Distance-from-the-wall scaling is widely understood to play a substantial role in shaping both the kinematics and mean dynamics of wall-bounded flows. Building models around such observations, however, requires a precise understanding of which quantities are affected, where in physical/scale space they are affected, and how the influence of wall-distance is manifested. In this study, we examine the scale-by-scale distribution of Reynolds normal and shear stresses and their dependence on wall-distance in order to better understand the emergence of distance-from-the-wall scaling. Coherence and phase distributions are considered in addition to auto- and co-spectral energy densities, as these reveal additional details regarding wall-distance scaling. It is found that the streamwise and wall-normal velocities are most coherent at zero phase lag, and that the scale at which this balance occurs is proportional to distance from the wall. Finally, results from a wavelet decomposition are shown to evaluate the degree to which self-similarity observed in a mean sense is also observed instantaneously.

A combined experimental and direct numerical simulation (DNS) investigation is undertaken to study the laminar boundary-layer (BL) flow adjacent to a melting vertical ice face at two far-field water salinities ( $S_\infty =0$ and 34 parts per thousand) and a range of far-field temperatures ( $T_\infty$ ). Wall-normal distributions of vertical velocity and temperature within the BL are measured by a modified molecular tagging velocimetry and thermometry technique. Experimental data match with DNS only when a nonlinear equation of state (EoS) for density is used rather than a linear EoS. For all $S_\infty =0$ , i.e. freshwater cases, the flow remains uni-directional, although the flow reverses direction at $T_\infty =4<^>{\,\circ} \text{C}$ . A bi-directional flow, however, exists for $S_\infty =$ 34 g kg-1, where an inner salinity-driven upward flow of fresher water is accompanied by a downward-flowing temperature-driven outer flow. Although the contribution of temperature to density relative to salinity is small $({\approx}1/40)$ , the thermal BL region is larger owing to higher diffusivity. This results in increased total buoyancy force when the buoyancy is integrated across the BL, which combined with effects of wall shear stress on salinity BL and a freer thermal BL growth reveals that buoyancy from temperature contributes almost equally to the overall flow. Melt rates ( $V$ ) also show differing features in uni- and bi-directional flows. The uni-directional flows exhibit the standard scaling of increasing velocity magnitude and BL thickness, and decreasing $V$ with distance along the flow direction. Such scalings are not followed in the bi-directional flows. These show a more uniform $V$ with height, which is attributed to the counteracting effects of an upward-growing salinity BL and a downward-growing temperature BL, combined with the necessity of maintaining salinity and temperature flux balance at the ice-water interface.

This study investigates necklace-vortex systems forming when a laminar shear-wake, generated by two streams merging at the trailing edge of a splitter plate, interacts with a circular cylinder placed downstream in the wake. Hydrogen-bubble flow visualisations were employed in a water channel capable of producing laminar shear-wake flows. In the absence of the cylinder, oppositely signed vorticity in the shear-wake undergoes mutual annihilation. The introduction of the cylinder interrupts this evolution, promoting off-wall flow separation upstream of the cylinder and vortex roll-up. The study primarily focuses on two non-dimensional parameters, the Reynolds number $ extit{Re}_m$ and the shear ratio $ extit{SR}$ , and presents a mapping of the observed vortex regimes. Increasing $ extit{Re}_m$ promotes either the formation of additional vortices or unsteadiness. Increasing $ extit{SR}$ generally suppresses vortex formation or attenuates unsteadiness, except near $ extit{SR}\approx 0$ at low to moderate $ extit{Re}_m$ , where the two-vortex system is unstable to additional vortex generation. Observed configurations range from no-vortex states to one- or two-vortex systems at low Reynolds numbers, and to three-, four- and five-vortex systems at larger Reynolds numbers, with unsteadiness becoming prominent beyond the three-vortex regime and predominant in four- and five-vortex systems. Beyond regime mapping, we delve into the structure of a steady two- and three-vortex system at low to moderate $ extit{Re}_m$ . This provides insights into the emergence and evolution of the vortex system, which is analysed in the context of the vorticity-transport equations.

The South Andes Mountain range is a global hotspot for stratospheric gravity waves. Here, we identify two selected case study flows over the South Andes, using data from the fifth generation ECMWF atmospheric reanalysis (ERA5) reanalysis, one featuring extensive vertical and downstream wave propagation with a characteristic cellular pattern and the other with little to no wave propagation. Core features of these two flow fields are identified and shown to correspond to different analytical solutions to the linear Taylor-Goldstein equation: one with vertical wave propagation (with and without rotation) and one with trapped waves propagating downstream. The importance of rotation is emphasized for its ability to suppress mountain waves at low altitudes, and wave trapping is identified as potentially contributing to the downstream propagation of waves in the stratosphere for one of the flow snapshots. The dimensionless number y, the ratio of the height scale of the Scorer parameter and the mountain's horizontal length scale, is highlighted as playing a critical role in the behavior of trapped waves in multiple analytical models.

A low-friction-Reynolds-number (Re tau approximate to 400-480) turbulent boundary layer perturbed by additional streamwise vortices is investigated using wall-resolved large-eddy simulation. To better understand how one might passively mimic and/or manipulate the formation of streamwise vortex pairs (SVPs) that naturally occur as part of the near-wall cycle, SVPs are artificially produced by small vortex generators (of height h+ = 30) within the near-wall region. Thus, the present study contrasts the more typical large vortex generators used in turbulent boundary layer manipulation and separation control. Analyses employing a triple decomposition of velocity and pressure and its extension to the incompressible Navier-Stokes equation is used to better understand the coherent and turbulence fields associated with the synthetic and naturally occurring SVPs. Overall, it is observed that the near-wall synthetic and natural streamwise vortices have similar interactions with the surrounding turbulent field, and exhibit similarities in statistical structure. In particular, it is found that the synthetic SVPs exhibit a remarkably similar signature of kinetic energy transport with their natural counterparts. Through the comparisons with the ensembleaveraged natural SVPs, it is found that the natural and synthetic SVPs are identified to own similar spatial scales and evolution characteristics such that when they are inner scaled they exhibit similar magnitudes at comparable streamwise stations. This indicates that the wall response to SVPs may in fact constitute a generic mechanism underlying turbulent transport near the wall. These similarities suggest that embedded small synthetic streamwise vortices are self-contained in the near-wall region and directly interact with the structures in this area to influence the associated turbulent transport.

A phenomenological description is presented to explain the intermediate and low-frequency/large-scale contributions to the wall-shear-stress ( ${\tau }_w$ ) and wall-pressure ( $\,{p}_w$ ) spectra of canonical turbulent boundary layers, both of which are well known to increase with Reynolds number, albeit in a distinct manner. The explanation is based on the concept of active and inactive motions (Townsend, J. Fluid Mech., vol. 11, issue 1, 1961, pp. 97–120) associated with the attached-eddy hypothesis. Unique data sets of simultaneously acquired ${\tau }_w$ , ${p}_w$ and velocity-fluctuation time series in the log region are considered, across a friction-Reynolds-number ( $Re_{\tau }$ ) range of $ {O}(10^3) \lesssim Re_{\tau } \lesssim {O}(10^6)$ . A recently proposed energy-decomposition methodology (Deshpande et al., J. Fluid Mech., vol. 914, 2021, A5) is implemented to reveal the active and inactive contributions to the ${\tau }_w$ - and $p_w$ -spectra. Empirical evidence is provided in support of Bradshaw's (J. Fluid Mech., vol. 30, issue 2, 1967, pp. 241–258) hypothesis that the inactive motions are responsible for the non-local wall-ward transport of the large-scale inertia-dominated energy, which is produced in the log region by active motions. This explains the large-scale signatures in the ${\tau }_w$ -spectrum, which grow with $Re_{\tau }$ despite the statistically weak signature of large-scale turbulence production, in the near-wall region. For wall pressure, active and inactive motions respectively contribute to the intermediate and large scales of the $p_w$ -spectrum. Both these contributions are found to increase with increasing $Re_{\tau }$ owing to the broadening and energization of the wall-scaled (attached) eddy hierarchy. This potentially explains the rapid $Re_{\tau }$ -growth of the $p_w$ -spectra relative to ${\tau }_w$ , given the dependence of the latter only on the inactive contributions.

Learning accurate numerical constants when developing algebraic models is a known challenge for evolutionary algorithms, such as Gene Expression Programming (GEP). This paper introduces the concept of adaptive symbols to the GEP framework by Weatheritt and Sandberg (2016) to develop advanced physics closure models. Adaptive symbols utilize gradient information to learn locally optimal numerical constants during model training, for which we investigate two types of nonlinear optimization algorithms. The second contribution of this work is implementing two regularization techniques to incentivize the development of implementable and interpretable closure models. We apply $L_2$ regularization to ensure small magnitude numerical constants and devise a novel complexity metric that supports the development of low complexity models via custom symbol complexities and multi-objective optimization. This extended framework is employed to four use cases, namely rediscovering Sutherland's viscosity law, developing laminar flame speed combustion models and training two types of fluid dynamics turbulence models. The model prediction accuracy and the convergence speed of training are improved significantly across all of the more and less complex use cases, respectively. The two regularization methods are essential for developing implementable closure models and we demonstrate that the developed turbulence models substantially improve simulations over state-of-the-art models.

This paper reports on the recent NATO Advanced Vehicle Technology (AVT) effort associated with smooth-wall two-dimensional turbulent boundary layer flows subjected to streamwise pressure gradients. The effort considered experiments, Reynolds Averaged Navier-Stokes (RANS) simulations, modelling, scaling and flow physics relative to the subject flows. Special attention is given to the current predictive capabilities and deficiencies of RANS simulations, and the interplay between experiments and RANS validation and development. In addition, the efficacy of the prediction of velocity field response and wall pressure statistics are respectively demonstrated via Resolvent and Gene Expression Programming based models. The persistence of a logarithmic mean velocity profile is evaluated and measures of non-equilibrium are described and discussed. A number of open issues are described and recommendations for future research are suggested.

Single-component molecular tagging velocimetry (1c-MTV) experiments are conducted in an oscillating grid turbulence facility to systematically clarify the trade-offs between maximizing measurement dynamic range and minimizing measurement uncertainty. The primary aim is to obtain reliable turbulence data with the maximum possible vector resolution ( x_1 ). Four optical magnifications ( M_0 =0.11, 0.22, 0.45, 1.11) with four interframe time delay ( t ) values, for each optical magnification case, ranging from 3 to 10 ms, are investigated. The grid oscillates with a frequency (f) of 3.2 and 4.1 Hz, and a single fixed stroke length (S) of 55 mm. The measurement quality is quantified using three important turbulence descriptors — the second-order transverse velocity correlation function (⟨g(r)⟩ ) , the second-order transverse velocity structure function ([Δ_gu]^2) , and the turbulence energy dissipation rate ( ⟨ϵ⟩ ). Other descriptors such as the longitudinal integral length scale (l) and the longitudinal Taylor microscale ( λ ) are also used to compare with reference values reported in literature. The degree of agreement with the reference values from literature, and insensitivity of the descriptor estimates to different MTV design parameters are used to determine the most robust means of obtaining high-resolution turbulence data from 1c-MTV. The experimental design parameters are contextualized using the Kolmogorov length ( η ), time ( τ _η ), and velocity ( u_η ) scales that are based on the most reliable ⟨ϵ⟩ estimates from the current experiments. The pixel-displacement corresponding to u_η (x_2_η) describes the interdependencies between M_0 and t . The data set with M_0=0.45 ( η =18x_1 for f=3.2 Hz, and η =13x_1 for f=4.1 Hz) represents the most reliable and the highest resolution case. This conclusion was deduced by taking the performance of all the turbulence descriptors into account. As far as the interframe time delay is concerned, t≥ 4 ms ( x_2_η≥ 0.5 pixels ) works well for all M_0 . To the authors’ knowledge, the present study documents the first instance of a series of 1c-MTV experiments conducted for a systematic clarification of the nature of balancing the available dynamic range and the measurement uncertainty. Additionally, since the study involves turbulent flow with negligible mean-flow, it serves as an adequate representation of 1c-MTV performance in a three-dimensional flow.

Utilizing the stress balance and a wall shear stress and pressure based velocity scale, u_hyb , introduced in Romero et al. (2022) non-equilibrium effects in adverse pressure gradient turbulent boundary layers are studied. Data from canonical flows, i.e. zero pressure gradient turbulent boundary layers, are used to provide context for this adverse pressure gradient turbulent boundary layer analysis. Adverse pressure gradient flows at varying Clauser pressure-gradient parameter β are studied. This parameter can be defined as the ratio of time-scales, β = t_Δ /t_PG , characteristic of the turbulence and pressure gradient. Here, t_Δ = Δ / u_τ , where Δ = (U_∞ /u_τ )δ ^* is the Rotta-Clauser thickness, δ ^* is the displacement thickness, and u_τ is the friction velocity. t_PG = -(dU_∞ /dx)^-1 , where U_∞ is the free-stream velocity. For the adverse pressure gradient flow to achieve self-similarity, β , the ratio of two time-scales, has to be a constant. Therefore, under a non-constant β the flow takes time to reach self-similarity and history effects become apparent. From studying non-constant β cases, this study hypothesizes a link between the cross-over from equilibrium to non-equilibrium with a corresponding qualitative change to the internal leading order stress balance.

Experiments at relatively large Reynolds number were conducted to better understand the influence of an adverse pressure gradient on the turbulence in two-dimensional boundary layer flows. One focus is on documenting the scale contributions to the Reynolds normal and shear stress profiles under the influence of a positive streamwise pressure gradient. Toward these aims, velocity spectra at friction Reynolds numbers (7100≤Reτ≤8600) are presented, analyzed, and compared to a zero pressure gradient case from the same facility at nominally matching Reynolds number. Distance-from-the-wall scaling is central to numerous theoretical and modeling considerations of the canonical flat plate flow. Consistently, the present spectral measurements reveal clear evidence for its preservation on the inertial sublayer under modest adverse pressure gradients. Increases are seen in the viscous-scaled premultiplied spectra in the outer region as the pressure gradient increases — commensurate with the increases observed in the outer peaks of the streamwise and wall-normal velocity variances, as well as the Reynolds shear stress. Effects of varying Reτ on the streamwise and wall-normal velocity variances and the Reynolds shear stress are studied by comparing the present data at varying Clauser pressure gradient parameter, β, with previous experiments having similarly varying β at significantly lower Reτ (≈1900). This allows to study the effects of changing Reynolds number at matched β. Cases at similar β and Reynolds number, but with varying flow history are also examined, wherein the same β value is arrived either by starting from a smaller or larger initial β. The effects of β>0 on the large and small scale motions are investigated by segregating the signal contributions according to a cut-off wavelength established in previous zero pressure gradient flows. Under a normalization that uses a hybrid velocity scale this analysis reveals similarity between the outer regions of the spectrally-decomposed variances and Reynolds shear stress. These analyses also reveal a more dramatic reduction in the near-wall large scale contribution to the Reynolds shear stress gradient in the β>0 flow when compared to the β=0 flow.

In their 2016 paper, Wei and Klewicki [Phys. Rev. Fluids 1, 082401 (2016)] developed an integral relation, UeVe/u2 tau = H12, which connects key parameters in zero-pressure-gradient (ZPG) boundary layer flows: mean velocity components Ue and Ve at the boundary layer edge, and friction velocity u tau to shape factor H12. While this relation holds exactly for ZPG laminar boundary layers featuring self-similar streamwise velocity profiles, it is an approximation for ZPG turbulent boundary layers (TBLs), with its accuracy improving as the Reynolds number increases. In this paper, we present a correction to the original integral relation, providing an exact integral relation that is applicable to ZPG boundary layer flows at arbitrary Reynolds numbers. The correction comprises two terms: one addressing deviations from self-similarity in mean streamwise velocity, and the other considering the impact of Reynolds normal stresses. Experimental and numerical data are shown to support the relative insignificance of the newly identified correction terms, except for Reynolds numbers in the transitional regime.

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.

This paper provides a framework for estimating appropriate turbulent velocity scales in adverse pressure gradient turbulent boundary layers (APG TBLs) via a study of the mean stress balance. We examine the velocity scales of APG TBLs using the relationship between the Reynolds shear stress and pressure stress. It is reasoned that as distance from the wall increases the velocity scaling transitions from one dominated by the wall-shear stress velocity scale, u(tau), to a scaling dominated by the pressure stress. A velocity scale, u(hyb), is proposed that varies with distance from the wall and combines the wall-shear-stress velocity with a pressure-stress-based velocity. This investigation uses new high Reynolds number (7000(sic)Re-tau(sic)7800) experimental measurements, existing lower Reynolds number experimental (600(sic)Re tau(sic)2000) and computational (Re-tau < 700) data sets. The proposed velocity scale realizes similarity in the turbulent stress profiles to a degree that is superior to that achievable via any wall-distance-independent velocity scale when considering the full extent of the flow domain.

Enormous efforts have been devoted to the prediction and control of turbulent wall flows. A primary consideration here is the Reynolds number scaling problem in which non-dimensional representations are sought that render the normalized variables of interest unchanged for varying Reynolds number. Relative to this objective and in contrast to the mean velocity, there remains a considerable lack of clarity associated with the apparent failure of inner normalization (i.e. using the friction velocity and kinematic viscosity) when applied to the statistical profiles of fluctuating quantities in the near-wall region. In their present work Chen & Sreenivasan (J. Fluid Mech., vol. 933, 2022, A20) generalize their earlier effort, Chen & Sreenivasan (J. Fluid Mech., vol. 908, 2021, R3), and present a rational framework for characterizing and describing the evolution of turbulence quantities that either attain a near-wall peak or have non-zero wall values. Their analysis enjoys considerable empirical support. Physically, the asymptotic boundedness of the inner-normalized dissipation is used to reason that there is a limiting state of near-wall turbulence at asymptotically large Reynolds numbers. The law of bounded dissipation arguments put forth by Chen and Sreenivasan prescribe the recovery of inner scaling and suggest new possibilities regarding the physics of how wall turbulence matures to its asymptotic state.

This paper introduces two novel concepts in data-driven turbulence modeling that enable the simultaneous development of multiple closure models and the training towards multiple objectives. The concepts extend the evolutionary framework by Weatheritt and Sandberg (2016), which derives interpretable and implementation-ready expressions from high-fidelity simulation data. By assigning a shared fitness value to the evolved closure models and utilizing the CFD-driven training approach by Zhao et al. (2020), the multi-expression training concept introduced here is able to account for the coupling between the trained models, i.e. Reynolds stress anisotropy, turbulent heat flux and turbulence production correction models. As a second concept, a multi-objective optimization algorithm is applied to the framework. The extension yields a diverse set of candidate models and allows a trade-off between the training objectives after analyzing the training results. In this study, the novel concepts are applied to a benchmark periodic hills case and a vertical natural convection flow. The predictions of mean flow quantities are improved compared to decoupled training strategies with distinct and robust improvements for strongly coupled momentum and thermal fields. The coupled training of closure models and the balancing of multiple training objectives are considered important capabilities on the path towards generalized data-driven turbulence models.

Finite Reynolds number behaviors of the asymptotically logarithmic mean velocity profile in fully developed turbulent channel flow are investigated. The scaling patch method of Fife et al. [“Multiscaling in the presence of indeterminacy: Wall-induced turbulence,” Multiscale Model. Simul. 4, 936 (2005)] is used to reveal invariance properties admitted by the appropriately simplified form of the mean momentum equation. These properties underlie the existence of a similarity solution to this equation over an interior inertial domain. The classical logarithmic mean velocity profile equation emerges from this similarity solution as the Reynolds number becomes large. Originally demonstrated via numerical integration, it is now shown that the solution to the governing nonlinear equation can be found by straight-forward analytical integration. The resulting solution contains both linear and logarithmic terms, but with the coefficient on the linear term decaying to zero as the Reynolds number tends to infinity. In this way, the universality of the classical logarithmic law comports with the existence of an invariant form of the mean momentum equation and is accordingly described by the present similarity solution. Existing numerical simulation data are used to elucidate Reynolds number dependent properties of the finite Reynolds number form of the similarity solution. Correspondences between these properties and those indicated by finite Reynolds number corrections to the classical overlap layer formulation for the mean velocity profile are described and discussed.

The focus of the present work is to characterize the features of the turbulent inertia term (the wall-normal gradient of Reynolds shear stress) through the mean momentum balance and the Reynolds shear stress correlation coefficient ( ρ _uv ). Effects of the Reynolds number and Clauser pressure-gradient parameter, β , are discussed. Large eddy simulations of low Reynolds number adverse pressure gradient turbulent boundary layers from Bobke et al. [1], low Reynolds number experimental data from Vila et al. [2] and Volino [3], and newly acquired experimental data at higher Reynolds number from the Flow Physics Facility at The University of New Hampshire are utilized for this analysis. Observations are compared to zero pressure gradient turbulent boundary layer direct numerical simulations of Schlatter and Örlu [4] and Sillero et al. [5], and experimental data from Zimmerman et al. [6] and Zimmerman [7]. These cases show that the correlation coefficient ( ρ _uv ) decreases in magnitude with increasing Reynolds number and β . However, from these initial observations we find that ρ _uv is more sensitive to changes in the Reynolds number in comparison to the examined range of β . We also find that the location of zero-crossing of the turbulent inertia term seems to scale with √(δ ^+) while the minimum of ρ _uv scales with δ .

Gravity currents produced by a lock-exchange flow are studied using high-resolution molecular tagging techniques. Instead of employing salt to produce density stratification, an initial temperature difference is introduced in the system to generate the ensuing gravity currents. The experiments focus on the interface between the hot and cold fluids to characterize the resultant mixing across the interface. The present measurements spatially resolve the flow to smaller than the Kolmogorov scale and close to the Batchelor scale. This enables reasonably accurate estimates of velocity and density gradients. The measured density (temperature) distribution allowed estimation of the background potential energy of the flow that is used to quantify mixing. These measurements yield a mixing efficiency of about 0.13 with a standard deviation of 0.05 for the present Reynolds number range [Re≤O(104)]. An analysis combining flow visualization and quantitative measurements reveals that spatially local values of high mixing efficiency occur after the occurrence of certain dissipative stirring events. These events, largely associated with vortical overturns, are commonly observed near the interface between the two fluids and are a precursor to locally efficient mixing.