Logarithmic Layer Research Articles

Effects of weak bed load on the Universal Law of the Wall

Two‐dimensional equilibrium boundary‐layer flows were investigated in an open water channel with a width/depth ratio of 6 for a smooth bed of 0.16 mm quartz grains (d50) and compared with those of an immobile smooth cemented bed with the same sand roughness. For flows at Reynolds numbers between 20,000, representing onset of erosion, and 28,000, before appearance of rhomboid ripples, quartz grains rolled over an otherwise smooth sand bed with a density of ≤40 grains cm−1 s−1. Then the universal law of the wall, as obtained for the fixed, smooth sand bed, could not be confirmed by the data. Instead, (1) a logarithmic layer was found that extended further into the wake region and had a reduced value of von Karman's constant κ = 0.32 ± 0.04, (2) the friction diagram indicated Reynolds number dependent drag reduction, and (3) the logarithmic layer extended down to the top of the rolling grains at ReU >25,000. These results are interpreted as a new class of wall‐bounded shear flow with different momentum transfer processes and a velocity‐defect law throughout the flow down to the top of the rolling grains. Some conclusions are discussed for sedimentological and engineering problems in which this type of flow is the rule rather than the exception.

The interaction of waves and a turbulent current: waves propagating against the current

The results of an experimental study of the interaction between waves and a current propagating in the same direction, have been reported by Kemp & Simons (1982). This paper describes the second part of the study, and considers the case of waves propagating against the current. Tests were performed in a laboratory flume with smooth and rough beds, and velocity measurements were made with a directionally sensitive laser anemometer as described in the previous paper. Analysis, including ensemble averaging of velocities and surface elevation, was performed by an on-line computer. Results indicate that the rate of wave attenuation is greatly increased by the addition of an opposing current, and reduced by a following current. Wave profiles remain closely described by Stokes second-order theory; orbital velocities are also found to be in agreement with a second-order wave theory modified to take account of the presence of the current. Certain results described occur regardless of the relative directions of current and wave. Mean velocities in the upper flow increase in the direction of the wave generator for increasing wave height. This suggests that the current is enhancing the wave-induced mass transport. Near the bed the velocity profiles so change that above the rough bed the current is retarded by the wave motion. In the logarithmic layer over the smooth bed velocities are increased with increasing wave height. However, all changes to velocity profiles have to be carefully interpreted, as the sidewall boundary layer decreases in thickness with even the smallest wave superimposed on the current. Turbulence intensities and Reynolds stresses near the rough bed are increased by the presence of the waves, most strongly in a layer two roughness heights above bed level, where fluctuations are periodic and effected by vortices ejected from the roughness troughs. Above this level, and over the smooth bed, turbulence levels are similar to those for the currents alone.

An Operational Large-Scale Marine Planetary Boundary Layer Model

A marine planetary boundary layer (PBL) model is presented and compared with data from sea-based experiments. The PBL model comprises two layers, the outer an Ekman-Taylor layer with stratification-dependent secondary flow, and the logarithmic surface layer corrected for stratification and humidity effects and variable surface roughness. Corrections are noted for air much warmer than water in stable conditions and for low wind speeds. The layers are analytically defined along with similarity relations and a resistance law for inclusion in a program. An additional interfacial layer correction is developed and shown to be significant for heat flux calculations. Experimental data from GOASEX were used to predict the windfield in the Gulf of Alaska, and JASIN data was used for windfields SE of Iceland. The JASIN-derived wind field predictions were accurate to within 1 m/sec and 10 deg in a 200 km triangle.

Observation of the bottom boundary layer on the continental shelf

Current meter data from various depths near the sea bottom collected for 31 days at time intervals of 10 minutes using a subsurface buoy system at a depth at 38 m on the continental shelf off Akita, Japan have been analyzed. The results show the existence of a stationary Ekman layer. The typical range of the characteristic parameters are estimated as follows; friction velocity: 0.38 cm s−1; Ekman layer thickness: 16 m; logarithmic layer thickness: 4 m–6 m; constant flux layer thickness: 0.4–0.6 m; Ekman veering: 28.7°; drag coefficient: 0.24×10−2−0.53×10−2. Veering was also observed in the logarithmic layer.

Some effects of suspended sediment stratification on an oceanic bottom boundary layer

A suspended‐sediment‐induced, stably stratified oceanic bottom boundary layer is examined with the Mellor‐Yamada level II turbulence closure model. The boundary layer equations are coupled through the eddy diffusion coefficients for momentum and mass. The bottom boundary layer of the Florida Current is modeled using flow and sediment properties as input parameters. Model results indicate that the boundary layer response to a suspended sediment concentration gradient is to reduce appreciably the level of turbulence. Turbulent kinetic energy and eddy conductivity are reduced by about 40% and 25%, respectively, and bottom stress is reduced by approximately 45%. Increases of veering angle of ≅5°, together with enhancement of the Ekman‐like nature of the current spiral with sediment‐induced stability, are noted. The boundary layer adjusts to the reduced turbulence level by a decrease in the slope of the velocity profile and by diminution of the thicknesses of the boundary layer and contained logarithmic layer by as much as 50%. The predictions are consistent with findings from studies of stably stratified atmospheric boundary layers and sediment‐carrying channel flows. The velocity field near the bottom of a neutrally stratified bottom boundary layer is governed by the relationship U = (u*/κ) ln z/z0. To recover the predicted velocity distribution from this relationship in the sediment‐stratified case, von Karman's constant κ must be reduced by about 15%. The model, however, treats κ as an internal constant. Similarity theory suggests that a more appropriate expression for the near‐bottom velocity field in the sediment‐stratified bottom boundary layer is U = (u*/κ)(ln z/z0 + A ∫Z0z dz/L(z)) where the Monin‐Obukhov length L = z/Rƒ, A = 5.5, and Rf is the flux Richardson number. For the thermally stratified atmospheric boundary layer, L(z) is nearly constant, which yields the familiar log‐linear profile. Our model predicts that in the sediment‐stratified bottom boundary layer, L is nearly linear in z (i.e., Rƒ ≅ const). This yields U ≅ (u*/k′) ln z/Z0, where κ′ = κ/(1 + ARƒ) and κ′ may be interpreted as a modified von Karman constant. The results suggest that u* values inferred from logarithmic speed profiles in sediment‐laden flows may be significant overestimates of the actual u*.

Organized Motion in Turbulent Flow

A review of organized motion in turbulent flow indicates that the transport properties of most shear flows are dominated by large-scale vortex nonrandom motions. The mean velocity profile of a turbulent boundary layer consists of a viscous sublayer, buffer layer, and a logarithmic outer layer; an empirical formula of Coles (1956) applies to various pressure gradients. The boundary layer coherent structure was isolated by the correlation methods of Townsend (1956) and flow visualization by direct observations of complex unsteady turbulent motions. The near-wall studies of Willmart and Wooldridge (1962) used the space-time correlation for pressure fluctuations at the wall under a thick turbulent boundary layer; finally, organized motion in free shear flows and transition-control of mixing demonstrated that the Reynolds number invariance of turbulence shows wide scatter.

Turbulent Wake of a Thin, Flat Plate

An analytical solution is developed in two near-wake regions using an inner and outer layer coordinate expansion technique. In region I, extending downstream approximately ten initial sublayer thicknesses, a laminar inner wake grows into the trailing-edge laminar sublayer yielding the Goldstein solution, where the centerline velocity and inner thickness grow as x. In region II, extending downstream approximately ten full boundarylayer thicknesses, a turbulent inner layer grows into the initial logarithmic layer; here the centerline velocity is shown to increase logarithmically with downstream distance. Comparisons of the solution are made with the near-wake data of Chevray and Kovasznay.

Tensorial volume of turbulence

A new simple equation is proposed, for a tensorial volume of turbulence, replacing the scale of turbulence equation. The equation was applied to two simple cases: a logarithmic boundary layer and the decay of isotropic homogeneous turbulence.

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.

Velocity Profiles and Minimum Stream Power

The velocity distribution of laminar and turbulent flows in a wide open channel was analyzed using the theory of minimum rate of energy expenditure or, equivalently, the theory of minimum stream power. The solution for the laminar flow is the classical parabolic velocity distribution. For turbulent flow it is necessary to assume the functional forms of the velocity distribution and the corresponding eddy viscosity distribution. A number of constants or parameters in the assumed functions were determined using the minimization theory and appropriate constraint equations. By assuming that the turbulent flow may consist of a logarithmic inner layer and a parabolic outer layer, the minimization theory determines that the thicknesses of the two layers must be equal. Two constants remain to be determined using experimental data. It is suggested that the two empirical constants are best determined by measured maximum and mean velocities. Theoretical results are verified with data available in the literature.

Turbulent velocity profiles from stability criteria

A velocity-defect law for channel flow is shown to result from the single requirement of Reynolds-stress spectral ‘smoothness’ for any mean profile maintained free of inflexions by transient instabilities. The deduced velocity is a logarithmic function of position near boundaries and parabolic in mid-regions of the flow, independent of the detailed mechanisms of momentum transport by the fluid. However, if the ‘smoothness’ of the spectral tail is decreased, a second logarithmic layer of steeper slope emerges inside the first layer. Profile data from drag-reduction experiments gathered by Virk (1975) exhibit the deduced inner log layer and its transition region to the usual outer flow.

Observations of shear in near-bed currents in the southern North Sea

Measurements of current speed and direction have been made at four heights between 0·6 m and 10 m above the sea-bed at a location in the southern North Sea during a 14-day period. Tidal ellipses are presented for the currents at each height for both neap and spring tides. Speed profiles indicate that there is a logarithmic layer in the lowest 5 m of the water column; both the roughness lengths and the drag coefficients determined from these profiles decrease with increasing current speed. The current directions were found to exhibit positive veering, some 8 ° between 0·6 m and 3·0 m above the bed, and a further 7 ° between 3·0 m and 10 m above the bed. The maximum currents when the flow was directed between north and north-west (the ebb) were consistently stronger than those when it was directed between south and south-east (the flood) at each of the levels, and hence the residual currents were in the ebb direction. The residual currents were generally greater during spring tides than during neap tides, and also exhibited positive veering: the 14-day vector-mean current 0·6 m above the bed was directed some 25 ° to the left of that at 10 m. The residual flow can be considered part of a clockwise circulation around the Well Bank.

Turbulent boundary layer bearing silt in suspension

Essential characteristics of a turbulent boundary layer, bearing silt flocks in suspension, are described and interpreted with the help of a simple steady state model calibrated for the test region of the Math Modelsea project off the Northern Belgian Coast under near critical conditions : no net flux of particles through the bottom boundary. The model emphasizes the existence between the viscous sublayer and the classical Prandtl–Kármán logarithmic layer, of an ’’elastic sublayer’’ where gravity acting on the suspended load provides the necessary restoring force.

Statistical Fluid Mechanics: The Mechanics of Turbulence

This book, originally published in Moscow in 1965, is of interest to a wide scientific and technical audience, including geophysicists, meteorologists, aerodynamicists, chemical, mechanical, and civil engineers--in short, all interested in the fundamental problems of flow, mass, and heat transfer. The authors deal with the theory of hydrodynamic instability and the development of turbulence, the application of dimensional analysis, and the theory of similarity to turbulent flow in pipes, ducts, and boundary layers, as well as free turbulence. They discuss semiempirical theories of turbulence, develop the similarity theory for turbulence in nonhomogeneous media, and present Lagrangian characteristics of turbulence and the theory of turbulent diffusion. Every effort has been made to present a wealth of experimental material; a large number of examples are drawn from physics of the atmosphere, permitting a generalization of results beyond that which can be obtained in the laboratory. Considerable attention has been given to Kolmogorov's theory of the local structure of developed turbulence and to the theory of turbulence in stratified media.Contents: I. Laminar and Turbulent Motion: Equations of dynamics of a fluid and their most important consequences; Hydrodynamic instability and development of turbulence. II. Mathematical Methods for Describing Turbulence, Mean Values and Correlation Functions: Methods for taking mean; The fields of hydrodynamic characteristics regarded as stochastic fields; The moments of hydrodynamic fields. III. The Reynolds Equation and Semiempirical Theories of Turbulence: Turbulent flow in pipes and in the boundary layer; Turbulent energy balance and results derived from it. IV. Turbulence in a Medium Stratified with Respect to Temperature: Generalization of the theory of the logarithmic boundary layer to the case of a medium stratified with respect to temperature; Comparison of the theory with experimental data on the atmospheric layer near the ground. V. Motion of Particles (or Elements) in a Turbulent Stream: Lagrangian description of Turbulence; Turbulent diffusion.

Influence of body forces on turbulent transport near a surface

By examining the asymptotic approach of turbulent flow to a surface, using a simplified second-order closure model of turbulent transport, the ‘law of the wall’ region is extended to include the influence of body forces in an equilibrium layer. The limit of large Reynolds number is considered so the flow is only a function of surface shear stress, surface roughness, and the nature of the body force. After development of the general formulation, application to two particular flows, a rotating flow adjacent to a stationary surface and the neutral planetary boundary layer, provide examples where the region of validity is an order of magnitude larger than the usual logarithmic wall layer.

Roughness length in the turbulent Ekman layer above the sea bed

Roughness lengths were determined from average current profiles in the logarithmic boundary layer at a site in the southern North Sea. Grab samples indicated that the bed configuration controlling the bed roughness was unlikely to change even at maximum spring tides, but the measured roughness lengths were found to decrease as the speed of the flow 10 m above the bed increased. This has been qualitatively interpreted in terms of the size of the turbulent eddies within the flow and their effectiveness at “feeling” the size of the elements forming the bed roughness. It is also shown that the dependence of the observed drag coefficient on the Rossby number follows closely the theoretical form when the roughness length decreases with increasing flow speed.

The influence of suspended cohesive sediments on boundary-layer structure and erosive activity of turbulent seawater flow

Velocity and suspension measurements in the logarithmic layer of hydraulically smooth turbulent tidal flow from the North Sea are reported. The data were not compatible with the assumption of Newtonian flow for the experimental seawater—clay suspension. Laboratory measurements were initiated with mud and seawater from the North Sea in which the boundary-layer structure of this two-phase flow was measured down into the viscous sublayer. The dilute seawater—clay suspension was a mixture of illite, kaolinite and chlorite minerals with concentrations less than 380 mg/l and exhibited turbulent drag reduction. By reviewing flow measurements of other authors it is suggested that turbulent drag reduction occurs on a geophysical scale if the flows transport cohesive sediments. It is proposed that drag reduction is caused by dynamic interaction between turbulent shear strain in the flow and deformation of aggregates. As a consequence, the values of the critical friction velocity u ∗ crit and of erosion rates must be reviewed for cohesive bottom materials. Normally they were obtained under the assumption of a Newtonian flow structure which is not applicable if the flow transports cohesive sediments. To detect the occurrence of drag reduction in geophysical boundary layers (hydraulically smooth), flow measurements must be performed down into the viscous sublayer. The adequate velocity sensors must have a diameter of ⩽1 mm.

Observations on turbulent-drag reduction in a dilute suspension of clay in sea-water

Hot-wire anemometer measurements have been made in a dilute sea-water/claymineral suspension. For fully developed turbulent flows in an open channel with a smooth mud (from the North Sea) bottom, mean streamwise velocity profiles were measured for Reynolds numbers between 5400 and 27 800 (i.e. non-eroding and eroding flow rates) and compared with Newtonian flows under the same experimental conditions. For the clay-mineral suspensions, measurements of the kinematic viscosityv, Kármán's constantkand the mean streamwise velocity$\overline{u}$of the logarithmic layer seemed to verify a Newtonian flow structure. Although the distributions of concentration showed no substantial increase towards the wall, it was found that beneath this Newtonian core there existed a viscous sublayer whose thickness was enhanced by a factor of 2–5. The friction velocityu*determined by the gradient method in the viscous sublayer was reduced by as much as 40 %. The mean flow structure exhibited an additional new layer in the region 10 <y+< 30.The measurements indicate that turbulent-drag reduction occurs for the experimental clay-mineral suspension at non-eroding and also at eroding velocities. Agglomeration of suspended clay-mineral particles is suggested as possible explanation of this phenomenon.

A Numerical Study of Time-Dependent Turbulent Ekman Layers over Horizontal and Sloping Bottoms

A numerical study is made of a time-dependent turbulent Ekman bottom boundary layer. Parameters for the model were chosen to simulate conditions near the bottom of the Florida Current in the Straits of Florida. The model used is that of Lykosov and Gutman. It allows the coefficient of turbulent viscosity v to vary with time t and height z and permits the effects of an imposed stable stratification and sloping bottom to be included. The variation of v with t and z is not preset but is determined in the course of solving the problem subject to the turbulent energy equation, the similarity arguments of Komolgoroff, and a mixing length hypothesis of Zilitinkevich and Laykhtman. The results of this preliminary study are compared to the author's observations. The agreement is good for the friction velocity values as well as for the mean total Ekman veering. However, most of the computed Ekman veering occurred above the logarithmic layer while most of the measured veering occurred within the logarithmic layer. The results suggest, as do the observations, that turbulent Ekman bottom layers varying on time scales of order the local inertial period are not quasi-stationary. Allowing the bottom to be inclined at a small angle transverse to the flow is found to modify significantly the temperature profile near the bottom, leading at times either to the formation of a homogeneous layer of depth order 10 m or to conditions marginally suitable for the formation of convectively mixed layer of comparable depth.

Rough-to-smooth transition of an equilibrium neutral constant stress layer

Purpose of research on rough-to-smooth transition of an equilibrium neutral constant stress layer is to develop a model for low-level atmospheric flow over terrains of abruptly changing roughness, such as those occurring near the windward end of a landing strip, and to use the model to derive functions which define the extent of the region affected by the roughness change and allow adequate prediction of wind and shear stress profiles at all points within the region. A model consisting of two bounding logarithmic layers and an intermediate velocity defect layer is assumed, and dimensionless velocity and stress distribution functions which meet all boundary and matching conditions are hypothesized. The functions are used in an asymptotic form of the equation of motion to derive a relation which governs the growth of the internal boundary layer. The growth relation is used to predict variation of surface shear stress.