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.     

Wave boundary layers in rough turbulent flow

A one-layer time-invariant eddy viscosity model is specified to develop a mathematical model for describing the essential features of the turbulent wave boundary layer over a rough bed. The functional form of the eddy viscosity is evaluated based on a modified one-equation turbulence model in which the eddy viscosity varies in time and space. The present eddy viscosity model simplifies much of the mathematical complexity in many existing models. Predictions from the present model have been compared favorably with a wide range of experimental data. It is found that the eddy viscosity model adopted in the present study is physically reasonable.     

Modelling wave-current interactions in rough turbulent bottom boundary layers

The aim of the present paper is to explain some of the differences between previously published analytical and numerical models of combined wave and current bottom boundary layer flow. To this end, the Grant and Madsen (1979) model for wave-current, rough turbulent flow is modified to include both first and second harmonic time variations in the eddy viscosity (K). The functional form of the coefficients controlling the amount of time variation is established by analysing the numerical model results of Davies (1990). The addition of time variation in K reduces the strong non-linearity exhibited by the mean stress in the original Grant and Madsen model for current dominated cases, and reproduces the veering of the current predicted by numerical turbulence closure models.     

A note on a theoretical model of oscillatory smooth turbulent boundary layers

An analytical theory which describes the motion in an oscillatory smooth turbulent boundary layer using a two-layer time invariant eddy viscosity model is presented. The eddy viscosity in the inner layer increases quadratically with the height above the wall. In the outer layer the eddy viscosity is taken as a constant.     

On a theoretical model of rough turbulent wave boundary layers

An analytical theory which describes the motion in a turbulent wave boundary layer near a rough sea bottom by using a two-layer time invariant eddy viscosity model is presented. The eddy viscosity in the inner layer increases quadratically with the height above the sea bottom. In the outer layer the eddy viscosity is taken as a constant. The mean velocity and shear stress profiles, the bottom shear stress and the bottom friction coefficient are presented, and comparisons are made with experimental results.     

Experimental study of turbulent oscillatory boundary layers in an oscillating water tunnel

A high-quality experimental study including a large number of tests which correspond to full-scale coastal boundary layer flows is conducted using an oscillating water tunnel for flow generations and a Particle Image Velocimetry system for velocity measurements. Tests are performed for sinusoidal, Stokes and forward-leaning waves over three fixed bottom roughness configurations, i.e. smooth, “sandpaper” and ceramic-marble bottoms. The experimental results suggest that the logarithmic profile can accurately represent the boundary layer flows in the very near-bottom region, so the log-profile fitting analysis can give highly accurate determinations of the theoretical bottom location and the bottom roughness. The first-harmonic velocities of both sinusoidal and nonlinear waves, as well as the second-harmonic velocities of nonlinear waves, exhibit similar patterns of vertical variation. Two dimensionless characteristic boundary layer thicknesses, the elevation of 1% velocity deficit and the elevation of maximum amplitude, are found to have power-law dependencies on the relative roughness for rough bottom tests. A weak boundary layer streaming embedded in nonlinear waves and a small but meaningful third-harmonic velocity embedded in sinusoidal waves are observed. They can be only explained by the effect of a time-varying turbulent eddy viscosity. The measured period-averaged vertical velocities suggest the presence of Prandtl's secondary flows of the second kind in the test channel. Among the three methods to infer bottom shear stress from velocity measurements, the Reynolds stress method underestimates shear stress due to missed turbulent eddies, and the momentum integral method also significantly underestimates bottom shear stress for rough bottom tests due to secondary flows, so only the log-profile fitting method is considered to yield the correct estimate. The obtained bottom shear stresses are analyzed to give the maximum and the first three harmonics, and the results are used to validate some existing theoretical models.     

On the structure of oscillatory boundary layers

An empirical analysis is performed on the most detailed, recent measurements of turbulent oscillatory boundary layer flow. The measurements show that throughout elevations where the flow can be considered horizontally uniform, one deficit model is sufficient for describing the fundamental mode. Some general properties of the non dimensional velocity deficit D₁(z) appear with striking consistency. First of all the identity $\text{ln}\text{¦D}{}_{\mathrm{<mtext>1</mtext>}}\text{¦ ≡}\text{Arg}\text{D}{}_{\mathrm{<mtext>1</mtext>}}$, which is a theoretical result for smooth laminar flow, seems to hold with great accuracy for a large range of turbulent flow conditions as well. This is of principal theoretical interest because all previous analytical eddy viscosity models as well as numerical mixing length models predict a consistent and fairly large difference between Arg D₁ and $\text{ln}\text{¦D}{}_{\mathrm{<mtext>1</mtext>}}\text{¦}$. If the identity between $\text{ln}\text{¦D}{}_{\mathrm{<mtext>1</mtext>}}\text{¦}$ and Arg D₁ extends all the way to the bed, it means that the bed shear stress leads the free stream velocity by 45 degrees. It is also found that the structure of smooth turbulent oscillatory flows as measured by Kalkanis (1964) corresponds to a sharp maximum in the normalized energy dissipation rate.     

Characteristics of turbulent boundary layers over a rough bed under saw-tooth waves and its application to sediment transport

A large number of studies have been done dealing with sinusoidal wave boundary layers in the past. However, ocean waves often have a strong asymmetric shape especially in shallow water, and net of sediment movement occurs. It is envisaged that bottom shear stress and sediment transport behaviors influenced by the effect of asymmetry are different from those in sinusoidal waves. Characteristics of the turbulent boundary layer under breaking waves (saw-tooth) are investigated and described through both laboratory and numerical experiments. A new calculation method for bottom shear stress based on velocity and acceleration terms, theoretical phase difference, φ and the acceleration coefficient, a_c expressing the wave skew-ness effect for saw-tooth waves is proposed. The acceleration coefficient was determined empirically from both experimental and baseline k–ω model results. The new calculation has shown better agreement with the experimental data along a wave cycle for all saw-tooth wave cases compared by other existing methods. It was further applied into sediment transport rate calculation induced by skew waves. Sediment transport rate was formulated by using the existing sheet flow sediment transport rate data under skew waves by Watanabe and Sato [Watanabe, A. and Sato, S., 2004. A sheet-flow transport rate formula for asymmetric, forward-leaning waves and currents. Proc. of 29th ICCE, ASCE, pp. 1703–1714.]. Moreover, the characteristics of the net sediment transport were also examined and a good agreement between the proposed method and experimental data has been found.     

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.     

Sheet flow and suspension of sand in oscillatory boundary layers

after revisionTime-dependent measurements of flow velocities and sediment concentrations were conducted in a large oscillating water tunnel. The measurements were aimed at the flow and sediment dynamics in and above an oscillatory boundary layer in plane bed and sheet-flow conditions. Two asymmetric waves and one sinusoidal wave were imposed using quartz sand with D₅₀ = 0.21 mm. A new electro-resistance probe with a large resolving power was developed for the measurement of the large sediment concentrations in the sheet-flow layer. The measurements revealed a three layer transport system consisting of a pick-up/deposition layer, an upper sheet flow layer and a suspension layer.In the asymmetric wave cases the total net transport was directed “onshore” and was mainly concentrated in the thin sheet flow layer (< 0.5 cm) at the bed. A small net sediment flux was directed “offhore” in the upper suspension layer. The measured flow velocities, sediment concentrations and sedimenl fluxes showed a good qualitative agreement with the results of a (numerical) 1DV boundary-layer flow and transport model. Although the model did not describe all the observed processes in the sheet-flow and suspension layer, the computational results showed a reasonable agreement with measured net transport rates in a wide range of asymmetric wave conditions.     

A general method for calculating three-dimensional laminar and turbulent boundary layers on ship hulls

A general method for representing the flow properties in the three-dimensional boundary layers around ship hulls of arbitrary shape is described. It makes use of an efficient two-point finite-diffirence schem to solve the boundary-layer equations and includes an algebraic eddy-viscosity representaion of the Reynolds-stress ternsor. The numericzal method contains novel and desirable features and allows the calculation of flows in which the circumferential velocity component contains regions of flow reversal across the boundary layer. The inviscid pressure distribution is determined with the Douglas-Neumann method which, if necessary, can conveniently allow for the boundary-layer displacement surface. To allow its application to ships, and particularly to those with double-elliptic and flat-bottomed hulls, a non-orthogonal coordinate system has been developed and is shown to be economical, precise and comparatively easy to use. Present calculations relate to zero Froude number but they can be extended to include the effects of a water wave and local regions of flow separation which may stem from bulbous-bow geometries.     

Numerical modeling investigation on turbulent oscillatory flow over a plane rough bed composed by randomly arrayed particles

A three-dimensional numerical model is established to simulate the turbulent oscillatory boundary layer over a fixed and rough bed composed by randomly arrayed solid spheres based on the lattice Boltzmann method and the large eddy simulation model.The equivalent roughness height,the location of the theoretical bed and the time variation of the friction velocity are investigated using the log-fit method.The time series of turbulent intensity and Reynolds stress are also investigated.The equivalent roughness height of cases with Reynolds numbers of 1×10~4–6×10~4 is approximately 2.81 d(grain size).The time variation of the friction velocity in an oscillatory cycle exhibits sinusoidal-like behavior.The friction factor depends on the relative roughness in the rough turbulent regime,and the pattern of solid particles arrayed as the rough bed in the numerical simulations has no obvious effect on the friction factor.     

Features of turbulent momentum and heat transfer in a stably stratified boundary layer over a rough surface

The features of the structure of a stable boundary layer over an urbanized surface are studied using a nonlocal model for the turbulent momentum and heat fluxes, which physically adequately takes into account the effect of buoyancy on turbulent transfer. The transformation of the structure of the boundary layer during transition from a state of convective mixing to a stable state is described by unified expressions for the turbulent momentum and heat fluxes. In some known schemes, different models are used for unstable and stable states. The model reproduces a stable dependence of the Prandtl number on the Richardson number and countergradient heat transfer in a strongly stable boundary layer. The results of numerical simulation are compared to the data of a laboratory experiment and the data obtained using the large-eddy simulation (LES) method.     

Comparison of theoretical and experimental wall pressure wavenumber-frequency spectra for axisymmetric and flat-plate turbulent boundary layers

The measurement and analysis of turbulent boundary layer wall pressure fluctuations using a wavenumber filter of sensors provide quantitative knowledge of turbulence physics. In addition, the sources of flow-induced noise and vibration for towed SONAR arrays can be determined. An axisymmetric turbulent boundary layer can have significantly different features than those of a comparable flat-plate boundary layer. Here, a detailed comparison of the distribution of wall pressure energy in both wavenumber and frequency between flat-plate and thick axisymmetric boundary layers is presented. The background theory of wavenumber-frequency spectra and state-of-the-art models for flat-plate boundary layers are discussed. The widely used model of Chase (1987), valid for flat-plate boundary layers over a wide range of Reynolds numbers, is used and combined with a sensor response function to allow the effects of spatial averaging to be considered. It is demonstrated that when measured boundary layer parameters for the axisymmetric case are used in the Chase flat-plate model, the results accurately predict the axisymmetric boundary layer wall pressure measurements.     

波浪运动在底边界层的湍流结构数值研究

本文基于$ k $-$ \varepsilon $ 模型研究了波流边界层内湍流结构特征。研究结果表明,时均流速分布数值解与实验结果高度吻合。一个波周期内湍流结构特征（如：涡量、湍动能、湍动能耗散率等）呈周期性变化规律,波浪作用引起涡量、湍动能及湍动能耗散率均在减速阶段减小,在波谷处达到最低值,而后在加速阶段增大,并在波峰处达到最大值。近壁面处湍流结构变化幅值较大（湍动能耗散率变化可达53%）,远离壁面处变化幅值较平均值较小（仅3%）。波流边界层厚度在减速阶段增加,在加速阶段减小。本文所建立的数值模型克服了现有模型因采用“高雷诺数方法”引起的近壁区精度不高问题,可较好地描述波浪作用下湍流结构演变过程的物理机制,为河口海岸地区泥沙运动、岸滩演变及海洋可再生能源的开发利用提供一些指导意义。     

A simplified approach to backscattering from a rough seafloor withsediment inhomogeneities

Current models used to predict the backscattering strength of the ocean floor are either very involved, requiring geoacoustic parameters usually unavailable for the site in practical applications, or overly simplistic, relying mainly on empirical terms such as Lambert's law. In any case, solutions are very approximate and the problem is still far from being solved. In this paper, a model is presented that avoids empirical functional forms yet requires only a few physical parameters to describe the surficial sediments, often tabulated for typical sediments. The aim of this paper is to develop a simple algorithm for operational prediction of bottom reverberation with only one free parameter, i.e., the volume scattering coefficient. The algorithm combines a two scale surface scattering model with scattered contributions originating from inhomogeneities within the sediments, talking into consideration the rough interface. No specific mechanism is assumed for scattering at the volume inhomogeneities; however, the inhomogeneities are assumed to be uniform and isotropic. The volume scattering coefficient, combined with the bottom attenuation and density and referenced to the surface, plays a role similar to the Lambert's constant in empirical models. The model is exercised on a variety of published datasets for low and moderately high frequency. In general, the model performs very well for both fast and slow sediments, showing a definite improvement over Lambert's law     

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.     

A note on boundary layers and wakes in rotating fluids

The flow induced by the two-dimensional line vortex moving in a rotating fluid is discussed. The governing vorticity equation is linearized adopting the Oseen approximation.First, the problem is considered on a constantf-plane. The solution shows that the Stewartson E^1/4 layer is transformed into the Oseen wake as the role of the advection becomes important.Second, the problem is considered on a-plane. When the line vortex moves westward, the solution shows a pattern of Rossby lee waves decaying downstream of the vortex and alternating flows far upstream. When the line vortex moves eastward, the inviscid solution shows definite alternating jets downstream. In a viscous case, however, the jets become less definite and identical with the above mentioned alternating flows in the far field. Far upstream, there are no disturbances because of the special propagation characteristics of Rossby waves.     

Numerical simulation of wave-induced laminar boundary layers

The three-dimensional numerical model with σ-coordinate transformation in the vertical direction is applied to the simulation of surface water waves and wave-induced laminar boundary layers. Unlike most of the previous investigations that solved the simplified one-dimensional boundary layer equation of motion and neglected the interaction between boundary layer and outside flow, the present model solves the full Navier–Stokes equations (NSE) in the entire domain from bottom to free surface. A non-uniform mesh system is used in the vertical direction to resolve the thin boundary layer. Linear wave, Stokes wave, cnoidal wave and solitary wave are considered. The numerical results are compared to analytical solutions and available experimental data. The numerical results agree favorably to all of the experimental data. It is found that the analytical solutions are accurate for both linear wave and Stokes wave but inadequate for cnoidal wave or solitary wave. The possible reason is that the existing analytical solutions for cnoidal and solitary waves adopt the first-order approximation for free stream velocity and thus overestimate the near bottom velocity. Besides velocity, the present model also provides accurate results for wave-induced bed shear stress.