To investigate the finite time singularity in three-dimensional (3D) Euler flows, the simplified model of 3D axisymmetric incompressible fluids (i.e., two-dimensional Boussinesq approximation equations) is studied numerically. The system describes a cap-like hot zone of fluid rising from the bottom, while the edges of the cap lag behind, forming eye-like vortices. The hot liquid is driven by the buoyancy and meanwhile attracted by the vortices, which leads to the singularity-forming mechanism in our simulation. In the previous 2D Boussinesq simulations, the symmetricial initial data is used. However, it is observed that the adoption of symmetry leads to coordinate singularity. Moreover, as demonstrated in this work that the locations of peak values for the vorticity and the temperature gradient becomes far apart as $t$ approaches the predicted blow-up time. This suggests that the symmetry assumption may be unreasonable for searching solution blow-ups. One of the main contributions of this work is to propose an appropriate asymmetric initial condition, which avoids coordinate singularity and also makes the blow-up to occur much earlier than that given by the previously simulations. The shorter simulation time suppresses the development of the round-off error. On the numerical side, the pseudo-spectral method with filtering technique is adopted. The resolutions adopted in this study vary from $1024^2$, $2048^2$, $4096^2$ to $6144^2$. With our proposed asymmetric initial condition, it is shown that the $4096^2$ and $6144^2$ runs yield convergent results when $t$ is fairly close to the predicted blow-up time. Moreover, as expected the locations of peak values for the vorticity and the temperature gradient are very close to each other as $t$ approaches the predicted blow-up time.
In this work we prove that the solution of the Serre-Green-Naghdi equation cannot be globally defined when the interface reaches the impervious bottom tangentially. As a consequence, our result complements the paper \emph{Camassa, R., Falqui, G., Ortenzi, G., Pedroni, M., \& Thomson, C. Hydrodynamic models and confinement effects by horizontal boundaries. Journal of Nonlinear Science, 29(4), 1445-1498, 2019.} Furthermore, we also prove that the solution to the $abcd-$Boussinesq system can change sign in finite time. Finally, we provide with a proof of a scenario of finite time singularity for …
In this paper, we perform a three-stage numerical relay to investigate the finite time singularity in the two-dimensional Boussinesq approximation equations. The initial asymmetric condition is the middle-stage output of a $2048^2$ run, the highest resolution in our study is $40960^2$, and some signals of numerical blowup are observed.
Two-dimensional Boussinesq convection is studied numerically with very fine spatial resolutions up to 4096^2. Our numerical study starts with a smooth asymmetric initial condition, which is chosen to make the flow field well confined in the computational domain until the blow-up time (T_c). Our study shows that the vorticity will blow up at a finite time with |ω|_{max} \~{(T_c-t)}^{-1.61} and |\nabla θ|_{max} ~ {(T_c - t)}^{-3.58}.
We establish stochastic functional integral representations for solutions of Oberbeck-Boussinesq equations in the form of McKean-Vlasov-type mean field equations, which can be used to design numerical schemes for calculating solutions and for implementing Monte-Carlo simulations of Oberbeck-Boussinesq flows. Our approach is based on the duality of conditional laws for a class of diffusion processes associated with solenoidal vector fields, which allows us to obtain a novel integral representation theorem for solutions of some linear parabolic equations in terms of the Green function and the pi…
The paper's main goal is to compare the motion of solitary surface waves resulting from two similar but slightly different approaches. In the first approach, the numerical evolution of soliton surface waves moving over the uneven bottom is obtained using single wave equations. In the second approach, the numerical evolution of the same initial conditions is obtained by the solution of a coupled set of the Boussinesq equations for the same Euler equations system. We discuss four physically relevant cases of relationships between small parameters $α,β,δ$. For the flat bottom, these cases imply t…
We present a new hybrid discrete exterior calculus (DEC) and finite difference (FD) method to simulate fully three-dimensional Boussinesq convection in spherical shells subject to internal heating and basal heating, relevant in the planetary and stellar phenomenon. We employ DEC to compute the surface spherical flows, taking advantage of its unique features of structure preservation (e.g., conservation of secondary quantities like kinetic energy) and coordinate system independence, while we discretize the radial direction using FD method. The grid employed for this novel method is free of prob…
This work concerns soliton-type numerical solutions for two Whitham-Boussinesq-type models. Solitary waves are computed using an iterative Newton-type and continuation methods with high accuracy. The method allow us to compute solitary waves with large amplitude and speed close to the singular limit. These solitary waves are set as initial data and overtaking collisions are considered for both systems. We show that both system satisfy the geometric Lax-categorization of two-soliton collision. Numerical evidences indicate that one of the systems also admits an algebraic Lax-categorization based…
Consideration is given to the influence of an underwater landslide on waves at the surface of a shallow body of fluid. The equations of motion which govern the evolution of the barycenter of the landslide mass include various dissipative effects due to bottom friction, internal energy dissipation, and viscous drag. The surface waves are studied in the Boussinesq scaling, with time-dependent bathymetry. A numerical model for the Boussinesq equations is introduced which is able to handle time-dependent bottom topography, and the equations of motion for the landslide and surface waves are solved …
For all $ε>0$, we prove the existence of finite-energy strong solutions to the axi-symmetric $3D$ Euler equations on the domains $ \{(x,y,z)\in\mathbb{R}^3: (1+ε|z|)^2\leq x^2+y^2\}$ which become singular in finite time. We further show that solutions with 0 swirl are necessarily globally regular. The proof of singularity formation relies on the use of approximate solutions at exactly the critical regularity level which satisfy a $1D$ system which has solutions which blow-up in finite time. The construction bears similarity to our previous result on the Boussinesq system \cite{EJB} though a nu…
Rayleigh-Taylor instability occurs when a heavier fluid overlies a lighter fluid, and the two seek to exchange positions under the effect of gravity. We present linearized theory for arbitrary 3D initial disturbances that grow in time, and calculate the evolution of the interface for early times. A new spectral method is introduced for the fully 3D non-linear problem in a Boussinesq fluid, where the interface between the light and heavy fluid is approximated with a smooth but rapid density change in the fluid. The results of large-scale numerical calculation are presented in fully 3D geometry,…
We numerically analyze Non-Oberbeck-Boussinesq (NOB) effects in two-dimensional Rayleigh-Benard flow in glycerol, which shows a dramatic change in the viscosity with temperature. The results are presented both as functions of the Rayleigh number (Ra) up to $10^8$ (for fixed temperature difference between the top and bottom plates) and as functions of "non-Oberbeck-Boussinesqness'' or "NOBness'' ($Δ$) up to 50 K (for fixed Ra). For this large NOBness the center temperature $T_c$ is more than 5 K larger than the arithmetic mean temperature $T_m$ between top and bottom plate and only weakly depen…
We consider a dissipative, dispersive system of Boussinesq type, describing wave phenomena in settings where dissipation has an effect. Examples include undular bores in rivers or oceans where dissipation due to turbulence is important for their description. We show that the model system admits traveling wave solutions known as diffusive-dispersive shock waves, and we categorize them into oscillatory and regularized shock waves depending on the relationship between dispersion and dissipation. Comparison of numerically computed solutions with laboratory data suggests that undular bores are accu…
A study of convection in a circular two dimensional cell is presented. The system is heated and cooled at two diametrically opposed points on the edge of the circle, which are parallel or anti-parallel to gravity. The latter's role in the plane of the cell can be changed by tilting the cell. When the system is in a horizontal position, a non-trivial analytic solution for the temperature distribution of the quiescent fluid can be found. For a slight inclination, the projection of gravity in the plane of the cell is used as a perturbation parameter in the full hydrodynamic description, as the Bo…
As shown in earlier work (Ahlers et al., J. Fluid Mech. 569, p.409 (2006)), non-Oberbeck Boussinesq (NOB) corrections to the center temperature in turbulent Rayleigh-Benard convection in water and also in glycerol are governed by the temperature dependences of the kinematic viscosity and the thermal diffusion coefficient. If the working fluid is ethane close to the critical point the origin of non-Oberbeck-Boussinesq corrections is very different, as will be shown in the present paper. Namely, the main origin of NOB corrections then lies in the strong temperature dependence of the isobaric the…
We present results of three-dimensional Direct Numerical Simulations (DNS) and Large Eddy Simulations (LES) of turbulent gravity currents with a Discontinuous Galerkin (DG) Finite Elements method. In particular, we consider the three-dimensional lock-exchange test case as a typical benchmark for gravity currents. Since, to the best of our knowledge, non-Boussinesq three-dimensional reference DNS are not available in the literature for this test case, we first perform a DNS experiment. The three-dimensional DNS allows to correctly capture the loss of coherence of the three-dimensional turbulent…
The equations of motion describing buoyant fluids are often simplified using a set of approximations proposed by J. Boussinesq one century ago. To resume, they consist in assuming constant fluid properties, incompressibility and conservation of calories during heat transport. Assuming fulfilment of the first requirement (constant fluid properties), we derive a set of 4 criteria for assessing the validity of the two other requirements in turbulent Rayleigh-Bénard convection. The first criterion $αΔ\ll 1 $ simply results from the incompressibility condition in the thermal boundary layer ($α$ and…
A novel parallel technique for Fourier-Galerkin pseudo-spectral methods with applications to two-dimensional Navier-Stokes equations and inviscid Boussinesq approximation equations is presented. It takes the advantage of the programming structure of the phase-shift de-aliased scheme for pseudo-spectral codes, and combines the task-distribution strategy [Yin, Clercx and Montgomery, Comput. Fluids, 33, 509 (2004)] and parallelized Fast Fourier Transform scheme. The performances of the resulting MPI Fortran90 codes with the new procedure on SGI 3800 are reported. For fixed resolution of the same …
This paper shows that the energetics of Boussinesq and anelastic fluids possesses a term that can be identified as the approximation $δW_{ba}$ to the compressible work of expansion/contraction $δW =-P {\rm d}\upsilon$, where $P$ is the pressure and $\upsilon$ is the specific volume. It follows that Boussinesq and anelastic fluids admit explicit compressible effects and conversions between internal energy and mechanical energy, under the form of apparent changes in gravitational potential energy resulting from changes in density by diabatic and adiabatic effects. From the knowledge of $δW_{ba}$…
In this paper we present numerical simulations of rapidly-rotating Rayleigh-Bénard convection in the Boussinesq approximation with stress-free boundary conditions. At moderately low Rossby number and large Rayleigh number, we show that a large-scale depth-invariant flow is formed, reminiscent of the condensate state observed in two-dimensional flows. We show that the large-scale circulation shares many similarities with the so-called vortex, or slow-mode, of forced rotating turbulence. Our investigations show that at a fixed rotation rate the large-scale vortex is only observed for a finite ra…
2D inviscid Boussinesq equations and 3D axisymmetric Euler equations: (1) A unification ($Em$), (2) Finite-time blow-up of two unified $(1+1)$D systems rigorously derived from ($Em$)
The linear stability threshold of the Rayleigh-Benard configuration is analyzed with compressible effects taken into account. It is assumed that the fluid obeys a Newtonian rheology and Fourier's law of thermal transport with constant, uniform (dynamic) viscosity and thermal conductivity in a uniform gravity field. Top and bottom boundaries are maintained at different constant temperatures and we consider here boundary conditions of zero tangential stress and impermeable walls. Under these conditions, and with the Boussinesq approximation, Rayleigh (1916) first obtained analytically the critic…
The evolution of three-dimensional, cellular convective flows in a plane horizontal layer of a Boussinesq fluid heated from below is well studied. Here we review results from the investigation of this system as well as a number of related and novel numerical findings. We present theoretical results for pattern formation in Rayleigh-Bénard convection with solving NS equation under the Boussinesq approximation . System of equations were reduced to 2-dimension form to simplify the analysis.The evolution of the flow agree with the idea of the flow achieving an optimal form.
Stochastic compressible Navier-Stokes equations under location uncertainty and their approximations for ocean modelling
INRIA Centre de l’Université de Rennes（法国国家信息与自动化研究中心雷恩大学中心）；IRMAR – UMR CNRS 6625（雷恩数学研究所-法国国家科学研究中心联合研究单位6625）；LOPS, Ifremer（海洋环境与资源研究所）；SHOM（法国海军水文与海洋学办公室）
An exact solution to the lock-exchange problem, which is a two-layer analogue of the classical dam-break problem, is obtained in the shallow-water (SW) approximation for two immiscible fluids with slightly different densities. The problem is solved by the method of characteristics using analytic expressions for the Riemann invariants. The obtained solution, which represents an inviscid approximation to the high-Reynolds-number limit, is in general discontinuous containing up to three hydraulic jumps which are due to either multivaluedness or instability of the continuous SW solution. Hydraulic…
In this paper we construct a weakly-nonlinear d'Alembert-type solution of the Cauchy problem for a Boussinesq-Klein-Gordon equation. Similarly to our earlier work based on the use of spatial Fourier series, we consider the problem in the class of periodic functions on an interval of finite length (including the limiting case of an "infinite" interval with zero boundary conditions), and work with the equation describing a deviation from the mean value. Unlike our earlier paper, here we develop a novel multiple-scales procedure involving fast characteristic variables and two slow time scales, wh…
Modeling late-time sensitivity to initial conditions in Boussinesq Rayleigh-Taylor turbulence
Université Paris Saclay（巴黎萨克雷大学）；Université Paris Cité（巴黎西岱大学）；ENS Paris Saclay（巴黎萨克雷高等师范学校）；CNRS（法国国家科学研究中心）；SSA（社会、精神与老龄化研究实验室）；INSERM（法国国家健康与医学研究院）；Centre Borelli（博尔勒中心）；CEA（法国原子能和替代能源委员会）；DAM（达莫坦分公司）；DIF（达莫坦信息技术部）；LMCE（力学与核化学实验室）
Convection in the Sun occurs at Rayleigh numbers, $Ra$, as high as $10^{22}$, molecular Prandtl numbers, $Pr$, as low as $10^{-6}$, under conditions that are far from satisfying the Oberbeck-Boussinesq (OB) idealization. The effects of these extreme circumstances on turbulent heat transport are unknown, and no comparable conditions exist on Earth. Our goal is to understand how these effects scale (since we cannot yet replicate the Sun's conditions faithfully). We study thermal convection by using direct numerical simulations, and determine the variation with respect to $Pr$, to values as low a…
We derive and analyze in the framework of the mild-slope approximation a new double-layer Boussinesq-type model which is linearly and nonlinearly accurate up to deep water. Assuming the flow to be irrotational, we formulate the problem in terms of the velocity potential thereby lowering the number of unknowns. The model derivation combines two approaches, namely the method proposed by Agnon et al. (Agnon et al. 1999, J. Fluid Mech., 399 pp. 319-333) and enhanced by Madsen et al. (Madsen et al. 2003, Proc. R. Soc. Lond. A, 459 pp. 1075-1104) which consists in constructing infinite-series Taylor…
In this paper we explore the fast rotation, nonhydrostatic limit of the rotating and stratified Boussinesq equations. We derive new reduced equations for the slow dynamics that describe Taylor-Proudman flows. One new aspect of the dynamics is a decoupling of the horizontal kinetic energy, described by 2D Navier-Stokes, from new dynamics that describe the coupling of vertical kinetic energy and buoyancy. We support the theory with high resolution numerical simulations of the full Boussinesq equations that, in this limit, reveal the spontaneous formation of Taylor-Proudman columns and their dyna…
