Numerical estimates of molecular mixing in confined RTI

Andrew G W Lawrie; SB Dalziel

Abstract

This paper examines the behaviour of a system in which a Rayleigh-Taylor unstable interface is confined between stable continuous stratifications. Recent experiments with linear stratifications (Lawrie & Dalziel 2011, J. Fluid Mech.) and extensions to curved stratifications presented in Davies-Wykes & Dalziel (2011) indicate the existence of a fundamental and intrinsic limit to a fluid’s ability to mix, which in this particular configuration can be measured robustly between quiescent initial and final states. However, the final state vertical density profile observed in experiments in salty water has not been satisfactorily replicated with standard incompressible ILES numerical methods. These mis-matches occur because the numerical method cannot respect the same balance of energy conversions as observed in experiment, and we conjecture that such discrepancies occur because the numerical scheme operates with a Sc=O(1) whereas Sc=700 in our experimental conditions. Lawrie & Dalziel (2011) discussed in detail the relation between the probability density function (pdf) of the density field and the availability of energy in the system. In this paper we extract from our numerical simulations the evolution of the pdf over the life-cycle of the instability, and use it to motivate a simple sub-grid model that improves the ILES representation of energy conversions at high Schmidt number. We compare both two- and three-dimensional Rayleigh-Taylor cases. In three dimensions energy cascades to small scales, so the stretching of material surfaces that it induces tends to occur at comparable scales (particularly so in low Schmidt number simulations) and this is the optimal condition for doing mixing. In two dimensions, however, energy accumulates at large scales and thus material surfaces do not become so rapidly stretched at scales associated with diffusive processes. We view the two-dimensional case as an analogue for high Schmidt number behaviour, and this informs our choice of sub-grid scale model.

Original language	English
Publication status	Published - 2012
Event	13th International Workshop on the Physics of Compressible Turbulent Mixing (2012) - Cranfield University (Woburn Abbey), Woburn, United Kingdom Duration: 16 Jul 2012 → 20 Jul 2012 Conference number: 13 http://www.iwpctm.org

Conference

Conference	13th International Workshop on the Physics of Compressible Turbulent Mixing (2012)
Abbreviated title	IWPCTM13
Country/Territory	United Kingdom
City	Woburn
Period	16/07/12 → 20/07/12
Internet address	http://www.iwpctm.org

Keywords

two-dimensional Rayleigh-Taylor
molecular mixing
direct numerical simulation

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Cite this

@conference{b4d704a516984e84ae9b77107bdfe194,

title = "Numerical estimates of molecular mixing in confined RTI",

abstract = "This paper examines the behaviour of a system in which a Rayleigh-Taylor unstable interface is confined between stable continuous stratifications. Recent experiments with linear stratifications (Lawrie & Dalziel 2011, J. Fluid Mech.) and extensions to curved stratifications presented in Davies-Wykes & Dalziel (2011) indicate the existence of a fundamental and intrinsic limit to a fluid{\textquoteright}s ability to mix, which in this particular configuration can be measured robustly between quiescent initial and final states. However, the final state vertical density profile observed in experiments in salty water has not been satisfactorily replicated with standard incompressible ILES numerical methods. These mis-matches occur because the numerical method cannot respect the same balance of energy conversions as observed in experiment, and we conjecture that such discrepancies occur because the numerical scheme operates with a Sc=O(1) whereas Sc=700 in our experimental conditions. Lawrie & Dalziel (2011) discussed in detail the relation between the probability density function (pdf) of the density field and the availability of energy in the system. In this paper we extract from our numerical simulations the evolution of the pdf over the life-cycle of the instability, and use it to motivate a simple sub-grid model that improves the ILES representation of energy conversions at high Schmidt number. We compare both two- and three-dimensional Rayleigh-Taylor cases. In three dimensions energy cascades to small scales, so the stretching of material surfaces that it induces tends to occur at comparable scales (particularly so in low Schmidt number simulations) and this is the optimal condition for doing mixing. In two dimensions, however, energy accumulates at large scales and thus material surfaces do not become so rapidly stretched at scales associated with diffusive processes. We view the two-dimensional case as an analogue for high Schmidt number behaviour, and this informs our choice of sub-grid scale model.",

keywords = "two-dimensional Rayleigh-Taylor, molecular mixing, direct numerical simulation",

author = "Lawrie, {Andrew G W} and SB Dalziel",

year = "2012",

language = "English",

note = "13th International Workshop on the Physics of Compressible Turbulent Mixing (2012), IWPCTM13 ; Conference date: 16-07-2012 Through 20-07-2012",

url = "http://www.iwpctm.org",

}

TY - CONF

T1 - Numerical estimates of molecular mixing in confined RTI

AU - Lawrie, Andrew G W

AU - Dalziel, SB

N1 - Conference code: 13

PY - 2012

Y1 - 2012

N2 - This paper examines the behaviour of a system in which a Rayleigh-Taylor unstable interface is confined between stable continuous stratifications. Recent experiments with linear stratifications (Lawrie & Dalziel 2011, J. Fluid Mech.) and extensions to curved stratifications presented in Davies-Wykes & Dalziel (2011) indicate the existence of a fundamental and intrinsic limit to a fluid’s ability to mix, which in this particular configuration can be measured robustly between quiescent initial and final states. However, the final state vertical density profile observed in experiments in salty water has not been satisfactorily replicated with standard incompressible ILES numerical methods. These mis-matches occur because the numerical method cannot respect the same balance of energy conversions as observed in experiment, and we conjecture that such discrepancies occur because the numerical scheme operates with a Sc=O(1) whereas Sc=700 in our experimental conditions. Lawrie & Dalziel (2011) discussed in detail the relation between the probability density function (pdf) of the density field and the availability of energy in the system. In this paper we extract from our numerical simulations the evolution of the pdf over the life-cycle of the instability, and use it to motivate a simple sub-grid model that improves the ILES representation of energy conversions at high Schmidt number. We compare both two- and three-dimensional Rayleigh-Taylor cases. In three dimensions energy cascades to small scales, so the stretching of material surfaces that it induces tends to occur at comparable scales (particularly so in low Schmidt number simulations) and this is the optimal condition for doing mixing. In two dimensions, however, energy accumulates at large scales and thus material surfaces do not become so rapidly stretched at scales associated with diffusive processes. We view the two-dimensional case as an analogue for high Schmidt number behaviour, and this informs our choice of sub-grid scale model.

AB - This paper examines the behaviour of a system in which a Rayleigh-Taylor unstable interface is confined between stable continuous stratifications. Recent experiments with linear stratifications (Lawrie & Dalziel 2011, J. Fluid Mech.) and extensions to curved stratifications presented in Davies-Wykes & Dalziel (2011) indicate the existence of a fundamental and intrinsic limit to a fluid’s ability to mix, which in this particular configuration can be measured robustly between quiescent initial and final states. However, the final state vertical density profile observed in experiments in salty water has not been satisfactorily replicated with standard incompressible ILES numerical methods. These mis-matches occur because the numerical method cannot respect the same balance of energy conversions as observed in experiment, and we conjecture that such discrepancies occur because the numerical scheme operates with a Sc=O(1) whereas Sc=700 in our experimental conditions. Lawrie & Dalziel (2011) discussed in detail the relation between the probability density function (pdf) of the density field and the availability of energy in the system. In this paper we extract from our numerical simulations the evolution of the pdf over the life-cycle of the instability, and use it to motivate a simple sub-grid model that improves the ILES representation of energy conversions at high Schmidt number. We compare both two- and three-dimensional Rayleigh-Taylor cases. In three dimensions energy cascades to small scales, so the stretching of material surfaces that it induces tends to occur at comparable scales (particularly so in low Schmidt number simulations) and this is the optimal condition for doing mixing. In two dimensions, however, energy accumulates at large scales and thus material surfaces do not become so rapidly stretched at scales associated with diffusive processes. We view the two-dimensional case as an analogue for high Schmidt number behaviour, and this informs our choice of sub-grid scale model.

KW - two-dimensional Rayleigh-Taylor

KW - molecular mixing

KW - direct numerical simulation

M3 - Conference Abstract

T2 - 13th International Workshop on the Physics of Compressible Turbulent Mixing (2012)

Y2 - 16 July 2012 through 20 July 2012