Simulations of elastic turbulence, the chaotic flow of highly elastic and inertialess polymer solutions, are plagued by numerical difficulties: the chaotically advected polymer conformation tensor develops extremely large gradients and can lose its positive-definiteness, which triggers numerical instabilities. While efforts to tackle these issues have produced a plethora of specialized techniques – tensor decompositions, artificial diffusion, and shock-capturing advection schemes – we still lack an unambiguous route to accurate and efficient simulations. In this work, we show that even when a simulation is numerically stable, maintaining positive-definiteness and displaying the expected chaotic fluctuations, it can still suffer from errors significant enough to distort the large-scale dynamics and flow structures. We focus on two-dimensional simulations of the Oldroyd-B and FENE-P equations, driven by a large-scale cellular body forcing. We first compare two positivity-preserving decompositions of the conformation tensor: symmetric square root (SSR) and Cholesky with a logarithmic transformation (Cholesky-log). While both simulations yield chaotic flows, only the latter preserves the pattern of the forcing, i.e. its fluctuating vortical cells remain ordered in a lattice. In contrast, the SSR simulation exhibits distorted vortical cells that shrink, expand and reorient constantly. To identify the accurate simulation, we appeal to a hitherto overlooked mathematical bound on the determinant of the conformation tensor, which unequivocally rejects the SSR simulation. Importantly, the accuracy of the Cholesky-log simulation is shown to arise from the logarithmic transformation. We also consider local artificial diffusion, a potential low-cost alternative to high-order advection schemes. Unfortunately, the artificially enhanced diffusive smearing of polymer stress in regions of intense stretching substantially modifies the global dynamics. We then show how the spurious large-scale motions, identified here, contaminate predictions of scalar mixing. Finally, we discuss the effects of spatial resolution, which controls the steepness of gradients in a non-diffusive simulation.

The seminal Bolgiano–Obukhov (BO) theory established the fundamental framework for turbulent mixing and energy transfer in stably stratified fluids. However, the presence of BO scalings remains debatable despite their being observed in stably stratified atmospheric layers and convective turbulence. In this study, we performed precise temperature measurements with 51 high-resolution loggers above the seafloor for 46 h on the continental shelf of the northern South China Sea. The temperature observation exhibits three layers with increasing distance from the seafloor: the bottom mixed layer (BML), the mixing zone and the internal wave zone. A BO-like scaling $\alpha =-1.34\pm 0.10$ is observed in the temperature spectrum when the BML is in a weakly stable stratified ($N\sim 0.0018$ rad s$^{-1}$) and strongly sheared ($Ri\sim 0.0027$) condition, whereas in the unstably stratified convective turbulence of the BML, the scaling $\alpha =-1.76\pm 0.10$ clearly deviated from the BO theory but approached the classical $-$5/3 scaling in isotropic turbulence. This suggests that the convective turbulence is not the promise of BO scaling. In the mixing zone, where internal waves alternately interact with the BML, the scaling follows the Kolmogorov scaling. In the internal wave zone, the scaling $\alpha =-2.12 \pm 0.15$ is observed in the turbulence range and possible mechanisms are provided.

This work reports an experimental study of the turbulent entrainment into the planar wake of a circular cylinder, exposed to various turbulent backgrounds, from the near- to the far-field. The background turbulence features independently varying turbulence intensity and integral length scale, thereby rendering different turbulent/turbulent interfaces (TTIs) between the background and the primary flow (wake). Combined, simultaneous particle image velocimetry and planar laser induced fluorescence measurements were conducted to quantify the entrainment characteristics across these various TTIs at an inlet Reynolds number of 3800. The primary focus was on understanding how turbulent entrainment evolves spatially in conjunction with the rapid development of the large-scale coherent vortices in the planar wake, and how such evolution is affected by the background turbulence. It is found that TTIs can establish two layers when the background turbulence is sufficiently intense, which distinguishes TTIs from the turbulent/non-turbulent interface (TNTI). The two layers are underpinned by different physical mechanisms but have the same thickness and appear to scale with the local Kolmogorov length scale after the wake spreading transition position (Chen & Buxton, J. Fluid Mech., vol. 969, 2023, A4). It is also found that the probability density functions of the entrainment velocity for both TTIs and a TNTI display power law tails, which are associated with extremely large entrainment velocities occurring more frequently than for a Gaussian process. These intermittent, extreme entrainment velocities make a remarkable contribution to the mean entrainment velocity, particularly in the near wake, which leads to a much higher mean entrainment velocity than farther downstream, for both a TNTI and the TTIs. Conditionally averaged analysis reveals that these extreme events of the entrainment velocity are directly associated with intense enstrophy structures close to the interface.

Understanding the solutal convection is a crucial step towards accurate prediction of CO$_2$ sequestration in deep saline aquifers. In this study, pore-scale resolved direct numerical simulations (DNS) are performed to analyse the scaling laws of the solutal convection in porous media. The porous media studied are composed of uniformly distributed square or circular elements. The Rayleigh numbers in the range $426 \le Ra \le 80\,000$, the Darcy numbers in the range $1.7\times 10^{-8} \le Da \le 8.8\times 10^{-6}$ and the Schmidt numbers in the range $250 \le Sc \le 10^4$ are considered in the DNS. The main time, length and velocity scales affecting the solutal convection are classified as the diffusive scales ($t_I$, $l_I$ and $u_I$), the convective scales ($t_{II}$, $l_{II}$ and $u_{II}$) and the shut-down scales ($t_{III}$, $l_{III}$ and $u_{III}$). These scales determine the pore-scale Rayleigh number $Ra_K$ and the Rayleigh number $Ra$. Based on the DNS results, the scaling laws for the transient dissolution flux are proposed in the different regimes of convection. It is found that the onset time of convection ($t_{oc}$) and the period of the flux-growth regime ($\Delta t_{fg}$) vary linearly with the convective time scale $t_{II}$. The merging of the original plumes and the re-initiation of the new plumes start in the same period, resulting in the merging re-initiation regime. Horizontal dispersion plays an important role in plume merging. The dissolution flux $F$ and the time since the onset of convection $t^{\ast }$ have an $F / u_{II} \sim (t^{\ast }/ t_{II})^{-0.2}$ scaling in the later stage of the merging re-initiation regime. This scaling is caused by the merging of the low-wavenumber plumes. It becomes more pronounced with decreasing porosity and leads to the nonlinear relationship between the Sherwood number $Sh$ and $Ra$ when the domain is not high enough for the plumes to fully develop. According to the DNS results, a regime is expected that is independent of both $Ra$ and $Ra_K$, while the dimensionless constant flux $F_{cf}/u_{II}$ in this regime decreases with decreasing porosity. When the mean solute concentration reaches approximately 30 %, convection enters the shut-down regime. For large $Ra$, the dimensionless flux in the shut-down regime follows the scaling law $F/u_{III}\sim (t/t_{III})^{-1.2}$ at large porosity ($\phi =0.56$) and $F/u_{III}\sim (t/t_{III})^{-1.5}$ at small porosity ($\phi =0.36$ or $0.32$). The study shows the significant pore-scale effect on the convection. The DNS cases in this study are mainly for simplified geometries and large $Ra_K$. This can lead to uncertainties of the obtained scaling coefficients. It is important to determine the scaling coefficients for real geological formations to predict a real CO$_2$ sequestration process.

Developing a model to describe the shock-accelerated cylindrical fluid layer with arbitrary Atwood numbers is essential for uncovering the effect of Atwood numbers on the perturbation growth. The recent model (J. Fluid Mech., vol. 969, 2023, p. A6) reveals several contributions to the instability evolution of a shock-accelerated cylindrical fluid layer but its applicability is limited to cases with an absolute value of Atwood numbers close to $1$, due to the employment of the thin-shell correction and interface coupling effect of the fluid layer in vacuum. By employing the linear stability analysis on a cylindrical fluid layer in which two interfaces separate three arbitrary-density fluids, the present work generalizes the thin-shell correction and interface coupling effect, and thus, extends the recent model to cases with arbitrary Atwood numbers. The accuracy of this extended model in describing the instability evolution of the shock-accelerated fluid layer before reshock is confirmed via direct numerical simulations. In the verification simulations, three fluid-layer configurations are considered, where the outer and intermediate fluids remain fixed and the density of the inner fluid is reduced. Moreover, the mechanisms underlying the effect of the Atwood number at the inner interface on the perturbation growth are mainly elucidated by employing the model to analyse each contribution. As the Atwood number decreases, the dominant contribution of the Richtmyer–Meshkov instability is enhanced due to the stronger waves reverberated inside the layer, leading to weakened perturbation growth at initial in-phase interfaces and enhanced perturbation growth at initial anti-phase interfaces.

The resolvent analysis reveals the worst-case disturbances and the most amplified response in a fluid flow that can develop around a stationary base state. The recent work by Padovan et al. (J. Fluid Mech., vol. 900, 2020, A14) extended the classical resolvent analysis to the harmonic resolvent analysis framework by incorporating the time-varying nature of the base flow. The harmonic resolvent analysis can capture the triadic interactions between perturbations at two different frequencies through a base flow at a particular frequency. The singular values of the harmonic resolvent operator act as a gain between the spatiotemporal forcing and the response provided by the singular vectors. In the current study, we formulate the harmonic resolvent analysis framework for compressible flows based on the linearized Navier–Stokes equation (i.e. operator-based formulation). We validate our approach by applying the technique to the low-Mach-number flow past an airfoil. We further illustrate the application of this method to compressible cavity flows at Mach numbers of 0.6 and 0.8 with a length-to-depth ratio of $2$. For the cavity flow at a Mach number of 0.6, the harmonic resolvent analysis reveals that the nonlinear cross-frequency interactions dominate the amplification of perturbations at frequencies that are harmonics of the leading Rossiter mode in the nonlinear flow. The findings demonstrate a physically consistent representation of an energy transfer from slow-evolving modes toward fast-evolving modes in the flow through cross-frequency interactions. For the cavity flow at a Mach number of 0.8, the analysis also sheds light on the nature of cross-frequency interaction in a cavity flow with two coexisting resonances.