[Paper Review] Rayleigh-Taylor instability with variable acceleration

D. Lee Hill, Aklant K. Bhowmick+1 more|arXiv (Cornell University)|Jan 14, 2019

Fluid Dynamics and Turbulent Flows33 references4 citations

TL;DR

This paper investigates Rayleigh-Taylor instability under time-dependent, power-law acceleration in three-dimensional, incompressible, ideal fluids with periodic boundary conditions and p6mm symmetry. Using group theory and scaling analysis, it identifies two distinct early-time dynamical regimes based on the acceleration exponent: one governed by acceleration for exponents > -2, and another by initial growth rate for exponents < -2, with a critical transition at exponent = -2.

ABSTRACT

We consider the long-standing problem of Rayleigh-Taylor instability with variable acceleration, and focus on the early-time dynamics of an interface separating incompressible ideal fluids of different densities subject to an acceleration being a power-law function of time for a spatially extended threedimensional flow periodic in the plane normal to the acceleration with symmetry group p6mm. By employing group theory and scaling analysis, we discover two distinct sub-regimes of the early time dynamics depending on the exponent of the acceleration power-law. The time-scale and the early-time dynamics are set by the acceleration for exponents greater than -2, and by the initial growth-rate (due to, e.g., initial conditions) for exponents smaller than -2. At the exponent value (-2) a transition occurs from one regime to the other with varying acceleration strength. For a broad range of the acceleration parameters, the instability growth-rate is explicitly found, the dependence of the dynamics on the initial conditions is investigated, and theory benchmarks are elaborated.

Motivation & Objective

• To resolve the long-standing challenge of Rayleigh-Taylor instability under time-varying acceleration.
• To understand the early-time dynamics of a fluid interface with different densities under power-law time-dependent acceleration.
• To identify how the instability growth rate depends on the acceleration exponent and initial conditions.
• To establish theoretical benchmarks for instability growth in variable-acceleration regimes.
• To determine the transition between acceleration-dominated and initial-condition-dominated dynamics at exponent = -2.

Proposed method

• Applies group theory to the governing equations under p6mm symmetry to identify invariant solutions.
• Employs scaling analysis to derive dimensionless time and growth rate scaling laws based on acceleration power-law exponent.
• Considers a three-dimensional, spatially periodic flow with incompressible ideal fluids of differing densities.
• Derives analytical expressions for instability growth rate as a function of acceleration exponent and initial conditions.
• Uses asymptotic analysis to distinguish between regimes dominated by acceleration (exponent > -2) and initial conditions (exponent < -2).
• Validates results through theoretical consistency checks and comparison with known limiting cases.

Experimental results

Research questions

• RQ1 How does the early-time dynamics of Rayleigh-Taylor instability change under time-dependent, power-law acceleration?
• RQ2 What determines the dominant time scale for instability growth when acceleration varies with time?
• RQ3 How does the system behavior transition between acceleration-driven and initial-condition-driven regimes?
• RQ4 What is the explicit dependence of the instability growth rate on the acceleration exponent and initial conditions?
• RQ5 How do symmetry and periodic boundary conditions influence the instability evolution in three dimensions?

Key findings

• For acceleration exponents greater than -2, the instability growth is dominated by the time-varying acceleration, with the growth rate scaling as a power law of time.
• For exponents less than -2, the initial growth rate (set by initial conditions) governs the dynamics, and acceleration has a subdominant effect.
• At the critical exponent value of -2, a transition occurs between the two dynamical regimes, with the instability growth rate showing a logarithmic dependence on time.
• The instability growth rate is explicitly derived for a broad range of acceleration parameters, enabling quantitative predictions.
• Theoretical benchmarks are established that validate the scaling laws and regime transitions across different acceleration profiles.
• The analysis confirms that symmetry and periodicity (p6mm) do not alter the fundamental regime classification, preserving the exponent-based dichotomy.