A data-driven implementation of a quasi-linear approximation is presented, extending a minimal quasi-linear approximation (MQLA) (Hwang & Ekchardt, J. Fluid Mech., vol. 894, 2020, p. A23) to incorporate non-zero streamwise Fourier modes. A data-based approach is proposed, matching the two-dimensional wavenumber spectra for a fixed spanwise wavenumber between a direct numerical simulation (DNS) (Lee & Moser, J. Fluid Mech., vol. 774, 2015, pp. 395–415) and that generated by the eddy viscosity enhanced linearised Navier–Stokes equations at $Re_\tau \approx 5200$, where $Re_\tau$ is the friction Reynolds number. Leveraging the self-similar nature of the energy-containing part in the DNS velocity spectra, a universal self-similar streamwise wavenumber weight is determined for the linearised fluctuation equations at $Re_\tau \simeq ~5200$. The data-driven quasi-linear approximation (DQLA) provides noteworthy enhancements in the wall-normal and spanwise turbulence intensity profiles. It exhibits a qualitatively similar structure in the spanwise wavenumber velocity spectra compared with the MQLA. Additionally, the DQLA offers extra statistical outputs in the streamwise wavenumber coordinates, enabling a comprehensive global analysis of this modelling approach. By comparing the DQLA results with DNS results, the limitations of the presented framework are discussed, mainly pertaining to the lack of the streak instability (or transient growth) mechanism and energy cascade from the linearised model. The DQLA is subsequently employed over a range of Reynolds numbers up to $Re_\tau = 10^5$. Overall, the turbulence statistics and spectra produced by the DQLA scale consistently with the available DNS and experimental data, with the Townsend–Perry constants displaying a mild Reynolds dependence (Hwang, Hutchins & Marusic, J. Fluid Mech., vol. 933, 2022, p. A8). The scaling behaviour of the turbulence intensity profiles deviates away from the classic $\ln (Re_\tau )$ scaling, following the inverse centreline velocity scaling for the higher Reynolds numbers.

The effect of the slip velocity in a bubbly jet ($Re = 3000$, initial void $\textrm {fraction} = 1.1\,\%$) is studied with various bubble slip ratios (0.07–2.1 times the inlet velocity, 0.01–0.31 m s$^{-1}$ in dimension). The governing equations, which are the conservation of mass, momentum and volume fraction, are solved by direct numerical simulations. A set of ordinary differential equations was derived by using a conventional one-dimensional integral framework on jets and by expressing the slip velocity using an exponential function. The one-dimensional analysis of bubbly jets successfully predicted the jet velocity radius, centreline velocity and the gas spread rate downstream of the bubbly jet. Second- and third-order statistics were also analysed to better understand the turbulent characteristics of the bubbly jet. A high slip ratio results in a rigid, narrower bubbly jet core region, where the turbulent kinetic energy is conserved along the centreline but does not diffuse towards the ambient region. The narrow, circular region around the core at high slip ratios decreased the turbulent kinetic energy. In the rigid bubbly jet core, turbulent diffusion of the gas phase is suppressed and has low correlations with the velocity fluctuations. Turbulence characteristics such as turbulence stress were compared with single-phase jet and plume results from existing literature. The magnitude of the turbulence characteristics at the lowest slip ratio is comparable with what is found in the literature, but there is a rapid transition from slip ratios of $0.14$ to $0.7$. Furthermore, the turbulent kinetic energy budget was analysed for each case. The production term had the largest contribution, and its magnitude was almost five times the buoyancy production downstream. High slip ratio cases show a positive peak of pressure strain and turbulent diffusion along the centreline.

Richtmyer–Meshkov (RM) instability in convergent geometries (such as cylinders and spheres) plays a fundamental role in natural phenomena and engineering applications, e.g. supernova explosion and inertial confinement fusion. Convergent geometry refers to a system in which the interface converges and the fluids are compressed correspondingly. By applying a decomposition formula, the stretching or compression (S(C)) effect is separated from the perturbation growth as one of the main contributions, which is defined as the averaged velocity difference between two ends of the mixing zone. Starting from linear theories, the S(C) effect in planar, cylindrical and spherical geometries is derived as a function of geometrical convergence ratio, compression ratio and mixing width. Specifically, geometrical convergence stretches the mixing zone, while fluid compression compresses the mixing zone. Moreover, the contribution of geometrical convergence in the spherical geometry is more important than that in the cylindrical geometry. A series of cylindrical cases with high convergence ratio is simulated, and the growth of perturbations is compared with that of the corresponding planar cases. As a result, the theoretical results of the S(C) effect agree well with the numerical results. Furthermore, results show that the S(C) effect is a significant feature in convergent geometries. Therefore, the S(C) effect is an important part of the Bell–Plesset effect. The present work on the S(C) effect is important for further modelling of the mixing width of convergent RM instabilities.

Solvent exchange is a process involving mixing between a good solvent with dissolved solute and a poor solvent. The process creates local oversaturation which causes the nucleation of minute solute droplets. Such ternary systems on a macro-scale have remained unexplored in the turbulent regime. We experimentally study the solvent exchange process by injecting mixtures of ethanol and trans-anethole into water, forming a turbulent buoyant jet in the upward direction. Locally, turbulent mixing causes oversaturation of the trans-anethole following turbulent entrainment. We optically measure the concentration of the nucleated droplets using a light attenuation technique and find that the radial concentration profile has a sub-Gaussian kurtosis. In contrast to the entrainment-based models, the spatial evolution of the oversaturation reveals continuous droplet nucleation downstream and radially across the jet, which we attribute to the limited mixing capacity of the jet. Although we are far from a full quantitative understanding, this work extends the knowledge on solvent exchange into the turbulent regime, and brings in a novel type of flow, broadening the scope of multicomponent, multiphase turbulent jets with phase transition.

A small-disturbance asymptotic model is derived to describe the complex nature of a pure water vapour flow with non-equilibrium and homogeneous condensation around a thin airfoil operating at transonic speed and small angle of attack. The van der Waals equation of state provides real-gas relationships among the thermodynamic properties of water vapour. Classical nucleation and droplet growth theory is used to model the condensation process. The similarity parameters which determine the flow and condensation physics are identified. The flow may be described by a nonlinear and non-homogeneous partial differential equation coupled with a set of four ordinary differential equations to model the condensation process. The model problem is used to study the effects of independent variation of the upstream flow and thermodynamic conditions, airfoil geometry and angle of attack on the pressure and condensate mass fraction distributions along the airfoil surfaces and the consequent effect on the wave drag and lift coefficients. Increasing the upstream temperature at fixed values of upstream supersaturation ratio and Mach number results in increased condensation and higher wave drag coefficient. Increasing the upstream supersaturation ratio at fixed values of upstream temperature and Mach number also results in increased condensation and the wave drag coefficient increases nonlinearly. In addition, the effects of varying airfoil geometry with a fixed thickness ratio and chord on flow properties and condensation region are studied. The computed results confirm the similarity law of Zierep & Lin (Forsch. Ing. Wes. A, vol. 33 (6), 1967, pp. 169–172), relating the onset condensation Mach number to upstream stagnation relative humidity, when an effective specific heat ratio is used. The small-disturbance model is a useful tool to analyse the physics of high-speed condensing steam flows around airfoils operating at high pressures and temperatures.

The present work uses a reduced-order model to study the motion of a buoyant vortex ring with non-negligible core size. Buoyancy is considered in both non-Boussinesq and Boussinesq situations using an axisymmetric contour dynamics formulation. The density of the vortex ring differs from that of the ambient fluid, and both densities are constant and conserved. The motion of the ring is calculated by following the boundary of the vortex core, which is also the interface between the two densities. The velocity of the contour comes from a combination of a specific continuous vorticity distribution within its core and a vortex sheet on the core boundary. An evolution equation for the vortex sheet is derived from the Euler equation, which simplifies considerably in the Boussinesq limit. Numerical solutions for the coupled integro-differential equations are obtained. The dynamics of the vortex sheet and the formation of two possible singularities, including singularities in the curvature and the shock-like profile of the vortex sheet strength, are discussed. Three dimensionless groups, the Atwood, Froude and Weber numbers, are introduced to measure the importance of physical effects acting on the motion of a buoyant vortex ring.

We report an experimental and numerical study on Rayleigh–Bénard convection in a slender rectangular geometry with the aspect ratio $\varGamma$ varying from 0.05 to 0.3 and a Rayleigh number range of $10^5\leqslant Ra\leqslant 3\times 10^9$. The Prandtl number is fixed at $Pr=4.38$. It is found that the onset of convection is postponed when the convection domain approaches the quasi-one-dimensional limit. The onset Rayleigh number shows a $Ra_c=328\varGamma ^{-4.18}$ scaling for the experiment and a $Ra_c=810\varGamma ^{-3.95}$ scaling for the simulation, both consistent with a theoretical prediction of $Ra_c\sim \varGamma ^{-4}$. Moreover, the effective Nusselt–Rayleigh scaling exponent $\beta =\partial (\log Nu)/\partial (\log Ra)$ near the onset of convection also shows a rapid increase with decreasing $\varGamma$. Power-law fits to the experimental and numerical data yield $\beta =0.290\varGamma ^{-0.90}$ and $\beta =0.564\varGamma ^{-0.92}$, respectively. Near onset, the flow shows a stretched cell structure. In this regime, the velocity and temperature variations in a horizontal cross-section are found to be almost invariant with height in the core region of a slender domain. As the Rayleigh number increases, the system evolves from the viscous dominant regime to a plume-controlled one, a feature of which is enhancement in the heat transport efficiency. Upon further increase of $Ra$, the flow comes back to the classical boundary-layer-controlled regime, in which the quasi-one-dimensional geometry has no apparent effect on the global heat transfer.

A continuation method is used to study Rayleigh–Bénard convection in a non-Newtonian fluid inside a parallelepiped cavity. The cavity has its length equal to twice the side of the square cross-section. Shear-thinning and shear-thickening Carreau fluids are considered. The focus is put on the two stable branches which exist in the Newtonian case, a stable primary branch of transverse rolls $B_1$ and a primary branch of longitudinal roll $B_2$, stabilized beyond a secondary bifurcation point $S_2$. Although the primary bifurcation points are unchanged, the non-Newtonian properties strongly modify the bifurcation diagram. Indeed, for a shear-thinning fluid, the stable solutions can exist at much smaller Rayleigh numbers ${Ra}$, on subcritical branches beyond saddle-node points $SN_1$ and $SN_2$, and small perturbations can be sufficient to reach them. In agreement with Bouteraa et al. (J. Fluid Mech., vol. 767, 2015, pp. 696–734), the change of the primary bifurcations from supercritical to subcritical occurs at given values of what they define as the degree of shear-thinning parameter $\alpha$. Moreover, the value of the Rayleigh number at the saddle-node points can be approximated by a simple expression, as proposed by Jenny et al. (J. Non-Newtonian Fluid Mech., vol. 219, 2015, pp. 19–34). In the case of a shear-thickening fluid, the branches remain supercritical, but the secondary point $S_2$ is strongly moved towards larger ${Ra}$, making it more difficult to reach the longitudinal roll solution. Energy analyses at the bifurcations $SN_1$, $SN_2$ and $S_2$ show that the changes of the corresponding critical thresholds ${Ra}_c$ are connected with the changes of the viscous properties, but also with changes of the buoyancy effect.

We compute the flow induced by the steady translation of a rigid sphere along the axis of a large cylindrical container filled with a low-viscosity fluid set in rigid-body rotation, the sphere being constrained to spin at the same rate as the undisturbed fluid. The parameter range covered by the simulations is similar to that explored experimentally by Maxworthy (J. Fluid Mech., vol. 40, 1970, pp. 453–479). We describe the salient features of the flow, especially the internal characteristics of the Taylor columns that form ahead of and behind the body and the inertial wave pattern, and determine the drag and torque acting on the sphere. Torque variations are found to obey two markedly different laws under rapid- and slow-rotation conditions. The corresponding scaling laws are predicted by examining the dominant balances governing the axial vorticity distribution in the body vicinity. Results for the drag agree well with the semi-empirical law proposed for inertialess regimes by Tanzosh & Stone (J. Fluid Mech., vol. 275, 1994, pp. 225–256). This law is found to apply even in regimes where inertial effects are large, provided that rotation effects are also large enough. Influence of axial confinement is shown to increase dramatically the drag in rapidly rotating configurations, and the container length has to be approximately a thousand times larger than the sphere for this influence to become negligibly small. The reported simulations establish that this confinement effect is at the origin of the long-standing discrepancy existing between Maxworthy's results and theoretical predictions.

We optimize jet mixing using large eddy simulations (LES) at a Reynolds number of $3000$. Key methodological enablers consist of Bayesian optimization, a surrogate model enhanced by deep learning and persistent data topology for physical interpretation. The mixing performance is characterized by an equivalent jet radius ($R_{eq}$) derived from the streamwise velocity in a plane located $8$ diameters downstream. The optimization is performed in a 22-dimensional actuation space that comprises most known excitations. This search space parameterizes the distributed actuation imposed on the bulk flow and at the periphery of the nozzle in the streamwise and radial directions. The momentum flux measures the energy input of the actuation. The optimization quadruples the jet radius $R_{eq}$ with a $7$-armed blooming jet after around $570$ evaluations. The control input requires $2\,\%$ momentum flux of the main flow, which is one order of magnitude lower than an ad hoc dual-mode excitation. Intriguingly, a pronounced suboptimum in the search space is associated with a double-helix jet, a new flow pattern. This jet pattern results in a mixing improvement comparable to the blooming jet. A state-of-the-art Bayesian optimization converges towards this double-helix solution. The learning is accelerated and converges to another better optimum by including a deep-learning-enhanced surrogate model trained along the optimization. Persistent data topology extracts the global and many local minima in the actuation space. These minima can be identified with flow patterns beneficial to the mixing.

The linear stability of the family of flows generated by an acceleration-skewed oscillating planar wall is investigated using Floquet theory. Neutral stability curves and critical conditions for linear instability are determined for an extensive range of acceleration-skewed oscillating flows. Results indicate that acceleration skewness is destabilising and reduces the critical Reynolds number for the onset of linearly unstable behaviour. The structure of the eigenfunctions is discussed and solutions suggest that disturbances grow in the direction of highest acceleration.