Clauser Pressure-gradient Parameter Research Articles

An adverse-pressure-gradient turbulent boundary layer with nearly constant up to

In this study, a new well-resolved large-eddy simulation of an incompressible near-equilibrium adverse-pressure-gradient (APG) turbulent boundary layer (TBL) over a flat plate is presented. In this simulation, we have established a near-equilibrium APG over a wide Reynolds-number range. In this so-called region of interest, the Rotta–Clauser pressure-gradient parameter $\beta$ exhibits an approximately constant value of around 1.4, and the Reynolds number based on momentum thickness reaches ${\textit {Re}}_{\theta }=8700$ . To the best of the authors’ knowledge, this is to date the highest ${\textit {Re}}_{\theta }$ achieved for a near-equilibrium APG TBL under an approximately constant moderate APG. We evaluated the self-similarity of the outer region using two scalings, namely the Zagarola–Smits and an alternative scaling based on edge velocity and displacement thickness. Our results reveal that outer-layer similarity is achieved, and the viscous scaling collapses the near-wall region of the mean flow in agreement with classical theory. Spectral analysis reveals that the APG displaces some small-scale energy from the near-wall to the outer region, an effect observed for all the components of the Reynolds-stress tensor, which becomes more evident at higher Reynolds numbers. In general, the effects of the APG are more noticeable at lower Reynolds numbers. For instance, the outer peak of turbulent-kinetic-energy (TKE) production is less prominent at higher $Re$ . Although the scale separation increases with ${\textit {Re}}$ in zero-pressure-gradient TBLs, this effect becomes accentuated by the APG. Despite the reduction of the outer TKE production at higher Reynolds numbers, the mechanisms of energisation of large scales are still present.

Pressure Gradient Effects on the Performance of Riblets Installed on Aircraft Surfaces

A method for estimating the drag reduction performance of riblets installed on the surfaces of an aircraft was extended to take the effect of the pressure gradient over the aircraft into account. This method simply and roughly estimates the effectiveness of riblets applied to arbitrary surfaces of any aircraft by fusing wind tunnel test data and computational fluid dynamics analysis. The drag reduction rate is calculated from the relationships between friction drag reduction , wall-normalized riblet spacing , and Clauser pressure gradient parameter . To obtain this relationship, a wind tunnel test with straight riblets in an adverse pressure gradient was carried out. The resulting shift in the value in the intercept of the log-law region of the mean velocity profile for riblet cases indicates that the drag reduction effect decreases as the Clauser parameter increases. Based on these results, an approximate surface for the relationship between , , and was proposed, and the drag reduction performance of riblets installed on the surfaces of a conceptual aircraft was estimated. The total drag reduction rates estimated by the extended method considering pressure gradient effects and the original method case were both about 2%, which indicates that the effects of the pressure gradient on the aircraft on overall riblet performance are small.

The turbulent/non-turbulent interface in an adverse pressure gradient turbulent boundary layer

The turbulent/non-turbulent interface (TNTI) in an adverse pressure gradient (APG, β = 1.45) turbulent boundary layer (TBL) is explored here by using direct numerical simulation (DNS) data; β is the Clauser pressure gradient parameter. For comparison, the DNS data for a zero pressure gradient (ZPG) TBL is included. The interface is extracted with an approach based on enstrophy criteria. Depending on the enstrophy, the outer boundary layer flow can be classified into the free stream, boundary layer wake, and intermittent flow regimes. The fractal dimension of the interface is obtained by using the box-counting algorithm, and was found to be constant over a long range of box sizes. The TNTI shows a monofractal behavior. The geometric complexity of a TNTI can be determined in terms of the genus, which is defined as the number of handles in a geometric object. We examine the volume and projection area of the genus of the TNTI to analyze the entrainment process. The geometric complexity of the APG TBL interface and the local entrainment are greater than those of the ZPG TBL, as is evident in the increases in the genus near the interface. The local entrainment velocity is dominantly affected by the viscous diffusion at the interface.

Effect of adverse pressure gradients on turbulent wing boundary layers

The characteristics of turbulent boundary layers (TBLs) subjected to adverse pressure gradients are analysed through well-resolved large-eddy simulations. The geometries under study are the NACA0012 and NACA4412 wing sections, at and angle of attack, respectively, both of them at a Reynolds number based on inflow velocity and chord length of . The turbulence statistics show that adverse pressure gradients (APGs) have a significant effect on the mean velocity, velocity fluctuations and turbulent kinetic energy budget, and this effect is more prominent on the outer region of the boundary layer. Furthermore, the effect of flow history is assessed by means of an integrated Clauser pressure-gradient parameter (Vinuesa et al., Flow Turbul. Combust., vol. 99, 2017, pp. 565–587), through the study of cases with matching local values of and the friction Reynolds number to isolate this effect. Our results show a noticeable effect of the flow history on the outer region, however the differences in the near-wall peak of the tangential velocity fluctuations appear to be mostly produced by the local APG magnitude. The one-dimensional power-spectral density shows energetic small scales in the outer region of APG TBLs, whereas these energetic scales do not appear in zero-pressure-gradient (ZPG) TBLs, suggesting that small scales near the wall are advected towards the outer layer by the APG. Moreover, the linear coherence spectra show that the spectral outer peak of high-Reynolds-number ZPG TBLs is highly correlated with the near-wall region (Baars et al., J. Fluid Mech., vol. 823, 2017, R2), unlike APG TBLs which do not show such a correlation. This result, together with the different two-dimensional spectra of APG and high-Reynolds-number ZPG TBLs, suggests different energisation mechanisms due to APG and increase in Reynolds number. To the authors’ knowledge, this is the first in-depth analysis of the TBL characteristics over wings, including detailed single-point statistics, spectra and coherence.

Flow organization in the wake of a rib in a turbulent boundary layer with pressure gradient

The effect of a streamwise pressure gradient on the wake developed by wall-attached square ribs in a turbulent boundary layer is investigated experimentally. Favourable-, adverse- and zero-pressure-gradient conditions (FPG, APG and ZPG, respectively) are reproduced at matched friction Reynolds number and non-dimensional rib height. Flow-field measurements are carried out by means of Particle Image Velocimetry (PIV). Turbulence statistics are extracted at high resolution using an Ensemble Particle Tracking Velocimetry approach. Modal analysis is performed with Proper Orthogonal Decomposition (POD). We demonstrate that a non-dimensional expression of the pressure gradient and shear stress is needed to quantify the pressure-gradient effects in the wake developing past wall-attached ribs. We suggest the Clauser pressure-gradient parameter β, commonly used in the literature for the characterization of turbulent boundary layers under the effect of a pressure gradient, as a suitable parameter. The results show that, in presence of an adverse pressure gradient, the recirculation region downstream of the rib is increased in size, thus delaying the reattachment, and that the peak of turbulence intensity and the shed eddies are shifted towards larger wall-normal distances than in the ZPG case. The observed changes with respect to the ZPG configuration appear more intense for larger magnitude of β, which are more likely to be obtained in APG than in FPG due to the reduced skin friction and increased displacement thickness.

Turbulent boundary layers around wing sections up to [formula omitted

Reynolds-number effects in the adverse-pressure-gradient (APG) turbulent boundary layer (TBL) developing on the suction side of a NACA4412 wing section are assessed in the present work. To this end, we analyze four cases at Reynolds numbers based on freestream velocity and chord length ranging from Rec=100,000 to 1,000,000, all of them with 5° angle of attack. The results of four well-resolved large-eddy simulations (LESs) are used to characterize the effect of Reynolds number on APG TBLs subjected to approximately the same pressure-gradient distribution (defined by the Clauser pressure-gradient parameter β). Comparisons of the wing profiles with zero-pressure-gradient (ZPG) data at matched friction Reynolds numbers reveal that, for approximately the same β distribution, the lower-Reynolds-number boundary layers are more sensitive to pressure-gradient effects. This is reflected in the values of the inner-scaled edge velocity Ue+, the shape factor H, the components of the Reynolds-stress tensor in the outer region and the outer-region production of turbulent kinetic energy. This conclusion is supported by the larger wall-normal velocities and outer-scaled fluctuations observed in the lower-Rec cases. Thus, our results suggest that two complementing mechanisms contribute to the development of the outer region in TBLs and the formation of large-scale energetic structures: one mechanism associated with the increase in Reynolds number, and another one connected to the APG. Future extensions of the present work will be aimed at studying the differences in the outer-region energizing mechanisms due to APGs and increasing Reynolds number.

Contribution of large-scale motions to the skin friction in a moderate adverse pressure gradient turbulent boundary layer

Direct numerical simulation of a turbulent boundary layer (TBL) subjected to a moderate adverse pressure gradient (APG,$\unicode[STIX]{x1D6FD}=1.45$) is performed to explore the contribution of large scales to the skin friction, where$\unicode[STIX]{x1D6FD}$is the Clauser pressure gradient parameter. The Reynolds number based on the momentum thickness develops from$Re_{\unicode[STIX]{x1D703}}\approx 110$to 6000 with an equilibrium region in$Re_{\unicode[STIX]{x1D703}}=4000$–5500. The spanwise wavelength ($\unicode[STIX]{x1D706}_{z}$) spectra of the streamwise and spanwise velocity fluctuations show that the large-scale energy is significantly enhanced throughout the boundary layer. We quantify the superposition and amplitude modulation effects of these enhanced large scales on the skin friction coefficient ($C_{f}$) by employing two approaches: (i) spanwise co-spectra of$\langle v\unicode[STIX]{x1D714}_{z}\rangle$and$\langle -w\unicode[STIX]{x1D714}_{y}\rangle$; (ii) conditionally averaged$\langle v\unicode[STIX]{x1D714}_{z}\rangle$and$\langle -w\unicode[STIX]{x1D714}_{y}\rangle$. The velocity–vorticity correlations$\langle v\unicode[STIX]{x1D714}_{z}\rangle$and$\langle -w\unicode[STIX]{x1D714}_{y}\rangle$are related to the advective transport and the vortex stretching, respectively. Although$\langle v\unicode[STIX]{x1D714}_{z}\rangle$negatively contributes to$C_{f}$, the positive contribution of the large scales ($\unicode[STIX]{x1D706}_{z}>0.5\unicode[STIX]{x1D6FF}$) is observed in the co-spectra of weighted$\langle v\unicode[STIX]{x1D714}_{z}\rangle$. For the co-spectra of weighted$\langle -w\unicode[STIX]{x1D714}_{y}\rangle$, we observe an outer peak at$\unicode[STIX]{x1D706}_{z}\approx 0.75\unicode[STIX]{x1D6FF}$and the superposition of the large scales in the buffer region, leading to the enhancement of$C_{f}$. The magnitude of$\langle v\unicode[STIX]{x1D714}_{z}\rangle$and$\langle -w\unicode[STIX]{x1D714}_{y}\rangle$depends on the large-scale streamwise velocity fluctuations ($u_{L}$). In particular, the negative-$u_{L}$events amplify$\langle v\unicode[STIX]{x1D714}_{z}\rangle$in the outer region, and$\langle -w\unicode[STIX]{x1D714}_{y}\rangle$is enhanced by the positive-$u_{L}$events. As a result, the skin friction induced by$\langle v\unicode[STIX]{x1D714}_{z}\rangle$and$\langle -w\unicode[STIX]{x1D714}_{y}\rangle$increases in the present APG TBL.

Adverse-Pressure-Gradient Effects on Turbulent Boundary Layers: Statistics and Flow-Field Organization

This manuscripts presents a study on adverse-pressure-gradient turbulent boundary layers under different Reynolds-number and pressure-gradient conditions. In this work we performed Particle Image Velocimetry (PIV) measurements supplemented with Large-Eddy Simulations in order to have a dataset covering a range of displacement-thickness-based Reynolds-number 2300 <Re_{delta ^{*}}< 34000 and values of the Clauser pressure-gradient parameter β up to 2.4. The spatial resolution limits of PIV for the estimation of turbulence statistics have been overcome via ensemble-based approaches. A comparison between ensemble-correlation and ensemble Particle Tracking Velocimetry was carried out to assess the uncertainty of the two methods. The effects of β, Re and of the pressure-gradient history on turbulence statistics were assessed. A modal analysis via Proper Orthogonal Decomposition was carried out on the flow fields and showed that about 20% of the energy contribution corresponds to the first mode, while 40% of the turbulent kinetic energy corresponds to the first four modes with no appreciable dependence on β and Re within the investigated range. The topology of the spatial modes shows a dependence on the Reynolds number and on the pressure-gradient strength, in line with the results obtained from the analysis of the turbulence statistics. The contribution of the modes to the Reynolds stresses and the turbulence production was assessed using a truncated low-order reconstruction with progressively larger number of modes. It is shown that the outer peaks in the Reynolds-stress profiles are mostly due to large-scale structures in the outer part of the boundary layer.

Revisiting History Effects in Adverse-Pressure-Gradient Turbulent Boundary Layers

The goal of this study is to present a first step towards establishing criteria aimed at assessing whether a particular adverse-pressure-gradient (APG) turbulent boundary layer (TBL) can be considered well-behaved, i.e., whether it is independent of the inflow conditions and is exempt of numerical or experimental artifacts. To this end, we analyzed several high-quality datasets, including in-house numerical databases of APG TBLs developing over flat-plates and the suction side of a wing section, and five studies available in the literature. Due to the impact of the flow history on the particular state of the boundary layer, we developed three criteria of convergence to well-behaved conditions, to be used depending on the particular case under study. (i) In the first criterion, we develop empirical correlations defining the Re\U0001d703-evolution of the skin-friction coefficient and the shape factor in APG TBLs with constant values of the Clauser pressure-gradient parameter β = 1 and 2 (note that β = δ∗/τwdPe/dx, where δ∗ is the displacement thickness, τw the wall-shear stress and dPe/dx the streamwise pressure gradient). (ii) In the second one, we propose a predictive method to obtain the skin-friction curve corresponding to an APG TBL subjected to any streamwise evolution of β, based only on data from zero-pressure-gradient TBLs. (iii) The third method relies on the diagnostic-plot concept modified with the shape factor, which scales APG TBLs subjected to a wide range of pressure-gradient conditions. These three criteria allow to ensure the correct flow development of a particular TBL, and thus to separate history and pressure-gradient effects in the analysis.

Pressure-Gradient Turbulent Boundary Layers Developing Around a Wing Section

A direct numerical simulation database of the flow around a NACA4412 wing section at Rec = 400,000 and 5∘ angle of attack (Hosseini et al. Int. J. Heat Fluid Flow 61, 117–128, 2016), obtained with the spectral-element code Nek5000, is analyzed. The Clauser pressure-gradient parameter β ranges from ≃ 0 and 85 on the suction side, and from 0 to − 0.25 on the pressure side of the wing. The maximum Re\U0001d703 and Reτ values are around 2,800 and 373 on the suction side, respectively, whereas on the pressure side these values are 818 and 346. Comparisons between the suction side with zero-pressure-gradient turbulent boundary layer data show larger values of the shape factor and a lower skin friction, both connected with the fact that the adverse pressure gradient present on the suction side of the wing increases the wall-normal convection. The adverse-pressure-gradient boundary layer also exhibits a more prominent wake region, the development of an outer peak in the Reynolds-stress tensor components, and increased production and dissipation across the boundary layer. All these effects are connected with the fact that the large-scale motions of the flow become relatively more intense due to the adverse pressure gradient, as apparent from spanwise premultiplied power-spectral density maps. The emergence of an outer spectral peak is observed at β values of around 4 for λz ≃ 0.65δ99, closer to the wall than the spectral outer peak observed in zero-pressure-gradient turbulent boundary layers at higher Re\U0001d703. The effect of the slight favorable pressure gradient present on the pressure side of the wing is opposite the one of the adverse pressure gradient, leading to less energetic outer-layer structures.

History effects and near equilibrium in adverse-pressure-gradient turbulent boundary layers

Turbulent boundary layers under adverse pressure gradients are studied using well-resolved large-eddy simulations (LES) with the goal of assessing the influence of the streamwise pressure-gradient development. Near-equilibrium boundary layers were characterized through the Clauser pressure-gradient parameter $\unicode[STIX]{x1D6FD}$. In order to fulfil the near-equilibrium conditions, the free stream velocity was prescribed such that it followed a power-law distribution. The turbulence statistics pertaining to cases with a constant value of $\unicode[STIX]{x1D6FD}$ (extending up to approximately 40 boundary-layer thicknesses) were compared with cases with non-constant $\unicode[STIX]{x1D6FD}$ distributions at matched values of $\unicode[STIX]{x1D6FD}$ and friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}$. An additional case at matched Reynolds number based on displacement thickness $Re_{\unicode[STIX]{x1D6FF}^{\ast }}$ was also considered. It was noticed that non-constant $\unicode[STIX]{x1D6FD}$ cases appear to approach the conditions of equivalent constant $\unicode[STIX]{x1D6FD}$ cases after long streamwise distances (approximately 7 boundary-layer thicknesses). The relevance of the constant $\unicode[STIX]{x1D6FD}$ cases lies in the fact that they define a ‘canonical’ state of the boundary layer, uniquely characterized by $\unicode[STIX]{x1D6FD}$ and $Re$. The investigations on the flat plate were extended to the flow around a wing section overlapping in terms of $\unicode[STIX]{x1D6FD}$ and $Re$. Comparisons with the flat-plate cases at matched values of $\unicode[STIX]{x1D6FD}$ and $Re$ revealed that the different development history of the turbulent boundary layer on the wing section leads to a less pronounced wake in the mean velocity as well as a weaker second peak in the Reynolds stresses. This is due to the weaker accumulated effect of the $\unicode[STIX]{x1D6FD}$ history. Furthermore, a scaling law suggested by Kitsios et al. (Intl J. Heat Fluid Flow, vol. 61, 2016, pp. 129–136), proposing the edge velocity and the displacement thickness as scaling parameters, was tested on two constant-pressure-gradient parameter cases. The mean velocity and Reynolds-stress profiles were found to be dependent on the downstream development. The present work is the first step towards assessing history effects in adverse-pressure-gradient turbulent boundary layers and highlights the fact that the values of the Clauser pressure-gradient parameter and the Reynolds number are not sufficient to characterize the state of the boundary layer.

On determining characteristic length scales in pressure-gradient turbulent boundary layers

In the present work, we analyze three commonly used methods to determine the edge of pressure gradient turbulent boundary layers: two based on composite profiles, the one by Chauhan et al. [“Criteria for assessing experiments in zero pressure gradient boundary layers,” Fluid Dyn. Res. 41, 021404 (2009)] and the one by Nickels [“Inner scaling for wall-bounded flows subject to large pressure gradients,” J. Fluid Mech. 521, 217–239 (2004)], and the other one based on the condition of vanishing mean velocity gradient. Additionally, a new method is introduced based on the diagnostic plot concept by Alfredsson et al. [“A new scaling for the streamwise turbulence intensity in wall-bounded turbulent flows and what it tells us about the ‘outer’ peak,” Phys. Fluids 23, 041702 (2011)]. The boundary layers developing over the suction and pressure sides of a NACA4412 wing section, extracted from a direct numerical simulation at chord Reynolds number Rec = 400 000, are used as the test case, besides other numerical and experimental data from favorable, zero, and adverse pressure-gradient flat-plate turbulent boundary layers. We find that all the methods produce robust results with mild or moderate pressure gradients, although the composite-profile techniques require data preparation, including initial estimations of fitting parameters and data truncation. Stronger pressure gradients (with a Rotta–Clauser pressure-gradient parameter β larger than around 7) lead to inconsistent results in all the techniques except the diagnostic plot. This method also has the advantage of providing an objective way of defining the point where the mean streamwise velocity is 99% of the edge velocity and shows consistent results in a wide range of pressure gradient conditions, as well as flow histories. Collapse of intermittency factors obtained from a wide range of pressure-gradient and Re conditions on the wing further highlights the robustness of the diagnostic plot method to determine the boundary layer thickness (equivalent to δ99) and the edge velocity in pressure gradient turbulent boundary layers.

Characterisation of a high Reynolds number boundary layer subject to pressure gradient and separation

The flow over a ramp model is characterised in detail in the present study. In the selected configuration, a turbulent boundary layer developed on a flat plate is first accelerated in a curved contraction. It is then submitted to a mild adverse pressure gradient on a flat plate followed by a separation above a flap. Inlet boundary conditions and pressure distribution are provided to allow numerical simulations. The flow in the mild adverse pressure gradient region is characterised with hot-wire anemometry. In this region, the boundary layer thickness is of the order of 20 cm, the momentum Reynolds number is about 11,000 and the Clauser pressure gradient parameter β in the stabilised region is about 0.4. Particular emphasis is laid on the separation to provide quantitative information to evaluate turbulence models. This is achieved through a large streamwise two-dimensional two-component particle image velocimetry (2D2C PIV) plane which contains all the separation bubble and part of the flow upstream and downstream of it. The separation border is detected using the backflow coefficient, resulting in a separation length of about 3.49 Hs (with Hs the flap step height) and a maximum height of about 0.17 Hs. The Reynolds stresses and their main production terms are also determined. A region of high turbulence intensity develops above the separation border for all the measured components. The production of dominates the production of turbulent kinetic energy which implies a redistribution from to to explain the increase observed. The production term drives the production of in the first part of the flap which is not the case for zero pressure gradient boundary layers. Finally, a high similarity is observed between and as the production of the latter is dominated by . This flow appears as a challenging test case for Reynolds averaged Navier–Stokes (RANS) and large eddy simulation (LES) validation.

Wake-boundary layer interaction subject to convex and concave curvatures and adverse pressure gradient

Measurements of mean velocity and turbulent quantities have been carried out when the wake of a symmetrical airfoil interacts with the boundary layer on the (i) walls of a straight duct/diffuser and (ii) convex and concave walls of a curved duct/diffuser. The effects of adverse pressure gradient and of curvatures on the interaction are studied separately and in combination. Six cases are considered, viz. with (i) neither pressure gradient nor curvature, (ii) adverse pressure gradient and no curvature, (iii) and (iv) convex curvature with zero and adverse pressure gradients, respectively, (v) and (vi) concave curvature with zero and adverse pressure gradients, respectively. For the flows with curvature, the curvature parameter δ/R is 0.023, and for the flows with adverse pressure gradient, the Clauser pressure gradient parameter β is 0.62. The individual influences of adverse pressure gradient and convex and concave curvatures on the boundary layer are similar to those observed by earlier investigations. It is further observed that the combined effect of concave/convex curvature and the adverse pressure gradient causes higher turbulence intensities than the sum of the individual effects. The effect of curvature is to make the wake asymmetric, and in combination with adverse pressure gradient the asymmetry increases. It is observed that the adverse pressure gradient causes faster wake–boundary-layer interaction. Comparing measurements in a straight duct, a curved duct, a curved diffuser and a straight diffuser, it is seen that the convex curvature reduces the boundary layer thickness. The asymmetry in wake development compensates for this effect and the wake–boundary-layer interaction on a convex surface is almost the same as that on a straight surface. In the case of interaction with the boundary layer on a concave surface, the curvature increases the boundary layer thickness and causes enhanced turbulence intensities. However, the asymmetry in wake is such that the extent of wake is lower towards the boundary layer side. As a result, the wake–boundary-layer interaction on concave surface is almost the same as on a straight surface. The interaction is faster in the presence of adverse pressure gradient.

Lateral straining of turbulent boundary layers. Part 2. Streamline convergence

Experimental measurements are presented showing the effects of streamline convergence on developing turbulent boundary layers. The longitudinal pressure-gradient in these experiments is nominally zero so the only extra rate-of-strain is the lateral convergence. Measurements have been made of mean flow and turbulence quantities at two different Reynolds numbers. The results show that convergence leads to a significant reduction in the skin-friction and an increase in the boundary layer thickness. There are also large changes in the Reynolds stresses with reductions occurring in the inner region and some increase in the outer flow. This is in contrast to the results of Saddoughi &amp; Joubert (1991) for a diverging flow of the same included angle and zero pressure-gradient which show much smaller changes in the stresses and an approach to equilibrium. A new non-dimensional parameter, βD, is proposed to characterize the local effect of the convergence and it is shown how this parameter is related to Clauser's pressure-gradient parameter, βx. It is suggested that this is an equilibrium parameter for turbulent boundary layers with lateral straining. In the present flow case βD increases rapidly with streamwise distance leading to a significant departure from equilibrium. Measurement of terms in the transport equations suggest that streamline convergence leads to a reduction in production and generation and large increases in mean advection. The recovery of the flow after the removal of convergence has been shown to be characterized by a significant increase in the turbulent transport of shear-stress and turbulent kinetic energy from the very near-wall region to the flow further out where the stresses have been depleted by convergence.

Experimental results for the transpired turbulent boundary layer in an adverse pressure gradient

An experimental investigation of the fluid mechanics of the transpired turbulent boundary layer in zero and adverse pressure gradients was carried out on the Stanford Heat and Mass Transfer Apparatus. Profiles of (a) the mean velocity, (b) the intensities of the three components of the turbulent velocity fluctuations and (c) the Reynolds stress were obtained by hot-wire anemometry. The wall shear stress was measured by using an integrated form of the boundary-layer equation to ‘extrapolate’ the measured shear-stress profiles to the wall.The two experimental adverse pressure gradients corresponded to free-stream velocity distributions of the type u∞ ∞ xm, where m = −0·15 and −0·20, x being the streamwise co-ordinate. Equilibrium boundary layers (i.e. flows with velocity defect profile similarity) were obtained when the transpiration velocity v0 was varied such that the blowing parameter B = pv0u∞/τ0 and the Clauser pressure-gradient parameter $\beta\equiv\delta_1\tau_0^{-1}\,dp/dx $ were held constant. (τ0 is the shear stress at the wall and δ1 is the displacement thickness.)Tabular and graphical results are presented.