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The highly nonlinear evolution of the single-mode stratified compressible Rayleigh–Taylor instability (RTI) is investigated via direct numerical simulation over a range of Atwood numbers ( $A_T=0.1$ – $0.9$ ) and Mach numbers ( $Ma=0.1$ – $0.7$ ) for characterising the isothermal background stratification. After the potential stage, it is found that the bubble is accelerated to a velocity which is well above the saturation value predicted in the potential flow model. Unlike the bubble re-acceleration behaviour in quasi-incompressible RTI with uniform background density, the characteristics in the stratified compressible RTI are driven by not only vorticity accumulation inside the bubble but also flow compressibility resulting from the stratification. Specifically, in the case of strong stratification and high $A_T$ , the flow compressibility dominates the bubble re-acceleration characters. To model the effect of flow compressibility, we propose a novel model to reliably describe the bubble re-acceleration behaviours in the stratified compressible RTI, via introducing the dilatation into the classical model that takes into account only vorticity accumulation.

Three-dimensional elastic turbulence in Taylor–Couette flows of dilute polymer solutions has been realized and thoroughly investigated via direct numerical simulations. A novel flow transition pathway from elastically dominated turbulence to solitary vortex pairs (or diwhirls) and eventually to elastic turbulence is observed by decreasing the fluid inertia ( $Re$ ) over seven orders of magnitude, i.e. from $Re=1000$ to $0.0001$ . The dominant spatio-temporal flow features in the elastic turbulence regime are those of large-scale unsteady diwhirls and small-scale axial and azimuthal travelling waves in the outer and inner halves of the gap, respectively. Moreover, it is conclusively shown that production of turbulent kinetic energy in purely elastic turbulence solely arises due to the stochastic nature of polymer stretch/relaxation. Overall, based on this comprehensive numerical investigation, the differences in the underlying fluid physics that give rise to turbulent fluctuations in elastically dominated and purely elastic turbulence have been delineated.

The width effect on contact angle hysteresis in a microchannel with patterned heterogeneous surfaces is systematically investigated. In the model, identical defects periodically appear on the background surface. To this end, a droplet's evaporation and condensation processes inside the microchannel are studied by theoretical analysis and numerical simulation based on a diffuse-interface lattice Boltzmann method. The microchannel width effect on the system's equilibrium properties is studied. The results demonstrate that the number of equilibrium configurations increases linearly with the microchannel width ( $b$ ), and has a quadratic relationship with the cosine of the reference contact angle and the heterogeneity strength ( $\varepsilon$ ). The average most stable contact angle is independent of $b$ and is always equal to the contact angle predicted by the Cassie–Baxter equation. For contact angle hysteresis ( $H$ ), when the microchannels are narrow and wide, there are individual-effect-dominated hysteresis (IDH) and collective-effect-dominated hysteresis (CDH), respectively. The IDH and CDH are hysteresis modes corresponding to the jumping behaviour of contact lines affected by individual defects and two neighbouring defects, respectively. Based on the graphical force balance approach, we establish a scaling law to quantify the connection between $H$ , $b$ and $\varepsilon$ . Specifically, in the IDH mode, $H\sim b \varepsilon ^2$ , while in the CDH mode, $H$ increases linearly with $\varepsilon$ but nonlinearly with $b$ .

Pressure fluctuations play an essential role in the transport of turbulent kinetic energy and vibrational loading. This study focuses on examining the effect of wall cooling on pressure fluctuations in compressible turbulent boundary layers by high-fidelity direct numerical simulations. Pressure fluctuations result from the vorticity mode and the acoustic mode that are both closely dependent on compressibility. To demonstrate the effects of wall cooling at various compressibility intensities, three free-stream Mach numbers are investigated, i.e. $M_\infty =0.5$ , 2.0 and 8.0, with real gas effects being absent for $M_\infty =8.0$ due to a low enthalpy inflow. Overall, opposite effects of wall cooling on pressure fluctuations are found between the subsonic/supersonic cases and the hypersonic case. Specifically, the pressure fluctuations normalized by wall shear stress $p^\prime _{rms}/\tau _w$ are suppressed in the subsonic and supersonic cases, while enhanced in the hypersonic case near the wall. Importantly, travelling-wave-like alternating positive and negative structures (APNS), which greatly contribute to pressure fluctuations, are identified within the viscous sublayer and buffer layer in the hypersonic cases. Furthermore, generating mechanisms of pressure fluctuations are explored by extending the decomposition based on the fluctuating pressure equation to compressible turbulent boundary layers. Pressure fluctuations are decomposed into five components, in which rapid pressure, slow pressure and compressible pressure are dominant. The suppression of pressure fluctuations in the subsonic and supersonic cases is due to both rapid pressure and slow pressure being suppressed by wall cooling. In contrast, wall cooling strengthens compressible pressure for all Mach numbers, especially in the hypersonic case, resulting in increased wall pressure fluctuations. Compressible pressure plays a leading role in the hypersonic case, mainly due to the APNS. Essentially, the main effects of wall cooling can be interpreted by the suppression of the vorticity mode and the enhancement of the acoustic mode.

Trailing-edge serrations inspired by owls are capable of reducing broadband noise. In this study, the wall-resolved large-eddy simulations (LES) are carried out on the flow over NACA-0012 airfoil with additional serrated trailing edges. The computations are conducted with the high-order flux reconstruction method on unstructured meshes. Three kinds of serrations with different lengths are studied and compared with the straight trailing-edge case, and all three types of serration achieved a certain degree of noise reduction. Presently, the medium-length serration achieves the best noise reduction effect. The maximum decrease of overall sound pressure level is approximately 2.4 dB, implying that the length of serration has a substantial impact on the noise reduction effect. The serration has no significant effect on the upstream turbulence statistics, but it changes the flow structure near the serration, such as inducing side vortex pairs attached to the serration edges. Moreover, dynamic mode decomposition shows that the pressure structures vary with the serration length. For the most unstable hydrodynamic wave, the spanwise coherence of the mode structure of pressure in the upstream boundary layer is weakened. In addition, serrations can redistribute the dipole sources on the surfaces of airfoil and serrations. The destructive interference is enhanced to some extent, which is favourable for noise reduction. In contrast with LES simulations, the pure dipole analysis shows that the longest serration case seems to be the best. Furthermore, a recently developed noise theory is used to evaluate the influence of serrations on the flow noise sources qualitatively and quantitatively. It is found that the serrations can mitigate noise source intensity near the serration edges but increase the source intensity in the near wake. The combined effect of serration on the dipole source and flow noise source determines the overall noise reduction effect. To conclude, destructive interference plays a primary role in suppressing noise radiation by serrated trailing edges, and the dual effect of flow noise sources should be considered in future serration designs. As the influence of turbulence structure will make it more difficult to find the optimal serration parameters, the position of high-fidelity simulation will become increasingly important.

We investigate by direct numerical simulation the active control of laminar-turbulent transition in a hypersonic flat-plate boundary layer at a freestream Mach number of 5.86. The control mechanism is a synthetic jet. Based upon the linear stability theory of Mack, in hypersonic flow the important path to transition involves a high-frequency, second-mode fundamental resonance. Through systematic investigation, we reveal that the forcing the boundary layer with a synthetic jet at appropriate combinations of amplitude and frequency suppresses the second mode and delays transition. To gain physical insights into the major control mechanism, we employ the momentum potential theory (MPT) to analyze the flows with and without control. Essentially, the underlying control mechanism relies on an intriguing effect of the synthetic jet via generating the outward radiated wave structures, which are identified to split the upstream acoustic and vortical components. The splitting treatment presents the second-mode energy to drop sharply after the flow passes through the synthetic jet slot. The MPT source-term analysis reveals that the significantly suppressed near-wall source terms are responsible for suppressing the second mode downstream. Compared with the vortical and thermal source terms, the acoustic source term is found to be suppressed most. The kinetic budget analysis further reveals that the splitting treatment is related to the non-parallel effect and the nonlinear interaction.

A direct numerical simulation database of a weakly compressible turbulent channel flow with bulk Mach number 1.56 is studied in detail, including the geometrical relationships between the pressure-Hessian tensor and the vorticity/strain-rate tensor, as well as the mechanism of the pressure-Hessian tensor contributing to the evolution of invariants of the velocity gradient tensor. The results show that the geometrical relationships between the pressure-Hessian tensor and the vorticity/strain-rate tensor in the central region of the channel are consistent with that of isotropic turbulence. However, in the buffer layer with relatively stronger inhomogeneity and anisotropy, the vorticity tends to be aligned with the first or second eigenvector of the pressure-Hessian tensor in the unstable focus/compressing topological region, and tends to be aligned with the first eigenvector of the pressure-Hessian tensor in the stable focus/stretching topological region. In the unstable node/saddle/saddle and stable node/saddle/saddle topological regions, the vorticity prefers to lie in the plane of the first and second eigenvectors of the pressure-Hessian tensor. The strain-rate and the pressure-Hessian tensors tend to share their second principal direction. Moreover, for the coupling between the pressure-Hessian tensor and the principal strain rates, we clarify the influence on dissipation, the nonlinear generation of dissipation and the enstrophy generation. The decomposition of the pressure-Hessian tensor further shows that the slow pressure-related term dominates the pressure-Hessian tensor's contribution, and the influence of inhomogeneity and anisotropy mainly originates from the inhomogeneity and anisotropy of the fluctuating velocity. These statistical properties would be instructive in formulating dynamical models of the velocity gradient tensor for wall turbulence.

Direct numerical simulation of spanwise-rotation-driven flow transitions in viscoelastic plane Couette flow from a drag-reduced inertial to a drag-enhanced elasto-inertial turbulent flow state followed by full relaminarization is reported for the first time. Specifically, this novel flow transition begins with a drag-reduced inertial turbulent flow state at a low rotation number $0\leqslant Ro \leqslant 0.1$ , and then transitions to a rotation/polymer-additive-driven drag-enhanced inertial turbulent regime, $0.1\leqslant Ro \leqslant 0.3$ . In turn, the flow transitions to a drag-enhanced elasto-inertial turbulent state, $0.3\leqslant Ro \leqslant 0.9$ , and eventually relaminarizes at $Ro=1$ . In addition, two novel rotation-dependent drag enhancement mechanisms are proposed and substantiated. (1) The formation of large-scale roll cells results in enhanced convective momentum transport along with significant polymer elongation and stress generated in the extensionally dominated flow between adjacent roll cells at $Ro\leqslant 0.2$ . (2) Coriolis-force-generated turbulent vortices cause strong incoherent transport and homogenization of significant polymer stress in the bulk via their vortical circulations at $Ro=0.5 - 0.9$ .

The stability of the two-layer film flow driven by an oscillatory plate under long-wave disturbances is studied. The influence of key factors, such as thickness ratio ( $n$ ), viscosity ratio ( $m$ ), density ratio ( $r$ ), oscillatory frequency ( $\beta$ ) and insoluble surfactants on the stability behaviours is studied systematically. Four special Floquet patterns are identified, and the corresponding growth rates are obtained by solving the eigenvalue problem of the fourth-order matrix. A small viscosity ratio ( $m\le 1$ ) may stabilize the flow but it depends on the thickness ratio. If the viscosity ratio is not very small ( $m>0.1$ ), in the $(\beta ,n)$ -plane, stable and unstable curved stripes appear alternately. In other words, under the circumstances, if the two-layer film flow is unstable, slightly adjusting the thickness of the upper film may make it stable. In particular, if the upper film is thin enough, even under high-frequency oscillation, the flow is always stable. The influence of density ratio is similar, i.e. there are curved stable and unstable stripes in the $(\beta ,r)$ -planes. Surface surfactants generally stabilize the flow of the two-layer oscillatory membrane, while interfacial surfactants may stabilize or destabilize the flow but the effect is mild. It is also found that gravity can generally stabilize the flow because it narrows the bandwidth of unstable frequencies.

The stability of the inverted flexible plate with non-uniform stiffness distribution in a free stream is studied by numerical simulation and mathematical theory. In our study, the bending stiffness distribution is expressed as the function of the leading edge's bending stiffness [Formula: see text] and the polynomial of the plate's coordinate. Based on the former theoretical work on the stability of inverted plates with uniform stiffness distribution, we derive the upper limit value of [Formula: see text] at which the zero-deflection equilibrium loses its stability for the plate with non-uniform stiffness distribution. The critical [Formula: see text] derived from the mathematical theory agrees well with that obtained from the numerical simulation. An effective bending stiffness is defined, which can be used to unify the regimes of the motion modes between uniform plates and non-uniform plates. Moreover, three orders of mass ratio [[Formula: see text], and [Formula: see text]] are investigated, and the underlying mechanism for large amplitude flapping is clarified for the inverted plate with different mass ratios. An appropriate bending stiffness distribution can greatly improve the deformation of the plate. The findings shed some light on the energy harvesting of the inverted plate.

When controlling the trailing-edge (TE) interference noise of airfoil, the design of the TE serration shape is still an open issue. To this end, the flow and noise generation for different TE serration shapes are explored by the wall-resolved implicit large-eddy simulation and acoustic analogy. The feather-like serrations are found to achieve the most prominent noise reduction among the four types of curved serrations, especially in the low-frequency range. With the aid of acoustic analogy, the coherence analysis of far-field noise produced by the dipole sources on the airfoil surface is performed. The results show that destructive interference is still the critical mechanism responsible for noise reduction. Considering only the dipole sources, we find that the feather-like serrated TE shape can obtain the best noise reduction performance among all the serrated cases. Furthermore, since flow structures are reorganized near the TE serrations, we investigated the flow noise sources quantitatively in the near field. In these cases, the noise source due to flow structures is suppressed to a greater extent in the feather-like serrated case near the TE serration roots. Consequently, the above findings indicate that the feather-like serration favors suppressing dipole and flow noise sources in the near field, which makes it an efficient configuration for reducing airfoil noise.

A high-order transition route from inertial to elasticity-dominated turbulence (EDT) in Taylor–Couette flows of polymeric solutions has been discovered via direct numerical simulations. This novel two-step transition route is realized by enhancing the extensional viscosity and hoop stresses of the polymeric solution via increasing the maximum chain extension at a fixed polymer concentration. Specifically, in the first step inertial turbulence is stabilized to a laminar flow much like the modulated wavy vortex flow. The second step destabilizes this laminar flow state to EDT, i.e. a spatially smooth and temporally random flow with a $-3.5$ scaling law of the energy spectrum reminiscent of elastic turbulence. The flow states involved are distinctly different to those observed in the reverse transition route from inertial turbulence via a relaminarization of the flow to elasto-inertial turbulence in parallel shear flows, underscoring the importance of polymer-induced hoop stresses in realizing EDT that are absent in parallel shear flows.

The flow physics of inertio-elastic turbulent Taylor–Couette flow for a radius ratio of $0.5$ in the Reynolds number ( $Re$ ) range of $500$ to $8000$ is investigated via direct numerical simulation. It is shown that as $Re$ is increased the turbulence dynamics can be subdivided into two distinct regimes: (i) a low $Re \leqslant 1000$ regime where the flow physics is essentially dominated by nonlinear elastic forces and the main contribution to transport and mixing of momentum, stress and energy comes from large-scale flow structures in the bulk region and (ii) a high $Re \geqslant 5000$ regime where inertial forces govern the flow physics and the flow dynamics is mainly governed by small-scale flow structures in the near-wall region. Flow–microstructure coupling analysis reveals that the elastic Görtler instability in the near-wall region is triggered via significant polymer extension and commensurately high hoop stresses. This instability gives rise to small-scale elastic vortical structures identified as elastic Görtler vortices which are present at all $Re$ considered. In fact, these vortices develop herringbone streaks near the inner wall that have a longer average life span than their Newtonian counterparts due to their elastic origin. Examination of the budgets of mean streamwise enstrophy, mean kinetic energy, turbulent kinetic energy and Reynolds shear stress demonstrates that increasing fluid inertia hinders the generation of elastic stresses, leading to a monotonic reduction of the elastic-related effects on the flow physics.

The topological evolution near the turbulent/non-turbulent interface (TNTI) in turbulent mixing layer is studied by means of statistical analysis of the invariants of velocity gradient tens...

Liquid films can be entrained when the dewetting velocity attains a threshold, and this dynamical wetting transition has been well studied in the situation of plane substrates. We investigate the forced dewetting in a capillary tube using diffuse-interface simulations and lubrication analysis, focusing on the onset of wetting transition and subsequent interface evolution. Results show that the meniscus remains stable when the displacing rate is below a threshold, beyond which film entrainment occurs and eventually leads to the formation of Taylor bubbles separated by liquid slugs, as has also been observed in the recent experiments of Zhao et al. (Phys. Rev. Lett., vol. 120, 2018, 084501). We derive an analytical solution of the critical capillary number, and demonstrate that the wetting transition is accompanied by a vanishing apparent contact angle and an abrupt drop of the contact-line velocity. Both the bubble and slug lengths are found to depend on the capillary number and the wettability of the wall. A theoretical formula for the bubble length is also proposed and compares favourably with numerical and experimental results.

In this work we extend the method of the constrained large-eddy simulation (CLES) to simulate the turbulent flow over inhomogeneous rough walls. In the original concept of CLES, the subgrid-scale (SGS) stress is constrained so that the mean part and the fluctuation part of the SGS stress can be modelled separately to improve the accuracy of the simulation result. Here in the simulation of the rough-wall flows, we propose to interpret the extra stress terms in the CLES formulation as the roughness-induced stress so that the roughness inhomogeneity can be incorporated by modifying the formulation of the constrained SGS stress. This is examined with the simulations of the channel flow with the spanwise alternating high/low roughness strips. Then the CLES method is employed to investigate the temporal response of the turbulence to the change of the wall condition from rough to smooth. We demonstrate that the temporal development of the internal boundary layer is just similar to that in a spatial rough-to-smooth transition process, and the spanwise roughness inhomogeneity has little impact on the transition process.

Direct numerical simulation of polymer-induced flow relaminarization of turbulent spanwise-rotating plane Couette flow (RPCF) is reported for the first time. Specifically, the reverse transition pathway from a Newtonian turbulent RPCF to a fully relaminarized drag enhanced viscoelastic flow has been elucidated. Evidently, this transition occurs gradually by weakening and eventual elimination of small-scale vortices as the Weissenberg number ( ) is enhanced, paving the way for a two-dimensional laminar flow consisting of large-scale and highly organized roll cells. The influence of polymer additives on convective momentum exchange by large-scale roll cells and small-scale turbulent vortices, namely, the drag reduction (DR) realized by elimination of turbulent vortices and the significant drag enhancement (DE) that results from polymer roll cell interactions has been identified as the mechanism of DE. The observed vortical changes point to a universal mechanism for the coupling of polymer chains and turbulent vortices in wall-bounded viscoelastic DE and DR flows.

We present an analytical model of nonlinear evolution of two-dimensional single-mode Rayleigh–Taylor instability (RTI) in cylindrical geometry at arbitrary Atwood number for the first time. Our model covers a full scenario of bubble evolution from the earlier exponential growth to the nonlinear regime with the bubbles growing in time as is formulated as a simplified function of the external acceleration, Atwood number and number of perturbation waves. This model's predictions are in good agreement with data from direct numerical simulations.

The subgrid effects on the dynamics of the filtered velocity gradient tensor (VGT) in compressible turbulence are studied by means of statistical analysis of the invariants of the filtered VGT in compressible mixing layers. The evolution of the filtered VGT is determined by the interaction among the invariants, the pressure effects, the viscous effects and the subgrid effects. Based on the probability fluxes in the plane of the second ( plane is identified in the locally expanded regions, which is determined by the non-normal effect. Further, an SGS model with the non-local effect is proposed to give a better prediction of the SGS energy dissipation.

The effect of trailing-edge shape on the self-propulsive performance of three-dimensional flexible plates is studied numerically. In our study, the trailing edges of the plates are symmetric chevron shapes, and the trailing-edge angle varies from (concave plate) to (convex plate). Under different bending stiffnesses , three regimes of the propulsive performance in terms of propulsive velocity and efficiency as a function of are identified. When is small, moderate and large, the square, convex and concave plate achieves the best performance, respectively. Analyses of vortical structures and velocity fields show that usually the jet behind the plate with the best performance is longest. Besides, the inclination angle of the jet may be small. The different propulsive performances at small and moderate are mainly attributed to the phase lag of the trailing edge. The force acting on the plate is analysed and it is found that the thrust force is mainly contributed by the normal force. If , and are rescaled by the normal force and the area moment of the plate, the curves for different almost collapse into a single curve when the bending stiffness coefficient is small or moderate. The scaling confirms that the normal force should be the characteristic fluid force at small or moderate and the effect is governed by the area moment. The findings may shed some light on the propulsive performance of aquatic animals.

Energetic benefit and enhanced performance are considered among the most fascinating achievements of collective behaviours, e.g. fish schools and flying formations. The collective locomotion of two self-propelled flapping plates initially in a side-by-side arrangement is investigated numerically. Both in-phase and antiphase oscillations for the two plates are considered. It is found that the plates will spontaneously form some stable configurations as a result of the flow-mediated interaction, specifically, the staggered-following (SF) mode and the alternate-leading (AL) mode for the in-phase scenario and the moving abreast (MA) mode and the AL mode for the antiphase scenario. In the SF mode, the rear plate follows the front one with a staggered configuration. In the AL mode, the plates chase each other side-by-side alternately. In terms of propulsive speed and efficiency, the performance of the plates in the SF mode with small lateral spacing $H$ is found to be better than those in the tandem following case ($H=0$) and the side-by-side case (i.e. the AL mode). To achieve higher propulsive efficiency, no matter in-phase or antiphase oscillations, the two plates with moderate bending stiffness, e.g. $K\approx O(1)$, are preferred and they should be close enough in the lateral direction. For the side-by-side configuration, the performance of each plate in the antiphase and in-phase scenarios is enhanced and weakened in comparison with that of the isolated plate, respectively. Besides the pressure and vorticity contours, the normal force and thrust acting on the plates are also analysed. It is revealed that the thrust is mainly contributed by the normal force at moderate bending stiffness. The normal force and thrust are critical to the propulsive speed and efficiency. For two self-propelled plates, in view of hydrodynamics, to achieve higher performance the in-phase SF mode and antiphase flappings in the side-by-side configuration are preferred.

Drag reduction (DR) in plane Couette flow (PCF) induced by the addition of flexible polymers has been studied via direct numerical simulation (DNS). The similarities and differences in the drag reduction features of PCF and plane Poiseuille flow (PPF) have been examined in detail, particularly in regard to the polymer-induced modification of large-scale structures (LSSs) in the near-wall turbulence. Specifically, it has been demonstrated that in the near-wall region, drag-reduced PCF has features similar to those of drag-reduced PPF; however, in the core region, intriguing differences are found between these two drag-reduced shear flows. Chief among these differences is the significant polymer stretch that arises from the enhanced exchanges between elastic potential energy and turbulent kinetic energy and the commensurate observation of peak values of the conformation tensor components $\unicode[STIX]{x1D60A}_{yy}$ and $\unicode[STIX]{x1D60A}_{zz}$ in this region. This finding is in stark contrast to that of drag-reduced PPF where the polymer stretch and the exchanges between elastic potential energy and turbulent kinetic energy in the core region are insignificant; to this end, in drag-reduced PPF, peak values of the conformation tensor components appear in the near-wall region. Therefore, this study paves the way for understanding the underlying flow physics in drag-reduced PCF, particularly in the context of elastic theory. Moreover, the longitudinal large-scale streaks at the channel centre of drag-reduced PCF are greatly strengthened due to the increased production/dissipation ratio; the LSS imprint effects on the near-wall flow of drag-reduced PCF monotonically increase as the Weissenberg number is enhanced.

We numerically investigate the mechanism leading to the entrapment of spheres at the gas–liquid interface after impact. Upon impact onto a liquid pool, a hydrophobic sphere is seen to follow one of the three regimes identified in the experiment (Lee & Kim, Langmuir, vol. 24, 2008, pp. 142–145): sinking, bouncing or being entrapped at the interface. It is important to understand the role of wettability in this process of flow–structure interaction with dynamic wetting, and in particular, to what extent the wettability can determine whether the sphere is entrapped at the interface. For this purpose, a diffuse-interface immersed boundary method is adopted in the numerical simulations. We expand the parameter space considered previously, provide the phase diagrams and identify the key phenomena in the impact dynamics. Then, we propose the scaling models to interpret the critical conditions for the occurrence of sphere entrapment, accounting for the wettability of the sphere. The models are shown to provide a good correlation among the impact inertia of the drop, the surface tension, the wettability and the density ratio of the sphere to the liquid.

The interaction of tandem inverted flexible flags in a uniform flow is investigated. For the inverted flags, their ends are fixed with their heads freely flapping. A direct numerical simulation is performed for which the Reynolds number is of order 200. Large flapping amplitude as well as large drag force is preferred because more energy may be harvested if more bending energy is generated. For the simple case of two tandem inverted flags, the drag force and flapping amplitude of the rear flag are found to be smaller than those of an isolated inverted flag due to the destructive merging mode of vortices. However, it is still unknown whether more bending energy can be generated when coupled inverted flags are arranged properly. To explore the possibility, inverted flags are proposed to be arranged as two rows, which indicate two lines of inverted flags perpendicular to the direction of the incoming flow, and flags in the front and rear rows are in-line or staggered. First the results for infinite flags with periodic boundary condition are presented. In both the in-line and the staggered arrangements, due to the interactions between the front–rear flags, the flapping amplitude or the maximum bending deformation and bending energy of a flag in the rear row can be enhanced, which may be significantly higher than those of an isolated case. Meanwhile, the bending energy of a flag in the front row is close to that of an isolated case. Second, results for finite inverted flag groups show that antiphase synchronization is preferred. When the group number is large enough, the bending energies of the front and rear flags in the inner groups are close to those in the infinite case. This finding may be helpful for the designing of an efficient energy harvesting device using inverted flags.

The locomotion of a flapping flexible plate with different shapes and non-uniform chordwise stiffness distribution in a stationary fluid is studied numerically. The normalized effective bending stiffness K∗ for three-dimensional plates with arbitrary stiffness distribution and shape parameters is proposed, and the overall bending stiffness of non-uniform plates with different shapes is reasonably characterized. It is found that the propulsion performance in terms of cruising speed and efficiency of the self-propelled flapping plate mainly depends on the effective bending stiffness. Plates with moderate flexibility K∗ show better propulsion performance. Meanwhile, both a large area moment of the plate and a flexible anterior are favorable to significantly improve their propulsive performance. The evolution of vortical structures and the pressure distribution on the upper and lower surfaces of the plate are analyzed, and the inherent mechanism is revealed. These findings are of great significance to the optimal design of propulsion systems with different fins or wings.

Experimental and numerical investigations on the interaction of a planar shock wave with two-dimensional (2-D) and three-dimensional (3-D) light gas cylinders are performed. The effects of initial interface curvature on flow morphology, wave pattern, vorticity distribution and interface movement are emphasized. In experiments, a wire-restriction method based on the soap film technique is employed to generate N$_{2}$ cylinders surrounded by SF$_{6}$ with well-characterized shapes, including a convex cylinder, a concave cylinder with a minimum-surface feature and a 2-D cylinder. The high-speed schlieren pictures demonstrate that fewer disturbance waves exist in the flow field and the evolving interfaces develop in a more symmetrical way relative to previous studies. By combining the high-order weighted essentially non-oscillatory construction with the double-flux scheme, numerical simulation is conducted to explore the detailed 3-D flow structures. It is indicated that the shape and the size of 3-D gas cylinders in different planes along the vertical direction change gradually due to the existence of both horizontal and vertical velocities of the flow. At very early stages, pressure oscillations in the vicinity of evolving interfaces induced by complex waves contribute much to the deformation of the 3-D gas cylinders. As time proceeds, the development of the shocked volume would be dominated by the baroclinic vorticity deposited on the interface. In comparison with the 2-D case, the oppositely (or identically) signed principal curvatures of the concave (or convex) SF$_{6}$/N$_{2}$ boundary cause complex high pressure zones and additional vorticity deposition, and the upstream interface from the symmetric slice of the concave (or convex) N$_{2}$ cylinder moves with an inhibition (or a promotion). Finally, a generalized 3-D theoretical model is proposed for predicting the upstream interface movements of different gas cylinders and the present experimental and numerical findings are well predicted.

A suspended ellipsoidal particle inside a Poiseuille flow with Reynolds number up to 360 is studied numerically. The effects of tube diameter ($D$), inertia of the particle and the flow, and the particle geometry (both prolate and oblate ellipsoids) are considered. When a prolate particle with $a/b=2$ is inside a wider tube (e.g. $D/A>1.9$), where $A=2a$ is the length of the major axis of the particle, the terminal stable state is tumbling. When the prolate particle is inside a narrower tube ($1.0), log-rolling or kayaking modes may appear. Which mode occurs depends on the competition between fluid and particle inertia. When the fluid inertia is dominant, the log-rolling mode appears, otherwise, the kayaking mode appears. Inclined and spiral modes may appear when $D/A and $D/A=1$, respectively. For a prolate ellipsoid with $a/b=4$, if $1, there is only the kayaking mode and the log-rolling mode is not observed. When an oblate particle is inside a wider tube (e.g. $D/A>3.5$), it may adopt the log-rolling mode. Inclined and intermediate modes are firstly identified in narrower tubes. The phase diagram of the modes is also provided. The modes in the phase diagrams were not found to be affected by the initial state of the particle based on limited observation.

The effect of non-uniform chordwise stiffness distribution on the self-propulsive performance of three-dimensional flexible plates is studied numerically. Some typical stiffness distributions, including uniform, declining, and growing distribution, are considered. First, the normalized bending stiffness K is derived, which can well represent the overall bending stiffness of the non-uniform plates. For different non-uniformly distributed plates with the same K, the maximum displacement difference between the trailing and leading edges of the plate during the flapping is almost identical. There exists a common optimal K at which all the plates achieve their optimal performance, i.e., the highest cruising speed and efficiency. Second, we reveal what kind of non-uniform distribution could be the best at a specific K in terms of the propulsive performance. The force analysis indicates that a larger bending deformation in the anterior part for the growing distribution leads to a larger thrust. Hence, the large local slope along the anterior flexible plate is preferred to enhance the propulsive performance. The results obtained in this study may shed some light on a better understanding of the hydrodynamic effect on the self-propulsion of the non-uniform stiffness wings or fins of animals.

We investigate the entrainment of liquid films on a partially wetting plate vertically withdrawn from a reservoir of viscous liquid using a combination of diffuse-interface numerical simulation and lubrication analysis. So far available theoretical investigations were commonly conducted by focusing on separate parameter regions, and a complete description of the flow regimes with increasing plate speed is still missing. By solving the full Stokes equations, we present a complete scenario of film transition in the presence of moving contact line. With increasing plate speed, we identify numerically four successive flow regimes in terms of the interfacial morphologies: (1) a stationary meniscus, (2) a speed-independent thick film connected to the liquid bath through a stationary dimple, (3) coexistence of a thick film and the classical Landau–Levich–Derjaguin (LLD) film connected by a propagating capillary shock and (4) a film with a monotonically varying thickness. The characteristics of the film profiles in different regions of the interfaces are analysed with lubrication theory as applicable, and satisfactory agreements with the numerical results are obtained. In particular, we confirm that the onset of film deposition occurs at a vanishing apparent contact angle, consistent with the predictions of lubrication theory. Numerical results suggest that the critical capillary number for the onset of film deposition is smaller than that for the onset of LLD film despite the fact that it is higher than the experimentally observed one, showing that the thick film can be realized in the two-dimensional model. We also demonstrated that the LLD film is triggered by the bifurcation of the stationary dimple, which is found to admit multiple branches of stable and unstable solutions.

The self-propulsion of a 3-D flapping flexible plate in a stationary fluid is numerically studied by an immersed boundary-lattice Boltzmann method for the fluid flow and a finite element method for the plate motion. When the leading-edge of the plate is forced to heave sinusoidally, the entire plate starts to move freely as a result of the fluid-structure interaction. Based on our simulation and analysis on the dynamical behaviors of the flapping flexible plate, we have found that the effect of plate aspect ratio on its propulsive properties can be divided into three typical regimes which are related to the plate flexibility, i.e. stiff, medium flexible, and more flexible regime. It is also identified that a suitable structure flexibility, corresponding to the medium flexible regime, can improve the propulsive speed and efficiency. The wake behind the flapping plate is investigated for several aspect ratios to demon- strate some typical vortical structures. The results obtained in this study can provide some physical insights into the understanding of the propulsive mechanisms in the flapping-based locomotion.

We investigate the cavity formation during the impact of spheres and cylinders into a liquid pool by using a combination of experiments, simulations and theoretical analysis, with particular interest in contact-line pinning and its relation with the subsequent cavity evolution. The flows are simulated by a Navier–Stokes diffuse-interface solver that allows for moving contact lines. On the basis of agreement on experimentally measured quantities such as the position of the pinned contact line and the interface shape, we investigate flow details that are not accessible experimentally, identify the interface regions in the cavity formation and examine the geometric effects of impact objects. We connect wettability, inertia, geometry of the impact object, interface bending and contact-line position with the contact-line pinning by analysing the force balance at a pinned meniscus, and the result compares favourably with those from simulations and experiments. In addition to adjusting the interface bending, the object geometry also has a significant effect on the magnitude of low pressure in the liquid and the occurrence of flow separation. As a result, it is easier for an object with sharp edges to generate a cavity than a smooth object. A theoretical model based on the Rayleigh–Besant equation is developed to provide a quantitative description of the radial expansion of the cavity after the pinning of the contact line. The accuracy of the solution is greatly affected by the geometrical information on the interface connected to the pinned meniscus, showing the dependence of the global cavity dynamics on the local flows around the pinned contact line. Vertical ripple propagation on the cavity wall is found to follow the dispersion relation for the perturbation evolution on a hollow jet.

Rectangular supersonic jets exist widely in propulsion systems of aircrafts. When they are imperfectly expanded under certain conditions, the upstream traveling waves referred to as screech tones will be produced, which may cause structural fatigue failure. In this work, high fidelity simulations are employed to investigate the bevelled effects due to the asymmetric lips of nozzles on shock structures and screech noise in planar supersonic jets. The present results are in agreement with previous experimental and numerical data for the symmetric case. For asymmetric cases, it is found that the bevelled effects will affect the shear layer transition, noise radiation, and shock cell oscillations. The level of screech noise generally decreases with increasing the length difference of two lips. The maximum 7.9 dB drop is identified, and the deflection angle of the mainstream of 9.35° is achieved when this length difference reaches the height of the nozzle. Moreover, dynamic mode decomposition (DMD) is specifically utilized to analyze shock cell oscillations. The results show that the bevelled effects suppress the most energetic DMD mode, corresponding to the dominant frequency of shock screech. The phenomenon of shock leakage is detected in the symmetric case, which is assumed to be an important screech noise source, while it seems to be weakened when the nozzle is bevelled. The longitudinal flapping motion of shock cells is substantially weakened due to the bevelled effects, which might be responsible for the suppression of shock leakage and the screech noise reduction.

Two deep learning (DL) models addressing the super-resolution (SR) reconstruction of turbulent flows from low-resolution coarse flow field data are developed. One is the static convolutional neural network (SCNN), and the other is the novel multiple temporal paths convolutional neural network (MTPC). The SCNN model takes instantaneous snapshots as an input, while the MTPC model takes a time series of velocity fields as an input, and it includes spatial and temporal information simultaneously. Three temporal paths are designed in the MTPC to fully capture features in different time ranges. A weight path is added to generate pixel-level weight maps of each temporal path. These models were first applied to forced isotropic turbulence. The corresponding high-resolution flow fields were reconstructed with high accuracy. The MTPC seems to be able to reproduce many important features as well, such as kinetic energy spectra and the joint probability density function of the second and third invariants of the velocity gradient tensor. As a further evaluation, the SR reconstruction of anisotropic channel flow with the DL models was performed. The SCNN and MTPC remarkably improve the spatial resolution in various wall regions and potentially grasp all the anisotropic turbulent properties. It is also shown that the MTPC supplements more under-resolved details than the SCNN. The success is attributed to the fact that the MTPC can extract extra temporal information from consecutive fluid fields. The present work may contribute to the development of the subgrid-scale model in computational fluid dynamics and enrich the application of SR technology in fluid mechanics.

We numerically study the impact of a compound drop on a hydrophobic substrate using a ternary-fluid diffuse-interface method, aiming to understand how the presence of the inner droplet affects the spreading dynamics and maximal spreading of the compound drop. First, it is interesting to see that the numerical results for an impacting pure drop agree well with the universal rescaling of maximal spreading ratio proposed by Lee et al. (J. Fluid Mech., vol. 786, 2016, R4). Second, two flow regimes have been identified for an impacting compound drop: namely jammed spreading and joint rim formation. The maximal spreading ratio of the compound drop is found to depend on the volume fraction of the inner droplet $\unicode[STIX]{x1D6FC}$, the surface tension ratio $\unicode[STIX]{x1D6FE}$, the Weber number and the flow regime. Moreover, we propose a universal rescaling of maximal spreading ratio for compound drops, by integrating the one for pure drops with a corrected Weber number that takes $\unicode[STIX]{x1D6FC}$, $\unicode[STIX]{x1D6FE}$ and the flow regime into account. The predictions of the universal rescaling are in good agreement with the numerical results for impacting compound drops.

In this paper, we investigate the ratchet mechanism of drops climbing a vibrated oblique plate based on three-dimensional direct numerical simulations, which for the first time reproduce the existing experiment (Brunet et al., Phys. Rev. Lett., vol. 99, 2007, 144501). With the help of numerical simulations, we identify an interesting and important wetting behaviour of the climbing drop; that is, the breaking of symmetry due to the inclination of the plate with respect to the acceleration leads to a hysteresis of the wetted area in one period of harmonic vibration. In particular, the average wetted area in the downhill stage is larger than that in the uphill stage, which is found to be responsible for the uphill net motion of the drop. A new hydrodynamic model is proposed to interpret the ratchet mechanism, taking account of the effects of the acceleration and contact angle hysteresis. The predictions of the theoretical analysis are in good agreement with the numerical results.

The yaw effect of the side flow around a high-speed train is studied by means of large eddy simulation at two typical yaw angles of φ = 30 ° and 60°, respectively. Both the mean and fluctuating values of lift force and side force coefficients increase obviously as the yaw angle increases. The spectral analysis indicates that the time-dependent aerodynamic forces are dominated by several energetic frequencies and the frequency range is broadened to a higher extent for the large yaw angle. To have a better understanding of the train aerodynamic behaviors, the dedicate three-dimensional vortical structures are analyzed for the flow at the two yaw angles. Moreover, the time-averaged flow patterns, turbulent statistics and the surface forces are also studied on sectional planes along the train.

Propulsive performance of a passively flapping plate in a uniform viscous flow has been studied numerically by means of a multiblock lattice Boltzmann method. The passively flapping plate is modeled by a rigid plate with a torsion spring acting about the pivot at the leading-edge of the plate, which is called a lumped-torsional-flexibility model. When the leading-edge is forced to take a vertical oscillation, the plate pitches passively due to the fluid-plate interaction. Based on our numerical simulations, various fundamental mechanisms dictating the propulsive performance, including the forces on the plate, power consumption, propulsive efficiency and vortical structures, have been studied. It is found that the torsional flexibility of the passively pitching plate can improve the propulsive performance. The results obtained in this study provide some physical insights into the understanding of the propulsive behaviors of swimming and flying animals.

The propulsive performance of two- and three-dimensional (2D and 3D) flapping flexible plates in a fluid at rest is investigated by a finite element method for the plate motion and an immersed boundary-lattice Boltzmann method for the fluid flow. We consider a model that as the leading-edge of the plate takes a vertical oscillation, the entire plate moves freely due to the fluid–structure interaction. The effects of flexibility on the dynamics of the 2D and 3D flapping plates are investigated. It is found that a suitable flexibility is benefit for improving the propulsive performance. The results obtained in this study provide physical insight into the understanding of the dynamics of the flapping-based locomotion.

The dynamics of viscous fluid flow over a circular flexible plate are studied numerically by an immersed boundary–lattice Boltzmann method for the fluid flow and a finite-element method for the plate motion. When the plate is clamped at its centre and placed in a uniform flow, it deforms by the flow-induced forces exerted on its surface. A series of distinct deformation modes of the plate are found in terms of the azimuthal fold number from axial symmetry to multifold deformation patterns. The developing process of deformation modes is analysed and both steady and unsteady states of the fluid–structure system are identified. The drag reduction due to the plate deformation and the elastic potential energy of the flexible plate are investigated. Theoretical analysis is performed to elucidate the deformation characteristics. The results obtained in this study provide physical insight into the understanding of the mechanisms on the dynamics of the fluid–structure system.

In this paper, drop impact onto a sphere is numerically investigated at moderate Reynolds and Weber numbers. It is naturally expected that the aspect ratio of the sphere to the drop, $\unicode[STIX]{x1D706}_{r}$, would make a big difference to drop spreading and retraction on the sphere, compared with drop impact onto a flat substrate. To quantitatively assess the effect of $\unicode[STIX]{x1D706}_{r}$, a diffuse-interface immersed-boundary method is adopted after being validated against experiments. With the help of numerical simulations, we identify the key regimes in the spreading and retraction, analyse the results by scaling laws, and quantitatively evaluate the effect of $\unicode[STIX]{x1D706}_{r}$ on the impact dynamics. We find that the thickness of the liquid film spreading on the sphere can be well approximated by $h_{L,\infty }(1+3/4\unicode[STIX]{x1D706}_{r}^{-3/2})$, where $h_{L,\infty }$ represents the film thickness of drop impact on a flat substrate. At the early stage of spreading, the temporal variation of the wetted area is independent of $\unicode[STIX]{x1D706}_{r}$ when the time is rescaled by the thickness of the liquid film. Drops are observed to retract on the sphere at a roughly constant speed, and the predictions of theoretical analysis are in good agreement with numerical results.

The unsteady dynamics of shock interactions on the crotch of two typical V-shaped blunt leading edges have been investigated numerically and experimentally with a freestream Mach number 6 for different ratios (R), defined as the rounding radius at the crotch to the blunt radius at the leading edge. The primary flow features observed in the shock tunnel experiments are reproduced by the large-eddy simulations. The time-averaged flow structures in the crotch are clearly shown as counterrotating vortices originating from the collision of jets near the stagnation point. These jets and vortices undergo unsteady motions coupled with the dynamics of shock interactions. Of great interest, two typical global oscillations, i.e., swing oscillation and arch-recover oscillation corresponding to the two values of R, are identified. The coherent structures of the oscillations are analyzed using the proper orthogonal decomposition technique. It is demonstrated that the swing oscillation and arch-recover oscillation are characterized by an antisymmetric pattern and a symmetrical pattern, respectively. These two oscillations are also characterized by the energetic middle-frequency components of the broadband wall pressure spectra. Two feedback models are proposed for the prediction of such middle-frequency components. The results show that the swing oscillation causes much more severe pressure load, and the local impingement of the transmitted shock on the crotch is responsible for the peak value of the pressure fluctuation. This study illustrates that the geometry of R has a key impact on the unsteady shock interactions and, therefore, should be considered critically in practical applications, such as the cowl lip of a hypersonic inward-turning inlet.The unsteady dynamics of shock interactions on the crotch of two typical V-shaped blunt leading edges have been investigated numerically and experimentally with a freestream Mach number 6 for different ratios (R), defined as the rounding radius at the crotch to the blunt radius at the leading edge. The primary flow features observed in the shock tunnel experiments are reproduced by the large-eddy simulations. The time-averaged flow structures in the crotch are clearly shown as counterrotating vortices originating from the collision of jets near the stagnation point. These jets and vortices undergo unsteady motions coupled with the dynamics of shock interactions. Of great interest, two typical global oscillations, i.e., swing oscillation and arch-recover oscillation corresponding to the two values of R, are identified. The coherent structures of the oscillations are analyzed using the proper orthogonal decomposition technique. It is demonstrated that the swing oscillation and arch-recover oscillation are c...

The role of the hydrodynamic effect in the collective locomotion of several birds or fish is an interesting topic. Taking a model of a self-propelled flapping plate, we numerically investigated the collective locomotion of a pair of plates with comparable but different propulsive capacities, e.g., one long and one short plates. The longer plate is supposed to have a stronger propulsive capacity. It is found that two typical equilibrium configurations, i.e., compact and sparse configurations, may emerge, which depend mainly on initial lateral and longitudinal gap spacing, i.e., H and G0, respectively. In the compact cases, when H is small, e.g., H < 0.6, in terms of cruising speed and efficiency, hydrodynamic advantages are found for both plates. In all sparse configurations, the propulsive performance of the leading plate is identical to that of the corresponding isolated one. The following short plate in the “long-short” (the longer in the front) sparse cases always takes hydrodynamic advantages in terms of cruising speed and efficiency. In the “short-long” (the shorter in the front) sparse cases, the follower’s propulsive capacity is suppressed because the cruising speed and input power decrease significantly compared to its isolated case. The analyses of hydrodynamic force and corresponding potential energy show that the staggered sparse configuration with H ∈ (0.4, 1.0) is more stable than that with other H. The “hydrodynamic drafting” analyzed here may shed some light on understanding the coordinated collective behaviors in biological and natural systems.

Topological evolution of compressible turbulent boundary layers at Mach 2 is investigated by means of statistical analysis of the invariants of the velocity gradient tensor based on the direct numerical simulation database. The probability density functions of the rate of change of the invariants exhibit the $- 3$ power-law distribution in the region of large Lagrangian derivative of the invariants in the inner and outer layers. The topological evolution is studied by conditional mean trajectories for the evolution of the invariants. The trajectories illustrate inward-spiralling orbits around and converging to the origin of the space of invariants in the outer layer, while they are repelled by the vicinity of the origin and converge towards a limit cycle in the inner layer. The compressibility effect on the mean topological evolution is studied in terms of the ‘incompressible’, compressed and expanding regions. It is found that the mean evolution of flow topologies is altered by the compressibility. The evolution equations of the invariants are derived and the relevant dynamics of the mean topological evolution are analysed. The compressibility effect is mainly related to the pressure effect. The mutual-interaction terms among the invariants are the root of the clockwise spiral behaviour of the local flow topology in the space of invariants.

The force and power of flapping plates are studied by vortex dynamic analysis. Based on the dynamic analysis of the numerical results of viscous flow past three-dimensional flapping plates, it is found that the force and power are strongly dominated by the vortical structures close to the body. Further, the dynamics of the flapping plate is investigated in terms of viscous vortex-ring model. It is revealed that the model can reasonably reflect the essential properties of the ring-like vortical structure in the wake, and the energy of the plate transferred to the flow for the formation of each vortical structure possesses a certain relation. Moreover, simplified formulae for the thrust and efficiency are proposed and verified to be reliable by the numerical solutions and experimental measurements of animal locomotion. The results obtained in this study provide physical insight into the understanding of the dynamic mechanisms relevant to flapping locomotion.

Flapping plates of typical fishlike tail shapes are simulated to investigate their locomotion performance using the multi-block Lattice Boltzmann Method (LBM) and Immersed Boundary (IB) method. Numerical results show that fishlike forked configurations have better locomotion performance compared with unforked plates. Based on our results, the caudal fin in carangi-form mode has greater thrust, and the lunate tail fin in thunniform mode has higher efficiency. These findings are qualitatively consistent with biological observations of fish swimming. Analysis of wake topology shows that the wake of the forked plate consists of a chain of alternating reverse horseshoe-like vortical structures. These structures induce a backward jet and generate a positive thrust. Moreover, this backward jet has a more favorable direction compared with that behind an unforked plate.