Zero-pressure Gradient Boundary Layer Research Articles

Role of Time Integration in Computing Transitional Flows Caused by Wall Excitation

Numerical investigation of receptivity and flow transition in spatio-temporal framework have shown the central role of spatio-temporal wave-front (STWF) created by wall excitation for transition of a two-dimensional (2D) zero pressure gradient boundary layer (ZPGBL) in Sengupta and Bhaumik (Phys Rev Lett 107:154501, 2011). Although the STWF is created by linear mechanism, it is the later nonlinear stage of evolution revealed by the solution of Navier---Stokes equation (NSE), which causes formation of turbulent spots merging together to create fully developed turbulent flow. Thus, computing STWF for ZPGBL from NSE is of prime importance, which has been reported by the present authors following earlier theoretical investigation. Similar computational efforts using NSE by other researchers do not report finding the STWF. In the present investigation we identify the main reason for other researchers to miss STWF, as due to taking a very short computational domain. Secondly, we show that even one takes a long enough domain and detect STWF, use of traditional low accuracy method will not produce the correct dynamics as reported by Sengupta and Bhaumik (2011). The role of time integration plays a very strong role in the dynamics of transitional flows. We have shown here that implicit methods are more error prone, as compared to explicit time integration methods during flow transition. For the present problem, it is noted that the classical Crank---Nicolson method is unstable for 2D NSE. Same error-prone nature will also be noted for hybrid implicit---explicit time integration methods (known as the IMEX methods). One of the main feature of present analysis is to highlight the accuracy of computations by compact schemes used by the present investigators over a significantly longer domain and over unlimited time, as opposed to those reported earlier in the literature for the wall excitation problem. A consequence of taking long streamwise domain enables one to detect special properties of STWF and its nonlinear growth. The main focus of the present research is to highlight the importance of STWF, which is a new class of spatio-temporal solution obtained from the linear receptivity by solving Orr---Sommerfeld equation and nonlinear analysis of NSE.

Pressure Gradient Effects on Smooth- and Rough-Surface Turbulent Boundary Layers—Part II: Adverse Pressure Gradient

An experimental investigation has been conducted to identify the effects of pressure gradient and surface roughness on turbulent boundary layers. In Part II, smooth- and rough-surface turbulent boundary layers with and without adverse pressure gradient (APG) are presented at a fixed Reynolds number (based on the length of flat plate) of 900,000. Flat-plate boundary layer measurements have been conducted using a single-sensor, hot-wire probe. For smooth surfaces, compared to the zero pressure gradient (ZPG) boundary layer, the APG boundary layer has a higher mean velocity defect throughout the boundary layer and lower friction coefficient. APG decreases the streamwise normal Reynolds stress for y less than 0.4 times the boundary layer thickness and increases it slightly in the outer region. For rough surfaces, APG reduces the roughness effects of increasing the mean velocity defect and normal Reynolds stress for y less than 23 and 28 times the average roughness height, respectively. Consistently, for the same roughness, APG decreases the integrated streamwise turbulent kinetic energy. APG also decreases the roughness effect on the friction coefficient, roughness Reynolds number, and roughness shift. Compared to the ZPG boundary layers, the roughness effects on integral boundary layer parameters—boundary layer thickness and momentum thickness—are weaker under APG. Thus, contrary to the favorable pressure gradient (FPG) in part I, APG reduces the roughness effects on turbulent boundary layers.

Effect of mean and fluctuating pressure gradients on boundary layer turbulence

AbstractThis study focuses on the effects of mean (favourable) and large-scale fluctuating pressure gradients on boundary layer turbulence. Two-dimensional (2D) particle image velocimetry (PIV) measurements, some of which are time-resolved, have been performed upstream of and within a sink flow for two inlet Reynolds numbers, ${Re}_{\theta }(x_{1})=3360$ and 5285. The corresponding acceleration parameters, $K$, are ${1.3\times 10^{-6}}$ and ${0.6\times 10^{-6}}$. The time-resolved data at ${Re}_{\theta }(x_{1})=3360$ enables us to calculate the instantaneous pressure distributions by integrating the planar projection of the fluid material acceleration. As expected, all the locally normalized Reynolds stresses in the favourable pressure gradient (FPG) boundary layer are lower than those in the zero pressure gradient (ZPG) domain. However, the un-scaled stresses in the FPG region increase close to the wall and decay in the outer layer, indicating slow diffusion of near-wall turbulence into the outer region. Indeed, newly generated vortical structures remain confined to the near-wall region. An approximate analysis shows that this trend is caused by higher values of the streamwise and wall-normal gradients of mean streamwise velocity, combined with a slightly weaker strength of vortices in the FPG region. In both boundary layers, adverse pressure gradient fluctuations are mostly associated with sweeps, as the fluid approaching the wall decelerates. Conversely, FPG fluctuations are more likely to accompany ejections. In the ZPG boundary layer, loss of momentum near the wall during periods of strong large-scale adverse pressure gradient fluctuations and sweeps causes a phenomenon resembling local 3D flow separation. It is followed by a growing region of ejection. The flow deceleration before separation causes elevated near-wall small-scale turbulence, while high wall-normal momentum transfer occurs in the ejection region underneath the sweeps. In the FPG boundary layer, the instantaneous near-wall large-scale pressure gradient rarely becomes positive, as the pressure gradient fluctuations are weaker than the mean FPG. As a result, the separation-like phenomenon is markedly less pronounced and the sweeps do not show elevated small-scale turbulence and momentum transfer underneath them. In both boundary layers, periods of acceleration accompanying large-scale ejections involve near-wall spanwise contraction, and a high wall-normal momentum flux at all elevations. In the ZPG boundary layer, although some of the ejections are preceded, and presumably initiated, by regions of adverse pressure gradients and sweeps upstream, others are not. Conversely, in the FPG boundary layer, there is no evidence of sweeps or adverse pressure gradients immediately upstream of ejections. Apparently, the mechanisms initiating these ejections are either different from those involving large-scale sweeps or occur far upstream of the peak in FPG fluctuations.

Direct numerical simulation of transitional mixed convection flows: Viscous and inviscid instability mechanisms

Receptivity studies using direct numerical simulation require computations of equilibrium flow and its response to deterministic excitation. Equivalent flow problem, without heat interaction for zero-pressure gradient boundary layer, has been studied with respect to wall-excitation by a finite difference high accuracy method based on the solution of Navier-Stokes equation in Sengupta and Bhaumik [Phys. Rev. Lett.107, 154501 (2011)] and Sengupta et al. [Phys. Rev. E85, 026308 (2012)]. One of the key features of this study has been that the same methodology is used for computing the equilibrium flow and the disturbance field. Computation of equilibrium flow was performed by solving Navier-Stokes equation to include the leading edge of the plate, so that the effects of leading edge singularity and the growth of the boundary layer is included in the nonlinear framework. When the same methodology is attempted for mixed convection flows past horizontal plate (with Boussinesq approximation to model heat transfer effects) some of the equilibrium flow features could not be explained with linear viscous instability theory results. For horizontal hot flat plate with adiabatic wall conditions, the equilibrium flow could be computed and its receptivity could be correlated with linear spatial theory for lower buoyancy parameter. Here, we focus on receptivity of mixed convection flows to wall excitation for the following cases which do not allow computing the equilibrium flows from the solution of Navier-Stokes equation: (i) Aadiabatic horizontal flat plate cooled significantly at the leading edge only and (ii) strongly heated isothermal wedge flow for a wedge angle of 60°. The cold plate case is particularly interesting as the linear spatial theory indicates enhanced stabilization for higher magnitude of the buoyancy parameter. Results presented for the cold plate case indicates disturbance growth outside the shear layer. This prompted us to re-investigate various mechanisms of instability present for mixed convection flows. There is no definitive study for inviscid mechanism in this respect and the present effort not only fills this void, but also explains the relative roles of viscous and inviscid mechanisms for strong heat transfer effects, within the Boussinesq approximation.

Breakdown mechanisms and heat transfer overshoot in hypersonic zero pressure gradient boundary layers

AbstractA laminar Mach 6 flat plate boundary layer is perturbed using three different types of disturbances introduced through blowing and suction. The linear and nonlinear development and eventual breakdown to turbulence are investigated using direct numerical simulation. The three different transition mechanisms compared are first mode oblique breakdown, second mode oblique breakdown and second mode fundamental resonance. The focus of the present work is to compare the nonlinear development and breakdown to turbulence for the different transition mechanisms and explain the heat transfer overshoot observed in experiments. First mode oblique breakdown leads to the shortest transition length and a clear peak in wall heat transfer in the transitional region. For all three transition mechanisms, the development of streamwise streaks precedes the breakdown to fully turbulent flow. The modal linear and nonlinear development are analysed including the breakdown of the streaks. The effect of wall cooling is investigated for second mode fundamental resonance and no qualitative differences in the nonlinear processes are observed. Finally, the development towards fully turbulent flow including mean flow, turbulent spectra, and turbulent fluctuations is shown and the first mode oblique breakdown simulation shows the furthest development towards a fully turbulent flow.

A reformulated synthetic turbulence generation method for a zonal RANS–LES method and its application to zero-pressure gradient boundary layers

A synthetic turbulence generation (STG) method for subsonic and supersonic flows at low and moderate Reynolds numbers to provide inflow distributions of zonal Reynolds-averaged Navier–Stokes (RANS) – large-eddy simulation (LES) methods is presented. The STG method splits the LES inflow region into three planes where a local velocity signal is decomposed from the turbulent flow properties of the upstream RANS solution. Based on the wall-normal position and the local flow Reynolds number, specific length and velocity scales with different vorticity content are imposed at the inlet plane of the boundary layer. The quality of the STG method for incompressible and compressible zero-pressure gradient boundary layers is shown by comparing the zonal RANS–LES data with pure LES, pure RANS, and direct numerical simulation (DNS) solutions. The distributions of the time and spanwise wall-shear stress, Reynolds stress distributions, and two point correlations of the zonal RANS–LES simulations are smooth in the transition region and in good agreement with the pure LES and reference DNS findings. The STG approach reduces the RANS-to-LES transition length to less than four boundary-layer thicknesses.

Thermal boundary layer profiles in turbulent Rayleigh-Bénard convection in a cylindrical sample

We numerically investigate the structures of the near-plate temperature profiles close to the bottom and top plates of turbulent Rayleigh-Bénard flow in a cylindrical sample at Rayleigh numbers Ra = 10(8) to Ra = 2 × 10(12) and Prandtl numbers Pr = 6.4 and Pr = 0.7 with the dynamical frame method [Zhou and Xia, Phys. Rev. Lett. 104, 104301 (2010)], thus extending previous results for quasi-two-dimensional systems to three-dimensional systems. The dynamical frame method shows that the measured temperature profiles in the spatially and temporally local frame are much closer to the temperature profile of a laminar, zero-pressure gradient boundary layer (BL) according to Pohlhausen than in the fixed reference frame. The deviation between the measured profiles in the dynamical reference frame and the laminar profiles increases with decreasing Pr, where the thermal BL is more exposed to the bulk fluctuations due to the thinner kinetic BL, and increasing Ra, where more plumes are passing the measurement location.

Application of Passive Surface Textures to Control the Growth of Turbulent Spots at Moderately High Reynolds Numbers

A spectral DNS code was used to study the growth and spreading of turbulent spots in a nominally laminar, zero-pressure gradient boundary layer. In addition to the flat wall case, the interaction of these spots with passive surface textures (riblets and fins) was examined. The dimensions of these textures were chosen based on parametric studies performed in past work, with the goal of achieving maximum reduction of spot spreading. The flat plate, surface textures, and initial spot perturbation were simulated via an immersed boundary method, and a “suction-wall” allowed the available channel code to model a boundary layer. In all cases, self-similar arrowhead shaped spots formed. The λ2 variable was used to visualize the vortical structures within a spot, and a developed spot was found to consist primarily of a multitude of hairpin vortices. The riblets and the fins both decreased the average spot spreading rate by 7-8% from the flat wall value.

Friction measurement in zero and adverse pressure gradient boundary layer using oil droplet interferometric method

The oil droplet interferometric technique has been used to investigate the skin friction distribution along a zero and adverse pressure gradient boundary layer developing in the Laboratoire de Mecanique de Lille wind tunnel. This experimental task was a part of the WALLTURB project, funded by the European Community, in order to bring significant progress in the understanding of near wall turbulence in boundary layers. Skin friction values close to 0.01 Pa have been measured with this optical method. A comparison with the results obtained with hot-wire anemometry and macro-PIV demonstrates the great potential of the oil droplet technique.

Nonlinear and Nonparallel Receptivity of Zero-pressure Gradient Boundary Layer

Nonlinear and nonparallel receptivity of zero pressure gradient boundary layer have been investigated by solving the full two-dimensional Navier-Stokes equation with respect to wall excitation at fixed frequency. The excitation is triggered through a blowing-suction strip at a location near the leading edge of the plate. Results are reported for cases which are linearly stable in a spatial formulation, for different amplitudes of the excitation of the strip.Presented results for a lower amplitude of excitation for the chosen frequency of excitation help explain the displayed nonlinearity of the streamwise component of the disturbance velocity at intermediate heights inside the shear layer. The disturbance field is noted to be neutrally stable at heights close to the plate and in the outer part of the boundary layer. Overall, such behaviour of the disturbance field is explained in terms of the presence of multiple modes and their spatiotemporal interactions and the importance of nonlinearity at specific he...

Instantaneous fluctuation velocity and skewness distributions upstream of transition onset

The development of streamwise orientated disturbances through the boundary layer thickness prior to transition onset for zero-pressure gradient boundary layer flow under the influence %Tu = 4.2 is presented. The analysis concentrates on the development of the maximum positive and negative of the fluctuation velocity in order to gain further insight into the transition process. The average location of the peak negative fluctuation velocity over a range of Reynolds numbers was measured in the upper portion of the boundary layer at y/ δ ≈ 0.6, whereas the location of the peak positive value was measured at y/ δ ≈ 0.3. The disturbance magnitude of the negative fluctuation velocity increased beyond that of the positive as transition onset approached. The distribution and disturbance magnitude of the maximum positive and negative fluctuation velocities indicate that the initiation of transition may occur on the low-speed components of the flow that are lifted up to the upper region of the boundary layer. This is in qualitative agreement with recent direct numerical simulations on the breakdown of the flow on the lifted low-speed streaks near the boundary layer edge. The results presented in this investigation also demonstrate the increased physical insight gained by examining the distributions of the maximum positive and negative of the streamwise fluctuation velocity component associated with the low- and high-speed streaks, compared to time-averaged values, in determining what structures cause the breakdown to turbulence.

The spatial relationships between dissipation and production rates and vortical structures in turbulent boundary and mixing layers

A novel approach for studying the spatial relationship between the production and dissipation rates of turbulent kinetic energy and vortical structures is presented. Two turbulent flows were investigated: the zero pressure gradient boundary layer and the two-stream mixing layer. In both flows, a multisensor hot-wire probe was used to measure the velocity components in all three coordinate directions, as well as six components of the velocity gradient tensor. The remaining three velocity gradients were determined using Taylor’s hypothesis. With these data, the “instantaneous” production and dissipation rates, defined by P=−(∂U¯i∕∂xj)uiuj and D=−ν[(∂ui∕∂xj)2+(∂ui∕∂xj)(∂uj∕∂xi)], respectively, were determined. Cross-correlating the fluctuations of these two signals reveals that they are not randomly distributed in time with respect to each other; rather they display significant levels of correlation. Plotting the cross-correlation coefficients versus a dimensionless length scale, defined as L′=sgn(τ)∣τ∣∕νU¯, reveals an asymmetric pattern that persists at several cross-stream locations for both flows. Furthermore, correlating both the dissipation and production rates with a vortex identifier, ωxy=[(ωx)2+(ωy)2]1∕2, also reveals consistent cross-stream patterns. The magnitude of these correlations and their persistent shapes across the flows suggest that the spatial separation between regions of concentrated dissipation and production rates is associated with the presence of quasistreamwise vortices in both of these flows. More specifically, they imply that regions of concentrated rates of dissipation are primarily in the cores of the vortices, whereas regions of rates of production are more concentrated on their periphery.

Self-similar profile of probability density functions in zero-pressure gradient turbulent boundary layers

The probability density function (pdf) of a streamwise velocity component is studied in zero-pressure gradient boundary layers. From analyzing the data up to R θ ≃ 13 , 000 , it is found that pdfs have self-similar profiles ranging from y + ≃ 180 to 0.049 Δ + , where Δ is Rotta–Clauser boundary layer thickness. Pdf profiles asymptote to the universal shape very close to the Gaussian, but are positively skewed at the core region, indicating smaller values in the tail parts. Based on this experimental fact, the mean velocity profile is reconsidered from the standpoint of pdf equation. The log-law profile is expected as the mean velocity distribution. The Kármán constant is evaluated to be 0.38, and the log-region starts at y s + = 180 and ends at y r + = 0.021 Δ + + 96.5 for R θ > 4000 . The end point locates in 0.15 δ + ⩽ y r + ⩽ 0.2 δ + . The relation to the new scaling law derived from Lie-group theory is also discussed.

Velocity scales in the near-wall layer beneath reattaching turbulent separated and boundary layer flows

This paper investigates the idea that each Reynolds stress has its own velocity scale – the mean shear stress cannot provide a velocity scale for the near-wall turbulence near reattachment, for example, and cannot by definition provide a velocity scale for ‘inactive’ motion. Beneath a separation bubble, the tangential velocity fluctuations scale on the r.m.s. of the respective wall shear stress fluctuation and on v / y , which is the viscous velocity scale in the usual way, independent of the other flow. This paper also shows that the wall-normal direct stress and the shear stress have respective, independent velocity scales. Moreover, and quite remarkably, the scaling functions for these latter stresses appear to be universal in that they are unchanged for the boundary layer upstream of separation or downstream of reattachment, and are as in the canonical zero-pressure gradient boundary layer. The streamwise direct stress does not exhibit this universality, raising questions about the near wall structures: it is inferred that the change in form is associated with the disappearance and reappearance of the streak-like structures.

Investigation into clustering of synthetic jet actuators for flow separation control applications

AbstractAn investigation into the behaviour of clustered synthetic jet Actuators for flow-control applications is described. Experiments have been undertaken with two small-scale synthetic jet actuators in a zero-pressure gradient boundary-layer, in order to investigate the effect of configuration yaw angle and relative input signal phase. Oil-flow visualisation and hotwire anemometry techniques were used, demonstrating that changes in the downstream flow structure could be observed. Compared to a streamwise configuration, in which a symmetrical counter-rotating vortex pair was produced by the synthetic jet-boundary-layer interaction, a broader asymmetric interaction was produced in a 15° yaw configuration. Streamwise velocity contour plots, illustrating the development of the interaction downstream, over four phase angles, were presented. Significant differences in the PSD analyses of downstream streamwise velocity time histories were found, suggesting that input signal phase could influence the stability and hence effectiveness of flow structures used in flow-control applications.

Similarity of instantaneous and filtered velocity fields in the near wall region of zero-pressure gradient boundary layer

Instantaneous velocity distributions in the viscous and roughness sublayers and in the lower parts of the logarithmic-law region of flat-plate boundary layers over smooth and rough surfaces are measured using a hot-wire rake consisting of two X-wire and nine single-sensor probes in order to investigate the turbulence structure and similarity of the instantaneous and filtered velocity profiles near the wall. The results can provide useful information as to the treatment of flow in this region in a large-eddy type simulation of high Reynolds-number flows over smooth and rough walls. The instantaneous velocity profiles feature large-scale structures but random fluctuations of considerable amplitude are superimposed and no similarity that is applicable to instantaneous unconditioned profiles can be derived. Conditionally averaged profiles based on the magnitudes of the wall shear and normal velocity in the buffer layer, however, show trends related to the sweep-ejection type events. The filtered profiles normalized by the filtered instantaneous friction velocity show similarity that progressively extends its region from the immediate neighborhood of the wall to the log layer as the filter size is increased. The minimum streamwise length of the filter size required to show logarithmic similarity is found to be as large as 1800 wall units. Similar results are obtained for rough-wall boundary layers if the instantaneous friction velocity is replaced by the equivalent velocity scale implied by the velocity just outside the roughness-influenced sublayer at about 10 viscous units from the wall, or about 1.5 times the height of the roughness bars from the wall. The sub-grid stress components have are obtained from the data and implication for modeling is discussed.

A Study of Hydrodynamic Characteristics of Boundary Layer With Algae Roughness

Filamentous algae fouling, such as Enteromorpha clathrata, is a soft and hairylike roughness that sometimes grows even thicker than a normal boundary layer. Typically, such fouling has been treated as traditional roughness functions to yield hydrodynamic characteristics. This technique has been successfully used for a thin fouling layer. However, it may not be applicable on a thicker layer, as the present study found substantial fluid flow within the layer. For such cases, the roughness cannot be treated simply as a passive geometric variable, but its kinematics and interactions with the flow must be considered. The inner law (log law) dynamics may be abnormal to yield any meaningful roughness function if it is calculated in the traditional way as the departure of a rough-wall log law profile over a smooth-wall log law profile. In the present research, velocity measurement of the E. clathrata roughness boundary layer using pitot-static tube and laser Doppler velocimeter (LDV) were compared. Large discrepancies in the velocity profiles within and in the vicinity of the roughness layer were observed between the two methods. The pitot-static tube data showed significantly high velocities (60% to 80% of the free stream) in the inner layer as compared to a smooth wall boundary layer. This local increase in velocity is believed to be the result of elastic transfer of free-stream energy to the near-wall motions by the E. clathrata filaments. Consequently, the usual assumption of the normal pressure gradient as a negligible second-order term for a normal zero-pressure gradient boundary layer may not be valid for the present kind of roughness. The LDV velocity measurements near and within the roughness layer have large uncertainties due to interference of the probe volume by the E. clathrata filaments. Above the roughness, the pitot-static tube and LDV profiles show relatively good agreement. It is concluded that for accurate prediction of the wall shear stress with E. clathrata-type of bio-fouling roughness, the Clauser velocity loss function should include a form drag factor instead of only the viscous drag factor.

Heat transfer in transitional and turbulent boundary layers with system rotation

The present paper reports on the influence of system rotation on the heat transfer characteristics of transitional and turbulent zero-pressure gradient boundary layers. A test plate is installed in a wind tunnel, which is rotatable around the axis parallel to the plate leading edge with constant speed of rotation. Local heat transfer coefficient during rotation is determined by employing a thermochromic liquid crystal. Effects of the Coriolis force and the centrifugal buoyancy force have been examined by comparing the heat transfer coefficient with different free-stream velocities, rotational speeds and wall temperatures. It has been revealed that the Coriolis force has significant effect on transitional heat transfer, while its effect on turbulent heat transfer is moderate. The centrifugal buoyancy exhibits additional effects if the thermal loading is high.

Vortex organization in the outer region of the turbulent boundary layer

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.

Use of k−ε−γ Model to Predict Intermittency in Turbulent Boundary-Layers

The intermittency profile in the turbulent flat-plate zero pressure-gradient boundary-layer and a thick axisymmetric boundary-layer has been computed using the Reynolds-averaged k−ε−γ model, where k denotes turbulent kinetic energy, ε its rate of dissipation, and γ intermittency. The Reynolds-averaged model is simpler compared to the conditional model used in the literature. The dissipation equation of the Reynolds-averaged model is modified to account for the effect of entrainment. It has been shown that the model correctly predicts the observed intermittency of the flows. [S0098-2202(00)02403-2]