Pressure Gradient Turbulent Boundary Layer Research Articles

Statistical analysis of low- and high-speed large-scale structures in the outer region of an adverse pressure gradient turbulent boundary layer

Two-point correlations and quadrant two-point correlations (termed and introduced by authors) have been employed in order to study low- and high-speed large-scale structures in the outer region of a turbulent boundary layer subjected to a strong adverse pressure gradient. They are computed with particle image velocimetry (PIV) data comprising of large sets of streamwise-spanwise instantaneous velocity fields at three wall-normal positions (0.2δ, 0.5δ, 0.8δ; δ is the boundary layer thickness) at three different streamwise locations in the adverse-pressure-gradient zone, from downstream of the strong suction peak up to detachment. Two-point correlations of the streamwise velocity fluctuation confirm the existence of alternating zones of high- and low-speed fluid in the middle (0.5δ) and lower (0.2δ) parts of the outer region similar to those observed in the log and wake regions of zero-pressure-gradient turbulent boundary layers. However, the negative-positive-negative pattern of these two-point correlations disappears in the lower part (0.2δ) of the large-velocity-defect zone, i.e. near detachment. In the latter region, the results support the existence of large-scale u-structures, many of which are streamwise elongated, that meander more or are less streamwise aligned than the u-structures elsewhere. The ratio of streamwise to spanwise length scales associated with the two-point correlations reveals that u-structures in the zero-pressure-gradient cases are more elongated in the streamwise direction than those in the adverse-pressure-gradient cases, especially in the lower part of the boundary layer. The quadrant two-point analysis allows us to discriminate events with respect to the sign of the streamwise velocity fluctuation at the reference and second points in the two-point correlation function. By using this technique, it is found that in the lower part of the outer region, i.e. 0.2δ (in the upper part of boundary layer, i.e. 0.8δ), large-scale high-speed events (low-speed events) are more intense than low-speed (high-speed) ones and also contribute more to the two-point correlations. In the middle of the boundary layer where the maximum turbulence activity occurs, large-scale low- and high-speed motions have comparable intensities and frequencies of occurrence. However, the low-speed events are slightly more intense and seem to be spatially more coherent.

A method to correct third and fourth order moments in turbulent flows

It is well known that spatial averaging, resulting from the finite size of a hot-wire probe, significantly affects the accuracy of such measurements in turbulent flows close to the wall. Here, a theoretical model which describes the effect of the spatial filtering of hot-wire probes on the third and fourth order moments of the streamwise velocity is presented. The model, which is based on the three (four) point velocity correlation function for the third (fourth) order moment, shows that the filtering can be related to a characteristic length scale which is an equivalent of the Taylor transverse micro-scale for the second order moment. The capacity of the model to accurately describe the attenuation is validated against direct numerical simulation (DNS) data of a zero pressure-gradient turbulent boundary layer. The DNS data allow the filtering effect to be appraised for different wire lengths and for the different moments. A procedure, based on the developed model, to correct the measured moments in turbulent flows is finally presented. The method is applied by combining the response of two single hot-wire sensors with different wire lengths. The technique has also been validated against spatially averaged DNS data showing a good capacity to reconstruct the actual profiles over the entire height of the boundary layer except, for the third order moment, in the region where the latter is close to zero.

A new formulation for the streamwise turbulence intensity distribution

Numerical and experimental data from zero pressure-gradient turbulent boundary layers over smooth walls have been analyzed by means of the so called diagnostic plot introduced by Alfredsson & Örlü [Eur. J. Fluid Mech. B/Fluids, 42, 403 (2010)]. In the diagnostic plot the local turbulence intensity is shown as a function of the local mean velocity normalized with a reference velocity scale. In the outer region of the boundary layer a universal linear decay of the turbulence intensity is observed independent of Reynolds number. The deviation from this linear region appears in the buffer region and seems to be universal when normalized with the friction velocity. Therefore, a new empirical fit for the streamwise velocity turbulence intensity distribution is proposed and the results are compared with up to date reliable high-Reynolds number experiments and extrapolated towards Reynolds numbers relevant to atmospherical boundary layers.

Inflow length and tripping effects in turbulent boundary layers

A recent assessment of available direct numerical simulation (DNS) data from turbulent boundary layer flows [Schlatter & Örlü, J. Fluid Mech. 659, 116 (2010)] showed surprisingly large differences not only in the skin friction coefficient or shape factor, but also in their predictions of mean and fluctuation profiles far into the sublayer. For the present paper the DNS of a zero pressure-gradient turbulent boundary layer flow by Schlatter et al. [Phys. Fluids 21, 051702 (2009)] serving as the baseline simulation, was re-simulated, however with physically different inflow conditions and tripping effects. The downstream evolution of integral and global quantities as well as mean and fluctuation profiles are presented and results indicate that different inflow conditions and tripping effects explain most of the differences observed when comparing available DNS. It is also found, that if transition is initiated at a low enough Reynolds number (based on the momentum-loss thickness) Reθ < 300, all data agree well for both inner and outer layer for Reθ > 2000; a result that gives a lower limit for meaningful comparisons between numerical and/or wind tunnel experiments.

Numerical prediction of turbulent flow structure generated by a synthetic cross‐jet into a turbulent boundary layer

SUMMARYA detailed numerical study using large‐eddy simulation (LES) and unsteady Reynolds‐averaged Navier–Stokes (URANS) was undertaken to investigate physical processes that are engendered in the injection of a circular synthetic (zero‐net mass flux) jet in a zero pressure gradient turbulent boundary layer. A complementary study was carried out and was verified by comparisons with the available experimental data that were obtained at corresponding conditions with the aim of achieving an improved understanding of fluid dynamics of the studied processes. The computations were conducted by OpenFOAM C++, and the physical realism of the incoming turbulent boundary layer was secured by employing random field generation algorithm. The cavity was computed with a sinusoidal transpiration boundary condition on its floor. The results from URANS computation and LES were compared and described qualitatively and quantitatively. There is a particular interest for acquiring the turbulent structures from the present numerical data. The numerical methods can capture vortical structures including a hairpin (primary) vortex and secondary structures. However, the present computations confirmed that URANS and LES are capable of predicting current flow field with a more detailed structure presented by LES data as expected. Copyright © 2011 John Wiley &amp; Sons, Ltd.

Aerodynamic Roughness Length of Fresh Snow

This study presents the results from a series of wind-tunnel experiments designed to investigate the aerodynamic roughness length z 0 of fresh snow under no-drift conditions. A two-component hot-film anemometer was employed to obtain vertical profiles of velocity statistics in a zero pressure gradient turbulent boundary layer for flow over naturally deposited snow surfaces. The roughness of these snow surfaces was measured by means of digital photography to capture characteristic length scales that can be related to z 0. Our results show that, under aerodynamically rough conditions, the mean value of the roughness length for fresh snow is $${\langle{z}_{0}\rangle= 0.24}$$ mm with a standard deviation σ(z 0) = 0.05 mm. In this study, we show that variations in z 0 are associated with variations in the roughness geometry. The roughness measurements suggest that the estimated values of z 0 are consistent with the presence of irregular roughness structures that develop during snowfalls that mimic ballistic deposition processes.

A study of the three-dimensional spectral energy distribution in a zero pressure gradient turbulent boundary layer

Time-resolved particle image velocimetry (PIV) measurements performed in wall parallel planes at three wall normal locations, y^+ = 34, 108, and 278, in a zero pressure gradient turbulent boundary layer at Re_τ = 470 are used to illuminate the distribution of streamwise velocity fluctuations in a three-dimensional energy spectrum (2D in space and 1D in time) over streamwise, spanwise, and temporal wavelengths. Two high-speed cameras placed side by side in the streamwise direction give a 10δ × 5δ streamwise by spanwise field of view with a vector spacing of Δx^+ = Δz^+ ≈ 37 and a time step of Δt^+ = 0.5. Although 3D wavenumber-frequency spectra have been calculated in acoustics studies, to the authors’ knowledge this is the first time they has been calculated and presented for a turbulent boundary layer. The calculation and normalization of this spectrum, its relation to 2D and 1D spectra, and the effects of the PIV algorithm on its shape are carefully analyzed and outlined.

A parametric study of adverse pressure gradient turbulent boundary layers

There are many open questions regarding the behaviour of turbulent boundary layers subjected to pressure gradients and this is confounded by the large parameter space that may affect these flows. While there have been many valuable investigations conducted within this parameter space, there are still insufficient data to attempt to reduce this parameter space. Here, we consider a parametric study of adverse pressure gradient turbulent boundary layers where we restrict our attention to the pressure gradient parameter, β, the Reynolds number and the acceleration parameter, K. The statistics analyzed are limited to the streamwise fluctuating velocity. The data show that the mean velocity profile in strong pressure gradient boundary layers does not conform to the classical logarithmic law. Moreover, there appears to be no measurable logarithmic region in these cases. It is also found that the large-scale motions scaling with outer variables are energised by the pressure gradient. These increasingly strong large-scale motions are found to be the dominant contributor to the increase in turbulence intensity (scaled with friction velocity) with increasing pressure gradient across the boundary layer.

A method to estimate turbulence intensity and transverse Taylor microscale in turbulent flows from spatially averaged hot-wire data

A new approach to evaluate turbulence intensity and transverse Taylor microscale in turbulent flows is presented. The method is based on a correction scheme that compensates for probe resolution effects and is applied by combining the response of two single hot-wire sensors with different wire lengths. Even though the technique, when compared to other correction schemes, requires two independent measurements, it provides, for the same data, an estimate of the spanwise Taylor microscale. The method is here applied to streamwise turbulence intensity distributions of turbulent boundary layer flows but it is applicable generally in any turbulent flow. The technique has been firstly validated against spatially averaged DNS data of a zero pressure-gradient turbulent boundary layer showing a good capacity to reconstruct the actual profiles and to predict a qualitatively correct and quantitatively agreeing transverse Taylor microscale over the entire height of the boundary layer. Finally, the proposed method has been applied to available higher Reynolds number data from recent boundary layer experiments where an estimation of the turbulence intensity and of the Taylor microscale has been performed.

Surface pressure and flow field behind an oscillating fence submerged in turbulent boundary layer

Phase-locked PSP and PIV measurements were used to study the evolution of three-dimensional disturbances produced by an oscillating fence actuator immersed in a zero pressure gradient turbulent boundary layer. For the single fence frequency studied, strong three-dimensionality is observed in the vortical structure that varies along the span of the fence soon after the fence enters the flow. At the midspan, the structure grows, weakens, and convects faster than at other locations. As the fence height increases, the data indicate that the vortical structure terminates near the edge of the fence. In contrast, the vortex structure terminates on the plate surface adjacent to the fence edge as the fence descends, similar to a wake vortex of a stationary obstacle. This study demonstrates that the combined use of surface and flow-field diagnostics provide a link between flow field and surface features, yielding an understanding of the flow that would have not been possible with any one technique.

A theoretical study on the overlap region in a flat plate turbulent boundary layer

This paper studies the wall-law defect-law overlap region in a zero pressure gradient turbulent boundary layer. By taking account of the non-parallel flow effect, a system of differential equations is derived, which describes the velocity profile for the overlap layer and the velocity and length scales for the outer layer. Two kinds of solutions are obtained for the velocity profile in combination with corresponding solutions for the outer layer scales. One is of log-law type and the other is of power-law type. These are possible solutions that represent the necessary condition for the existence of the overlap layer. In the light of the present solutions, recent experimental data for a zero pressure gradient turbulent boundary layer are examined. It is found that the observed velocity data follow the log-law solution for momentum thickness Reynolds numbers Rθ above 20 000.

The rough favourable pressure gradient turbulent boundary layer

Laser Doppler anemometry measurements of the mean velocity and Reynolds stresses are carried out for a rough-surface favourable pressure gradient turbulent boundary layer. The experimental data is compared with smooth favourable pressure gradient and rough zero-pressure gradient data. The velocity and Reynolds stress profiles are normalized using various scalings such as the friction velocity and free stream velocity. In the velocity profiles, the effects of roughness are removed when using the friction velocity. The effects of pressure gradient are not absorbed. When using the free stream velocity, the scaling is more effective absorbing the pressure gradient effects. However, the effects of roughness are almost removed, while the effects of pressure gradient are still observed on the outer flow, when the mean deficit velocity profiles are normalized by the U∞ δ∗/δ scaling. Furthermore, when scaled with U2∞, the 〈u2〉 component of the Reynolds stress augments due to the rough surface despite the imposed favourable pressure gradient; when using the friction velocity scaling u∗2, it is dampened. It becomes ‘flatter’ in the inner region mainly due to the rough surface, which destroys the coherent structures of the flow and promotes isotropy. Similarly, the pressure gradient imposed on the flow decreases the magnitude of the Reynolds stress profiles especially on the 〈v2〉 and -〈uv〉 components for the u∗2 or U∞2 scaling. These effects are reflected in the boundary layer parameter δ∗/δ, which increase due to roughness, but decrease due to the favourable pressure gradient. Additionally, the pressure parameter Λ found not to be in equilibrium, describes the development of the turbulent boundary layer, with no influence of the roughness linked to this parameter. These measurements are the first with an extensive number of downstream locations (11). This makes it possible to compute the required x-dependence for the production term and the wall shear stress from the full integrated boundary layer equation. The finding indicates that the skin friction coefficient depends on the favourable pressure gradient condition and surface roughness.

Determination of skin friction in strong pressure-gradient equilibrium and near-equilibrium turbulent boundary layers

The conventional Clauser-chart method for determination of local skin friction in zero or weak pressure-gradient turbulent boundary layer flows fails entirely in strong pressure-gradient situations. This failure occurs due to the large departure of the mean velocity profile from the universal logarithmic law upon which the conventional Clauser-chart method is based. It is possible to extend this method, even for strong pressure-gradient situations involving equilibrium or near-equilibrium turbulent boundary layers by making use of the so-called non-universal logarithmic laws. These non-universal log laws depend on the local strength of the pressure gradient and may be regarded as perturbations of the universal log law. The present paper shows that the modified Clauser-chart method, so developed, yields quite satisfactory results in terms of estimation of local skin friction in strongly accelerated or retarded equilibrium and near-equilibrium turbulent boundary layers that are not very close to relaminarization or separation.

Experimental analysis of a turbulent boundary layer at high Reynolds numbers

In this work, time-resolved two-dimensional particle image velocimetry measurements in a zero pressure gradient turbulent boundary layer at high Reynolds numbers (Re θ = 3940 and 7257, based on momentum thickness) are presented. The boundary layer is obtained on the flat wall of a very large recirculating water channel and is investigated on a streamwise–wall-normal plane. Statistical moments of velocity are determined with the aim to investigate high-Reynolds-number effects in profiles obtained in a direction orthogonal to the wall. A careful analysis of such a profile for the measured average velocity indicates departures from the classical logarithmic law towards the power law or parametric models. Such a departure seems to be independent on the specific parameters used in model laws. On the other hand, instantaneous velocity fields are considered to derive information on the dynamic processes involving near-wall vortical structures. Different vortex eduction methods are implemented including Reynolds decomposition, vorticity, invariants of velocity gradient tensor and wavelet tools and compared. The agreement between these methods is rather good, so all of them are used to derive information on wall dynamics. At both Reynolds numbers, packets of vortical structures are observed to form and to evolve aligned more or less at an angle of 30° to the wall. In order to quantify the importance of such ‘coherent’ phenomena over the entire range of turbulent structure, probability density functions of vortical structure size (obtained from wavelet analysis) are determined. The results show the existence of a hierarchical relation between coherent vortices, the average vortex size being almost equal to 1/10 of the boundary layer displacement thickness. The measured data are consistent with the formation of about seven large ‘children’ vortices from each ‘parent’ vortex.

Theoretical evaluation of the Reynolds shear stress and flow parameters in transitionally rough turbulent boundary Layers

The theory by W.K. George and L. Castillo (Zero-pressure gradient turbulent boundary layers, Appl. Mech. Rev. 50 (1997), pp. 689–729) is extended for rough surfaces and numerically implemented on zero pressure gradient turbulent boundary layers. The method is based on the similarity transformations of the Navier–Stokes equations. From these equations, a composite profile for the Reynolds shear stress-⟨ uv⟩ is obtained over the entire boundary layer. The solution is in good agreement with the experiments in the inner and outer regions, for hydraulically smooth (k+ < 5) and transitionally rough surfaces up to roughness parameter of k+ ≈ 55. Beyond this limit, the accuracy of the solutions decreases with k+, especially in the inner region of the mean velocity. However, the accuracy always increases with the Reynolds number, Re_θ. In addition, the eddy viscosity ⟨ ν_T⟩/ν and flow parameters, including the x-dependence, have also been computed and tested with experimental data. Furthermore, the friction power law for smooth/rough surfaces has been used for all calculations and comparisons with direct methods and velocity-based methods are shown to be in good agreement with the theory.

Mean momentum balance in moderately favourable pressure gradient turbulent boundary layers

Moderately favourable pressure gradient turbulent boundary layers are investigated within a theoretical framework based on the unintegrated two-dimensional mean momentum equation. The present theory stems from an observed exchange of balance between terms in the mean momentum equation across different regions of the boundary layer. This exchange of balance leads to the identification of distinct physical layers, unambiguously defined by the predominant mean dynamics active in each layer. Scaling domains congruent with the physical layers are obtained from a multi-scale analysis of the mean momentum equation. Scaling behaviours predicted by the present theory are evaluated using direct measurements of all of the terms in the mean momentum balance for the case of a sink-flow pressure gradient generated in a wind tunnel with a long development length. Measurements also captured the evolution of the turbulent boundary layers from a non-equilibrium state near the wind tunnel entrance towards an equilibrium state further downstream. Salient features of the present multi-scale theory were reproduced in all the experimental data. Under equilibrium conditions, a universal function was found to describe the decay of the Reynolds stress profile in the outer region of the boundary layer. Non-equilibrium effects appeared to be manifest primarily in the outer region, whereas differences in the inner region were attributed solely to Reynolds number effects.

Comparison of mean flow similarity laws in zero pressure gradient turbulent boundary layers

The paper presents an in depth assessment of different similarity laws for the mean velocity profile in zero pressure gradient (ZPG) turbulent boundary layers (TBL's) in comparison with mostly experimental and few computational data. The emphasis is on the descriptions which are complete in the sense that a full representation of the mean velocity profile, its streamwise evolution and all integral parameters, including the friction factor and the shape factor, are provided as a function of Reynolds number. The first such complete description is the classical two-layer theory with its characteristic logarithmic mean velocity profile in the region where the two layers overlap, henceforth referred to as the “log law.” The main alternative scalings which have been proposed over the last decade have led to power law descriptions of the turbulent mean velocity profile. Since the different descriptions were calibrated with different data sets, the controversy over the relative merits of the different approaches has lingered on. The purpose of the present paper is to measure the principal competing theories against the same vast data set of more than 300 mean velocity profiles from more than twenty different sources. The results confirm the conclusions of numerous authors that the log law provides a fully self-consistent and accurate description of all the mean quantities and demonstrates conclusively that the same cannot be achieved by the competing power law theories. Along the way, it is also argued that the traditional description of the outer velocity profile in terms of a wall-normal coordinate normalized to unity at a hypothetical boundary layer “edge” δ and a “wake parameter” Π is not robust with respect to the fit of the outer velicity profile and should therefore not be used in theoretical arguments.

Similarity analysis of favorable pressure gradient turbulent boundary layers with eventual quasilaminarization

Using similarity analysis, the scales and similarity constraints for a favorable pressure gradient (FPG) turbulent boundary layer with eventual quasilaminarization are obtained. In order to achieve equilibrium in the boundary layer, the pressure parameter Λ must be a constant; thus, a power relation between the boundary layer thickness δ and the free-stream velocity U∞ exists. Consequently, the power is given by the pressure parameter Λ as δ∼U∞−1/Λ. Through an analysis using the pressure parameter, two quadrants are found: quadrant I describes FPG turbulent flows and quadrant II corresponds to quasilaminar flows. Moreover, a horizontal line exists for zero pressure gradient flows. Different values of the pressure parameter are found for equilibrium FPG flows, contrary to the findings of Castillo and George [“Similarity analysis for turbulent boundary layer with pressure gradient: Outer flow,” AIAA J. 39, 41 (2001)]. In the case of strong FPG flows with quasilaminarization, the pressure parameter reaches a maximum value of 0.47. At this point, a sudden reduction in the skin friction of about 57% is observed and a redistribution of the Reynolds stresses throughout the boundary layer is achieved. The mean velocity deficit profiles are also found to be attenuated when scaled using the free-stream velocity U∞ or U∞δ∗/δ. For flows in quadrant II, a reduction in the outer flow of the ⟨u2⟩ component of the Reynolds stress is observed, whereas the ⟨v2⟩ and −⟨uv⟩ components nearly vanish impending quasilaminarization. Due to the presence of the ⟨u2⟩ component in the boundary layer, the flow never reaches a full laminar state and a more uniform redistribution of the component is observed as the skin friction decreases due to the imposed FPG. Furthermore, the shape of the profile remains the same until a quasilaminar state is reached, where the profile no longer shows high values of the stress on the proximity to the wall. In addition, the production term −⟨uv⟩dU/dy is nearly zero for flows in quadrant II. Also, the boundary layer parameters such as the shape factor H and the ratio of the displacement thickness to the boundary layer thickness, δ∗/δ, increase as the flow achieves a quasilaminar state.

Variations of von Kármán coefficient in canonical flows

The overlap parameters for the logarithmic law are obtained for available turbulent pipe and channel flow data using composite profiles fitted to the mean velocity. The composite profile incorporates κ, B, and Π as the varying parameters and their resulting behavior with Reynolds number is examined for these flows and compared to results from boundary layers. The von Kármán coefficient in channel flow is smaller than the well-established value for zero pressure gradient turbulent boundary layers of 0.384, while in pipe flows it is consistently higher. In contrast, the estimates of the wake parameter Π are the smallest for channel flows and largest for boundary layers. Further, the Superpipe data are reanalyzed to reveal that κ=0.41 is a better value for the von Kármán constant in pipe flow. The collective behavior of κ in boundary layers, pipes, and channels reveals that the von Kármán coefficient is not universal and exhibits dependence not only on the pressure gradient but also on the flow geometry.

The effects of pulse frequency and duty cycle on the skin friction downstream of pulsed jet vortex generators in an adverse pressure gradient turbulent boundary layer

An investigation on the viability of pulsed jets as active vortex generator devices was conducted. The devices were installed and tested on an adverse pressure gradient turbulent boundary layer designed to simulate the suction side of a conventional aircraft wing. Both co-rotating and counter-rotating jet geometries were used. The duty cycle and frequency of pulsation were varied and their effects were investigated by measuring the skin friction gains at a predefined location (the location of the minimum skin friction for the un-actuated situation) on the adverse pressure gradient turbulent boundary layer. Pulsing the jets proved to be successful in increasing the wall skin friction and therefore potentially delaying separation. The improvements in wall shear stress were approximately proportional to the duty cycle. The frequency of jet pulsation was found to be important for attaining optimal gains, however no clear relationship between frequency and shear stress gain was observed. Phase averaged wall shear stress measurements far downstream of actuation indicate that quasi-steady structures are introduced by the vortex generators when actuating with a sufficiently high pulse frequency. In this situation interactions between successive structures produced by the jets were likely to be occurring.