Recent zbMATH articles in MSC 76Ehttps://zbmath.org/atom/cc/76E2022-09-13T20:28:31.338867ZWerkzeugSecondary flows from a linear array of vortices perturbed principally by a Fourier modehttps://zbmath.org/1491.353122022-09-13T20:28:31.338867Z"Chen, Zhi-Min"https://zbmath.org/authors/?q=ai:chen.zhiminSummary: In the understanding of primary bifurcating flows of a linear array of electromagnetically forced vortices in an experimental fluid motion, a theoretical study on the nonlinear instability is presented. The existence of the bifurcating flows is obtained from a Fourier mode perturbation. This large-scale perturbation, leading to the primary bifurcation observed in a laboratory experiment, was found to be generated principally from a single vortex mode.Rayleigh-Taylor instability for viscous incompressible capillary fluidshttps://zbmath.org/1491.353262022-09-13T20:28:31.338867Z"Zhang, Zhipeng"https://zbmath.org/authors/?q=ai:zhang.zhipengSummary: We investigate the linear and nonlinear instability of a smooth Rayleigh-Taylor steady state solution to the three-dimensional incompressible Navier-Stokes-Korteweg equations in the presence of a uniform gravitational field. We first analyze the linearized equations around the steady state solution and find that for any capillary coefficient \(\kappa >0\), we can construct the solutions of the linearized problem that grow in time in Sobolev space \(H^m\), thus leading to the linear instability. However, with the help of the constructed unstable solutions of the linearized problem, we just establish the nonlinear instability for small enough capillary coefficient \(\kappa >0\).Nonlinear stability of planar steady Euler flows associated with semistable solutions of elliptic problemshttps://zbmath.org/1491.353282022-09-13T20:28:31.338867Z"Wang, Guodong"https://zbmath.org/authors/?q=ai:wang.guodongSummary: This paper is devoted to the study of nonlinear stability of steady incompressible Euler flows in two dimensions. We prove that a steady Euler flow is nonlinearly stable in \(L^p\) norm of the vorticity if its stream function is a semistable solution of some semilinear elliptic problem with nondecreasing nonlinearity. The idea of the proof is to show that such a flow has strict local maximum energy among flows whose vorticities are rearrangements of a given function, with the help of an improved version of Wolansky and Ghil's stability theorem. The result can be regarded as an extension of Arnol'd's second stability theorem.On the effect of fast rotation and vertical viscosity on the lifespan of the \(3D\) Primitive equationshttps://zbmath.org/1491.353422022-09-13T20:28:31.338867Z"Lin, Quyuan"https://zbmath.org/authors/?q=ai:lin.quyuan"Liu, Xin"https://zbmath.org/authors/?q=ai:liu.xin.5|liu.xin.3|liu.xin.2|liu.xin|liu.xin.4|liu.xin.1"Titi, Edriss S."https://zbmath.org/authors/?q=ai:titi.edriss-salehSummary: We study the effect of the fast rotation and vertical viscosity on the lifespan of solutions to the three-dimensional primitive equations (also known as the hydrostatic Navier-Stokes equations) with impermeable and stress-free boundary conditions. Firstly, for a short time interval, independent of the rate of rotation \(|\Omega|\), we establish the local well-posedness of solutions with initial data that is analytic in the horizontal variables and only \(L^2\) in the vertical variable. Moreover, it is shown that the solutions immediately become analytic in all the variables with increasing-in-time (at least linearly) radius of analyticity in the vertical variable for as long as the solutions exist. On the other hand, the radius of analyticity in the horizontal variables might decrease with time, but as long as it remains positive the solution exists. Secondly, with fast rotation, i.e., large \(|\Omega|\), we show that the existence time of the solution can be prolonged, with ``well-prepared'' initial data. Finally, in the case of two spatial dimensions with \(\Omega =0\), we establish the global well-posedness provided that the initial data is small enough. The smallness condition on the initial data depends on the vertical viscosity and the initial radius of analyticity in the horizontal variables.Cattaneo-LTNE effects on the stability of Brinkman convection in an anisotropic porous layerhttps://zbmath.org/1491.760262022-09-13T20:28:31.338867Z"Hema, M."https://zbmath.org/authors/?q=ai:hema.m"Shivakumara, I. S."https://zbmath.org/authors/?q=ai:shivakumara.i-s"Ravisha, M."https://zbmath.org/authors/?q=ai:ravisha.mSummary: The stability of Brinkman local thermal nonequilibrium anisotropic porous convection under the impact of Cattaneo law of heat conduction in solid is investigated. In the analysis, anisotropies in permeability and thermal (solid and fluid phases) conductivities are highlighted. Condition for stationary onset and oscillatory onset is obtained by carrying out linear instability analysis. A novel result is that the instability occurs through oscillatory mode against the stationary convection perceived in the absence of Cattaneo effect. The relative magnitudes of governing parameters on the initiation of oscillatory instability are delineated in detail. The thermal and mechanical anisotropies inflict stabilizing and destabilizing effects on the onset, respectively. The influence of mechanical anisotropy, thermal anisotropy of fluid, thermal relaxation time parameters and the Darcy number is to broaden the size of convection cells whereas thermal anisotropy of the solid and the Darcy-Prandtl number demonstrate a mixed behaviour. A first order amplitude equation is derived separately for steady and overstable modes by performing a weak nonlinear stability analysis using a modified multiscale method. Depending on the values of governing parameters, it is seen that the stationary mode bifurcates subcritically and supercritically, while the oscillatory mode always bifurcates supercritically.Finite rotating and translating vortex sheetshttps://zbmath.org/1491.760272022-09-13T20:28:31.338867Z"Protas, Bartosz"https://zbmath.org/authors/?q=ai:protas.bartosz"Llewellyn Smith, Stefan G."https://zbmath.org/authors/?q=ai:llewellyn-smith.stefan-g"Sakajo, Takashi"https://zbmath.org/authors/?q=ai:sakajo.takashiSummary: We consider the rotating and translating equilibria of open finite vortex sheets with endpoints in two-dimensional potential flows. New results are obtained concerning the stability of these equilibrium configurations which complement analogous results known for unbounded, periodic and circular vortex sheets. First, we show that the rotating and translating equilibria of finite vortex sheets are linearly unstable. However, while in the first case unstable perturbations grow exponentially fast in time, the growth of such perturbations in the second case is algebraic. In both cases the growth rates are increasing functions of the wavenumbers of the perturbations. Remarkably, these stability results are obtained entirely with analytical computations. Second, we obtain and analyse equations describing the time evolution of a straight vortex sheet in linear external fields. Third, it is demonstrated that the results concerning the linear stability analysis of the rotating sheet are consistent with the infinite aspect ratio limit of the stability results known for Kirchhoff's ellipse [\textit{A. E. H. Love}, Proc. Lond. Math. Soc. 25, 18--42 (1894; JFM 25.1467.02); \textit{T. B. Mitchell} and \textit{L. F. Rossi}, Phys. Fluids 20, No. 5, Paper No. 054103, 12 p. (2008; Zbl 1182.76523)] and that the solutions we obtained accounting for the presence of external fields are also consistent with the infinite aspect ratio limits of the analogous solutions known for vortex patches.Advection versus diffusion in Richtmyer-Meshkov mixinghttps://zbmath.org/1491.760282022-09-13T20:28:31.338867Z"Doss, Forrest W."https://zbmath.org/authors/?q=ai:doss.forrest-wSummary: The Richtmyer-Meshkov (RM) instability is one of the most severe degradation mechanisms for inertial confinement fusion (ICF), and mitigating it has been a priority for the global ICF effort. In this Letter, the instability's ability to atomically mix is linked to its background decay of residual turbulent energy. We show how recently derived inequalities from the mathematical theory of PDEs constrain the evolution. A model RM process at leading order may diffusively mix or retain imprints of its initial structures indefinitely, depending on initial conditions, and there exists a theoretical range of zero-mixing for certain values of parameters. The results may apply to other systems resembling scalar transport in decaying turbulence.Effects of head loss, surface tension, viscosity and density ratio on the Kelvin-Helmholtz instability in different types of pipelineshttps://zbmath.org/1491.760292022-09-13T20:28:31.338867Z"Yang, X. C."https://zbmath.org/authors/?q=ai:yang.xuechun|yang.xu-chen|yang.xia-chun|yang.xuecheng|yang.xuchi|yang.xiaochun|yang.xuechao|yang.xiaochen|yang.xiaochuan|yang.xiaochao|yang.xuechang|yang.xinchao|yang.xiuchun|yang.xiaocheng"Cao, Y. G."https://zbmath.org/authors/?q=ai:cao.yanguang|cao.yonggang|cao.yigangSummary: We report the effects of head loss, surface tension, viscosity and density ratio on the Kelvin-Helmholtz instability (KHI) in two typical pipelines, i.e., straight pipeline with different cross-sections and bend pipeline. The dynamic governing equations for upper and lower fluids in the two pipes are solved analytically. We find in the straight pipeline with different cross-sections that the relative tangential velocity of fluid decreases with the increase of the head loss, viscosity and density ratio of upper and lower fluids, but it increases with the surface tension; the amplification factor decreases with the increase of the head loss and surface tension but increases with the density ratio of upper and lower fluids; the higher the height of fluid interface is, the more both the relative tangential velocity of fluid and the amplification factor are depressed. In the bend pipeline, the critical tangential velocity of fluid is found to decrease with the increase of the head loss, viscosity and density ratio of upper and lower fluids, but it increases with the surface tension; the amplification factor increases with the head loss and density ratio of upper and lower fluids, but it decreases with the increase of the surface tension; when the elbow angle is close to \(80^\circ\), the head loss reaches its maximum. The results provide guidance for pipeline design and theoretical prediction for flooding velocity in different types of tubes.Rayleigh-Taylor and Richtmyer-Meshkov instabilities: a journey through scaleshttps://zbmath.org/1491.760302022-09-13T20:28:31.338867Z"Zhou, Ye"https://zbmath.org/authors/?q=ai:zhou.ye"Williams, Robin J. R."https://zbmath.org/authors/?q=ai:williams.robin-j-r"Ramaprabhu, Praveen"https://zbmath.org/authors/?q=ai:ramaprabhu.praveen"Groom, Michael"https://zbmath.org/authors/?q=ai:groom.michael"Thornber, Ben"https://zbmath.org/authors/?q=ai:thornber.ben"Hillier, Andrew"https://zbmath.org/authors/?q=ai:hillier.andrew"Mostert, Wouter"https://zbmath.org/authors/?q=ai:mostert.wouter"Rollin, Bertrand"https://zbmath.org/authors/?q=ai:rollin.bertrand"Balachandar, S."https://zbmath.org/authors/?q=ai:balachandar.s-raja"Powell, Phillip D."https://zbmath.org/authors/?q=ai:powell.phillip-d"Mahalov, Alex"https://zbmath.org/authors/?q=ai:mahalov.alex"Attal, N."https://zbmath.org/authors/?q=ai:attal.nSummary: Hydrodynamic instabilities such as Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) instabilities usually appear in conjunction with the Kelvin-Helmholtz (KH) instability and are found in many natural phenomena and engineering applications. They frequently result in turbulent mixing, which has a major impact on the overall flow development and other effective material properties. This can either be a desired outcome, an unwelcome side effect, or just an unavoidable consequence, but must in all cases be characterized in any model. The RT instability occurs at an interface between different fluids, when the light fluid is accelerated into the heavy. The RM instability may be considered a special case of the RT instability, when the acceleration provided is impulsive in nature such as that resulting from a shock wave. In this pedagogical review, we provide an extensive survey of the applications and examples where such instabilities play a central role. First, fundamental aspects of the instabilities are reviewed including the underlying flow physics at different stages of development, followed by an overview of analytical models describing the linear, nonlinear and fully turbulent stages. RT and RM instabilities pose special challenges to numerical modeling, due to the requirement that the sharp interface separating the fluids be captured with fidelity. These challenges are discussed at length here, followed by a summary of the significant progress in recent years in addressing them. Examples of the pivotal roles played by the instabilities in applications are given in the context of solar prominences, ionospheric flows in space, supernovae, inertial fusion and pulsed-power experiments, pulsed detonation engines and Scramjets. Progress in our understanding of special cases of RT/RM instabilities is reviewed, including the effects of material strength, chemical reactions, magnetic fields, as well as the roles the instabilities play in ejecta formation and transport, and explosively expanding flows. The article is addressed to a broad audience, but with particular attention to graduate students and researchers who are interested in the state-of-the-art in our understanding of the instabilities and the unique issues they present in the applications in which they are prominent.Effect of water vorticity on wind-generated gravity waves in finite depthhttps://zbmath.org/1491.760312022-09-13T20:28:31.338867Z"Abid, Malek"https://zbmath.org/authors/?q=ai:abid.malek"Kharif, Christian"https://zbmath.org/authors/?q=ai:kharif.christianSummary: The generation of wind waves at the surface of an established underlying vertically sheared water flow, of constant vorticity, is considered. A particular attention is paid to the role of the vorticity in water on wind-wave generation in finite depth. The present theoretical results are compared with experimental data obtained by \textit{I. R. Young} and \textit{L. A. Verhagen} [``The growth of fetch limited waves in water of finite depth. I: Total energy and peak frequency'', Coastal Eng. 29, No. 1--2, 47--78 (1996; \url{doi:10.1016/S0378-3839(96)00006-3})], in the shallow Lake George (Australia), and the least squares fit of these data by \textit{I. R. Young} [``The growth rate of finite depth wind-generated waves'', ibid. 32, No. 2--3, 181--195 (1997; \url{doi:10.1016/S0378-3839(97)81749-8})]. It is shown that without vorticity in water, there is a deviation between theory and experimental data. However, a good agreement between the theory and the fit of experimental data is obtained when negative vorticity is taken into account. Furthermore, it is shown that the amplitude growth rate increases with vorticity and depth. A limit to the wave energy growth, corresponding to the vanishing of the growth rate, is obtained. The corresponding limiting wave age is derived in a closed form showing its explicit dependence on vorticity and depth. The limiting wave age is found to increase with both vorticity and depth.Transverse bifurcation of viscous slow MHD shockshttps://zbmath.org/1491.760322022-09-13T20:28:31.338867Z"Barker, Blake"https://zbmath.org/authors/?q=ai:barker.blake"Monteiro, Rafael"https://zbmath.org/authors/?q=ai:monteiro.rafael-a"Zumbrun, Kevin"https://zbmath.org/authors/?q=ai:zumbrun.kevin-rSummary: We study by a combination of analytical and numerical Evans function techniques multi-D viscous and inviscid stability and associated transverse bifurcation of planar slow Lax MHD shocks in a channel with periodic boundary conditions. Notably, this includes the first multi-D numerical Evans function study for viscous MHD. Our results suggest that, rather than a planar shock, a nonplanar traveling wave with the same normal velocity is the typical mode of propagation in the slow Lax mode. Moreover, viscous and inviscid stability transitions appear to agree, answering (for this particular model and setting) an open question of \textit{K. Zumbrun} and \textit{D. Serre} [Indiana Univ. Math. J. 48, No. 3, 937--992 (1999; Zbl 0944.76027)].Convective, absolute and global azimuthal magnetorotational instabilitieshttps://zbmath.org/1491.760332022-09-13T20:28:31.338867Z"Mishra, A."https://zbmath.org/authors/?q=ai:mishra.amrutmayee|mishra.abinash|mishra.arijit|mishra.ankit|mishra.akshat|mishra.arvind-kumar|mishra.anwesha|mishra.awadhesh-kumar|mishra.amardeep|mishra.amrita|mishra.amarendra|mishra.aditya-mani|mishra.amit-kumar|mishra.abhishek-c|mishra.amitabh|mishra.apurva|mishra.aseem-k|mishra.ashok-kumar|mishra.asim-kumar|mishra.arti|mishra.ajit|mishra.akansha|mishra.alok-kumar|mishra.amiya|mishra.aditi|mishra.aurosish|mishra.anju|mishra.apoorva|mishra.anshuman|mishra.arunima|mishra.anurag|mishra.anil-kumar|mishra.asitav|mishra.arabinda|mishra.amruta|mishra.anand-k|mishra.ambuj-kumar|mishra.akash-k|mishra.avdesh|mishra.akshaya-kumar|mishra.ashish|mishra.asmita|mishra.asha-s|mishra.amar-p|mishra.alpna|mishra.arindam|mishra.aashwin-ananda|mishra.ajay-k|mishra.akhilesh-k|mishra.arunodaya-raj"Mamatsashvili, G."https://zbmath.org/authors/?q=ai:mamatsashvili.g-r"Galindo, V."https://zbmath.org/authors/?q=ai:galindo.vladimir"Stefani, F."https://zbmath.org/authors/?q=ai:stefani.frankSummary: We study the convective and absolute forms of azimuthal magnetorotational instability (AMRI) in a cylindrical Taylor-Couette (TC) flow with an imposed azimuthal magnetic field. We show that the domain of the convective AMRI is wider than that of the absolute AMRI. Actually, it is the absolute instability which is the most relevant and important for magnetic TC flow experiments. The absolute AMRI, unlike the convective one, stays in the device, displaying a sustained growth that can be experimentally detected. We also study the global AMRI in a TC flow of finite height using direct numerical simulation and find that its emerging butterfly-type structure -- a spatio-temporal variation in the form of axially upward and downward travelling waves -- is in a very good agreement with the linear analysis, which indicates the presence of two dominant absolute AMRI modes in the flow giving rise to this global butterfly pattern.Breathers, cascading instabilities and Fermi-Pasta-Ulam-Tsingou recurrence of the derivative nonlinear Schrödinger equation: effects of `self-steepening' nonlinearityhttps://zbmath.org/1491.760342022-09-13T20:28:31.338867Z"Yin, H. M."https://zbmath.org/authors/?q=ai:yin.khin-m|yin.huiming|yin.hui-min"Chow, K. W."https://zbmath.org/authors/?q=ai:chow.kwok-wing|chow.ka-wing|chow.kong-wingSummary: Breathers, modulation instability and recurrence phenomena are studied for the derivative nonlinear Schrödinger equation, which incorporates second order dispersion, cubic nonlinearity and self-steepening effect. By insisting on periodic boundary conditions, a cascading process will occur where initially small higher order Fourier modes can grow alongside with lower order modes. Typically a breather is first observed when all modes attain roughly the same order of magnitude. Beyond the formation of the first breather, analytical formula of spatially periodic but temporally localized breather ceases to be a valid indicator. However, numerical simulations display Fermi-Pasta-Ulam-Tsingou type recurrence. Self-steepening effect plays a crucial role in the dynamics, as it induces motion of the breather and generates chaotic behavior of the Fourier coefficients. Theoretically, correlation between breather motion and the Lax pair formulation is made. Physically, quantitative assessments of wave profile evolution are made for different initial conditions, e.g. random noise versus modulation instability mode of maximum growth rate. Potential application to fluid mechanics is discussed.Nozzle dynamics and wavepackets in turbulent jetshttps://zbmath.org/1491.760352022-09-13T20:28:31.338867Z"Kaplan, Oğuzhan"https://zbmath.org/authors/?q=ai:kaplan.oguzhan"Jordan, Peter"https://zbmath.org/authors/?q=ai:jordan.peter"Cavalieri, André V. G."https://zbmath.org/authors/?q=ai:cavalieri.andre-v-g"Brès, Guillaume A."https://zbmath.org/authors/?q=ai:bres.guillaume-aSummary: We study a turbulent jet issuing from a cylindrical nozzle to characterise coherent structures evolving in the turbulent boundary layer. The analysis is performed using data from a large-eddy simulation of a Mach 0.4 jet. Azimuthal decomposition of the velocity field in the nozzle shows that turbulent kinetic energy predominantly resides in high azimuthal wavenumbers; the first three azimuthal wavenumbers, that are important for sound generation, contain much lower, but non-zero amplitudes. Using two-point statistics, low azimuthal modes in the nozzle boundary layer are shown to exhibit significant correlations with modes of the same order in the free-jet region. Spectral proper orthogonal decomposition is used to distill a low-rank approximation of the flow dynamics. This reveals the existence of tilted coherent structures within the nozzle boundary layer and shows that these are coupled with wavepackets in the jet. The educed nozzle boundary-layer structures are modelled using a global resolvent analysis of the mean flow inside the nozzle to determine the most amplified flow responses using the linearised Navier-Stokes system. It is shown that the most-energetic nozzle structures can be successfully described with optimal resolvent response modes, whose associated forcing modes are observed to tilt against the nozzle boundary layer, suggesting that the Orr mechanism underpins these organised, turbulent, boundary-layer structures.A route to chaos in Rayleigh-Bénard heat convectionhttps://zbmath.org/1491.760672022-09-13T20:28:31.338867Z"Hsia, Chun-Hsiung"https://zbmath.org/authors/?q=ai:hsia.chun-hsiung"Nishida, Takaaki"https://zbmath.org/authors/?q=ai:nishida.takaakiSummary: We use numerical methods to study the global bifurcation diagrams of the Bénard convection problem. In our computations, we include a huge number of Fourier modes of stream function and temperature function so that our results reflect more reality of the dynamics of the Rayleigh-Bénard heat convection. Our results confirm that the period doubling scenario is a route to chaos.Dynamo action between two rotating discshttps://zbmath.org/1491.760932022-09-13T20:28:31.338867Z"Arslan, A."https://zbmath.org/authors/?q=ai:arslan.atakan|arslan.a-muzaffer|arslan.ayse-n|arslan.ahmet-faruk|arslan.a-v|arslan.abdullah-n|arslan.ali|arslan.aykut"Mestel, A. J."https://zbmath.org/authors/?q=ai:mestel.a-jonathanSummary: Dynamo action is considered in the region between two differentially rotating infinite discs. The boundaries may be insulating, perfectly conducting or ferromagnetic. In the absence of a magnetic field, various well-known self-similar flows arise, generalising that of von Kármán. Magnetic field instabilities with the same similarity structure are sought. The kinematic eigenvalue problem is found to have growing modes for \(Re_m > R_c\simeq 100\). The growth rate is real for the perfectly conducting and ferromagnetic cases, but may be complex for insulating boundaries. As \(Re_m\to\infty\) it is shown that the dynamo can be fast or slow, depending on the flow structure. In the slow case, the growth rate is governed by a magnetic boundary layer on one of the discs. The growing field saturates in a solution to the nonlinear dynamo problem. The bifurcation is found to be subcritical and nonlinear dynamos are found for \(Re_m\gtrsim0.7R_c\). Finally, the flux of magnetic energy to large \(r\) is examined, to determine which solutions might generalise to dynamos between finite discs. It is found that the fast dynamos tend to have inward energy flux, and so are unlikely to be realised in practice. Slow dynamos with outward flux are found. It is suggested that the average rotation rate should be non-zero in practice.