Spanwise wall forcing in the form of streamwise-travelling waves is applied to the suction side of a transonic airfoil with a shock wave to reduce aerodynamic drag. The study, conducted using direct numerical simulations, extends earlier findings by Quadrio et al. (J. Fluid Mech. vol. 942(R2), 2022, pp. 1–10) and confirms that the wall manipulation shifts the shock wave on the suction side towards the trailing edge of the profile, thereby enhancing its aerodynamic efficiency. A parametric study over the parameters of wall forcing is carried out for the Mach number set at 0.7 and the Reynolds number at 300 000. Similarities and differences with the incompressible plane case are discussed; for the first time, we describe how the interaction between the shock wave and the boundary layer is influenced by flow control via spanwise forcing. With suitable combinations of control parameters, the shock is delayed, which results in a separated region whose length correlates well with friction reduction. The analysis of the transient process following the sudden application of control is used to link flow separation with the intensification of the shock wave.

Opposition control (OC) is a reactive flow-control approach that mitigates the near-wall fluctuations by imposing blowing and suction at the wall, being opposite to the off-wall observations. We carried out high-resolution large-eddy simulations to investigate the effects of OC on turbulent boundary layers (TBLs) over a wing at a chord-based Reynolds number (${Re}_c$) of $200 \ 000$. Two cases were considered: flow over the suction sides of the NACA0012 wing section at an angle of attack of $0^{\circ }$, and the NACA4412 wing section at an angle of attack of $5^{\circ }$. These cases represent TBLs subjected to mild and strong non-uniform adverse pressure gradients (APGs), respectively. First, we assessed the control effects on the streamwise development of TBLs and the achieved drag reduction. Our findings indicate that the performance of OC in terms of friction-drag reduction significantly diminishes as the APG intensifies. Analysis of turbulence statistics subsequently reveals that this is directly linked to the intensified wall-normal convection caused by the strong APG: it energizes the control intensity to overload the limitation that guarantees drag reduction. The formation of the so-called virtual wall that reflects the mitigation of wall-normal momentum transport is also implicitly affected by the pressure gradient. Control and pressure-gradient effects are clearly apparent in the anisotropy invariant maps, which also highlight the relevance of the virtual wall. Finally, spectral analyses indicate that the wall-normal transport of small-scale structures to the outer region due to the APG has a detrimental impact on the performance of OC. Uniform blowing and body-force damping were also examined to understand the differences between the various control schemes. Despite the distinct performance of friction-drag reduction, the effects of uniform blowing are akin to those induced by a stronger APG, while the effects of body-force damping exhibit similarities to those of OC in terms of the streamwise development of the TBL although there are differences in the turbulent statistics. To authors’ best knowledge, the present study stands as the first in-depth analysis of the effects of OC applied to TBL subjected to non-uniform APGs with complex geometries.

Scalar dissipation rate (SDR) evolution in a stopping turbulent jet was analysed using direct numerical simulations and a theoretical approach. After the jet is stopped, a deceleration wave for the SDR propagates downstream with a speed similar to that for axial velocity. Upstream of the deceleration wave, mean centreline SDR becomes proportional to axial distance, and inversely proportional to the square of time. After passing of the deceleration wave, normalised radial profiles of SDR and its axial, radial and azimuthal components reach self-similar states, denoted decelerating self-similar profiles, which are different from their steady-state counterparts. Production and destruction terms in the mean SDR transport equation remain dominant in the decelerating self-similar state. The theoretical approach provides an explicit prediction for the radial profile of a turbulent fluctuation term of the mean SDR transport equation. Three turbulent SDR models are validated, and modifications suitable for the decelerating jet are proposed, based on a self-similarity analysis.

An isolated Leidenfrost droplet levitating over its own vapour above a superheated flat substrate is considered theoretically, the superheating for water being up to several hundred degrees above the boiling temperature. The focus is on the limit of small, practically spherical droplets of several tens of micrometres or less. This may occur when the liquid is sprayed over a hot substrate, or just be a late life stage of an initially large Leidenfrost droplet. A rigorous numerically assisted analysis is carried out within verifiable assumptions such as quasi-stationarities and small Reynolds/Péclet numbers. It is considered that the droplet is surrounded by its pure vapour. Simple formulae approximating our numerical data for the forces and evaporation rates are preliminarily obtained, all respecting the asymptotic behaviours (also investigated) in the limits of small and large levitation heights. They are subsequently used within a system of ordinary differential equations to study the droplet dynamics and take-off (drastic height increase as the droplet vapourises). A previously known quasi-stationary inverse-square-root law for the droplet height as a function of its radius (at the root of the take-off) is recovered, although we point out different prefactors in the two limits. Deviations of a dynamic nature therefrom are uncovered as the droplet radius further decreases due to evaporation, improving the agreement with experiment. Furthermore, we reveal that, if initially large enough, the droplets vanish at a universal finite height (just dependent on the superheat and fluid properties). Scalings in various distinguished cases are obtained along the way.

An exact solution is developed for bubble-induced acoustic microstreaming in the case of a gas bubble undergoing asymmetric oscillations. The modelling is based on the decomposition of the solenoidal, first- and second-order, vorticity fields into poloidal and toroidal components. The result is valid for small-amplitude bubble oscillations without restriction on the size of the viscous boundary layer $(2\nu /\omega )^{1/2}$ in comparison to the bubble radius. The non-spherical distortions of the bubble interface are decomposed over the set of orthonormal spherical harmonics $Y_{n}^{m}(\theta , \phi )$ of degree $n$ and order $m$. The present theory describes the steady flow produced by the non-spherical oscillations $(n,\pm m)$ that occur at a frequency different from that of the spherical oscillation, as in the case of a parametrically excited surface oscillation. The three-dimensional aspect of the streaming pattern is revealed as well as the particular flow signatures associated with different asymmetric oscillations.

Oscillatory flows induced by a monochromatic forcing frequency $\omega$ close to a planar surface are present in many applications involving fluid–matter interaction such as ultrasound, vibrational spectra by microscopic pulsating cantilevers, nanoparticle oscillatory magnetometry, quartz crystal microbalance and more. Numerical solution of these flows using standard time-stepping solvers in finite domains present important drawbacks. First, hydrodynamic finite-size effects scale as $1/L_{\parallel }^2$ close to the surface and extend several times the penetration length $\delta \sim \omega ^{-1/2}$ in the normal $z$ direction and second, they demand rather long transient times $O(L_z^2)$ to allow vorticity to diffuse over the computational domain. We present a new frequency-based scheme for doubly periodic (DP) domains in free or confined spaces which uses spectral-accurate solvers based on fast Fourier transform in the periodic $(xy)$ plane and Chebyshev polynomials in the aperiodic $z$ direction. Following the ideas developed for the steady Stokes solver (Hashemi et al. J. Chem. Phys. vol. 158, 2023, p. 154101), the computational system is decomposed into an ‘inner’ domain (where forces are imposed) and an outer domain (where the flow is solved analytically using plane-wave expansions). Matching conditions leads to a solvable boundary value problem. Solving the equations in the frequency domain using complex phasor fields avoids time-stepping and permits a strong reduction in computational time. The spectral scheme is validated against analytical results for mutual and self-mobility tensors, including the in-plane Fourier transform of the Green function. Hydrodynamic couplings are investigated as a function of the periodic lattice length. Applications are finally discussed.