Bénard–Marangoni convection at low Prandtl number

1999 ◽  
Vol 399 ◽  
pp. 251-275 ◽  
Author(s):  
THOMAS BOECK ◽  
ANDRÉ THESS
Keyword(s):  
Free Surface ◽  
Prandtl Number ◽  
Marangoni Number ◽  
Marangoni Convection ◽  
Three Dimensional ◽  
Bottom Wall ◽  
Liquid Sodium ◽  
Small Scale ◽  
Quiescent State ◽  
Two Dimensional

Surface-tension-driven Bénard convection in low-Prandtl-number fluids is studied by means of direct numerical simulation. The flow is computed in a three-dimensional rectangular domain with periodic boundary conditions in both horizontal directions and either a free-slip or no-slip bottom wall using a pseudospectral Fourier–Chebyshev discretization. Deformations of the free surface are neglected. The smallest possible domain compatible with the hexagonal flow structure at the linear stability threshold is selected. As the Marangoni number is increased from the critical value for instability of the quiescent state to approximately twice this value, the initially stationary hexagonal convection pattern becomes quickly time-dependent and eventually reaches a state of spatio-temporal chaos. No qualitative difference is observed between the zero-Prandtl-number limit and a finite Prandtl number corresponding to liquid sodium. This indicates that the zero-Prandtl-number limit provides a reasonable approximation for the prediction of low-Prandtl-number convection. For a free-slip bottom wall, the flow always remains three-dimensional. For the no-slip wall, two-dimensional solutions are observed in some interval of Marangoni numbers. Beyond the Marangoni number for onset of inertial convection in two-dimensional simulations, the convective flow becomes strongly intermittent because of the interplay of the flywheel effect and three-dimensional instabilities of the two-dimensional rolls. The velocity field in this intermittent regime is characterized by the occurrence of very small vortices at the free surface which form as a result of vortex stretching processes. Similar structures were found with the free-slip bottom at slightly smaller Marangoni number. These observations demonstrate that a high numerical resolution is necessary even at moderate Marangoni numbers in order to properly capture the small-scale dynamics of Marangoni convection at low Prandtl numbers.

Inertial Bénard–Marangoni convection

1997 ◽  
Vol 350 ◽  
pp. 149-175 ◽  
Author(s):  
THOMAS BOECK ◽  
ANDRÉ THESS
Keyword(s):  
Prandtl Number ◽  
Marangoni Number ◽  
Marangoni Convection ◽  
Pseudospectral Method ◽  
Critical Value ◽  
Two Dimensional ◽  
Conductive Temperature ◽  
Reasonable Approximation ◽  
Convection Rolls ◽  
Dimensional Surface

Two-dimensional surface-tension-driven Bénard convection in a layer with a free-slip bottom is investigated in the limit of small Prandtl number using accurate numerical simulations with a pseudospectral method complemented by linear stability analysis and a perturbation method. It is found that the system attains a steady state consisting of counter-rotating convection rolls. Upon increasing the Marangoni number Ma the system experiences a transition between two typical convective regimes. The first one is the regime of weak convection characterized by only slight deviations of the isotherms from the linear conductive temperature profile. In contrast, the second regime, called inertial convection, shows significantly deformed isotherms. The transition between the two regimes becomes increasingly sharp as the Prandtl number is reduced. For sufficiently small Prandtl number the transition from weak to inertial convection proceeds via a subcritical bifurcation involving weak hysteresis. In the viscous zero-Prandtl-number limit the transition manifests itself in an unbounded growth of the flow amplitude for Marangoni numbers beyond a critical value Mai. For Ma<Mai the zero-Prandtl-number equations provide a reasonable approximation for weak convection at small but finite Prandtl number. The possibility of experimental verification of inertial Bénard–Marangoni convection is briefly discussed.

scholarly journals Two- and three-dimensional numerical simulations of the transition to oscillatory convection in low-Prandtl-number fluids

1998 ◽  
Vol 374 ◽  
pp. 145-171 ◽  
Author(s):  
DANIEL HENRY ◽  
MARC BUFFAT
Keyword(s):  
Prandtl Number ◽  
Relative Concentration ◽  
Three Dimensional ◽  
Power Spectra ◽  
Hadley Circulation ◽  
Time Dependent ◽  
Two Dimensional ◽  
General Behaviour ◽  
Shallow Cavities ◽  
Dependent Flow

The convective flows which arise in shallow cavities filled with low-Prandtl-number fluids when subjected to a horizontal temperature gradient are studied numerically with a finite element method. Attention is focused on a rigid cavity with dimensions 4×2×1, for which experimental data are available. The three-dimensional results indicate that, after a relative concentration of the initial Hadley circulation, a transition to time-dependent flows occurs in the form of a roll oscillation with a purely dynamical origin. This transition corresponds to a Hopf bifurcation with a breaking of symmetry that gives some specific properties to the time evolution of the flow: these properties are shown to be the result of the general behaviour of the dynamical systems. Calculations performed in the case of mercury compare well with the experiments with similar power spectra of the temperature, and this validates the analysis of the nature of the global flow performed in the limiting case Pr=0. All these results are discussed with respect to the linear and nonlinear analyses and to other computational experiments. Numerical results obtained in the corresponding two-dimensional situation give a different transition to the time-dependent flow: it is shown that in the three-dimensional cavity this type of two-dimensional transition is less probable than the observed transition with breaking of symmetry.

The anatomy of the mixing transition in homogeneous and stratified free shear layers

2000 ◽  
Vol 413 ◽  
pp. 1-47 ◽  
Author(s):  
C. P. CAULFIELD ◽  
W. R. PELTIER
Keyword(s):  
Shear Layer ◽  
Three Dimensional ◽  
Streamwise Vortex ◽  
Shear Layers ◽  
Finite Amplitude ◽  
Small Scale ◽  
Two Dimensional ◽  
Streamwise Vortices ◽  
Free Shear Layers ◽  
Nonlinear Amplification

We investigate the detailed nature of the ‘mixing transition’ through which turbulence may develop in both homogeneous and stratified free shear layers. Our focus is upon the fundamental role in transition, and in particular the associated ‘mixing’ (i.e. small-scale motions which lead to an irreversible increase in the total potential energy of the flow) that is played by streamwise vortex streaks, which develop once the primary and typically two-dimensional Kelvin–Helmholtz (KH) billow saturates at finite amplitude.Saturated KH billows are susceptible to a family of three-dimensional secondary instabilities. In homogeneous fluid, secondary stability analyses predict that the stream-wise vortex streaks originate through a ‘hyperbolic’ instability that is localized in the vorticity braids that develop between billow cores. In sufficiently strongly stratified fluid, the secondary instability mechanism is fundamentally different, and is associated with convective destabilization of the statically unstable sublayers that are created as the KH billows roll up.We test the validity of these theoretical predictions by performing a sequence of three-dimensional direct numerical simulations of shear layer evolution, with the flow Reynolds number (defined on the basis of shear layer half-depth and half the velocity difference) Re = 750, the Prandtl number of the fluid Pr = 1, and the minimum gradient Richardson number Ri(0) varying between 0 and 0.1. These simulations quantitatively verify the predictions of our stability analysis, both as to the spanwise wavelength and the spatial localization of the streamwise vortex streaks. We track the nonlinear amplification of these secondary coherent structures, and investigate the nature of the process which actually triggers mixing. Both in stratified and unstratified shear layers, the subsequent nonlinear amplification of the initially localized streamwise vortex streaks is driven by the vertical shear in the evolving mean flow. The two-dimensional flow associated with the primary KH billow plays an essentially catalytic role. Vortex stretching causes the streamwise vortices to extend beyond their initially localized regions, and leads eventually to a streamwise-aligned collision between the streamwise vortices that are initially associated with adjacent cores.It is through this collision of neighbouring streamwise vortex streaks that a final and violent finite-amplitude subcritical transition occurs in both stratified and unstratified shear layers, which drives the mixing process. In a stratified flow with appropriate initial characteristics, the irreversible small-scale mixing of the density which is triggered by this transition leads to the development of a third layer within the flow of relatively well-mixed fluid that is of an intermediate density, bounded by narrow regions of strong density gradient.

On the Hydrodynamic Interaction Between Ship and Free-Surface Motions on Vessels With Moonpools

2019 ◽  
Author(s):  
Senthuran Ravinthrakumar ◽  
Trygve Kristiansen ◽  
Babak Ommani
Keyword(s):  
Free Surface ◽  
Flow Separation ◽  
Coupling Effect ◽  
Three Dimensional ◽  
Dimensional Case ◽  
Navier Stokes ◽  
Linear Potential ◽  
Two Dimensional ◽  
Sloshing Mode ◽  
Vessel Motion

Abstract Coupling between moonpool resonance and vessel motion is investigated in two-dimensional and quasi three-dimensional settings, where the models are studied in forced heave and in freely floating conditions. The two-dimensional setups are with a recess, while the quasi three-dimensional setups are without recess. One configuration with recess is presented for the two-dimensional case, while three different moonpool sizes (without recess) are tested for the quasi three-dimensional setup. A large number of forcing periods, and three wave steepnesses are tested. Boundary Element Method (BEM) and Viscous BEM (VBEM) time-domain codes based on linear potential flow theory, and a Navier–Stokes solver with linear free-surface and body-boundary conditions, are implemented to investigate resonant motion of the free-surface and the model. Damping due to flow separation from the sharp corners of the moonpool inlets is shown to matter for both vessel motions and moonpool response around the piston mode. In general, the CFD simulations compare well with the experimental results. BEM over-predicts the response significantly at resonance. VBEM provides improved results compared to the BEM, but still over-predicts the response. In the two-dimensional study there are significant coupling effects between heave, pitch and moonpool responses. In the quasi three-dimensional tests, the coupling effect is reduced significantly as the moonpool dimensions relative to the displaced volume of the ship is reduced. The first sloshing mode is investigated in the two-dimensional case. The studies show that damping due to flow separation is dominant. The vessel motions are unaffected by the moonpool response around the first sloshing mode.

scholarly journals Bilge-Keel Influence on Free Decay of Roll Motion of a Realistic Hull1

2017 ◽  
Vol 139 (4) ◽  
Author(s):  
Yichen Jiang ◽  
Ronald W. Yeung
Keyword(s):  
Free Surface ◽  
Fluid Motion ◽  
Three Dimensional ◽  
Vortex Method ◽  
Roll Motion ◽  
Two Dimensional ◽  
Desktop Computer ◽  
Roll Damping ◽  
Discrete Vortex ◽  
Bilge Keel

The prediction of roll motion of a ship with bilge keels is particularly difficult because of the nonlinear characteristics of the viscous roll damping. Flow separation and vortex shedding caused by bilge keels significantly affect the roll damping and hence the magnitude of the roll response. To predict the ship motion, the Slender-Ship Free-Surface Random-Vortex Method (SSFSRVM) was employed. It is a fast discrete-vortex free-surface viscous-flow solver developed to run on a standard desktop computer. It features a quasi-three-dimensional formulation that allows the decomposition of the three-dimensional ship-hull problem into a series of two-dimensional computational planes, in which the two-dimensional free-surface Navier–Stokes solver Free-Surface Random-Vortex Method (FSRVM) can be applied. In this paper, the effectiveness of SSFSRVM modeling is examined by comparing the time histories of free roll-decay motion resulting from simulations and from experimental measurements. Furthermore, the detailed two-dimensional vorticity distribution near a bilge keel obtained from the numerical model will also be compared with the existing experimental Digital Particle Image Velocimetry (DPIV) images. Next, we will report, based on the time-domain simulation of the coupled hull and fluid motion, how the roll-decay coefficients and the flow field are altered by the span of the bilge keels. Plots of vorticity contour and vorticity isosurface along the three-dimensional hull will be presented to reveal the motion of fluid particles and vortex filaments near the keels.

Thermocapillary Convection of Low Prandtl Number Fluid in a Shallow Cylindrical Pool

2006 ◽  
Author(s):  
Fang Ling ◽  
You-Rong Li ◽  
Peng Lan ◽  
Shuang-Ying Wu ◽  
Qing-Hua Chen
Keyword(s):  
Prandtl Number ◽  
Marangoni Number ◽  
Oscillatory Flow ◽  
Thermocapillary Convection ◽  
Three Dimensional ◽  
Temperature Fields ◽  
Critical Conditions ◽  
Three Dimensional Flow ◽  
Dimensional Flow ◽  
Oscillation Frequencies

In order to understand the characteristics of thermocapillary convection, we conducted a series of unsteady three-dimensional numerical simulations of thermocapillary convection of low Prandtl number fluid in a shallow cylindrical pool with an azimuthal nonuniform temperature, an adiabatic solid bottom and free surface. The simulation results indicate that thermocapillary convection is steady three-dimensional flow at the small Marangoni number. But when Marangoni number exceeds some critical value, the flow will undergo a transition to oscillatory three-dimensional flow. The critical conditions for the onset of oscillatory flow are determined. Details of the flow and temperature fields are discussed, and oscillation frequencies are also exhibited.

scholarly journals Numerical simulation of mixing by Rayleigh–Taylor and Richtmyer–Meshkov instabilities

1994 ◽  
Vol 12 (4) ◽  
pp. 725-750 ◽  
Author(s):  
D.L. Youngs
Keyword(s):  
Numerical Simulation ◽  
Turbulence Model ◽  
Turbulent Mixing ◽  
Inertial Confinement Fusion ◽  
Three Dimensional ◽  
Small Scale ◽  
Inertial Confinement ◽  
Two Dimensional ◽  
Confinement Fusion ◽  
Icf Target

Rayleigh-Taylor (RT) and Richtmyer–Meshkov (RM) instabilities at the pusher–fuel interface in inertial confinement fusion (ICF) targets may significantly degrade thermonuclear burn. Present-day supercomputers may be used to understand the fundamental instability mechanisms and to model the effect of the ensuing mixing on the performance of the ICF target. Direct three-dimensional numerical simulation is used to investigate turbulent mixing due to RT and RM instability in simple situations. A two-dimensional turbulence model is used to assess the effect of small-scale turbulent mixing in the axisymmetric implosion of an idealized ICF target.

Small-scale structure of two-dimensional magnetohydrodynamic turbulence

1979 ◽  
Vol 90 (1) ◽  
pp. 129-143 ◽  
Author(s):  
Steven A. Orszag ◽  
Cha-Mei Tang
Keyword(s):  
Numerical Simulation ◽  
Direct Numerical Simulation ◽  
Three Dimensional ◽  
Scale Structure ◽  
Magnetohydrodynamic Flow ◽  
Small Scale ◽  
Two Dimensional ◽  
Magnetohydrodynamic Turbulence ◽  
Hydrodynamic Turbulence ◽  
Small Scale Structure

The formation of singularities in two-dimensional magnetohydrodynamic flow is investigated by direct numerical simulation. It is shown that two-dimensional magnetohydrodynamic turbulence is not as singular as three-dimensional hydrodynamic turbulence (in the sense that it has a less highly excited small-scale structure) but that it is more singular than two-dimensional hydrodynamic turbulence.

Secondary instability of a temporally growing mixing layer

1987 ◽  
Vol 184 ◽  
pp. 207-243 ◽  
Author(s):  
Ralph W. Metcalfe ◽  
Steven A. Orszag ◽  
Marc E. Brachet ◽  
Suresh Menon ◽  
James J. Riley
Keyword(s):  
Growth Rates ◽  
Stokes Equations ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Mixing Layers ◽  
Navier Stokes ◽  
Small Scale ◽  
Two Dimensional ◽  
Mean Flow ◽  
Direct Numerical Integration

The three-dimensional stability of two-dimensional vortical states of planar mixing layers is studied by direct numerical integration of the Navier-Stokes equations. Small-scale instabilities are shown to exist for spanwise scales at which classical linear modes are stable. These modes grow on convective timescales, extract their energy from the mean flow and exist at moderately low Reynolds numbers. Their growth rates are comparable with the most rapidly growing inviscid instability and with the growth rates of two-dimensional subharmonic (pairing) modes. At high amplitudes, they can evolve into pairs of counter-rotating, streamwise vortices, connecting the primary spanwise vortices, which are very similar to the structures observed in laboratory experiments. The three-dimensional modes do not appear to saturate in quasi-steady states as do the purely two-dimensional fundamental and subharmonic modes in the absence of pairing. The subsequent evolution of the flow depends on the relative amplitudes of the pairing modes. Persistent pairings can inhibit three-dimensional instability and, hence, keep the flow predominantly two-dimensional. Conversely, suppression of the pairing process can drive the three-dimensional modes to more chaotic, turbulent-like states. An analysis of high-resolution simulations of fully turbulent mixing layers confirms the existence of rib-like structures and that their coherence depends strongly on the presence of the two-dimensional pairing modes.

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