We study the energy exchanges between coherent structures and the mean flow in the wake of a cylinder in the presence of a leeward control rod using particle image velocimetry data at Reynolds number ($Re$) $20\times 10^3$. The shedding of the control rod depends on the oncoming shear layer and hence the downstream interaction of the main cylinder's and control rod's wake strongly depends on the control rod's setting angle ($\theta$). In this work we study this interaction between the shedding modes from the cylinder and control rod at different $\theta$. New secondary coherent motions with distinct characteristic frequencies appear in the flow field aside from the frequencies associated with the sheddings of the control rod, the main cylinder and its harmonics. A multiscale triple decomposition method is applied to extract the coherent modes associated with each of these frequencies, and the dynamics of the modes are studied using kinetic energy budget equations. The primary shedding modes of the control rod and main cylinder, as well as the harmonics of the main cylinder's shedding modes, are found to be primarily energised by the mean flow at this $Re$, while the secondary modes are almost entirely energised by the primary modes, similar to the findings of Baj & Buxton (Phys. Rev. Fluids, vol. 2, issue 11, 2017, 114607) for a different multiscale flow configuration. The remarkable similarity in the energy exchange process forming the secondary coherent modes, observed in two different studies, hints at a possible universality in the formation process of these secondary structures in a multiscale flow.

The effect of adverse pressure gradient (APG) on near-wall turbulence is studied, with a particular attention to the turbulence production mechanism. A plane turbulent Couette flow is considered for several values of constant APG in the lower wall region. A direct numerical simulation (DNS) in a large computational domain shows that turbulence near the lower wall continues to exist even at sufficiently large APGs. On increasing the APG, the cross-streamwise turbulence intensities increase, and the near-wall streaks gradually disappear. A linear analysis using the optimal transient growth indicates that the APG inhibits the generation of near-wall streaks due to the significant reduction of the mean shear in the region near the lower wall. The turbulent fluctuation dynamics beyond the linear regime is studied with a DNS in a minimal flow unit. The near-wall self-sustaining process involving streaks is significantly weakened or destroyed as APG increases, while the turbulent fluctuations become more isotropic and localised. Using a conditional averaging analysis, a new mechanism of near-wall turbulence production under strong APG is uncovered. This mechanism is initiated by the wall-normal nonlinear transport of an outer wall-normal velocity fluctuation to the near-wall region. The transported wall-normal velocity fluctuation is subsequently amplified via the Orr mechanism, resulting in the non-zero turbulence production involving spatially localised vortical structures. This mechanism is also confirmed by DNS of the flow in a large computational domain, where strong correlation between the wall-normal nonlinear transport and turbulence production is observed.

We perform a theoretical study to understand vortex-acoustic lock-in in the presence of an additive Gaussian white noise using a lower-order model. The acoustic field is realized as an external harmonic excitation. Frequency, harmonic excitation and noise amplitudes are varied over ranges encountered in practical and lab-scale combustors. Probability density functions (PDFs) for shedding time periods and phase instants are obtained in the Fokker–Planck framework. Unlike the general scenario, where stochastic bifurcation is identified by a qualitative change in the stationary PDFs, in this case, stochastic lock-in is identified by a qualitative change in the spectrum of the transition probability matrix. The effect of the noise is to reduce the extent of lock-in while preserving the underlying deterministic dynamical features. Although various orders of lock-in are identified, 1 : 1 stochastic lock-in with the excitation frequency close to the natural vortex shedding frequency is found to be the most favourable condition for combustion instability to occur. We show that lock-in can be accompanied by both instability and amplitude suppression. Stochastic resonance is also observed; however, its contribution to instability is marginal.

We use well-resolved numerical simulations to study the combined effects of buoyancy, pressure-driven shear and rotation on the melt rate and morphology of a layer of pure solid overlying its liquid phase in three dimensions at a Rayleigh number $Ra=1.25\times 10^5$. During thermal convection, we find that the rate of melting of the solid phase varies non-monotonically with the strength of the imposed shear flow. In the absence of rotation, depending on whether buoyancy or shear dominates the flow, we observe either domes or ridges aligned in the direction of the shear flow, respectively. Furthermore, we show that the geometry of the phase boundary has important effects on the magnitude and evolution of the heat flux in the liquid layer. In the presence of rotation, the strength of which is characterized by the Rossby number, $Ro$, we observe that for $Ro={O}(1)$, the mean flow in the interior is perpendicular to the direction of the constant horizontal applied pressure gradient. As the magnitude of this pressure gradient increases, the geometry of solid–liquid interface evolves from the voids characteristic of melting by rotating convection, to grooves oriented perpendicular or obliquely to the direction of the pressure gradient.

The unstable fluid–fluid displacement patterns in porous media with rough invasion fronts and trapping of the defending phase are often observed in drainage, i.e. when the solid is non-wetting to the invading phase. On the other hand, during imbibition, compact and faceted growth is expected in regular porous media with geometrically homogeneous pore structure. Here, we report the irregular growth of invading fluid during the imbibition process in two-dimensional regular porous media. The ramified invasion morphology associated with thin fingers is reminiscent of capillary fingering. Through examining the capillary pressure signals and the type of pore-scale invasion mechanisms, the fundamental differences between faceted growth and irregular invasion are revealed. By analysing the pore-scale invasion mechanisms, a phase diagram describing the dominance of different invasion events is proposed. Through conducting systematic quasi-static radial injection simulations across a wide range of porosity and wettability, excellent agreement is observed on the transition boundary from faceted and compact displacement patterns to irregular and dendritic invasion morphologies. This is reflected by the overlap of the transition boundaries from analytical prediction, type of pore-scale invasion events, and macroscopic morphology quantified by the fractal dimension. This work provides new insights on the role of geometrical features of solid structures during multiphase flow with emphasis on the porosity and wettability. The findings could assist in guiding the design of microfluidic devices to control deterministically the multiphase flow, transport and reaction processes.

We develop a new model, to our knowledge, for the many-body hydrodynamics of amphiphilic Janus particles suspended in a viscous background flow. The Janus particles interact through a hydrophobic attraction potential that leads to self-assembly into bilayer structures. We adopt an efficient integral equation method for solving the screened Laplace equation for hydrophobic attraction and for solving the mobility problem for hydrodynamic interactions. The integral equation formulation accurately captures both interactions for near touched boundaries. Under a linear shear flow, we observe the tank-treading deformation in a two-dimensional vesicle made of Janus particles. The results yield measurements of intermonolayer friction, membrane permeability and, at large shear rates, membrane rupture. The simulation studies include a Janus particle vesicle in both linear and parabolic shear flows, and interactions between two Janus particle vesicles in shear and extensional flows. The hydrodynamics of the Janus particle vesicle is similar to the behaviour of an inextensible, elastic vesicle membrane with permeability.

This study proposes a method for simultaneous measurements of time-resolved three-dimensional velocity fields of the dispersed and continuous phases of a turbulent bubbly jet at a low void fraction using Lagrangian particle tracking (LPT) velocimetry with the Shake-The-Box algorithm. Four high-speed cameras are used to acquire time series of images that include both bubbles and fluid tracer particles. Bubbles are firstly tracked using intensity differences between tracer particles and bubbles, then the bubble images are removed from the camera images and all tracer particles are tracked using the residual images. Subsequently, FlowFit interpolation is applied to the LPT results obtained by phase separation to investigate flow characteristics of a bubbly jet. The bubbly jet was divided into two regions along the vertical direction: jet-like and plume-like regions. Streamwise vortex structures of continuous phase were generated mainly by the rising bubbles. The Gaussian and top-hat velocity profiles matched well with the ensemble-averaged fluid and bubble velocities, respectively. The measured slip velocity in the radial direction was not constant but linearly increased. The classical assumption of self-similarity with Gaussian profiles for fluid velocity and bubble concentration is experimentally verified. The fluid volume flux and entrainment coefficient are obtained as a function of the slip velocity, void fraction, plume and bubble width based on three-dimensional measurements. We found that the classical integral theory agrees well with experiments in the plume region.

A methodology to generate sparse Galerkin models of chaotic/unsteady fluid flows containing a minimal number of active triadic interactions is proposed. The key idea is to find an appropriate set of basis functions for the projection representing elementary flow structures that interact minimally one with the other, resulting in a triadic interaction coefficient tensor with sparse structure. Interpretable and computationally efficient Galerkin models are obtained, since a reduced number of triadic interactions are computed to evaluate the right-hand side of the model. To find the basis functions, a subspace rotation technique is used, whereby a set of proper orthogonal decomposition (POD) modes is rotated into a POD subspace of larger dimension using coordinates associated with low-energy dissipative scales to alter energy paths and the structure of the triadic interaction coefficient tensor. This rotation is obtained as the solution of a non-convex optimisation problem that maximises the energy captured by the new basis, promotes sparsity and ensures long-term temporal stability of the sparse Galerkin system. We demonstrate the approach on two-dimensional lid-driven cavity flow at $Re = 2 \times 10^4$ where the motion is chaotic and energy interactions are scattered in modal space. We show that the procedure generates Galerkin models with a reduced set of active triadic interactions, distributed in modal space according to established knowledge of scale interactions in two-dimensional flows. This property, however, is observed only if long-term temporal stability is included explicitly in the formulation, indicating that a dynamical constraint is necessary to obtain a physically consistent sparsification.

Experiments and spatiotemporal stability analysis are carried out to study the global oscillations in laminar low-density round jets with parabolic velocity profiles. The experimental results of laminar low-density jets with parabolic velocity profiles exhibit global axisymmetric oscillations. The spatiotemporal stability results based on base profiles from numerical simulations are consistent with the present experimental results. These results differ from the prediction of stability study by Coenen et al. (Phys. Fluids, vol. 20, 2008, p. 074104). They reported that the low-density jets with near parabolic velocity profiles show global helical oscillations. It is observed that the method used by Coenen et al. is not able to predict the nature of global oscillations observed in experiments for low-density jets with near parabolic profiles. The present spatiotemporal stability results demonstrate that the base flows from Navier–Stokes equations are required to predict the critical conditions observed in experiments for low-density jets with near parabolic velocity profiles. The breakdown distance of globally unstable low-density jets scales with $(Re-Re_c)^{-1/2}$ ($Re_c$ is the critical Reynolds number), consistent with the scaling law obtained from the nonlinear global mode of the Ginzburg–Landau equation.

Type IV shock interference has two triple points which produce two sliplines, forming a narrow jet that penetrates into the subsonic flow region behind the bow shock. Type IV shock interference in the case that the jet is supersonic has been extensively studied in the past. In this paper, type IV shock interference where the jet is transonic is studied. The transition condition from a supersonic to a transonic jet is identified and the influence of compression Mach waves, created inside the leading portion of the jet due to the pressure gradient outside the jet, on the transition and jet shape is analysed. It is found that these compression waves advance the transition from a supersonic to a transonic jet and make the flow inside the transonic jet have a two-branch structure: the lower branch is subsonic, the upper branch is supersonic and is composed of a combination of compressive and expansive waves. The mechanism by which the two-branch structure is produced is explained.

Usually, the load on lifting bodies in incompressible, inviscid flow is determined by integration of the pressure on the body surface, once the flow is solved. Prandtl (Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch- Physikalische Klasse, vol. 1918, pp. 451–477) proposed an opposite method in which the body force field is the source term in the equation of motion to solve the flow. This force field method has not been used intensively but has regained importance in modern wind energy research. However, an analysis of which type of body force field generates vorticity and converts energy, and which body force field does not, is lacking. Prandtl's method is adopted here, but with the addition that the force field is allowed to be conservative or non-conservative. The relation between conservative/non-conservative body forces, vorticity generation, potential and kinetic energy and Helmholtz's vorticity theorems are derived. Similarly, the load on lifting bodies in two and three dimensions is classified as (non-)conservative, with some examples. To show that the force field method is consistent with the method where the load is output of an analysis, the expression for the Kutta–Joukowsky load and the relation between bound and trailing vorticity of a wing have been rederived using the force field method. The analysis confirms that this relation between bound and trailing vorticity is not governed by Helmholtz's theorems, as is often assumed, but by the non-conservative force field generating vorticity.

Direct numerical simulations (DNS) are used to systematically investigate the applicability of the minimal-channel approach (Chung et al., J. Fluid Mech., vol. 773, 2015, pp. 418–431) for the characterization of roughness-induced drag on irregular rough surfaces. Roughness is generated mathematically using a random algorithm, in which the power spectrum (PS) and probability density function (p.d.f.) of the surface height can be prescribed. Twelve different combinations of PS and p.d.f. are examined, and both transitionally and fully rough regimes are investigated (roughness height varies in the range $k^+ = 25$–100). It is demonstrated that both the roughness function (${\rm \Delta} U^+$) and the zero-plane displacement can be predicted with ${\pm }5\,\%$ accuracy using DNS in properly sized minimal channels. Notably, when reducing the domain size, the predictions remain accurate as long as 90 % of the roughness height variance is retained. Additionally, examining the results obtained from different random realizations of roughness shows that a fixed combination of p.d.f. and PS leads to a nearly unique ${\rm \Delta} U^+$ for deterministically different surface topographies. In addition to the global flow properties, the distribution of time-averaged surface force exerted by the roughness onto the fluid is calculated. It is shown that patterns of surface force distribution over irregular roughness can be well captured when the sheltering effect is taken into account. This is made possible by applying the sheltering model of Yang et al. (J. Fluid Mech., vol. 789, 2016, pp. 127–165) to each specific roughness topography. Furthermore, an analysis of the coherence function between the roughness height and the surface force distributions reveals that the coherence drops at larger streamwise wavelengths, which can be an indication that very large horizontal scales contribute less to the skin-friction drag.

Estimation of the initial state of turbulent channel flow from spatially and temporally resolved wall data is performed using adjoint-variational data assimilation. The estimated flow fields satisfy the Navier–Stokes equations and minimize a cost function defined as the difference between model predictions and the available observations. The accuracy of the predicted flow deteriorates with distance from the wall, most precipitously across the buffer layer beyond which only large-scale structures are reconstructed. To explain this trend, we examine the domain of dependence of the observations and the Hessian matrix of the state-estimation cost function, both of which are efficiently evaluated using adjoint fields initiated from impulses at wall sensors. Eigenanalysis of the Hessian is performed, and the eigenvalues are related to the capacity to reconstruct flow structures represented by the eigenvectors. Most of the eigenmodes decay beyond the buffer layer, thus demonstrating weak sensitivity of wall observations to the turbulence in the bulk. However, when the measurement time $t_m^{+} \gtrsim 20$, some streamwise-elongated Hessian eigenfunctions remain finite in the outer flow, and correspond to the sensitivity of wall observations to outer large-scale motions. At much longer observation times, the adjoint field becomes chaotic as it amplifies exponentially, which is indicative of extreme gradients of the cost function and an ill-conditioned Hessian matrix, and both exacerbate the difficulty of estimating turbulence from wall observations.

We consider a phase-field model for the incompressible flow of two immiscible fluids. This model extends widespread models for two fluid phases by including a third, solid phase, which can evolve due to e.g. precipitation and dissolution. We consider a simple, two-dimensional geometry of a thin strip, which can still be seen as the representation of a single pore throat in a porous medium. Under moderate assumptions on the Péclet number and the capillary number, we investigate the limit case when the ratio between the width and the length of the strip goes to zero. In this way, and employing transversal averaging, we derive an upscaled model. The result is a multi-scale model consisting of the upscaled equations for the total flux and the ion transport, while the phase-field equation has to be solved in cell problems at the pore scale to determine the position of interfaces. We also investigate the sharp-interface limit of the multi-scale model, in which the phase-field parameter approaches 0. The resulting sharp-interface model consists only of Darcy-scale equations, as the cell problems can be solved explicitly. Notably, we find asymptotic consistency, that is, the upscaling process and the sharp-interface limit commute. We use numerical results to investigate the validity of the upscaling when discontinuities are formed in the upscaled model.

The fluid mechanics problem of an initially spherical drop subjected to a two-directional nonlinear cubic shear flow, under creeping flow conditions, is theoretically studied. This physical situation is described by four dimensionless numbers: the capillary number (Ca), the viscosity ratio (λ), the linear velocity ratio (G) and the nonlinear intensity of the flow (E). Analytical expressions for the external and internal flows of a spherical drop as well as for the shape of the deformed drop were obtained by applying Lamb's general solution to the Stokes equations. The internal and external streamlines of a spherical drop (Ca = 0) suggest interesting and unusual shapes which are sometimes also accompanied by many stagnation points. When λ = ∞, the drop becomes a solid sphere with an angular velocity $\omega ={-} (1 - G)/2$ independent of E. For the familiar simple shear flow in the x-direction (G = 0, E = 0), Taylor's theory, valid at $Ca \ll 1$ and λ = O(1), predicts an ellipsoidal drop oriented at an azimuthal angle ϕ = 45°; as Ca increases, the deformation increases and the drop attains a peanut shape. For other linear or nonlinear cases, the drop is aligned at ϕ = ±45°, and a spherical drop with no deformation is possible at Ca ≠ 0. In a highly viscous drop $(\lambda \gg 1)$, the drop remains almost spherical even at Ca = O(1), oriented at ϕ = 0° or 90° (G ≠ 1), and its deformation is independent of Ca.

Steep streams involve shallow, supercritical turbulent flows over a permeable bed made up of coarse particles. They usually exhibit higher flow resistance and stronger mass and momentum exchanges between the stream and subsurface flow than low-gradient streams. Describing their flow dynamics using generalised Manning–Strickler equations has led to empirical relationships with weak predictive power (errors between predictions and data of over one order of magnitude). We studied shallow turbulent flows by employing a mesoscopic approach based on the double-averaged Navier–Stokes equations. More specifically, we were concerned with the possibility of modelling the turbulent and dispersive shear stress equations using simple algebraic equations. To that end, we studied shallow, supercritical turbulent flows over a sloping bed made up of randomly packed spherical particles. Using visualisation techniques based on particle velocimetry imaging and refractive index matched scanning, we were able to reconstruct the velocity field throughout the bed and stream, far from the sidewalls, and estimate the contributions of the dispersive and turbulent shear stresses to the total shear stress. The dispersive shear stress represented less than 20 % of the turbulent shear stress, but because it was concentrated within a thin layer (called the roughness layer) where it outweighed the turbulent shear stress, it had a significant influence on the mean velocity profile. We proposed an algebraic closure equation for dispersive shear stress, based on the mixing-length model used for turbulent shear stress, and we found that it captured closely the mean-velocity and turbulence-intensity profiles of shallow flows over horizontal or sloping permeable beds. Our data suggest that flow dynamics was affected largely by turbulence damping, drag forces and dispersion within the roughness layer, which may explain why steep streams differ from low-gradient streams.

This paper presents the response and the wake modes of a freely vibrating D-section prism with varying angles of attack ($\alpha = 0^\circ \text {--}180^\circ$) and reduced velocity ($U^* = 2\text {--}20$) by a numerical investigation. The Reynolds number, based on the effective diameter, is fixed at 100. The vibration of the prism is allowed only in the transverse direction. We found six types of response with increasing angle of attack: typical vortex-induced vibration (VIV) at $\alpha = 0^\circ \text {--}35^\circ$; extended VIV at $\alpha = 40^\circ \text {--}65^\circ$; combined VIV and galloping at $\alpha = 70^\circ \text {--}80^\circ$; narrowed VIV at $\alpha = 85^\circ \text {--}150^\circ$; transition response, from narrowed VIV to pure galloping, at $\alpha = 155^\circ \text {--}160^\circ$; and pure galloping at $\alpha = 165^\circ \text {--}180^\circ$. The typical and narrowed VIVs are characterized by linearly increasing normalized vibration frequency with increasing $U^*$, which is attributed to the stationary separation points of the boundary layer. On the other hand, in the extended VIV, the vortex shedding frequency matches the natural frequency in a large $U^*$ range with increasing $\alpha$ generally. The galloping is characterized by monotonically increasing amplitude with enlarging $U^*$, with the largest amplitude being $A^* = 3.2$. For the combined VIV and galloping, the vibration amplitude is marginal in the VIV branch while it significantly increases with $U^*$ in the galloping branch. In the transition from narrowed VIV to pure galloping, the vibration frequency shows a galloping-like feature, but the amplitude does not monotonically increase with increasing $U^*$. Moreover, a partition of the wake modes in the $U^*$–$\alpha$ parametric plane is presented, and the flow physics is elucidated through time variations of the displacement, drag and lift coefficients and vortex dynamics. The angle-of-attack range of galloping is largely predicted by performing a quasi-steady analysis of the galloping instability. Finally, the effects of $m^*$ and ${\textit {Re}}$, the roles of afterbody and the roles of separation point in determining vibration responses and vortex shedding frequency are further discussed.

Asymptotic homogenization is employed to formulate upscaled effective boundary conditions at a smooth virtual surface for a natural-convection flow over a periodically roughened vertical wall, to bypass the expensive numerical resolution of flow and temperature fields near and within wall corrugations. Microscale problems are found by expanding near-wall variables in terms of a small parameter $\epsilon$, the ratio between the microscopic and the macroscopic length scales. The expressions of the upscaled velocity and temperature boundary conditions are provided up to second-order accuracy in $\epsilon$. The case of transverse square ribs is considered as a representative example. The classical Navier-slip condition for the streamwise and the spanwise velocity components is modified at second order by the gradient of the normal stress and the time derivative of the shear stress. The streamwise slip velocity is additionally corrected by a buoyancy term at first order and a temperature-gradient term at second order. The normal velocity at the virtual surface appears only as a second-order transpiration condition. A Robin-like condition for the temperature is found, where the wall temperature is corrected with a temperature-gradient term representing thermal slip. The accuracy levels and the applicability range of the effective conditions to mimic the macroscopic flow behaviour are investigated under laminar flow conditions, in comparison with results of full feature-resolving simulations. A formal validity limit for the approximation is sought in terms of a single accuracy criterion ($C$), which combines the effects of the Grashof number and the ribs’ density. The introduced model is further tested on different rib geometries.

The deformation and breakup of droplets in airflows is important in spray and atomisation processes, but the shear effect in non-uniform airflow is rarely reported. In this study, the deformation and breakup of droplets in a shear flow of air is investigated experimentally using high-speed imaging, digital image processing and particle image velocimetry. The results show that in airflow with a strong shear effect, the droplet breakup exhibits unique features due to the uplift and stretching produced by the interaction between the deformed droplet and the shear layer. The breakup process can be divided into three stages according to the droplet morphology and the breakup mechanism, namely the sheet breakup, the swing breakup and the rim breakup stages. Theoretical analysis reveals that the swing breakup is governed by the transverse Rayleigh–Taylor instability. A regime map of the droplet breakup is produced, and the transitions between different regimes are obtained theoretically. The stretching liquid film during the droplet deformation and the fragment size distribution after droplet breakup are analysed quantitatively, and the results show that they are determined by the competition of breakup at different stages affected by the shear. Finally, the effect of the droplet viscosity is investigated, and the viscosity inhibits the droplet breakup in a strong shear airflow.

Direct numerical simulations of a three-dimensional cylindrical Richtmyer–Meshkov instability with and without chemical reactions are carried out to explore the chemical reaction effects on the statistical characteristics of transition and turbulent mixing. We adopt 9-species and 19-reaction models of non-premixed hydrogen and oxygen separated by a multimode perturbed cylindrical interface. A new definition of mixing width suitable for a chemical reaction is introduced, and we investigate the spatio-temporal evolution of typical flow parameters within the mixing regions. After reshock with a fuller mixing of fuels and oxygen, the chemical reaction becomes sufficiently apparent at affecting the evolution of the flow fields. Because of the generation of a combustion wave within the combustion regions and propagation, the growth of the mixing width with a chemical reaction is accelerated, especially around the outer radius with large temperature gradient profiles. However, the viscous dissipation rate in the early stage of the chemical reaction is greater because of heat release, which results in weakened turbulent mixing within the mixing regions. We confirm that small-scale structures begin to develop after reshock and then decay over time. During the developing process, helicity also begins to develop, in addition to kinetic energy, viscous dissipation rate, enstrophy, etc. In the present numerical simulations with cylindrical geometry, the fluctuating flow fields evolve from quasi-two-dimensional perturbations, and the generations of helicity can capture this transition process. The weakened fluctuations during shock compression can be explained as the inverse energy cascade, and the chemical reaction can promote this inverse energy cascade process.