The influence of turbulence on drag

The use of the Reynolds number as the only correlating factor for drag force measurements may be inadequate in circumstances involving highly turbulent flows. The results of previous investigations relating to the effects of turbulence scale and intensity are examined. Of special interest is the possibility of a drag minimum, even at low Reynolds number, for a free-stream turbulence intensity of about 5%. This appears to be the result of interaction between the free stream and the boundary layer. As intensity increases beyond 5%, the minimum may be succeeded by an increase in drag to values exceeding the laminar flow values. Further elucidation of the subject is required, particularly because of its importance in various problems related to geophysical flows.     

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.     

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.     

浅化波浪层流边界层流速分布特性的数值分析

建立了同时考虑波致雷诺应力和时均水平压强梯度影响的二阶波浪边界层数学模型,模型计算得到的浅化波浪层流边界层内瞬时流速剖面、振荡速度幅值和时均流速剖面均与水槽实验数据吻合较好,在此基础上探讨了浅化波浪边界层流速分布特性及其影响机制。随着波浪的浅化变形,边界层内时均流速剖面"底部向岸、上部离岸"的变化特征越来越明显。这是二阶对流项引起的波致雷诺应力和离岸回流引起的时均水平压强梯度共同作用的结果,在床面附近由波致雷诺应力占主导作用并趋于引起向岸流动,在上部区域由时均水平压强梯度占主导作用并趋于引起离岸流动。     

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.     

Effects of time dependence in unstratified tidal boundary layers: results from large eddy simulations

A three-dimensional Large Eddy Simulation (LES) model is used to simulate oscillating tidal boundary layers and test previous results obtained from one-dimensional boundary layer models and turbulence measurements in tidal channels. The LES model produces low-order turbulence statistics in agreement with the semi-analytic theory and observations. It shows a logarithmic layer in the mean velocity profile and a linear distribution of Reynolds stress with water depth. However, the eddy viscosity profile predicted by the LES model is not parabolic but better matches a parabolic profile modified by wake effect observed in the outer part of depth-limited steady boundary layers. Low-order turbulence statistics can be scaled by the instantaneous friction velocity at the bottom boundary. Although turbulence intensities in three directions fluctuate over a tidal cycle, their normalized values are in good agreement with those determined from laboratory experiments of steady open-channel flows. The LES model confirms that tidal turbulence is in quasi-equilibrium. However, it also demonstrates the importance of flow acceleration/deceleration term in the depth-integrated momentum balance for the mean flow. Phase differences are found between flows at different heights above the bottom boundary.     

Bottom friction and its effects on periodic long wave propagation

A new set of Boussinesq-type equations describing the free surface evolution and the corresponding depth-integrated horizontal velocity is derived with the bottom boundary layer effects included. Inside the boundary layer the eddy viscosity gradient model is employed to characterize Reynolds stresses and the eddy viscosity is further approximated as a linear function of the distance measured from the seafloor. Boundary-layer velocities are coupled with the irrotational velocity in the core region through boundary conditions. The leading order boundary layer effects on wave propagation appear in the depth-integrated continuity equation to account for the velocity deficit inside the boundary layer. This formulation is different from the conventional approach in which a bottom stress term is inserted in the momentum equation. An iterative scheme is developed to solve the new model equations for the free surface elevation, depth-integrated velocity, the bottom stress, the boundary layer thickness and the magnitude of the turbulent eddy viscosity. A numerical example for the evolution of periodic waves propagating in one-dimensional channel is discussed to illustrate the numerical procedure and physics involved. The differences between the conventional approach and the present formulation are discussed in terms of the bottom frictional stress and the free surface profiles.     

Measurements of the wall pressure spectra on a full-scale experimental towed array

Turbulent wall pressure data acquired during tests of a full-scale experimental towed array over a range of tow speeds in straight tows and turns is presented. The experimental towed array contained a linear array of sensors mounted at the fluid–solid interface to measure the spectra of the wall pressure fluctuations due to the cylindrical turbulent boundary layer. The physics are dominated by the growth of a thick, high Reynolds number turbulent boundary layer at arc length Reynolds numbers as high as 9×10⁸. The measured wavenumber-frequency spectra, autospectra, cross-spectral decay and convection velocities are presented. A well-defined convective ridge exists in the wavenumber-frequency spectra obtained during straight tows and turns. Turns give rise to a complicated fluid–structure interaction problem, but do not lead to the separation of the turbulent boundary layer. As the array moves through a turn, flow-induced vibrations of the array are shown to dominate the spectra at low frequencies, with more rapid decay in the measured coherence occurring at higher frequencies. The use of tow speed as a velocity scale is shown to collapse autospectra and convection velocities.     

Inleakage of a vortex-ring bunch onto a plane screen

A method is suggested for simulating axisymmetric laminar or turbulent flows formed during the motion of a vortex-ring bunch of given geometry and circulation toward a plane screen. Earlier, similar problems were simulated with the numerical solution of the Navier-Stokes equations for laminar flows. Turbulent flows have remained unconsidered until now. When a vortex ring approaches the screen, the secondary nonstationary flow is induced near the screen’s surface and this secondary flow causes the formation of the radial boundary layer (provided that air viscosity is taken into account). First, the medium spreads out from the critical point at the screen’s center with the negative pressure gradient along the radial coordinate and then detaches in the region of the positive pressure gradient. This radial wall flow and the corresponding boundary layer are considered in the quasi-stationary approximation. When the boundary layer detaches at successive instances, the flow is replenished with the radially moving secondary vortex rings whose circulations have the sign opposite to that of the circulation of the primary vortex ring. It is the interaction of the primary and secondary vortices that governs process dynamics, which differs substantially from that in the case when the formation of secondary vortices is disregarded. The suggested method is based on the method of discrete vortices (a perfect liquid) and the boundary-layer (laminar or turbulent) theory. During the development of the flow under investigation, the nonstationary ascending flow in the direction perpendicular to the screen’s plane is formed and then this flow decays and dissipates. Simulations for large Reynolds numbers corresponding to the formation of the turbulent boundary layer show that the velocity of ascending vortices in the plane of the initial vortex bunch is less than one-tenth of the initial velocity of the descending vortex ring. The boundary layer is introduced into calculations with the sole goal of determining the parameters of the secondary vortex rings formed during boundary-layer detachments. The interaction of the primary and secondary vortices is then considered within the framework of a perfect medium. Simulations for large Reynolds numbers corresponding to the formation of the turbulent boundary layer on the screen were correlated with the available data obtained in laboratory experiments for small Reynolds numbers. Qualitative agreement between the simulations and experiments is fairly satisfactory. The simulation for one combination of the circulation and vortex-ring geometry takes at most 10–15 min with the use of an average PC.     

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.     

Turbulence-induced steady streaming in an oscillating boundary layer: On the reliability of turbulence closure models

The accuracy of several closure models of the Reynolds-Averaged Navier–Stokes Equations in predicting the characteristics of an oscillating turbulent wall boundary layer is analyzed. The analysis involves four low Reynolds number k − ε models and a k − ω model and it is carried out by comparing the model results both with experimental data and with data obtained by a Direct Numerical Simulation (DNS) of the Navier–Stokes equations. The boundary layer is generated by a spatially constant time-oscillating pressure gradient given by the sum of two harmonic components characterized by angular frequencies Ω^∗ and 2Ω^∗ respectively, which generates a steady streaming because of the asymmetry of turbulence intensity during the cycle. Thus the results are relevant to the boundary layer at the bottom of nonlinear sea waves. The attention is therefore focused on the accuracy of the models in reproducing the period averaged profiles of the hydrodynamic characteristics of the steady streaming. The instantaneous quantities, such as time development of the wall shear stress, profiles of the streamwise velocity, Reynolds stresses and turbulent kinetic energy are also considered and analyzed. The results shows that a model can be judged better or worse than other models depending on the specific flow characteristic under investigation. However, an approach has been adopted which allowed to rank the models according to their accuracy in predicting the values of the hydrodynamic quantities involved in the present study.     

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.     

Numerical simulation of flow around a circular cylinder close to a flat seabed at high Reynolds numbers using a k–ε model

High Reynolds number flows around a circular cylinder close to a flat seabed have been computed using a two-dimensional standard high Reynolds number k–ε turbulence model. The effects of gap to diameter ratio, Reynolds number and flat seabed roughness for a given boundary layer thickness of the inlet flow upstream of the cylinder have been investigated. Hydrodynamic quantities and the resulting bedload transport have been predicted, and the vortex shedding mechanisms have been investigated. Predictions of hydrodynamic quantities around a cylinder located far away from the bed (so that the effect of the bed is negligible) are in satisfactory agreement with published experimental data and numerical results obtained for the flow around an isolated cylinder. Results for lower Reynolds number flows have also been computed for comparison with the high Reynolds number flow results. Overall it appears that the present approach is suitable for design purposes at high Reynolds numbers which are present near the seabed in the real ocean.     

A simplified model for the transformation of the atmospheric planetary boundary layer overlying a thermal front in the sea

This paper discusses a simplified model for the evolution of the atmospheric planetary boundary layer overlying a thermal front in the sea. The model provides local values of the friction/heat transfer geostrophic coefficients and the direction of surface wind stress, as well as the wind/temperature profiles at any point on the front. With the running over a warm front, the baroclinicity of the internal boundary layer leads to the generation of a near-surface current of air directed down the front. The model can be used to interpret radar imagery of the sea surface with the purpose of determining its mesoscale variability. Translated by Vladimir A. Puchkin.     

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.     

Mixing processes relevant to phytoplankton dynamics in lakes

Active turbulence in lakes is confined to the surface mixed layer, to boundary layers on the lake sides and bottom, and to turbulent patches in the interior. The density stratification present in most lakes fundamentally alters the pathways connecting external mechanical energy inputs, for example by wind, with its ultimate fate as dissipation to heat; the density stratification supports internal waves and intrusions that distribute the input energy throughout the lake. Intrusions may be viewed as internal waves with zero horizontal wavenumber and are formed each time localised mixing occurs in a stratified fluid. Intrusions are also formed in the epilimnion by differential heating or cooling and by differential deepening. The fraction of lake volume below the diurnal mixed layer that is subject to active turbulence is very small, probably of the order of 1% or less in small to medium‐sized lakes. By contrast, in the surface mixed layer, turbulence is less intermittent and maintains phytoplankton in suspension and controls their exposure to the underwater solar flux. Nutrient transport to individual cells depends not only on the cell Reynolds number but also on the Peclet number, which, if large, implies enhanced mass transfer above purely diffusive values.     

Validation of a new generation system for bottom boundary layer beneath solitary wave

Understanding of sea bottom boundary layer characteristics, especially bottom shear stress acting on the sea bed, is an important step needed in sediment transport modeling for practical application purposes. In the present study, a new generation system for bottom boundary layer under solitary wave is proposed. Applicability of this system is examined by comparing measured and numerical solution velocities. Moreover, transitional behavior from laminar to turbulence was investigated. It is concluded that the critical Reynolds number in the experiments shows good agreement with DNS result of Vittori and Blondeaux (2008) and laboratory data of Sumer et al. (2010), indicating validity of the generation system. Since the present generation system enables continuous measurement to obtain ensemble averaged quantities, it can be effectively utilized for future experimental studies on solitary wave boundary layers, including sediment transport experiments with movable bed.     

An experimental study of turbulent boundary layer with slit suction

Experiments are performed on a flat plate with a transverse suction slit in the Reynolds number range 5 × 10⁵ < R_e < 1.1 × 10⁶. Mean velocity profiles, RMS values are measured with hot wire anemometry. Friction velocity is numerically calculated. The experiments showed that a classical boundary layer parameter α is related to the suction coefficient S_c with an equation of the form: .The value of A seems to depend strongly on the relative location with respect to suction slit and possibly weakly on Reynolds number.     

Observational and numerical modeling methods for quantifying coastal ocean turbulence and mixing

In this review paper, state-of-the-art observational and numerical modeling methods for small scale turbulence and mixing with applications to coastal oceans are presented in one context. Unresolved dynamics and remaining problems of field observations and numerical simulations are reviewed on the basis of the approach that modern process-oriented studies should be based on both observations and models. First of all, the basic dynamics of surface and bottom boundary layers as well as intermediate stratified regimes including the interaction of turbulence and internal waves are briefly discussed. Then, an overview is given on just established or recently emerging mechanical, acoustic and optical observational techniques. Microstructure shear probes although developed already in the 1970s have only recently become reliable commercial products. Specifically under surface waves turbulence measurements are difficult due to the necessary decomposition of waves and turbulence. The methods to apply Acoustic Doppler Current Profilers (ADCPs) for estimations of Reynolds stresses, turbulence kinetic energy and dissipation rates are under further development. Finally, applications of well-established turbulence resolving particle image velocimetry (PIV) to the dynamics of the bottom boundary layer are presented. As counterpart to the field methods the state-of-the-art in numerical modeling in coastal seas is presented. This includes the application of the Large Eddy Simulation (LES) method to shallow water Langmuir Circulation (LC) and to stratified flow over a topographic obstacle. Furthermore, statistical turbulence closure methods as well as empirical turbulence parameterizations and their applicability to coastal ocean turbulence and mixing are discussed. Specific problems related to the combined wave-current bottom boundary layer are discussed. Finally, two coastal modeling sensitivity studies are presented as applications, a two-dimensional study of upwelling and downwelling and a three-dimensional study for a marginal sea scenario (Baltic Sea). It is concluded that the discussed methods need further refinements specifically to account for the complex dynamics associated with the presence of surface and internal waves.     

Laboratory observation of boundary layer flow under spilling breakers in surf zone using particle image velocimetry

A boundary layer flow under spilling breakers in a laboratory surf zone with a smooth bottom is investigated using a high resolution particle image velocimetry (PIV) technique. By cross-correlating the images, oscillatory velocity profiles within a viscous boundary layer of O(1) mm in thickness are resolved over ten points. Using PIV measurements taken for an earlier study and the present study, flow properties in the wave bottom boundary layer (WBBL) over the laboratory surf zone are obtained, including the mean velocities, turbulence intensity, Reynolds stresses, and intermittency of coherent events. The data are then used to estimate the boundary layer thickness, phase variation, and bottom shear stress. It is found that while the time averaged mass transport inside the WBBL is onshore in the outer surf zone, it changes to offshore in the inner surf zone. The zero Eulerian mass transport occurs at h/h_b ≈ 0.92 in the outer surf zone. The maximum overshoot of the streamwise velocity and boundary layer thickness are not constant across the surf zone. The bottom shear stress is mainly contributed by the viscous stress through mean velocity gradient while the Reynolds stress is small and negligible. The turbulence level is higher in the inner surf zone than that in the outer surf zone, although only a slight increase of turbulent intensity is observed inside the WBBL from the outer surf zone to the inner surf zone. The variation of phase inside and outside the WBBL was examined through the spatial velocity distribution. It is found the phase lead is not constant and its value is significantly smaller than previous thought. By analyzing instantaneous velocity and vorticity fields, a remarkable number of intermittent turbulent eddies are observed to penetrate into the WBBL in the inner surf zone. The size of the observed large eddies is about 0.11 to 0.16 times the local water depth. Its energy spectra follow the − 5/3 slope in the inertial subrange and decay exponentially in the dissipation subrange.