We examine the full ‘life cycle’ of miscible viscous fingering from onset to shutdown with the aid of high-resolution numerical simulations. We study the injection of one fluid into a planar two-dimensional porous medium containing another, more viscous fluid. We find that the dynamics are distinguished by three regimes: an early-time linearly unstable regime, an intermediate-time nonlinear regime and a late-time single-finger exchange-flow regime. In the first regime, the flow can be linearly unstable to perturbations that grow exponentially. We identify, using linear stability theory and numerical simulations, a critical Péclet number below which the flow remains stable for all times. In the second regime, the flow is dominated by the nonlinear coalescence of fingers which form a mixing zone in which we observe that the convective mixing rate, characterized by a convective Nusselt number, exhibits power-law growth. In this second regime we derive a model for the transversely averaged concentration which shows good agreement with our numerical experiments and extends previous empirical models. Finally, we identify a new final exchange-flow regime in which a pair of counter-propagating diffusive fingers slow exponentially. We derive an analytic solution for this single-finger state which agrees well with numerical simulations. We demonstrate that the flow always evolves to this regime, irrespective of the viscosity ratio and Péclet number, in contrast to previous suggestions.

Advective–diffusive transport of passive or reactive scalars in confined environments (e.g. tubes and channels) is often accompanied by diffusive losses/gains through the confining walls. We present analytical solutions for transport of a reactive solute in a tube, whose walls are impermeable to flow but allow for solute diffusion into the surrounding medium. The solute undergoes advection, diffusion and first-order chemical reaction inside the tube, while diffusing and being consumed in the surrounding medium. These solutions represent a leading-order (in the radius-to-length ratio) approximation, which neglects the longitudinal variability of solute concentration in the surrounding medium. A numerical solution of the full problem is used to demonstrate the accuracy of this approximation for a physically relevant range of model parameters. Our analysis indicates that the solute delivery rate can be quantified by a dimensionless parameter, the ratio of a solute’s residence time in a tube to the rate of diffusive losses through the tube’s wall.

The vertical heat transfer in Bénard–Marangoni convection of a fluid layer with infinite Prandtl number is studied by means of upper bounds on the Nusselt number $Nu$ as a function of the Marangoni number $Ma$. Using the background method for the temperature field, it has recently been proved by Hagstrom & Doering (Phys. Rev. E, vol. 81, 2010, art. 047301) that $Nu\leqslant 0.838Ma^{2/7}$. In this work we extend previous background method analysis to include balance parameters and derive a variational principle for the bound on $Nu$, expressed in terms of a scaled background field, that yields a better bound than Hagstrom & Doering’s formulation at a given $Ma$. Using a piecewise-linear, monotonically decreasing profile we then show that $Nu\leqslant 0.803Ma^{2/7}$, lowering the previous prefactor by 4.2 %. However, we also demonstrate that optimisation of the balance parameters does not affect the asymptotic scaling of the optimal bound achievable with Hagstrom & Doering’s original formulation. We subsequently utilise convex optimisation to optimise the bound on $Nu$ over all admissible background fields, as well as over two smaller families of profiles constrained by monotonicity and convexity. The results show that $Nu\leqslant O(Ma^{2/7}(\ln Ma)^{-1/2})$ when the background field has a non-monotonic boundary layer near the surface, while a power-law bound with exponent $2/7$ is optimal within the class of monotonic background fields. Further analysis of our upper-bounding principle reveals the role of non-monotonicity, and how it may be exploited in a rigorous mathematical argument.

The variation of transitional flow features past a micro-ramp is investigated when the Reynolds number is decreased approaching the critical regime. Experiments are conducted in the incompressible flow spanning from supercritical to subcritical roughness-height-based Reynolds number ($Re_{h}=1170$, 730, 460 and 320) with tomographic particle image velocimetry. The effect of $Re_{h}$ on three-dimensional flow behaviour is analysed in a domain encompassing 73 ramp heights in the streamwise direction. Above the critical $Re_{h}$, the primary vortex pair and induced central low-speed region in the mean flow field are active over longer range when decreasing $Re_{h}$. In the instantaneous flow, at $Re_{h}<1000$, the hairpin vortices induced by Kelvin–Helmholtz (K–H) instability progress gradually from close to the micro-ramp into the region where the overall shear layer is destabilized, indicating the correlation between the K–H instability and the onset of transition. The breakdown of K–H vortices as observed at $Re_{h}=1170$, does not occur at lower $Re_{h}$. Decreasing $Re_{h}$, the secondary vortex structures make their first appearance significantly downstream, postponing the formation of sideward disturbances, which destabilize the local shear layer by ejection events. Two major types of eigenmodes with symmetric and asymmetric spatial distribution of velocity fluctuations in the near wake are clearly identified by proper orthogonal decomposition. The symmetric and asymmetric modes correspond to the presence of vortex shedding and a sinuous wiggling motion respectively. It is found that $Re_{h}$ is the key factor determining the importance of the symmetric mode. At $Re_{h}=1170$, the disturbance energy of the symmetric mode decays before the onset of transition, suggesting that it is relatively insignificant in the process. However, decreasing $Re_{h}$ to 730 and 460, the symmetric mode produces continuous growth of high level disturbance energy, leading to transition.

This paper examines flow-separation lines on axisymmetric bodies with tapered tails, where the separating flow takes into account the effect of local body radius $r(x)$, incidence angle $\unicode[STIX]{x1D713}$ and the body-length Reynolds number $\mathit{Re}_{L}$. The flow is interpreted as a transient problem which relates the longitudinal distance $x$ to time $t^{\ast }=x\,\text{tan}(\unicode[STIX]{x1D713})/r(x)$, similar to the approach of Jeans & Holloway (J. Aircraft, vol. 47 (6), 2010, pp. 2177–2183) on scaling separation lines. The windward and leeward sides correspond to the body azimuth angles $\unicode[STIX]{x1D6E9}=0$ and $180^{\circ }$, respectively. From China-clay flow visualisation on axisymmetric bodies and from a literature review of slender-body flows, the present study shows three findings. (i) The time scale $t^{\ast }$ provides a collapse of the separation-line data, $\unicode[STIX]{x1D6E9}$, for incidence angles between $6$ and $35^{\circ }$, where the data fall on a power law $\unicode[STIX]{x1D6E9}\sim (t^{\ast })^{k}$. (ii) The data suggest that the separation rate $k$ is independent of the Reynolds number over the range $2.1\times 10^{6}\leqslant \mathit{Re}_{L}\leqslant 23\times 10^{6}$; for a primary separation $k_{1}\simeq -0.190$, and for a secondary separation $k_{2}\simeq 0.045$. (iii) The power-law curve fits trace the primary and secondary lines to a characteristic start time $t_{s}^{\ast }\simeq 1.5$.

The propagation of wave disturbances in water of varying depth bounded above by ice sheets is discussed, accounting for gravity, compressibility and elasticity effects. Considering the more realistic scenario of elastic ice sheets reveals a continuous spectrum of acoustic–gravity modes that propagate even below the cutoff frequency of the rigid surface solution where surface (gravity) waves cannot exist. The balance between gravitational forces and oscillations in the ice sheet defines a new dimensionless quantity $\mathfrak{Ka}$. When the ice sheet is relatively thin and the prescribed frequency is relatively low ($\mathfrak{Ka}\ll 1$), the free-surface bottom-pressure solution is retrieved in full. However, thicker ice sheets or propagation of relatively higher frequency modes ($\mathfrak{Ka}\gg 1$) alter the solution fundamentally, which is reflected in an amplified asymmetric signature and different characteristics of the eigenvalues, such that the bottom pressure is amplified when acoustic–gravity waves are transmitted to shallower waters. To analyse these scenarios, an analytical solution and a depth-integrated equation are derived for the cases of constant and varying depths, respectively. Together, these are capable of modelling realistic ocean geometries and an inhomogeneous distribution of ice sheets.

All previous experiments in open turbulent flows (e.g. downstream of grids, jets and the atmospheric boundary layer) have produced quantitatively consistent values for the scaling exponents of velocity structure functions (Anselmet et al., J. Fluid Mech., vol. 140, 1984, pp. 63–89; Stolovitzky et al., Phys. Rev. E, vol. 48 (5), 1993, R3217; Arneodo et al., Europhys. Lett., vol. 34 (6), 1996, p. 411). The only measurement of scaling exponents at high order (${>}6$) in closed turbulent flow (von Kármán swirling flow) using Taylor’s frozen flow hypothesis, however, produced scaling exponents that are significantly smaller, suggesting that the universality of these exponents is broken with respect to change of large scale geometry of the flow. Here, we report measurements of longitudinal structure functions of velocity in a von Kármán set-up without the use of the Taylor hypothesis. The measurements are made using stereo particle image velocimetry at four different ranges of spatial scales, in order to observe a combined inertial subrange spanning approximately one and a half orders of magnitude. We found scaling exponents (up to ninth order) that are consistent with values from open turbulent flows, suggesting that they might be in fact universal.

High-Rayleigh-number ($Ra$) convection in an inclined two-dimensional porous layer is investigated using direct numerical simulations (DNS) and stability and variational upper-bound analyses. When the inclination angle $\unicode[STIX]{x1D719}$ of the layer satisfies $0^{\circ }<\unicode[STIX]{x1D719}\lesssim 25^{\circ }$, DNS confirm that the flow exhibits a three-region wall-normal asymptotic structure in accord with the strictly horizontal ($\unicode[STIX]{x1D719}=0^{\circ }$) case, except that as $\unicode[STIX]{x1D719}$ is increased the time-mean spacing between neighbouring interior plumes also increases substantially. Both DNS and upper-bound analysis indicate that the heat transport enhancement factor (i.e. the Nusselt number) $Nu\sim CRa$ with a $\unicode[STIX]{x1D719}$-dependent prefactor $C$. When $\unicode[STIX]{x1D719}>\unicode[STIX]{x1D719}_{t}$, however, where $30^{\circ }<\unicode[STIX]{x1D719}_{t}<32^{\circ }$ independently of $Ra$, the columnar flow structure is completely broken down: the flow transitions to a large-scale travelling-wave convective roll state, and the heat transport is significantly reduced. To better understand the physics of inclined porous medium convection at large $Ra$ and modest inclination angles, a spatial Floquet analysis is performed, yielding predictions of the linear stability of numerically computed, fully nonlinear steady convective states. The results show that there exist two types of instability when $\unicode[STIX]{x1D719}\neq 0^{\circ }$: a bulk-mode instability and a wall-mode instability, consistent with previous findings for $\unicode[STIX]{x1D719}=0^{\circ }$ (Wen et al., J. Fluid Mech., vol. 772, 2015, pp. 197–224). The background flow induced by the inclination of the layer intensifies the bulk-mode instability during its subsequent nonlinear evolution, thereby favouring increased spacing between the interior plumes relative to that observed in convection in a horizontal porous layer.

The objective of the present work is to study the drying of a quasi-two-dimensional model porous medium, hereafter called the micromodel, initially filled with a pure liquid. The micromodel consists of cylinders measuring $50~\unicode[STIX]{x03BC}\text{m}$ in both height and diameter, radially arranged as a set of neighbouring spirals and sandwiched between two horizontal flat plates. As drying proceeds, air invades the pore space and elongated liquid films trapped by capillary forces form along the spirals. These films consist of ‘chains’ of liquid bridges connecting neighbouring cylinders. They provide hydraulic connectivity between the central bulk liquid cluster and the external rim of the cylinder pattern, where evaporation takes place during a first constant-evaporation-rate drying stage. The first goal of the present paper is to describe experimentally the phase distribution during drying, notably the evolution of liquid films, which controls the evaporation kinetics (e.g. the depinning of the films from the external rim signals the end of the constant-evaporation-rate period). Then, a viscocapillary model for the drying process is presented. It is based on numerical simulations of a liquid film capillary shape and viscous flow within a film. The model shows a reasonably good agreement with the experimental data. Thus, the present study is a step towards direct modelling of the effect of films on the drying of more complex porous media (e.g. packing of beads) and should be of interest for multiphase flow applications in porous media, involving transport within liquid films.

For the first time, an experiment has been conducted to investigate synthetic jet laminar vortex rings impinging onto porous walls with different geometries by time-resolved particle image velocimetry. The geometry of the porous wall is changed by varying the hole diameter on the wall (from 1.0 mm to 3.0 mm) when surface porosity is kept constant ($\unicode[STIX]{x1D719}=75\,\%$). The finite-time Lyapunov exponent and phase-averaged vorticity field derived from particle image velocimetry data are presented to reveal the evolution of the vortical structures. A mechanism associated with vorticity cancellation is proposed to explain the formation of downstream transmitted vortex rings; and both the vortex ring trajectory and the time-mean flow feature are compared between different cases. It is found that the hole diameter significantly influences the evolution of the flow structures on both the upstream and downstream sides of the porous wall. In particular, for a porous wall with a small hole diameter ($d_{h}^{\ast }=0.067$, 0.10 and 0.133), the transmitted finger-type jets will reorganize into a well-formed transmitted vortex ring in the downstream flow. However, for the case of a large hole diameter of $d_{h}^{\ast }=0.20$, the transmitted vortex ring is not well formed because of insufficient vorticity cancellation. Additionally, the residual vorticity gradually evolves into discrete jet-like structures downstream, which further weaken the intensity of the transmitted vortex ring. Consequently, the transmitted flow structures for the $d_{h}^{\ast }=0.20$ case would lose coherence more easily (or probably even transition to turbulence), resulting in a faster decay of the axial velocity and stronger entrainment of the transmitted jet. For all porous wall cases, the velocity profile of the transmitted jet exhibits self-similar behaviour in the far field ($z/D_{0}\geqslant 6.03$), which agrees well with the velocity distribution of free synthetic jets. With the help of the control-volume approach, the time-mean drag of the porous wall is evaluated experimentally for the first time. It is shown that the porous wall drag increases with the decrease in the hole diameter. Moreover, for a porous wall with a small hole diameter ($d_{h}^{\ast }=0.067$, 0.10 and 0.133), it appears that the porous wall drag mainly derives from the viscous effect. However, as $d_{h}^{\ast }$ increases to 0.20, the form drag associated with the porous wall geometry becomes significant.

The recent development of the adaptive-anisotropic wavelet-collocation method, which incorporates the use of coordinate transforms, opens new horizons for wavelet-based simulations of wall-bounded turbulent flows. The new wavelet-based adaptive unsteady Reynolds-averaged Navier–Stokes approach for computational modelling of turbulent flows is presented. The proposed methodology that is integrated with anisotropic wavelet-based mesh refinement is demonstrated for a two-equation eddy-viscosity turbulence model. The performance of the method is assessed by conducting numerical simulations of the turbulent flow past a circular cylinder at subcritical Reynolds number. The present study demonstrates both the feasibility and the effectiveness of the new wavelet-based adaptive unsteady Reynolds-averaged turbulence modelling procedure for external flows.

The sloshing of water waves in a vertical cylindrical tank is an archetypal damped oscillator in fluid mechanics. The wave frequency is traditionally derived in the potential flow limit (Lamb, Hydrodynamics, Cambridge University Press, 1932), and the damping rate results from the combined effects of the viscous dissipation at the wall, in the bulk and at the free surface (Case & Parkinson, J. Fluid Mech., vol. 2, 1957, pp. 172–184). Still, the classic theoretical prediction accounting for these effects significantly underestimates the damping rate when compared to careful laboratory experiments (Cocciaro et al., J. Fluid Mech., vol. 246, 1993, pp. 43–66). Specifically, theory provides a unique value for the damping rate, while experiments reveal that the damping increases as the sloshing amplitude decreases. Here, we investigate theoretically the effects of capillarity at the contact line on the decay time of capillary–gravity waves. To this end, we marry a model for the inviscid waves to a nonlinear empiric law for the contact line that incorporates contact angle hysteresis. The resulting system of equations is solved by means of a weakly nonlinear analysis using the method of multiple scales. Capillary effects have a dramatic influence on the calculated damping rate, especially when the sloshing amplitude gets small: this nonlinear interfacial term increases in the limit of zero wave amplitude. In contrast to viscous damping, where the wave motion decays exponentially, the contact angle hysteresis can act as Coulomb solid friction, thus yielding the arrest of the contact line in a finite time.

A static rivulet is subject to disturbances in shape, velocity and pressure fields. Disturbances to interfacial shape accommodate a contact line that is either (i) fixed (pinned) or (ii) fully mobile (free) and preserves the static contact angle. The governing hydrodynamic equations for this inviscid, incompressible fluid are derived and then reduced to a functional eigenvalue problem on linear operators, which are parametrized by axial wavenumber and base-state volume. Solutions are decomposed according to their symmetry (varicose) or anti-symmetry (sinuous) about the vertical mid-plane. Dispersion relations are then computed. Static stability is obtained by setting growth rate to zero and recovers existing literature results. Critical growth rates and wavenumbers for the varicose and sinuous modes are reported. For the varicose mode, typical capillary break-up persists and the role of the liquid/solid interaction on the critical disturbance is illustrated. There exists a range of parameters for which the sinuous mode is the dominant instability mode. The sinuous instability mechanism is shown to correlate with horizontal centre-of-mass motion and illustrated using a toy model.

We derive and assess the sharpness of analytic upper bounds for the instantaneous growth rate and finite-time amplification of palinstrophy in solutions of the two-dimensional incompressible Navier–Stokes equations. A family of optimal solenoidal fields parametrized by initial values for the Reynolds number $Re$ and palinstrophy ${\mathcal{P}}$ which maximize $\text{d}{\mathcal{P}}/\text{d}t$ is constructed by numerically solving suitable optimization problems for a wide range of $Re$ and ${\mathcal{P}}$, providing numerical evidence for the sharpness of the analytic estimate $\text{d}{\mathcal{P}}/\text{d}t\leqslant (a+b\sqrt{\ln Re+c}){\mathcal{P}}^{3/2}$ with respect to both $Re$ and ${\mathcal{P}}$. This family of instantaneously optimal fields is then used as initial data in fully resolved direct numerical simulations, and the time evolution of different relevant norms is carefully monitored as the palinstrophy is transiently amplified before decaying. The peak values of the palinstrophy produced by these initial data, i.e. $\sup _{t>0}{\mathcal{P}}(t)$, are observed to scale with the magnitude of the initial palinstrophy ${\mathcal{P}}(0)$ in accord with the corresponding a priori estimate. Implications of these findings for the question of finite-time singularity formation in the three-dimensional incompressible Navier–Stokes equation are discussed.

Trailing vortices are generated in aeronautical and maritime applications and produce a variety of adverse effects that remain difficult to control. A stability analysis can direct flow control designers towards pertinent frequencies, wavelengths and locations that may lead to the excitation of instabilities, resulting in the eventual breakup of the vortex. Most models for trailing vortices, however, are far-field models, making implementation of the findings from stability analyses challenging. As such, we perform a stability analysis in the formative region where the numerically computed base flow contains both a two-dimensional wake and a tip vortex generated from a NACA0012 at a $5^{\circ }$ angle of attack and a chord-based Reynolds number of $Re_{c}=1000$. The parallel temporal and spatial analyses show that at three chord lengths downstream of the trailing edge, seven unstable modes are present: three stemming from the temporal analysis and four arising in the spatial analysis. The three temporal instabilities are analogues to three unstable modes in the spatial analysis, with the wake instability dominating in both analyses. The helical mode localized to the vortex co-rotates with the base flow, which is converse with the counter-rotating $m=-1$ instabilities of a Batchelor vortex model, which may be a result of the formative nature of the base-flow vortex. The fourth spatial mode is localized to the tip vortex region. The continuous part of the spectrum contains oscillatory and wavepacket solutions prompting the utilization of a wavepacket analysis to analyse the flow field and group velocity. The structure and details of the full bi-global spectrum will help navigate the design space of effective control strategies to hasten decay of persistent wingtip vortices.

Wake visualization experiments were conducted on a finite curved cylinder whose plane of curvature is aligned with the free stream. The stagnation face of the cylinder is oriented concave or convex to the flow at $230\leqslant Re_{D}\leqslant 916$, where $Re_{D}$ is the cylinder Reynolds number and the curvature is constant and ranges from a straight cylinder to a quarter-ring. While the magnitude of the local angle of incidence to the flow is the same for both orientations, the contrast in their wakes demonstrates a violation of a common approximation known as the ‘independence principle’ for curved cylinders. Vortex shedding always occurred for the convex-oriented cylinder for the Reynolds-number range investigated, along most of the cylinder span, at a constant vortex shedding angle. In contrast, a concave-oriented cylinder could exhibit multiple concurrent wake regimes along its span: two shedding regimes (oblique, normal) and two non-shedding regimes. The occurrence of these wake regimes depended on the curvature, aspect ratio and Reynolds number. In some cases, vortex shedding was entirely suppressed, particularly at higher curvatures. In the laminar wake regime, increasing the curvature or decreasing the aspect ratio restricts vortex shedding to smaller regions along the span of the cylinder. Furthermore, the local angle of incidence where vortex shedding occurs is self-similar across cylinders of the same aspect ratio and varying curvature. After the wake transitions to turbulence, the vortex shedding extends along most of the cylinder span. The difference in the wakes between the concave and convex orientations is attributed to the spanwise flow induced by the finite end conditions, which reduces the generation of spanwise vorticity and increases the incidence of non-shedding and obliquely shedding wakes for the concave cylinder.

The free-stream coherent structure theory developed by Deguchi & Hall (J. Fluid Mech., vol. 752, 2014, pp. 602–625), valid in the large-Reynolds-number asymptotic limit, is extended and applied to jet flows. It is shown that a nonlinear exact coherent structure can be supported at the edge of the jet, and the structure induces a much bigger streaky flow in the centre of the jet. The lambda-shaped vortices that characterise the coherent structure are qualitatively consistent with those seen in experimental observations. Here a planar incompressible jet is investigated for the sake of simplicity, but the structure we describe could be used as a basis of more complex theories for incompressible and compressible jets of practical importance.

Flow-induced vibration of an elastically mounted sphere was investigated computationally for the classic case where the sphere motion was constrained to move in a direction transverse to the free stream. This study, therefore, provides additional insight into, and comparison with, corresponding experimental studies of transverse motion, and distinction from numerical and experimental studies with specific constraints such as tethering (Williamson & Govardhan, J. Fluids Struct., vol. 11, 1997, pp. 293–305) or motion in all three directions (Behara et al., J. Fluid Mech., vol. 686, 2011, pp. 426–450). Two sets of simulations were conducted by fixing the Reynolds number at $Re=300$ or 800 over the reduced velocity ranges $3.5\leqslant U^{\ast }\leqslant 100$ and $3\leqslant U^{\ast }\leqslant 50$ respectively. The reduced mass of the sphere was kept constant at $m_{r}=1.5$ for both sets. The flow satisfied the incompressible Navier–Stokes equations, while the coupled sphere motion was modelled by a spring–mass–damper system, with damping set to zero. The sphere showed a highly periodic large-amplitude vortex-induced vibration response over a lower reduced velocity range at both Reynolds numbers considered. This response was designated as branch A, rather than the initial/upper or mode I/II branch, in order to allow it to be discussed independently from the observed experimental response at higher Reynolds numbers which shows both similarities and differences. At $Re=300$, it occurred over the range $5.5\leqslant U^{\ast }\leqslant 10$, with a maximum oscillation amplitude of ${\approx}0.4D$. On increasing the Reynolds number to 800, this branch widened to cover the range $4.5\leqslant U^{\ast }\leqslant 13$ and the oscillation amplitude increased (maximum amplitude ${\approx}0.6D$). In terms of wake dynamics, within this response branch, two streets of interlaced hairpin-type vortex loops were formed behind the sphere. The upper and lower sets of vortex loops were disconnected, as were their accompanying tails. The wake maintained symmetry relative to the plane defined by the streamwise and sphere motion directions. The topology of this wake structure was analogous to that seen experimentally at higher Reynolds numbers by Govardhan & Williamson (J. Fluid Mech., vol. 531, 2005, pp. 11–47). At even higher reduced velocities, the sphere showed distinct oscillatory behaviour at both Reynolds numbers examined. At $Re=300$, small but non-negligible oscillations were found to occur (amplitude of ${\approx}0.05D$) within the reduced velocity ranges $13\leqslant U^{\ast }\leqslant 16$ and $26\leqslant U^{\ast }\leqslant 100$, named branch B and branch C respectively. Moreover, within these reduced velocity ranges, the centre of motion of the sphere shifted from its static position. In contrast, at $Re=800$, the sphere showed an aperiodic intermittent mode IV vibration state immediately beyond branch A, for $U^{\ast }\geqslant 14$. This vibration state was designated as the intermittent branch. Interestingly, the dominant frequency of the sphere vibration was close to the natural frequency of the system, as observed by Jauvtis et al. (J. Fluids Struct., vol. 15(3), 2001, pp. 555–563) in higher-mass-ratio higher-Reynolds-number experiments. The oscillation amplitude increased as the reduced velocity increased and reached a value of ${\approx}0.9D$ at $U^{\ast }=50$. The wake was irregular, with multiple vortex shedding cycles during each cycle of sphere oscillation.

A tiled approach to rough surface simulation is used to explore the full range of roughness Reynolds numbers, from the limiting case of hydrodynamic smoothness up to fully rough conditions. The surface is based on a scan of a standard grit-blasted comparator, subsequently low-pass filtered and made spatially periodic. High roughness Reynolds numbers are obtained by increasing the friction Reynolds number of the direct numerical simulations, whereas low roughness Reynolds numbers are obtained by scaling the surface down and tiling to maintain a constant domain size. In both cases, computational requirements on box size, resolution in wall units and resolution per minimum wavelength of the rough surface are maintained. The resulting roughness function behaviour replicates to good accuracy the experiments of Nikuradse (1933 VDI-Forschungsheft, vol. 361), suggesting that the processed grit-blasted surface can serve as a surrogate for his sand-grain roughness, the precise structure of which is undocumented. The present simulations also document a monotonic departure from hydrodynamic smooth-wall results, which is fitted with a geometric relation, the exponent of which is found to be inconsistent with both the Colebrook formula and an earlier theoretical argument based on low-Reynolds-number drag relations.

We present an investigation into the influence of upstream shear on the viscous flow around a steady two-dimensional (2-D) symmetric airfoil at zero angle of attack, and the corresponding loads. In this computational study, we consider the NACA 0012 airfoil at a chord Reynolds number $1.2\times 10^{4}$ in an approach flow with uniform positive shear with non-dimensional shear rate varying in the range 0.0–1.0. Results show that the lift force is negative, in the opposite direction to the prediction from Tsien’s inviscid theory for lift generation in the presence of positive shear. A hypothesis is presented to explain the observed sign of the lift force on the basis of the asymmetry in boundary layer development on the upper and lower surfaces of the airfoil, which creates an effective airfoil shape with negative camber. The resulting scaling of the viscous effect with shear rate and Reynolds number is provided. The location of the leading edge stagnation point moves increasingly farther back along the airfoil’s upper surface with increased shear rate, a behaviour consistent with a negatively cambered airfoil. Furthermore, the symmetry in the location of the boundary layer separation point on the airfoil’s upper and lower surfaces in uniform flow is broken under the imposed shear, and the wake vortical structures exhibit more asymmetry with increasing shear rate.