Turbulent Wall Layer Research Articles

Near-wall response in turbulent shear flows subjected to imposed unsteadiness

Rapid-distortion theory is adapted to introduce a truly unsteady closure into a simple phenomenological turbulence model in order to describe the unsteady response of a turbulent wall layer exposed to a temporarily oscillating pressure gradient. The closure model is built by taking the ratio of turbulent shear stress to turbulent kinetic energy to be a function of the effective strain. The latter accounts for the history of the flow. The computed unsteady velocity fluctuations and modulated turbulent stresses compare favourably in the ‘non-quasi-steady’ frequency range, where quasi-steady assumptions would fail. This suggests that the concept of rapid distortion is especially appropriate for unsteady flows. This paper forms the basis for acoustical studies of the problem to be reported elsewhere.

Similarity Behavior in Transitional Boundary Layers Over a Range of Adverse Pressure Gradients and Turbulence Levels

Boundary layer transition has been investigated experimentally under low, moderate, and high free-stream turbulence levels and varying adverse pressure gradients. Under high turbulence levels and adverse pressure gradients a pronounced subtransition was present. A strong degree of similarity in intermittency distributions was observed, for all conditions, when the Narasimha procedure for determination of transition inception was used. Effects of free-stream turbulence on the velocity profile are particularly strong for the laminar boundary layer upstream of the transition region. This could reflect the influence of the turbulence on the shear stress distribution throughout the layer and this matter needs further attention. The velocity profiles in wall coordinates undershoot the turbulent wall layer asymptote near the wall over most of the transition region. The rapidity with which transition occurs under adverse pressure gradients produces strong lag effects on the velocity profile; the starting turbulent boundary layer velocity profile may depart significantly from local equilibrium conditions. The practice of deriving integral properties and skin friction for transitional boundary layers by a linear combination of laminar and turbulent values for equilibrium layers is inconsistent with the observed lag effects. The velocity profile responds sufficiently slowly to the perturbation imposed by transition that much of the anticipated drop in form factor will not have occurred prior to the completion of transition. This calls into question both experimental techniques, which rely on measured form factor to characterize transition, and boundary layer calculations, which rely on local equilibrium assumptions in the vicinity of transition.

On the etiology of shear layer vortices

In an effort to isolate the mechanism by which streamwise structures form in turbulent wall layers, evolution equations were derived for the streamwise velocity and vorticity perturbations about a mean turbulent fully developed channel flow. The stability of these equations, which take their most concise form when derived from the Generalized Lagrangian mean equations of Andrews and McIntyre, are studied assuming normal modes and infinitesimal disturbances. The resulting stability diagram yields, inter alia, the spanwise periodicity of the resulting structures, which we term shear layer vortices. If streaks are thought of as the footprints of these vortices, we then have a formal way of determining the spacing of streaks. The first three modes of instability are determined; at the first not just two vortices form per period, but four. It is also evident that an intense local shear layer forms about the plane in which the convection velocity equals the mean Eulerian velocity.

Scaling Turbulent Wall Layers

The two-layer concept is a framework for interpreting events and constructing mathematical models of turbulent wall layers. In this paper an asymptotic theory is constructed employing the idea that the interaction between the layers is the most important aspect. It is shown that the matching process for the layers can be used to define a characteristic scale, u*, and to produce an equation that relates u* to the known parameters; U∞, v, h, e, and dp/dx. At infinite Reynolds number the scale u* is equal to uτ, the friction velocity, but they are distinct at moderate Reynolds numbers. The theory produces very simple results. For instance, the overlap velocity laws are logarithmic with an invariant von Ka´rma´n constant; at low Reynolds numbers the additive constant changes while the slope remains the same. The effect of low Reynolds numbers on the Reynolds stress in the overlap layer is also analyzed. A composite expansion explains the strong Reynolds number effect on the stress profiles. This occurs because the mixing of outer and inner layer phenomena take place at different locations as the size of the overlap region changes. The location of the maximum Reynolds stress is given by y+max = (Re/k)1/2. The overlap region was not found to be a region of constant stress, as put forth in many heuristic arguments.

Suspension of solid particles in agitated vessels—I. Archimedes numbers ≲ 40

Complete suspension of fine-grained particles is achieved at mean circulation velocities of the fluid exceeding the settling velocities of the particles by several orders of magnitude. It is therefore argued that boundary layer effects are significant. In this paper, through an analysis of the boundary layer flow in an agitated vessel two theoretical criteria for the required minimum stirrer angular velocities are derived. The evaluation of own experiments proves the significance of one of them for Archimedes numbers Ar ≲ 40, i.e. for particles completely immersed in the viscous sublayer. For higher Archimedes numbers, i.e. for particles protruding into the buffer layer and into the turbulent near wall layer, criteria based on the boundary layer flow turn out not to be suitable.

Wall-layer structure and drag reduction

When drag-reducing additives are confined entirely to the linear sublayer of a turbulent channel flow of water, both the spanwise spacing and bursting rate of the wall-layer structure are the same as those for a water flow and there is no evidence of drag reduction. Drag reduction is measured downstream of the location where the additives injected into the sublayer begin to mix in significant quantities with the buffer region (10 < y+ < 100)The superscript + denotes a dimensionless quantity scaled with the kinematic viscosity ν and the wall shear velocity v* = (τw/ρ)½. of the channel flow. At streamwise locations where drag reduction does occur and where the injected fluid is not yet uniformly mixed with the channel flow, the dimensionless spanwise streak spacing increases and the average bursting rate decreases. The decrease in bursting rate is larger than the corresponding increase in streak spacing. The wall-layer structure is like the structure in the flow of a homogeneous, uniformly mixed, drag-reducing solution. Thus, the additives have a direct effect on the flow processes in the buffer region and the linear sublayer appears to have a passive role in the interaction of the inner and outer portions of a turbulent wall layer.

On the evolution of the system of wind drift currents and Langmuir circulations in the ocean. Part 1. Theory and averaged current

A theory for the evolution of the wind drift current and of the Langmuir circulations in infinitely deep water of constant density is presented. The model improves and extends a recent quasi-steady theory of Craik & Leibovich which asserts that the Langmuir circulations arise from a nonlinear interaction between surface waves and the frictional wind drift current. In turn, the development of the wind drift should be strongly influenced by Langmuir circulations, when they are present, and the two current systems are therefore treated here as a single inseparable system driven by a prescribed wind stress and surface wave field. Mixing by the vertical motions in the Langmuir circulations is shown to yield solutions for the wind drift, obtained both analytically and numerically, which are consistent with experiments and with field observations. The model yields a streaky flow pattern with a mean motion much like a turbulent wall layer, although the model is deterministic. In particular, it is found that a ‘viscous sublayer’ joins surface water to a logarithmic ‘inertial sublayer’ below. The scaling rules that emerge from the theory allow the surface speed of the wind drift to reach nearly full development in a matter of minutes.

Characteristics of a turbulent boundary layer with an external turbulent uniform shear flow

Measurements are presented of both mean and fluctuating velocity components in a turbulent boundary layer subjected to a nearly homogeneous external turbulent shear flow. The Reynolds shear stress in the external shear flow is small compared with the wall shear stress. Its transverse mean velocity gradient λ (≃ 6 s−l) is also small compared with typical gradients based on outer variables (say Uw/δ, where Uwis the value of the linear velocity profile extrapolated to the wall and δ is the boundary-layer thickness), but is of the same order as Ut/δ (Ur is the friction velocity). The influence of both positive and negative transverse velocity gradients on the turbulent wall layer is investigated over a streamwise region where the normal Reynolds stresses in the external flow are approximately equal and constant in the streamwise direction. In this region, the integral length scale of the external flow is of the same order of magnitude as that of the wall layer. Measurements in the boundary layer are also given for an un-sheared external turbulent flow (λ = 0) with a turbulence level Tu of 1.5%, approximately the same as that for h = ± 6 s−1. (Tu, is defined as the ratio of the r.m.s. longitudinal velocity fluctuation to Uw.) The measurements are in good agreement with those available in the literature for a similar free-stream turbulence level and show that the external turbulence level and length scale exert a large influence on the turbulence structure in the boundary layer. The additional effect of the external shear on the mean velocity and turbulent energy budget distributions in the inner region of the boundary layer is found to be small. In the outer region, the ‘wake’ component of the mean velocity defect is lowered by the presence of free-stream turbulence and one extra effect due to the external shear is an increase in the Reynolds shear stress when h is positive and a decrease when h is negative. Another interesting effect due to the shear is the appearance near the edge of the layer of a small but distinct region where the local mean velocity is constant and the Reynolds shear stress is negligible.

Structure of the Turbulent Boundary Layer

The turbulent boundary layer can be subdivided into four distinct regions, sublayer, turbulent wall layer, outer region, and superlayer. Each one of these regions is briefly discussed and recent theoretical approaches dealing with the Reynolds stress are examined.

Wall layers with non-uniform shear stress

The empirical description of turbulent wall layers across which the shear stress varies is considered. The description given by Townsend for zero-stress layers is found to be inapplicable to uniform pressure flows in pipes and two-dimensional channels, and to a boundary layer developing in the absence of a pressure gradient.