Physics

We present a steady analytical solution of the incompressible Navier-Stokes equation for arbitrary viscosity in an arbitrary dimension d of space. It represents a d?? dimensional vortex "sheet" with an asymmetric profile of vorticity as a function of the normal coordinate z . This profile is related to the Hermite polynomials H μ (z) which are analytically continued to the negative fractional index μ=??d d?? . In d=2 dimensions, the solution degenerates to a constant vorticity flow. In d?? dimensions, the vorticity is confined to the thin layer around the hyperplane with Gaussian decay on one side of the hyperplane and the power decay on another side. One can adjust the common scale of velocity so that the dissipation will stay finite at vanishing viscosity. In this limit, the width w of the viscous lawyer will shrink to zero as ν 3 5 for arbitrary dimension d>3 . In d=3 dimensions, this power law is also accompanied by powers of the logarithm.

Scaling of turbulent wall-bounded flows is revealed in the gradient structures, for each of the Reynolds stress components. Within the dissipation structure, an asymmetrical order exists, that we can deploy to unify the scaling and transport dynamics within and across these flows. There are subtle differences in the outer boundary conditions between channel and flat-plate boundary-layer flows, which modify the turbulence structure far from the wall. The self-similarity exhibited in the dissipation structures and corresponding transport dynamics establish capabilities and encompassing knowledge of wall-bounded turbulent flows.

The friction f is the property of wall-bounded flows that sets the pumping cost of a pipeline, the draining capacity of a river, and other variables of practical relevance. For highly turbulent rough-walled pipe flows, f depends solely on the roughness length scale r, and the f -- r relation may be expressed by the Strickler empirical scaling f ??r 1 3 . Here, we show experimentally that for soap film flows that are the two-dimensional (2D) equivalent of highly turbulent rough-walled pipe flows, f ??r and the f -- r relation is not the same in 2D as in 3D. Our findings are beyond the purview of the standard theory of friction but consistent with a competing theory in which f is linked to the turbulent spectrum via the spectral exponent α : In 3D, α = 5/3 and the theory yields f ??r 1 3 ; in 2D, α = 3 and the theory yields f ??r.

Modelling river physical processes is of critical importance for flood protection, river management and restoration of riverine environments. Developments in algorithms and computational power have led to a wider spread of river simulation tools. However, the use of two-dimensional models can still be hindered by complexity in the setup and the high computational costs. Here we present the freeware BASEMENT version 3, a flexible tool for two-dimensional river simulations that bundles solvers for hydrodynamic, morphodynamic and scalar advection-diffusion processes. BASEMENT leverages different computational platforms (multi-core CPUs and graphics processing units GPUs) to enable the simulation of large domains and long-term river processes. The adoption of a fully costless worflow and a light GUI facilitate its broad utilization. We test its robustness and efficiency in a selection of benchmarks. Results confirm that BASEMENT could be an efficient and versatile tool for research, engineering practice and education in river modelling.

We perform a bifurcation analysis of the steady state solutions of Rayleigh--Bénard convection with no-slip boundary conditions in two dimensions using a numerical method called deflated continuation. By combining this method with an initialisation strategy based on the eigenmodes of the conducting state, we are able to discover multiple solutions to this non-linear problem, including disconnected branches of the bifurcation diagram, without the need of any prior knowledge of the dynamics. One of the disconnected branches we find contains a s-shape bifurcation with hysteresis, which is the origin of the flow pattern that may be related to the dynamics of flow reversals in the turbulent regime. Linear stability analysis is also performed to analyse the steady and unsteady regimes of the solutions in the parameter space and to characterise the type of instabilities.

Although street artists have the know-how to blow bubbles over one meter in length, the bubble width is typically determined by the size of the hoop, or wand they use. In this article we explore a regime in which, by blowing gently, we generate bubbles with radius up to ten times larger than the wand. We observe the big bubbles at lowest air speeds, analogous to the dripping mode observed in droplet formation. We also explore the impact of the surfactant chosen to stabilize the bubbles. We are able to create bubbles of comparable size using either Fairy liquid, a commercially available detergent often used by street artists, or sodium dodecyl sulfate (SDS) solutions. The bubbles obtained from Fairy liquid detach from the wand and are stable for several seconds, however those from SDS tend to burst just before detachment.

Interesting analogies between shallow water dynamics and astrophysical phenomena have offered valuable insight from both the theoretical and experimental point of view. To help organize these efforts, here we analyze systematically the hydrodynamic properties of backwater profiles of the shallow water equations with 2D radial symmetry. In contrast to the more familiar 1D case typical of hydraulics, %already for {even in} isentropic conditions, a solution with minimum-radius horizon for the flow emerges, similar to the black hole and white hole horizons, where the critical conditions of unitary Froude number provide a unidirectional barrier for surface waves. Beyond these time-reversible solutions, a greater variety of cases arises, when allowing for dissipation by turbulent friction and shock waves (i.e., hydraulic jumps) for both convergent and divergent flows. The resulting taxonomy of the base-flow cases may serve as a starting point for a more systematic analysis of higher-order effects linked, e.g., to wave propagation and instabilities, capillarity, variable bed slope, and rotation.

Recently, in Zhang et al. (2020), it was found that in rapidly rotating turbulent Rayleigh-Bénard convection (RBC) in slender cylindrical containers (with diameter-to-height aspect ratio Γ=1/2 ) filled with a small-Prandtl-number fluid ( Pr≈0.8 ), the Large Scale Circulation (LSC) is suppressed and a Boundary Zonal Flow (BZF) develops near the sidewall, characterized by a bimodal PDF of the temperature, cyclonic fluid motion, and anticyclonic drift of the flow pattern (with respect to the rotating frame). This BZF carries a disproportionate amount ( >60% ) of the total heat transport for Pr<1 but decreases rather abruptly for larger Pr to about 35% . In this work, we show that the BZF is robust and appears in rapidly rotating turbulent RBC in containers of different Γ and in a broad range of Pr and Ra . Direct numerical simulations for 0.1≤Pr≤12.3 , 10 7 ≤Ra≤5× 10 9 , 10 5 ≤1/Ek≤ 10 7 and Γ = 1/3, 1/2, 3/4, 1 and 2 show that the BZF width δ 0 scales with the Rayleigh number Ra and Ekman number Ek as δ 0 /H∼ Γ 0 Pr {−1/4,0} R a 1/4 E k 2/3 ( Pr<1,Pr>1 ) and the drift frequency as ω/Ω∼ Γ 0 P r −4/3 RaE k 5/3 , where H is the cell height and Ω the angular rotation rate. The mode number of the BZF is 1 for Γ≲1 and 2Γ for Γ = {1,2} independent of Ra and Pr . The BZF is quite reminiscent of wall mode states in rotating convection.

The mean vertical heat transport ?�wT??in convection between isothermal plates driven by uniform internal heating is investigated by means of rigorous bounds. These are obtained as a function of the Rayleigh number R by constructing feasible solutions to a convex variational problem, derived using a formulation of the classical background method in terms of quadratic auxiliary functions. When the fluid's temperature relative to the boundaries is allowed to be positive or negative, numerical solution of the variational problem shows that best previous bound ?�wT?�≤1/2 can only be improved up to finite R . Indeed, we demonstrate analytically that ?�wT?�≤ 2 ??1/5 R 1/5 and therefore prove that ?�wT??1/2 for R<65536 . However, if the minimum principle for temperature is invoked, which asserts that internal temperature is at least as large as the temperature of the isothermal boundaries, then numerically optimised bounds are strictly smaller than 1/2 until at least R=3.4? 10 5 . While the computational results suggest that the best bound on ?�wT??approaches 1/2 asymptotically from below as R?��? , we prove that typical analytical constructions cannot be used to prove this conjecture.

Directing a jet of humid air to impinge on a surface that is cooled below the dew point results in micro-sized water droplets. Lord Rayleigh discussed the phenomenon called such behavior Breath Figures (BF). Historically, utilizing dew as a water source was investigated by several scientists dating back to Aristotle. However, due to the degrading effects of air as a non-condensable gas (NCG) such efforts are limited to small scale water production systems. Recently, the concept of BF has been utilized extensively in the generation of micro-scale polymer patterns as a self-assembly process. However, the generation of BF on surfaces while being impinged by a humid air jet has not been quantified. In this work, we illustrate that a BF spot generated on a cooled surface is a manifestation of a recovery concentration. The concept is analogous to the concept of adiabatic-wall temperature defined for heat transfer applications. Upon closer examination of the vapor concentration distribution on a cooled impinged surface, we found that the distribution exhibits distinct regimes depending on the radial location from the center of the impingement region. The first regime is confined within the impingement region, whereas the second regime lies beyond this radial location including the wall jet region. Scaling analysis as well as numerical solution of the former regime shows that the maximum concentration on the surface is equivalent to its counterpart of a free unbounded jet with similar geometrical conditions. Additionally, the scaling analysis of the latter regime reveals that the jet speed and standoff distance are not important in determining the recovery concentration. However, the recovery concentration is found to vary monotonically with the radial location. Our conclusions are of great importance in optimizing jet impingement where condensation phase change is prevalent.