We explore the application of the reference map technique, originally developed for Eulerian simulation of solid mechanics, in Lagrangian kinematics of turbulent flows. Unlike traditional methods based on explicit particle tracking, the reference map facilitates the calculation of flow maps and gradients without the need for particles. This is achieved through an Eulerian update of the reference map, which records the take-off positions of fluid particles. This approach is found to be mathematically equivalent to the work of Leung (J. Comput. Phys., vol. 230, issue 9, 2011, pp. 3500–3524), who computed the flow map of simple two-dimensional flows using an Eulerian approach. We discuss important modifications necessary for its first application to complex three-dimensional turbulent flows, including the conservative, low-dissipation update of the flow map and the treatment of periodic boundary conditions. We first demonstrate the accuracy of finite-time Lyapunov exponent (FTLE) calculations based on the reference map against the standard particle-based approach in a two-dimensional Taylor–Green vortex. Then we apply it to turbulent channel flow at $Re_\tau =180$, where Lagrangian coherent structures identified as ridges of the backward-time FTLE are found to bound vortical regions of flow, consistent with Eulerian coherent structures from the $Q$-criterion. The reference map also proves suitable for material surface tracking despite not explicitly tracking particles. This capability can provide valuable insights into the Lagrangian landscape of turbulent momentum transport, complementing Eulerian velocity field analysis. The evolution of initially wall-normal material surfaces in the viscous sublayer, buffer layer and log layer sheds light on the Reynolds stress-generating events from a Lagrangian perspective.

We investigate the effect of high wind speeds on the breakup mechanisms that govern the formation of a spray from nozzles that form liquid sheets, which subsequently break up. The fragmentation mechanism of liquid sheets from spray nozzles has recently been described in detail under quiescent conditions. With high wind speeds, measurements of the droplet size distribution reveal two rather than one characteristic drop sizes, suggesting the existence of two distinct breakup mechanisms. High-speed images of the spray are used to identify these two mechanisms. We show that the smaller droplets result from the breakup of ‘bags’ formed in the spray sheet by the wind, while the larger droplets result from the breakup of the remaining perforated sheet. Based on the two mechanisms, a probability density function is constructed and fitted to the measured droplet size distributions. We show that the spray sheet destabilises due to the Rayleigh–Taylor instability induced by the airflow, and that the experimentally observable breakup length and size of the holes blown in the sheet are predicted by the fastest growing wavenumber. From this, a theoretical prediction for the droplet size from bag breakup and remaining sheet breakup is derived.

The impact of a chemical reaction, $A+B \rightarrow C$, on the stability of a miscible radial displacement in a porous medium is established. Our study involves a comprehensive analysis employing both linear stability analysis and nonlinear simulations. Through linear stability analysis, the onset of instability for monotonic as well as non-monotonic viscosity profiles corresponding to the same end-point viscosity are discussed and compared. We establish a $(R_b,R_c)$ phase plane for a wide range of Damköhler number ($Da$) and Péclet number ($Pe$) into stable and unstable regions. Here, $R_b=\ln (\mu _B/ \mu _A)$ and $R_c=\ln (\mu _C/ \mu _A)$ and $\mu _{i}$ is the viscosity of fluid $i$ $\in \{A$, $B$, $C$}. The stable zone in the $(R_b, R_c)$ phase plane contracts with increased $Da$ and $Pe$ but never vanishes. It exists even for $Da \rightarrow \infty$. Interestingly, we obtain a $Da$ independent stable region in the neighbourhood of $R_c=R_b$ where no transition occurs in stability despite changes in reaction rate. The study allows us to acquire knowledge about the transition of the stability for varying $Da, Pe$ and different reactions classified using $R_b, R_c$.

The breaking and energy distribution of mode-1 depression internal solitary wave interactions with Gaussian ridges are examined through laboratory experiments. A series of processes, such as shoaling, breaking, transmission and reflection, are captured completely by measuring the velocity field in a large region. It is found that the maximum interface descent ($a_{max}$) during wave shoaling is an important parameter for diagnosing the type of wave–ridge interaction and energy distribution. The wave breaking on the ridge depends on the modified blockage parameter $\zeta _m$, the ratio of the sum of the upper layer depth and $a_{max}$ to the water depth at the top of the ridge. As $\zeta _m$ increases, the interaction type transitions from no breaking to plunging and mixed plunging–collapsing breaking. Within the scope of this experiment, the energy distribution can be characterized solely by $\zeta _m$. The transmission energy decreases monotonically with increasing $\zeta _m$, and there is a linear relationship between $\zeta _m^2$ and the reflection coefficient. The value of $a_{max}$ can be determined from the basic initial parameters of the experiment. Based on the incident wave parameters, the depth of the upper and lower layers, and the topographic parameters, two new simple methods for predicting $a_{max}$ on the ridge are proposed.

We theoretically and experimentally study gravity currents of a Newtonian fluid advancing in a two-dimensional, infinite and saturated porous domain over a horizontal impermeable bed. The driving force is due to the density difference between the denser flowing fluid and the lighter, immobile ambient fluid. The current is taken to be in the Darcy–Forchheimer regime, where a term quadratic in the seepage velocity accounts for inertial contributions to the resistance. The volume of fluid of the current varies as a function of time as $\sim T^{\gamma }$, where the exponent parameterizes the case of constant volume subject to dam break ($\gamma =0$), of constant ($\gamma =1$), waning ($\gamma <1$) and waxing inflow rate ($\gamma >1$). The nonlinear governing equations, developed within the lubrication theory, admit self-similar solutions for some combinations of the parameters involved and for two limiting conditions of low and high local Forchheimer number, a dimensionless quantity involving the local slope of the current profile. Another parameter $N$ expresses the relative importance of the nonlinear term in Darcy–Forchheimer's law; values of $N$ in practical applications may vary in a large interval around unity, e.g. $N\in [10^{-5},10^{2}]$; in our experiments, $N\in [2.8,64]$. Sixteen experiments with three different grain sizes of the porous medium and different inflow rates corroborate the theory: the experimental nose speed and current profiles are in good agreement with the theory. Moreover, the asymptotic behaviour of the self-similar solutions is in excellent agreement with the numerical results of the direct integration of the full problem, confirming the validity of a relatively simple one-dimensional model.

Compound flows consist of two or more parallel compressible streams in a duct and their theoretical treatment has gained attention for the analysis and modelling of ejectors. Recent works have shown that these flows can experience choking upstream of the geometric throat. While it is well known that friction can push the sonic section downstream of the throat, no mechanism has been identified yet to explain its displacement in the opposite direction. This study extends the existing compound flow theory and proposes a one-dimensional (1-D) model, including friction between the streams and the duct walls. The model captures the upstream and downstream displacements of the sonic section. Through an analytical investigation of the singularity at the sonic section, it is demonstrated that friction between the streams is the primary driver of upstream displacement. The 1-D formulation is validated against axisymmetric Reynolds averaged Navier–Stokes simulations of a compound nozzle for various inlet pressures and geometries. The effect of friction is investigated using an inviscid simulation for the isentropic case and viscous simulations with both slip and no-slip conditions at the wall. The proposed extension accurately captures the displacement of the sonic section, offering a new tool for in-depth analysis and modelling of internal compound flows.

The motion of a sphere freely rising or falling in a 5d (d is the diameter of the sphere) square tube was numerically studied for the sphere-to-fluid density ratio ranging from 0.1 to 2.3 (0.1 ≤ ρs/ρ ≤ 2.3, ρs is the density of spheres and ρ the fluid density) and Galileo number from 140 to 230 (140 ≤ Ga ≤ 230). We report that Hopf bifurcation occurs at Gacrit ≈ 157, where both the heavy and light spheres lose stability. The helical motion is widely seen for all spheres at Ga > 160 resulting from a double-threaded vortex interacting with the tube walls, which becomes irregular at Ga ≥ 190 where heavy spheres act differently from their counterparts; that is, heavy spheres change their helical directions alternately while light spheres exhibit helical trajectories with jaggedness in connection with the shedding of the double-threaded vortices. This is because of the difference in inertia between the heavy and light spheres. We also checked the oscillation periods for the helical motion of the spheres. They show opposite variations with ρs/ρ for the two types of spheres. Light spheres (ρs/ρ ≤ 0.7) reach a zigzagging regime at Ga ≥ 200 where a vortex loop (hairpin-like vortical structure) is formed which may develop into a vortex ring downstream at small ρs/ρ. This might be the first time a transition from the helical motion to the zigzagging motion for heavy spheres (ρs/ρ ≥ 1.8) has been reported. Finally, we examined the dependence of both the terminal Reynolds number and the drag coefficient of the spheres on the Galileo number.

The flow-induced oscillation of an S-shaped buckled flexible filament was explored using the penalty immersed boundary method. As the length and bending rigidity of the filament were varied, three distinct modes emerged: the equilibrium mode, streamwise oscillation (SO) mode and transverse oscillation (TO) mode. A transition region between the SO and TO modes was identified. Notably, the filament exhibited a 3P wake pattern under SO and a 2S wake pattern under TO. The former was induced by fluid–elastic instability, while the latter was attributed to vortex-induced oscillation. The interaction between the filament's motion and vortex shedding was examined for both modes. To elucidate the disparity between the TO of the S-shaped buckled filament and snap-through oscillation (STO), a ball-on-a-hill analogy was introduced. The performance of energy harvesting was evaluated using metrics including the elastic energy and power coefficient. The TO mode was found to show significantly higher energy harvesting performance than the SO and STO modes. The majority of the strain energy was concentrated at the upper and lower midpoints of the filament.

We study numerically the flow around a spherical droplet set fixed in a linear shear flow with moderate shear rates ($Sr\leq 0.5$, $Sr$ being the ratio between the velocity difference across the drop and the relative velocity) over a wide range of external Reynolds numbers ($0.1<{{Re}}\leq 250$, ${{Re}}$ based on the slip velocity and the viscosity of the external fluid) and drop-to-fluid viscosity ratios ($0.01\leq \mu ^\ast \leq 100$). The flow structure, the vorticity field and their intrinsic connection with the lift force are analysed. Specifically, the results on lift force are compared with the low-${{Re}}$ solution derived for droplets of arbitrary $\mu ^\ast$, as well as prior data at finite ${{Re}}$ available in both the clean-bubble limit ($\mu ^\ast \to 0$) and the solid-sphere limit ($\mu ^\ast \to \infty$). Notably, at ${{Re}}=O(100)$, the lift force exhibits a non-monotonic transition from $\mu ^\ast \to 0$ to $\mu ^\ast \to \infty$, peaking at $\mu ^\ast \approx 1$. This behaviour is related to an internal three-dimensional flow bifurcation also occurring under uniform-flow conditions, which makes the flow to evolve from axisymmetric to biplanar symmetric. This flow bifurcation occurs at low-but-finite $\mu ^\ast$ when the internal Reynolds number (${{Re}}^i$, based on the viscosity of the internal fluid) exceeds approximately 300. In the presence of shear, the corresponding imperfect bifurcation enhances the extensional rate of the flow in the wake. Consequently, the streamwise vortices generated behind the droplet can be more intense compared with those behind a clean bubble. Given the close relation between the lift and these vortices, a droplet with ${{Re}}=O(100)$ and $\mu ^\ast \approx 1$ typically experiences a greater lift force than that in the inviscid limit.

The flow near a moving contact line depends on the dynamic contact angle, viscosity ratio and capillary number. We report experiments involving immersing a plate into a liquid bath, concurrently measuring the interface shape, interfacial velocity and fluid flow using digital image processing and particle image velocimetry. All experiments were performed at low plate speeds to maintain small Reynolds and capillary numbers for comparison with viscous theories. The dynamic contact angle, measured in the viscous phase, was kept below $90^{\circ }$ and the viscosity ratio, $\lambda < 1$. This region of parameter space is largely unexplored for advancing contact lines. An important aim of the present study is to provide new experimental data against which new contact line models can be developed. The flow field is directly compared against the prediction from the viscous theory of Huh & Scriven (J. Colloid Interface Sci., vol. 35, issue 1, 1971, pp. 85–101) but with a slight modification involving the curved interface. Remarkable agreement is found between experiments and theory across a wide parameter range. The prediction for interfacial speed from Huh & Scriven is also in excellent agreement with experiments except in the vicinity of the contact line. Material points along the interface were found to rapidly slow down near the contact line, thus alleviating the singularity at the moving contact line. To the best of our knowledge, such a detailed test of theoretical models has not been performed before and we hope the present study will spur new modelling efforts in the field.

We introduce a novel recursive procedure to a neural-network-based subgrid-scale (NN-based SGS) model for large eddy simulation (LES) of high-Reynolds-number turbulent flow. This process is designed to allow an SGS model to be applicable to a hierarchy of different grid sizes without requiring expensive filtered direct numerical simulation (DNS) data: (1) train an NN-based SGS model with filtered DNS data at a low Reynolds number; (2) apply the trained SGS model to LES at a higher Reynolds number; (3) update this SGS model with training data augmented with filtered LES (fLES) data, accommodating coarser filter size; (4) apply the updated NN to LES at a further higher Reynolds number; (5) go back to Step (3) until a target (very coarse) filter size divided by the Kolmogorov length scale is reached. We also construct an NN-based SGS model using a dual NN architecture whose outputs are the SGS normal stresses for one NN and the SGS shear stresses for the other NN. The input is composed of the velocity gradient tensor and grid size. Furthermore, for the application of an NN-based SGS model trained with one flow to another flow, we modify the NN by eliminating bias and introducing a leaky rectified linear unit function as an activation function. The present recursive SGS model is applied to forced homogeneous isotropic turbulence (FHIT) and successfully predicts FHIT at high Reynolds numbers. The present model trained from FHIT is also applied to decaying homogeneous isotropic turbulence and shows an excellent prediction performance.

We provide scaling relations for the Nusselt number $Nu$ and the friction coefficient $C_{S}$ in sheared Rayleigh–Bénard convection, i.e. in Rayleigh–Bénard flow with Couette- or Poiseuille-type shear forcing, by extending the Grossmann & Lohse (J. Fluid Mech., vol. 407, 2000, pp. 27–56, Phys. Rev. Lett., vol. 86, 2001, pp. 3316–3319, Phys. Rev. E, vol. 66, 2002, 016305, Phys. Fluids, vol. 16, 2004, pp. 4462–4472) theory to sheared thermal convection. The control parameters for these systems are the Rayleigh number $Ra$, the Prandtl number $Pr$ and the Reynolds number $Re_S$ that characterises the strength of the imposed shear. By direct numerical simulations and theoretical considerations, we show that, in turbulent Rayleigh–Bénard convection, the friction coefficients associated with the applied shear and the shear generated by the large-scale convection rolls are both well described by Prandtl's (Ergeb. Aerodyn. Vers. Gött., vol. 4, 1932, pp. 18–29) logarithmic friction law, suggesting some kind of universality between purely shear-driven flows and thermal convection. These scaling relations hold well for $10^6 \leq Ra \leq 10^8$, $0.5 \leq Pr \leq 5.0$, and $0 \leq Re_S \leq 10^4$.