Adverse Pressure Gradient Turbulent Boundary Layer Research Articles

A modification of production term for the shear stress transport turbulence model based on a new wall-law for adverse pressure gradient

This paper presents an attempt to improve the predictive accuracy of the Menter shear stress transport (SST) turbulence model for adverse pressure gradient (APG) flows. The foundation of this work lies in the newly modified law of the wall and turbulent kinetic energy (TKE) transport equation within the APG turbulent boundary layer. The modification of the law of the wall is achieved through a newly proposed variable λ, which is derived theoretically and calibrated using direct numerical simulation data. Based on the new law of the wall, we analyze the TKE transport equation of the SST model at APG and demonstrate that the analytical solution satisfying the law of the wall cannot achieve equilibrium in the current SST model. Further theoretical analysis reveals that this imbalance is caused by the TKE production term Pk, and the decomposition of the wall friction coefficient Cf indicates that Pk directly affects the prediction of flow separation. Based on the analysis and the new law of the wall, an additional term for Pk is suggested to correct Pk and improve the model's predictive accuracy for APG flows. Numerical validation results show that the correction leads to an increase in Pk within the APG turbulent boundary layer, resulting in more accurate predictions of Cf and the mean velocity profile in the inner layer of the turbulent boundary layer.

Consistent outer scaling and analysis of adverse pressure gradient turbulent boundary layers

Under adverse pressure gradient (APG) conditions, the outer regions of turbulent boundary layers (TBLs) are characterized by an increased velocity defect $U_{e}-U$ , an outwards shift of the peak value of the Reynolds shear stress $-\langle uv\rangle$ and an appearance of the outer peak value of the Reynolds normal stress $\langle uu\rangle$ . Here $U_{e}$ is the TBL edge velocity. Scaling APG TBLs is challenging due to the non-equilibrium effects caused by changes in the APG. To address this, the response distance of TBLs to non-equilibrium conditions is utilized to extend the Zagarola–Smits scaling $U_{zs} = U_{e}({\delta ^{*} }/{\delta })$ and ensure that the original properties of the Zagarola–Smits scaling are maintained as $Re \to \infty$ . Here $\delta ^{*}$ is the displacement thickness and $\delta$ is the boundary layer thickness. Based on the established correlation between $U_{e}-U$ and $-\langle uv\rangle$ , the scaling is extended to $-\langle uv\rangle$ . Furthermore, considering the coupling relationship between Reynolds stress components, the scaling is extended to encompass each Reynolds stress component. The proposed consistent scaling is verified using five non-equilibrium databases and five near-equilibrium databases, successfully collapsing the data of the TBL outer region. The pressure gradient parameter $\beta =({\delta ^{*} }/{\rho u_{\tau }^{2} }) ({\mathrm {d} P_{e} }/{\mathrm {d}\kern0.7pt x})$ of these databases spans two orders of magnitude. Here $P_{e}$ is the boundary layer edge pressure, $u_{\tau }$ is the friction velocity and $\rho$ is the density. Finally, the influence of the APG on the inner and outer regions of TBLs is analysed using the mean momentum balance equation. The analysis suggests that the shift of the $-\langle uv\rangle$ peak to the outer region under APG conditions is due to an insufficient inertia term near the inner region to balance the APG. It is observed that the APG promotes interaction between the inner and outer regions of TBLs, but the inner and outer regions still retain distinctive properties.

A family of adverse pressure gradient turbulent boundary layers with upstream favourable pressure gradients

A flat-plate turbulent boundary layer (TBL) is experimentally subjected to a family of 22 favourable–adverse pressure gradients (FAPGs) using a ceiling panel of variable convex curvature. We define FAPGs as a sequence of streamwise pressure gradients in the order of favourable followed by adverse, similar in sequence to the pressure gradients over the suction side of an airfoil. For the strongest pressure gradient case, the acceleration parameter, $K$ , varied spatially from $6 \times 10^{-6}$ to $- 4.8 \times 10^{-6}$ . The adverse pressure gradient (APG) region of this configuration is studied using particle image velocimetry in the streamwise–wall-normal plane. The statistics of the APG TBL show that the upstream favourable pressure gradient (FPG) exerts a strong and lasting downstream influence, and that the rapid spatial changes in the pressure gradients imposed cause an internal boundary layer to grow within the TBL for 15 of the cases studied. The internal layer typically occupies 20 $\%$ of the boundary layer thickness and dominates the boundary layer response, containing the peak turbulent production, peak strength and population of vortices, and most of the spectral energy content of the flow. The outer layer, on the other hand, develops in the APG region without considerable changes to the state dictated by the upstream FPG. These trends are in contrast to APG TBLs that originate from a zero pressure gradient region, where the outer/wake region is known to dominate TBL response. The observed changes are quantified across the family of FAPGs imposed.

Outer scaling of the mean momentum equation for turbulent boundary layers under adverse pressure gradient

A new scaling of the mean momentum equation is developed for the outer region of turbulent boundary layers (TBLs) under adverse pressure gradient (APG). The maximum Reynolds shear stress location, denoted as $y_{m}$ , is employed to determine the proper scales for the outer region of an APG TBL. An outer length scale is proposed as $\delta _e - y_{m}$ , where $\delta _e$ is the boundary layer thickness. An outer velocity scale for the mean streamwise velocity deficit is proposed as $U_e - U_{m}$ , where $U_e$ and $U_m$ are the mean streamwise velocities at the boundary layer edge and $y_{m}$ , respectively. An outer velocity scale for the mean wall-normal velocity deficit is proposed as $V_e - V_{m}$ , where $V_e$ and $V_{m}$ are the wall-normal velocities at $\delta _e$ and $y_{m}$ , respectively. The maximum Reynolds shear stress is found to scale as $(\delta _e - y_{m}) U_e \,{\rm d}U_e/{{\rm d}x}$ . The new outer scaling collapses well the experimental and numerical data on APG TBLs over a wide range of Reynolds numbers and strengths of pressure gradient. Approximations of the new scaling are developed for TBLs under strong APG and at high Reynolds numbers. The relationships between the new scales and previously proposed scales are discussed.

Energy transfer mechanisms in adverse pressure gradient turbulent boundary layers: production and inter-component redistribution

Production and inter-component redistribution of turbulence in adverse pressure gradient (APG) turbulent boundary layers (TBLs) with small and large velocity defects are investigated, along with the structures playing a role in these energy transfer mechanisms. We examine the wall-normal and spectral distributions of energy, production and pressure-strain in APG TBLs, and compare these distributions with those in canonical flows. It is found that the spectral distributions of production and pressure-strain are not affected profoundly by an increase of the velocity defect, although the energy spectra change drastically in the inner layer of the large-defect APG TBL. In the latter, the signature of the inner-layer streaks is absent from the energy spectra. In the outer layer, energetic, production and pressure-strain structures appear to change from wall-attached to wall-detached structures with increasing velocity defect. Despite this, the two-dimensional spectral distributions have similar shapes and wavelength aspect ratios of the peaks in all these flows. Therefore, the conclusion is that the mechanisms responsible for turbulence production and inter-component energy transfer may remain the same within each layer in all these flows. It is the intensity of these mechanisms within one layer that changes with velocity defect, because of the local mean shear variation.

Intense large-scale motions in zero and adverse pressure gradient turbulent boundary layers

Proper orthogonal decomposition (POD) is used to study coherent structures in wall-bounded turbulent flows. The present study uses POD in turbulent boundary layers to determine the contributions of the intense large-scale motions (LSMs) to the Reynolds stresses. This study uses the 2C-2D PIV measurements of zero pressure gradient turbulent boundary layers (ZPG-TBL) at Re_{δ2} = 7750, and adverse pressure gradient turbulent boundary layer (APG-TBL) at β = 2.27 and Re_{δ2}= 16240, where Re_{δ2} is the momentum thickness based Reynolds number and β is the Clauser’s pressure gradient parameter. The measurements were obtained in the Laboratoire de Mécanique des Fluides de Lille (LMFL) High-Reynolds-Number (HRN) Boundary Layer Wind Tunnel, Lille, France. The snapshots of the flow field are segregated into those dominated by the intense and mild LSMs based on the intensity of the temporal coefficients of the first POD mode. The intense LSMs are further decomposed into high-momentum (HM) and low-momentum (LM) motions. The relative contributions of the HM motions to the Reynolds stresses are larger near the wall as compared to the LM motions. At the wall-normal distance of the displacement thickness (δ1), HM and LM motions have similar contributions. Beyond δ1, the LM motions have larger contributions with their peaks located closer to the displacement thickness height. This shows that in the presence of an APG, the turbulence activity is shifted closer to the displacement thickness height.

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.

Properties of the inertial sublayer in adverse pressure-gradient turbulent boundary layers

The inertial sublayer of adverse pressure-gradient (APG) turbulent boundary layers is investigated using new experimental measurements ( $7000 \lesssim \delta ^+ \lesssim 7800$ ), existing lower Reynolds number experimental ( $\delta ^+ \approx 1000$ ) and computational ( $\delta ^+<800$ ) data sets, where $\delta ^+$ is the friction Reynolds number. In the present experimental set-up the boundary layer is under modest APG conditions, where the Clauser PG parameter $\beta$ is ${\leq }1.8$ . Well-resolved hot-wire measurements are obtained at the Flow Physics Facility at the University of New Hampshire in the region of an APG ramp. Comparisons are made with zero pressure-gradient turbulent boundary layer (ZPG TBL) experimental data at similar Reynolds number and numerical simulation data at lower Reynolds number. The main aims of the present study centre on the inertial sublayer of the APG TBL and the degree to which its characteristics are similar to those of the ZPG TBL. This investigation utilizes equation-based analyses and empirical approaches. Among other results, the data suggest that even though the APG TBL streamwise variance does not exhibit a logarithmic profile (unlike the ZPG TBL) both ZPG and APG TBLs exhibit distance-from-the-wall scaling on the inertial sublayer. Theoretical arguments suggest that wall-distance scaling resulting from a self-similar dynamics is consistent with both a single velocity scale leading to a log-law in mean velocity profile as well as multiple velocity scales leading to a power-law mean velocity profile.

Investigation of Large Scale Motions in Zero and Adverse Pressure Gradient Turbulent Boundary Layers Using High-Spatial-Resolution PIV

Particle image velocimetry (PIV) has been used to capture the high-spatial-resolution (HSR) two-component, two-dimensional (2C-2D) velocity fields of a zero-pressure-gradient (ZPG) turbulent boundary layer (TBL) and of an adverse-pressure-gradient (APG) TBL. Proper Orthogonal Decomposition (POD) is performed on the measured velocity fields to characterize the velocity fields as large or small scale motions (LSMs or SSMs), with further characterisation of the LSMs into high and low momentum events. This paper reports the findings of the PIV experiment and the subsequent analysis of the high Reynolds number ZPG and APG TBLs

Reynolds shear-stress carrying structures in shear-dominated flows

Four direct numerical simulation (DNS) databases are examined to understand the effect of the wall and near-wall turbulence on the Reynolds shear-stress carrying structures in shear-driven flows. The first DNS database is of a non-equilibrium adverse-pressure-gradient (APG) turbulent boundary layer (TBL) with momentum thickness Reynolds number (Reg) reaching 8000. The second one is the same flow as the previous, but turbulence activity in the inner layer (y/S < 0.1) is artificially eliminated. The last two DNS databases are homogeneous shear turbulence (HST) with Taylor microscale Reynolds numbers (Re\\) are 104 and 248. Results show that outer layer turbulence in the APG TBLs with large velocity defect is only slightly affected by the near-wall region turbulence which suggests outer layer turbulence sustains itself without necessitating near-wall turbulence. The Corrsin length scale (Lc) scales the size of the Reynolds shear-stress carrying structures in both APG TBLs and HSTs. The streamwise length of these structures is 1LC or larger in all cases. The aspect ratio of the structures behaves similarly in both APG TBLs and HSTs when the size of the structures are normalized with Lc. Sweeps and ejections tend to form side-by-side pairs in both flow types. The spatial properties of sweeps and ejections, such as aspect ratios or relative positions are not affected by near-wall turbulence activity or presence of the wall. This suggests that the structures mostly dependent on the local mean strain rates.

Analysis of the spanwise extent and time persistence of uniform momentum zones in zero pressure gradient and adverse pressure gradient turbulent boundary layers

Time-resolved velocity fields from direct numerical simulations (DNS) are used to investigate the spanwise extent and the time persistence of uniform momentum zones (UMZs) in a zero pressure gradient turbulent boundary layer (ZPG-TBL) and a self-similar adverse pressure gradient turbulent boundary layer (APG-TBL) at the verge of separation. The instantaneous detection methodology introduced by Adrian et al. [1] is used to detect the UMZs and is extended to take into account the spanwise domain length and the temporal evolution of the UMZs. The Reynolds number based on friction velocity (Reτ) ranges from 1176 to 1277 for the ZPG-TBL and from 1652 to 1745 for the self-similar APG-TBL within the domain of interest. For both the TBL cases, probability density functions (PDFs) of the number of UMZs are computed as a function of the streamwise extent, spanwise extent and time extent. For the ZPG-TBL, when the streamwise length of the domain is greater than or equal to 3 boundary layer thickness, the probability of finding 4 UMZs becomes almost negligible. For the APG-TBL, even when the streamwise domain length is taken as large as 1.3 boundary layer thickness, the probability of finding 4 UMZs is still significant. The spanwise extent of the UMZs is found to be shorter than their streamwise extent regardless of the pressure gradient in the flow. In the ZPG-TBL flow, the majority of the UMZs have a spanwise extent of the order of one-tenth of a boundary layer thickness while for the APG-TBL, it is found to be on the order of one-hundredth of a boundary layer thickness. In the ZPG-TBL, the probability of finding 2 UMZs that persist over a time period of 2 integral time scale is around 50%. Similarly, for the APG-TBL, the probability of finding 2 UMZs with a time persistence of 0.4 integral time scale is over 50%. In the case of the ZPG-TBL, it is observed that some of the UMZs with higher persistence in time have higher streamwise momentum and are found to be closer to the free-stream in general. This result is consistent with the previous observations by Laskari et al. [2]. In contrast, for the APG-TBL, UMZs with longer time persistence are found closer to the wall with lower streamwise momentum.

Analysis of the factors contributing to the skin friction coefficient in adverse pressure gradient turbulent boundary layers and their variation with the pressure gradient

This paper reports on a study of the various factors contributing to the skin friction in incompressible adverse pressure gradient turbulent boundary layer (APG-TBL) flows. Specifically, it deals with the contributions to the skin friction coefficient from the Reynolds stresses and the viscous effects and the role of the pressure gradient. The skin friction coefficient is calculated based on the theoretical decomposition for mean skin friction generation introduced by Renard and Deck (2016). This decomposition is compatible with spatially developing flows as it is applicable to every local streamwise position. The turbulent flows are generated through the direct numerical simulation of a TBL on a smooth flat plate with the desired farfield pressure gradient. It is observed that the Reynolds shear stress provides the dominant positive contribution to the skin friction coefficient for all the pressure gradient cases. However, with increasing adverse pressure gradient, the skin friction coefficient continues to decrease and approaches zero as the positive contribution from the Reynolds shear stress is diminished by the negative contribution of the pressure gradient. When the flow reaches the verge of separation, the predominant Reynolds shear stress contribution to the skin friction coefficient is from a spatially localized outer peak at an approximate height of the displacement thickness (y=δ1) which coincides with the inflection point of the mean streamwise velocity. Even though, the decompositions in Renard and Deck (2016) and Fukagata et al. (2002) give a different distribution for the skin friction coefficient in the zero pressure gradient (ZPG) and the mild APG cases, both of the identities capture the dominant outer peak of the Reynolds shear stress contribution when the flow reaches the verge of separation. This emphasizes the growing importance of the outer layer dynamics with increasing pressure gradient as it pertains to skin friction generation.

Characterisation of uniform momentum zones in adverse pressure gradient turbulent boundary layers

Uniform momentum zones (UMZs) for an adverse pressure gradient turbulent boundary layer (APG-TBL) are characterised in this paper.The database described by Cuvier et al.(2017), which was obtained using 2-component 2-dimensional (2C-2D) particle image velocimetry (PIV) at two free-stream velocities of 5 m/s and 9 m/s, is used for this study.The methodology described by Adrian et al.(2000) is used to identify and hence, statistically characterise the UMZs and UMZ interfaces in the APG-TBL. This study shows that the number of detected UMZs increases in the streamwise direction for both free-stream velocities.This increase is primarily thought to be due to the adverse pressure gradient (APG) since a comparison of the results between the two free-stream velocity APG-TBLs did not show any statistically significant difference with Reθ for the range of Reynolds number examined in this study.The thickness of the UMZs, δw and the jump in streamwise velocity across the UMZ interface were found to increase in the wall-normal direction, whereas the width of the UMZ interface, Δu, was found to decrease.The number of UMZs, Numz, identified appeared to influence the modal velocities/uniform velocities, geometrical properties of the UMZs and statistics of the flow.With increasing Numz, the modal velocities of the UMZs moved to higher values to accommodate new UMZs that developed closer to the wall and the thickness of each UMZ andΔu across the UMZ interface decreased, whereas δw increased. A slight difference in the magnitude of streamwise Reynolds stress and Reynolds shear stress was observed in the inner and outer regions, and a slight difference in the magnitude of turbulence production was also observed in the inner region between the velocity fields having a low or high number of UMZs indicating that the flow statistics are influenced by the number of UMZs present in the APG-TBL.

Experimental realisation of near-equilibrium adverse-pressure-gradient turbulent boundary layers

A new experimental database of adverse-pressure-gradient (APG) turbulent boundary layers (TBLs) obtained through hot-wire anemometry and oil-film interferometry covering a momentum–loss Reynolds number 450<Reθ<23450 and Clauser pressure-gradient-parameter range up to β≈2.4 is presented. Both increasing and approximately constant β distributions with the same upstream history are characterised. Turbulence statistics are compared among the different pressure-gradient distributions with additional numerical and experimental zero-pressure-gradient (ZPG) TBL data. Cases at approximately constant β, which can be considered as canonical representations of the boundary layer under a certain pressure-gradient magnitude, exhibit skin-friction and shape-factor curves consistent with the ones proposed by Vinuesa et al. (2017). These curves show a similar scaling behaviour as those proposed by Nagib et al. (2007) for ZPG TBLs. The pre-multiplied power-spectral density is employed to study the differences in the large-scale energy content throughout the boundary layer. Two different large-scale phenomena are identified, the first one related to the pressure gradient and the second one (also present in high-Re ZPG TBLs) due to the Reynolds number. Recently proposed scaling laws by Kitsios et al. (2016) and Maciel et al. (2018) are tested over a wider Reynolds-number range and for different β cases. The mean velocity and streamwise velocity fluctuation profiles are found to be dependent on the upstream development. The mean velocity profile is found to be self-similar only in the outer region, in agreement with classical theory. The mean and higher-order statistics of the new APG TBL database are made available under www.flow.kth.se.

Volumetric measurements of a self-similar adverse pressure gradient turbulent boundary layer using single-camera light-field particle image velocimetry

As a novel volumetric particle image velocimetry technique, single-camera light-field PIV (LF-PIV) is able to acquire three-dimensional flow fields through a single camera. Compared with other multi-camera 3D PIV techniques, LF-PIV has distinct advantages, including concise hardware setup and low optical access requirements. Its capability has proven effective in many experimental investigations. In this study, the use of LF-PIV in measuring a self-similar adverse pressure-gradient turbulent boundary layer (APG-TBL) is demonstrated. Experiments are performed in a large water tunnel at the Laboratory for Turbulence Research in Aerospace and Combustion (LTRAC), Monash University. Sets of 250 light-field PIV image pairs are captured covering both the inner and outer regions of the boundary layer. Instantaneous 3D velocity fields are reconstructed using a GPU accelerated density ray tracing multiplicative reconstruction technique (DRT-MART) and three-dimensional cross-correlation methods. The LF-PIV results are compared with two-dimensional PIV (2D-PIV) measurements of the same flow. Comparable accuracy to 2D-PIV is achieved for first- and second-order velocity statistics above approximately $$ y/\delta_{1} = 1 $$ .

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.

Reynolds stress structures in a self-similar adverse pressure gradient turbulent boundary layer at the verge of separation.

Mean Reynolds stress profiles and instantaneous Reynolds stress structures are investigated in a self-similar adverse pressure gradient turbulent boundary layer (APG-TBL) at the verge of separation using data from direct numerical simulations. The use of a self-similar APG-TBL provides a flow domain in which the flow gradually approaches a constant non-dimensional pressure gradient, resulting in a flow in which the relative contribution of each term in the governing equations is independent of streamwise position over a domain larger than two boundary layer thickness. This allows the flow structures to undergo a development that is less dependent on the upstream flow history when compared to more rapidly decelerated boundary layers. This APG-TBL maintains an almost constant shape factor of H = 2.3 to 2.35 over a momentum thickness based Reynolds number range of Reδ2 = 8420 to 12400. In the APG-TBL the production of turbulent kinetic energy is still mostly due to the correlation of streamwise and wall-normal fluctuations, 〈uv〉, however the contribution form the other components of the Reynolds stress tensor are no longer negligible. Statistical properties associated with the scale and location of sweeps and ejections in this APG-TBL are compared with those of a zero pressure gradient turbulent boundary layer developing from the same inlet profile, resulting in momentum thickness based range of Reδ2 = 3400 to 3770. In the APG-TBL the peak in both the mean Reynolds stress and the production of turbulent kinetic energy move from the near wall region out to a point consistent with the displacement thickness height. This is associated with a narrower distribution of the Reynolds stress and a 1.6 times higher relative number of wall-detached negative uv structures. These structures occupy 5 times less of the boundary layer volume and show a similar reduction in their streamwise extent with respect to the boundary layer thickness. A significantly lower percentage of wall-attached structures is observed in the present case when compared with a similar investigation of a rapidly decelerating APG-TBL, suggesting that these wall-attached features could be the remanent from the lower pressure gradient domain upstream.

Direct numerical simulation of a self-similar adverse pressure gradient turbulent boundary layer at the verge of separation

The statistical properties are presented for the direct numerical simulation of a self-similar adverse pressure gradient (APG) turbulent boundary layer (TBL) at the verge of separation. The APG TBL has a momentum thickness-based Reynolds number range from $Re_{\unicode[STIX]{x1D6FF}_{2}}=570$ to 13 800, with a self-similar region from $Re_{\unicode[STIX]{x1D6FF}_{2}}=10\,000$ to 12 300. Within this domain the average non-dimensional pressure gradient parameter $\unicode[STIX]{x1D6FD}=39$, where for a unit density $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FF}_{1}P_{\!e}^{\prime }/\unicode[STIX]{x1D70F}_{w}$, with $\unicode[STIX]{x1D6FF}_{1}$ the displacement thickness, $\unicode[STIX]{x1D70F}_{w}$ the mean shear stress at the wall and $P_{\!e}^{\prime }$ the far-field pressure gradient. This flow is compared with previous zero pressure gradient and mild APG TBL ($\unicode[STIX]{x1D6FD}=1$) results of similar Reynolds number. All flows are generated via the direct numerical simulation of a TBL on a flat surface with far-field boundary conditions tailored to apply the desired pressure gradient. The conditions for self-similarity, and the appropriate length and velocity scales, are derived. The mean and Reynolds stress profiles are shown to collapse when non-dimensionalised on the basis of these length and velocity scales. As the pressure gradient increases, the extent of the wake region in the mean streamwise velocity profiles increases, whilst the extent of the log-layer and viscous sublayer decreases. The Reynolds stress, production and dissipation profiles of the APG TBL cases exhibit a second outer peak, which becomes more pronounced and more spatially localised with increasing pressure gradient. This outer peak is located at the point of inflection of the mean velocity profiles, and is suggestive of the presence of a shear flow instability. The maximum streamwise velocity variance is located at a wall normal position of $\unicode[STIX]{x1D6FF}_{1}$ of spanwise wavelength of $2\unicode[STIX]{x1D6FF}_{1}$. In summary as the pressure gradient increases the flow has properties less like a zero pressure gradient TBL and more akin to a free shear 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.

Direct numerical simulation of a self-similar adverse pressure gradient turbulent boundary layer

The statistical properties of a self-similar adverse pressure gradient (APG) turbulent boundary layer (TBL) are presented. The flow is generated via the direct numerical simulation of a TBL on a flat surface with a farfield boundary condition designed to apply the desired pressure gradient. The conditions for self-similarity and appropriate scaling are derived, with the mean profiles, Reynolds stress profiles, and turbulent kinetic energy budgets non-dimensionalised using this scaling. The APG TBL has a momentum thickness based Reynolds number range from Reδ2=300 to 6000, with a self-similar region spanning a Reynolds number range from Reδ2=3500 to 4800. Within this range the non-dimensional pressure gradient parameter β=1. Two-point correlations of each of the velocity components in the streamwise/wall-normal plane are also presented, which illustrate the statistical imprint of the structures in this plane for the APG TBL.