Alexander J. Smits

Wall-pressure fluctuations beneath turbulent boundary layers (BLs) drive noise and structural fatigue through interactions between fluid and structural modes. Conventional predictive models for the spectrum – such as the widely accepted Goody model (2004 AIAA J. , vol. 42 (9), pp. 1788–1794) – fail to capture the energetic growth in the low-frequency range that occurs at high Reynolds number, while at the same time over-predicting the variance. To address these shortcomings, two semi-empirical models are proposed for the wall-pressure spectrum in canonical turbulent BLs, pipes and channels for friction Reynolds numbers $\delta ^+$ ranging from 180 to 47 000. Consistent with the approach outlined modelling the streamwise Reynolds stress in the recent work of Gustenyov et al. (2025 J. Fluid Mech., vol. 1016, A23), the models are based on consideration of two spectral components that represent the contributions to the wall-pressure fluctuations from inner-scale motions and outer-scale motions. The first model expresses the pre-multiplied spectrum as the sum of two overlapping log-normal components: an inner-scaled term that is $\delta ^+$ -invariant and an outer-scaled term whose amplitude broadens smoothly with $\delta ^+$ . Calibrated against large-eddy simulations, direct numerical simulations and recent high- $\delta ^+$ pipe data, it reproduces the inner-scaled peak and the emergence of an outer-scaled peak at large $\delta ^+$ . The second model, developed around newly available pipe data, uses theoretical arguments to prescribe the spectral shapes of the inner and outer components. Embedding the $\delta ^+$ -dependence in smooth asymptotic functions yields a formulation that varies continuously with $\delta ^+$ and generalises beyond the calibration range. Both models capture the full spectrum and recover the observed logarithmic growth of its variance, providing a compact, physics-informed empirical representation for more accurate engineering predictions of wall-pressure fluctuations.

A model is proposed for the one-dimensional spectrum and streamwise Reynolds stress in pipe flow for arbitrarily large Reynolds numbers. Constructed in wavenumber space, the model comprises four principal contributions to the spectrum: streaks, large-scale motions, very-large-scale motions and incoherent turbulence. It accounts for the broad and overlapping spectral content of these contributions from different eddy types. The model reproduces well the broad structure of the premultiplied one-dimensional spectrum of the streamwise velocity, although the bimodal shape that has been observed at certain wall-normal locations, and the $-5/3$ slope of the inertial subrange, are not captured effectively because of the simplifications made within the model. Regardless, the Reynolds stress distribution is well reproduced, even within the near-wall region, including key features of wall-bounded flows such as the Reynolds number dependence of the inner peak, the formation of a logarithmic region, and the formation of an outer peak. These findings suggest that many of these features arise from the overlap of energy content produced by both inner- and outer-scaled eddy structures combined with the viscous-scaled influence of the wall. The model is also used to compare with canonical turbulent boundary layer and channel flows, and despite some differences being apparent, we speculate that with only minor modifications to its coefficients, the model can be adapted to these flows as well.

Direct numerical simulations are performed for turbulent forced convection in a half-channel flow with a wall oscillating either as a spanwise plane oscillation or to generate a streamwise travelling wave. The friction Reynolds number is fixed at $Re_{\tau _0} = 590$ , but the Prandtl number $Pr$ is varied from 0.71 to 20. For $Pr\gt 1$ , the heat transfer is reduced by more than the drag, 40 % compared with 30 % at $Pr=7.5$ . This outcome is related to the different responses of the velocity and thermal fields to the Stokes layer. It is shown that the Stokes layer near the wall attenuates the large-scale energy of the turbulent heat flux and the turbulent shear stress, but amplifies their small-scale energy. At higher Prandtl numbers, the thinning of the conductive sublayer means that the energetic scales of the turbulent heat flux move closer to the wall, where they are exposed to a stronger Stokes layer production, increasing the contribution of the small-scale energy amplification. A predictive model is derived for the Reynolds and Prandtl number dependence of the heat-transfer reduction based on the scaling of the thermal statistics. The model agrees well with the computations for Prandtl numbers up to 20.

A turbulent pipe flow experiment was conducted where the surface of the pipe was oscillated azimuthally over a wide range of frequencies, amplitudes, and Reynolds numbers. The drag was reduced by as much as 35%. Past work has suggested that the drag reduction scales with the velocity amplitude of the motion, its period, and/or the Reynolds number. Here, we find that the key parameter is the acceleration, which greatly simplifies the complexity of the phenomenon. This result is shown to apply to channel flows with spanwise surface oscillation as well. This insight opens potential avenues for reducing fuel consumption by large vehicles and for reducing energy costs in large piping systems.

The physics and computational prediction of turbulent boundary layer flow over axisymmetric and three-dimensional bodies are examined. Three cases were considered for which extensive experimental results and companion Reynolds-averaged Navier Stokes (RANS) solutions were obtained and/or available in the open literature. These cases all have Reynolds numbers based upon the freestream velocity and body geometric scale on the order of 105 to 106, which is large for laboratory scales but small compared to the maximum scales observed for full-scale vehicles. Despite significant differences in approach flow fields and geometries for these three cases, some common themes emerged in the findings. All cases involved complications due to pressure gradients combined with streamwise curvature, and all exhibited regions of turbulence reduction due to accelerated flow. These complications led to discrepancies in computed results even in attached flow regions where it is often assumed that RANS models provide reliable predictions. The authors recommend further work on modelling approaches that can capture rapid distortion effects on turbulence transport that can be incorporated into industry-useful frameworks. Two cases involving laterally symmetric, three-dimensional wall-mounted hills with aft-body separation revealed that asymmetric mean flow fields are likely to result. This finding has been observed in experiments conducted in multiple facilities and in computations using multiple solvers and turbulence models. It is concluded that non-unique and asymmetric global flow solutions are fundamental to flow cases with lateral geometric symmetry involving turbulent boundary layer separation. Further work is also needed to accurately predict low-frequency unsteadiness due to geometries that produce non-unique mean flow fields. For such flows, it remains to be definitively determined whether experimentally observed modes of the mean flow are equivalent, or nearly equivalent, to asymmetric mean flow solutions obtained using RANS approaches.

Nailfold capillary density is lower in patients with pulmonary arterial hypertension (PAH). It is unclear whether this observation signifies a unique systemic manifestation of PAH, or reflects microcirculatory dysfunction secondary to pulmonary hypertension (PH). Capillary density and loop dimensions were measured by nailfold-capillaroscopy (NC) in 30 PAH (23 idiopathic, or iPAH, 7 hereditary, or hPAH), 17 chronic thromboembolic PH (CTEPH) patients and 48 controls. NC-Measurements were repeated after pulmonary endarterectomy (PEA) or balloon pulmonary angioplasty (BPA) in CTEPH patients. We examined whether NC-measurements were related to markers of disease severity and predictive of time to clinical worsening (TTCW) as tested by univariate linear/logistic regression and cox-regression analysis, respectively. Capillary density was significantly lower in PAH (7.5 ± 1.1, p < 0.001) and in CTEPH (8.4 ± 1.5, p < 0.001) compared to asymptomatic controls (10.3 ± 1.0 capillaries/mm). Capillary density was similar in iPAH and hPAH and unrelated to hemodynamics in either PAH or CTEPH. A lower capillary density was predictive of clinical worsening in PAH (p 0.05). After normalization of pulmonary artery pressures by PEA or BPA, capillary density remained reduced in CTEPH patients. Capillary loop apex, capillary and venous- and arterial limb diameter were increased in patients with PAH and CTEPH compared to controls. Nailfold capillary density is reduced to a similar extent in iPAH, hPAH and CTEPH. Normalization of hemodynamics by PEA or BPA does not lead to a restoration of capillary density in CTEPH. Capillary dimensions were increased in both patients with PAH and CTEPH. Lower capillary density was predictive of clinical worsening in PAH. Our findings indicate that a loss of peripheral capillaries is not specific to PAH and is not related to the hemodynamic disturbance per se, but that shared mechanisms may account for a simultaneous development of a systemic microangiopathy and pulmonary vascular remodeling.

Many flows that are expected to be symmetric are actually observed to be asymmetric. The appearance of asymmetry in the face of no particular cause is a widespread although underappreciated occurrence. This rather puzzling and sometimes frustrating phenomenon can occur in wide-angle diffusers, over the forebody of axisymmetric bodies at high angles of attack, in the wake downstream of streamlined as well as bluff bodies, and in the flow over three-dimensional bumps and ramps. We review some notable examples and highlight the extreme sensitivity of many such flows to small disturbances in the body geometry or the incoming flow. Some flows appear to be permanently asymmetric, while others are bistable on timescales that are orders of magnitude longer than any convective timescale. Convective or global instabilities can occur, bistability is common, and mode interactions become important when multiple similar but distinct timescales and length scales are present. Our understanding of these phenomena is still very limited, and further research is urgently required; asymmetries in otherwise symmetric flows can have serious real-world consequences on vehicle control and performance.

A new wall-wake law is proposed for the streamwise turbulence in the outer region of a turbulent boundary layer. The formulation pairs the logarithmic part of the profile (with a slope $A_1$ and additive constant $B_1$ ) to an outer linear part, and it accurately describes over 95 % of the boundary layer profile at high Reynolds numbers. Once the slope $A_1$ is fixed, $B_1$ is the only free parameter determining the fit. Most importantly, $B_1$ is shown to follow the same trend with Reynolds number as the wake factor in the wall-wake law for the mean velocity, which is tied to changes in scaling of the mean flow and the turbulence that occur at low Reynolds number.

Direct numerical simulations are performed of turbulent forced convection in a half channel flow with wall oscillating either as a spanwise plane oscillation or to generate a streamwise travelling wave. The friction Reynolds number is fixed at $Re_{τ_0} = 590$, but the Prandtl number $Pr$ is varied from 0.71 to 20. For $Pr&gt;1$, the heat transfer is reduced by more than the drag, 40\% compared to 30\% at $Pr=7.5$. This outcome is related to the different responses of the velocity and thermal fields to the Stokes layer. It is shown that the Stokes layer near the wall attenuates the large-scale energy of the turbulent heat-flux and the turbulent shear-stress, but amplifies their small-scale energy. At higher Prandtl numbers, the thinning of the conductive sublayer means that the energetic scales of the turbulent heat-flux move closer to the wall, where they are exposed to a stronger Stokes layer production, increasing the contribution of the small-scale energy amplification. A predictive model is derived for the Reynolds and Prandtl number dependence of the heat-transfer reduction based on the scaling of the thermal statistics. The model agrees well with the computations for Prandtl numbers up to 20.

Abstract When acquiring quantitative data using cameras, calibration is required to establish the mapping relation between the image space and physical space. Calibration targets with known dimensions are often used, with the most popular being physical targets. In setups where physical access is a challenge, using physical targets may not be possible, and so we develop an adaptive non-intrusive calibration target capable of conducting volumetric calibrations in free space. The calibration target is formed by two intersecting laser beams traversed in the test domain. A novel algorithm is presented for accurately finding the beam intersections, even at small crossing angles. The error sources are assessed along with their scaling behavior with respect to key parameters. The performance of the calibration method is evaluated by using it to examine a test object with known dimensions.

To investigate the existence of geometrically self-similar eddies in fully developed turbulent pipe flow, stereoscopic particle image velocimetry measurements were performed in two parallel cross-sectional planes, for friction Reynolds numbers Re${}_{\ensuremath{\tau}}$ = 1310, 2430, and 3810. The instantaneous turbulence structures are sorted by width using an azimuthal Fourier decomposition, then azimuthally aligned to create a set of average eddy velocity profiles. The streamwise similarity is investigated using two-point correlations. Over the range of scales examined, the candidate structures establish full three-dimensional geometric self-similarity.

In 1955, Klebanoff published the first full set of turbulence stress measurements in a zero-pressure-gradient boundary layer (Klebanoff characteristics of turbulence in a boundary layer with zero-pressure gradient. NACA Report 1247, 1955). These results have achieved landmark status, and they are still widely used for comparisons with measurements and computations. The purpose of this paper is to show that these data are inaccurate in a number of ways, and that future comparisons should avoid using these results.

A new wall-wake law is proposed for the streamwise turbulence in the outer region of a turbulent boundary layer. The formulation pairs the logarithmic part of the profile (with a slope A_1 and additive constant B_1) to an outer linear part, and it accurately describes over 95% of the boundary layer profile at high Reynolds numbers. Once the slope A_1 is fixed, B_1 is the only free parameter determining the fit. Most importantly, B_1 is shown to be proportional to the wake factor in the wall-wake law for the mean velocity, revealing a previously unsuspected connection between the turbulence and the mean flow.

Abstract In 1955, Klebanoff published the first full set of turbulence stress measurements in a zero pressure gradient boundary layer (Klebanoff 1955). These results have achieved landmark status, and they are still widely used for comparisons with measurements and computations. The purpose of this paper is to show that these data are inaccurate in a number of ways, and that more recent data drawn from experiments and DNS should be used instead for future comparisons.

A turbulent pipe flow experiment was conducted where the surface of the pipe was oscillated azimuthally over a wide range of frequencies, amplitudes and Reynolds number. The drag was reduced by as much as 30\%. Past work has suggested that the drag reduction scales with the velocity amplitude of the motion, its period, or the Reynolds number. Here, we find that the key parameter is simply the acceleration, which reduces the complexity of the phenomenon by two orders of magnitude. This insight opens new potential avenues for reducing fuel consumption by large vehicles and for reducing energy costs in large piping systems.

We investigate the role of inter-scale interactions in the high-Reynolds-number skin-friction drag reduction strategy reported by Marusic et al. ( Nat. Commun. , vol. 12, 2021). The strategy involves imposing relatively low-frequency streamwise travelling waves of spanwise velocity at the wall to actuate the drag generating outer scales. This approach has proven to be more energy efficient than the conventional method of directly targeting the drag producing inner scales, which typically requires actuation at higher frequencies. Notably, it is observed that actuating the outer scales at low frequencies leads to a substantial attenuation of the major drag producing inner scales, suggesting that the actuations affect the nonlinear inner–outer coupling inherently existing in wall-bounded flows. In the present study, we find that increased drag reduction, through imposition of spanwise wall oscillations, is always associated with an increased coupling between the inner and outer scales. This enhanced coupling emerges through manipulation of the phase relationships between these triadically linked scales, with the actuation forcing the entire range of energy-containing scales, from the inner (viscous) to the outer (inertial) scales, to be more in phase. We also find that a similar enhancement of this nonlinear coupling, via manipulation of the inter-scale phase relationships, occurs with increasing Reynolds number for canonical turbulent boundary layers. This indicates improved efficacy of the energy-efficient drag reduction strategy at very high Reynolds numbers, where the energised outer scales are known to more strongly superimpose and modulate the inner scales. Leveraging the inter-scale interactions, therefore, offers a plausible mechanism for achieving energy-efficient drag reduction at high Reynolds numbers.

We present measurements of turbulent drag reduction (DR) in boundary layers at high friction Reynolds numbers in the range of $4500 \le Re_\tau \le 15\ 000$ . The efficacy of the approach, using streamwise travelling waves of spanwise wall oscillations, is studied for two actuation regimes: (i) inner-scaled actuation (ISA), as investigated in Part 1 of this study, which targets the relatively high-frequency structures of the near-wall cycle, and (ii) outer-scaled actuation (OSA), which was recently presented by Marusic et al. ( Nat. Commun. , vol. 12, 2021) for high- $Re_\tau$ flows, targeting the lower-frequency, outer-scale motions. Multiple experimental techniques were used, including a floating-element balance to directly measure the skin-friction drag force, hot-wire anemometry to acquire long-time fluctuating velocity and wall-shear stress, and stereoscopic particle image velocimetry to measure the turbulence statistics of all three velocity components across the boundary layer. Under the ISA pathway, DR of up to 25 % was achieved, but mostly with net power saving (NPS) losses due to the high-input power cost associated with the high-frequency actuation. The low-frequency OSA pathway, however, with its lower input power requirements, was found to consistently result in positive NPS of 5–10 % for moderate DRs of 5–15 %. The results suggest that OSA is an attractive pathway for energy-efficient DR in high-Reynolds-number applications.

Turbulent drag reduction (DR) through streamwise travelling waves of the spanwise wall oscillation is investigated over a wide range of Reynolds numbers. Here, in Part 1, wall-resolved large-eddy simulations in a channel flow are conducted to examine how the frequency and wavenumber of the travelling wave influence the DR at friction Reynolds numbers $Re_\tau = 951$ and $4000$ . The actuation parameter space is restricted to the inner-scaled actuation (ISA) pathway, where DR is achieved through direct attenuation of the near-wall scales. The level of turbulence attenuation, hence DR, is found to change with the near-wall Stokes layer protrusion height $\ell _{0.01}$ . A range of frequencies is identified where the Stokes layer attenuates turbulence, lifting up the cycle of turbulence generation and thickening the viscous sublayer; in this range, the DR increases as $\ell _{0.01}$ increases up to $30$ viscous units. Outside this range, the strong Stokes shear strain enhances near-wall turbulence generation leading to a drop in DR with increasing $\ell _{0.01}$ . We further find that, within our parameter and Reynolds number space, the ISA pathway has a power cost that always exceeds any DR savings. This motivates the study of the outer-scaled actuation pathway in Part 2, where DR is achieved through actuating the outer-scaled motions.

The wall dependence of length scales used to describe large- and small-scale structures of turbulence is examined using highly resolved experiments in zero-pressure-gradient turbulent boundary layers and pipe flows spanning the range $2000&lt; Re_\tau &lt;37\ 700$ . Of particular interest is the influence of external intermittency on the scaling of these length scales. It is found that when suitable scaling parameters are selected and external intermittency is accounted for, the dissipative motions follow inner scaling even into the outer-scaled regions of the flow, and that certain large-scale descriptions follow outer scaling even in the inner-scaled regions of the flow. The wall dependence is the same for both internal pipe and external boundary layer flows, and the different length scales can be related to recognizable features in the longitudinal wavenumber spectrum.

This paper deals with discerning outer-layer universality in the mean velocity profiles of turbulent boundary layers. For that purpose we derive an objective criterion which generalizes the classical wall scaling, as well as the Rotta-Clauser and Zagarola-Smits scaling. Outer-layer universality in the present scaling clearly emerges at displacement thickness Reynolds number exceeding about 3000. We propose universal velocity distributions based on simple patched logarithmic/parabolic fitting functions, which are accurate to within less than 1%

Taylor–Couette flow with a low aspect ratio cylinder suffers from end effects due to the finite-span of the gap between the cylinder sides and the secondary flow in the region below the inner cylinder. We experimentally explore these end effects by varying the cylinder aspect ratio between 6.67 and 40 for a range of wall gap widths and bottom gap heights. For these geometries, end effects (i.e. non-ideal Taylor–Couette flow) can be substantial due to both features of the finite-span and the bottom secondary flow. In some cases, the finite-span effects extended between 20% and 30% of the way into the Taylor–Couette flow region, and the secondary flow at the bottom accounted for nearly half of the total measured torque. By taking these effects into consideration, our high aspect ratio results agreed well with those obtained by Taylor (Taylor 1936 Proc. R. Soc. Lond. A 157 , 546–564. (doi: 10.1098/rspa.1936.0215 )) at considerably higher aspect ratios. This article is part of the theme issue ‘Taylor–Couette and related flows on the centennial of Taylor’s seminal Philosophical Transactions paper (part 1)’.

Digitized Data from G.I. Taylor's 1936 Paper "Fluid friction between rotating cylinders'