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This study investigates the mean flow structure of two shock-wave boundary-layer interactions generated by moderately swept compression ramps in a Mach 2 flow. The ramps have a compression angle of either $19^{\circ }$ or $22.5^{\circ }$ and a sweep angle of $30^{\circ }$ . The primary diagnostic methods used for this study are surface-streakline flow visualization and particle image velocimetry. The shock-wave boundary-layer interactions are shown to be quasi-conical, with the intermittent region, separation line and reattachment line all scaling in a self-similar manner outside of the inception region. This is one of the first studies to investigate the flow field of a swept ramp using particle image velocimetry, allowing more sensitive measurements of the velocity flow field than previously possible. It is observed that the streamwise velocity component outside of the separated flow reaches the quasi-conical state at the same time as the bulk surface flow features. However, the streamwise and cross-stream components within the separated flow take longer to recover to the quasi-conical state, which indicates that the inception region for these low-magnitude velocity components is actually larger than was previously assumed. Specific scaling laws reported previously in the literature are also investigated and the results of this study are shown to scale similarly to these related interactions. Certain limiting cases of the scaling laws are explored that have potential implications for the interpretation of cylindrical and quasi-conical scaling.
The behaviour of a laser-induced cavitation bubble near two perpendicular rigid walls and its dependence on the distance between bubble and walls is investigated experimentally. It was shown by means of high-speed photography with $100\,000~\text{frames}~\text{s}^{-1}$ that an inclined jet is formed during bubble collapse and the bubble migrates in the direction of the jet. At a given position of the bubble with respect to the horizontal wall, the inclination of the jet increases with decreasing distance between the bubble and the second, vertical wall. A bubble generated at equal distances from the walls develops a jet that is directed in their bisection. The penetration of the jet into the opposite bubble surface leads to the formation of an asymmetric toroidal bubble that is perpendicular to the jet direction. At a large distance from the rigid walls, the toroidal bubble collapses in the radial direction, eventually disintegrating into tiny microbubbles. When the bubble is in contact with the horizontal wall at its maximum expansion, the toroidal ring collapses in both radial and toroidal directions, starting from the bubble part opposite to the vertical wall, and the bubble achieves a crescent shape at the moment of second collapse. The bubble oscillation is accompanied by a strong migration along the horizontal wall.
The scattering of acoustic waves by a moving vortex is studied in two dimensions to bring further insight into the physical mechanisms responsible for the spectral broadening caused by a region of turbulence. When propagating through turbulence, a monochromatic sound wave will be scattered over a range of frequencies, resulting in typical spectra with broadband sidelobes on either side of the tone. This spectral broadening, also called ‘haystacking’, is of importance for noise radiation from jet exhausts and for acoustic measurements in open-jet wind tunnels. A semianalytical model is formulated for a plane wave scattered by a vortex, including the influence of the convection of the vortex. This allows us to perform a detailed parametric study of the properties and evolution of the scattered field. A time-domain numerical model for the linearised Euler equations is also used to consider more general sound fields, such as that radiated by a point source in a uniform flow. The spectral broadening stems from the combination of the spatial scattering of sound due to the refraction of waves propagating through the vortex, and two Doppler shifts induced by the motion of the vortex relative to the source and of the observer relative to the vortex. The fact that the spectrum exhibits sidebands is directly explained by the directivity of the scattered field which is composed of several beams radiating from the vortex. The evolution of the acoustic spectra with the parameters considered in this paper is compared with the trends observed in previous experimental work on acoustic scattering by a jet shear layer.
This work investigates the three-dimensional, spatio-temporal flow development in the aft portion of a laminar separation bubble. The bubble is forming on a flat plate geometry, subjected to an adverse pressure gradient, featuring maximum reverse flow of approximately 2 % of the local free-stream velocity. Time-resolved velocity measurements are performed by means of planar and tomographic particle image velocimetry, in the vicinity of the reattachment region. The measurements are complemented with a numerical solution of the boundary layer equations in the upstream field. The combined numerical and measured boundary layer is used as a baseline flow for linear stability theory analysis. The results provide insight into the dynamics of dominant coherent structures that form in the separated shear layer and deform along the span. Stability analysis shows that the flow becomes unstable upstream of separation, where both normal and oblique modes undergo amplification. While the shear layer roll up is linked to the amplification of the fundamental normal mode, the oblique modes at angles lower than approximately $30^{\circ }$ are also amplified substantially at the fundamental frequency. A model based on the stability analysis and experimental measurements is employed to demonstrate that the spanwise deformations of rollers are produced due to a superposition of normal and oblique instability modes initiating upstream of separation. The degree of the initial spanwise deformations is shown to depend on the relative amplitude of the dominant normal and oblique waves. This is confirmed by forcing the normal mode through a controlled impulsive perturbation introduced by a spanwise invariant dielectric-barrier-discharge plasma actuator, resulting in the formation of spanwise coherent vortices. The findings elucidate the link between important features in the bubble shedding dynamics and stability characteristics and provide further clarification on the differences in the development of coherent structures seen in recent experiments. Moreover, the results present a handle on the development of effective control strategies that can be used to either promote or suppress shedding in separation bubbles, which is of interest for system performance improvement and control of aeroacoustic emissions in relevant applications.
The free surface and flow field structure generated by the uniform acceleration (with dimensionless acceleration $\unicode[STIX]{x1D70E}$ ) of a rigid plate, inclined at an angle $\unicode[STIX]{x1D6FC}\in (0,\unicode[STIX]{x03C0}/2)$ to the exterior horizontal, as it advances ( $\unicode[STIX]{x1D70E}>0$ ) or retreats ( $\unicode[STIX]{x1D70E}<0$ ) from an initially stationary and horizontal strip of inviscid incompressible fluid under gravity, are studied in the small-time limit via the method of matched asymptotic expansions. This work generalises the case of a uniformly accelerating plate advancing into a fluid as studied by Needham et al. (Q. J. Mech. Appl. Maths, vol. 61 (4), 2008, pp. 581–614). Particular attention is paid to the innermost asymptotic regions encompassing the initial interaction between the plate and the free surface. We find that the structure of the solution to the governing initial boundary value problem is characterised in terms of the parameters $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D707}$ (where $\unicode[STIX]{x1D707}=1+\unicode[STIX]{x1D70E}\tan \unicode[STIX]{x1D6FC}$ ), with a bifurcation in structure as $\unicode[STIX]{x1D707}$ changes sign. This bifurcation in structure leads us to question the well-posedness and stability of the governing initial boundary value problem with respect to small perturbations in initial data in the innermost asymptotic regions, the discussion of which will be presented in the companion paper Gallagher et al. (J. Fluid Mech. vol. 841, 2018, pp. 146–166). In particular, when $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/2)\times \mathbb{R}^{+}$ , the free surface close to the initial contact point remains monotone, and encompasses a swelling jet when $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/2)\times [1,\infty )$ or a collapsing jet when $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/2)\times (0,1)$ . However, when $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/2)\times \mathbb{R}^{-}$ , the collapsing jet develops a more complex structure, with the free surface close to the initial contact point now developing a finite number of local oscillations, with near resonance type behaviour occurring close to a countable set of critical plate angles $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}_{n}^{\ast }\in (0,\unicode[STIX]{x03C0}/2)$ ( $n=1,2,\ldots$ ).
We consider the problem of a rigid plate, inclined at an angle $\unicode[STIX]{x1D6FC}\in (0,\unicode[STIX]{x03C0}/2)$ to the horizontal, accelerating uniformly from rest into, or away from, a semi-infinite strip of inviscid, incompressible fluid under gravity. Following on from Gallagher et al. (J. Fluid Mech., vol. 841, 2018, pp. 109–145) (henceforth referred to as GNB), it is of interest to analyse the well-posedness and stability of the principal flow with respect to perturbations in the initially horizontal free surface close to the plate contact point. In particular we find that the solution to the principal unperturbed problem, denoted by [IBVP] in GNB, is well-posed and stable with respect to perturbations in initial data in the region of interest, localised close to the contact point of the free surface and the plate, when the plate is accelerated with dimensionless acceleration $\unicode[STIX]{x1D70E}\geqslant -\cot \,\unicode[STIX]{x1D6FC}$ , while the solution to [IBVP] is ill-posed with respect to such perturbations in the initial data, when the plate is accelerated with dimensionless acceleration $\unicode[STIX]{x1D70E}<-\cot \,\unicode[STIX]{x1D6FC}$ . The physical source of the ill-posedness of the principal problem [IBVP], when $\unicode[STIX]{x1D70E}<-\cot \,\unicode[STIX]{x1D6FC}$ , is revealed to be due to the leading-order problem in the innermost region localised close to the initial contact point being in the form of a local Rayleigh–Taylor problem. As a consequence of this mechanistic interpretation we anticipate that, when the plate is accelerated with $\unicode[STIX]{x1D70E}<-\cot \,\unicode[STIX]{x1D6FC}$ , the inclusion of weak surface tension effects will restore well-posedness of the problem [IBVP] which will, however, remain temporally unstable.
A numerical investigation of unsteady hydrodynamic forces on the particle bed in an oscillatory flow environment is performed by means of direct numerical simulations. Statistical descriptions of drag and lift forces for two particle sizes of diameter 372 and 125 in wall units in a very rough turbulent flow regime are reported. Characterization of unsteady forces in terms of spatial distribution, temporal autocorrelation, force spectrum as well as cross-correlations with measurable flow variables is carried out. Based on the concept of impulse, intermittency in the drag and lift forces is also investigated. Temporal correlations show drag and lift to be positively correlated with a time delay that is approximately equal to the Taylor micro-scale related to the drag/lift fluctuations. The force spectra for drag and lift reveal roughly two scaling regions, $-11/3$ and $-7/3$ ; the former typically represents turbulence–mean-shear interactions, whereas the latter indicates dominance of turbulence–turbulence interactions. Particle forces are strongly correlated with streamwise velocity and pressure fluctuations in the near-bed region for both flow cases. In comparison to the large-diameter particle case, the spatial extent of these correlations is 2–3 times larger in homogeneous directions for the small sized particle, a feature that is reminiscent of longer near-bed structures. For both large- and small-particle cases, it is shown that the distributions of drag (lift) fluctuations, in particular, peakedness and long tails, match remarkably well with fourth-order Gram–Charlier distributions of velocity (pressure) fluctuations. Furthermore, it is demonstrated that the intermittency is larger in the case of the lift force compared to that for the drag in both flow cases. Distributions of impulse events are heavily and positively skewed and are well described by a generalized extreme value distribution.
Radionuclide scanning images published in Nature by Di Chiro in 1964 showed a downward migration along the spinal canal of particle tracers injected in the brain ventricles while also showing an upward flow of tracers injected in the lumbar region of the canal. These observations, since then corroborated by many radiological measurements, have been the basis for the hypothesis that there must be an active circulation mechanism associated with the transport of cerebrospinal fluid (CSF) deep down into the spinal canal and subsequently returning a portion back to the cranial vault. However, to date, there has been no physical explanation for the mechanism responsible for the establishment of such a bulk recirculating motion. To investigate the origin and characteristics of this recirculating flow, we have analyzed the motion of the CSF in the subarachnoid space of the spinal canal. Our analysis accounts for the slender geometry of the spinal canal, the small compliance of the dura membrane enclosing the CSF in the canal, and the fact that the CSF is confined to a thin annular subarachnoid space surrounding the spinal cord. We apply this general formulation to study the characteristics of the flow generated in a simplified model of the spinal canal consisting of a slender compliant cylindrical pipe with a coaxial cylindrical inclusion, closed at its distal end, and subjected to small periodic pressure pulsations at its open entrance. We show that the balance between the local acceleration and viscous forces produces a leading-order flow consisting of pure oscillatory motion with axial velocities on the order of a few centimetres per second and amplitudes monotonically decreasing along the length of the canal. We then demonstrate that the nonlinear term associated with the convective acceleration contributes to a second-order correction consisting of a steady streaming that generates a bulk recirculating motion of the CSF along the length of the canal with characteristic velocities two orders of magnitude smaller than the leading-order oscillatory flow. The results of the analysis of this idealized geometry of the spinal canal are shown to be in good agreement not only with experimental measurements in an in-vitro model but also with radiological measurements conducted in human adults.
The dynamics of light spheres rising freely under buoyancy in a large expanse of viscous fluid at rest at infinity is investigated numerically. For this purpose, the computational approach developed by Mougin & Magnaudet (Intl J. Multiphase Flow, vol. 28, 2002, pp. 1837–1851) is improved to account for the instantaneous viscous loads induced by the translational and rotational sphere accelerations, which play a crucial role in the dynamics of very light spheres. A comprehensive map of the rise regimes encountered up to Reynolds numbers (based on the sphere diameter and mean rise velocity) of the order of $10^{3}$ is set up by varying independently the body-to-fluid density ratio and the relative magnitude of inertial and viscous effects in approximately 250 distinct combinations. These computations confirm or reveal the presence of several distinct periodic regions on the route to chaos, most of which only exist within a finite range of the sphere relative density and Reynolds number. The wake structure is analysed in these various regimes, evidencing the existence of markedly different shedding modes according to the style of path. The variation of the drag force with the flow parameters is also examined, revealing that only one of the styles of path specific to very light spheres yields a non-standard drag behaviour, with drag coefficients significantly larger than those measured on a fixed sphere under equivalent conditions. The outcomes of this investigation are compared with available experimental and numerical results, evidencing points of consensus and disagreement.
Laminar flow in devices fabricated from soft materials causes deformation of the passage geometry, which affects the flow rate–pressure drop relation. For a given pressure drop, in channels with narrow rectangular cross-section, the flow rate varies as the cube of the channel height, so deformation can produce significant quantitative effects, including nonlinear dependence on the pressure drop (Gervais et al., Lab on a Chip, vol. 6, 2006, pp. 500–507). Gervais et al. proposed a successful model of the deformation-induced change in the flow rate by heuristically coupling a Hookean elastic response with the lubrication approximation for Stokes flow. However, their model contains a fitting parameter that must be found for each channel shape by performing an experiment. We present a perturbation approach for the flow rate–pressure drop relation in a shallow deformable microchannel using the theory of isotropic quasi-static plate bending and the Stokes equations under a lubrication approximation (specifically, the ratio of the channel’s height to its width and of the channel’s height to its length are both assumed small). Our result contains no free parameters and confirms Gervais et al.’s observation that the flow rate is a quartic polynomial of the pressure drop. The derived flow rate–pressure drop relation compares favourably with experimental measurements.
This work focuses on using the power of a collapsing bubble in ice breaking. We experimentally validated the possibility and investigated the mechanism of ice breaking with a single collapsing bubble, where the bubble was generated by underwater electric discharge and collapsed at various distances under ice plates with different thicknesses. Characteristics of the ice fracturing, bubble jets and shock waves emitted during the collapse of the bubble were captured. The pattern of the ice fracturing is related to the ice thickness and the bubble–ice distance. Fractures develop from the top of the ice plate, i.e. the ice–air interface, and this is attributed to the tension caused by the reflection of the shock waves at the interface. Such fracturing is lessened when the thickness of the ice plate or the bubble–ice distance increases. Fractures may also form from the bottom of the ice plate upon the shock wave incidence when the bubble–ice distance is sufficiently small. The ice plate motion and its effect on the bubble behaviour were analysed. The ice plate motion results in higher jet speed and greater elongation of the bubble shape along the vertical direction. It also causes the bubble initiated close to the ice plate to split and emit multiple shock waves at the end of the collapse. The findings suggest that collapsing bubbles can be used as a brand new way of ice breaking.
Numerous experimental studies have documented that injecting low-salinity water into an oil reservoir can increase the amount of oil recovered. However, owing to the complexity of the chemical interactions involved in this process, there has been much debate over the dominant mechanism causing this effect. In order to further understand one proposed mechanism, multicomponent ionic exchange, we study the motion of an oil slug through a clay pore throat filled with saline water. The pore throat is modelled as a capillary tube connecting two bulk regions of water. We assume that the surfaces of the oil and the capillary are negatively charged and that, due to repulsion between these surfaces, the oil slug is separated from the capillary surface by a thin film of water. Ion interactions at the oil–water and clay–water interfaces are modelled using the law of mass action. By using lubrication theory to describe the thin-film flow in the water layer separating the oil from the clay surface, and the macroscopic flow through the capillary, we derive expressions for the thickness of the wetting film, and the velocity of the oil slug, given a pressure difference across the ends of the capillary. Numerical results show that the thickness of the water layer and the velocity of the oil slug increase as the salinity of the water is reduced, suggesting that this mechanism contributes to the low-salinity effect. An analytical solution is presented in the limit in which the applied pressure is small.
This paper reports turbulent boundary layer measurements made over open-cell reticulated foams with varying pore size and thickness, but constant porosity ( $\unicode[STIX]{x1D716}\approx 0.97$ ). The foams were flush-mounted into a cutout on a flat plate. A laser Doppler velocimeter (LDV) was used to measure mean streamwise velocity and turbulence intensity immediately upstream of the porous section, and at multiple measurement stations along the porous substrate. The friction Reynolds number upstream of the porous section was $Re_{\unicode[STIX]{x1D70F}}\approx 1690$ . For all but the thickest foam tested, the internal boundary layer was fully developed by ${<}10\unicode[STIX]{x1D6FF}$ downstream from the porous transition, where $\unicode[STIX]{x1D6FF}$ is the boundary layer thickness. Fully developed mean velocity profiles showed the presence of a substantial slip velocity at the porous interface ( ${>}30\,\%$ of the free-stream velocity) and a mean velocity deficit relative to the canonical smooth-wall profile further from the wall. While the magnitude of the mean velocity deficit increased with average pore size, the slip velocity remained approximately constant. Fits to the mean velocity profile suggest that the logarithmic region is shifted relative to a smooth wall, and that this shift increases with pore size until it becomes comparable to substrate thickness $h$ . For all foams, the turbulence intensity was found to be elevated further into the boundary layer to $y/\unicode[STIX]{x1D6FF}\approx 0.2$ . An outer peak in intensity was also evident for the largest pore sizes. Velocity spectra indicate that this outer peak is associated with large-scale structures resembling Kelvin–Helmholtz vortices that have streamwise length scale $2\unicode[STIX]{x1D6FF}{-}4\unicode[STIX]{x1D6FF}$ . Skewness profiles suggest that these large-scale structures may have an amplitude-modulating effect on the interfacial turbulence.
We report on small-scale instabilities in a thermally driven rotating annulus filled with a liquid with moderate Prandtl number. The study is based on direct numerical simulations and an accompanying laboratory experiment. The computations are performed independently with two different flow solvers, that is, first, the non-oscillatory forward-in-time differencing flow solver EULAG and, second, a higher-order finite-difference compact scheme (HOC). Both branches consistently show the occurrence of small-scale patterns at both vertical sidewalls in the Stewartson layers of the annulus. Small-scale flow structures are known to exist at the inner sidewall. In contrast, short-period waves at the outer sidewall have not yet been reported. The physical mechanisms that possibly trigger these patterns are discussed. We also debate whether these small-scale structures are a gravity wave signal.
Electrohydrodynamics of a leaky dielectric suspended drop subjected to the combined influence of a uniform electric field and linear velocity field is analysed analytically and numerically. In the limit of small charge convection and small shape deformation, an analytical solution is obtained for the deformed drop shape when the imposed linear flow is of uniaxial extensional type with the extensional component aligned in the direction of the electric field. This perturbation approach is then applied towards obtaining the effect of a uniform electric field on the effective extensional rheology of a dilute emulsion. Key results indicate that the magnitude and sense of drop deformation not only depends on the material properties of the drop and medium but is also governed by the strength of the applied electric field relative to the applied flow field. The interfacial charge convection is found to increase or decrease the drop deformation depending on the direction of electrohydrodynamic flow and relative strength of electric field. The electrohydrodynamic flow and drop deformation modulates the effective extensional viscosity of the emulsion. Importantly, the presence of the electric field leads to strain-rate-dependent effective extensional viscosity of the emulsion. The emulsion is found to exhibit strain-rate thinning/thickening behaviour depending on the drop to medium charge relaxation time scale. The analytically obtained drop shape and deformation are in excellent agreement with numerical simulations for the small deformation ranges.
Using a large number of numerical simulations we examine the steady state of rotating turbulent flows in triple periodic domains, varying the Rossby number $Ro$ (that measures the inverse rotation rate) and the Reynolds number $Re$ (that measures the strength of turbulence). The examined flows are sustained by either a helical or a non-helical Roberts force, that is invariant along the axis of rotation. The forcing acts at a wavenumber $k_{f}$ such that $k_{f}L=4$ , where $2\unicode[STIX]{x03C0}L$ is the size of the domain. Different flow behaviours were obtained as the parameters are varied. Above a critical rotation rate the flow becomes quasi-two-dimensional and transfers energy to the largest scales of the system, forming large coherent structures known as condensates. We examine the behaviour of these condensates and their scaling properties close to and away from this critical rotation rate. Close to the critical rotation rate the system transitions supercritically to the condensate state, displaying a bimodal behaviour oscillating randomly between an incoherent-turbulent state and a condensate state. Away from the critical rotation rate, it is shown that two distinct mechanisms can saturate the growth of the large-scale energy. The first mechanism is due to viscous forces and is similar to the saturation mechanism observed for the inverse cascade in two-dimensional flows. The second mechanism is independent of viscosity and relies on the breaking of the two-dimensionalization condition of the rotating flow. The two mechanisms predict different scaling with respect to the control parameters of the system (Rossby and Reynolds), which are tested with the present results of the numerical simulations. A phase space diagram in the $Re,Ro$ parameter plane is sketched.
We revisit the classical but as yet unresolved problem of predicting the breaking onset of 2D and 3D irrotational gravity water waves. Based on a fully nonlinear 3D boundary element model, our numerical simulations investigate geometric, kinematic and energetic differences between maximally tall non-breaking waves and marginally breaking waves in focusing wave groups. Our study focuses initially on unidirectional domains with flat bottom topography and conditions ranging from deep to intermediate depth (depth to wavelength ratio from 1 to 0.2). Maximally tall non-breaking (maximally recurrent) waves are clearly separated from marginally breaking waves by their normalised energy fluxes localised near the crest tip region. The initial breaking instability occurs within a very compact region centred on the wave crest. On the surface, this reduces to the local ratio of the energy flux velocity (here the fluid velocity) to the crest point velocity for the tallest wave in the evolving group. This provides a robust threshold parameter for breaking onset for 2D wave packets propagating in uniform water depths from deep to intermediate. Further targeted study of representative cases of the most severe laterally focused 3D wave packets in deep and intermediate depth water shows that the threshold remains robust. These numerical findings for 2D and 3D cases are closely supported by our companion observational results. Warning of imminent breaking onset is detectable up to a fifth of a carrier wave period prior to a breaking event.
The dynamics of a spherical elastic capsule, containing a Newtonian fluid bounded by an elastic membrane and immersed in another Newtonian fluid, in a uniform DC electric field is investigated. Discontinuity of electrical properties, such as the conductivities of the internal and external fluid media as well as the capacitance and conductance of the membrane, leads to a net interfacial Maxwell stress which can cause the deformation of such an elastic capsule. We investigate this problem considering well-established membrane laws for a thin elastic membrane, with fully resolved hydrodynamics in the Stokes flow limit, and describe the electrostatics using the capacitor model. In the limit of small deformation, the analytical theory predicts the dynamics fairly satisfactorily. Large deformations at high capillary number, though, necessitate a numerical approach (axisymmetric boundary element method in the present case) to solve this highly nonlinear problem. Akin to vesicles, at intermediate times, highly nonlinear biconcave shapes along with squaring and hexagon-like shapes are observed when the outer medium is more conducting. The study identifies the essentiality of parameters such as high membrane capacitance, low membrane conductance, low hydrodynamic time scales and high capillary number (the ratio of the destabilizing electric force to the stabilizing elastic force) for observation of these shape transitions. The transition is due to large compressive Maxwell stress at the poles at intermediate times. Thus such shape transition can be seen in spherical globules admitting electrical capacitance, possibly irrespective of the nature of the interfacial restoring force.
This paper proposes a new two-point closure model that is compatible with the Kolmogorov $-5/3$ power law for homogeneous isotropic turbulence in an incompressible fluid using the Lagrangian specification of the flow field. A closed set of three equations was derived from the Navier–Stokes equation with no adjustable free parameters. The Kolmogorov constant and the skewness of the longitudinal velocity derivative were evaluated to be 1.779 and $-0.49$ , respectively, using the proposed model. The bottleneck effect was also reproduced in the near-dissipation range.
Separating turbulent boundary layers over smooth and rough flat plates are studied by large-eddy simulations. A suction–blowing velocity distribution imposed at the top boundary of the computation domain produces an adverse-to-favourable pressure gradient and creates a closed separation bubble. The Reynolds number based on the momentum thickness and the free-stream velocity before the pressure gradient begins is 2500. Virtual sand grain roughness in the fully rough regime is modelled by an immersed boundary method. Compared with a smooth-wall case, streamline detachment occurs earlier and the separation region is substantially larger for the rough-wall case, due to the momentum deficit caused by the roughness. The adverse pressure gradient decreases the form drag, so that the point where the wall stress vanishes does not coincide with the detachment of the flow from the surface. A thin reversed-flow region is formed below the roughness crest; the presence of recirculation regions behind each roughness element also affects the intermittency of the near-wall flow, so that upstream of the detachment point the flow can be reversed half of the time, but its average velocity can still be positive. The separated shear layer exhibits higher turbulent kinetic energy (TKE) in the rough-wall case, the growth of the TKE there begins earlier relative to the separation point, and the peak TKE occurs close to the separation point. The momentum deficit caused by the roughness, again, plays a critical role in these changes.