Momentum Thickness Research Articles

Reynolds-number dependence of wall-pressure fluctuations in a pressure-induced turbulent separation bubble

Direct numerical simulations are used to examine the behaviour of wall-pressure fluctuations$p_{w}$in a flat-plate turbulent boundary layer with large adverse and favourable pressure gradients, involving separation and reattachment. The Reynolds number$Re_{\unicode[STIX]{x1D703}}$based on momentum thickness is equal to 300, 600 and 900. Particular attention is given to effects of Reynolds number on root-mean-square (r.m.s.) values, frequency/power spectra and instantaneous fields. The possible scaling laws are also examined as compared with the existing direct numerical simulation and experimental data. The r.m.s. value of$p_{w}$normalized by the local maximum Reynolds shear stress$-\unicode[STIX]{x1D70C}\overline{uv}_{max}$(Simpson et al. J. Fluid Mech.vol. 177, 1987, pp. 167–186; Na & MoinJ. Fluid Mech.vol. 377, 1998b, pp. 347–373) leads to near plateau (i.e.$p_{w\,rms}/-\unicode[STIX]{x1D70C}\overline{uv}_{max}=2.5\sim 3$) in the adverse pressure gradient and separated regions in which the frequency spectra exhibit good collapse at low frequencies. The magnitude of$p_{w\,rms}/-\unicode[STIX]{x1D70C}\overline{uv}_{max}$is however reduced down to 1.8 near reattachment where good collapse is also obtained with normalization by the local maximum wall-normal Reynolds stress$\unicode[STIX]{x1D70C}\overline{vv}_{max}$. Near reattachment,$p_{w\,rms}/-\unicode[STIX]{x1D70C}\overline{vv}_{max}=1.2$is attained unambiguously independently of the Reynolds number and pressure gradient. The present magnitude (1.2) is smaller than (1.35) obtained for step-induced separation by Ji & Wang (J. Fluid Mech.vol. 712, 2012, pp. 471–504). The reason for this difference is intrinsically associated with convective nature of a pressure-induced separation bubble near reattachment where the magnitude of$p_{w\,rms}$depends essentially on the favourable pressure gradient. The resulting mean flow acceleration leads to delay of the r.m.s. peak after reattachment. Attention is also given to structures of$p_{w}$. It is shown that large-scale spanwise rollers of low pressure fluctuations are formed above the bubble, whilst changing to large-scale streamwise elongated structures after reattachment. These large-scale structures become more prominent with increasing$Re_{\unicode[STIX]{x1D703}}$and affect$p_{w}$significantly.

Transient energy growth of optimal streaks in parallel round jets

We present a linear optimal perturbation analysis of streamwise invariant disturbances evolving in parallel round jets. The potential for transient energy growth of perturbations with azimuthal wavenumber m≥1 is analyzed for different values of Reynolds number Re. Two families of steady (frozen) and unsteady (diffusing) base flow velocity profiles have been used, for different aspect ratios α = R/θ, where R is the jet radius and θ is the shear layer momentum thickness. Optimal initial conditions correspond to infinitesimal streamwise vortices, which evolve transiently to produce axial velocity streaks, whose spatial structure and intensity depend on base flow and perturbation parameters. Their dynamics can be characterized by a maximum optimal value of the energy gain Gopt, reached at an optimal time τopt after which the perturbations eventually decay. Optimal energy gain and time are shown to be, respectively, proportional to Re2 and Re, regardless of the frozen or diffusing nature of the base flow. Besides, it is found that the optimal gain scales like Gopt∝1/m3 for all m except m = 1. This quantitative difference for azimuthal wavenumber m = 1 is shown to be based on the nature of transient mechanisms. For m = 1 perturbations, the shift-up effect [J. I. Jiménez-González et al., “Modal and non-modal evolution of perturbations for parallel round jets,” Phys. Fluids 27, 044105-1–044105-19 (2015)] is active: an initial streamwise vorticity dipole induces a nearly uniform velocity flow in the jet core, which shifts the whole jet radially. By contrast, optimal perturbations with m≥2 are concentrated along the shear layer, in a way that resembles the classical lift-up mechanism in wall-shear flows. The m = 1 shift-up effect is more energetic than the m≥2 lift-up, but it is slower, with optimal times considerably shorter in the case of m≥2 disturbances. This suggests that these perturbations may emerge very quickly in the flow when injected as initial conditions. When the base flow diffuses, the large time scale for m = 1 disturbances allows the shear layer to spread and the jet core velocity to decrease substantially, thus lowering the values of corresponding optimal gain and time. For m≥2, results are less affected, since the shorter transient dynamics does not leave room for significant modifications of the base flow velocity profiles, and the scaling laws obtained in the frozen case are recovered. Nevertheless, base flow diffusion hinders the transient growth, as a consequence of a weaker component-wise non-normality and a smoother, radially spread structure of optimal disturbances.

Streamwise Vortices and Velocity Streaks in a Locally Drag-Reduced Turbulent Boundary Layer

This work aims to understand the changes associated with the near-wall streaky structures in a turbulent boundary layer (TBL) where the local skin-friction drag is substantially reduced. The Reynolds number is R e 𝜃 = 1000 based on the momentum thickness or R e τ = 440 based on the friction velocity of the uncontrolled flow. The TBL is perturbed via a local surface oscillation produced by an array of spanwise-aligned piezo-ceramic (PZT) actuators and measurements are made in two orthogonal planes using particle image velocimetry (PIV). Data analyses are conducted using the vortex detection, streaky structure identification, spatial correlation and proper orthogonal decomposition (POD) techniques. It is found that the streaky structures are greatly modified in the near-wall region. Firstly, the near-wall streamwise vortices are increased in number and swirling strength but decreased in size, and are associated with greatly altered velocity correlations. Secondly, the velocity streaks grow in number and strength but contract in width and spacing, exhibiting a regular spatial arrangement. Other aspects of the streaky structures are also characterized; they include the spanwise gradient of the longitudinal fluctuating velocity and both streamwise and spanwise integral length scales. The POD analysis indicates that the turbulent kinetic energy of the streaky structures is reduced. When possible, our results are compared with those obtained by other control techniques such as a spanwise-wall oscillation, a spanwise oscillatory Lorentz force and a transverse traveling wave.

A numerical study of a variable-density low-speed turbulent mixing layer

Direct numerical simulations of a temporally developing, low-speed, variable-density, turbulent, plane mixing layer are performed. The Navier–Stokes equations in the low-Mach-number approximation are solved using a novel algorithm based on an extended version of the velocity–vorticity formulation used by Kim et al. (J. Fluid Mech., vol 177, 1987, 133–166) for incompressible flows. Four cases with density ratios $s=1,2,4$ and 8 are considered. The simulations are run with a Prandtl number of 0.7, and achieve a $Re_{\unicode[STIX]{x1D706}}$ up to 150 during the self-similar evolution of the mixing layer. It is found that the growth rate of the mixing layer decreases with increasing density ratio, in agreement with theoretical models of this phenomenon. Comparison with high-speed data shows that the reduction of the growth rates with increasing density ratio has a weak dependence with the Mach number. In addition, the shifting of the mixing layer to the low-density stream has been characterized by analysing one-point statistics within the self-similar interval. This shifting has been quantified, and related to the growth rate of the mixing layer under the assumption that the shape of the mean velocity and density profiles do not change with the density ratio. This leads to a predictive model for the reduction of the growth rate of the momentum thickness, which agrees reasonably well with the available data. Finally, the effect of the density ratio on the turbulent structure has been analysed using flow visualizations and spectra. It is found that with increasing density ratio the longest scales in the high-density side are gradually inhibited. A gradual reduction of the energy in small scales with increasing density ratio is also observed.

Three-dimensional plasma actuation for faster transition to turbulence

We demonstrate that a 3D non-linear plasma actuation method creates secondary instabilities by forming lambda vortices for a spatially developing turbulent boundary layer flow over a flat plate. Both bypass transition and controlled transition processes are numerically investigated using wall resolved modal discontinuous Galerkin based implicit large eddy simulation. The largest momentum thickness based Reynolds numbers tested are 1250 and 1100 for the bypass transition and the controlled transition, respectively. The 3D actuation method is based on a square serpentine plasma actuator (Durscher and Roy 2012 J. Phys. D: Appl. Phys. 45 035202). The transition is achieved via oblique wave transition by perturbing the flow at a frequency of 1 kHz with amplitude of 10% of the freestream velocity. Although the flow is perturbed at a single frequency, the instabilities arising due to the nonlinear interaction between the consecutive lambda vortices, creates subharmonic lambda vortices (half of the fundamental frequency), which finally break down into fully turbulent flow. These interactions have been thoroughly studied and discussed. Since the actuation creates oblique wave transition it will allow faster transition compared to the standard secondary instability mechanism with similar disturbance amplitude reducing the amount of energy input required for flow control.

Changes in flat plate wake characteristics obtained with decreasing plate thickness

The near and very near wake of a flat plate with a circular trailing edge is investigated with data from direct numerical simulations. Computations were performed for four different Reynolds numbers based on plate thickness (D) and at constant plate length. The value of θ/D varies by a factor of approximately 20 in the computations (θ being the boundary layer momentum thickness at the trailing edge). The separating boundary layers are turbulent in all the cases. One objective of the study is to understand the changes in wake characteristics with changing θ/D (as obtained by decreasing D). Vortex shedding is vigorous in the low θ/D cases with a substantial decrease in shedding intensity in the largest θ/D case (for all practical purposes shedding becomes almost intermittent). Other characteristics that are significantly altered with increasing θ/D are the roll-up of the detached shear layers and the magnitude of fluctuations in shedding period. These effects are explored in depth. The effects of changing θ/D on the distributions of the time-averaged, near-wake velocity statistics are discussed.

A Relatively Simple Integral Method for Turbulent Flow Over Rough Surfaces

The integral form of the equation for x momentum is solved for the skin friction coefficient, in external thin boundary layer flow, on surfaces whose technical roughness elements' size is given. This is done by using a “roughness depression function” in the law of the wall and wake which serves as the needed velocity profile. The method uses the equivalent sand grain size concept in its calculations. Predictions are made of the friction coefficient, Cf, as a function of momentum thickness Reynolds number and also, of Cf's dependence on the ratio of momentum thickness to the size of the technical (actual) roughness elements. In addition, boundary layer thicknesses and velocity profiles on rough surfaces are calculated and, when available, comparisons are made with the experimental data from a number of sources in the literature. Also, comparisons are made with the results of another major predictive scheme which does not use the equivalent sand grain concept.

Experimental investigation of Görtler vortices in hypersonic ramp flows

A sharp leading-edge ramp model with 15°, 20°, and 25° ramp angles is experimentally investigated at a freestream Mach number 7.7 and a unit Reynolds number 4.2 × 106 m−1 in the Aachen Shock Tunnel TH2. The objective of this paper is to analyze Gortler vortices in terms of shear layer length, flow curvature, and vortex related parameters. The spanwise heat flux variation caused by it and the streamwise heat flux enhancement during its tenure are also studied. Thermocouples, pressure sensors, schlieren imagery, and infrared imaging are used for the investigation. The boundary layer on the flat plate until separation is laminar. An important outcome is that the arc length at reattachment is constant irrespective of the ramp angle. Characteristic boundary layer thicknesses at different Mach number show that the momentum thickness is insensitive to compressibility and is used to determine the Gortler number. A model is presented to determine the Gortler number in terms of ramp angle and a constant based on separation angle, arc length, momentum thickness, and Reynolds number. The half of the vortex wavelength is equal to the boundary layer thickness just before reattachment. The length scale required for breakdown of Gortler vortices decreases with rising ramp angle and is analogous to peak heating length. The streamwise heat flux enhancement occurs during the tenure of Gortler vortices and the enhancement rises with ramp angle. Although the visibility of Gortler vortices through temperature variation is distinct, the spanwise heat flux variation is not too high. Moreover, the spanwise heat flux variation rises marginally with ramp angle.

Onset of global instability in low-density jets

In low-density axisymmetric jets, the onset of global instability is known to depend on three control parameters, namely the jet-to-ambient density ratio $S$, the initial momentum thickness $\unicode[STIX]{x1D703}_{0}$ and the Reynolds number $Re$. For sufficiently low values of $S$ and $\unicode[STIX]{x1D703}_{0}$, these jets bifurcate from a steady state (a fixed point) to a self-excited oscillatory state (a limit cycle) when $Re$ increases above a critical value corresponding to the Hopf point, $Re_{H}$. In the literature, this Hopf bifurcation is often regarded as supercritical. In this experimental study, however, we find that under some conditions, there exists a hysteretic bistable region at $Re_{SN}<Re<Re_{H}$, where $Re_{SN}$ denotes a saddle-node point. This shows that, contrary to expectations, the Hopf bifurcation can also be subcritical, which we explore by evaluating the coefficients of a truncated Landau model. The existence of subcritical bifurcations implies the potential for triggering and the need for weakly nonlinear analyses to be performed to at least fifth order if one is to be able to predict saturation and bistability. We conclude by proposing a universal scaling for $Re_{H}$ in terms of $S$ and $\unicode[STIX]{x1D703}_{0}$. This scaling, which is insensitive to the super/subcritical nature of the bifurcations, can be used to predict the onset of self-excited oscillations, providing further evidence to support Hallberg & Strykowski’s concept (J. Fluid Mech., vol. 569, 2006, pp. 493–507) of universal global modes in low-density jets.

Effect of plasma actuator in boundary layer on flat plate model with turbulent promoter

This paper shows experimental results for velocity measurement in the boundary layer with the use of a flat plate model. The flat plate model is disrupted with a wire trip and the effect of the plasma actuator to alter the flow in the boundary layer is then observed. The purpose of this research is to characterize the performance of the plasma actuator in a no-flow condition and with the use of a 2 m/s flow and also to theoretically analyze the performance of actuator in the boundary layer namely, displacement thickness, momentum thickness, and energy thickness. This is all done to acquire a deeper understanding of the capabilities of plasma actuator as one of the alternative active flow control equipment and to increase the effect of aerodynamic drag reduction. One of the ways to decrease the aerodynamic drag is to manipulate the flow to have a low boundary layer thickness value in order to prevent an adverse pressure gradient from happening, which then may lead to the formation of a flow separation. From experimental results, it is known that plasma actuator could decrease the thickness of the boundary layer by 9 mm.

An Assessment of the Two-Layer Quasi-Laminar Theory of Relaminarization Through Recent High-Re Accelerated Turbulent Boundary Layer Experiments

The phenomenon of relaminarization is observed in many flow situations, including that of an initially turbulent boundary layer (TBL) subjected to strong favorable pressure gradients (FPG). As several experiments on relaminarizing flows have indicated, TBLs subjected to high pressure gradients do not follow the universal log-law, and (for this and other reasons) the prediction of boundary layer (BL) parameters using current turbulence models has not been successful. However, a quasi-laminar theory (QLT; proposed in 1973), based on a two-layer model to explain the later stages of relaminarization, showed good agreement with the experimental data available at that time. These data were mostly at relatively low Re and hence left the precise role of viscosity undefined. QLT, therefore, could not be assessed at high-Re. Recent experiments, however, have provided more comprehensive data and extended the Reynolds number range to nearly 5 × 103 in momentum thickness. These data provide a basis for a reassessment of QLT, which is revisited here with an improved predictive code. It is demonstrated that even for these high-Re flows subjected to high acceleration, QLT provides good agreement with experimental results, and therefore, has the potential to substitute for Reynolds-averaged Navier–Stokes (RANS) simulations in high FPG regions.

Extremal-point density of scaling processes: From fractional Brownian motion to turbulence in one dimension

In recent years several local extrema-based methodologies have been proposed to investigate either the nonlinear or the nonstationary time series for scaling analysis. In the present work, we study systematically the distribution of the local extrema for both synthesized scaling processes and turbulent velocity data from experiments. The results show that for the fractional Brownian motion (fBm) without intermittency correction the measured extremal-point-density (EPD) agrees well with a theoretical prediction. For a multifractal random walk (MRW) with the lognormal statistics, the measured EPD is independent of the intermittency parameter μ, suggesting that the intermittency correction does not change the distribution of extremal points but changes the amplitude. By introducing a coarse-grained operator, the power-law behavior of these scaling processes is then revealed via the measured EPD for different scales. For fBm the scaling exponent ξ(H) is found to be ξ(H)=H, where H is Hurst number, while for MRW ξ(μ) shows a linear relation with the intermittency parameter μ. Such EPD approach is further applied to the turbulent velocity data obtained from a wind tunnel flow experiment with the Taylor scale λ-based Reynolds number Re_{λ}=720, and a turbulent boundary layer with the momentum thickness θ based Reynolds number Re_{θ}=810. A scaling exponent ξ≃0.37 is retrieved for the former case. For the latter one, the measured EPD shows clearly four regimes, which agrees well with the corresponding sublayer structures inside the turbulent boundary layer.

The generation of turbulent inflow boundary conditions using precursor channel flow simulations

The use of a precursor simulation of fully developed turbulent channel flow for the generation of turbulent boundary layer (TBL) inflow data is investigated. Based on the desired properties of the TBL, a complete procedure is described for how to specify the precursor simulation. The key feature of the specification is to match the momentum thickness of the precursor to that of the inflow TBL. The inflow data is then constructed from time- and space-dependent flow data in a cross-plane of the precursor. The proposed procedure removes the need to rescale the flow data and thus violate the governing equations, as is common practice in other state-of-the-art inflow generation methods for TBLs. The adaption length of the generated TBL is investigated using wall-resolved large-eddy simulation (WRLES) for a zero-pressure gradient (ZPG-) TBL, with a momentum thickness Reynolds number in the interval 830–2 400. The results are compared with a solution obtained using a standard rescaling procedure for the inflow data. The adaption length is shown to be similar for the two methods. Practical differences and advantages of the proposed of method, as compared to other inflow generation techniques, are assessed. These involve the role of the auxiliary simulation, channel flow in the present case, in the overall computational procedure, as well as data handling, initial transients and adaption lengths.

On the identification of well-behaved turbulent boundary layers

This paper introduces a new method based on the diagnostic plot (Alfredsson et al., Phys. Fluids, vol. 23, 2011, 041702) to assess the convergence towards a well-behaved zero-pressure-gradient (ZPG) turbulent boundary layer (TBL). The most popular and well-understood methods to assess the convergence towards a well-behaved state rely on empirical skin-friction curves (requiring accurate skin-friction measurements), shape-factor curves (requiring full velocity profile measurements with an accurate wall position determination) or wake-parameter curves (requiring both of the previous quantities). On the other hand, the proposed diagnostic-plot method only needs measurements of mean and fluctuating velocities in the outer region of the boundary layer at arbitrary wall-normal positions. To test the method, six tripping configurations, including optimal set-ups as well as both under- and overtripped cases, are used to quantify the convergence of ZPG TBLs towards well-behaved conditions in the Reynolds-number range covered by recent high-fidelity direct numerical simulation data up to a Reynolds number based on the momentum thickness and free-stream velocity $Re_{\unicode[STIX]{x1D703}}$ of approximately 4000 (corresponding to 2.5 m from the leading edge) in a wind-tunnel experiment. Additionally, recent high-Reynolds-number data sets have been employed to validate the method. The results show that weak tripping configurations lead to deviations in the mean flow and the velocity fluctuations within the logarithmic region with respect to optimally tripped boundary layers. On the other hand, a strong trip leads to a more energized outer region, manifested in the emergence of an outer peak in the velocity-fluctuation profile and in a more prominent wake region. While established criteria based on skin-friction and shape-factor correlations yield generally equivalent results with the diagnostic-plot method in terms of convergence towards a well-behaved state, the proposed method has the advantage of being a practical surrogate that is a more efficient tool when designing the set-up for TBL experiments, since it diagnoses the state of the boundary layer without the need to perform extensive velocity profile measurements.

Constrained large-eddy simulation of supersonic turbulent boundary layer over a compression ramp

ABSTRACTA supersonic turbulent boundary layer over a compression ramp is numerically investigated using the constrained large-eddy simulation (CLES) method. The compression corner is characterised by a deflection angle of 24°. The free-stream Mach number is Ma∞ = 2.9, and the Reynolds number based on the momentum thickness of inlet boundary layer is Reθ = 2300. The mean and statistical quantities, such as mean velocity, wall pressure and Reynolds stresses, are thoroughly analysed and compared with those from traditional large-eddy simulation (LES), experimental measurement and direct numerical simulation (DNS). It turns out that CLES can predict the friction coefficient, wall-pressure distribution, size of separation bubble, Reynolds stresses, etc. more accurately than traditional LES, and the results are in reasonable agreement with the experimental and/or DNS data. Also discussed are the effects of specific parameterisations of the Reynolds constraint and interfacial positions separating the constrained and unconstrained regions on the performance of the CLES method.

Measurement and modelling of the turbulent boundary layer near the attachment line of a swept wing

ABSTRACTThis work is motivated by the need for low-order aerodynamic models to predict accurately the effect on profile drag of controlling attachment line transition. Head's entrainment method(1), a rapid integral boundary layer technique used for design studies on swept wings, suffers from the governing swept-tapered turbulent integral boundary layer equations being ill-posed in the vicinity of the attachment line. This singularity has been treated using crude extrapolations of the attachment-line similarity solution for over half a century, but this approach is unlikely to deliver accurate predictions of the effect of changes in the attachment line flow on profile drag. An experimental study has been carried out to explore the nature of the turbulent flow in the vicinity of a highly swept swept attachment line and has revealed a quite complex, non-monotonic development of the momentum thickness in this region. It has also revealed lower levels of twist in the boundary layer velocity profiles than anticipated from the highly curved character of the inviscid flow streamlines. These observations have prompted an alternative approach to the modelling of the flow in this region which not only successfully eliminates the lack of robustness in the swept-tapered equations but which also matches the experimental results to within ±5%.

Adiabatic Film Cooling Effectiveness Measurements Throughout Multirow Film Cooling Arrays

Adiabatic film cooling effectiveness measurements are obtained using pressure-sensitive paint (PSP) on a flat film cooled surface. The effects of blowing ratio and hole spacing are investigated for four multirow arrays comprised of eight rows containing 52 holes of 3.8 mm diameter with 20 deg inclination angles and hole length-to-diameter ratio of 11.2. The four arrays investigated have two different hole-to-hole spacings composed of cylindrical and diffuser holes. For the first case, lateral and streamwise pitches are 7.5 times the diameter. For the second case, pitch-to-diameter ratio is 14 in lateral direction and 10 in the streamwise direction. The holes are in a staggered arrangement. Adiabatic effectiveness measurements are taken for a blowing ratio range of 0.3–1.2 and a density ratio of 1.5, with CO2 injected as the coolant. A thorough boundary layer analysis is presented, and data were taken using hotwire anemometry with air injection, with boundary layer, and turbulence measurements taken at multiple locations in order to characterize the boundary layer. Local effectiveness, laterally averaged effectiveness, boundary layer thickness, momentum thickness, turbulence intensity, and turbulence length scale are presented. For the cylindrical holes, at the first row of injection, the film jets are still attached at a blowing ratio of 0.3. By a blowing ratio of 0.5, the jet is observed to lift off, and then impinge back onto the test surface. At a blowing ratio of 1.2, the jets lift off, but reattach much further downstream, spreading the coolant further along the test surface. A thorough uncertainty analysis has been conducted in order to fully understand the presented measurements and any shortcomings of the measurement technique. The maximum uncertainty of effectiveness and blowing ratio is 0.02 counts of effectiveness and 3%, respectively.

Numerical Investigation of Shock-Train Response to Inflow Boundary-Layer Variations

A dataset of normal shock trains in a rectangular cross-section channel has been created from direct numerical simulations in an effort to quantify the impact of inflow confinement ratio on the shock-train structure. To this end, the inlet boundary-layer momentum thickness was varied while the bulk inflow and outflow conditions remained constant. The simulations show that the shock train is displaced upstream as the inflow confinement ratio increases. Also, an increase in boundary-layer momentum thickness results in a reduction of the normal-like portion of the lambda-shock structures in the channel core. This leads to more numerous but weaker bifurcating shocks as well as an increase of the shock-train length. When the inflow boundary-layer thickness is varied temporally, the complex shock-train response depends on the excitation frequency. A resonant frequency is observed when different components of the shock train exhibit the highest amplification in terms of pressure jumps. Further, the domain upstream of the leading shock acts as a low-pass filter, which has the end result of limiting the axial shock-train motion. Nevertheless, even a small oscillation in boundary-layer momentum thickness of 0.45 mm (0.6% of the channel height) is seen to increase the shock-train length by two orders of magnitude (4 cm).

Wall shear stress fluctuations: Mixed scaling and their effects on velocity fluctuations in a turbulent boundary layer

The present work investigates numerically the statistics of the wall shear stress fluctuations in a turbulent boundary layer (TBL) and their relation to the velocity fluctuations outside of the near-wall region. The flow data are obtained from a Direct Numerical Simulation (DNS) of a zero pressure-gradient TBL using the high-order flow solver Incompact3D [S. Laizet and E. Lamballais, “High-order compact schemes for incompressible flows: A simple and efficient method with quasi-spectral accuracy,” J. Comput. Phys. 228(16), 5989 (2009)]. The maximum Reynolds number of the simulation is Re𝜃≈2000, based on the free-stream velocity and the momentum thickness of the boundary layer. The simulation data suggest that the root-mean-squared fluctuations of the streamwise and spanwise wall shear-stress components τx and τz follow a logarithmic dependence on the Reynolds number, consistent with the empirical correlation of Örlü and Schlatter [R. Örlü and P. Schlatter, “On the fluctuating wall-shear stress in zero pressure-gradient turbulent boundary layer flows,” Phys. Fluids 23, 021704 (2011)]. These functional dependencies can be used to estimate the Reynolds number dependence of the wall turbulence dissipation rate in good agreement with reference DNS data. Our results suggest that the rare negative events of τx can be associated with the extreme values of τz and are related to the presence of coherent structures in the buffer layer, mainly quasi-streamwise vortices. We also develop a theoretical model, based on a generalisation of the Townsend-Perry hypothesis of wall-attached eddies, to link the statistical moments of the filtered wall shear stress fluctuations and the second order structure function of fluctuating velocities at a distance y from the wall. This model suggests that the wall shear stress fluctuations may induce a higher slope in the turbulence energy spectra of streamwise velocities than the one predicted by the Townsend-Perry attached-eddy model.

Direct numerical simulation of turbulent boundary layer over a compliant wall

The performance of compliant wall with different parameters in a spatially developing turbulent boundary layer is investigated by direct numerical simulation. The Reynolds number based on the external flow velocity and the momentum thickness ranges from 300 to approximately 860. The compliant wall is placed in the region starting from 200 to 600 in the streamwise direction, scaled by the inlet momentum thickness. The monoharmonic analysis is carried out for selection of the parameters of compliant wall in numerical simulations. The instantaneous flow fields and the turbulence statistics are presented to show the impact of the compliant wall. A dynamical decomposition of the skin-friction coefficient Cf is performed for examining the sources bringing the variations in Cf. The spatial evolution of compliant wall motion is disclosed, which is a special feature of the boundary layer flow. The spatial–temporal self-correlation of the wall displacement is calculated for obtaining the convection velocity of the wavy motion of compliant wall. The frequency–wavenumber power spectrum of wall pressure fluctuation is presented for studying the interaction of the compliant wall and the above turbulent flow. The effect of the fixed boundaries of the compliant wall is also examined.