of the sequence of processes (formation of ripples), the ripple steepness (normalized with the angle of repose) for sediment with liquefaction history is found to be the same as that in sediment with no liquefaction history.

Jørgen Fredsøe

Papers

Scour Below Pipelines in Waves

The horseshoe vortex and vortex shedding around a vertical wall-mounted cylinder exposed to waves

Turbulent combined oscillatory flow and current in a pipe

Distribution of suspended sediment in large waves

The sequence of sediment behaviour during wave-induced liquefaction