Reynolds number variation in oscillatory boundary layers: Part I. Purely oscillatory motion

A numerical model based upon a low Reynolds number turbulence closure is proposed to study Reynolds number variation in reciprocating oscillatory boundary layers. The model is used to compute the boundary layer for flow regimes ranging from smooth laminar to rough turbulent. Criteria for fully developed turbulence are derived for walls of the smooth and rough types. In particular, a new criterion to identify the rough turbulent regime is determined based on the time-averaged turbulence intensity. The reliability of the present model is assessed through comparisons with detailed experimental data collected by other investigators. The model globally improves upon standard high Reynolds number closures. Variation through the wave cycle of the main flow variables (ensemble-averaged velocity, shear stress, turbulent kinetic energy) is remarkably well-predicted for smooth walls. Predictions are satisfactory for rough walls as well. Yet, the turbulence level in the rough turbulent regime is overpredicted in the vicinity of the bed.     

A new approach to oscillatory rough turbulent boundary layers

Velocity measurements have been performed in an oscillatory turbulent boundary layer over a rough wall, using a large oscillating water tunnel. These together with measurements by Kalkanis (1964) over an oscillating wall indicate the existence of universal wall and defect laws for velocity. A logarithmic overlap layer is predicted and observed as in a steady turbulent boundary layer, and this results in a new relationship between friction factor and relative boundary layer thickness. The phase lead of the defect velocity relative to the wall ditto seems to follow a universal law over the whole defect layer. A method is suggested for the calculation of the phase lead of wall shear stress over velocity in the free stream for large amplitude to roughness ratios. Apart from the inner layer, it is in principle possible to construct the velocity profiles in a turbulent oscillatory boundary layer at a rough wall, using the findings of this report. A review of experimental and theoretical investigations of the stability of the oscillatory boundary layer is also given.     

Direct bed shear stress measurements in bore-driven swash

Direct measurements of bed shear in the swash zone are presented. The data were obtained using a shear plate in medium and large-scale laboratory bore-driven swash and cover a wide range of bed roughness. Data were obtained across the full width of the swash zone and are contrasted with data from the inner surf zone. Estimates of the flow velocities through the full swash cycle were obtained through numerical modelling and calibrated against measured velocity data. The measured stresses and calculated flow velocities were subsequently used to back-calculate instantaneous local skin friction coefficients using the quadratic drag law. The data show rapid temporal variation of the bed shear stress through the leading edge of the uprush, which is typically two–four times greater than the backwash shear stresses at corresponding flow velocity. The measurements indicate strong temporal variation in the skin friction coefficient, particularly in the backwash. The general behaviour of the skin friction coefficient with Reynolds number is consistent with classical theory for certain stages of the swash cycle. A spatial variation in skin friction coefficient is also identified, which is greatest across the surf-swash boundary and likely related to variations in local turbulent intensities. Skin friction coefficients during the uprush are approximately twice those in the backwash at corresponding Reynolds number and cross-shore location. It is suggested that this is a result of the no-slip condition at the tip leading to a continually developing leading edge and boundary layer, into which high velocity fluid and momentum are constantly injected from the flow behind and above the tip region. Finally, the measured stress data are used to determine the asymmetry and cross-shore variation in potential sediment transport predicted by three forms of sediment transport formulae.     

Modification of the damping function in the k–ε model to analyse oscillatory boundary layers

A simple relationship has been developed between the wall coordinate y⁺ and Kolmogorov's length scale using direct numerical simulation (DNS) data for a steady boundary layer. This relationship is then utilized to modify two popular versions of low Reynolds number k–ε model. The modified models are used to analyse a transitional oscillatory boundary layer. A detailed comparison has been made by virtue of velocity profile, turbulent kinetic energy, Reynolds stress and wall shear stress with the available DNS data. It is observed that the low Reynolds number models used in the present study can predict the boundary layer properties in an excellent manner.     

Numerical study on scale effect of form factor

The scale effect of form factor is investigated via a numerical approach in this paper, where the turbulent ship flow is computed by solving the steady and incompressible Reynolds-averaged Navier-Stokes and continuity equations. A wall function approach is employed to bridge the near-wall and outer turbulent flow region. The numerical scheme based on a finite-volume formulation is applied to discretize the coupled governing equation. For the sake of numerical stability, accuracy and economy, an identical grid is employed to compute ship flow at different Reynolds number, where the grid is optimized for the medium Reynolds number of the investigated range. Four surface ships and two sub-bodies with notably different geometrical characteristics are chosen as the investigated cases, where double-model flow without appendages is computed. The calculated total resistance coefficient shows a decreasing tendency against Reynolds number among all studied hulls. Similar to the calculated total resistance coefficient, the calculated friction resistance coefficient decreases with the Reynolds number and varies relatively little for a given Reynolds number among different hulls. The viscous pressure resistance coefficient is less insensitive to the Reynolds number but apparently depends on hull form. Compared with the form factor calculation based on empirical friction lines, the flat-plate friction prediction based on CFD approach clearly gives smaller Re-dependent form factor, which should more realistically reflect the scale effect of form factor. The form factor exhibits a near linear and increasing dependence on Reynolds number. The numerical results show that the dependence of r_P on Reynolds number mainly governs the scale effect of form factor.     

A model of oscillatory rough turbulent boundary layer flow

Approximate analytical solutions of the boundary layer equation are obtained in closed form for oscillatory rough turbulent flow. The solutions are based on a time-varying eddy viscosity, and the aim of the study is to assess the effects of these time variations on the properties of the wave boundary layer. The flow and the eddy viscosity are made interdependent by a closure assumption which relates the peak value of viscosity in the wave cycle to the peak value of shear velocity. Instantaneous vertical profiles of horizontal velocity and shear stress, and time series of the bed shear stress, are presented for a typical case study. In addition, the wave drag coefficient, the boundary layer thickness and the phase lead of peak bed shear stress over peak free-stream velocity, are determined as functions of both the relative roughness and the parameter governing the magnitude of the time variations in viscosity. Reasonable agreement is demonstrated with previous experimental and theoretical results.     

Calculation of wave-induced turbulent flows in estuaries

The k-ε turbulence model which relates the eddy viscosity to turbulent kinetic energy, k, and to the rate of its dissipation, ε, and determines the distribution of these two quantities from modeled transport equations, is employed in calculating the vertical structure of wave-induced turbulent flows in two-dimensional estuaries. The empirical constants in this model are given the standard values cited in the literature and used successfully for calculating a large variety of steady flows; they are not tuned to the oscillating flows. The free surface elevation and the eddy-viscosity distribution are also calculated as a part of the solution. First, calculations performed for oscillatory laminar flows are compared with analytic solutions to ensure the proper performance of the numerical scheme. The turbulent flow in a laboratory-estuary model with one end closed, and in the Humber Estuary is simulated with the numerical model, and the results are compared with the corresponding flume and field measurements. The influence of frequency and roughness is thereby investigated. The results indicate that the time and space variation of the velocity field and the phase lag between the surface slope and the horizontal velocity can be predicted satisfactorily in wave-induced turbulent flows.     

A traversing system to measure bottom boundary layer hydraulic properties

This study describes a new convenient and robust system developed to measure benthic boundary layer properties, with emphasis placed on the determination of bed shear stress and roughness height distribution within estuarine systems by using velocity measurements. This system consisted of a remotely operated motorised traverser that allowed a single ADV to collect data between 0 and 1 m above the bed. As a case study, we applied the proposed traversing system to investigate bottom boundary layer (BBL) hydraulic properties within Coombabah Creek, Queensland, Australia. Four commonly-employed techniques: (1) Log-Profile (LP); (2) Reynolds stress (RS); (3) Turbulent Kinetic Energy (TKE); and (4) Inertial Dissipation (ID) used to estimate bed shear stresses from velocity measurements were compared. Bed shear stresses estimated with these four methods agreed reasonably well; of these, the LP method was found to be most useful and reliable. Additionally, the LP method permits the calculation of roughness height, which the other three methods do not. An average value of bed shear stress of 0.46 N/m², roughness height of 4.3 mm, and drag coefficient of 0.0054 were observed within Coombabah Creek. Results are consistent with that reported for several other silty bed estuaries.     

Study of the Neutral Ekman Flow Using an Algebraic Reynolds Stress Model

A recently developed fully explicit algebraic model of Reynolds stress and turbulent heat flux in a thermally stratified planetary atmospheric boundary layer without stratification has been used for a numerical study of the Ekman turbulent boundary layer over a homogeneous rough surface for different dimensionless surface Rossby numbers. A comparative analysis has been conducted for a closure model of the transport term in the prognostic equation of turbulent kinetic energy dissipation including third-order moments. Dependences of the total wind rotation angle on the Rossby number have been obtained. The calculated vertical profiles of mean velocity, turbulent stress, turbulent kinetic energy, surface-friction velocity, and boundary-layer height agree satisfactorily with observational and earlier obtained LES data.     

Shear stress statistics and wall similarity analysis in turbulentboundary layers using a high-resolution 3-D ADVP

Three-dimensional (3-D) quasi-instantaneous acoustic Doppler velocity profiles at the center of uniform, turbulent open-channel flow over smooth and rough beds have been analyzed for the dynamics of coherent structures. The qualitative aspects of simultaneously measured Eulerian velocity and shear stress signatures identify coherent structures in the water column. A cumulant discard method is applied to describe the statistical properties of the covariance terms u'w' along the mean flow and v'w' across the mean flow relative to their time means. Conditional statistics and conditional sampling are used to compare the theoretical and experimental relative covariance contributions from the four quadrants in the longitudinal and transverse planes. The results in the (u', w') plane show the dominance of ejections (quadrant 2; u'<0, w'>0) and sweeps (quadrant 4; u'>0, w'<0). In contrast, the distribution of fractional v'w' events in the transverse plane is quasi-uniform over the four quadrants. Based on these experimentally determined statistical properties of the covariance terms in different flow conditions, a simplified form of the vertical turbulent energy flux in the intermediate flow region is given and the concept of wall similarity in turbulent boundary layers is validated. Since the validity of the wall similarity concept over a wide range of bed roughness has been shown, it is proposed to determine the mean bed friction velocity from the evaluation of the vertical turbulent energy flux     

槽道湍流的展向振荡电磁力壁面减阻

采用直接数值模拟方法,对槽道湍流的展向振荡电磁力的减阻效果和减阻机理进行了研究,讨论了电磁力强度和振荡频率对湍流猝发事件以及壁面减阻率的影响.结果表明,电磁力强度或振荡频率变化时,湍流猝发频率和猝发强度的变化趋势是相反的,所以存在最优参数使得减阻效果最好.等价壁面展向速度可以很好地描述电磁力强度和振荡频率的变化对减阻效果的综合效应.     

Force and flow characteristics of a circular cylinder with uniform surface roughness at subcritical Reynolds numbers

The main purpose of this study is to establish a better understanding of the relationship between drag reduction and surface roughness. Experiments were conducted to measure the force and flow characteristics of a circular cylinder with different types of artificial surface roughness over the range 6 × 10³ < Re < 8 × 10⁴ (Re is based on the cylinder diameter D). The roughness cylinder was formed by covering the exterior surface of the cylinder with uniformly distributed (1) sandpaper, (2) netting, and (3) dimples. The roughness coefficient ranged from k/D = 0.0028 to 0.025 (k is the roughness height). A detailed quantitative measurement of the flow field around the cylinder using Particle Imaging Velocimetry (PIV) was carried out. The hydrodynamic force coefficients (drag and lift) of the rough cylinders are compared against those of a smooth cylinder measured under the same flow conditions. It is found that certain configuration of surface roughness significantly reduces the mean drag coefficient of the cylinder, particularly at large Reynolds numbers. In addition, the root-mean-square (r.m.s.) lift coefficient of the rough cylinders is considerably lower than that of a smooth cylinder.     

Tidal boundary layer measurements in the presence of waves

Measurements of tidal current and wave velocity made at 0.69 and 1.85 m above a rough seafloor exhibit large current gradients (boundary layer) in the water column. The logarithmic boundary layer flow model was fitted to the measurements, and thus roughness (z₀) and friction velocity (u^*) parameters were derived. The roughness parameter values were generally consistent with the observed upstream physical roughness. The values of both parameters for conditions in the rough turbulence flow regime are generally larger (much larger for ebb) than earlier published values for similar measurements of currents in the absence of significant waves but are comparable to values from recent measurements of currents in the presence of storm waves. The high parameter values here appear to relate more to the magnitude of the current and to the upstream physical bottom roughness than to the magnitude of the seastate. Large boundary layers in the flow at the seabed have a profound effect on the design of offshore structures such as offshore pipelines.     

Boundary layer approach in the modeling of breaking solitary wave runup

The boundary layer is very important in the relation between wave motion and bed stress, such as sediment transport. It is a known fact that bed stress behavior is highly influenced by the boundary layer beneath the waves. Specifically, the boundary layer underneath wave runup is difficult to assess and thus, it has not yet been widely discussed, although its importance is significant. In this study, the shallow water equation (SWE) prediction of wave motion is improved by being coupled with the k–ω model, as opposed to the conventional empirical method, to approximate bed stress. Subsequently, the First Order Center Scheme and Monotonic Upstream Scheme of Conservation Laws (FORCE MUSCL), which is a finite volume shock-capturing scheme, is applied to extend the SWE range for breaking wave simulation. The proposed simultaneous coupling method (SCM) assumes the depth-averaged velocity from the SWE is equivalent to free stream velocity. In turn, free stream velocity is used to calculate a pressure gradient, which is then used by the k–ω model to approximate bed stress. Finally, this approximation is applied to the momentum equation in the SWE. Two experimental cases will be used to verify the SCM by comparing runup height, surface fluctuation, bed stress, and turbulent intensity values. The SCM shows good comparison to experimental data for all before-mentioned parameters. Further analysis shows that the wave Reynolds number increases as the wave propagates and that the turbulence behavior in the boundary layer gradually changes, such as the increase of turbulent intensity.     

Measurement and modeling of bed shear stress under solitary waves

Direct measurements of bed shear stresses (using a shear cell apparatus) generated by non-breaking solitary waves are presented. The measurements were carried out over a smooth bed in laminar and transitional flow regimes (~ 10⁴ < R_e < ~ 10⁵). Measurements were carried out where the wave height to water depth (h/d) ratio varied between 0.12 and 0.68; maximum near bed velocity varied between 0.16 m/s and 0.51 m/s and the maximum total shear stress (sum of skin shear stress and Froude–Krylov force) varied between 0.386 Pa and 2.06 Pa. The total stress is important in determining the stability of submarine sediment and in sheet flow regimes. Analytical modeling was carried out to predict total and skin shear stresses using convolution integration methods forced with the free stream velocity and incorporating a range of eddy viscosity models. Wave friction factors were estimated from skin shear stress at different instances over the wave (viz., time of maximum positive total shear stress, maximum skin shear stress and at the time of maximum velocity) using both the maximum velocity and the instantaneous velocity at that phase of the wave cycle. Similarly, force coefficients obtained from total stress were estimated at time of maximum positive and negative total stress and at maximum velocity. Maximum positive total shear stress was approximately 1.5 times larger than minimum negative total stress. Modeled and measured positive bed shear stresses are well correlated using the best convolution model, but the model underestimates the data by about 4%. Friction factors are dependent on the choice of normalizing using the maximum velocity, as is conventional, or the instantaneous velocity. These differ because the stress is not in phase with the velocity in general. Friction factors are consistent with previous data for monochromatic waves, and vary inversely with the square-root of the Reynolds number. The total shear stress leads the free stream fluid velocity by approximately 50°, whereas the skin friction shear stress leads by about 30°, which is similar to that reported by earlier researchers.     

Study on the Log-Linear Velocity Profile of Near-Bed Tidal Current in Estuarine and Coastal Waters

Many observed data show that the near-bed tidal velocity profile deviates from the usual logarithmic law. The amount of deviation may not be large, but it results in large errors when the logarithmic velocity profile is used to calculate the bed roughness height and friction velocity （or shear stress）. Based on their investigation, Kuo et al. （1996） indicate that the deviation amplitude may exceed 100%. On the basis of fluid dynamic principle, the profile of the near-bed tidal velocity in estuarine and coastal waters is established by introducing Prandtl＇ s mixing length theory and Von Kannan selfsimilarity theory. By the fitting and calculation of the near-bed velocity profde data observed in the west Solent, England, the results are compared with those of the usual logarithmic model, and it is shown that the present near-bed tidal velocity profile model has such advantages as higher fitting precision, and better inner consistency between the roughness height and friction velocity. The calculated roughness height and friction velocity are closer to reality. The conclusions are validated that the logarithmic model underestimates the roughness height and friction velocity during tidal acceleration and overestimates them during tidal deceleration.     

Propagation of solitary waves over a submerged permeable breakwater

We study the interactions between a non-breaking solitary wave and a submerged permeable breakwater experimentally and numerically. The particle image velocimetry (PIV) technique is employed to measure instantaneous free surface displacements and velocity fields in the vicinity of a porous dike. The porous medium, consisting of uniform glass spheres, is mounted on the seafloor. Due to the limited size of each field of view (FOV) for high spatial resolution purposes, four FOVs are set in order to form a continuous flow field around the structure. Quantitative mean properties are obtained by ensemble averaging 30 repeated instantaneous measurements. The Reynolds decomposition method is then adopted to separate the velocity fluctuations for each trial to estimate the turbulent kinetic energy. In addition, a highly accurate two-dimensional model with the volume of fluid interface tracking technique is used to simulate an idealized volume-averaged porous medium. The model is based on the Volume-Averaged Reynolds Averaged Navier–Stokes equations coupled with the non-linear k–ε turbulence closure solver. Comparisons are performed between measurements and numerical results for the time histories of the free surface elevation recorded by wave gauges and the spatial distributions of free surface displacement with the corresponding velocity and turbulent kinetic energy around the permeable object imaged by the PIV system. Fairly good agreements are obtained. It is found that the measured and modeled turbulent intensities on the weather side are much larger than those on the lee side of the object, and that the magnitude of the turbulent intensity increases with increasing wave height of a solitary wave at a constant water depth. The verified numerical model is then used to estimate the energy reflection, transmission and dissipation using the energy integral method by varying the aspect ratio and the grain size of the permeable obstacle.