Asymptotics of the $$\phi ^4_1$$ Measure in the Sharp Interface Limit

Side weirs with triangular channel are used as flow controlling devices in draining and irrigation networks. By installing a side weir on the main conduits side walls, the runoff overflows from the weir and are conducted toward the diversion channel. In this study, changing of the flow free surface and the turbulence of the flow field in triangular channels with side weir are numerically simulated using volume of fluid (VOF) scheme and RNG k–ε turbulence model. In the present paper, the pattern of the spatially varied flow with decreasing discharge in both subcritical and supercritical flow regimes for triangular channels with side weirs was simulated. The present numerical model has precisely predicted the changes of the water surface and the specific energy. In subcritical regime, the flow depth is from the beginning of the weir toward its end is followed by an increase and in supercritical conditions is followed by a reduction in depth. For both subcritical and supercritical regimes, a drop in the surface in the first third of the weir’s opening and a surface jump in the final third of its length has occurred. Along the mentioned surface jump the amount of the kinetic energy increases and the potential energy reduces. According to results of the simulation, the maximum longitudinal velocity for subcritical flow regime occurs in the first third of the length of the side weir and for supercritical flow regime, almost in the middle of the weir opening happens. In both subcritical and supercritical regimes, the maximum transverse velocity has occurred in the final third of the length of the side weir.

We consider the problem of nonlinear buoyant flow in a horizontal mushy layer during alloy solidification. We study the nonlinear evolution of such flow based on a recently developed realistic model for the mushy layer. The evolution approach is based on a Landau type equation for the amplitude of the secondary nonlinear solution, which is derived in this article. Using both analytical and computational methods, we calculate the solution to the evolution equation for both subcritical and supercritical regimes. We find, in particular, that for a passive mush, where the permeability is constant, and supercritical regime, the primary solution is linearly unstable to the secondary solution which becomes a steady stable solution for sufficiently large time, while the secondary solution decays to zero for the subcritical regime. On the other hand, for a realistic reactive mush, where the permeability is variable, the secondary flow can break down in a finite time for either supercritical or subcritical regime, which indicates existence of some kind of bursting behavior. These results are then compared to the corresponding ones based on the weakly nonlinear theory.

Three-dimensional dynamics of vortex-induced vibration of a pipe with internal flow in the subcritical and supercritical regimes

The present work investigates the linear instability of three-dimensional boundary layers in thermodynamically non-ideal regimes. As a representative fluid, we consider carbon dioxide at supercritical pressure (80 bar). The flow set-up is matched to the redesigned DLR (German Aerospace Center) experiment on cross-flow instability, with identical pressure-coefficient distribution (accelerating the flow), sweep angle and Reynolds number, at a low Mach number. The flow temperature relative to the Widom line – also known as the pseudocritical line – thus characterises the non-ideality of the flow. We consider supercritical (gas-like), subcritical (liquid-like) and transcritical (pseudoboiling) regimes, where the flow temperature remains above, below or crosses the Widom line. The stability analyses of the parabolised Navier–Stokes baseflows indicate that wall heating destabilises the flow in the supercritical regime while wall cooling stabilises both effects similar to the ideal-fluid situation but being stronger. On the contrary, wall heating/cooling exhibits reversed effects in the subcritical regime, like for an ideal liquid. In the transcritical regime, with its sharp gradients of the thermodynamic and transport properties, wall heating stabilises the flow. Most substantially, however, wall cooling provokes a changeover of the leading instability mechanism: the accelerated streamwise flow attains inflectional wall-normal profiles, and the invoked inviscid Tollmien–Schlichting instability prevails with growth rates up to one order of magnitude larger than those of the cross-flow mode. We establish a two-fold mathematical relation from the momentum equation that explains the consequence of non-ideality and wall heating/cooling. The streamwise perturbation patterns of the flows in their linear instability regime are shown by mimicking wave trains emanating from virtual point-disturbance sources. From the viewpoint of keeping laminar flows, the transcritical thermodynamic state with a cooling wall must be avoided.

The objective of this work is to investigate linear modal and algebraic instability in Poiseuille flows with fluids close to their vapour–liquid critical point. Close to this critical point, the ideal gas assumption does not hold and large non-ideal fluid behaviours occur. As a representative non-ideal fluid, we consider supercritical carbon dioxide ($\text{CO}_{2}$) at a pressure of 80 bar, which is above its critical pressure of 73.9 bar. The Poiseuille flow is characterized by the Reynolds number ($Re=\unicode[STIX]{x1D70C}_{w}^{\ast }u_{r}^{\ast }h^{\ast }/\unicode[STIX]{x1D707}_{w}^{\ast }$), the product of the Prandtl ($Pr=\unicode[STIX]{x1D707}_{w}^{\ast }C_{pw}^{\ast }/\unicode[STIX]{x1D705}_{w}^{\ast }$) and Eckert numbers ($Ec=u_{r}^{\ast 2}/C_{pw}^{\ast }T_{w}^{\ast }$) and the wall temperature that in addition to pressure determine the thermodynamic reference condition. For low Eckert numbers, the flow is essentially isothermal and no difference with the well-known stability behaviour of incompressible flows is observed. However, if the Eckert number increases, the viscous heating causes gradients of thermodynamic and transport properties, and non-ideal gas effects become significant. Three regimes of the laminar base flow can be considered: the subcritical (temperature in the channel is entirely below its pseudo-critical value), transcritical and supercritical temperature regimes. If compared to the linear stability of an ideal gas Poiseuille flow, we show that the base flow is modally more unstable in the subcritical regime, inviscid unstable in the transcritical regime and significantly more stable in the supercritical regime. Following the principle of corresponding states, we expect that qualitatively similar results will be obtained for other fluids at equivalent thermodynamic states.

A data-driven approach to estimate the global spectrum of gravitational planar liquid jets (sheet or curtain flows) is presented in this work. The investigation is carried out by means of two-dimensional numerical simulations performed through the solver BASILISK, based on the one-fluid formulation and the volume-of-fluid approach. The dynamic mode decomposition technique is applied to extract the underlying linear operator, considering random perturbations of the base flow. The effectiveness of this procedure is first evaluated comparing results with those of a simplified one-dimensional curtain model in terms of spectrum and eigenfunctions. The methodology is then applied to a two-dimensional configuration obtaining the BiGlobal spectra for both supercritical (Weber number We > 1) and subcritical (We < 1) regimes. Results highlight that in supercritical regime, the spectrum presents three branches: the upper and lower ones exhibit a purely sinuous behavior with frequencies quite close to those predicted by the one-dimensional model; the middle branch presents a predominant varicose component, increasing with the frequency. The subcritical spectrum, instead, shows that the first two less stable eigenvalues, sorted by increasing frequency, exhibit, respectively, a sinuous and a varicose behavior, while their growth rate is almost the same. As expected, the subcritical regime does not reveal the slow branch. The effect of the density ratio, rρ, between the two phases is investigated, revealing that the flow system is unstable for rρ>0.05. Topological inspections of the leading modes in this unstable configuration show that the predominance of a varicose behavior is related to the rupture of the curtain.

We present an analytical solution and numerical tests of the epidemic‐type aftershock (ETAS) model for aftershocks, which describes foreshocks, aftershocks, and main shocks on the same footing. In this model, each earthquake of magnitude m triggers aftershocks with a rate proportional to 10αm. The occurrence rate of direct aftershocks triggered by a single main shock decreases with the time from the main shock according to the “local” modified Omori law K/(t + c)p with p = 1 + θ. Contrary to the usual definition, the ETAS model does not impose an aftershock to have a magnitude smaller than the main shock. Starting with a main shock at time t = 0 that triggers aftershocks according to the local Omori law, which in turn trigger their own aftershocks and so on, we study the seismicity rate of the global aftershock sequence composed of all the secondary and subsequent aftershock sequences. The effective branching parameter n, defined as the mean aftershock number triggered per event, controls the transition between a subcritical regime n < 1 and a supercritical regime n > 1. A characteristic time t*, function of all the ETAS parameters, marks the transition from the early time behavior to the large time behavior. In the subcritical regime, we recover and document the crossover from an Omori exponent 1 − θ for t < t* to 1 + θ for t > t* found previously in the work of Sornette and Sornette for a special case of the ETAS model. In the supercritical regime n > 1 and θ > 0, we find a novel transition from an Omori decay law with exponent 1 − θ for t < t* to an explosive exponential increase of the seismicity rate for t > t*. The case θ < 0 yields an infinite n‐value. In this case, we find another characteristic time τ controlling the crossover from an Omori law with exponent 1 − |θ| for t < τ, similar to the local law, to an exponential increase at large times. These results can rationalize many of the stylized facts reported for aftershock and foreshock sequences, such as (1) the suggestion that a small p‐value may be a precursor of a large earthquake, (2) the relative seismic quiescence sometimes observed before large aftershocks, (3) the positive correlation between b and p values, (4) the observation that great earthquakes are sometimes preceded by a decrease of b‐value, and (5) the acceleration of the seismicity preceding great earthquakes.

Experimental studies of the unsteady hydrodynamic loads on a tension-leg platform at high Reynolds numbers

We made a detailed study of the timing and spectral properties of the X-ray pulsar 1A 0535+262 during the recent giant outburst in 2020 November and December. The flux of the pulsar reached a record value of ∼12.5 Crab as observed by Swift/BAT (15–50 keV) and the corresponding mass accretion rate was ∼6.67 × 1017 g s−1 near the peak of the outburst. There was a transition from the subcritical to the supercritical accretion regime which allows exploring different properties of the source in the supercritical regime. A q-like feature was detected in the hardness–intensity diagram during the outburst. We observed high variability and strong energy dependence of pulse profiles during the outburst. Cyclotron Resonant Scattering Feature (CRSF) was detected at ∼44 keV from the NuSTAR energy spectrum in the subcritical regime and the corresponding magnetic field was B ≃ 4.9 × 1012 G. The energy of the CRSF was shifted towards lower energy in the supercritical regime. The luminosity dependence of the CRSF was studied and during the supercritical regime, a negative correlation was observed between the line energy and luminosity. The critical luminosity was ∼6 × 1037erg s−1 above which a state transition occurred. A reversal of correlation between the photon index and luminosity was observed near the critical luminosity. The NuSTAR spectra can be described by a composite model with two continuum components, a blackbody emission, cut-off power law, and a discrete component to account for the iron emission line at 6.4 keV. An additional cyclotron absorption feature was included in the model.

The aim of this study is to numerically investigate the effect of bed slope on the hydraulic performance of side weirs in supercritical and subcritical flow regimes. Increasing the bed slope decreases the weir efficiency and discharge coefficient in the subcritical flow regime up to 14.54% and in the supercritical flow regime up to 9.26%. By examining the longitudinal velocity distribution, it was observed that in the subcritical flow regime, the maximum longitudinal velocity occurs at the beginning of the upstream of the side weir, and by moving towards the downstream of the weir, the flow velocity decreases and increases again after leaving the length of the weir. In the supercritical flow regime, it has an increasing trend when the flow enters from the upstream of the side weir. The increase in discharge coefficient with the increase of the weir height varied between 4.6% and 7.8%. With increase of the slope of the channel bed, the velocity along the length of the weir increases by 10.88% and 6.17%, respectively, in the subcritical and supercritical regimes, and the transverse velocity along the transverse direction for the subcritical and supercritical flow regimes decreased by 22.23% and 4.80%, respectively.

Notwithstanding the increasing industrial interest toward unbaffled tanks, available experimental information on their behavior is still scant, even for basic quantities such as the mechanical power drawn. In this work, the influence of the Reynolds and Froude numbers on the power consumption characteristics is presented for unbaffled stirred tanks operating both in nonaerated conditions (subcritical regime) and in aerated conditions (supercritical regime), i.e., when the free surface vortex has reached the impeller and the gas phase is ingested and dispersed inside the reactor. Experimental results obtained at various liquid viscosities show that power numbers obtained in subcritical conditions do line up quite well on a smooth Np versus Re function, with no need to involve the Froude number in the correlation. At rotational speeds involving air entrapment and dispersion inside the reactor (supercritical regime), a steep reduction of the power number is observed. A novel overall correlation for power number prediction, able to deal with both the subcritical and supercritical regimes, is finally proposed.

Large eddy simulation of flow past a circular cylinder of low aspect ratio ( $AR=1$ and $3$ ), spanning subcritical, critical and supercritical regimes, is carried out for $2\times 10^3 \le Re \le 4\times 10^5$ . The end walls restrict three-dimensionality of the flow. The critical $Re$ for the onset of the critical regime is significantly lower for small aspect ratio cylinders. The evolution of secondary vortex (SV), laminar separation bubble (LSB) and the related transition of boundary layer with $Re$ is investigated. The plateau in the surface pressure due to LSB is modified by the presence of SV. Proper orthogonal decomposition of surface pressure reveals that although the vortex shedding mode is most dominant throughout the $Re$ regime studied, significant energy of the flow lies in a symmetric mode that corresponds to expansion–contraction of the vortex formation region and is responsible for bursts of weak vortex shedding. A triple decomposition of the time signals comprising of contributions from shear layer vortices, von Kármán vortex shedding and low frequency modulation due to the symmetric mode of flow is proposed. A moving average, with appropriate size of window, is utilized to estimate the component due to vortex shedding. It is used to assess the variation, with $Re$ , of strength of vortex shedding as well as its coherence along the span. Weakening of vortex shedding in the high subcritical and critical regime is followed by its rejuvenation in the supercritical regime. Its spanwise correlation is high in the subcritical regime, decreases in the critical regime and improves again in the supercritical regime.

This paper explains how to correctly measure the drag coefficient of a circular cylinder in wind tunnels with large blockage ratios and for the sub-critical to the super-critical flow regimes. When dealing with large blockage ratios, the drag has to be corrected for wall constraints. Different formulations for correcting blockage effect are compared for each flow regime based on drag measurements of smooth circular cylinders performed in a wind tunnel for three different blockage ratios. None of the correction model known in the literature is valid for all the flow regimes. To optimize the correction and reduce the scatter of the results, different correction models should be combined depending on the flow regime. In the sub-critical regime, the best results are obtained using Allen and Vincenti's formula or Maskell's theory with <TEX>${\varepsilon}$</TEX>=0.96. In the super-critical regime, one should prefer using Glauert's formula with G=0.6 or the model of Modi and El-Sherbiny. The change in the formulations appears at the flow transition with a variation of the wake pattern when passing from sub-critical to super-critical flow regimes. This parameter being not considered in the known blockage corrections, these theories are not valid for all the flow regimes.

The unsteady dynamics of a gravitational liquid sheet, driven by a continuous harmonic perturbation in the lateral velocity component applied at the inlet section, is analyzed. The topology and the dynamics of the relevant flow structures are characterized by applying POD (Proper Orthogonal Decomposition) and spectral POD (SPOD) modal decompositions on two-dimensional two-phase numerical simulation data obtained with the volume-of-fluid approach. The investigation is carried out by varying the Weber number, the forcing frequency (Strouhal number), and the Reynolds number. The supercritical regime (We &amp;gt; 1) features a traveling perturbation, exhibiting a spatial structure with leading sinuous modes. SPOD spectra confirm the occurrence of a discontinuity in frequency response between the supercritical and subcritical regimes. In the subcritical regime (We &amp;lt; 1), the investigation highlights the excitation of a combined sinuous–varicose motion when the system is driven at resonance frequency for a relatively high Reynolds number (approaching the inviscid limit). The emergence of varicose modes is favored by low Weber numbers. The excitation of these modes occurs when the Weber number is decreased from We = 0.90 down to 0.75, with a progressive shift of the varicose mode from higher harmonics toward the main frequency; it can be considered as a possible mechanism of breakup observed in experiments when the inlet flow rate is progressively reduced. The flow reconstruction based on both POD and SPOD confirms the good capability of SPOD modes to capture dynamically relevant features of the fluid motion in subcritical conditions.

The flow field around a sphere in an uniform flow has been analyzed numerically for conditions corresponding to the subcritical (laminar separation) and supercritical (turbulent separation) regimes spanning a wide range of Reynolds numbers (104–106). Particular attention has been devoted to assessing predictions of the pressure distribution, skin friction, and drag as well as to understanding the changes in the wake organization and vortex dynamics with the Reynolds number. The unsteady turbulent flow is computed using detached-eddy simulation, a hybrid approach that has Reynolds-averaged Navier–Stokes behavior near the wall and becomes a large eddy simulation in the regions away from solid surfaces. For both the subcritical and supercritical solutions, the agreement with experimental measurements for the mean drag and pressure distribution over the sphere is adequate; differences in skin friction exist due to the simplistic treatment of the attached boundary layers in the computations. Improved agreement in the skin-friction distribution is obtained for the supercritical flows in which boundary layer transition is fixed at the position observed in experiments conducted at the same Reynolds numbers. For the subcritical flows the Strouhal number, St, associated with the large-scale shedding is predicted at St∼0.195 along with a higher frequency component associated with the development of the Kelvin–Helmholtz instabilities in the detached shear layers. If in the subcritical regime the wake assumes a helical-like form due to the shedding of hairpin-like vortices at different azimuthal angles, in the supercritical regime the wake structure is characterized by “regular” shedding of hairpin-like vortices at approximately the same azimuthal angle and at a much higher frequency (St∼1.3) that is practically independent of the Reynolds number and not sensitive to the position of laminar-to-turbulent transition.