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The turbulence behaviour of current-dominated pulsating flows has been investigated. Direct numerical simulations have been carried out for Stokes lengths over a range of l+ s = 5-26, and amplitudes spanning 90% of the current-dominated regime, about a mean flow of Re = 6275. The results show that the turbulence response in intermediate and low-frequency pulsations is governed by a multistage turbulent-turbulent transition process, which bears a strong similarity to the multistage response of non-periodic acceleration. During the early acceleration period, the flow enters a pretransition stage, in which a new laminar perturbation boundary layer forms at the wall, and the streamwise velocity streaks are stretched. If the low-speed streaks destabilise prior to the deceleration period, then the flow enters a transition stage in which the perturbation boundary layer undergoes a bypass-like transition process. A unique feature of pulsating flows is the ongoing mechanism of turbulence decay, which initiates during the deceleration period and constitutes the main transient turbulence mechanism for much of the cycle. For high-frequency pulsations, the perturbation boundary layer fails to reach the pretransition stage prior to the deceleration period. Instead, the flow alternates between two inertial stages which are characterised by two layers of amplified viscous force; one growing at the wall, and one detached and moving towards the core. [ABSTRACT FROM AUTHOR]

This study introduces an analytical solution for the laminar entrance flow in circular pipes, aiming to confirm the occurrence of velocity overshoot. Velocity overshoot is characterised by the maximum axial velocity appearing near the pipe wall instead of the central axis. Similar to the previous studies, the analytical solution is derived from the parabolised Navier-Stokes equation; however, the specific approach used in linearising the momentum equation has not been attempted before. The accuracy of this analytical solution has been verified through a comprehensive comparison with various published experimental data. The existence of velocity overshoot at a short distance from the inlet, which is evident in numerous numerical calculations based on the full Navier-Stokes equations and corroborated by recent magnetic resonance (MR) velocimetry experiments, is identified analytically for the first time. The parabolised Navier-Stokes equation has inherent self-similarity with respect to the Reynolds number, implying that Re is incorporated into the dimensionless variables rather than serving as an independent flow parameter. According to both MR velocimetry measurements and the present analytical solution, the self-similarity is not valid immediately following the pipe inlet, and this becomes more evident as Re decreases; hence, the analytical solution derived from the parabolised Navier-Stokes equation cannot accurately predict the evolution of the velocity profile within this region near the pipe inlet. [ABSTRACT FROM AUTHOR]

Measurement data sets are presented for the turbulence intensity profile of three velocity components ($u$ , $v$ and $w$) and turbulent kinetic energy (TKE, $k$) over a wide range of Reynolds numbers from $Re_\tau =990$ to 20 750 in a pipe flow. The turbulence intensity profiles of the $u$ - and $w$ -component show logarithmic behaviour, and that of the $v$ -component shows a constant region at high Reynolds numbers, $Re_\tau >10\,000$. Furthermore, a logarithmic region is also observed in the TKE profile at $y/R=0.055\unicode{x2013}0.25$. The Reynolds number dependences of peak values of $u$ -, $w$ -component and TKE fit to both a logarithmic law (Marusic et al. , Phys. Rev. Fluids , vol. 2, 2017, 100502) and an asymptotic law (Chen and Sreenivasan, J. Fluid Mech. , vol. 908, 2020, R3), within the uncertainty of measurement. The Reynolds number dependence of the bulk TKE $k^+_{bulk}$ , which is the total amount of TKE in the cross-sectional area of the pipe also fits to both laws. When the asymptotic law is applied to the $k^+_{bulk}$ , it asymptotically increases to the finite value $k^+_{bulk}=11$ as the Reynolds number increases. The contribution ratio $\langle u'^2\rangle /k$ reaches a plateau, and the value tends to be constant within $100 at $Re_\tau >10\ 000$. Therefore, the local balance of each velocity component also indicates asymptotic behaviour. The contribution ratios are balanced in this region at high Reynolds numbers as $\langle u'^2\rangle /k\simeq 1.25$ , $\langle w'^2\rangle /k\simeq 0.5$ and $\langle v'^2\rangle /k \simeq 0.25$. [ABSTRACT FROM AUTHOR]

We study the statistics of passive scalars (either temperature or concentration of a diffusing substance) at friction Reynolds number Reτ = 1140, for turbulent flow within a smooth straight pipe of circular cross-section, in the range of Prandtl numbers from Pr = 0.00625, to Pr = 16, using direct numerical simulations (DNS) of the Navier-Stokes equations. Whereas the organization of passive scalars is similar to the axial velocity field at Pr = O(1), similarity is impaired at low Prandtl number, at which the buffer-layer dynamics is filtered out, and at high Prandtl number, at which the passive scalar fluctuations become confined to the near-wall layer. The mean scalar profiles at Pr ≳ 0.0125 are found to exhibit logarithmic overlap layers, and universal parabolic distributions in the core part of the flow. Near-universality of the eddy diffusivity is exploited to derive accurate predictive formulas for the mean scalar profiles, and for the corresponding logarithmic offset function. Asymptotic scaling formulas are derived for the thickness of the conductive (diffusive) layer, for the peak scalar variance, and its production rate. The DNS data are leveraged to synthesize a modified form of the classical predictive formula of Kader & Yaglom (Intl J. Heat Mass Transfer, vol. 15, 1972, pp. 2329-2351), which is capable of accounting accurately for the dependence on both Reynolds and Prandtl numbers, for Pr ≳ 0.25. [ABSTRACT FROM AUTHOR]

Extensive studies of the hydraulics of pipes have focused on limiting cases, such as fully-developed laminar or turbulent flow through long conduits and the accelerating flow through an orifice, for which there exist laws relating pressure drop and flow rate. We carry out experiments on smooth, circular pipes for dimensions and flow rates that interrogate intermediate conditions between the well-studied limits. Organizing this information in terms of dimensionless friction factor, Reynolds number and pipe aspect ratio yields a surface $f_D(Re,\alpha)$ that is shown to match the three laws associated with developed laminar, developed turbulent, and orifice flows. While each law fails outside its applicable range of $(Re,\alpha)$ , we present a hybrid theoretical–empirical model that includes inlet, development and transition effects, and that proves accurate to approximately 10 % over wide ranges of $Re$ and $\alpha$. We also present simple formulas for the boundaries between the three hydraulic regimes, which intersect at a triple point. Measurements show that sipping through a straw is an everyday example of such intermediate conditions not accounted for by existing laws but described accurately by our model. More generally, our findings provide formulas for predicting frictional resistance for intermediate- $Re$ flows through finite-length pipes. [ABSTRACT FROM AUTHOR]

A numerical investigation of turbulent flows in straight pipes with a cross-section in the form of a circular sector at different values of the apex angle α is carried out. Main attention is paid to the cases α > π when the cross-section boundary contains a convex external corner. An analytical solution to the problem of laminar flow in the geometry under consideration is given. The law of resistance is determined, and velocity distributions over the pipe cross-section are obtained. The singularity of wall shear stress τw ∼ rπ/α−1 at α > π is shown which persists in turbulent flows as well. Turbulent flows are simulated at Re = 2000 and four apex-angle values α = 5π/4, 3π/2, 7π/4 and 2π, as well as at a larger Re = 4000 and α = 3π/2. The characteristic features of the mean-velocity distributions are outlined, and their relationship with secondary flows is shown. The velocity of the secondary flow in the vicinity of the external corner reaches the value 6.3% of the bulk flow velocity at the maximum α = 2π. A characteristic feature of the considered flows is the presence of intense transverse velocity fluctuations in the region above the apex corner. It is hypothesized that the cause of these fluctuations may be the linear instability of the mean flow field. The studies of linear stability confirm this hypothesis. Considerable attention is paid to the study of secondary flows in the vicinity of the external corner. In general, the formation of these flows fits into the scheme proposed earlier (Nikitin, J. Fluid Mech., vol. 917, 2021, A24; Nikitin et al., Fluid Dyn., vol. 56, issue 4, 2021, pp. 513–538), according to which the main mechanism is the centrifugal force arising from the fluctuating flow over the corner in the transverse plane. It is shown that the shape of secondary flows is reproduced to a certain extent by the vortical part of the Reynolds-stress force field. The latter may be found from the Helmholtz decomposition. The features of the mean pressure distribution associated with the secondary flow and with the presence of a region of concentrated fluctuations above the external corner are explained. [ABSTRACT FROM AUTHOR]

Our understanding of turbulent boundary layer scaling and structure has advanced greatly in the past 20 to 30 years. On the computational side, direct numerical simulations and large-eddy simulations have made extraordinary contributions as numerical methods and computational resources have advanced, while on the experimental side major advances in instrumentation have made available new imaging and quantitative techniques that provide unprecedented accuracy and detail. Here, I illustrate how the development of such experimental methods have aided our progress by reference to some particular topics related to the structure of turbulent boundary layers: the power law scaling of the mean velocity and its relationship to the mesolayer; the scaling of the outer layer with regard to the log law in turbulence; the development of the outer peak; and the scaling of the turbulent stresses in the near-wall region, with an emphasis on the streamwise component. [ABSTRACT FROM AUTHOR]

Large-eddy simulations of turbulent flow in partially filled pipes are conducted to investigate the effect of secondary currents on the friction factor, first- and second-order statistics and large-scale turbulent motion. The method is validated first and simulated profiles of the mean streamwise velocity, normal stresses and turbulent kinetic energy (TKE) are shown to be in good agreement with experimental data. The secondary flow is stronger in half- and three-quarters full pipes compared with quarter full or fully filled pipe flows, respectively. The origin of the secondary flow is examined by both the TKE budget and the steamwise vorticity equation, providing evidence that secondary currents originate from the corner between the free surface and the pipe walls, which is where turbulence production is larger than the sum of the remaining terms of the TKE budget. An extra source of streamwise vorticity production is found at the free surface near the centreline bisector, due to the two-component asymmetric turbulence there. The occurrence of dispersive stresses (due to secondary currents) reduces the contribution of the turbulent shear stress to the friction factor, which results in a reduction of the total friction factor of flows in half and three-quarters full pipes in comparison to a fully filled pipe flow. Furthermore, the presence of significant secondary currents inhibits very-large-scale motion (VLSM), which in turn reduces the strength and scales of near-wall streaks. Subsequently, near-wall coherent structures generated by streak instability and transient growth are significantly suppressed. The absence of VLSM and less coherent near-wall turbulence structures is supposedly responsible for the drag reduction in partially filled pipe flows relative to a fully filled pipe flow at an equivalent Reynolds number. [ABSTRACT FROM AUTHOR]

The dominant paradigm in turbulent wall flows is that the mean velocity near the wall, when scaled on wall variables, is independent of the friction Reynolds number Reτ. This paradigm faces challenges when applied to fluctuations but has received serious attention only recently. Here, by extending our earlier work (Chen & Sreenivasan, J. Fluid Mech., vol. 908, 2021, p. R3) we present a promising perspective, and support it with data, that fluctuations displaying non-zero wall values, or near-wall peaks, are bounded for large values of Reτ, owing to the natural constraint that the dissipation rate is bounded. Specifically, Φ∞ - Φ = CΦ Re-1/4τ, where Φ represents the maximum value of any of the following quantities: energy dissipation rate, turbulent diffusion, fluctuations of pressure, streamwise and spanwise velocities, squares of vorticity components, and the wall values of pressure and shear stresses; the subscript ∞ denotes the bounded asymptotic value of Φ, and the coefficient CΦ depends on Φ but not on Reτ. Moreover, there exists a scaling law for the maximum value in the wall-normal direction of high-order moments, of the form (ϕ2q)1/q max = αq - βq Re-1/4τ, where ϕ represents the streamwise or spanwise velocity fluctuation, and αq and βq are independent of Reτ. Excellent agreement with available data is observed. A stochastic process for which the random variable has the form just mentioned, referred to here as the 'linear q-norm Gaussian', is proposed to explain the observed linear dependence of αq on q. [ABSTRACT FROM AUTHOR]

The reference length is Graph $\lambda ^*=\theta g R^*$ (Graph $\lambda ^*/R^* = O(1)$), that is, the circumferential wavelength of the gust at Graph $r^*=R^*$, where Graph $\theta g$ is the azimuthal angle corresponding to the wavelength Graph $\lambda ^*$. The radial velocity is never singular at the centreline despite Graph $J m$ being divided by Graph $\bar {r}$ because, given Graph $J m(\bar {r}) \sim (\bar {r}/2)^m/\varGamma (m+1)$ for Graph $\bar {r}\ll 1$ (where Graph $\varGamma$ is the gamma function), the radial average Graph $\hat {u} {r,0n}^\infty$ is zero. Figure 16( I b i ) confirms that the pressure Graph $|\bar {p} 0|$ is closely related to Graph $|\bar {u} {\theta,0}|$ as the near-wall maximum of Graph $|\bar {u} {\theta,0}|$ is located at the same radial position of the local maximum of Graph $|\bar {p} 0|$, although the overall maximum pressure disturbance is found at the pipe wall. Differently from the leading-order components where Graph $\bar {u} {\theta,out}$ is needed to determine the amplitude of Graph $\{\bar {u} {x,in},\bar {u} {r,in}\}$, it is not necessary to compute Graph $\bar {u}^{(0)} {\theta,out}$ as we have already found Graph $\{\bar {u} x^{(0)},\bar {u} r^{(0)}\}$ to start the integration of the boundary-region equations (2.11) and (2.11). In the pipe core, the axisymmetric outer flow is described by the inviscid Stokes stream function Graph $\psi$, 3.6 Graph \begin{equation} \psi(x,r)=\frac{r^2}{2}+Re \lambda^{{-}1/2}\psi 2(x,r), \end{equation} i.e. Graph $\bar {U} {out}=r^{-1}\partial \psi /\partial r$ and Graph $\bar {V} {out}=-r^{-1}\partial \psi /\partial x$. [Extracted from the article]

A new scaling is derived that yields a Reynolds-number-independent profile for all components of the Reynolds stress in the near-wall region of wall-bounded flows, including channel, pipe and boundary layer flows. The scaling demonstrates the important role played by the wall shear stress fluctuations and how the large eddies determine the Reynolds number dependence of the near-wall turbulence behaviour. [ABSTRACT FROM AUTHOR]

The high-Strouhal-number pulsatile flow in a curved pipe is studied numerically. A general force analysis is developed for the bend force, where the new contribution from flow acceleration is identified and analysed. The mechanisms of secondary flow production are studied by extending Hawthorne's (Proc. R. Soc. Lond. A , vol. 206, issue 1086, 1951, pp. 374–387) model to account for viscous effects and applied to assess the distinct contributions from an inviscid stretching and no-slip condition. A detailed comparison is made between the numerical simulations and models for a pipe flow characterised by a volume flux $Q=U_b A |\sin \varOmega _p t|$ (where $U_b$ is the maximum bulk velocity, $\varOmega _p$ is the angular frequency and $A$ is the pipe cross-sectional area). For high-Reynolds-number ($Re_b$) and high-Strouhal-number ($St$), the bend force predictions are in good agreement with the numerical results over a wide range of bend curvature ($R_c/D$ ; where $R_c$ is the bend radius of curvature and $D$ is the pipe diameter) owing to the influence of the streamwise flow acceleration on the pressure field. At high- $St$ , the streamwise vorticity (secondary flow) distribution is steady and close to the low- $St$ case, which is explained using a linear secondary flow model. [ABSTRACT FROM AUTHOR]

The purpose of this research is to provide a new understanding of the turbulence dynamics in a heated flow of fluid at supercritical pressure. A unified explanation has been established for the laminarisation mechanisms due to the variations of thermophysical properties, buoyancy and inertia, the last of which plays a significant role in a developing flow. In the new understanding, the various factors can all be treated similarly as (pseudo-)body forces, the effect of which is to cause a reduction in the so-called apparent Reynolds number. The partially laminarising flow is represented by an equivalent-pressure-gradient reference flow plus a perturbation flow. Full laminarisation is used in the paper referring to a region where no new vortical structures are generated. This region is akin to the pre-transition region of a boundary layer bypass transition, and in both cases, the free-stream or pipe-core turbulence decays exponentially, but elongated streaks are formed in the boundary layer. Turbulence kinetic energy in this region may still be significant due to the decaying turbulence as well as newly generated streaks. The latter lead to an increase in streamwise velocity fluctuations near the wall. Later, re-transition occurs when the streaks break down and multi-scale vortices are generated, leading to an increase in the radial and circumferential velocity fluctuations. The structural effect of buoyancy on turbulence is weak and negative in the partially laminarising flow, but is dominant in the full laminarisation and re-transition regions. [ABSTRACT FROM AUTHOR]

The instability and transition to turbulence and its evolution in pulsatile flows, which involve reverse flows and unsteady flow separations, is the primary focus of this experimental work. A piston driven by a programmable DC servo motor was used to set-up a water flow system and provide the pulsation characteristics. Time-resolved particle image velocimetry data were acquired in a refractive index matching set-up by using a continuous wave laser and a high-frame-rate digital camera. The position of the piston was continuously recorded by a laser proximity sensor. Five different experiments were carried out with Reynolds numbers in the range of 535–4825 and Womersley numbers from 11.91 to 23.82. The non-stationarity of the data was addressed by incorporating trend removal methods involving low- and high-pass filtering of the data, and using empirical mode decomposition together with the relevant Hilbert–Huang transform to determine the intrinsic mode functions. This latter method is more appropriate for nonlinear and non-stationary cases, for which traditional analysis involving classical Fourier decomposition is not directly applicable. It was found that transition to turbulence is a spontaneous event covering the whole near-wall region. The instantaneous vorticity profiles show the development of a large-scale ring-like attached wall vortical layer (WVL) with smaller vortices of higher frequencies than the pulsation frequency superimposed, which point to a shear layer Kelvin–Helmholtz (K–H) type of instability. Inflectional instability leads to flow separation and the formation of a major roll-up structure with the K–H vortices superimposed. This structure breaks down in the azimuthal direction into smaller turbulence patches with vortical content, which appears to be the prevailing structural content of the flow at each investigated Reynolds number (Re). At higher Re numbers, the strength and extent of the vortices are larger and substantial disturbances appear in the free stream region of the flow, which are typical of pipe flows at transitional Re numbers. Turbulence appears to be produced at the locations of maximum or minimum vorticity within the attached WVL, in the ridges between the K–H vortices around the separated WVL and the upstream side of the secondary vortex where the flow impinges on the wall. This wall turbulence breaks away into the middle section of the pipe, at approximately Re ≥ 2200, by strong eruptions of the K–H vortices. [ABSTRACT FROM AUTHOR]

A three-layer asymptotic structure for turbulent pipe flow is proposed revealing, in terms of intermediate variables, the existence of a Reynolds-number-invariant logarithmic region for the streamwise mean velocity and variance. The formulation proposes a local velocity scale (which is not the friction velocity) for the intermediate layer and results in two overlap layers. We find that the near-wall overlap layer is governed by a power law for the pipe for all Reynolds numbers, whereas the log law emerges in the second overlap layer (the inertial sublayer) for sufficiently high Reynolds numbers ($Re_{\tau }$). This provides a theoretical basis for explaining the presence of a power law for the mean velocity at low $Re_{\tau }$ and the coexistence of power and log laws at higher $Re_{\tau }$. The classical von Kármán ($\kappa$) and Townsend–Perry ($A_1$) constants are determined from the intermediate-scaled log-law constants; $\kappa$ shows a weak trend at sufficiently high $Re_{\tau }$ but falls within the commonly accepted uncertainty band, whereas $A_1$ exhibits a systematic Reynolds-number dependence until the largest available $Re_{\tau }$. The key insight emerging from the analysis is that the scale separation between two adjacent layers in the pipe is proportional to $\sqrt {Re_{\tau }}$ (rather than $Re_{\tau }$) and therefore the approach to an asymptotically invariant state can be expected to be slow. [ABSTRACT FROM AUTHOR]

The celebrated wall-scaling works for many statistical averages in turbulent flows near smooth walls, but the streamwise velocity fluctuation, $u^{\prime }$ , is thought to be among the few exceptions. In particular, the near-wall mean-square peak, $\overline {u'u'}^+_p$ – where the superscript $+$ indicates normalization by the friction velocity $u_\tau$ , the subscript $p$ indicates the peak value and the overbar indicates time averaging – is known to increase with increasing Reynolds number. The existing explanations suggest a logarithmic growth with respect to $Re$ , where $Re$ is the Reynolds number based on $u_\tau$ and the thickness of the wall flow. We show that this boundless growth calls into question the veracity of wall-scaling and so cannot be sustained, and we establish an alternative formula for the peak magnitude that approaches a finite limit $\overline {u'u'}^+_\infty$ owing to the natural constraint of boundedness on the dissipation rate at the wall. This new formula agrees well with the existing data and, in contrast to the logarithmic growth, supports the classical wall-scaling for turbulent intensity at asymptotically high Reynolds numbers. [ABSTRACT FROM AUTHOR]

Dense, small particles suspended in turbulent smooth-wall flow are known to migrate towards the wall. It is, however, not clear if the particle migration continues in a rough-wall flow and what the responsible mechanism is, especially with changing roughness parameters. Here, we address this using direct numerical simulation of a turbulent pipe flow of a fixed friction Reynolds numbers and changing the roughness size as well as the Stokes number of the particles. The transport and deposition mechanisms of particles are segregated into three different regimes dictated by the Stokes number. Particles with small Stokes number follow the carrier fluid and are affected by the turbulent structures of the rough wall. Flow separation in the wake of the roughness and stagnant flow in the trough of the roughness causes these particles to be trapped in the roughness canopy. Particles with very large Stokes number, on the other hand, are attracted to the wall due to turbophoresis and collide with the rough wall where the frequency of wall collision increases with increasing Stokes number. These ballistic particles are unaffected by the turbulent fluctuations of the flow and their trajectory is determined by the roughness topography. At intermediate Stokes numbers, the transport of the particles is influenced by both the wall collisions and also the turbulent flow. Particles in this range of Stokes number occasionally collide with the wall and are entrained by the turbulent flow. In this regime, the particles may have a mean streamwise velocity that is larger than the bulk flow rate of the fluid. Finally, we observe that bulk particle velocity scale better with a time scale based on the roughness elements rather than the usual viscous time scale. [ABSTRACT FROM AUTHOR]

This work presents a detailed analysis of the flow structures relevant to extreme wall shear stress events for turbulent pipe flow direct numerical simulation data at a friction Reynolds number $\textit {Re}_{\tau} \approx 1000$. The results reveal that extreme positive wall-friction events are located below an intense sweep (Q4) event originated from a strong quasi-streamwise vortex at the buffer region. This vortex transports high streamwise momentum from the overlap and the outer layers towards the wall, giving rise to a high-speed streak within the inner region. This vortical structure also relates to regions with extreme wall-normal velocity. Consequently, the conditional fields of turbulence production and viscous dissipation exhibit peaks whose magnitudes are approximately 25 times higher than the ensemble mean quantities in the vicinity of the extreme positive events. An analysis of the turbulent inertia force reveals that the energetic quasi-streamwise vortex acts as an essential source of momentum at the near-wall region. Similarly, extremely rare backflow events are studied. An examination of the wall-normal vorticity and velocity vector fields shows an identifiable oblique vortical structure along with two other large-scale roll modes. These counter-rotating motions contribute to the formation of backflow events by transporting streamwise momentum from the inner to the outer region, creating a large-scale meandering low-speed streak. It is found that extreme events are clustered below large-scale structures of positive streamwise momentum that interact with near-wall low-speed streaks, related to regions densely populated with vortical structures. Finally, a three-dimensional model is proposed to conceptualise the flow dynamics associated with extreme events. [ABSTRACT FROM AUTHOR]

In this paper, we use the well-known background method to obtain a rigorous lower bound on the volume flow rate through a helical pipe driven by a pressure differential in the limit of large Reynolds number. As a consequence, we also obtain an equivalent upper bound on the friction factor. These bounds are also valid for toroidal and straight pipes as limiting cases. By considering a two-dimensional background flow with varying boundary layer thickness along the circumference of the pipe, we obtain these bounds as a function of the curvature and torsion of the pipe and therefore capture the geometrical aspects of the problem. In this paper, we also present a sufficient criterion for determining which pressure-driven flow and surface-velocity-driven flow problems can be tackled using the background method. [ABSTRACT FROM AUTHOR]

This paper provides an experimental investigation on the internal shear layers and the edges of the uniform momentum zones (UMZs) in a turbulent pipe flow. The time-resolved stereoscopic particle image velocimetry data are acquired in the cross-section of the pipe, and span the range of Reynolds number $\textit {Re}_\tau =340\text {--}1259$. In the first part of the study, internal shear layers are detected using a three-dimensional detection method, and both their geometry as well as their fingerprint in the flow statistics are examined. Three-dimensional conditional mean flow analysis revealed a strong low-speed region beneath the average shear layers. This low-speed region is associated with positive wall-normal fluctuations, and it is accompanied by two swirling motions having opposite signs on either side in the azimuthal direction. Moreover, the shear layers are stretched by the two opposite azimuthal motions. In the second part of the study, the shear layers are treated as the continuous edges of the UMZs, which are detected using the histogram method following Adrian et al. (J. Fluid Mech., vol. 422, 2000, pp. 1–54) and de Silva et al. (J. Fluid Mech., vol. 786, 2016, pp. 309–331). For this part, two different orientation of the planes are used, i.e. the wall-normal–streamwise plane and the wall-normal–spanwise plane (cross-section of the pipe). Comparison of the detected structures shows that the shear layers mostly overlap with a UMZ edge (in either plane). [ABSTRACT FROM AUTHOR]

Direct numerical simulations, performed with a high-order spectral-element method, are used to study coherent structures in turbulent pipe flow at friction Reynolds numbers Reτ = 180 and 550. The database was analysed using spectral proper orthogonal decomposition (SPOD) to identify energetically dominant coherent structures, most of which turn out to be streaks and quasi-streamwise vortices. To understand how such structures can be modelled, the linear flow responses to harmonic forcing were computed using the singular value decomposition of the resolvent operator, using the mean field as a base flow. The SPOD and resolvent analysis were calculated for several combinations of frequencies and wavenumbers, allowing the mapping out of similarities between SPOD modes and optimal responses for a wide range of relevant scales in turbulent pipe flows. In order to explore physical reasons behind the agreement between both methods, an indicator of lift-up mechanism in the resolvent analysis was introduced, activated when optimal forcing is dominated by the wall-normal and azimuthal components, and associated response corresponds to streaks of streamwise velocity. Good agreement between leading SPOD and resolvent modes is observed in a large region of parameter space. In this region, a significant gain separation is found in resolvent analysis, which may be attributed to the strong amplification associated with the lift-up mechanism, here understood as nonlinear forcing terms leading to the appearance of streamwise vortices, which in turn form high-amplitude streaks. For both Reynolds numbers, the observed concordances were generally for structures with large energy in the buffer layer. The results highlight resolvent analysis as a pertinent reduced-order model for coherent structures in wall-bounded turbulence, particularly for streamwise elongated structures corresponding to near-wall streamwise vortices and streaks. [ABSTRACT FROM AUTHOR]

We present findings on scalar flashes, turbulent spots and transition statistics in a direct numerical simulation of a 500 radius-long pipe flow with a radial-mode inlet disturbance. Transitional spots are found to contain two different types of eddies. The wall region consists primarily of 'reverse hairpin vortices'. This unusual structure is related to the high-speed streaks arising from the prescribed inlet perturbation. The core region is populated by the normally observed 'forward' hairpin vortices. Number density of the reverse hairpins is quantified indirectly and conservatively by measuring the number density of negative skin-friction patches. During the late stages of transition, second-order statistics such as Reynolds stresses and the rate of dissipation of turbulent kinetic energy exhibit substantial overshoot. This is associated with mid-to-high frequency content in the energy spectra that exceeds the corresponding levels in fully developed turbulence. Flow visualizations reveal bursts of small-scale vortex motions, including the reverse hairpins, that probably account for the enhanced mid-to-high frequency spectral content. A passive scalar is injected at the centreline of the inlet plane, mimicking laboratory injection of dye through a needle, to investigate the mysterious phenomenon of turbulent scalar patches residing in fully developed turbulent pipe flow. At several hundred radii downstream of transition where the velocity field is genuine fully developed turbulence, the scalar patches retain persistent memory of events far upstream. Comparing the present flow with a similar pipe flow disturbed by a significantly different inlet condition suggests that the foregoing observations are insensitive to the form of the disturbances. [ABSTRACT FROM AUTHOR]

Pulsatile rough-wall turbulent pipe flow is compared against its non-pulsatile counterpart using data obtained from direct numerical simulation. Results are presented at a mean friction Reynolds number of 540 for a set of three geometrically scaled roughness topographies at a single forcing condition, which, based on existing classifications, falls into the current-dominated very-high-frequency regime. By comparing the pulsatile data against an equivalent non-pulsatile dataset (Chan et al., J. Fluid Mech., vol. 854, 2018, pp. 5–33), the key differences (and similarities) between the forced and unforced configurations are identified. A major finding of this study is that the flow in the outer region retains its self-similar functional form under pulsatile rough-wall conditions, and, as a result, Townsend's outer-layer similarity hypothesis holds for the roughness-forcing combinations considered here. On the other hand, the unsteady cases exhibit a rich array of flow physics in the region beneath the roughness crests not observed in the steady case. These differences are examined using a Moody chart, which encapsulates how the hydraulic properties of pulsatile rough-wall pipe flow differ from their non-pulsatile counterpart. [ABSTRACT FROM AUTHOR]

In searching for the mechanisms of boundary layer bypass transition, Schubauer & Klebanoff (NACA-TR-1289, 1956) conducted an extensive search for negative skin-friction events during the laminar-to-turbulent transition in a zero-pressure-gradient, smooth flat-plate boundary layer (ZPGSFPBL) under a wide range of free-stream turbulence intensity levels. They concluded that no evidence of incipient flow separation could be found in any part of the transition region. Although it has been known that extremely rare backflow events can occur in a fully turbulent ZPGSFPBL and in fully developed turbulent pipe flow, the Schubauer–Klebanoff conclusion regarding the total absence of negative skin friction in ZPGSFPBL transition has remained unchallenged over the past six decades. Here we report our discovery of negative skin-friction events during the bypass transition in an extensively researched representative ZPGSFPBL case, and in pipe flows with no-slip and slip boundary conditions under circumferential mode inlet disturbance, respectively. The peak probability of such events is located in the late stage of transition between the streamwise stations where the minimum and maximum mean skin friction are attained. The magnitude of the peak probability is substantially larger than the corresponding probability in the downstream fully turbulent region: in the case of no-slip pipe flow, the difference is two orders of magnitude and is primarily due to an enhanced number density of the backflow events coupled with a notable increase in the wall footprint size of such events. For both the boundary layer and pipe flows considered here, and in both the transitional and fully turbulent regions, the incipient negative skin friction is found, through instantaneous flow visualization and conditional averaging, to be induced by the head element of a reverse hairpin vortex. [ABSTRACT FROM AUTHOR]

Streamwise velocity and wall-shear stress are acquired simultaneously with a hot-wire and an array of azimuthal/spanwise-spaced skin friction sensors in large-scale pipe and boundary layer flow facilities at high Reynolds numbers. These allow for a correlation analysis on a per-scale basis between the velocity and reference skin friction signals to reveal which velocity-based turbulent motions are stochastically coherent with turbulent skin friction. In the logarithmic region, the wall-attached structures in both the pipe and boundary layers show evidence of self-similarity, and the range of scales over which the self-similarity is observed decreases with an increasing azimuthal/spanwise offset between the velocity and the reference skin friction signals. The present empirical observations support the existence of a self-similar range of wall-attached turbulence, which in turn are used to extend the model of Baars et al. (J. Fluid Mech. , vol. 823, p. R2) to include the azimuthal/spanwise trends. Furthermore, the region where the self-similarity is observed correspond with the wall height where the mean momentum equation formally admits a self-similar invariant form, and simultaneously where the mean and variance profiles of the streamwise velocity exhibit logarithmic dependence. The experimental observations suggest that the self-similar wall-attached structures follow an aspect ratio of $7:1:1$ in the streamwise, spanwise and wall-normal directions, respectively. [ABSTRACT FROM AUTHOR]

We present well-resolved large-eddy simulations of turbulent flow through a straight, high aspect ratio cooling duct operated with water at a bulk Reynolds number of Reb = 110 X 10³ and an average Nusselt number of Nuxz = 371. The geometry and boundary conditions follow an experimental reference case and good agreement with the experimental results is achieved. The current investigation focuses on the influence of asymmetric wall heating on the duct flow field, specifically on the interaction of turbulence-induced secondary flow and turbulent heat transfer, and the associated spatial development of the thermal boundary layer and the inferred viscosity variation. The viscosity reduction towards the heated wall causes a decrease in turbulent mixing, turbulent length scales and turbulence anisotropy as well as a weakening of turbulent ejections. Overall, the secondary flow strength becomes increasingly less intense along the length of the spatially resolved heated duct as compared to an adiabatic duct. Furthermore, we show that the assumption of a constant turbulent Prandtl number is invalid for turbulent heat transfer in an asymmetrically heated duct. [ABSTRACT FROM AUTHOR]

The occurrence of secondary flows is investigated for three-dimensional sinusoidal roughness where the wavelength and height of the roughness elements are systematically altered. The flow spanned from the transitionally rough regime up to the fully rough regime and the solidity of the roughness ranged from a wavy, sparse roughness to a dense roughness. Analysing the time-averaged velocity, secondary flows are observed in all of the cases, reflected in the coherent stress profile which is dominant in the vicinity of the roughness elements. The roughness sublayer, defined as the region where the coherent stress is non-zero, scales with the roughness wavelength when the roughness is geometrically scaled (proportional increase in both roughness height and wavelength) and when the wavelength increases at fixed roughness height. Premultiplied energy spectra of the streamwise velocity turbulent fluctuations show that energy is reorganised from the largest streamwise wavelengths to the shorter streamwise wavelengths. The peaks in the premultiplied spectra at the streamwise and spanwise wavelengths are correlated with the roughness wavelength in the fully rough regime. Current simulations show that the spanwise scale of roughness determines the occurrence of large-scale secondary flows. [ABSTRACT FROM AUTHOR]

Pressure-driven laminar and turbulent flow in a horizontal partially filled pipe was investigated using stereoscopic particle imaging velocimetry (S-PIV) in the cross-stream plane. Laminar flow velocity measurements are in excellent agreement with a recent theoretical solution in the literature. For turbulent flow, the flow depth was varied independently of a nominally constant Reynolds number b and kinematic viscosity ν> of ReH = UbDHν ≈ 30 000 ± 5 %. When running partially full, the inferred friction factor is no longer a simple function of Reynolds number, but also depends on the Froude number Fr = Ub/√gDm where g is gravitational acceleration and Dm is hydraulic mean depth. S-PIV measurements in turbulent flow reveal the presence of secondary currents which causes the maximum streamwise velocity to occur below the free surface consistent with results reported in the literature for rectangular cross-section open channel flows. Unlike square duct and rectangular open channel flow the mean secondary motion observed here manifests only as a single pair of vortices mirrored about the vertical bisector and these rollers, which fill the half-width of the pipe, remain at a constant distance from the free surface even with decreasing flow depth for the range of depths tested. Spatial distributions of streamwise Reynolds normal stress and turbulent kinetic energy exhibit preferential arrangement rather than having the same profile around the azimuth of the pipe as in a full pipe flow. Instantaneous fields reveal the signatures of elements of canonical wall-bounded turbulent flows near the pipe wall such as large-scale and very-large-scale motions and associated hairpin packets whilst near the free surface, the signatures of free surface turbulence in the absence of imposed mean shear such as 'upwellings', 'downdrafts' and 'whirlpools' are present. Two-point spatio-temporal correlations of streamwise velocity fluctuation suggest that the large-scale coherent motions present in full pipe flow persist in partially filled pipes but are compressed and distorted by the presence of the free surface and mean secondary motion. [ABSTRACT FROM AUTHOR]

We develop predictive formulae for frictional resistance in ducts with complex crosssectional shape based on the use of the log law and neglect of wall shear stress nonuniformities. The traditional hydraulic diameter naturally emerges from the analysis as the controlling length scale for common duct shapes such as triangles and regular polygons. The analysis also suggests that a new effective diameter should be used in more general cases, yielding corrections of a few percent to friction estimates based on the traditional hydraulic diameter. Fair, but consistent, predictive improvement is shown for duct geometries of practical relevance, including rectangular and annular ducts, and circular rod bundles. [ABSTRACT FROM AUTHOR]

We report new laboratory experiments of a flow accelerating from an initially turbulent state following the opening of a valve, together with large eddy simulations of the experiments and extended Stokes first problem solutions for the early stages of the flow. The results show that the transient flow closely resembles an accelerating laminar flow superimposed on the original steady turbulent flow. The primary consequence of the acceleration is the temporal growth of a boundary layer from the wall, gradually leading to a strong instability causing transition. This extends the findings of previous direct numerical simulations of transient flow following a near-step increase in flow rate. In this interpretation, the initial turbulence is not the primary characteristic of the resulting transient flow, but can be regarded as noise, the evolution of which is strongly influenced by the development of the boundary layer. We observe the spontaneous appearance of turbulent spots and discontinuities in the velocity signals in time and space, revealing rich detail of the transition process, including a striking contrast between streamwise and wall-normal fluctuating velocities. [ABSTRACT FROM AUTHOR]

Swirl-switching is a low-frequency oscillatory phenomenon which affects the Dean vortices in bent pipes and may cause fatigue in piping systems. Despite thirty years worth of research, the mechanism that causes these oscillations and the frequencies that characterise them remain unclear. Here we show that a three-dimensional wave-like structure is responsible for the low-frequency switching of the dominant Dean vortex. The present study, performed via direct numerical simulation, focuses on the turbulent flow through a 90° pipe bend preceded and followed by straight pipe segments. A pipe with curvature 0.3 (defined as ratio between pipe radius and bend radius) is studied for a bulk Reynolds number Re = 11 700, corresponding to a friction Reynolds number Reτ ≈ 360. Synthetic turbulence is generated at the inflow section and used instead of the classical recycling method in order to avoid the interference between recycling and swirl-switching frequencies. The flow field is analysed by three-dimensional proper orthogonal decomposition (POD) which for the first time allows the identification of the source of swirl-switching: a wave-like structure that originates in the pipe bend. Contrary to some previous studies, the flow in the upstream pipe does not show any direct influence on the swirl-switching modes. Our analysis further shows that a three-dimensional characterisation of the modes is crucial to understand the mechanism, and that reconstructions based on two-dimensional POD modes are incomplete. [ABSTRACT FROM AUTHOR]

This paper reports on near-wall two-component–two-dimensional (2C–2D) particle image velocimetry (PIV) measurements of a turbulent pipe flow at shear Reynolds numbers up to Reτ = 40 000 acquired in the CICLoPE facility of the University of Bologna. The 111.5 m long pipe of 900 mm diameter offers a well-established turbulent flow with viscous length scales ranging from 85 mm at Rτ = 5000 down to 11 mm at Reτ = 40 000. These length scales can be resolved with a high-speed PIV camera at image magnification near unity. Statistically converged velocity profiles were determined using multiple sequences of up to 70 000 PIV recordings acquired at sampling rates of 100 Hz up to 10 kHz. Analysis of the velocity statistics shows a well-resolved inner peak of the streamwise velocity fluctuations that grows with increasing Reynolds number and an outer peak that develops and moves away from the inner peak with increasing Reynolds number. [ABSTRACT FROM AUTHOR]

Direct numerical simulations of turbulent pipe flow of power-law fluids at Reι = 323 are analysed in order to understand the way in which shear thinning or thickening affects first- and second-order flow statistics including turbulent kinetic energy production, transport and dissipation in such flows. The results show that with shear thinning, near-wall streaks become weaker and the axial and azimuthal correlation lengths of axial velocity fluctuations increase. Viscosity fluctuations give rise to an additional shear stress term in the mean momentum equation which is negative for shear-thinning fluids and which increases in magnitude as the fluid becomes more shear thinning: for an equal mean wall shear stress, this term increases the mean velocity gradient in shear-thinning fluids when compared to a Newtonian fluid. Consequently, the mean velocity profile in power-law fluids deviates from the law of the wall Uz+ =+ in the viscous sublayer when traditional near-wall scaling is used. Consideration is briefly given to an alternative scaling that allows the law of wall to be recovered but which results in loss of a common mean stress profile. With shear thinning, the mean viscosity increases slightly at the wall and its profile appears to be approximately logarithmic in the velocity log layer. Through analysis of the turbulent kinetic energy budget, undertaken here for the first time for generalised Newtonian fluids, it is shown that shear thinning decreases the overall turbulent kinetic energy production but widens the wall-normal region where it is generated. Additional dissipation terms in the mean flow and turbulent kinetic energy budget equations arise from viscosity fluctuations; with shear thinning, these result in a net decrease in the total viscous dissipation. The overall effect of shear thinning on the turbulent kinetic energy budget is found to be largely confined to the inner layers, y+ ≲ 60. [ABSTRACT FROM AUTHOR]

When a fluid enters a rotating circular pipe, an angular momentum or swirl boundary layer appears at the wall and interacts with the axial momentum boundary layer. In the centre of the pipe, the fluid is free of swirl and is accelerated due to boundary layer growth. Below a critical flow number, defined as the ratio of average axial velocity to circumferential velocity of the pipe, there is flow separation, known in the turbomachinery context as part load recirculation. To describe this phenomenon analytically, we extended boundary layer theory to a swirl boundary layer interacting with the axial momentum boundary layer. The solution of the resulting generalized von Kármán momentum equation takes into account the influence of the Reynolds number and flow number. We show the impact of swirl on the axial boundary layer and conduct experiments in which we vary Reynolds number, flow number and surface roughness to validate the analytical results. The extended boundary layer theory predicts a critical flow number which is analytically derived and validated. Below this critical flow number, separation is expected. [ABSTRACT FROM AUTHOR]

It is well established that when a turbulent flow is subjected to a non-uniform body force, the turbulence may be significantly suppressed in comparison with that of the flow of the same flow rate and hence the flow is said to be laminarised. This is the situation in buoyancy-aided mixed convection when severe heat transfer deterioration may occur. Here we report results of direct numerical simulations of flow with a linear or a step-change profile of body force. In contrast to the conventional view, we show that applying a body force to a turbulent flow while keeping the pressure force unchanged causes little changes to the key characteristics of the turbulence. In particular, the mixing characteristics of the turbulence represented by the turbulent viscosity remain largely unaffected. The so-called flow laminarisation due to a body force is in effect a reduction in the apparent Reynolds number of the flow, based on an apparent friction velocity associated with only the pressure force of the flow (i.e. excluding the contribution of the body force). The new understanding allows the level of the flow 'laminarisation' and when the full laminarisation occurs to be readily predicted. In terms of the near-wall turbulence structure, the numbers of ejections and sweeps are little influenced by the imposition of the body force, whereas the strength of each event may be enhanced if the coverage of the body force extends significantly away from the wall. The streamwise turbulent stress is usually increased in accordance with the observation of more and stronger elongated streaks, but the wall-normal and the circumferential turbulent stresses are largely unchanged. [ABSTRACT FROM AUTHOR]

We use direct numerical simulations to study the effect of thermal boundary conditions on developing turbulent pipe flows with fluids at supercritical pressure. The Reynolds number based on pipe diameter and friction velocity at the inlet is Reτ0 = 360 and Prandtl number at the inlet is Pr0 = 3.19. The thermodynamic conditions are chosen such that the temperature range within the flow domain incorporates the pseudo-critical point where large variations in thermophysical properties occur. Two different thermal wall boundary conditions are studied: one that permits temperature fluctuations and one that does not allow temperature fluctuations at the wall (equivalent to cases where the thermal effusivity ratio approaches infinity and zero, respectively). Unlike for turbulent flows with constant thermophysical properties and Prandtl numbers above unity - where the effusivity ratio has a negligible influence on heat transfer - supercritical fluids shows a strong dependency on the effusivity ratio. We observe a reduction of 7% in Nusselt number when the temperature fluctuations at the wall are suppressed. On the other hand, if temperature fluctuations are permitted, large property variations are induced that consequently cause an increase of wall-normal velocity fluctuations very close to the wall and thus an increased overall heat flux and skin friction. [ABSTRACT FROM AUTHOR]

Townsend's attached eddy hypothesis assumes the existence of a set of energetic and geometrically self-similar eddies in the logarithmic layer in wall-bounded turbulent flows, which can be scaled with their distance to the wall. To examine the possible self-similarity of the energetic eddies in fully developed turbulent pipe flow, we performed stereo particle image velocimetry measurements together with a proper orthogonal decomposition analysis. For two Reynolds numbers, Reτ =1330 and 2460, the resulting modes/eddies were shown to exhibit self-similar behaviour for eddies with wall-normal length scales spanning a decade. This single length scale provides a complete description of the cross-sectional shape of the self-similar eddies. [ABSTRACT FROM AUTHOR]

Wall shear stress measurements employing a hot-film sensor along with laser Doppler velocimetry measurements of the axial and tangential velocity and turbulence profiles in a pulsating turbulent pipe flow are presented. Time-mean and phase-averaged results are derived from measurements performed at pulsation frequencies ω+=ωv/ū2 τ over the range of 0.003-0.03, covering the low-frequency, intermediate and quasi-laminar regimes. In addition to the base case of a single pulsation imposed on the mean flow, the study also investigates the flow response when two pulsations are superimposed simultaneously. The measurements from the base case show that, when the pulsation belongs to the quasi-laminar regime, the oscillating flow tends towards a laminar state in which the velocity approaches the purely viscous Stokes solution with a low level of turbulence. For ω+ < 0:006, the oscillating flow is turbulent and exhibits a region with a logarithmic velocity distribution and a collapse of the turbulence intensities, similar to the time-averaged counterparts. In the low-frequency regime, the oscillating wall shear stress is shown to be directly proportional to the Stokes length normalized in wall units l+s (= √2ω+, as predicted by quasi-steady theory. The base case measurements are used as a reference when evaluating the data from the double-frequency case and the oscillating quantities are shown to be close to superpositions from the base case. The previously established view that the time-averaged quantities are unaffected by the imposition of small-amplitude pulsed unsteadiness is shown to hold also when two pulsations are superposed on the mean flow. [ABSTRACT FROM AUTHOR]

The inertial-subrange scaling of the axial velocity component is examined for the centreline of turbulent pipe flow for Reynolds numbers in the range 249⩽Reλ ⩽986. Estimates of the dissipation rate are made by both integration of the one-dimensional dissipation spectrum and the third-order moment of the structure function. In neither case does the non-dimensional dissipation rate asymptote to a constant; rather than decreasing, it increases indefinitely with Reynolds number. Complete similarity of the inertial range spectra is not evident: there is little support for the hypotheses of Kolmogorov (Dokl. Akad. Nauk SSSR, vol. 32, 1941a, pp. 16-18; Dokl. Akad. Nauk SSSR, vol. 30, 1941b, pp. 301-305) and the effects of Reynolds number are not well represented by Kolmogorov's 'extended similarity hypothesis' (J. Fluid Mech., vol. 13, 1962, pp. 82-85). The second-order moment of the structure function does not show a constant value, even when compensated by the extended similarity hypothesis. When corrected for the effects of finite Reynolds number, the third-order moments of the structure function accurately support the 'four-fifths law', but they do not show a clear plateau. In common with recent work in grid turbulence, non-equilibrium effects can be represented by a heuristic scaling that includes a global Reynolds number as well as a local one. It is likely that non-equilibrium effects appear to be particular to the nature of the boundary conditions. Here, the principal effects of the boundary conditions appear through finite turbulent transport at the pipe centreline, which constitutes a source or a sink at each wavenumber. [ABSTRACT FROM AUTHOR]

Direct numerical simulation was used to study laminar and turbulent flows in circular pipes with smoothly corrugated walls. The corrugation wavelength was kept constant at 0:419D, where D is the mean diameter of the wavy-wall pipe and the corrugation height was varied from zero to 0:08D. Flow rates were varied in steps between low values that generate laminar flow and higher values where the flow is in the post-transitional turbulent regime. Simulations in the turbulent regime were also carried out at a constant Reynolds number, Reτ = 314, for all corrugation heights. It was found that even in the laminar regime, larger-amplitude corrugations produce flow separation. This leads to the proportion of pressure drop attributable to pressure drag being approximately 50 %, and rising to approximately 85% in transitional rough-wall flow. The near-wall structure of turbulent flow is seen to be heavily influenced by the effects of flow separation and reattachment. Farther from the wall, the statistical profiles examined exhibit behaviours characteristic of smooth-wall flows or distributed roughness rough-wall flows. These observations support Townsend's wall-similarity hypothesis. The organized nature of the present roughness allows the mean pressure drop to be written as a function of the corrugation height. When this is exploited in an analysis of the mean dynamical equation, the scaling problem is explicitly revealed to result from the combined influences of roughness and Reynolds number. The present results support the recent analysis and observations of Mehdi et al. (J. Fluid Mech., vol. 731, 2013, pp. 682-712), indicating that the length scale given by the distance from the wall at which the mean viscous force loses leading order is important to describing these combined influences, as well as providing a dynamically self-consistent connection to the scaling structure of smooth-wall pipe flow. [ABSTRACT FROM AUTHOR]

The spectral model of Perry et al. (J. Fluid Mech., vol. 165, 1986, pp. 163–199) predicts that the integral length scale varies very slowly with distance to the wall in the intermediate layer. The only way for the integral length scale’s variation to be more realistic while keeping with the Townsend–Perry attached eddy spectrum is to add a new wavenumber range to the model at wavenumbers smaller than that spectrum. This necessary addition can also account for the high-Reynolds-number outer peak of the turbulent kinetic energy in the intermediate layer. An analytic expression is obtained for this outer peak in agreement with extremely high-Reynolds-number data by Hultmark et al. (Phys. Rev. Lett., vol. 108, 2012, 094501; J. Fluid Mech., vol. 728, 2013, pp. 376–395). Townsend’s (The Structure of Turbulent Shear Flows, 1976, Cambridge University Press) production–dissipation balance and the finding of Dallas et al. (Phys. Rev. E, vol. 80, 2009, 046306) that, in the intermediate layer, the eddy turnover time scales with skin friction velocity and distance to the wall implies that the logarithmic derivative of the mean flow has an outer peak at the same location as the turbulent kinetic energy. This is seen in the data of Hultmark et al. (Phys. Rev. Lett., vol. 108, 2012, 094501; J. Fluid Mech., vol. 728, 2013, pp. 376–395). The same approach also predicts that the logarithmic derivative of the mean flow has a logarithmic decay at distances to the wall larger than the position of the outer peak. This qualitative prediction is also supported by the aforementioned data. [ABSTRACT FROM PUBLISHER]

One-dimensional energy spectra in flat plate zero pressure gradient boundary layers and pipe flows are examined over a wide range of Reynolds numbers (2600 ≤ Reτ 72 500). The spectra show excellent collapse with Kolmogorov scaling at high wavenumbers for both flows at all Reynolds numbers. The peaks associated with the large-scale motions (LSMs) and superstructures (SS) in boundary layers behave as they do in pipe flows, with some minor differences. The location of the outer spectral peak, associated with SS or very large-scale motions (VLSMs) in the turbulent wall region, displays only a weak dependence on Reynolds number, and it occurs at the same wall-normal distance where the variances establish a logarithmic behaviour and where the amplitude modulation coefficient has a zero value. The results suggest that with increasing Reynolds number the energy is largely confined to a thin wall layer that continues to diminish in physical extent. The outer-scaled wavelength of the outer spectral peak appears to decrease with increasing Reynolds number. However, there is still significant energy content in wavelengths associated with the SS and VLSMs. The location of the outer spectral peak appears to mark the start of a plateau that is consistent with a kx-1 slope in the spectrum and the logarithmic variation in the variances. This kx-1 region seems to occur when there is sufficient scale separation between the locations of the outer spectral peak and the outer edge of the log region. It does not require full similarity between outer and wall-normal scaling on the wavenumber. The extent of kx-1 region depends on the wavelength of the outer spectral peak (λOSP), which appears to emerge as a new length scale for the log region. Finally, based on the observations from the spectra together with the statistics presented in Vallikivi et al. (J. Fluid Mech., 2015 (submitted)), five distinct wall-normal layers are identified in turbulent wall flows. [ABSTRACT FROM AUTHOR]

Direct numerical simulations (DNS) are conducted for turbulent flow through pipes with three-dimensional sinusoidal roughnesses explicitly represented by body-conforming grids. The same viscous-scaled roughness geometry is first simulated at a range of different Reynolds numbers to investigate the effects of low Reynolds numbers and low R0/h, where R0 is the pipe radius and h is the roughness height. Results for the present class of surfaces show that the Hama roughness function ΔU+ is only marginally affected by low Reynolds numbers (or low R0/h), and observations of outer-layer similarity (or lack thereof) show no signs of sensitivity to Reynolds number. Then, building on this, a systematic approach is taken to isolate the effects of roughness height h+ and wavelength λ+ in a turbulent wall-bounded flow in both transitionally rough and fully rough regimes. Current findings show that while the effective slope ES (which for the present sinusoidal surfaces is proportional to h+/λ+) is an important roughness parameter, the roughness function ΔU+ must also depend on some measure of the viscous roughness height. A simplistic linear-log fit clearly illustrates the strong correlation between ΔU+ and both the roughness average height ka+ (which is related to h+) and ES for the surfaces simulated here, consistent with published literature. Various definitions of the virtual origin for rough-wall turbulent pipe flow are investigated and, for the surfaces simulated here, the hydraulic radius of the pipe appears to be the most suitable parameter, and indeed is the only virtual origin that can ever lead to collapse in the total stress. First- and second-order statistics are also analysed and collapses in the outer layer are observed for all cases, including those where the largest roughness height is a substantial proportion of the reference radius (low R0/h). These results provide evidence that turbulent pipe flow over the present sinusoidal surfaces adheres to Townsend's notion of outer-layer similarity, which pertains to statistics of relative motion. [ABSTRACT FROM AUTHOR]

We generate predictions for the fluctuating pressure field in turbulent pipe flow by reformulating the resolvent analysis of McKeon and Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382) in terms of the so-called primitive variables. Under this analysis, the nonlinear convective terms in the Fourier-transformed Navier–Stokes equations (NSE) are treated as a forcing that is mapped to a velocity and pressure response by the resolvent of the linearized Navier–Stokes operator. At each wavenumber–frequency combination, the turbulent velocity and pressure field are represented by the most-amplified (rank-1) response modes, identified via a singular value decomposition of the resolvent. We show that these rank-1 response modes reconcile many of the key relationships among the velocity field, coherent structure (i.e. hairpin vortices), and the high-amplitude wall-pressure events observed in previous experiments and direct numerical simulations (DNS). A Green’s function representation shows that the pressure fields obtained under this analysis correspond primarily to the fast pressure contribution arising from the linear interaction between the mean shear and the turbulent wall-normal velocity. Recovering the slow pressure requires an explicit treatment of the nonlinear interactions between the Fourier response modes. By considering the velocity and pressure fields associated with the triadically consistent mode combination studied by Sharma and McKeon (J. Fluid Mech., vol. 728, 2013, pp. 196–238), we identify the possibility of an apparent amplitude modulation effect in the pressure field, similar to that observed for the streamwise velocity field. However, unlike the streamwise velocity, for which the large scales of the flow are in phase with the envelope of the small-scale activity close to the wall, we expect there to be a $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\pi /2$ phase difference between the large-scale wall-pressure and the envelope of the small-scale activity. Finally, we generate spectral predictions based on a rank-1 model assuming broadband forcing across all wavenumber–frequency combinations. Despite the significant simplifying assumptions, this approach reproduces trends observed in previous DNS for the wavenumber spectra of velocity and pressure, and for the scale-dependence of wall-pressure propagation speed. [ABSTRACT FROM PUBLISHER]

Marusic et al. (J. Fluid Mech., vol. 716, 2013, R3) show the first clear evidence of universal logarithmic scaling emerging naturally (and simultaneously) in the mean velocity and the intensity of the streamwise velocity fluctuations about that mean in canonical turbulent flows near walls. These observations represent a significant advance in understanding of the behaviour of wall turbulence at high Reynolds number, but perhaps the most exciting implication of the experimental results lies in the agreement with the predictions of such scaling from a model introduced by Townsend (J. Fluid Mech., vol. 11, 1961, pp. 97–120), commonly termed the attached eddy hypothesis. The elegantly simple, yet powerful, study by Marusic et al. should spark further investigation of the behaviour of all fluctuating velocity components at high Reynolds numbers and the outstanding predictions of the attached eddy hypothesis. [ABSTRACT FROM PUBLISHER]

Direct numerical simulations (DNS) are performed of a transient channel flow following a rapid increase of flow rate from an initially turbulent flow. It is shown that a low-Reynolds-number turbulent flow can undergo a process of transition that resembles the laminar–turbulent transition. In response to the rapid increase of flow rate, the flow does not progressively evolve from the initial turbulent structure to a new one, but undergoes a process involving three distinct phases (pre-transition, transition and fully turbulent) that are equivalent to the three regions of the boundary layer bypass transition, namely, the buffeted laminar flow, the intermittent flow and the fully turbulent flow regions. This transient channel flow represents an alternative bypass transition scenario to the free-stream-turbulence (FST) induced transition, whereby the initial flow serving as the disturbance is a low-Reynolds-number turbulent wall shear flow with pre-existing streaky structures. The flow nevertheless undergoes a ‘receptivity’ process during which the initial structures are modulated by a time-developing boundary layer, forming streaks of apparently specific favourable spacing (of about double the new boundary layer thickness) which are elongated streamwise during the pre-transitional period. The structures are stable and the flow is laminar-like initially; but later in the transitional phase, localized turbulent spots are generated which grow spatially, merge with each other and eventually occupy the entire wall surfaces when the flow becomes fully turbulent. It appears that the presence of the initial turbulent structures does not promote early transition when compared with boundary layer transition of similar FST intensity. New turbulent structures first appear at high wavenumbers extending into a lower-wavenumber spectrum later as turbulent spots grow and join together. In line with the transient energy growth theory, the maximum turbulent kinetic energy in the pre-transitional phase grows linearly but only in terms of ${u}^{\ensuremath{\prime} } $, whilst ${v}^{\ensuremath{\prime} } $ and ${w}^{\ensuremath{\prime} } $ remain essentially unchanged. The energy production and dissipation rates are very low at this stage despite the high level of ${u}^{\ensuremath{\prime} } $. The pressure–strain term remains unchanged at that time, but increases rapidly later during transition along with the generation of turbulent spots, hence providing an unambiguous measure for the onset of transition. [ABSTRACT FROM PUBLISHER]

Acoustics within mean flows are governed by the linearized Euler equations, which possess a singularity wherever the local mean flow velocity is equal to the phase speed of the disturbance. Such locations are termed critical layers, and are usually ignored when estimating the sound field, with their contribution assumed to be negligible. This paper studies fully both numerically and analytically a simple yet typical sheared ducted flow in order to investigate the influence of the critical layer, and shows that the neglect of critical layers is sometimes, but certainly not always, justified. The model is that of a linear-then-constant velocity profile with uniform density in a cylindrical duct, allowing exact Green’s function solutions in terms of Bessel functions and Frobenius expansions. For sources outside the sheared flow, the contribution of the critical layer is found to decay algebraically along the duct as $O(1/ {x}^{4} )$, where $x$ is the distance downstream of the source. For sources within the sheared flow, the contribution from the critical layer is found to consist of a non-modal disturbance of constant amplitude and a disturbance decaying algebraically as $O(1/ {x}^{3} )$. For thin boundary layers, these disturbances trigger the inherent convective instability of the flow. Extra care is required for high frequencies as the critical layer can be neglected only in combination with a particular downstream pole. The advantages of Frobenius expansions over direct numerical calculation are also demonstrated, especially with regard to spurious modes around the critical layer. [ABSTRACT FROM PUBLISHER]

Turbulent Couette–Poiseuille and Couette flows at different mean strain rates (velocity ratio of Couette wall to bulk flow, $r= 0. 6$–$3. 15$), in a square duct at a bulk Reynolds number $\approx $10 000 are investigated by large eddy simulation. The numerical framework consists of a finite-volume method with a staggered-grid arrangement of dependent variables. Spatial derivatives are approximated using second-order centred differencing, and a fractional-step method is used for temporal integration. Simulations are conducted with $160\ensuremath{\times} 160\ensuremath{\times} 256$ grids. Secondary flow near the Couette wall shows a gradual change of vortex size and position as the moving wall velocity increased, where the two clockwise rotating vortices gradually merge in tandem with speed of the moving wall and form a large clockwise vortex. A linear relation is observed to exist between the angle of the two vortices and the parameter $r$, and the angle saturates beyond $r\ensuremath{\sim} 2. 06$. Also, at $0. 6\lt r\lt 1. 6$, together with a small counter-clockwise corner vortex, this vortex pattern is similar to that observed in the corner region of the duct flow with a free surface. The change of the vortex patterns also influences the dominant transport terms in the streamwise vorticity transport equation. Near the moving wall due to the reduction of the streamwise velocity fluctuation at the moving wall, turbulence structure gradually moves towards a rod-like axisymmetric turbulence, and as $r$ increases beyond $1. 2$, turbulence reverts to the disc-like structure. [ABSTRACT FROM AUTHOR]

This paper numerically investigates the effects of a harmonic volume forcing of prescribed frequency on the turbulent pipe flow at a Reynolds number, based on bulk velocity and pipe diameter, of 5900. The thickness of the Stokes layer, resulting from the oscillatory flow component, is a small fraction of the pipe radius and therefore the associated vorticity is confined within a few wall units. The harmonic forcing term is prescribed so that the ratio of the oscillating to the mean bulk velocity ($\ensuremath{\beta} $) ranges between 1 and 10.6. In all cases the oscillatory flow obeys the Stokes analytical velocity distribution while remarkable changes in the current component are observed. At intermediate values $\ensuremath{\beta} = 5$, a relaminarization process occurs, while for $\ensuremath{\beta} = 10. 6$, turbulence is affected so much by the harmonic forcing that the near-wall coherent structures, although not fully suppressed, are substantially weakened. The present study focuses on the analysis of the time- and space-averaged statistics of the first- and second-order moments, vorticity fluctuations and Reynolds stress budgets. Since the flow is unsteady not only locally but also in its space-averaged dynamics, it can be analysed using phase-averaged and time-averaged statistics. While the former gives information about the statistics of the fluctuations about the mean, the latter, postponed to a subsequent paper, shows how the mean is affected by the fluctuations. Clearly, the two phenomena are connected and both of them deserve investigation. [ABSTRACT FROM PUBLISHER]

Fully developed incompressible turbulent pipe flow at Reynolds number ${\mathit{Re}}_{D} = 24\hspace{0.167em} 580$ (based on bulk velocity) and Kármán number ${R}^{+ } = 684. 8$ is simulated in a periodic domain with a length of $30$ pipe radii $R$. While single-point statistics match closely with experimental measurements, questions have been raised of whether streamwise energy spectra calculated from spatial data agree with the well-known bimodal spectrum shape in premultiplied spectra produced by experiments using Taylor’s hypothesis. The simulation supports the importance of large- and very large-scale motions (VLSMs, with streamwise wavelengths exceeding $3R$). Wavenumber spectral analysis shows evidence of a weak peak or flat region associated with VLSMs, independent of Taylor’s hypothesis, and comparisons with experimental spectra are consistent with recent findings (del Álamo & Jiménez, J. Fluid Mech., vol. 640, 2009, pp. 5–26) that the long-wavelength streamwise velocity energy peak is overestimated when Taylor’s hypothesis is used. Yet, the spectrum behaviour retains otherwise similar properties to those documented based on experiment. The spectra also reveal the importance of motions of long streamwise length to the $uu$ energy and $uv$ Reynolds stress and support the general conclusions regarding these quantities formed using experimental measurements. Space–time correlations demonstrate that low-level correlations involving very large scales persist over $40R/ {U}_{\mathit{bulk}} $ in time and indicate that these motions convect at approximately the bulk velocity, including within the region approaching the wall. These very large streamwise motions are also observed to accelerate the flow near the wall based on force spectra, whereas smaller scales tend to decelerate the mean streamwise flow profile, in accordance with the behaviour observed in net force spectra of prior experiments. Net force spectra are resolved for the first time in the buffer layer and reveal an unexpectedly complex structure. [ABSTRACT FROM PUBLISHER]