The structure of the turbulent boundary layer over a mobile and deformable boundary

The structure of the velocity field above a propagating water wave of fixed frequency was investigated in order to evaluate the transport of wind momentum to water waves and the influence of a mobile and deformable boundary on the bursting cycle. The vertical and horizontal velocities were measured in a transformed Eulerian wave-following frame of reference with the aid of a cross hot film, in a wind-wave research facility at Stanford University.The mean velocity profiles have a log-linear form with a wake free-stream characteristic. The wave-coherent motion in the free-stream is irrotational; in the boundary layer, it has a strong shear behavior related to the wave-associated stress. The wave-induced velocity field and the wave-perturbed turbulence depend strongly on the ratio of the wave-speed to the mean free-stream velocity, c/U ₀.The presence of the propagating waves affects the bursting cycle, making the contribution of sweeps and ejections almost equal and dependent on the ratio c/U ₀. The magnitudes of the contribution of the bursting events are generally enhanced by the presence of water waves. The time interval between ejections or sweeps does not scale with either the inner and/or outer flow variables.This paper was presented at the Ninth Symposium on Turbulence, University of Missouri-Rolla, October 1–3, 1984     

Higher-order and length-scale statistics of velocity and temperature fluctuations in turbulent boundary layer along a heated vertical flat plate

Time-developing direct numerical simulation (DNS) was performed to clarify the higher-order turbulent behaviors in the thermally-driven boundary layers both in air and water along a heated vertical flat plate. The predicted statistics of the heat transfer rates and the higher-order turbulent behaviors such as skewness factors, flatness factors and spatial correlation coefficients of the velocity and temperature fluctuations in the natural-convection boundary layer correspond well with those obtained from experiments for space-developing flows. The numerical results reveal that the turbulent structures of the buoyancy-driven boundary layers are mainly controlled by the fluid motions in the outer region of the boundary layer, and these large-scale structures are strongly connected with the generation of turbulence in the thermally-driven boundary layers, in accordance with the actual observations for space-developing flows. Moreover, to specify the turbulence structures of the boundary layers, the cross-correlation coefficients and the characteristic length scales are examined for the velocity and thermal fields. Consequently, it is found that with a slight increase in freestream velocity, the cross-correlation coefficient for the Reynolds shear stress and turbulent heat flux increases for opposing flow and decreases for aiding flow, and the integral scales for the velocity and temperature fields become larger for opposing flow and smaller for aiding flow compared with those for the pure natural-convection boundary layer.     

Computational studies of inertia-gravity waves radiated from upper tropospheric jets

The generation and physical characteristics of inertia-gravity waves radiated from an unstable forced jet at the tropopause are investigated through high-resolution numerical simulations of the three-dimensional Navier–Stokes anelastic equations. Such waves are induced by Kelvin–Helmholtz instabilities on the flanks of the inhomogeneously stratified jet. From the evolution of the averaged momentum flux above the jet, it is found that gravity waves are continuously radiated after the shear-stratified flow reaches a quasi-equilibrium state. The time–vertical coordinate cross-sections of potential temperature show phase patterns indicating upward energy propagation. The sign of the momentum flux above and below the jet further confirms this, indicating that the group velocity of the generated waves is pointing away from the jet core region. Space–time spectral analysis at the upper flank level of the jet shows a broad spectral band, with different phase speeds. The spectra obtained in the stratosphere above the jet show a shift toward lower frequencies and larger spatial scales compared to the spectra found in the jet region. The three-dimensional character of the generated waves is confirmed by analysis of the co-spectra of the spanwise and vertical velocities. Imposing the background rotation modifies the polarization relation between the horizontal wind components. This out-of-phase relation is evidenced by the hodograph of the horizontal wind vector, further confirming the upward energy propagation. The background rotation also causes the co-spectra of the waves high above the jet core to be asymmetric in the spanwise modes, with contributions from modes with negative wavenumbers dominating the co-spectra. Dedicated to the memory of our colleague Dr. Binson Joseph     

Die turbulente Grenzschicht an einer stark geheizten ebenen Platte bei Unterschallströmung

Experiments for air flowing over a flat plate heated up to 250°C with velocities of 10 to 30 m/s, which have been made at the DFVLR-AVA, are briefly reviewed and a new analysis of the data is given. The analysis is based on an analytical representation of the velocity and temperature profiles. Close to the wall, a law of the wall approximation is used, which includes the effect of density and viscosity variation. The whole velocity profile is constructed by adding Coles' law of the wake to the law of the wall. In a similar way, the temperature profile is obtained from the law of the wall and an auxiliary distribution. The integrals of momentum and heat flux for two-dimensional flow are used in conjunction with a similarity assumption, to derive a relation between rate of heat transfer from the plate and skin friction. A maximum likelihood procedure has been applied to determine skin friction and rate of heat transfer from the measured dynamic pressure profiles.—The analytical velocity and temperature profiles are found in good agreement with the experimental data, except for the stations near the leading edge of plate. The skin friction coefficients and the Stanton numbers decrease slightly in downstream direction as a consequence of growing local Reynolds number, and decrease with increasing ratio of plate to free stream temperature. The latter fact is in qualitative agreement with the behavior of turbulent boundary layers in supersonic flow. The ratio of Stanton number to half of skin friction coefficient (Reynolds analogy factor) varies with increasing local boundary layer Reynolds number from 1.23 to 1.16.     

A general model for unconfined bubble plumes from extended sources

A model for unconfined bubble plumes is developed without assumptions as to the form of the velocity and gas-fraction profiles or as to the fraction of the momentum flux associated with the mean flow. Dimensionless solutions for axisymmetric and plane-symmetric extended sources indicate an initial contraction followed by an almost linear expansion which closely resembles the single-phase case. A second contraction is predicted near the surface of deep bodies of liquid. Gas and liquid velocity measurements in laboratory-scale plumes are presented, providing information on entrainment coefficients, velocity profiles, bubble velocities and the contribution of velocity fluctutations to the total momentum flux. The latter effect is considerable, providing an explanation of the “too-low” plume velocities found by other investigators.     

Passive Scalar Transport in a Turbulent Mixing Layer

This paper describes a combined experimental and numerical study of scalar transport in spatially developing, two-stream, turbulent mixing layers with velocity ratios of approximately 2:1. The experimental mixing layer was created by an S-shaped splitter plate mounted in a wind tunnel, and the concentration field was realized by releasing incense smoke into the high-speed side boundary layer above the splitter plate. Simultaneous measurements of the velocity and concentration fields were performed. A 12-sensor hot-wire probe was used to measure the velocity field and its gradients, while the concentration field was recorded with digital photographs of the laser-illuminated smoke. In parallel, a large-eddy simulation (LES) of the spatially developing mixing layer was carried out. Auxiliary turbulent boundary layer LES were used to provide high quality inflow boundary conditions for the velocity and concentration fields. By synchronizing the velocity and concentration measurements, concentration fluxes were also determined. Octant analysis based on the sign combinations of the velocity and concentration fuctuations was performed on the flux data to investigate the scalar transport processes. It was found that octants compatible with mean gradient transport of the scalar contribute most to the scalar fluxes. Conditional planar averages of scalar and momentum fluxes were obtained to determine their spatial distribution with respect to the organized roller and rib vortices of the mixing layer, and distinct patterns were observed. The simulation provided additional insight about the flow and scalar flux distribution topology. This topology was found to be partially compatible with simple models of roller and rib vortices that transport the scalar in a mean gradient sense.     

On the behaviour of solid particles in a horizontal boundary layer with turbulence and saltation effects

 A horizontal turbulent boundary layer of air carrying heavy solid particles is investigated experimentally. Mean and r.m.s. velocities of air and particles are measured by LDA, and particle mass flux distributions are obtained by means of a sampling method. The influence of the saltation mechanism is revealed by the large particle r.m.s. velocity in the near-wall region, and by the velocity lag of the particles in the outer region of the boundary layer, which is shown to be closely related to their free fall velocity. The present original results are discussed and compared with available experimental data concerning other kinds of horizontal flows. Received: 17 June 1996/Accepted: 3 April 1997     

Transient flow rate behaviour in an external natural convection boundary layer

A theoretical approach is proposed to investigate the transient dynamic behaviour of a free convection boundary layer-type flow. The set of continuity, momentum and energy equations are solved with the classical Boussinesq approximation using the Karman–Pohlhausen integral method. Applying a step variation of the uniform heat flux on a vertical wall, the boundary layer thickness and velocity profiles within the viscous layer, streamline patterns and volumetric flow rate are evaluated as a function of time. In addition, corresponding fully analytical asymptotic solutions are derived to be readily used in engineering applications.     

Features of the turbulent wall pressure field on a long towed cylinder

Turbulent wall pressure fluctuation correlation functions were measured in water on a towed cylindrical model of length 129.8 m and diameter 3.8 cm for steady speeds ranging from 6.2 to 15.5 m/s. The drag on the model was measured with a strut-mounted load cell to provide estimates of the momentum thickness and friction velocity that are used for scaling the correlation functions. Very high momentum thickness Reynolds numbers Re_θ were achieved, and varied from 4.8 × 10⁵ to 1.1 × 10⁶. The ratio of boundary layer thickness to cylinder radius was approximately 24, which is an order of magnitude greater than previous laboratory investigations. The ratio of momentum thickness to viscous length scale is significantly greater than for flat plate cases at comparable Re_θ. A similarity scaling is shown to be more effective than outer or inner boundary layer scalings for collapsing the correlation functions. Comparisons with the early streamwise and transverse correlation measurements of Willmarth and Yang are favorable, and show consistent trends of a more rapid loss of correlated energy for cylindrical turbulent boundary layers than for flat plate cases. Convection velocities are also presented and shown to collapse well with separation scaled on outer variables. A simple model that relates the peak of the correlation function to the average coherence levels is shown to be valid for spatial separations less than the approximate momentum thickness.     

Vertical two-dimensional non-hydrostatic pressure model with single layer

The vertical two-dimensional non-hydrostatic pressure models with multiple layers can make prediction more accurate than those obtained by the hydrostatic pres-sure assumption. However, they are time-consuming and unstable, which makes them unsuitable for wider application. In this study, an efficient model with a single layer is developed. Decomposing the pressure into the hydrostatic and dynamic components and integrating the x-momentum equation from the bottom to the free surface can yield a horizontal momentum equation, in which the terms relevant to the dynamic pressure are discretized semi-implicitly. The convective terms in the vertical momentum equation are ignored, and the rest of the equation is approximated with the Keller-box scheme. The velocities expressed as the unknown dynamic pressure are substituted into the continuity equation, resulting in a tri-diagonal linear system solved by the Thomas algorithm. The validation of solitary and sinusoidal waves indicates that the present model can provide comparable results to the models with multiple layers but at much lower computation cost.     

Turbulent structure of air flow over wind-induced gravity waves

This is the second paper in a group of three that reports the systematic measurements of wind-generated water waves in a wind tunnel experiment. Here, the structure of the boundary layer on the air side of the water?Cair interface was analysed and compared with the boundary layer over a smooth plane rigid wall. The contribution of the wave-induced Reynolds stress was detected through filtering the spectrum of velocity fluctuations. Wave-induced Reynolds stresses became negligible for z?>?5 H _rms. The intermittency factor in the boundary layer over water waves was similar to that in a boundary layer over a rigid plane wall, with several differences near the interface. Here, the presence/absence of water damps out the turbulence. The quadrant analyses revealed that ejection and sweep events were dominant and more concentrated. At small fetches, the large-amplitude negative streamwise perturbations were preferentially lifted. Turbulence energy production peaked at z/???=?0.2 and had a distribution similar to that observed for a self-preserving boundary layer with a strong adverse gradient pressure. The quadrant analysis contribution to the energy production revealed that ejections still dominated the balance and that the production was spatially modulated in the wind direction with a couple of cells and with a minimum in the area of the free surface wave height reduction.     

Development of the interfacial air layer in the non-aerated region of high-velocity spillway flows. Instabilities growth,entrapped air and influence on the self-aeration onset

Self-aeration is traditionally explained by the water turbulent boundary layer outer edge intersection with the free surface. This paper presents a discussion on the commonly accepted hypothesis behind the computation of the critical point of self-aeration in spillway flows and a new formulation is proposed based on the existence of a developing air flow over the free surface. Upstream of the inception point of self-aeration, some surface roughening has been often reported in previous studies which consequently implies some entrapped air transport and air–water flows coupling. Such air flow is proven in this study by presenting measured air velocities and computing the air boundary layer thickness for a 1V:2H smooth chute flow. Additionally, the growth rate of free surface waves has been analysed by means of Ultrasonic Sensors measurements, obtaining also the entrapped air concentration. High-speed camera imaging has been used for qualitative study of the flow perturbations.     

Numerical investigation of a spatially developing turbulent natural convection boundary layer along a vertical heated plate

Large-eddy simulation (LES) on a spatially developing natural convection boundary layer along a vertical heated plate was conducted. The heat transfer rate, friction velocity, mean velocity and temperature, and second-order turbulent properties both in the wall-normal and the stream-wise direction showed reasonable agreement with the findings of past experiments. The spectrum of velocity and temperature fluctuation showed a -2/3-power decay slope and -2-power decay slope respectively. Quadrant analysis revealed the inclination on Q1 and Q3 in the Reynolds stress and turbulent heat flux, changing their contribution along the distance from the plate surface. Following the convention, we defined the threshold region where the stream-wise mean velocity takes local maximum, the inner layer which is closer to the plate than the threshold region, the outer layer which is farther to the plate than the threshold region. The space correlation of stream-wise velocity tilted the head toward the wall in the propagating direction in the outer layer; on the other hand, the correlated motion had little inclination in the threshold region. The time history of the second invariant of gradient tensor Q revealed that the vortex strength oscillates both in the inner and the outer layers in between the laminar and the transition region. In the turbulent region, the vortex was often dominant in the outer layer. Instantaneous three-dimensional visualization of Q revealed the existence of high-speed fluid parcels associated with arch-shape vortices. These results were considered as an intrinsic structure in the outer layer, which is symmetrical to the structure of canonical smooth/rough wall bounded layer flow in forced convection.     

Semi‐coupled air/water immersed boundary approach for curvilinear dynamic overset grids with application to ship hydrodynamics

For many problems in ship hydrodynamics, the effects of air flow on the water flow are negligible (the frequently called free surface conditions), but the air flow around the ship is still of interest. A method is presented where the water flow is decoupled from the air solution, but the air flow uses the unsteady water flow as a boundary condition. The authors call this a semi‐coupled air/water flow approach. The method can be divided into two steps. At each time step the free surface water flow is computed first with a single‐phase method assuming constant pressure and zero stress on the interface. The second step is to compute the air flow assuming the free surface as a moving immersed boundary (IB). The IB method developed for Cartesian grids (Annu. Rev. Fluid Mech. 2005; 37 :239–261) is extended to curvilinear grids, where no‐slip and continuity conditions are used to enforce velocity and pressure boundary conditions for the air flow. The forcing points close to the IB can be computed and corrected under a sharp interface condition, which makes the computation very stable. The overset implementation is similar to that of the single‐phase solver (Comput. Fluids 2007; 36 :1415–1433), with the difference that points in water are set as IB points even if they are fringe points. Pressure–velocity coupling through pressure implicit with splitting of operators or projection methods is used for water computations, and a projection method is used for the air. The method on each fluid is a single‐phase method, thus avoiding ill‐conditioned numerical systems caused by large differences of fluid properties between air and water. The computation is only slightly slower than the single‐phase version, with complete absence of spurious velocity oscillations near the free surface, frequently present in fully coupled approaches. Validations are performed for laminar Couette flow over a wavy boundary by comparing with the analytical solution, and for the surface combatant model David Taylor Model Basin (DTMB) 5512 by comparing with Experimental Fluid Dynamics (EFD) and the results of two‐phase level set computations. Complex flow computations are demonstrated for the ONR Tumblehome DTMB 5613 with superstructure subject to waves and wind, including 6DOF motions and broaching in SS7 irregular waves and wind. Copyright © 2008 John Wiley & Sons, Ltd.     

The effect of Reynolds number on boundary layer turbulence

A new facility for studying high Reynolds number incompressible turbulent boundary layer flows has been constructed. It consists of a moderately sized wind tunnel, completely enclosed by a pressure vessel, which can raise the ambient air pressure in and around the wind tunnel to 8 atmospheres. This results in a Reynolds number range of about 20:1, while maintaining incompressible flow. Results are presented for the zero pressure gradient flat plate boundary layer over a momentum thickness Reynolds number range 1500–15?000. Scaling issues for high Reynolds number non-equilibrium boundary layers are discussed, with data comparing the three-dimensional turbulent boundary layer flow over a swept bump at Reynolds numbers of 3800 and 8600. It is found that successful prediction of these types of flows must include length scales which do not scale on Reynolds number, but are inherent to the geometry of the flow.     

薄膜涂层材料中的广义瑞利波

基于弹性动力学的线性理论，建立了涂层材料中广义瑞利波传播的理论分析模型，并 且由波动方程和边界条件推导了波的频散方程．分析了慢层和快层对相速度频散的影响，给 出了不同层厚-波长比和不同涂层-基体密度比情况下广义瑞利波相速度的理论解．算例分 析分别比较了慢层和快层结构中波的相速度、群速度，以及随深度衰减的位移与应力振 幅．另外，相速度曲线和位移振幅曲线与文献中给出的结果吻合，验证了理论模型和分析过 程的正确性．     

An Analysis of Turbulent Motions In and Around a Differentially Forced Density Interface

The Rapid-Distortion-Theory-based analysis proposed by Fernando and Hunt [1] is extended to study the nature of turbulence in and around a density interface sandwiched between turbulent layers with dissimilar properties. It is shown that interfacial motions consist of low-frequency, resonantly excited, nonlinear internal waves and high-frequency, linear internal waves driven by background turbulence. Based on the assumptions that (i) all resonant waves and some nonresonant waves having frequencies close to the resonant frequencies grow rapidly, break, and cause interfacial mixing, (ii) the spectral amplitude of the vertical velocity in the wave-breaking regime is constant, and (iii) kinetic energy is equipartitioned between linear and nonlinear breaking wave regimes, the r.m.s. vertical velocity at the interface and the turbulent kinetic energy flux into the interface are calculated. The migration velocity of the interface is calculated using the additional assumption that the buoyancy flux into a given turbulent layer is a fixed fraction of the turbulent kinetic energy flux supplied to the interface by the same layer. The calculations are found to be in good agreement with the entrainment data obtained in previous laboratory experiments in the parameter regime where the interface is dominated by internal wave dynamics. Received 23 July 1997 and accepted 8 January 1999     

Some equilibrium turbulent boundary layers

Using the Coles additive law of the wall and law of the wake for the mean velocity profile of a two-dimensional turbulent boundary layer, a differential equation for the friction and wake parameters is derived from the momentum integral equation with a view to finding out the conditions under which the boundary layer can exhibit equilibrium. It is predicted that equilibrium is possible for boundary layers in favorable pressure gradient over smooth as well as k-type rough walls. When the roughness height is allowed to increase linearly with the streamwise distance, equilibrium exists also in zero pressure gradient. For a d-type rough wall, equilibrium is possible for a certain range of pressure gradients, from favorable to adverse. Most of the predictions are verified by evaluating the friction and wake parameters from the available experimental data on mean velocity measurements.