Application of an All-Speed Implicit Finite-Volume Algorithm to Rayleigh–Taylor Instability

2015 ◽  
Vol 12 (03) ◽  
pp. 1550018 ◽  
Author(s):  
Ilyas Yilmaz ◽  
Firat Oguz Edis ◽  
Hasan Saygin
Keyword(s):  
Mach Number ◽  
Finite Volume ◽  
Turbulent Mixing ◽  
Stokes Equations ◽  
Baroclinic Instability ◽  
Three Dimensional ◽  
Taylor Instability ◽  
Fully Implicit ◽  
Energy Conserving ◽  
Rayleigh Taylor Instability

We present a three-dimensional Direct Numerical Simulation (DNS) study of Rayleigh–Taylor Instability (RTI) using an all-speed, fully implicit, nondissipative and discrete kinetic energy conserving algorithm. In order to perform this study, an in-house, fully parallel, finite-volume, DNS solver, iDNS, which solves the set of time-dependent, compressible Navier–Stokes equations with gravity was developed based on the present algorithm and the PETSc parallel library. It is shown that the algorithm is able to capture the correct physics of the baroclinic instability and turbulent mixing. Compressibility (i.e., high Mach number) has been found more effective on the development of the flow after the diffusive growth phase passed. An increase in bubble growth rate together with a decrease in turbulent mixing was also observed at Mach number 1.1.

scholarly journals Variable density and viscosity, miscible displacements in horizontal Hele-Shaw cells. Part 2. Nonlinear simulations

2013 ◽  
Vol 721 ◽  
pp. 295-323 ◽  
Author(s):  
M. O. John ◽  
R. M. Oliveira ◽  
F. H. C. Heussler ◽  
E. Meiburg
Keyword(s):  
Oil Recovery ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Viscous Fingering ◽  
Variable Density ◽  
Taylor Instability ◽  
Constant Density ◽  
Streamwise Vorticity ◽  
Miscible Displacements ◽  
Rayleigh Taylor Instability

AbstractDirect numerical simulations of the variable density and viscosity Navier–Stokes equations are employed, in order to explore three-dimensional effects within miscible displacements in horizontal Hele-Shaw cells. These simulations identify a number of mechanisms concerning the interaction of viscous fingering with a spanwise Rayleigh–Taylor instability. The dominant wavelength of the Rayleigh–Taylor instability along the upper, gravitationally unstable side of the interface generally is shorter than that of the fingering instability. This results in the formation of plumes of the more viscous resident fluid not only in between neighbouring viscous fingers, but also along the centre of fingers, thereby destroying their shoulders and splitting them longitudinally. The streamwise vorticity dipoles forming as a result of the spanwise Rayleigh–Taylor instability place viscous resident fluid in between regions of less viscous, injected fluid, thereby resulting in the formation of gapwise vorticity via the traditional, gap-averaged viscous fingering mechanism. This leads to a strong spatial correlation of both vorticity components. For stronger density contrasts, the streamwise vorticity component increases, while the gapwise component is reduced, thus indicating a transition from viscously dominated to gravitationally dominated displacements. Gap-averaged, time-dependent concentration profiles show that variable density displacement fronts propagate more slowly than their constant density counterparts. This indicates that the gravitational mixing results in a more complete expulsion of the resident fluid from the Hele-Shaw cell. This observation may be of interest in the context of enhanced oil recovery or carbon sequestration applications.

Rayleigh–Taylor instability of an inclined buoyant viscous cylinder

2011 ◽  
Vol 671 ◽  
pp. 313-338 ◽  
Author(s):  
JOHN R. LISTER ◽  
ROSS C. KERR ◽  
NICK J. RUSSELL ◽  
ANDREW CROSBY
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Maximum Growth ◽  
Propagation Rate ◽  
Taylor Instability ◽  
Polar Coordinates ◽  
Rayleigh Taylor Instability

The Rayleigh–Taylor instability of an inclined buoyant cylinder of one very viscous fluid rising through another is examined through linear stability analysis, numerical simulation and experiment. The stability analysis represents linear eigenmodes of a given axial wavenumber as a Fourier series in the azimuthal direction, allowing the use of separable solutions to the Stokes equations in cylindrical polar coordinates. The most unstable wavenumber k∗ is long-wave if both the inclination angle α and the viscosity ratio λ (internal/external) are small; for this case, k∗ ∝ max{α, (λ ln λ−1)1/2} and thus a small angle in experiments can have a significant effect for λ ≪ 1. As α increases, the maximum growth rate decreases and the upward propagation rate of disturbances increases; all disturbances propagate without growth if the cylinder is sufficiently close to vertical, estimated as α ≳ 70°. Results from the linear stability analysis agree with numerical calculations for λ = 1 and experimental observations. A point-force numerical method is used to calculate the development of instability into a chain of individual plumes via a complex three-dimensional flow. Towed-source experiments show that nonlinear interactions between neighbouring plumes are important for α ≳ 20° and that disturbances can propagate out of the system without significant growth for α ≳ 40°.

scholarly journals Phase-field model for the Rayleigh–Taylor instability of immiscible fluids

2009 ◽  
Vol 622 ◽  
pp. 115-134 ◽  
Author(s):  
ANTONIO CELANI ◽  
ANDREA MAZZINO ◽  
PAOLO MURATORE-GINANNESCHI ◽  
LARA VOZELLA
Keyword(s):  
Phase Field ◽  
Field Model ◽  
Stokes Equations ◽  
Phase Field Model ◽  
Terminal Velocity ◽  
Immiscible Fluids ◽  
Taylor Instability ◽  
Upper And Lower Bounds ◽  
Weakly Nonlinear ◽  
Rayleigh Taylor Instability

The Rayleigh–Taylor instability of two immiscible fluids in the limit of small Atwood numbers is studied by means of a phase-field description. In this method, the sharp fluid interface is replaced by a thin, yet finite, transition layer where the interfacial forces vary smoothly. This is achieved by introducing an order parameter (the phase-field) continuously varying across the interfacial layers and uniform in the bulk region. The phase-field model obeys a Cahn–Hilliard equation and is two-way coupled to the standard Navier–Stokes equations. Starting from this system of equations we have first performed a linear analysis from which we have analytically rederived the known gravity–capillary dispersion relation in the limit of vanishing mixing energy density and capillary width. We have performed numerical simulations and identified a region of parameters in which the known properties of the linear phase (both stable and unstable) are reproduced in a very accurate way. This has been done both in the case of negligible viscosity and in the case of non-zero viscosity. In the latter situation, only upper and lower bounds for the perturbation growth rate are known. Finally, we have also investigated the weakly nonlinear stage of the perturbation evolution and identified a regime characterized by a constant terminal velocity of bubbles/spikes. The measured value of the terminal velocity is in agreement with available theoretical prediction. The phase-field approach thus appears to be a valuable technique for the dynamical description of the stages where hydrodynamic turbulence and wave-turbulence come into play.

Jetting of viscous droplets from cavitation-induced Rayleigh–Taylor instability

2018 ◽  
Vol 846 ◽  
pp. 916-943 ◽  
Author(s):  
Qingyun Zeng ◽  
Silvestre Roberto Gonzalez-Avila ◽  
Sophie Ten Voorde ◽  
Claus-Dieter Ohl
Keyword(s):  
Bubble Dynamics ◽  
Stokes Equations ◽  
Droplet Surface ◽  
Taylor Instability ◽  
Bubble Collapse ◽  
Fluid Viscosity ◽  
Analytic Model ◽  
Radial Acceleration ◽  
Rayleigh Taylor Instability ◽  
Good Agreement

Liquid jetting and fragmentation are important in many industrial and medical applications. Here, we study the jetting from the surface of single liquid droplets undergoing internal volume oscillations. This is accomplished by an explosively expanding and collapsing vapour bubble within the droplet. We observe jets emerging from the droplet surface, which pinch off into finer secondary droplets. The jetting is excited by the spherical Rayleigh–Taylor instability where the radial acceleration is due to the oscillation of an internal bubble. We study this jetting and the effect of fluid viscosity experimentally and numerically. Experiments are carried out by levitating the droplet in an acoustic trap and generating a laser-induced cavitation bubble near the centre of the droplet. On the simulation side, the volume of fluid method (OpenFOAM) solves the compressible Navier–Stokes equations while accounting for surface tension and viscosity. Both two-dimensional (2-D) axisymmetric and 3-D simulations are performed and show good agreement with each other and the experimental observation. While the axisymmetric simulation reveals how the bubble dynamics results destabilizes the interface, only the 3-D simulation computes the geometrically correct slender jets. Overall, experiments and simulations show good agreement for the bubble dynamics, the onset of disturbances and the rapid ejection of filaments after bubble collapse. Additionally, an analytic model for the droplet surface perturbation growth is developed based on the spherical Rayleigh–Taylor instability analysis, which allows us to evaluate the surface stability over a large parameter space. The analytic model predicts correctly the onset of jetting as a function of Reynolds number and normalized internal bubble energy.

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