Rayleigh-Taylor Instability: A Status Review of Experimental Designs and Measurement Diagnostics

Rayleigh-Taylor instability (RTI) [1,2] with large density differences occurs in high-speed applications such as supersonic aviation (ramjet engines) [3,4] and high-energy-density processes like inertial confinement fusion (ICF) [5–8], which is key to achieving energy from fusion. In the realm of Nature, astrophysical flows (on planetary and galactic scales) involve compressible variable density (VD) effects [9,10]. In geophysics, density changes due to variations in temperature and salinity are known to drive ocean flows [11]; changes in moisture (which affect the density) drive atmospheric flows [12,13]; and thermal abnormalities in the earth's crust lead to mantle plumes [14] which result in volcanic island formation [15]. Together with the shear-driven Kelvin-Helmholtz instability (KHI) [16,17] and the shock-driven Ritchmyer-Meshkov instability (RMI) [18,19], the acceleration-driven RTI [1,2] are some of the most fundamental but still open problems in fluid mechanics [20].

The practical significance of RTI has led to recent comprehensive reviews [21–23]. The focus of experiments employed in RTI induced mixing studies have evolved considerably due to improvements in diagnostic capabilities and simultaneous improvements in predictive modeling for such flows. Several experimental designs have been used to create an RTI unstable interface between two materials of different densities. The early experiments developed between 1950–1990 though innovative, were arduous to diagnose and provided little information on the internal, turbulent structure and initial conditions of the RT mixing layer. Coupled with the development of high-fidelity diagnostics, the experimental designs in the last three decades have allowed detailed measurements of various turbulence statistics matching the scope and capabilities of shear-driven mixing experiments. In addition, these studies have allowed broadly to validate and verify turbulence mix-models [24–32] for buoyancy-driven flows and provide insights to solve several long-standing problems in the field. The developments in modeling and simulations in RTI can be found in a companion review by Schilling [33], a review of the theoretical modeling techniques and issues related to variable-density flows with large thermal and density fluctuations can be found in [34]. No attempt is also made to cover the experiments related to the formation of RTI in the high-energy-density regime or RTI due to change in viscosity or chemical reactions; the reader can find those in other recent reviews [35,36]. Furthermore, RTI in solids is also not discussed and can be found in reviews by Zhou [21,22]. This review describes experiments and diagnostics that have been developed in the last 30 years (1990–2020) to provide valuable insights into the RTI driven mixing process. Besides, the scope of the current review is restricted to experiments that allow investigation of low to high Atwood number (>0.1) RT mixing; a summary of the small Atwood number experiments can be found elsewhere [37]. This review is organized as follows. In Sec. 2, a brief overview of the flow fundamentals is presented with short descriptions of the experiments and the measurement diagnostics that are covered in this review. In Sec. 3, an overview of the measurements and the current state-of-the-art is discussed. In Sec. 4, open questions are discussed to provoke thoughts for future experimental designs and facilities. Section 5 concludes the paper and discusses future opportunities.

where h₀ is the initial perturbation amplitude, $γ=Atgk$ the classical growth rate; t is the time; g is the acceleration, and the Atwood number [$At=(ρ1−ρ2)/(ρ1+ρ2)$] is the nondimensional density contrast ranging from 0 to 1. For example, in a scenario where air interpenetrates helium, the density ratio is seven and the A_t ∼ 0.75; for air–hydrogen combination, A_t ∼ 0.85. In contrast, the Boussinesq approximation corresponds to A_t → 0, and the value of ∼ 0.05–0.1 is usually taken to define this limit. Since the validity of the linear theory is limited to the early stages of the instability, it is the nonlinear behavior at later times that holds the most interest from the perspectives of both the basic science and the technological applications. When the flow transitions to nonlinearity (h ∼ 1/k), the amplitude increases linearly with time as the growth slows down. In the nonlinear regime, bubbles of light fluid rise through the heavier fluid and attain a terminal velocity $∝ λ$ for a single-mode initial perturbation (see Fig. 1(a)); and corresponding spikes of heavier fluid fall through the lighter fluid, and later on, into mushroom-like shapes [40] (see Fig. 1(b)).

where, a_b,s is the growth constant and h_b,s is the bubble/spike heights. The quadratic growth is a dimensionally consistent result confirmed by experiments and simulations [44]. Although the second-order term dominates the growth at late-time, the onset of self-similarity of the mixing layer growth occurs much earlier [45]. Equation (2) can be viewed as the late-time approximation of a more general second-order polynomial defined by Ristorcelli and Clark [45] based on a self-similar analysis for small-Atwood RT mixing; a more detailed discussion is provided later in this paper (Sec. 4.1). Equation (2) also implies that the flow has lost memory of the initial conditions (ICs), and the only relevant length scale Agt² is the self-similar scale. However, experiments [37,46–50] report values of α_b = 0.04–0.07, whereas simulations [41,43,45,51–53] have reported lower values of α_b = 0.02–0.03 [51]. This discrepancy is believed to be due to the presence of long wavelengths in the experiments that are not captured in those earlier numerical simulations [41,43,54]. When initialized with long wavelengths, numerical simulations give a higher α_b value and exhibit sensitivity to ICs [41,43,55] (see Fig. 2). Equation (2) can be interpreted as the quadratic envelope of growth curves of all individual modes. By careful selection of the initial amplitudes of these modes, the slope of this envelope (α) can be changed [42] as the instability may saturate earlier (or later). The disturbances grow exponentially up to saturation, so the time to nonlinearity depends logarithmically on the amplitudes and the spectrum of mode(s) seeded in the ICs. Thus, the ability to control and measure the ICs seeded into an RT mixing layer is central to accurately understand the late time mixing dynamics and for the development and validation of predictive models.

It is a challenging task to create an unstable stratification of a heavy fluid over a light one under the action of gravity to study RTI induced mixing. Experimental configurations in RTI can be broadly classified under three categories based on mechanisms to initiate the instability (see Fig. 3). These include approaches to (a) accelerate the interface/tank with a stably stratified (i.e., light fluid over heavy fluid) configuration; (b) create an unstable stratification (i.e., heavy fluid over light fluid) by using a barrier or magnetorheological fluids in a stationary tank; and (c) the flow channel approach with two coflowing unstably stratified streams separated by a splitter plate. The experiments developed under each of these categories are briefly described next. Table 1 provides an overview of the range of various parameters explored using these three configurations.

The experiments in this category can be further subdivided into two groups. The first group uses rarefaction waves to accelerate a stably stratified interface. The second group uses different mechanisms to accelerate the container/tank that holds the stably stratified fluid combination. Initial conditions were generated by shaking the setup using various mechanisms.

The first group consists of experiments where the interface is accelerated using rarefaction waves. Lewis [56] was the first to perform such experiments, probably instigated by G.I. Taylor to verify and validate the analytical theory [2]. The experimental apparatus consisted of a rectangular tube connected to a large air reservoir (Fig. 4(a)). A flange holding a thin diaphragm was placed at the center of the rectangular tube. Supported on the diaphragm was water, whose height varied from 0.01 to 0.508 m. The top and bottom reservoirs were isolated, and air pressures were adjusted separately. When the foil was ruptured, a rarefaction wave is created, and the unbalanced pressure led to the creation of RTI at the interface that drove the liquid down the tube. The pressure in the reservoir and the tube could be varied to obtain a range of accelerations from 3 g to 140 g with run times of ∼ 10⁻²s. High-speed shadow photography was used to diagnose the flow (see Fig. 4(b)). Lewis [56] used immiscible fluid combinations like air-benzene and air–water to study the instability; the effects of surface tension and viscosity on the RTI growth rate were negligible due to the high acceleration used in the experiment. Allred and Blount [57] used a similar apparatus as Lewis [56] and chose a combination of water and n-heptane/iso-amyl alcohol/n-octyl alcohol as the immiscible working fluids to obtain a large range of Atwood numbers (0.1–0.99) as the interfacial tension between the two fluids was lowered by a factor of 20 using surfactants. An experimental facility that uses rarefaction waves to drive RTI has been developed recently at the U. Arizona (see Fig. 4(b)) [58,59] (henceforth referred to as RW). Similar in principle to the experimental design of Lewis, a diaphragm separates a vacuum tank beneath the test section; the diaphragm is ruptured to generate a rarefaction wave that travels outward and accelerates the interface downwards (see Fig. 5(a)). Well-controlled, 2D and 3D single-mode diffuse perturbations were generated by oscillating the gases either side to side for 2D, or vertically for 3D-perturbations. Several miscible gas combinations, including CO₂/SF₆, air/SF₆, He/CO₂, and He/SF_6, were used to generate a range of Atwood numbers between 0.4 and 0.94. The diffuse initial interface thicknesses were measured using Rayleigh scattering and the instability was visualized using planar laser-induced Mie scattering. An image sequence from a 2D CO₂/SF₆ experiment up to t = 7.2 ms is shown in Fig. 5(b). It can be observed that small scales are produced due to secondary instabilities, and the initial symmetry of the perturbations are not retained at a late time when the flow is nonlinear.

Rayleigh-Taylor instability has also been studied by accelerating an initially stable stratified layer of two fluids by using mechanisms like rubber cords [60], bungee/elastic cords [61], and compressed air [62]. Emmons et al. [60] was the first to use the method of accelerating a container to create RTI. An approximately single-mode perturbation was generated at the density interface by oscillating a paddle, and the growth of individual bubbles and spikes was measured using high-speed photography. These early experiments [60–62] mostly aimed to excite a particular eigenmode [63] and accelerate the stratified mixture when the ICs had a prescribed amplitude and phase; the experiments, unfortunately, were not easily repeatable. In addition, the experiments of Emmons et al. [60] show severe edge effects due to surface tension. Popil and Curzon [63] used an electrostatic generator to accurately generate single-mode standing waves on the water interface and increase repeatability. By controlling the number of electrodes, multimode excitations were also induced at the interface. Kucherenko and colleagues at VNIITF (Russian Federal Nuclear Center) [64,65] used a gas gun to accelerate a tank and achieved accelerations between 100–350 g, an aqueous solution of glycerin and benzene were used in those experiments to obtain A_t numbers in the range 0.23–0.5. Similar techniques using a gas gun has also been used by Nevmerzhitskiy and colleagues at VNIIEF (Russian Scientific Research Institute for Experimental Physics) [66,67]. Andrews and Spalding [68] created an unstable stratification by quickly inverting a box of the stably stratified mixture. However, the experiments were restricted to two-dimensions to prevent slopping during the overturn, and the mixing layer was susceptible to vibrations due to the catching mechanism used [37]. A recent A major drawback of these early (before 1980) experiments was that they were two-dimensional as the instability was studied in long-narrow cavities.

To examine three-dimensional turbulent mixing layers, Read [69], and Youngs [70] used rockets to accelerate an initially stable stratified mixture downwards in a facility that is known as the rocket rig (henceforth referred to as RR) (see Fig. 6(a)). Accelerations of 25–75 g were obtained in the RR; a variety of miscible and immiscible fluid combinations was used to vary the Atwood number between 0.23–0.997. Figure 6b shows the mix-evolution of compressed SF₆ and Pentane (A_t = 0.789); the bubble spike asymmetry is visible at t = 79 ms. The RR has also been used for exploring complex stratifications where a tilt in the test section gave rise to an angled two-dimensional interface [69,71]; the experiment has been used as a test-case for validating models and comparing different computational techniques [70,72]. Jacobs and Catton [73] accelerated a small volume of water down a vertical tube (round and square) using compressed air and used high-speed motion-picture photography for 3D RTI studies with acceleration varying between 5–10 g. Waddell et al. [74] accelerated a drop-tank using weights and used planar laser-induced fluorescence (PLIF) to obtained measures of mixing layer width of a single-mode, three-dimensional RTI. Dimonte and Schneider [49] studied a wide range of A_t (0.1304–0.961) using a linear electric motor (henceforth referred to as the LEM) to accelerate the tank (see Fig. 6(c)). In these multimode cases, high-speed imaging techniques were used to measure the spatial growth of the turbulent mixing layer (see Fig. 6(d)). The facility also allowed investigation of complex acceleration histories, both impulsive acceleration (which represents the RM limit) and time-varying acceleration, which represents the accel-decel-accel (ADA) time history have been investigated [75,76]. These experiments though innovative, were arduous to diagnose and provided little information on the internal, turbulent structure (statistics) of the RT mixing layer. In addition, surface tension between the immiscible fluids caused unwanted meniscus perturbation in this otherwise unperturbed RR and LEM experiments.

To avoid difficulties associated with the imaging of a flow in an accelerating tank in the early 1980s, various researchers have designed RTI experiments that were stationary in space. In these experiments, an unstable stratification was created by separating two fluids using a plate or using magnetic fluids to create the initially unstable stratification. These techniques eliminate the complications from artificial acceleration; however, they do give rise to other issues regarding control and precise knowledge of the ICs.

The sliding barrier approach was developed in the early 1960s at Los Alamos National Laboratory by Duff et al. [77] to study the instability; an unstable interface was created by withdrawing a plate that separated a heavier fluid (argon-bromine mixture) situated above a lighter fluid (air or helium) in a tank. The experiments were conducted primarily to study the effect of diffusion on the early time growth of the instability. A similar setup was later used at the University of Cambridge (see Fig. 7(a)). These first-generation experiments used a rigid sheet to separate the two fluids and had the disadvantage of viscous boundary layers forming on their upper and lower surfaces as the barrier was withdrawn [78]. The wake left behind by the barrier due to these shear layers introduced a long-wave disturbance to the ICs, and pair of strong vortices were also formed as the boundary layers were stripped by the tank end-walls. This deficiency was addressed by using a hollow barrier and two pieces of nylon fabric, which were stretched on both sides of the barrier (see Fig. 7(b)). The nylon fabric remained motionless as the barrier was withdrawn, leading to the elimination of the viscous boundary layers Ref. [48]. The low Atwood number combinations (A_t < 0.01) allowed for refractive index matching and advanced optical diagnostics to be employed that include particle imaging velocimetry (PIV) to measure velocity statistics and planar laser-induced fluorescence (PLIF) to measure the degree of molecular mixing. Figure 7(c) shows density isosurface during RTI evolution at A_t = 7 × 10⁻⁴ at t = 3.5 and 12.5 s [79], the visualization was possible by using a cocktail of semi-opaque dyes. The experiment has also been used for studying RTI unstable interface sitting between stably stratified layers with linear density profiles [80], an RTI in a system of three fluids [81] in which the stratification consists of one stable and one unstable interface. Similar experimental configurations have been developed elsewhere to study passive and reactive scalar measurements in a high Schmidt number (∼1000) mixing layer [82] and elastoplastic fluids [83]. Molecular mixing estimates were determined from a passive scalar (Nigrosine) and reactive scalar (Phenolphthalein) measurements; snapshots of the passive scalar measurements are shown in Fig. 7(d) [82,84].

Another approach that uses a stationary tank is using the technique of magnetic levitation to hold a heavier paramagnetic mixture on top of a light fluid. The technique was first proposed by Huang et al. [85]; the authors used precisely shaped wire of magnetically permeable materials fixed outside the fluid cell to perturb the magnetic force and provide the ICs for the instability (see Fig. 8(a)). When the magnetic field was switched off, the total magnetic force disappeared, and gravity-driven RTI was formed. The fluid combinations used included a moderately strong paramagnetic mixture of water and MnCl₂ • 4H₂O with a surfactant to reduce surface tension as the heavy fluid (ρ₁ = 1.394 g/cm³). The lighter fluid was diamagnetic hexadecane (ρ₂ = 0.773 g/cm³) giving an A_t = 0.29. PLIF images of spike growth at late times obtained after switching off the magnetic field is shown in Fig. 8(b) with growth rates reported as α_s = 0.06–0.07. Pacitto et al. [86] developed another experiment where the unstable stratification was setup) inside a vertical Hele-Shaw cell using a combination of cobalt ferrite particles (MF) in a mixture of water and glycerol (ρ₁ = 1.686±0.085 g/cm³) and low-density oil/white spirit (ρ₂ = 0.8 g/cm³). An external magnetic field (H_ext) was applied perpendicular to gravity (see Fig. 8(c)). The cell is then rotated around a horizontal axis so that H_ext is both horizontal and perpendicular to the cell, creating a dynamic situation such that the magnetic force and gravitational forces destabilize the interface. In contrast, the capillary forces help to stabilize it. Viscous fingering patterns of the destabilizing interface between MF and oil for H_ext = 7.9 kA/m is shown in Fig. 8(d). For H_ext = 40 kA/m that was much larger than the threshold magnetic field, the tip of the rising bubbles split into two-fingers, a phenomenon that was described as “tip-splitting” when the new fingers being at angles of roughly 90 deg (see Fig. 8(e)). This phenomenon is similar to the radial viscous fingering that is observed in Saffman-Taylor instability [87]. White et al. [88] used magnetorheological (MR) fluids (ρ₁ = 2.735 g/cm³) for better control of ICs [88]. An aluminum plunger with the desired ICs was inserted into the test section (see Fig. 8(f) for a schematic of the facility). Water was poured over the plunger, which was then removed after the water was frozen. The MR fluid is then poured over ice (in the inverse shape of the ICs) and held in place by a magnetic field. The ice is then melted, and the magnetic field is switched off to generate RTI instability. Images of the instability evolution that use MR and water (A_t = 0.46) and MR and air (A_t = 1) is shown in Fig. 8(g) and 8(f). Measurements of the late time RT spike reveal poor agreement with analytical models [89,90].

The experiments in the first two categories though innovative, were arduous to diagnose and provided limited information on the internal, turbulent structure and initial conditions of the RT mixing layer. Most experiments were restricted to short data capture times, thereby requiring a large number of repeated experiments to obtain statistically significant data sets. The last three decades have also witnessed advances in the modeling of variable density turbulence [24,55,91–94]. Validation of these predictive models consisting of inhomogeneous, anisotropic, and variable-density flows requires a priori knowledge of various turbulent statistics that are used in those models. Thus, starting the early 2000s, there was a push in using diagnostics that would allow experimentally to determine the internal structure of the RTI mixing layer and allow measurement of turbulent statistics for model validation and verification. Although significant progress has been made in this regard, detailed measurements inside an RTI driven mixing layer still lacks the sophistication of their shear-driven or boundary layer counterparts.

Motivated by shear-flow experiments of Brown and Roshko [95], a flow channel type setup for generating an RT unstable interface was pioneered by Snider and Andrews [96]; the experiment consisted of two parallel coflowing streams (cold and hot water) initially separated by a thin plate (see Fig. 9(a), henceforth referred to as the WC). Two 500-gallon tanks fed water to the facility; the cold-water stream was fed on top-stream, and warm water was fed into the lower-stream. The cold and warm water streams entered the channel at a mean velocity of ∼4.4 cm/s, thereby eliminating any shear (KHI) roll-ups. For the flow channel configuration, the distance downstream (x) is converted to time, t = x/U, using Taylor's hypothesis [97,98]. The flow-channel configuration allows for long data-collection times and is significantly easier to diagnose than the transient experiments. The density difference was achieved through a temperature difference of 5–10 °C in water. Temperature data were collected through a thermocouple system and converted to density through an equation of state [99]. After termination of the splitter plate, the resulting flow configuration was RT unstable, and a mixing layer develops downstream. Various diagnostics have been developed over the last decade which includes: (a) optical techniques to measure the volume fraction profiles and the growth of the mixing layer (see Fig. 9(b)); (b) high-resolution thermocouples and PIV/PLIF [100] (PIV) to investigate the velocity and density fluctuations within the mixing layer and the degree of molecular mixing within the layer (see Fig. 9(c)) [47,48,101,102]. Besides, detailed turbulence statistics, which include measurements of density and velocity variance (energy) spectra [47,102], Reynolds stress tensor at various downstream locations; and statistics of a high Schmidt number reacting mixing layer [103] has also been measured. The facility was also used to study the combined effects of buoyancy and shear at small Atwood numbers [101,104] and a buoyant plane wake [105]. However, due to the usage of hot- and cold-water as the two fluids, the WC facility is limited to a maximum Atwood number of ∼10⁻³

To address the Atwood number limitation of WC, a gas channel facility that uses air and helium as the two streams was subsequently developed [46,106–108] at Texas A&M (henceforth referred to as GC1). Two gas streams, one containing air and the other a helium–air mixture, flow parallel to each other separated by a thin splitter plate. The streams meet at the termination of the splitter plate leading to the formation of RTI induced mixing (see Fig. 10(a)). Like the WC experiment, the gas channel allowed for long data collection times and was statistically steady. The facility design allows for large Atwood number studies (up to A_t ∼ 0.75). The flow was diagnosed using backlight imaging techniques and hot-wire anemometry. Two hot-wire anemometry techniques were developed to measure a variety of turbulence statistics in the air-helium RTI mixing layer. The first diagnostic used a multiposition, multi-overheat (MPMO) single-wire technique. A single-wire response function was estimated to variations in density, velocity, and orientation; the method provides time-averaged statistics but was restricted to moderate A_t < 0.25. The second diagnostic works on the concept of temperature as a fluid marker. It uses a four-wire configuration; a simultaneous three-wire/cold-wire anemometry technique (S3WCA) was developed to measure instantaneous statistics. Both methods were validated in a low Atwood number (A_t ≤ 0.04) by comparing various statistics with PIV measurements in the water channel facility. Good agreement was found for the measured growth parameters, velocity fluctuation anisotropy, probability density function (PDF) and molecular mix parameters. The techniques were then used to measure density and velocity statistics using both regular and conditional averaging methods in a large Atwood number RT mixing layers (A_t ∼ 0.6) [46,108–110]. Asymmetries in the mixing layer were observed at the high Atwood number experiments (A_t > 0.45), Fig. 10(b) shows a snapshot of an A_t ∼ 0.6 experiment. A similar coflowing system with improved features and the ability to explore Reynolds numbers beyond the mixing transition threshold [20,111,112] was subsequently built at Georgia Tech to study RTI mixing at high Atwood numbers A_t > 0.5 (see Fig. 10(c), henceforth referred to as GC2). The GC2 facility has been designed with the option to perform multilayer experiments. Visualization and Mie-scattering techniques were implemented to obtain volume fraction profiles and mix-widths. Asymmetry was evident in the flow field, with the spike side of the mixing layer growing 50% faster than the bubble side [50]. PIV was implemented for the first time in these high-Atwood-number experiments to obtain velocity and density statistics across the mixing layer. Besides, the dynamics of the coupled shear (KHI) and RTI has also been explored in recent experiments [113]. The flow was governed by KHI dynamics at early times and RTI dynamics at late times with the transition from KHI to RTI dynamics occurring at Richardson number [Ri $=−2ghAt/(ΔU)2$] range of 0.17–0.56.

where $f1¯$ is the plane averaged volume fraction of the heavy fluid, and the integral was taken over a domain (H) that includes the mixing region. The factor of six derives from considering a linear volume fraction distribution across the mixing region. As observed in Fig. 11(a), the mix profiles are linear except at the edges where they are slightly rounded. Thus, $WbandWs$ are reasonable measures of bubbles and spikes at low to moderate Atwood numbers (A_t ≤ 0.5). However, interchanging mix widths by integral mix widths should be done carefully at high Atwood numbers (A_t > 0.5) as the volume fraction profiles at the edges are rounded. Using a linear approximation (as in Eq. (3)) leads to similar measures for $WbandWs$ with $Wb/Ws∼1$⁠, whereas $hbandhs$ based on 5% and 95% edge detection gives $hb/hs<1$ due to asymmetry between the spike and bubble interpenetration, which occurs at large Atwood numbers (see Fig. 11(b)).

On differentiating Eq. (5) and equating $h˙b$⁠, at late time, the third term in the quadratic expression dominates, and one can conclude that $αCL≈αb,RC$⁠. Thus based on this reasoning, one may conclude that although the above definitions may give different values of the growth parameter at early times, once the flow becomes self-similar, they all give similar and closely related values of the growth-constants. Banerjee et al. [46] tested the viability of this assumption based on measurements for GC1 at A_t = 0.04. At late-time, when the flow has attained self-similarity, the different definitions of α satisfyingly converge to $αb,VO$ = $αb,RC$ = $αCL$= 0.066 ± 0.002 [46]. As a note of caution, this conclusion is appropriate for low Atwood RTI when α_b ∼ α_s. However, for large A_t$≥$ 0.1, the results are generally valid only for the bubble side as the mixing layer becomes asymmetric. Although the role of ICs on the late-time value of α_b and α_s is relatively well understood; ICs in the experiments are relatively difficult to measure. Detailed characterization of ICs are available for a few experiments, notably the SB (Cambridge experiment) [48], WC [102,114], and GC1 [115]. Other studies have estimated the long-wavelength perturbations based on the early time photographs from the LEM [51] and the RR [116].

The exponent Dα was found as 0.33 ± 0.05 for the GC2 experiments. A comparison of the growth-constants measured using the different experiments shown in Fig. 12(a). To keep the comparison simple, data from the LEM, GC1, and GC2 are compared. The value of α_b was measured as 0.064 ± 0.005 for the in the range 0.04 $≤$A_t$≤$ 0.6 in GC1 [46]. In the same experiment, α_s was found to increase with A_t number from a value of 0.065 at A_t = 0.04 to a value 0.088 at A_t = 0.6. At a slightly larger A_t = 0.73 runs in GC2, the reported values of α_b were 0.040, while α_s was estimated as 0.077 using the RC method in Eq. (6) [50]. These values of α_b and α_s obtained in GC1 and GC2 (plotted in Fig. 12) are comparable with the LEM experiments [49] where an average value for α_b of 0.053 ± 0.006 was reported when A_t < 0.5, and value for α_b of 0.0496 ± 0.003 was found for A_t > 0.8. Considering both immiscible data from LEM and miscible date from GC1 and GC2, the exponent Dα is re-evaluated and decreases slightly to a value of 0.3 over the range of Atwood numbers tested in those three experiments (see Fig. 12(b)). The GC1, GC2, and LEM experiments show an increase of α_s with an increase in Atwood number; however, the measurements of α_b and α_s are slightly different for the entire Atwood number range, with three possible explanations: (a) the ICs for the GC1, GC2 and LEM are not the same, as a result, a small (∼20%) variation of alpha with ICs is expected; (b) the difference may be due to the use of surfactants in the immiscible experiments in LEM which might yield different growth parameters [49]; and (c) the difference is due to lack of statistical convergence as not enough structures are averaged for estimating the growth rate parameters [107] in the high-speed transient (LEM) experiments. It should be noted that the value of Dα reported here is based on α_s/α_b is larger than earlier estimates based on direct measurements of h_s/h_b in the RR experiment (which gives a value of Dα = 0.231) or recently published numerical simulations where a value of Dα 0.18 is reported for three-dimensional numerical simulations [117]. A much higher asymmetry at high Atwood numbers between the bubbles and spikes are reported in experimental measurements in LEM, GC1, and GC2 (see volume fraction profiles and mix widths in Fig. 11). This can be attributed to uncertainties with image analysis techniques or with the experimental configurations; both GC1 and GC2 use Taylor Hypothesis whose validity needs closer scrutiny at large Atwood numbers.

An asymptotic value of the global mixing parameter may be reached at a Re_h ∼ 8000–10000 indicating self-similarity if the mixing mechanism of mixing is similar to that of shear layers at the asymptotic limit. A comparison between the WC and the SB experiment (see Fig. 13(a)) demonstrate a delay in mixing transition for buoyancy-driven flows, even though the experiments of Mueschke et al. [103] were closer to the self-similar state. This implies that Re_h for mixing transition needs to be > 2 × 10⁴ or higher and presents a significant challenge to DNS studies of high-Sc number RTI as computational cost scales ∼ Re³Sc² [112].

where Fr is the bubble/spike Froude number and λ is the wavelength of the single model interfacial perturbation. Numerical simulations have shown that Eq. (15) holds for bubbles for all Atwood numbers [124] with Fr_b,s approaching $(3π)−12$ for two-dimensions and $(π)−12∼0.56$ for three-dimensional RTI. The models also give prescription of spike behavior which instead of approaching terminal velocity approaches a free fall with α_s ∼ 0.5. The early experiments [56,60] were mostly restricted to the linear stage of the instability; a few accelerated tank experiments in the 1970s [61,63] studied RTI into the nonlinear regime. Following developments in the late time models in the early 2000s [89,121], Wilkinson and Jacobs [125] used the drop-tank experiment to investigate the nonlinear regime of low A_t (=0.15) miscible fluids and compared their experimental results to the nonlinear models. The late-time interfacial velocity, represented in a nondimensional form as the Froude number, showed an initial increase up to Fr ∼ 0.62 at h/λ ∼ 0.5 before it decreases slightly to Fr ∼ 0.56 before increasing significantly to Fr > 0.8 at late time (see Fig. 14(a)). This behavior was in contrast to the generally accepted models [89,121] which assume a terminal velocity with Fr = $(π)−12∼0.56$⁠. The deviation was attributed to secondary instabilities and vortex roll-up which was not accounted for in the models. In contradiction to potential flow theory, single-mode RTI exhibits late-time acceleration as the bubble penetration exceeds the bubble diameter, i.e., h_b/λ > 1. Ramaprabhu et al. [123] explored the limits of the potential flow theory computationally using a long-aspect ratio (λ × λ × 8λ) domain. Froude numbers > 0.8 were reported over a large range of A_t (0.1–0.9) for h_b/λ > 1 and was attributed to the formation of KHI vortices on the bubble-spike interface, which propelled the bubbles forward. Mikaelian [122,126] proposed new scaling of h_b and h_s in the linear and nonlinear regimes to derive a model based on the limitations of the potential flow models. This new model captures the transition of bubbles as it overshoots the region of constant Fr number. Recent experiments by Morgan et al. [58] using the RW facility at U. Arizona over a large range of A_t (0.49–0.94) have provided some insights into the late-time bubble/spike behavior. The facility uses a square test-section of with 8.89 cm with a length of 80.6 cm which gives an aspect ratio of ∼ λ × λ × 9λ and allows the study of later time three-dimensional single model RTI up to h_b/λ ∼ 2 . The experiments were compared with Mikaelian's models [122,126] and with large eddy simulations (LES) [111,127]. At moderate A_t ≤ 0.63, the bubble and spikes show KHI roll-ups, and the bubble/spike velocities overshot the Fr ∼ 0.56 value similar to earlier low A_t (= 0.15) experiments [125]. However, the late-time bubble velocities matched well with Mikaelian's models [122,126], as can be seen in Fig. 14(b). The symmetry of the bubbles in the experiments was broken, probably due to the formation of boundary layers on the test section. This feature is not captured by the computations [40] or the models. Froude number trends for large A_t (=0.94) two-dimensional RTI experiments using He/SF6 show a delayed roll-up and a free-fall behavior; however, both bubbles and spikes do not approach a constant Froude number (see Figs. 14(c)–14(e)). Most importantly, the Reynolds number achieved (3 × 10⁵) well exceeds the mixing transition Reynolds number of 10⁴ [111,128]. The authors [58] thus concluded that a mixing transition Reynolds number based on self-similar arguments may not be ideal to accurately describe the transition of individual modes.

Detailed measurements of various turbulence statistics have been made in the last decade using the Gas Channel experiments (GC1 and GC2). Banerjee et al. [46] reported the experimental measurements over a wide range of Atwood numbers (0.03 ≤ A_t ≤ 0.06) by using the GC1 facility [107] and a variety of hot-wire anemometry techniques [109,110]. Data collected included density profiles (see Fig. 11(a)), growth rate parameters (see Fig. 12), various turbulence statistics, density, and velocity PDFs, as well as spectra of density, velocity, and mass flux. The spectral measurements were the first reported measurements at A_t ∼ 0.6. More importantly, the measurements (α_b, θ, velocity and mass-flux) at the mixing layer centerline at A_t ∼ 0.6 showed a strong similarity to the low A_t (=0.03) measurements when properly normalized using the methods for small A_t (=7.5 × 10⁻⁴) experiments performed in the WC facility [47,54,119] (see Fig. 15(a)). Hot-wire anemometry techniques were developed [110] to measure high-frequency instantaneous information for the evaluation of PDFs of velocity and density fluctuations (see Fig. 15(b) and 15(c)) as well as mass-flux at the centerline of the RTI mixing layer. Skewness and Kurtosis were measured at the centerline, and the Kurtosis of the cross-stream velocity fluctuations were found to be 40% higher than the vertical velocity fluctuations indicating significant anisotropy inside the mixing layer. Measurements of different components of the anisotropy tensor (B_ij) [129] at the centerline and across the mix-width was also made with the turbulent transport (indicated by b_vv) in the vertical direction. The density fluctuation PDF shown in Fig. 15(c), shows a bi-modal distribution as seen at the low A_t = 0.03 measurements. Negative values of turbulent mass fluxes in the vertical direction were observed as bubbles of the light fluid rise and spikes of mostly heavy fluid fall through the mixing layer; the negative correlation of vertical velocity and density fluctuations results in a broad tail of the PDF toward negative values of the mass flux (see Fig. 15(d)). The negative tail of the PDF extended to larger nondimensional values of the turbulent mass flux due to asymmetries between the bubbles and spikes at large A_t flows. The measured energy density spectra of $v′$⁠, $ρ′$ and $ρ′v′$ (see Fig. 15(e)) limited by the sampling frequency (83 Hz) and a spatial resolution of ∼3.5 cm, which was done to ensure that both the hot- and cold-wire probes measured the same fluid inside the RT mixing layer. The energy spectra did not exhibit an inertial range as the measurements were made at early time (τ = 0.76), where the turbulence was not fully mature. The measurements were also used to evaluate several turbulence-model constants [24].

The Reynolds number and measurement limitations were addressed in the GC2 facility developed at Georgia Tech [50,113] to explore Reynolds numbers beyond the mixing transition threshold [20,111,112] and study RTI mixing at high Atwood numbers A_t > 0.5. Besides, to hot-wire techniques, PIV was used in such flows for the first time to provide details beyond the mixing layer centerline at A_t ∼ 0.73 [50]. Figure 16 shows results from PIV measurements of vertical velocity across the mixing layer at four downstream locations. The velocity was nondimensionalized using terminal bubble-saturation velocity, which has been used for low Atwood number flows [47] and was modified for higher A_t numbers as (⁠$v∞≈0.54Atghx$⁠, where h_x = (h_b + h_s)/2 at the downstream location) in the GC2 experiments. The velocity profiles were observed to be self-similar with a Gaussian-like shape and centered along the centerline of the mixing layer. PDFs of the velocity fluctuation, density fluctuation, and velocity-density (mass flux) fluctuations were evaluated across the mix-width (see Figs. 16(b)–16(d)). Although both u′ and v′ PDFs showed a Gaussian-like shape for the Re = 8100 experiments, at Re = 17,380, the v′ PDF showed a plateau-like distribution showing equal contribution from different v′ scales (see Fig. 16(b)). Density PDF observed at three different cross-stream locations show bubble- and spike-domination at locations away from the centerline where the PDF shows two small peaks similar to low-Atwood number experiments in GC1 (see Fig. 16(c)). The distribution of the mixed fluid is relatively flat in the middle as the probability of finding a fluid pocket with any volume fraction is the same. The mass-flux PDF distributions at two downstream locations (see Fig. 16(d)) showed good collapse with small-Atwood-number experiments in GC1 when the same normalization parameter ΔρA_t gt was used. However, the turbulent mass flux when normalized with the same parameter results in values that are a little higher than those reported in the GC1 experiments [46]. This led the authors to conclude that either a different scaling parameter might be required for higher-At experiments, or an asymptotic value of mass flux has not been reached. This indicates that different parameters may reach self-similarity at different Reynolds numbers; this has been further validated by recent computations of a triply periodic homogeneous variable-density mixing layer using direct numerical simulations [130–132]. Spectra of ρ′, v′ and ρ′ v′ were measured using hot-wire anemometry, both ρ′ and v′ show approximately half to a decade of scales in the inertial range, the ρ′ v′ spectra did not show a clear indication of the—5/3 slope and were similar to the spectra at smaller Atwood numbers in GC1 [46]. In addition to these measurements, both GC1 and GC2 facilities have been used to measure energy budgets and conditional statistics (based on ρ′ and v′) in the RTI mixing layer. Those measurements are not discussed here for brevity; the reader is referred to Refs. [46] and [50] for additional details.

The last three decades have seen developments of RTI experiments over a large range of Atwood numbers. The development of the experiments and the diagnostics have gone hand in hand. In their small Atwood number review, Andrews and Dalziel [37] noted a lack of data sets at A_t > 0.1, especially with the lack of miscible high A_t experiments. Some of those concerns have been addressed by the development of GC1, GC2, and RW facilities, which has allowed investigation at A_t > 0.5. However, several open questions remain. Initial conditions in experiments need to be well-controlled and properly diagnosed as it is critical for better comparison with simulations and models. The mixing transition threshold has proved elusive with recent experiments at Re_h > 10,000, illustrating that further studies at higher Reynolds numbers are required to address a limit/threshold beyond which RTI mixing would be independent of the flow Reynolds number. Limited attention has been given to experimental RTI studies that have coupled physics that may affect instability growth. These include (a) the coupled shear-buoyancy problem where RTI and KHI are competing [101], [113], and [133]; (b) the tilted rig RTI problem where the tilt gives rise to an angled (two-dimensional) interface concerning an acceleration history due to rockets (as in RR) attached at the top of the tank [69,71]; (c) suppression of RTI due to rotational effects [134,135]; (d) RTI with variable acceleration history [49,75,76,136–138]; (e) RTI in solids where the material strength needs to be overcome before the growth of the instability [139–141]; and (f) compressible RTI in cylindrical geometry [142]. RTI experiments are extremely difficult to build, operate, run, and diagnose; as a result, the number of experimental studies pales in comparison to computational and modeling efforts. The current thrust in several research groups across the world aims at solving some of the topics mentioned above and are showing considerable promise.

The author would also like to dedicate this review to the memory of Professor Malcolm J. Andrews, who was his Ph.D. and postdoctoral advisor. The author with Malcolm put the outline of this review together in 2009, and a decision was then made to postpone writing a paper until 2020 to accomodate high Atwood number experiments. It is thus befitting that a focused review on RTI experiments is included in this special issue to commemorate the life and scientific legacy of Malcolm J. Andrews. Many know Malcolm as a brilliant scientist who has contributed immensely to RTI mixing using his innovative experiments as well as his computational approaches; others know him through his service and editorship for the ASME Journal of Fluids Engineering. To a few fortunate individuals like the author, he was a mentor, colleague, and friend at different stages of their career. His untimely passing is a big loss for all in the community. The author would like to thank Devesh Ranjan, David Youngs, and Andrew G.W. Lawrie for providing some figures for their experiments and Denis Aslangil for helping several plots in this review. Finally, the author would also like to thank the DOE-SSAA Award (Grant No. DE-NA0003195), LANL subcontract (370333), and NSF Awards (CBET Fluid Dynamics Grants 1453056, 1305512) that have made possible some of the experimental studies that are discussed in this review.