scholarly journals Relaxation of the Boussinesq system and applications to the Rayleigh–Taylor instability

2021 ◽  
Vol 29 (1) ◽  
Author(s):  
Björn Gebhard ◽  
József J. Kolumbán
Keyword(s):  
Boussinesq Equations ◽  
Initial Energy ◽  
Boussinesq System ◽  
Quadratic Growth ◽  
Limit States ◽  
Two Fluids ◽  
Long Time ◽  
Total Mixture ◽  
Rayleigh Taylor Instability ◽  
Full Separation

AbstractWe consider the evolution of two incompressible fluids with homogeneous densities $$\rho _{-}<\rho _+$$ ρ - < ρ + subject to gravity described by the inviscid Boussinesq equations and provide the explicit relaxation of the associated differential inclusion. The existence of a subsolution to the relaxation allows one to conclude the existence of turbulently mixing solutions to the original Boussinesq system. As a specific application we investigate subsolutions emanating from the classical Rayleigh-Taylor initial configuration where the two fluids are separated by a horizontal interface with the heavier fluid being on top of the lighter. It turns out that among all self-similar subsolutions the criterion of maximal initial energy dissipation selects a linear density profile and a quadratic growth of the mixing zone. The subsolution selected this way can be extended in an admissible way to exist for all times. We provide two possible extensions with different long-time limits. The first one corresponds to a total mixture of the two fluids, the second corresponds to a full separation with the lighter fluid on top of the heavier. There is no motion in either of the limit states.

HOMOTOPY ANALYSIS METHOD TO SOLVE BOUSSINESQ EQUATIONS

2015 ◽  
Vol 10 (3) ◽  
pp. 2825-2833
Author(s):  
Achala Nargund ◽  
R Madhusudhan ◽  
S B Sathyanarayana
Keyword(s):  
Homotopy Analysis Method ◽  
Degrees Of Freedom ◽  
Initial Approximation ◽  
Boussinesq Equations ◽  
Convergent Series ◽  
Boussinesq System ◽  
Analysis Method ◽  
Analytical Technique ◽  
Homotopy Analysis ◽  
Region Of Convergence

In this paper, Homotopy analysis method is applied to the nonlinear coupleddifferential equations of classical Boussinesq system. We have applied Homotopy analysis method (HAM) for the application problems in [1, 2, 3, 4]. We have also plotted Domb-Sykes plot for the region of convergence. We have applied Pade for the HAM series to identify the singularity and reflect it in the graph. The HAM is a analytical technique which is used to solve non-linear problems to generate a convergent series. HAM gives complete freedom to choose the initial approximation of the solution, it is the auxiliary parameter h which gives us a convenient way to guarantee the convergence of homotopy series solution. It seems that moreartificial degrees of freedom implies larger possibility to gain better approximations by HAM.

scholarly journals A conservative fully-discrete numerical method for the regularised shallow water wave equations

2021 ◽  
Author(s):  
Dimitrios Mitsotakis ◽  
Hendrik Ranocha ◽  
David I Ketcheson ◽  
Endre Süli
Keyword(s):  
Finite Element ◽  
Shallow Water ◽  
Numerical Method ◽  
Solitary Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Mixed Finite Element ◽  
Boussinesq System ◽  
Fully Discrete ◽  
Long Time

The paper proposes a new, conservative fully-discrete scheme for the numerical solution of the regularised shallow water Boussinesq system of equations in the cases of periodic and reflective boundary conditions. The particular system is one of a class of equations derived recently and can be used in practical simulations to describe the propagation of weakly nonlinear and weakly dispersive long water waves, such as tsunamis. Studies of small-amplitude long waves usually require long-time simulations in order to investigate scenarios such as the overtaking collision of two solitary waves or the propagation of transoceanic tsunamis. For long-time simulations of non-dissipative waves such as solitary waves, the preservation of the total energy by the numerical method can be crucial in the quality of the approximation. The new conservative fully-discrete method consists of a Galerkin finite element method for spatial semidiscretisation and an explicit relaxation Runge--Kutta scheme for integration in time. The Galerkin method is expressed and implemented in the framework of mixed finite element methods. The paper provides an extended experimental study of the accuracy and convergence properties of the new numerical method. The experiments reveal a new convergence pattern compared to standard Galerkin methods.

scholarly journals Experimental investigation into inertial properties of Rayleigh–Taylor turbulence

1997 ◽  
Vol 15 (1) ◽  
pp. 25-31 ◽  
Author(s):  
Yu.A. Kucherenko ◽  
S.I. Balabin ◽  
R. Cherret ◽  
J.F. Haas
Keyword(s):  
Experimental Investigation ◽  
Kinetic Energy ◽  
Turbulent Mixing ◽  
Average Density ◽  
Taylor Instability ◽  
X Ray ◽  
Two Fluids ◽  
Time Space ◽  
Rayleigh Taylor Instability ◽  
Region Size

An experimental investigation into inertial properties of the developed Rayleigh–Taylor instability with the different initial values of the kinetic energy of turbulence has been performed. The experiments were performed by using two fluids having different densities with density ration n = 3. Fluids were placed in an ampoule. At the unstable stage of motion, the ampoule was moving under an acceleration. At a certain instant of time the acceleration was removed and the ampoule moved under the force of inertia. By means of pulsed X-ray photography, the mixing region size and the time-space distributionof the average density of matter in the turbulent mixing region have been determined at different instants of time. The time-space distributions are compared with those obtained by semiempirical theories of mixing.

ON THE REGULARITY OF THREE-DIMENSIONAL ROTATING EULER–BOUSSINESQ EQUATIONS

1999 ◽  
Vol 09 (07) ◽  
pp. 1089-1121 ◽  
Author(s):  
A. BABIN ◽  
A. MAHALOV ◽  
B. NICOLAENKO
Keyword(s):  
Initial Data ◽  
Three Dimensional ◽  
Fluid Flows ◽  
Boussinesq Equations ◽  
Stable Stratification ◽  
Primitive Equations ◽  
Time Interval ◽  
Regular Solutions ◽  
Time Intervals ◽  
Long Time

The 3-D rotating Boussinesq equations (the "primitive" equations of geophysical fluid flows) are analyzed in the asymptotic limit of strong stable stratification. The resolution of resonances and a nonstandard small divisor problem are the basis for error estimates for such fast singular oscillating limits. Existence on infinite time intervals of regular solutions to the viscous 3-D "primitive" equations is proven for initial data in Hα, α≥ 3/4. Existence on a long-time interval T*of regular solutions to the 3-D inviscid equations is proven for initial data in Hα, α > 5/2 (T*→∞ as the frequency of gravity waves →∞).

Clusters of sedimenting high-Reynolds-number particles

2009 ◽  
Vol 625 ◽  
pp. 371-385 ◽  
Author(s):  
W. BRENT DANIEL ◽  
ROBERT E. ECKE ◽  
G. SUBRAMANIAN ◽  
DONALD L. KOCH
Keyword(s):  
Reynolds Number ◽  
Mass Conservation ◽  
Characteristic Size ◽  
Quadratic Growth ◽  
Growth Exponent ◽  
Theoretical Predictions ◽  
Large Tank ◽  
Long Time ◽  
Self Similar ◽  
High Reynolds

We report experiments wherein groups of particles were allowed to sediment in an otherwise quiescent fluid contained in a large tank. The Reynolds number of the particles, defined as Re = aU/ν, ranged from 93 to 425; here, a is the radius of the spherical particle, U its settling velocity and ν the kinematic viscosity of the fluid. The characteristic size of a cluster, in a plane transverse to gravity, was measured by a ‘cluster variance’(〈r2t〉); the latter is defined as the mean square of the transverse coordinates of all constituent particles, averaged over a series of runs. The cluster variance, when plotted as a function of time, exhibited two regimes. There was a quadratic growth in the variance at short times(〈r2t〉 ∝ t2), while for long times, the cluster variance exhibited a slower sublinear growth with 〈r2t〉 ∝ t0.67. A theory, based on isotropic repulsive hydrodynamic interactions between particles, predicts the cluster variance to grow as t2/3 in the limit of long times. The theoretical framework was originally proposed to describe the long-time self-similar evolution of dilute clusters in the limit Re ≪ 1 Subramanian & Koch (J. Fluid Mech., vol. 603, 2008, p. 63), when the probability of wake-mediated interactions between particles remains asymptotically small; the latter requirement is satisfied for homogeneous spherical clusters larger than a critical radius, and is evidently satisfied for planar clusters oriented transversely to gravity. The isotropy of the interactions therefore stems from the isotropy, at large distances, of the disturbance velocity field produced by a single sedimenting particle outside its wake(which contains the compensating inflow to satisfy mass conservation). Herein, the theory is extended to large Re using an empirical correlation for the drag on a sedimenting particle. This allows one to predict, as a function of Re, the numerical prefactors in the expressions for the cluster variance of both spherical and planar clusters; the predictions for the growth exponent remain unchanged. The agreement between the theoretical and experimental growth exponents supports the hypothesis of a self-similar expansion at long times. The prefactor determined from the experimental observations is found to lie between the theoretical predictions for planar and spherical clusters.

Numerical studies of some nonlinear hydrodynamic problems by discrete vortex element methods

1978 ◽  
Vol 84 (3) ◽  
pp. 433-453 ◽  
Author(s):  
J. C. S. Meng ◽  
J. A. L. Thomson
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Function Method ◽  
Gravity Current ◽  
Vortex Sheet ◽  
Taylor Instability ◽  
Green’S Function Method ◽  
Discrete Vortex ◽  
Two Fluids ◽  
Rayleigh Taylor Instability

A class of nonlinear hydrodynamic problems is studied. Physical problems such as shear flow, flow with a sharp interface separating two fluids of different density and flow in a porous medium all belong to this class. Owing to the density difference across the interface, vorticity is generated along it by the interaction between the gravitational pressure gradient and the density gradient, and the motion consists of essentially two processes: the creation of a vortex sheet and the subsequent mutual induction of different portions of this sheet.Two numerical methods are investigated. One is based upon the well-known Green's function method, which is a Lagrangian method using the Biot-Savart law, while the other is the vortex-in-cell (VIC) method, which is a Lagrangian-Eulerian method. Both methods treat the interface as sharp and represent it by a distribution of point vortices. The VIC method applies the FFT (fast Fourier transform) to solve the stream-function/vorticity equation on an Eulerian grid, and computational efficiency is further improved by using the reality properties of the physical variables.Four specific problems are investigated numerically in this paper. They are: the Rayleigh-Taylor instability, the Saffman-Taylor instability, transport of aircraft trailing vortices in a wind shear, and the gravity current. All four problems are solved using the VIC method and the results agree well with results obtained by previous investigators. The first two problems, the Rayleigh-Taylor instability and the Saffman-Taylor instability, are also solved by the Green's function method. Comparisons of results obtained by the two methods show good agreement, but, owing to its computational economy, the VIC method is concluded to be the better method for treating the class of hydrodynamic problems considered here.

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