Adverse Pressure Gradient Boundary Layers Research Articles

Studying turbulent flows with physics-informed neural networks and sparse data

Physics-informed neural networks (PINNs) have recently become a viable modelling method for the scientific machine-learning community. The appeal of this network architecture lies in the coupling of a deep neural network (DNN) with partial differential equations (PDEs): the DNN can be considered a universal function approximator, and the physical knowledge from the embedded PDEs regularises the network during training. This regularisation improves robustness of the network and means the network only requires sparse training data in the domain. We apply PINNs with embedded Reynolds-averaged Navier–Stokes (RANS) equations to a spatially developing adverse-pressure-gradient (APG) boundary layer 1000<Reθ<3000, and also to the periodic hill problem 5600≤Reb≤37,000. We do not use a turbulence model to close the RANS equations, rather the network infers the Reynolds-stress fields during training as a means to satisfy closure. As PINNs are able to produce robust predictions with sparse training data, this paper demonstrates how PINNs can be trained with data typical of experimental campaigns. For the APG boundary layer, we seek to quantify the amount and location of data required to model a time-averaged turbulent flow with PINNs, and show that PINNs can accurately model wall shear stress and wall pressure. We demonstrate how PINNs perform when trained without information of a critical feature of the flow (such as a separation bubble), and we highlight how the cost of training the network does not increase with Reynolds number. Finally, we undertake a series of benchmarks to demonstrate how the method scales with network size and the number of residual points in the problem. This work seeks to establish the quantity and type of data required to successfully apply PINNs to time-averaged flow without a closure model, and seeks to set expectations on the prediction quality of the network when trained on an available data set.

A universal velocity profile for turbulent wall flows including adverse pressure gradient boundary layers

A recently developed mixing length model of the turbulent shear stress in pipe flow is used to solve the streamwise momentum equation for fully developed channel flow. The solution for the velocity profile takes the form of an integral that is uniformly valid from the wall to the channel centreline at all Reynolds numbers from zero to infinity. The universal velocity profile accurately approximates channel flow direct numerical simulation (DNS) data taken from several sources. The universal velocity profile also provides a remarkably accurate fit to simulated and experimental flat plate turbulent boundary layer data including zero and adverse pressure gradient data. The mixing length model has five free parameters that are selected through an optimization process to provide an accurate fit to data in the range $R_\tau = 550$ to $R_\tau = 17\,207$ . Because the velocity profile is directly related to the Reynolds shear stress, certain statistical properties of the flow can be studied such as turbulent kinetic energy production. The examples presented here include numerically simulated channel flow data from $R_\tau = 550$ to $R_\tau =8016$ , zero pressure gradient (ZPG) boundary layer simulations from $R_\tau =1343$ to $R_\tau = 2571$ , zero pressure gradient turbulent boundary layer experimental data between $R_\tau = 2109$ and $R_\tau = 17\,207$ , and adverse pressure gradient boundary layer data in the range $R_\tau = 912$ to $R_\tau = 3587$ . An important finding is that the model parameters that characterize the near-wall flow do not depend on the pressure gradient. It is suggested that the new velocity profile provides a useful replacement for the classical wall-wake formulation.

Integral measures of the zero pressure gradient boundary layer over the Reynolds number range ≤Rτ&amp;lt;∞

A recently developed mixing length model of the turbulent shearing stress has been shown to generate a universal velocity profile that provides an accurate approximation to incompressible pipe flow velocity profiles over a wide Reynolds number range [B. J. Cantwell, “A universal velocity profile for smooth wall pipe flow,” J. Fluid Mech. 878, 834–874 (2019)]. More recently, the same profile was shown to accurately approximate velocity profiles in channel flow, the zero pressure gradient boundary layer, and the boundary layer in an adverse pressure gradient [M. A. Subrahmanyam, B. J. Cantwell, and J. J. Alonso, “A universal velocity profile for turbulent wall flows,” AIAA Paper No. 2021-0061, 2021 and M. A. Subrahmanyam, B. J. Cantwell, and J. J. Alonso, “A universal velocity profile for turbulent wall flows including adverse pressure gradient boundary layers,” J. Fluid Mech. (unpublished) (2021)] The universal velocity profile is uniformly valid from the wall to the free stream at all Reynolds numbers from zero to infinity. At a low Reynolds number, the profile approaches the laminar channel/pipe flow limit. The primary measure of the Reynolds number used in this work is the friction Reynolds number Rτ=uτδ/ν. It is a little unusual to use Rτ for the boundary layer since it requires that the velocity profile be cutoff using an arbitrarily defined overall boundary layer thickness, δ. Because of the slow approach of the velocity to the free stream, different conventions used to define the thickness lead to different values of Rτ assigned to a given flow. It will be shown in this paper that, through its connection to channel/pipe flow, the universal velocity profile can be used to define a practically useful, unambiguous, measure of overall boundary layer thickness, called here the equivalent channel half height, δh. For Rτ&amp;gt;≈5000, the universal velocity profile defines a Reynolds number independent shape function that can be used to generate explicit expressions for the infinite Reynolds number behavior of all the usual integral boundary layer measures; displacement thickness, momentum thickness, energy thickness, overall boundary layer thickness, and skin friction. The friction coefficient Cf(Rδ2) generated by the universal velocity profile accurately approximates data over a wide range of momentum thickness Reynolds numbers collected by Nagib et al. [“Can we ever rely on results from wall-bounded turbulent flows without direct measurements of wall shear stress?,” AIAA Paper No. 2004-2392, 2004]. The universal velocity profile is used to integrate the von Kaŕmań boundary layer integral equation [T. von Kármán, “Uber laminaire und turbulente reibung,” Z. Angew. Math. Mech. 1, 233–252 (1921)] in order to generate the various thicknesses and friction velocity as functions of the spatial Reynolds number, Rx=uex/ν.

Features of the Flow of an Adverse Pressure Gradient Boundary Layer

Vertical velocity profiles are obtained experimentally for the boundary layer of a flow with an inverse pressure gradient in different phases of the periodic process of fluid deceleration. At the greatest drop in velocity, the profile has a local minimum with two inflection points at the upper boundary of the viscous layer. Chains of vortices then form at this location, testifying to a loss of stability of the laminar motion.

Flow modulation by a mushroom-like coating around the separation region of a wind-turbine airfoil section

The flow over a mushroom-shaped microscale coating was experimentally inspected over a diverging channel that followed the pressure side of a wind turbine blade (S835). High-resolution particle image velocimetry was used to obtain in-plane velocity measurements in a refractive-index-matching flume at Reynolds number Reθ ≈ 1200 based on the momentum thickness. The results show that the evolution of the boundary layer thickness, displacement thickness, and shape factor change with the coating, contrary to the expected behavior of an adverse pressure gradient boundary layer over a canonical rough surface. Comparison of the flow with that over a smooth wall revealed that the turbulence production exhibited similar levels in both cases, suggesting that the coating does not behave like a typical rough wall, which increases the Reynolds stresses. Proper orthogonal decomposition was used to decompose the velocity field to investigate the possible structural changes introduced by the wall region. It suggests that large-scale motions in the wall region lead to high-momentum flow over the coated case compared to the smooth counterpart. This unique behavior of this surface coating can be useful in wind-turbine applications, with great potential to increase the power production.

Empirical Model of Wall Pressure Spectra in Adverse Pressure Gradients

An empirical model of wall pressure spectra for adverse pressure gradient boundary layers is proposed. The effect of adverse pressure gradients on wall pressure spectra and the relationship between the boundary-layer parameters and the wall pressure are studied and demonstrated. A dataset for five experiments at four different test facilities, covering Reynolds number between and based on the boundary-layer momentum thickness, is selected to develop the new model. Goody’s model is served as the basis for the development. Predictions of the new model and comparisons with other published wall pressure spectral models for adverse pressure gradient boundary layers are made for the selected dataset. The new model shows good prediction accuracy for the selected dataset and a significant improvement compared to the other published models.

Miniature Vortex Generators for Flow Control in Falkner–Skan Boundary Layers

Vortex generators with heights comparable to displacement thickness are an effective means of producing persistent mean-flow streaks in laminar boundary layers. Inducing streaky base flows can suppress the growth of unsteady disturbances that would otherwise incite laminar-to-turbulent transition. Previous experimental and numerical works demonstrated the versatility of these miniature vortex generators in zero-pressure-gradient boundary layers. In this work, mean-flow disturbances developing from miniature vortex generators in adverse and favorable pressure-gradient boundary layers are measured systemically to assess the possibility of extending miniature vortex generator-based flow control to these scenarios. Boundary-layer streak amplitudes are measured across a range of Falkner–Skan values, and an empirical scaling is found based on existing results. The effect of streaks on transition in an adverse pressure-gradient boundary layer is also tested, and moderate increases to laminar flow extents are observed.

Large-scale motions in turbulent boundary layers subjected to adverse pressure gradients

It is known that large-scale streamwise velocity-fluctuating structures ($u^{\prime }$) are frequently observed in the log region of a zero pressure gradient turbulent boundary layer, and that these motions significantly influence near-wall small-scale $u^{\prime }$-structures by modulating the amplitude (Hutchins &amp; Marusic, J. Fluid Mech., vol. 579, 2007, pp. 1–28; Mathis et al., J. Fluid Mech., vol. 628, 2009, pp. 311–337). In the present study, we provide evidence that the spatial organization of large-scale structures in the log region is significantly influenced by the strength of adverse pressure gradients in turbulent boundary layers based on a direct numerical simulation dataset. For a mild adverse pressure gradient boundary layer flow, groups of hairpin vortices are coherently aligned in the streamwise direction to form hairpin vortex packets, and streamwise merging events of the induced large-scale $u^{\prime }$-structures create a larger streamwise length scale of structures than that for a zero pressure gradient boundary layer flow. As the pressure gradient strength increases further, however, the formation of hairpin packets is continuously suppressed, and large-scale motions are consequently not concatenated to create a longer motion, resulting in a significant reduction of the streamwise coherence of large-scale structures in the log layer. Although energy spectrum maps for $u^{\prime }$-structures show that the large-scale energy is continuously intensified above the log layer with an increase in the pressure gradient, amplitude modulation of the near-wall small-scale motions is dominantly induced by log region large-scale structures for adverse pressure gradient flows. Conditional averaged flow fields with large-scale Q2 and Q4 events indicate that large-scale counter-rotating roll modes play an important role in organizing the flows under the pressure gradients, and the large-scale roll modes associated with Q4 events are more enhanced in the outer layer than those associated with Q2 events, reducing the streamwise coherence of the vortices in a packet.

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

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

Extension of Lysak’s Approach to Evaluate the Wall Pressure Spectrum for Boundary Layer Flows

An analytical model proposed by Lysak to evaluate the wall pressure fluctuation spectrum and variance for pipe flows is extended to use it as a post-processor of RANS computations, for attached boundary layers, whatever the turbulence model. Model extension also deals with three-dimensional flows. The model fairly reproduces key features of the wall pressure spectrum such as the f 2 behaviour in the low frequency range, the inertial f −1 region and the viscous cut-off. Wall pressure spectra and variance can be fairly well predicted. The model allows identifying the contribution of each part of the boundary layer to the wall pressure spectrum. The model was validated for zero pressure gradient boundary layer flows. Sources of differences in the predictions according to the turbulence model and ways of improvement were identified. For adverse pressure gradient boundary layers, departures from experiments are observed and several plausible causes are pointed out.

Amplitude modulated near-wall cycle in a turbulent boundary layer under an adverse pressure gradient ∗

This paper presents the effect of amplitude modulation of small-scale turbulence by large-scale structures in a turbulent boundary layer subjected to an adverse pressure gradient. The results have been compared with the literature data for high Reynolds number, zero-pressure gradient cases (Mathis et al . [1]). It was observed that for relatively low-Reynolds-number turbulent boundary layer (Re τ ≈ 1000) subjected to the adverse pressure gradient, apart from the inner peak of streamwise velocity fluctuations u , a second maximum located in the outer zone of turbulent boundary layer appears. It was found that the large-scale motions are much more energetic for adverse pressure gradients compared with the zero gradient case and the pressure gradient affects mostly the outer region of the flow. Using the method proposed by Mathis et al . [1], based on the correlation function and Hilbert transform, we obtained a clear evidence of a strong correlation between small inner-layer and large log-layer structures, which is similar to that observed for high Reynolds number zeropressure gradient cases. A simple model of modulation process occurring in adverse pressure gradient boundary layer has been proposed.

LES of the adverse-pressure gradient turbulent boundary layer

We describe large-eddy simulations (LES) of the flat-plate turbulent boundary layer in the presence of an adverse pressure gradient. The stretched-vortex subgrid-scale model is used in the domain of the flow coupled to a wall model that explicitly accounts for the presence of a finite pressure gradient. The LES are designed to match recent experiments conducted at the University of Melbourne wind tunnel where a plate section with zero pressure gradient is followed by section with constant adverse pressure gradient. First, LES are described at Reynolds numbers based on the local free-stream velocity and the local momentum thickness in the range 6560–13,900 chosen to match the experimental conditions. This is followed by a discussion of further LES at Reynolds numbers at approximately 10 times and 100 times these values, which are well out of range of present day direct numerical simulation and wall-resolved LES. For the lower Reynolds number runs, mean velocity profiles, one-point turbulent statistics of the velocity fluctuations, skin friction and the Clauser and acceleration parameters along the streamwise, adverse pressure-gradient domain are compared to the experimental measurements. For the full range of LES, the relationship of the skin-friction coefficient, in the form of the ratio of the local free-stream velocity to the local friction velocity, to both Reynolds number and the Clauser parameter is explored. At large Reynolds numbers, a region of collapse is found that is well described by a simple log-like empirical relationship over two orders of magnitude. This is expected to be useful for constant adverse-pressure gradient flows. It is concluded that the present adverse pressure gradient boundary layers are far from an equilibrium state.

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

AbstractResearch into high-Reynolds-number turbulent boundary layers in recent years has brought about a renewed interest in the larger-scale structures. It is now known that these structures emerge more prominently in the outer region not only due to increased Reynolds number (Metzger &amp; Klewicki, Phys. Fluids, vol. 13(3), 2001, pp. 692–701; Hutchins &amp; Marusic, J. Fluid Mech., vol. 579, 2007, pp. 1–28), but also when a boundary layer is exposed to an adverse pressure gradient (Bradshaw, J. Fluid Mech., vol. 29, 1967, pp. 625–645; Lee &amp; Sung, J. Fluid Mech., vol. 639, 2009, pp. 101–131). The latter case has not received as much attention in the literature. As such, this work investigates the modification of the large-scale features of boundary layers subjected to zero, adverse and favourable pressure gradients. It is first shown that the mean velocities, turbulence intensities and turbulence production are significantly different in the outer region across the three cases. Spectral and scale decomposition analyses confirm that the large scales are more energized throughout the entire adverse pressure gradient boundary layer, especially in the outer region. Although more energetic, there is a similar spectral distribution of energy in the wake region, implying the geometrical structure of the outer layer remains universal in all cases. Comparisons are also made of the amplitude modulation of small scales by the large-scale motions for the three pressure gradient cases. The wall-normal location of the zero-crossing of small-scale amplitude modulation is found to increase with increasing pressure gradient, yet this location continues to coincide with the large-scale energetic peak wall-normal location (as has been observed in zero pressure gradient boundary layers). The amplitude modulation effect is found to increase as pressure gradient is increased from favourable to adverse.

Evaluation of a Vortex Generator Model in Adverse Pressure Gradient Boundary Layers

The use of a two-dimensional statistical passive vortex generator model, applied to an adverse pressure gradient boundary-layer flow, is evaluated qualitatively against experimental and fully resol ...

Structures in turbulent boundary layers subjected to adverse pressure gradients

The effects of adverse pressure gradients on turbulent structures were investigated by carrying out direct numerical simulations of turbulent boundary layers subjected to adverse and zero pressure gradients. The equilibrium adverse pressure gradient flows were established by using a power law free-stream distribution U∞ ~ xm. Two-point correlations of velocity fluctuations were used to show that the spanwise spacing between near-wall streaks is affected significantly by a strong adverse pressure gradient. Low-momentum regions are dominant in the middle of the boundary layer as well as in the log layer. Linear stochastic estimation was used to provide evidence for the presence of low-momentum regions and to determine their statistical properties. The mean width of such large-scale structures is closely associated with the size of the hairpin-like vortices in the outer layer. The conditionally averaged flow fields around events producing Reynolds stress show that hairpin-like vortices are the structures associated with the production of outer turbulence. The shapes of the vortices beyond the log layer were found to be similar when their length scales were normalized according to the boundary layer thickness. Estimates of the conditionally averaged velocity fields associated with the spanwise vortical motion were obtained by using linear stochastic estimation. These results confirm that the outer region of the adverse pressure gradient boundary layer is populated with streamwise-aligned vortex organizations, which are similar to those of the vortex packet model proposed by Adrian, Meinhart &amp; Tomkins (J. Fluid Mech., vol. 422, 2000, pp. 1–54). The adverse pressure gradient augments the inclination angles of the packets and the mean streamwise spacing of the vortex heads in the packets.

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

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

The flow structure behind vortex generators embedded in a decelerating turbulent boundary layer

The objective of the present work is to analyse the behaviour of a turbulent decelerating boundary layer under the effect of both passive and active jets vortex generators (VGs). The stereo PIV database of Godard and Stanislas [1,2] obtained in an adverse pressure gradient boundary layer is used for this study. After presenting the effect on the mean velocity field and the turbulent kinetic energy, the line of analysis is extended with two points spatial correlations and vortex detection in instantaneous velocity fields. It is shown that the actuators concentrate the boundary layer turbulence in the region of upward motion of the flow, and segregate the near-wall streamwise vortices of the boundary layer based on their vorticity sign.

Reynolds number scaling in a non-equilibrium turbulent boundary layer with mild adverse pressure gradient

High resolution laser Doppler anemometer measurements were acquired in a two-dimensional turbulent boundary layer over a four degree ramp at three momentum thickness Reynolds numbers: 3300, 14,100 and 20,600. The goal was to examine an adverse pressure gradient boundary layer far from separation in order to develop turbulent stress scalings to collapse the adverse pressure gradient and flat plate stress profiles over a range of Reynolds numbers. The flow develops as a flat plate boundary layer before being subjected to a varying pressure gradient along the length of a four degree straight ramp. Mean velocity measurements show a log law region for all velocity profiles. The stresses however are perturbed by the pressure gradient with the streamwise normal stress developing an extended outer layer plateau and the wall-normal stress displaying a growing outer layer peak. Scaling parameters are proposed to collapse the inner and outer layer regions of the normal stresses for both the adverse pressure gradient and flat plate profiles over the range of Reynolds numbers examined.

Control of a decelerating boundary layer. Part 3: Optimization of round jets vortex generators

In the previous two parts of this study [G. Godard, M. Stanislas, Control of a decelerating boundary layer. Part 1: Optimization of passive vortex generators, Aerospace Sci. Technol. 10 (3) (2006) 181–191; G. Godard, J.M. Foucaut, M. Stanislas, Control of a decelerating boundary layer. Part 2: Optimization of slotted jets vortex generators, Aerospace Sci. Technol. 10 (5) (2006) 394–400], two different types of vortex generators were characterized and optimized in an adverse pressure gradient boundary layer. The model used was a bump in a boundary layer wind tunnel, which mimics the adverse pressure gradient on the suction side of an airfoil at the verge of separation. The present contribution describes the results of a test campaign performed in the same facility to optimize round jets devices with continuous blowing. The optimization was done as previously with hot film shear stress probes. The results show that the optimized jet devices give performances comparable to standard passive vortex generators in terms of skin friction. They also allow a quantitative comparison between three different types of vortex generators: passive devices, slotted and round jets. This comparison is performed in both co- and counter-rotating configurations.

Control of a decelerating boundary layer. Part 2: Optimization of slotted jets vortex generators

In the first part of this contribution [G. Godard, M. Stanislas, Control of a decelerating boundary layer. Part 1: Optimization of passive vortex generators, Aerospace Sci. Technol. 10 (3) (2006) 181–191], an optimization of passive vortex generators was performed in an adverse pressure gradient boundary layer. The model used was a bump in a boundary layer wind tunnel, which mimics the adverse pressure gradient on the suction side of an airfoil at the verge of separation. The present contribution describes the next step of the study: a test campaign was performed in the same facility to optimize slotted jets devices with both continuous and pulsed blowing. The optimization was done using hot film shear stress probes. The results show quantitatively the improvement brought by the slotted jets devices in terms of skin friction increase. They also show that the tested devices are less effective than equivalent passive devices.