Pressure gradient effects on the large-scale structure of turbulent boundary layers

Research into high-Reynolds-number turbulent boundary layers in recent years has brought about a renewed interest in the larger-scale structures. It is now known that these structures emerge more prominently in the outer region not only due to increased Reynolds number (Metzger & Klewicki, , 2001,  Hutchins & Marusic,  2007, pp. 1–28), but also when a boundary layer is exposed to an adverse pressure gradient (Bradshaw,  1967,  Lee & Sung,  2009, pp. 101–131). The latter case has not received as much attention in the literature. As such, this work investigates the modification of the large-scale features of boundary layers subjected to zero, adverse and favourable pressure gradients. It is first shown that the mean velocities, turbulence intensities and turbulence production are significantly different in the outer region across the three cases. Spectral and scale decomposition analyses confirm that the large scales are more energized throughout the entire adverse pressure gradient boundary layer, especially in the outer region. Although more energetic, there is a similar spectral distribution of energy in the wake region, implying the geometrical structure of the outer layer remains universal in all cases. Comparisons are also made of the amplitude modulation of small scales by the large-scale motions for the three pressure gradient cases. The wall-normal location of the zero-crossing of small-scale amplitude modulation is found to increase with increasing pressure gradient, yet this location continues to coincide with the large-scale energetic peak wall-normal location (as has been observed in zero pressure gradient boundary layers). The amplitude modulation effect is found to increase as pressure gradient is increased from favourable to adverse.

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title: Wall turbulence closure based on classical similarity laws and the attached eddy hypothesis
abstract: A new look at the closure problem of turbulent boundary layers is taken here using recently derived analytical expressions for the shear stress distributions. These expressions are based on the logarithmic law of the wall and law of the wake formulation of Coles [ (1956)] with the mean continuity and the mean momentum differential and integral equations. The concept of equilibrium layers of Clauser [ (1956)] is extended and using similar ideas as Rotta [ (1962)] for self‐similarity, a closure scheme is proposed for layers developing in arbitrary pressure gradients for the case where the streamwise derivative of the Coles wake factor is not too large. For a given flow case, this Coles wake condition can be tested with internal consistency checks. The mathematical framework is most suitable for incorporating Townsend’s attached eddy hypothesis as recently developed by Perry, Li, and Marusic [ A 336, 67 (1991)] for closure. This gives an opportunity to incorporate coherent structure concepts into closure schemes. Possible ways of handling the difficult case where the streamwise derivative of the Coles wake factor is significant are discussed.

title: On the streamwise evolution of turbulent boundary layers in arbitrary pressure gradients
abstract: A new approach to the classic closure problem for turbulent boundary layers is presented. This involves, first, using the well-known mean-flow scaling laws such as the log law of the wall and the law of the wake of Coles (1956) together with the mean continuity and the mean momentum differential and integral equations. The important parameters governing the flow in the general non-equilibrium case are identified and are used for establishing a framework for closure. Initially closure is achieved here empirically and the potential for achieving closure in the future using the wall-wake attached eddy model of Perry & Marusic (1995) is outlined. Comparisons are made with experiments covering adverse-pressure-gradient flows in relaxing and developing states and flows approaching equilibrium sink flow. Mean velocity profiles, total shear stress and Reynolds stress profiles can be computed for different streamwise stations, given an initial upstream mean velocity profile and the streamwise variation of free-stream velocity. The attached eddy model of Perry & Marusic (1995) can then be utilized, with some refinement, to compute the remaining unknown quantities such as Reynolds normal stresses and associated spectra and cross-power spectra in the fully turbulent part of the flow.

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title: Reynolds-number scaling of the flat-plate turbulent boundary layer
abstract: Despite extensive study, there remain significant questions about the Reynolds-number scaling of the zero-pressure-gradient flat-plate turbulent boundary layer. While the mean flow is generally accepted to follow the law of the wall, there is little consensus about the scaling of the Reynolds normal stresses, except that there are Reynolds-number effects even very close to the wall. Using a low-speed, high-Reynolds-number facility and a high-resolution laser-Doppler anemometer, we have measured Reynolds stresses for a flat-plate turbulent boundary layer from Reθ = 1430 to 31 000. Profiles of u′2, v′2, and u′v′ show reasonably good collapse with Reynolds number: u′2 in a new scaling, and v′2 and u′v′ in classic inner scaling. The log law provides a reasonably accurate universal profile for the mean velocity in the inner region.

title: The effects of a favourable pressure gradient and of the Reynolds number on an incompressible axisymmetric turbulent boundary layer - Part 1. The turbulent boundary layer
abstract: The effects of a favourable pressure gradient ( K [les ]4×10 −6 ) and of the Reynolds number
 (862[les ] Re δ 2 [les ]5800) on the mean
 and fluctuating quantities of four turbulent boundary layers were studied experimentally and are presented in this paper
 and a companion paper (Part 2). The measurements consist of extensive hot-wire
 and skin-friction data. The former comprise mean and fluctuating velocities, their correlations
 and spectra, the latter wall-shear stress measurements obtained by four
 different techniques which allow testing of calibrations in both laminar-like and
 turbulent flows for the first time. The measurements provide complete data sets,
 obtained in an axisymmetric test section, which can serve as test cases as specified
 by the 1981 Stanford conference. Two different types of accelerated boundary layers were investigated
 and are described: in this paper (Part 1) the fully turbulent, accelerated boundary
 layer (sometimes denoted laminarescent) with approximately local equilibrium
 between the production and dissipation of the turbulent energy and with relaxation
 to a zero pressure gradient flow (cases 1 and 3); and in Part 2 the strongly accelerated
 boundary layer with ‘inactive’ turbulence, laminar-like mean flow behaviour
 (relaminarized), and reversion to the turbulent state (cases 2 and 4). In all four cases
 the standard logarithmic law fails but there is no single parametric criterion which
 denotes the beginning or the end of this breakdown. However, it can be demonstrated
 that the departure of the mean-velocity profile is accompanied by characteristic
 changes of turbulent quantities, such as the maxima of the Reynolds stresses or the
 fluctuating value of the skin friction. The boundary layers described here are maintained in the laminarescent
 state just up to the beginning of relaminarization and then relaxed to the turbulent
 state in a zero pressure gradient. The relaxation of the turbulence structure occurs
 much faster than in an adverse pressure gradient. In the accelerating boundary layer
 absolute values of the Reynolds stresses remain more or less constant in the outer
 region of the boundary layer in accordance with the results of Blackwelder &
 Kovasznay (1972), and rise both in the vincinity of the wall in conjunction with the rising
 wall shear stress and in the centre region of the boundary layer with the increase
 of production.

title: An experimental investigation of a highly accelerated turbulent boundary layer
abstract: A canonical flat-plate turbulent boundary layer with Reθ = 4590 is exposed to a favourable mean streamwise pressure gradient sufficient to cause relaminarization. The favourable pressure gradient is generated by a linear contraction, yielding a peak value of the acceleration parameter of K = 4.4 × 10−6 which is sustained for approximately 13 local boundary layer thicknesses. The relaminarization process is characterized by an extensive series of mean flow and turbulence measurements obtained at several representative streamwise locations. In anticipation of the loss of standard log-law behaviour, the local wall shear stress is directly measured using the oil-film interferometry technique. Mean flow measurements show a systematic variation in the Kármán and additive constants with applied streamwise strain rate. The series of measurements also indicate an apparent decoupling of the outer and near-wall regions of the accelerating boundary layer. In accord with this, conditional measurements show that fourth-quadrant sweep events are virtually eliminated, while much less frequent but larger-amplitude near-wall second-quadrant ejection events remain. The reduction in fourth-quadrant sweep events is matched by an observed increase in near-wall third-quadrant events. The consequent reduction in near-wall Reynolds stress correlation and associated cross-stream momentum transport results in a large reduction in cf for the relaminarized flow.

title: Direct Numerical Simulation of Self-Similar Turbulent Boundary Layers in Adverse Pressure Gradients
abstract: Direct numerical simulations of the Navier–Stokes equations have been carried out with the objective of studying turbulent boundary layers in adverse pressure gradients. The boundary layer flows concerned are of the equilibrium type which makes the analysis simpler and the results can be compared with earlier experiments and simulations. This type of turbulent boundary layers also permits an analysis of the equation of motion to predict separation. The linear analysis based on the assumption of asymptotically high Reynolds number gives results that are not applicable to finite Reynolds number flows. A different non-linear approach is presented to obtain a useful relation between the freestream variation and other mean flow parameters. Comparison of turbulent statistics from the zero pressure gradient case and two adverse pressure gradient cases shows the development of an outer peak in the turbulent energy in agreement with experiment. The turbulent flows have also been investigated using a differential Reynolds stress model. Profiles for velocity and turbulence quantities obtained from the direct numerical simulations were used as initial data. The initial transients in the model predictions vanished rapidly. The model predictions are compared with the direct simulations and low Reynolds number effects are investigated.

title: Turbulence development in a non-equilibrium turbulent boundary layer with mild adverse pressure gradient
abstract: High-resolution laser-Doppler anemometer measurements were acquired in a two-dimensional turbulent boundary layer over a $4^\circ$ ramp. The goals were to provide a detailed data set for an adverse pressure gradient boundary layer far from separation and to examine near-wall behaviour of the Reynolds stresses as compared to flat-plate boundary layers. The flow develops over a flat plate, reaching a momentum thickness Reynolds number of 3350 at an upstream reference location. The boundary layer is then subjected to a varying pressure gradient along the length of the ramp and partially redevelops on a downstream flat plate. Mean velocity measurements show a log law region in all velocity profiles, but the outer layer does not collapse in deficit coordinates indicating that the boundary layer is not in equilibrium. Measurements of non-dimensional stress ratios and quadrant analysis of the two-component data indicate relatively small changes to the turbulence structure. However, the streamwise normal stress has an extended outer layer plateau, and the shear stress and wall-normal stress have outer layer peaks. Near the wall, the streamwise normal stress and shear stress collapse with flat-plate data using standard scaling, but the wall normal stress is substantially larger than flat-plate cases.

title: High Reynolds number effects in wall turbulence
abstract: A review of recent advances in the study of high Reynolds number turbulent boundary layers is given. The emergent regime of very large-scale structures in the logarithmic region and their subsequent influence on the near-wall cycle challenges many of the previously held assumptions regarding scaling of turbulent boundary layers at high Reynolds numbers. Experimental results are presented to illustrate the superimposition of large-scale energy onto the near-wall cycle, together with an interaction well described by an amplitude modulation effect. Both phenomena are shown to increase in magnitude (as compared to viscous-scaled events) as Reynolds number increases. These observations lead to a possible model for a statistically representative near-wall velocity signal (giving accurate energy spectra) based on a given filtered velocity signal from the log region of a high Reynolds number turbulent flow.

title: Vortex organization in the outer region of the turbulent boundary layer
abstract: The structure of energy-containing turbulence in the outer region of a zero-pressure- gradient boundary layer has been studied using particle image velocimetry (PIV) to measure the instantaneous velocity fields in a streamwise-wall-normal plane. Experiments performed at three Reynolds numbers in the range 930 < Reθ < 6845 show that the boundary layer is densely populated by velocity fields associated with hairpin vortices. (The term ‘hairpin’ is here taken to represent cane, hairpin, horseshoe, or omega-shaped vortices and deformed versions thereof, recognizing these structures are variations of a common basic flow structure at different stages of evolution and with varying size, age, aspect ratio, and symmetry.) The signature pattern of the hairpin consists of a spanwise vortex core located above a region of strong second-quadrant fluctuations (u < 0 and v > 0) that occur on a locus inclined at 30–60° to the wall. In the outer layer, hairpin vortices occur in streamwise-aligned packets that propagate with small velocity dispersion. Packets that begin in or slightly above the buffer layer are very similar to the packets created by the autogeneration mechanism (Zhou, Adrian & Balachandar 1996). Individual packets grow upwards in the streamwise direction at a mean angle of approximately 12°, and the hairpins in packets are typically spaced several hundred viscous lengthscales apart in the streamwise direction. Within the interior of the envelope the spatial coherence between the velocity fields induced by the individual vortices leads to strongly retarded streamwise momentum, explaining the zones of uniform momentum observed by Meinhart & Adrian (1995). The packets are an important type of organized structure in the wall layer in which relatively small structural units in the form of three-dimensional vortical structures are arranged coherently, i.e. with correlated spatial relationships, to form much longer structures. The formation of packets explains the occurrence of multiple VITA events in turbulent ‘bursts’, and the creation of Townsend's (1958) large-scale inactive motions. These packets share many features of the hairpin models proposed by Smith (1984) and co-workers for the near-wall layer, and by Bandyopadhyay (1980), but they are shown to occur in a hierarchy of scales across most of the boundary layer. In the logarithmic layer, the coherent vortex packets that originate close to the wall frequently occur within larger, faster moving zones of uniform momentum, which may extend up to the middle of the boundary layer. These larger zones are the induced interior flow of older packets of coherent hairpin vortices that originate upstream and over-run the younger, more recently generated packets. The occurence of small hairpin packets in the environment of larger hairpin packets is a prominent feature of the logarithmic layer. With increasing Reynolds number, the number of hairpins in a packet increases.

title: Large- and very-large-scale motions in channel and boundary-layer flows
abstract: Large-scale motions (LSMs; having wavelengths up to 2–3 pipe radii) and very-LSMs (having wavelengths more than 3 pipe radii) have been shown to carry more than half of the kinetic energy and Reynolds shear stress in a fully developed pipe flow. Studies using essentially the same methods of measurement and analysis have been extended to channel and zero-pressure-gradient boundary-layer flows to determine whether large structures appear in these canonical wall flows and how their properties compare with that of the pipe flow. The very large scales, especially those of the boundary layer, are shorter than the corresponding scales in the pipe flow, but otherwise share a common behaviour, suggesting that they arise from similar mechanism(s) aside from the modifying influences of the outer geometries. Spectra of the net force due to the Reynolds shear stress in the channel and boundary layer flows are similar to those in the pipe flow. They show that the very-large-scale and main turbulent motions act to decelerate the flow in the region above the maximum of the Reynolds shear stress.

title: The structure of turbulent boundary layers
abstract: Extensive visual and quantitative studies of turbulent boundary layers are described. Visual studies reveal the presence of surprisingly well-organized spatially and temporally dependent motions within the so-called ‘laminar sublayer’. These motions lead to the formation of low-speed streaks in the region very near the wall. The streaks interact with the outer portions of the flow through a process of gradual ‘lift-up’, then sudden oscillation, bursting, and ejection. It is felt that these processes play a dominant role in the production of new turbulence and the transport of turbulence within the boundary layer on smooth walls. Quantitative data are presented providing an association of the observed structure features with the accepted ‘regions’ of the boundary layer in non-dimensional co-ordinates; these data include zero, negative and positive pressure gradients on smooth walls. Instantaneous spanwise velocity profiles for the inner layers are given, and dimensionless correlations for mean streak-spacing and break-up frequency are presented. Tentative mechanisms for formation and break-up of the low-speed streaks are proposed, and other evidence regarding the implications and importance of the streak structure in turbulent boundary layers is reviewed.

title: Estimation of turbulent convection velocities and corrections to Taylor's approximation
abstract: A new method is introduced for estimating the convection velocity of individual modes in turbulent shear flows that, in contrast to most previous ones, only requires spectral information in the temporal or spatial direction over which a modal decomposition is desired, while only using local derivatives in other directions. If no spectral information is desired, the method provides a natural definition for the average convection velocity, as well as a way to estimate the accuracy of the frozen-turbulence approximation. Existing data from numerical turbulent channels at friction Reynolds numbers Reτ ≤ 1900 are used to validate the new method against classical ones, and to characterize the dependence of the convection velocity on the eddy wavelength and wall distance. The results indicate that the small scales in turbulent channels travel at the local mean velocity, while large ‘global’ modes travel at a more uniform speed proportional to the bulk velocity. To estimate the systematic deviations introduced in experimental spectra by the use of Taylor's approximation with a wavelength-independent convection velocity, a semi-empirical fit to the computed convection velocities is provided. It represents well the data throughout the Reynolds number range of the simulations. It is shown that Taylor's approximation not only displaces the large scales near the wall to shorter apparent wavelengths but also modifies the shape of the spectrum, giving rise to spurious peaks similar to those observed in some experiments. To a lesser extent the opposite is true above the logarithmic layer. The effect increases with the Reynolds number, suggesting that some of the recent challenges to the kx−1 energy spectrum may have to be reconsidered.

title: Large-eddy simulation of large-scale structures in long channel flow
abstract: We investigate statistics of large-scale structures from large-eddy simulation (LES) of turbulent channel flow at friction Reynolds numbers Reτ = 2K and 200K (where K denotes 1000). In order to capture the behaviour of large-scale structures properly, the channel length is chosen to be 96 times the channel half-height. In agreement with experiments, these large-scale structures are found to give rise to an apparent amplitude modulation of the underlying small-scale fluctuations. This effect is explained in terms of the phase relationship between the large- and small-scale activity. The shape of the dominant large-scale structure is investigated by conditional averages based on the large-scale velocity, determined using a filter width equal to the channel half-height. The conditioned field demonstrates coherence on a scale of several times the filter width, and the small-scale–large-scale relative phase difference increases away from the wall, passing through π/2 in the overlap region of the mean velocity before approaching π further from the wall. We also found that, near the wall, the convection velocity of the large scales departs slightly, but unequivocally, from the mean velocity.

title: On the limitations of Taylor's hypothesis in constructing long structures in a turbulent boundary layer
abstract: Taylor's hypothesis of frozen flow has frequently been used to convert temporal experimental measurements into a spatial domain. This technique has led to the discovery of long meandering structures in the log-region of a turbulent boundary layer. There is some contention over whether Taylor's approximation is valid over large distances. This paper presents an experiment that compares velocity fields constructed using Taylor's approximation with those obtained from particle image velocimetry (PIV), i.e. spatial data, obtained in the logarithmic region of a turbulent boundary layer.

title: Self-Similarity in the Outer Region of Adverse-Pressure-Gradient Turbulent Boundary Layers
abstract: This paper presents a consistent theory of self-similarity and of equilibrium in the outer region of turbulent boundary layers that explains recent experimental findings on the subject, including new ones presented here. The theory is first presented in a general form where the outer scales are left unspecified and it is not assumed that the mean velocity defect and the Reynolds stresses share a common velocity scale. It is shown that the main results of the traditional similarity theory remain valid even in this case. Common outer scaling with the Zagarola-Smits length and velocity scales is then chosen. A new pressure gradient parameter is introduced to characterize the local effect of the pressure gradient in all flow conditions including strong adverse-pressure-gradient conditions. By analyzing several adverse-pressure-gradient flow cases, it is shown that self-similarity of the mean velocity defect profile is reached in all cases in localized but significant flow regions. The same is, however, not true of the Reynolds stress profiles. In agreement with the similarity analysis, the self-similar velocity defect profile is found to be a function of the pressure gradient and most flows studied here are only in an approximate state of equilibrium in the region of self-similar defect profiles despite the excellent collapse of the profiles.