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 critical review of the intrinsic nature of vortex-induced vibrations

The numerical simulation of high Reynolds number flows is hampered by model accuracy if the Reynolds-averaged Navier–Stokes (RANS) equations are used, and by computational cost if direct or large-eddy simulations (LES) that resolve the near-wall layer are employed. The cost of a calculation scales like the Reynolds number to the power 3 for direct numerical simulations, or 2.4 for LES, making the resolution of the wall layer at high Reynolds number infeasible even with the most advanced computers. In LES, an attractive alternative to compute high-Re flows is the use of wall-layer models, in which only the outer layer is resolved, while the near-wall region is modeled. Three broad classes of approaches are presently used: bypassing this region altogether using wall functions, solving a separate set of equations in the near-wall region, weakly coupled to the outer flow, or simulating the near-wall region in a global, Reynolds-averaged, sense. These approaches are discussed and their ranges of applicability are highlighted. Various unresolved issues in wall-layer modeling are presented.

Wall-layer models for large-eddy simulations

New parameterizations for the spectral dissipation of wind-generated waves are proposed. The rates of dissipation have no predetermined spectral shapes and are functions of the wave spectrum and wind speed and direction, in a way consistent with observations of wave breaking and swell dissipation properties. Namely, the swell dissipation is nonlinear and proportional to the swell steepness, and dissipation due to wave breaking is nonzero only when a nondimensional spectrum exceeds the threshold at which waves are observed to start breaking. An additional source of short-wave dissipation is introduced to represent the dissipation of short waves due to longer breaking waves. A reduction of the wind-wave generation of short waves is meant to account for the momentum flux absorbed by longer waves. These parameterizations are combined and calibrated with the discrete interaction approximation for the nonlinear interactions. Parameters are adjusted to reproduce observed shapes of directional wave spect...

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Semiempirical Dissipation Source Functions for Ocean Waves. Part I: Definition, Calibration, and Validation

Hydraulic transients in closed conduits have been a subject of both theoretical stud intense practical interest for more than one hundred years. While straightforward in te of the one-dimensional nature of pipe networks, the full description of transient fluid fl pose interesting problems in fluid dynamics. For example, the response of the turbu structure and strength to transient waves in pipes and the loss of flow axisymme pipes due to hydrodynamic instabilities are currently not understood. Yet, such u standing is important for modeling energy dissipation and water quality in transient flows. This paper presents an overview of both historic developments and presen research and practice in the field of hydraulic transients. In particular, the paper cusses mass and momentum equations for one-dimensional Flows, wavespeed, nu solutions for one-dimensional problems, wall shear stress models; two-dimensional and momentum equations, turbulence models, numerical solutions for two-dimen problems, boundary conditions, transient analysis software, and future practical an search needs in water hammer. The presentation emphasizes the assumptions and tions involved in various governing equations so as to illuminate the range of applic ity as well as the limitations of these equations. Understanding the limitations of cu models is essential for (i) interpreting their results, (ii) judging the reliability of the da obtained from them, (iii) minimizing misuse of water-hammer models in both research practice, and (iv) delineating the contribution of physical processes from the contribu of numerical artifacts to the results of waterhammer models. There are 134 refrences in this review article.@DOI: 10.1115/1.1828050 #

A Review of Water Hammer Theory and Practice

This study deals with turbulent oscillatory boundary-layer flows over both smooth and rough beds. The free-stream flow is a purely oscillating flow with sinusoidal velocity variation. Mean and turbulence properties were measured mainly in two directions, namely in the streamwise direction and in the direction perpendicular to the bed. Some measurements were made also in the transverse direction. The measurements were carried out up to Re = 6 × 106 over a mirror-shine smooth bed and over rough beds with various values of the parameter a/ks covering the range from approximately 400 to 3700, a being the amplitude of the oscillatory free-stream flow and ks the Nikuradse's equivalent sand roughness. For smooth-bed boundary-layer flows, the effect of Re is discussed in greater detail. It is demonstrated that the boundary-layer properties change markedly with Re. For rough-bed boundary-layer flows, the effect of the parameter a/ks is examined, at large values (O(103)) in combination with large Re.

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Turbulent oscillatory boundary layers at high Reynolds numbers

Sheet-flow layer of movable bed in steady currents has been investigated experimentally, using four kinds of sediment. Velocity and concentration profiles as well as flow resistance were measured inside and outside the sheet-flow layer. The range of fall-velocity–to–friction-velocity ratio ( w /U f) achieved in the study is 0.2 0.8–1). In the suspension-mode sheet flow, the flow resistance is found to depend not only on the Shields parameter but also on a parameter involving w . Measured velocity profiles are found to follow the logarithmic law near the bed, outside the sheet-flow layer, whereas inside the sheet-flow layer, they satisfy a power law. Measured concentration profiles, on the other hand, indicate a linear variation with the distance from the bed inside the sheet flow-layer near the bed, whereas away from the bed, the familiar Rouse distribution is sat...

Velocity and concentration profiles in sheet-flow layer of movable bed

Onset of scour below pipelines and self-burial

Sinking/floatation of pipelines and other objects in liquefied soil under 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).

B. M. Sumer

Papers

Turbulent oscillatory boundary layers at high Reynolds numbers

Velocity and concentration profiles in sheet-flow layer of movable bed

Onset of scour below pipelines and self-burial

Sinking/floatation of pipelines and other objects in liquefied soil under waves

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