Session Z08: Flow Instability: Rayleigh-Taylor (12:15pm - 1:00pm CST)

Weak Nonlinear Analysis of the Rayleigh-Taylor Instability in Linear Viscoelastic Fluids

Pattern formation at the free surface of compliant media when subject to gravity is of relevance in the design of soft devices with tuneable shapes and in the dynamics of mucus films in pulmonary capillaries. These interfacial patterns owe their origin to an instability arising from the interplay between elasticity, viscosity, surface tension and gravity. In this work, we focus on the stability of a linear viscoelastic fluid layer attached to a rigid surface in the presence of gravity. The stability of this system is investigated by carrying out a linear stability and nonlinear analysis. From this analysis, we identify the critical parameter space for the onset of the instability. The weakly nonlinear analysis indicates that the elasticity of the soft-gel layer plays a key role in the supercritical to subcritical transition. To glean the physics of this transition, we focus on a soft-gel layer whose thickness in infinite. This infinite layer configuration reveals that the elastic component of normal force balance alters the nature of the bifurcation from subcritical rupture to supercritical saturation of the free surface. The subcritical rupture of the interface is attributed to the elastic shear stresses and surface tension.   

Scaling of vertical mixing in two-species buoyancy-driven instabilities

A miscible horizontal interface separating two solutions of different solutes can deform into convective finger-like structures due to buoyancy-driven instabilities like the classical Rayleigh-Taylor instability or the double-diffusive instabilities, triggered by differential diffusion of the solutes in the solutions. We analyse numerically for porous media flows the scaling of the fingers vertical speed, defined as the slope of the temporal evolution of the mixing length of the fingers. In the parameter space of the problem, spanned by the buoyancy ratio R, and the ratio $\delta$ of diffusion coefficients of the two species, the vertical speed is found to scale linearly with the adverse density difference that drives the convective mixing in these flows. The adverse density difference is the density jump across the spatial domain where the density gradient of the diffusive base-state is negative along the direction of gravity. It can be computed analytically from the diffusive base-state density profile and can be significantly different from the initial density difference when differential diffusion of the solutes are at play. Our results evidence the possibility of controlling the nonlinear evolution of mixing of buoyancy-driven instabilities in two-species stratifications   

Buoyancy--Shear--Drag--Scalar Turbulence Modeling for Rayleigh--Taylor Mixing

A buoyancy--shear--drag model [O. Schilling, \textit{Physica D} \textbf{402}, 132238 (2020)] is extended to include scalar variance to describe scalar (i.e., molecular) mixing in addition to mechanical mixing. The two coefficients in the scalar variance equation are calibrated to predict specific values of the scalar variance decay exponent and molecular mixing parameter for Rayleigh--Taylor mixing. An ordinary differential equation for the normalized scalar fluctuation $\Theta(t)=\phi^{\prime}/\overline{\phi}$ with terms representing production and destruction of scalar variance is coupled to the buoyancy--shear--drag equations. Analytic solutions of the resulting coupled equations for Rayleigh--Taylor mixing are obtained, which modify the classical expression $h(t)=\alpha\,At\,g\,t^2$ for the mixing layer width. Applications of the buoyancy--shear--drag--scalar model to Rayleigh--Taylor turbulent mixing are briefly described.   

Experimental investigation on elastic and plastic regimes in Rayleigh-Taylor instability with soft materials

Rayleigh Taylor instability (RTI) is a hydrodynamic instability that can also be observed in materials that have significant resistance to yield. The majority of the past studies have focused on estimating the instability threshold as it is critical to several high energy density applications. The elastic to plastic (EP) transition threshold is also of significance in those applications and has received limited attention in the scientific literature. We explore the EP threshold and the stable regimes of RTI under complex acceleration profiles using the rotating wheel RTI experiment. The test section with the soft-solid (mayonnaise) is placed on the wheel such that the free surface of the soft material is driven radially (outwards) by the centrifugal acceleration. The growth of the perturbation is recorded using a high-speed camera for the entire experiment. In the current study, we explore the initial perturbation geometry and the acceleration rate effects on the EP threshold. Additionally, stable EP regimes are investigated in detail by employing four different acceleration profiles. The perturbation amplitude response and the mass ejection data as a function of the instantaneous driving accelerations will be discussed. Finally, the results for the EP thresholds will be compared to the existing analytical formulations for RTI in solids.   

Rayleigh-Taylor Instability with Variable Durations of Acceleration Removal

In this study, we explore the effects of variable duration of acceleration removal on the Rayleigh Taylor Instability (RTI) by using implicit large-eddy simulations. For our test case, RTI undergoes a period of constant acceleration (A) followed by a period when the acceleration is removed (Z) followed by a period of second acceleration (A). We call this acceleration time-history as the AZA case. Acceleration removal leads to a rapid decay in kinetic energy within the RTI mixing layer, whereas the mixing state stays similar during the Z stage. The duration of the zero acceleration (Z) period is varied, and it did not influence the re-growth of RTI during the second acceleration stage. In this talk, we will discuss our findings and the similarities of RTI-AZA with the Richtmyer-Meshkov Instability (RMI). The mixing state evolution and the growth of the mixing layer were observed to be similar for both RMI and RTI during acceleration removal stage.   

Expansion-compression motions in the single-mode Rayleigh-Taylor instability.

In order to study the influence of compressibility on Rayleigh-Taylor (RT) instability, we used a high-order central compact finite difference scheme to numerically simulate the late-time evolution of two-dimensional single-mode compressible Rayleigh-Taylor instability for isothermal background stratification. The simulations were presented for different stratification strengths, corresponding to different isothermal Mach numbers ($M)$ at Atwood numbers ($A_{t} )$ 0.1 and 0.5. We studied the solenoidal component and compressible component of the velocity field employing the Helmholtz decomposition. At low Mach number, the expansion-compression motion is very weak and flow field is close to the incompressible state. For the case of $A_{t} \mbox{=}0.1$ and $M\mbox{=}0.5$, the rising light fluid expands and the falling heavy fluid is compressed, but the expansion and compression motions are weak. The expansion-compression motion at $A_{t} \mbox{=}0.5$ is significantly stronger than that at $A_{t} \mbox{=}0.1$ for the same Mach number $M\mbox{=}0.5$ in the mixing zone. The fluid outside the mixing zone also has stronger expansion and compression at $A_{t} \mbox{=}0.5$. The expansion motion inside the bubble can promote the development of the bubble.   

Rayleigh-Taylor instabilities in dry granular flows, a stability analysis

Rayleigh-Taylor instabilities also occur in dry granular flows (D'Ortona \& Thomas, PRL 2020). In an assembly of dense particles lying above light particles, an instability develops when the system is tilted and set into motion. More surprisingly, if the system is initially homogeneous and dense particles are larger than the light ones, granular segregation first induces the formation of an upper layer of large dense particles, which subsequently destabilizes. Both initial conditions eventually evolve into a pattern of rolls aligned with the mean motion, analogous to Rayleigh-Benard convection rolls. This movement is sustained thanks to granular segregation. The stability analysis of an initially sinusoidal perturbation of the interface between two layers of dense and light particles is performed by numerical simulations. As in fluids, the amplitude of the perturbation grows exponentially and the growth rate varies linearly with the Atwood number. As in systems confined between two horizontal walls, the most unstable wavelength ($l$) is proportional to the flow thickness ($H$), with $l=1.9\,H$. In the case of very thin systems (below $H=10$ particles), a transition occurs where the cross diffusion of particles prevents the RT instability.   

Confinement-induced stabilization of the Rayleigh-Taylor instability and transition to the unconfined limit

The Rayleigh-Taylor instability, which arises when a fluid sinks into and displaces a lighter one below it, is relevant in many situations: modeling type Ia supernovae in universal expansion, stabilizing ignition in inertial confinement fusion, and understanding the formation of salt fingers in the ocean. We prepare a density inversion between two miscible fluids in the thin gap between two plates, creating a clean initial stationary interface. Under these conditions, we find no Rayleigh-Taylor fingers are formed below a critical plate spacing. As we increase the plate separation, the system transitions from a stable regime where the diffusion of mass dominates the buoyant forces, through a regime where the gap sets the wavelength of the instability, and finally to the unconfined regime governed by the competition between buoyancy and momentum diffusion. We compare this miscible case to its immiscible counterpart, where diffusion is negligible and the effects of surface tension dominate. Our study, including experiment, simulation and linear-stability analysis, characterizes all regimes of confinement for the miscible and immiscible Rayleigh-Taylor instability.   

Variable Transport Property Effects on the Compressible Rayleigh-Taylor Instability

In the Rayleigh-Taylor instability, how the presence of large transport property differences, which for example may develop in response to heating, can affect the flow development remains an important open question. Using the PadeOps compressible flow solver, we explore an idealized problem of two compressible, RT unstable fluid layers beginning at different temperatures, specifically a hotter, lower density gas layer pushing against a colder, denser layer. Three transport property configurations are considered: simulations that use constant properties, ones that use variable temperature-dependent properties which obey a plasma-type power law and ones that begin with constant properties then transition to variable properties. Results are presented for simulations with temperature ratios up to 10 and Atwood numbers up to 0.7. Heat conduction due to the initial thermodynamic nonequilibrium and transport property variations caused by temperature increases delay instability development, influence the overall amount of molecular mixing and suppress turbulent behavior in the hotter fluid.   

Imaging Improvement of Miscible Experiments on the Rayleigh-Taylor Instability in the Linear Induction Motor Drop Tower

Incompressible miscible experiments on the Rayleigh-Taylor Instability (RTI) using Planar Laser Induced Fluorescence (PLIF) on a Linear Induction Motor Drop Tower are presented. The vertical tower guides a test sled that is accelerated using linear induction motors. Experimental fluid pairs are prepared and placed into a test chamber attached to the sled. The sled is accelerated downward at a rate of approximately 15g. Upon acceleration, the stratified initially stable fluid pair within the chamber becomes unstable allowing the RTI to develop. The resulting RTI is imaged using PLIF by seeding the heavier fluid with fluorescein dye that is illuminated by a scanning 445nm laser beam. The indices of refraction (IOR) for the two fluids are initially matched before experimental runs, however, mixing of the fluids causes variation of IOR. Variation of IOR within the fluids causes laser beam wander, negatively impacting PLIF imaging, and resulting in image blurriness. Modeling IOR as a nonlinear fluid property is performed. Variation of IOR is minimized using this model, allowing preparation of optimized fluid pairs that reduce image blurriness. Resulting images and information on RTI growth are presented.   

Turbulent/non-turbulent interface in flows affected by Rayleigh-Taylor instability

Simultaneous velocity-density measurements (PIV/LIF) are used to study turbulent/non-turbulent interface (TNTI) in flows affected by Rayleigh-Taylor Instability (RTI). Experiments are conducted in a gas tunnel facility with air as heavy fluid and helium$+$nitrogen mixture as light fluid giving Atwood number \textasciitilde 0.1. The nature of the TNTI on bubble front, as well as the change in mean and turbulent quantities across this TNTI are investigated. The molecular mixing is also studied relative to the TNTI. The TNTI shows a complex conditionally averaged volume fraction profile in its vicinity. In the external layer, the fluid is mostly pure heavy fluid, leading to no concentration gradients and a nearly zero measurement of the scalar dissipation. At the interface, there is a very large magnitude of scalar dissipation. In the adjustment layer, the scalar dissipation is nearly constant. These results challenge the conventional shape of profiles of turbulent quantities in RTI flows which are typically assumed parabolic. A better way to interpret the variation of turbulent quantities across the mixing width is to assume them to be nearly uniform in the core of the flow and be modulated by the location of the TNTI. If a stochastic model for the variation of the TNTI could be found which also described the value of these turbulence parameters in the core, the description of the RTI flow could be simplified to only be determined by values at a single point in the core and knowledge about the TNTI location.