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This paper examines the Reynolds number ( $Re$ ) dependence of a zero-pressure-gradient (ZPG) turbulent boundary layer (TBL) which develops over a two-dimensional rough wall with a view to ascertaining whether this type of boundary layer can become independent of $Re$ . Measurements are made using hot-wire anemometry over a rough wall that consists of a periodic arrangement of cylindrical rods with a streamwise spacing of eight times the rod diameter. The present results, together with those obtained over a sand-grain roughness at high Reynolds number, indicate that a $Re$ -independent state can be achieved at a moderate $Re$ . However, it is also found that the mean velocity distributions over different roughness geometries do not collapse when normalised by appropriate velocity and length scales. This lack of collapse is attributed to the difference in the drag coefficient between these geometries. We also show that the collapse of the $U_{\unicode[STIX]{x1D70F}}$ -normalised mean velocity defect profiles may not necessarily reflect $Re$ -independence. A better indicator of the asymptotic state of $Re$ is the mean velocity defect profile normalised by the free-stream velocity and plotted as a function of $y/\unicode[STIX]{x1D6FF}$ , where $y$ is the vertical distance from the wall and $\unicode[STIX]{x1D6FF}$ is the boundary layer thickness. This is well supported by the measurements.
In a study motivated by considerations associated with heart murmurs and cardiac auscultation, numerical simulations are used to analyse the haemodynamics in a simple model of an aorta with an aortic stenosis. The aorta is modelled as a curved pipe with a $180^{\circ }$ turn, and three different stenoses with area reductions of 50 %, 62.5 % and 75 % are examined. A uniform steady inlet velocity with a Reynolds number of 2000 is used for all of the cases and direct numerical simulation is employed to resolve the dynamics of the flow. The poststenotic flow is dominated by the jet that originates from the stenosis as well as the secondary flow induced by the curvature, and both contribute significantly to the flow turbulence. On the anterior surface of the modelled aorta, the location with maximum pressure fluctuation, which may be considered as the source location for the murmurs, is found to be located around $60^{\circ }$ along the aortic arch, and this location is relatively insensitive to the severity of the stenosis. For all three cases, this high-intensity wall pressure fluctuation includes contributions from both the jet and the secondary flow. Spectral analysis shows that for all three stenoses, the Strouhal number of the vortex shedding of the jet shear layer is close to 0.93, which is higher than the shedding frequency of a corresponding free jet or a jet confined in a straight pipe. This frequency also appears in the pressure spectra at the location postulated as the source of the murmurs, in the form of a ‘break frequency.’ The implications of these findings for cardiac auscultation-based diagnosis of aortic stenosis are also discussed.
The role of the seam in the ‘swing’ of a cricket ball is investigated via unsteady force and surface-pressure measurements and oil-flow visualization in a low-turbulence wind tunnel. Various seam angles of the ball and flow speeds are considered. Static tests are carried out on a new ‘SG Test’ cricket ball as well as its idealized models: a smooth sphere with one and five trips. To study the effect of surface roughness of the ball as the game progresses, force measurements are also carried out on a cricket ball that is manually roughened, on one-half and completely, to model a ball that has been in play for approximately 40 overs (240 deliveries/balls). The Reynolds number ( $Re$ ) is based on the free-stream speed and diameter of the respective model. A new cricket ball experiences three flow states with increase in $Re$ : no swing (NS), conventional swing (CS) and reverse swing (RS). At relatively low $Re$ , in the NS regime, the seam does not have any significant effect on the flow. The separation of the laminar boundary layer, with no subsequent reattachment, is almost axisymmetric with respect to the free-stream flow. Therefore, the ball does not experience any significant lateral force. Beyond a certain $Re$ , the boundary layer on the seam side of the ball undergoes transition. The boundary layer on the non-seam side, however, continues to undergo a laminar separation with no reattachment, thereby creating a lateral force in the direction of the seam, leading to CS. The onset of the CS regime is marked by intermittent formation of a laminar separation bubble (LSB) on the surface of the ball in the region between the laminar separation of the boundary layer and its reattachment at a downstream location. Owing to the varying azimuthal location of the seam, with respect to the front stagnation point on the ball, the transition via LSB formation is localized to a specific region over the seam side. In other regions, the boundary layer either transitions directly without the formation of an LSB, or separates on encountering the seam with no further reattachment. The spatial extent of the region where the flow directly transitions to a turbulent state increases with increase in $Re$ , while that of the LSB decreases. Interestingly, the flow dynamics is such that the magnitude of the swing force coefficient stays relatively constant with increase in $Re$ . With further increase in $Re$ , the boundary layer on the non-seam side undergoes a transition via formation of an LSB. This, along with an upstream shift of the separation point on the seam side, leads to a switch in the direction of the lateral force. It now acts away from the seam, and leads to RS. The transition from CS to RS occurs over a very narrow range of $Re$ wherein the flow intermittently switches between the two flow states. It is observed that the transition of the boundary layer on the seam side leads to an upstream shift of the separation point on the non-seam side at the onset of CS. A complementary effect is observed at the onset of RS. Experiments on a ball that is manually roughened bring out the relative effect of the seam and roughness on the transition of the boundary layer. Compared to a new ball, the magnitude of the maximum swing force coefficient for a rough ball is smaller during the CS regime, and larger during the RS regime. Unlike other models, the ball with roughened non-seam side and smooth seam side, for certain seam orientations, exhibits RS at relatively lower speeds and CS at higher speeds. The forces measured on the cricket ball are utilized to estimate the trajectory of the ball bowled at various initial speeds and seam angles. The lateral movement of the ball depends very significantly on the seam angle, surface roughness and speed of the ball at its delivery. The maximum lateral deviation of a new ball during RS is found to be less than half of that observed in CS. On the other hand, the lateral movement of a roughened ball during RS may significantly exceed its movement during CS. The range of the speed of the ball, for various seam orientations and surface roughnesses, are estimated wherein it undergoes CS, RS or one followed by the other. Optimal conditions are estimated for the desired lateral movement of the ball.
To investigate the effects of the nozzle-exit conditions on jet flow and sound fields, large-eddy simulations of an isothermal Mach 0.9 jet issued from a convergent-straight nozzle are performed at a diameter-based Reynolds number of $1\times 10^{6}$ . The simulations feature near-wall adaptive mesh refinement, synthetic turbulence and wall modelling inside the nozzle. This leads to fully turbulent nozzle-exit boundary layers and results in significant improvements for the flow field and sound predictions compared with those obtained from the typical approach based on laminar flow in the nozzle. The far-field pressure spectra for the turbulent jet match companion experimental measurements, which use a boundary-layer trip to ensure a turbulent nozzle-exit boundary layer to within 0.5 dB for all relevant angles and frequencies. By contrast, the initially laminar jet results in greater high-frequency noise. For both initially laminar and turbulent jets, decomposition of the radiated noise into azimuthal Fourier modes is performed, and the results show similar azimuthal characteristics for the two jets. The axisymmetric mode is the dominant source of sound at the peak radiation angles and frequencies. The first three azimuthal modes recover more than 97 % of the total acoustic energy at these angles and more than 65 % (i.e. error less than 2 dB) for all angles. For the main azimuthal modes, linear stability analysis of the near-nozzle mean-velocity profiles is conducted in both jets. The analysis suggests that the differences in radiated noise between the initially laminar and turbulent jets are related to the differences in growth rate of the Kelvin–Helmholtz mode in the near-nozzle region.
Collision between two identical counterflowing gravity currents was studied in the laboratory with the goal of understanding the fundamental turbulent mixing physics of flow collisions in nature, for example katabatic flows and thunderstorm outflows. The ensuing turbulent mixing is a subgrid process in mesoscale forecasting models, and needs to be parameterized using eddy diffusivity. Laboratory gravity currents were generated by simultaneously removing two identical locks, located at both ends of a long rectangular tank, which separated dense and lighter water columns with free surfaces of the same depth $H$ . The frontal velocity $u_{f}$ and the velocity and density fields of the gravity currents were monitored using time-resolved particle image velocimetry and planar laser-induced fluorescence imaging. Ensemble averaging of identical experimental realizations was used to compute turbulence statistics, after removing inherent jitter via phase alignment of successive data realizations by iteratively maximizing the cross-correlation of each realization with the ensemble average. Four stages of flow evolution were identified: initial (independent) propagation of gravity currents, their approach while influencing one another, collision and resulting updraughts, and postcollision slumping of collided fluid. The collision stage, in turn, involved three phases, and produced the strongest turbulent mixing as quantified by the rate of change of density. Phase I spanned $-0.2\leqslant tu_{f}/H<0.5$ , where collision produced a rising density front (interface) with strong shear and intense turbulent kinetic energy production ( $t$ is a suitably defined time coordinate such that gravity currents make the initial contact at $tu_{f}/H=-0.2$ ). In Phase II ( $0.5\leqslant tu_{f}/H<1.2$ ), the interface was flat and calm with negligible vertical velocity. Phase III ( $1.2\leqslant tu_{f}/H<2.8$ ) was characterized by slumping which led to hydraulic bores propagating away from the collision area. The measurements included root mean square turbulent velocities and their decay rates, interfacial velocity, rate of change of fluid-parcel density, and eddy diffusivity. These measures depended on the Reynolds number $Re$ , but appeared to achieve Reynolds number similarity for $Re>3000$ . The eddy diffusivity $K_{T}$ , space–time averaged over the spatial extent ( $H\times H$ ) and the lifetime ( $t\approx 3H/u_{f}$ ) of collision, was $K_{T}/u_{f}H=0.0036$ for $Re>3000$ , with the area $A$ of active mixing being $A/H^{2}=0.037$ .
We study the turbulent square duct flow of dense suspensions of neutrally buoyant spherical particles. Direct numerical simulations (DNS) are performed in the range of volume fractions $\unicode[STIX]{x1D719}=0{-}0.2$ , using the immersed boundary method (IBM) to account for the dispersed phase. Based on the hydraulic diameter a Reynolds number of 5600 is considered. We observe that for $\unicode[STIX]{x1D719}=0.05$ and 0.1, particles preferentially accumulate on the corner bisectors, close to the corners, as also observed for laminar square duct flows of the same duct-to-particle size ratio. At the highest volume fraction, particles preferentially accumulate in the core region. For plane channel flows, in the absence of lateral confinement, particles are found instead to be uniformly distributed across the channel. The intensity of the cross-stream secondary flows increases (with respect to the unladen case) with the volume fraction up to $\unicode[STIX]{x1D719}=0.1$ , as a consequence of the high concentration of particles along the corner bisector. For $\unicode[STIX]{x1D719}=0.2$ the turbulence activity is reduced and the intensity of the secondary flows reduces to below that of the unladen case. The friction Reynolds number increases with $\unicode[STIX]{x1D719}$ in dilute conditions, as observed for channel flows. However, for $\unicode[STIX]{x1D719}=0.2$ the mean friction Reynolds number is similar to that for $\unicode[STIX]{x1D719}=0.1$ . By performing the turbulent kinetic energy budget, we see that the turbulence production is enhanced up to $\unicode[STIX]{x1D719}=0.1$ , while for $\unicode[STIX]{x1D719}=0.2$ the production decreases below the values for $\unicode[STIX]{x1D719}=0.05$ . On the other hand, the dissipation and the transport monotonically increase with $\unicode[STIX]{x1D719}$ . The interphase interaction term also contributes positively to the turbulent kinetic energy budget and increases monotonically with $\unicode[STIX]{x1D719}$ , in a similar way as the mean transport. Finally, we show that particles move on average faster than the fluid. However, there are regions close to the walls and at the corners where they lag behind it. In particular, for $\unicode[STIX]{x1D719}=0.05,0.1$ , the slip velocity distribution at the corner bisectors seems correlated to the locations of maximum concentration: the concentration is higher where the slip velocity vanishes. The wall-normal hydrodynamic and collision forces acting on the particles push them away from the corners. The combination of these forces vanishes around the locations of maximum concentration. The total mean forces are generally low along the corner bisectors and at the core, also explaining the concentration distribution for $\unicode[STIX]{x1D719}=0.2$ .
We investigate the dynamics of a swimming microorganism inside a surfactant-laden drop for axisymmetric configurations under the assumptions of small Reynolds number and small surface Péclet number $(Pe_{s})$ . Expanding the variables in $Pe_{s}$ , we solve the Stokes equations for the concentric configuration using Lamb’s general solution, while the dynamic equation for the stream function is solved in the bipolar coordinates for the eccentric configurations. For a two-mode squirmer inside a drop, the surfactant redistribution can either increase or decrease the magnitude of swimmer and drop velocities, depending on the value of the eccentricity. This was explained by analysing the influence of surfactant redistribution on the thrust and drag forces acting on the swimmer and the drop. The far-field representation of a surfactant-covered drop enclosing a pusher swimmer at its centre is a puller; the strength of this far field is reduced due to the surfactant redistribution. The advection of surfactant on the drop surface leads to a time-averaged propulsion of the drop and the time-reversible swimmer that it engulfs, thereby causing them to escape from the constraints of the scallop theorem. We quantified the range of parameters for which an eccentrically stable configuration can be achieved for a two-mode squirmer inside a clean drop. The surfactant redistribution shifts this eccentrically stable position towards the top surface of the drop, although this shift is small.
The evaporation of sessile droplets is analysed when the influence of the thermal properties of the system is strong. We obtain asymptotic solutions for the evolution, and hence explicit expressions for the lifetimes, of droplets when the substrate has a high thermal resistance relative to the droplet and when the saturation concentration of the vapour depends strongly on temperature. In both situations we find that the lifetimes of the droplets are significantly extended relative to those when thermal effects are weak.
The unstable evolution of an elongated elliptically shaped inhomogeneity that is embedded in ambient air and aligned both normal and at an angle to an incident plane blast wave of impact Mach number 2.15 is investigated both experimentally and numerically. The elliptic inhomogeneities and the blast waves are generated using gas heating and exploding wire technique and their interaction is captured optically using shadowgraph method. While two symmetric counter-rotating vortices due to Richtmyer–Meshkov instability are observed for the straight interaction, the formation of a train of vortices similar to Kelvin–Helmholtz instability, introducing asymmetry into the flow field, are observed for an inclined interaction. During the early phase of the interaction process in the straight case, the growth of the counter-rotating vortices (based on the sequence of images obtained from the high-speed camera) and circulation (calculated with the aid of numerical data) are found to be linear in both space and time. Moreover, the normalized circulation is independent of the inhomogeneity density and the ellipse thickness, enabling the formulation of a unique linear fit equation. Conversely, the circulation for an inclined case follows a quadratic function, with each vortex in the train estimated to move with a different velocity directly related to its size at that instant. Two factors influencing the quadratic nature are identified: the reduction in strength of the transmitted shock thereby generating vortices with reduced vorticity, along with the gradual loss of vorticity of the earlier-generated vortices.
The distribution of kinetic helicity in a dipolar planetary dynamo is central to the success of that dynamo. Motivated by the helicity distributions observed in numerical simulations of the Earth’s dynamo, we consider the relationship between the kinetic helicity, $h=\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\times \boldsymbol{u}$ , and the buoyancy field that acts as a source of helicity, where $\boldsymbol{u}$ is velocity. We show that, in the absence of a magnetic field, helicity evolves in accordance with the equation $\unicode[STIX]{x2202}h/\unicode[STIX]{x2202}t=-\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{F}+S_{h}$ , where the flux, $\boldsymbol{F}$ , represents the transport of helicity by inertial waves, and the helicity source, $S_{h}$ , involves the product of the buoyancy and the velocity fields. In the numerical simulations it is observed that the helicity outside the tangent cylinder is predominantly negative in the north and positive in the south, a feature which the authors had previously attributed to the transport of helicity by waves (Davidson & Ranjan, Geophys. J. Intl, vol. 202, 2015, pp. 1646–1662). It is also observed that there is a strong spatial correlation between the distribution of $h$ and of $S_{h}$ , with $S_{h}$ also predominantly negative in the north and positive in the south. This correlation tentatively suggests that it is the in situ generation of helicity by buoyancy that establishes the distribution of $h$ outside the tangent cylinder, rather than the dispersal of helicity by waves, as had been previously argued by the authors. However, although $h$ and $S_{h}$ are strongly correlated, there is no such correlation between $\unicode[STIX]{x2202}h/\unicode[STIX]{x2202}t$ and $S_{h}$ , as might be expected if the distribution of $h$ were established by an in situ generation mechanism. We explain these various observations by showing that inertial waves interact with the buoyancy field in such a way as to induce a source $S_{h}$ which has the same sign as the helicity in the local wave flux, and that the sign of $h$ is simply determined by the direction of that flux. We conclude that the observed distributions of $h$ and $S_{h}$ outside the tangent cylinder are consistent with the transport of helicity by waves.
Direct numerical simulations (DNSs) are performed to analyse the secondary flow of Prandtl’s second kind in fully developed spanwise-periodic channels with in-plane sinusoidal walls. The secondary flow is characterized for different combinations of wave parameters defining the wall geometry at $Re_{h}=2500$ and 5000, where $h$ is the half-height of the channel. The total cross-flow rate in the channel $Q_{yz}$ is defined along with a theoretical model to predict its behaviour. Interaction between the secondary flows from opposite walls is observed if $\unicode[STIX]{x1D706}\simeq h\simeq A$ , where $A$ and $\unicode[STIX]{x1D706}$ are the amplitude and wavelength of the sinusoidal function defining the wall geometry. As the outer-scaled wavelength ( $\unicode[STIX]{x1D706}/h$ ) is reduced, the secondary vortices become smaller and faster, increasing the total cross-flow rate per wall. However, if the inner-scaled wavelength ( $\unicode[STIX]{x1D706}^{+}$ ) is below 130 viscous units, the cross-flow decays for smaller wavelengths. By analysing cases in which the wavelength of the wall is much smaller than the half-height of the channel $\unicode[STIX]{x1D706}\ll h$ , we show that the cross-flow distribution depends almost entirely on the separation between the scales of the instantaneous vortices, where the upper and lower bounds are determined by $\unicode[STIX]{x1D706}/h$ and $\unicode[STIX]{x1D706}^{+}$ , respectively. Therefore, the distribution of the secondary flow relative to the size of the wave at a given $Re_{h}$ can be replicated at higher $Re_{h}$ by decreasing $\unicode[STIX]{x1D706}/h$ and keeping $\unicode[STIX]{x1D706}^{+}$ constant. The mechanisms that contribute to the mean cross-flow are analysed in terms of the Reynolds stresses and using quadrant analysis to evaluate the probability density function of the bursting events. These events are further classified with respect to the sign of their instantaneous spanwise velocities. Sweeping events and ejections are preferentially located in the valleys and peaks of the wall, respectively. The sweeps direct the instantaneous cross-flow from the core of the channel towards the wall, turning in the wall-tangent direction towards the peaks. The ejections drive the instantaneous cross-flow from the near-wall region towards the core. This preferential behaviour is identified as one of the main contributors to the secondary flow.
While it has been known that an afterbody (i.e. the structural part of a bluff body downstream of the flow separation points) plays an important role affecting the wake characteristics and even may change the nature of the flow-induced vibration (FIV) of a structure, the question of whether an afterbody is essential for the occurrence of one particular common form of FIV, namely vortex-induced vibration (VIV), still remains. This has motivated the present study to experimentally investigate the FIV of an elastically mounted forward- or backward-facing D-section (closed semicircular) cylinder over the reduced velocity range $2.3\leqslant U^{\ast }\leqslant 20$ , where $U^{\ast }=U/(f_{nw}D)$ . Here, $U$ is the free-stream velocity, $D$ the cylinder diameter and $f_{nw}$ the natural frequency of the system in quiescent fluid (water). The normal orientation with the body’s flat surface facing upstream is known to be subject to another common form of FIV, galloping, while the reverse D-section with the body’s curved surface facing upstream, due to the lack of an afterbody, has previously been reported to be immune to VIV. The fluid–structure system was modelled on a low-friction air-bearing system in conjunction with a recirculating water channel facility to achieve a low mass ratio (defined as the ratio of the total oscillating mass to that of the displaced fluid mass). Interestingly, through a careful overall examination of the dynamic responses, including the vibration amplitude and frequency, fluid forces and phases, our new findings showed that the D-section exhibits a VIV-dominated response for $U^{\ast }<10$ , galloping-dominated response for $U^{\ast }>12.5$ , and a transition regime with a VIV–galloping interaction in between. Also observed for the first time were interesting wake modes associated with these response regimes. However, in contrast to previous studies at high Reynolds number (defined by $Re=UD/\unicode[STIX]{x1D708}$ , with $\unicode[STIX]{x1D708}$ the kinematic viscosity), which have showed that the D-section was subject to ‘hard’ galloping that required a substantial initial amplitude to trigger, it was observed in the present study that the D-section can gallop softly from rest. Surprisingly, on the other hand, it was found that the reverse D-section exhibits pure VIV features. Remarkable similarities were observed in a direct comparison with a circular cylinder of the same mass ratio, in terms of the onset $U^{\ast }$ of significant vibration, the peak amplitude (only approximately 6 % less than that of the circular cylinder), and also the fluid forces and phases. Of most significance, this study shows that an afterbody is not essential for VIV at low mass and damping ratios.
The effect of a leading-edge vortex (LEV) on the lift, thrust and moment of a two-dimensional heaving and pitching thin airfoil is analysed within the unsteady linear potential theory. First, general expressions that take into account the effect of any set of unsteady point vortices interacting with the oscillating foil and unsteady wake are derived. Then, a simplified analysis, based on the Brown–Michael model, of the initial stages of the growing LEV from the sharp leading edge during each half-stroke is used to obtain simple expressions for its main contribution to the unsteady lift, thrust and moment. It is found that the LEV contributes to the aerodynamic forces and moment provided that a pitching motion exists, while its effect is negligible, in the present approximation, for a pure heaving motion, and for some combined pitching and heaving motions with large phase shifts which are also characterized in the present work. In particular, the effect of the LEV is found to decrease with the distance of the pivot point from the trailing edge. Further, the time-averaged lift and moment are not modified by the growing LEVs in the present approximation, and only the time-averaged thrust force is corrected, decreasing slightly in most cases in relation to the linear potential results by an amount proportional to $a_{0}^{2}k^{3}$ for large $k$ , where $k$ is the reduced frequency and $a_{0}$ is the pitching amplitude. The time-averaged input power is also modified by the LEV in the present approximation, so that the propulsion efficiency changes by both the thrust and the power, these corrections being relevant only for pivot locations behind the midchord point. Finally, the potential results modified by the LEV are compared with available experimental data.
We critically analyse the different ways to evaluate the dependence of the Nusselt number ( $\mathit{Nu}$ ) on the Rayleigh number ( $\mathit{Ra}$ ) in measurements of the heat transport in turbulent Rayleigh–Bénard convection under general non-Oberbeck–Boussinesq conditions and show the sensitivity of this dependence to the choice of the reference temperature at which the fluid properties are evaluated. For the case when the fluid properties depend significantly on the temperature and any pressure dependence is insignificant we propose a method to estimate the centre temperature. The theoretical predictions show very good agreement with the Göttingen measurements by He et al. (New J. Phys., vol. 14, 2012, 063030). We further show too the values of the normalized heat transport $\mathit{Nu}/\mathit{Ra}^{1/3}$ are independent of whether they are evaluated in the whole convection cell or in the lower or upper part of the cell if the correct reference temperatures are used.
Fully resolved measurements of turbulent boundary layers are reported for the Reynolds number range $Re_{\unicode[STIX]{x1D70F}}=6000{-}20\,000$ . Despite several decades of research in wall-bounded turbulence there is still controversy over the behaviour of streamwise turbulence intensities near the wall, especially at high Reynolds numbers. Much of it stems from the uncertainty in measurement due to finite spatial resolution. Conventional hot-wire anemometry is limited for high Reynolds number measurements due to limited spatial resolution issues that cause attenuation in the streamwise turbulence intensity profile near the wall. To address this issue we use the nano-scale thermal anemometry probe (NSTAP), developed at Princeton University to conduct velocity measurements in the high Reynolds number boundary layer facility at the University of Melbourne. The NSTAP has a sensing length almost one order of magnitude smaller than conventional hot-wires. This enables us to acquire fully resolved velocity measurements of turbulent boundary layers up to $Re_{\unicode[STIX]{x1D70F}}=20\,000$ . Results show that in the near-wall region, the viscous-scaled streamwise turbulence intensity grows with $Re_{\unicode[STIX]{x1D70F}}$ in the Reynolds number range of the experiments. A second outer peak in the streamwise turbulence intensity is also shown to emerge at the highest Reynolds numbers. Moreover, the energy spectra in the near-wall region show excellent inner scaling over the small to moderate wavelength range, followed by a large-scale influence that increases with Reynolds number. Outer scaling in the outer region is found to collapse the energy spectra over high wavelengths across various Reynolds numbers.
The influence of the large scale organisation of free-stream turbulence on a turbulent boundary layer is investigated experimentally in a wind tunnel through hot-wire measurements. An active grid is used to generate high-intensity free-stream turbulence with turbulence intensities and local turbulent Reynolds numbers in the ranges $7.2\,\%\leqslant u_{\infty }^{\prime }/U_{\infty }\leqslant 13.0\,\%$ and $302\leqslant Re_{\unicode[STIX]{x1D706},\infty }\leqslant 760$ , respectively. In particular, several cases are produced with fixed $u_{\infty }^{\prime }/U_{\infty }$ and $Re_{\unicode[STIX]{x1D706},\infty }$ , but up to a 65 % change in the free-stream integral scale $L_{u,\infty }/\unicode[STIX]{x1D6FF}$ . It is shown that, while qualitatively the spectra at various wall-normal positions in the boundary layer look similar, there are quantifiable differences at the large wavelengths all the way to the wall. Nonetheless, profiles of the longitudinal statistics up to fourth order are well collapsed between cases at the same $u_{\infty }^{\prime }/U_{\infty }$ . It is argued that a larger separation of the integral scale would not yield a different result, nor would it be physically realisable. Comparing cases across the wide range of turbulence intensities and free-stream Reynolds numbers tested, it is demonstrated that the near-wall spectral peak is independent of the free-stream turbulence, and seemingly universal. The outer peak was also found to be described by a set of global scaling laws, and hence both the near-wall and outer spectral peaks can be predicted a priori with only knowledge of the free-stream spectrum, the boundary layer thickness ( $\unicode[STIX]{x1D6FF}$ ) and the friction velocity ( $U_{\unicode[STIX]{x1D70F}}$ ). Finally, a conceptual model is suggested that attributes the increase in $U_{\unicode[STIX]{x1D70F}}$ as $u_{\infty }^{\prime }/U_{\infty }$ increases to the build-up of energy at large wavelengths near the wall because that energy cannot be transferred to the universal near-wall spectral peak.
We experimentally study the influence of wall roughness on bubble drag reduction in turbulent Taylor–Couette flow, i.e. the flow between two concentric, independently rotating cylinders. We measure the drag in the system for the cases with and without air, and add roughness by installing transverse ribs on either one or both of the cylinders. For the smooth-wall case (no ribs) and the case of ribs on the inner cylinder only, we observe strong drag reduction up to DR $=33\,\%$ and DR $=23\,\%$ , respectively, for a void fraction of $\unicode[STIX]{x1D6FC}=6\,\%$ . However, with ribs mounted on both cylinders or on the outer cylinder only, the drag reduction is weak, less than DR $=11\,\%$ , and thus quite close to the trivial effect of reduced effective density. Flow visualizations show that stable turbulent Taylor vortices – large-scale vortical structures – are induced in these two cases, i.e. the cases with ribs on the outer cylinder. These strong secondary flows move the bubbles away from the boundary layer, making the bubbles less effective than what had previously been observed for the smooth-wall case. Measurements with counter-rotating smooth cylinders, a regime in which pronounced Taylor rolls are also induced, confirm that it is really the Taylor vortices that weaken the bubble drag reduction mechanism. Our findings show that, although bubble drag reduction can indeed be effective for smooth walls, its effect can be spoiled by e.g. biofouling and omnipresent wall roughness, as the roughness can induce strong secondary flows.
Thermodynamic fluctuations of pressure, density, temperature or entropy $\{p^{\prime },\unicode[STIX]{x1D70C}^{\prime },T^{\prime },s^{\prime }\}$ in compressible aerodynamic turbulence, although generated by the flow, are fundamentally related to one another by the thermodynamic equation of state. Ratios between non-dimensional root-mean-square (r.m.s.) levels ( $\text{CV}_{p^{\prime }}:=\bar{p}^{-1}\,p_{rms}^{\prime }$ , $\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}:=\bar{\unicode[STIX]{x1D70C}}^{-1}\,\unicode[STIX]{x1D70C}_{rms}^{\prime }$ , $\text{CV}_{T^{\prime }}:=\bar{T}^{-1}\,T_{rms}^{\prime }$ ), along with all possible 2-moment correlation coefficients $\{c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }},c_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }},c_{p^{\prime }T^{\prime }},c_{s^{\prime }\unicode[STIX]{x1D70C}^{\prime }},c_{s^{\prime }T^{\prime }},c_{s^{\prime }p^{\prime }}\}$ , represent, in the sense of Bradshaw (Annu. Rev. Fluid Mech., vol. 9, 1977, pp. 33–54), the thermodynamic turbulence structure of the flow. We use direct numerical simulation (DNS) data, both for plane channel flow and for sustained homogeneous isotropic turbulence, to determine the range of validity of the leading-order, formally $O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }})$ , approximations of the exact relations between thermodynamic turbulence structure parameters. Available DNS data are mapped on the $(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{-1}\,\text{CV}_{T^{\prime }},c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }})$ -plane and their loci, identified using the leading-order approximations, highlight specific behaviour for different flows or flow regions. For the particular case of sustained compressible homogeneous isotropic turbulence, it is shown that the DNS data collapse onto a single curve corresponding to $c_{s^{\prime }T^{\prime }}\approxeq 0.2$ (for air flow), while the approximation $c_{s^{\prime }p^{\prime }}\approxeq 0$ fits reasonably well wall turbulence DNS data, providing building blocks towards the construction of simple phenomenological models.
In this paper we study the wall pressure and vorticity fields of the Stokes boundary layer in the intermittently turbulent regime through direct numerical simulation (DNS). The DNS results are compared to experimental measurements and a good agreement is found for the mean and fluctuating velocity fields. We observe maxima of the turbulent kinetic energy and wall shear stress in the early deceleration stage and minima in the late acceleration stage. The wall pressure field is characterized by large fluctuations with respect to the root mean square level, while the skewness and kurtosis of the wall pressure show significant deviations from their Gaussian values. The wall vorticity components show different behaviours during the cycle: for the streamwise component, positive and negative fluctuations have the same probability of occurrence throughout the cycle while the spanwise fluctuations favour negative extrema in the acceleration stage and positive extrema in the deceleration stage. The wall vorticity flux is a function of the wall pressure gradients. Vorticity creation at the wall reaches a maximum at the beginning of the deceleration stage due to the increase of uncorrelated wall pressure signals. The spanwise vorticity component is the most affected by the oscillations of the outer flow. These findings have consequences for the design of wave energy converters. In extreme seas, wave induced fluid velocities can be very high and extreme wall pressure fluctuations may occur. Moreover, the spanwise vortical fields oscillate violently in a wave cycle, inducing strong interactions between vortices and the device that can enhance the device motion.
Unconfined three-dimensional gravity currents generated by lock exchange using a small dividing gate in a sufficiently large tank are investigated by means of large eddy simulations under the Boussinesq approximation, with Grashof numbers varying over five orders of magnitudes. The study shows that, after an initial transient, the flow can be separated into an axisymmetric expansion and a globally translating motion. In particular, the circular frontline spreads like a constant-flow-rate, axially symmetric gravity current about a virtual source translating along the symmetry axis. The flow is characterised by the presence of lobe and cleft instabilities and hydrodynamic shocks. Depending on the Grashof number, the shocks can either be isolated or produced continuously. In the latter case a typical ring structure is visible in the density and velocity fields. The analysis of the frontal spreading of the axisymmetric part of the current indicates the presence of three regimes, namely, a slumping phase, an inertial–buoyancy equilibrium regime and a viscous–buoyancy equilibrium regime. The viscous–buoyancy phase is in good agreement with the model of Huppert (J. Fluid Mech., vol. 121, 1982, pp. 43–58), while the inertial phase is consistent with the experiments of Britter (Atmos. Environ., vol. 13, 1979, pp. 1241–1247), conducted for purely axially symmetric, constant inflow, gravity currents. The adoption of the slumping model of Huppert & Simpson (J. Fluid Mech., vol. 99 (04), 1980, pp. 785–799), which is here extended to the case of constant-flow-rate cylindrical currents, allows reconciling of the different theories about the initial radial spreading in the context of different asymptotic regimes. As expected, the slumping phase is governed by the Froude number at the lock’s gate, whereas the transition to the viscous phase depends on both the Froude number at the gate and the Grashof number. The identification of the inertial–buoyancy regime in the presence of hydrodynamic shocks for this class of flows is important, due to the lack of analytical solutions for the similarity problem in the framework of shallow water theory. This fact has considerably slowed the research on variable-flow-rate axisymmetric gravity currents, as opposed to the rapid development of the knowledge about cylindrical constant-volume and planar gravity currents, despite their own environmental relevance.