Growth Of Mixing Width Research Articles

Mixing and inter-scale energy transfer in Richtmyer–Meshkov turbulence

This work investigates the compressible turbulence induced by Richtmyer–Meshkov (RM) instability using high-resolution Navier–Stokes simulations. Special attention is paid to the characteristics of RM turbulence including the mixing width growth, the turbulent kinetic energy (TKE) decay, the mixing degree, inhomogeneity and anisotropy. Three distinct initial perturbation spectra are designed at the interface to reveal the initial condition imprint on the RM turbulence. Results show that cases with initial large-scale perturbations present a stronger imprint on statistical characteristics and also a quicker growth of mixing width, whereas cases with small-scale perturbations present a faster TKE decay, greater mixing level, higher isotropy and homogeneity. A thorough analysis on the inter-scale energy transfer in RM turbulence is also presented with the coarse-graining approach that exposes the two subfilter-scale (SFS) energy fluxes (i.e. deformation work and baropycnal work). A strong correlation between the nonlinear model of baropycnal work (Fluids, 4(2), 2019) and the simulation results is confirmed for the first time, demonstrating its potential in modelling the RM turbulence. Two primary mechanisms of baropycnal work (the straining and baroclinic generation processes) are explored with this nonlinear model. The evolutions of two SFS energy fluxes exhibit distinct behaviours at various filter scales, in different flow regions and under various flow motions (strain and rotation). It is found that all three cases share the common inter-scale energy transfer dynamics, which is important for modelling the RM turbulence.

Experimental and theoretical studies on heavy fluid layers with reshock

The Richtmyer–Meshkov instability of a heavy fluid layer with reshock is investigated experimentally and theoretically, to reveal the mechanisms of the interfacial instabilities and motions of waves and interfaces under reshock conditions. Three kinds of heavy fluid layers with different thicknesses in unperturbed and perturbed cases are considered, highlighting the effects of initial layer thickness on the interface evolution. A general one-dimensional theory for describing the interaction between unperturbed interfaces and waves under reshock conditions is constructed, which supports the instability analysis and theoretical modeling in the perturbed cases. The perturbed fluid-layer results indicate that as the fluid-layer thickness decreases, the inhibition of squeezing effects on perturbation growth is gradually stronger than the promotion of rarefaction-wave effects on perturbation growth. The whole mixing width of fluid layers after reshock acceleration exhibits an approximately linear growth feature, and the reshocked mixing width growth rate is weakly dependent on the initial layer thickness, which is different from the finding before reshock. The linear amplitude growth of the right interface after reshock is promoted by the interface coupling effects but is still smaller than the prediction of the empirical linear model. When the squeezing effects are weak, the nonlinear model, which considers the modification of rarefaction-wave effects, can well predict the perturbation growth during and after the acceleration stage of rarefaction waves.

Shock-tube studies of single- and quasi-single-mode perturbation growth in Richtmyer–Meshkov flows with reshock

The Richtmyer–Meshkov instability of heavy/light single-mode (SM), trapezoid (TR) and sawtooth (ST) interfaces is studied experimentally by considering the reshock. The TR and ST interfaces can be expanded into Fourier series with a dominant fundamental mode and more high-order modes, recognized as quasi-single-mode ones. In experiments, the distorted interfaces at the time of first reshock arrival develop in the weakly nonlinear stage, ensuring an approximate single-scale function of evolving interface. The results show an evident memory of initial interface shapes: the bubbles and spikes of ST interface after reshock mainly develop in the streamwise direction with sharp heads, while the counterparts of TR interface tend to grow in the spanwise direction. The influences of high-order modes are amplified by the reshock, resulting in significant interface shape dependence of mixing width growths. The amplitude superposition of major odd-order modes promotes the growth rates of mixing widths for the SM and ST cases, different from the TR one. The ST interface has larger mixing width growth rates in comparison with the SM interface, since high-order modes play a great role in promoting the increase of ST amplitudes, while the TR interface has a relatively small one. The linear and nonlinear mixing width growths of SM, TR and ST interfaces before and after reshock are further analysed theoretically, indicating that the fundamental mode still has a predominant influence on the interface evolution after reshock, and the growth behaviours exhibit strong similarities to those for the singly shocked cases.

Turbulent mixing and energy transfer of reshocked heavy gas curtain

The interaction between the shock wave and the air-SF6-air gas curtain and the associated instability of the reshocked interface are numerically studied using a high-resolution finite volume method with minimum dispersion and controllable dissipation reconstruction. The evolution of flow structures shows that, although the initial perturbation of the interface is two-dimensional, the reshock process will lead to an asymmetrical instability and enhance the mixing of fluids dramatically. The exponential scaling laws for the growth of mixing width are observed both before and after the reshock. The nearly homogeneous fluids’ mixing at the late stage after the reshock is confirmed by the temporal asymptotic behaviors of mixing parameters such as local anisotropy and density-specific volume correlation. Both the decoupled length scales and the broadened inertial range of the turbulent kinetic energy spectrum and density fluctuation spectrum imply that the turbulent mixing transition is triggered at the late stage. The analysis of sub-grid stress energy transfer manifests the energy cascade in physical space. Furthermore, the investigation of the role of coherent structures in sub-grid stress energy transfer indicates that the backscatter of sub-grid stress energy is most prominent in the regions with washing motion induced by nearly two-dimensional vortical structures. However, the intense forward scatter of sub-grid stress energy occurs in the regions where two washing motions collide with each other leading to highly three-dimensional motions.

Late-time mixing and turbulent behavior in high-energy-density shear experiments at high Atwood numbers

The LANL Shear Campaign uses millimeter-scale initially solid shock tubes on the National Ignition Facility to conduct high-energy-density hydrodynamic plasma experiments, capable of reaching energy densities exceeding 100 kJ/cm3. These shock-tube experiments have for the first time reproduced spontaneously emergent coherent structures due to shear-based fluid instabilities [i.e., Kelvin-Helmholtz (KH)], demonstrating hydrodynamic scaling over 8 orders of magnitude in time and velocity. The KH vortices, referred to as “rollers,” and the secondary instabilities, referred to as “ribs,” are used to understand the turbulent kinetic energy contained in the system. Their evolution is used to understand the transition to turbulence and that transition's dependence on initial conditions. Experimental results from these studies are well modeled by the RAGE (Radiation Adaptive Grid Eulerian) hydro-code using the Besnard-Harlow-Rauenzahn turbulent mix model. Information inferred from both the experimental data and the mix model allows us to demonstrate that the specific Turbulent Kinetic Energy (sTKE) in the layer, as calculated from the plan-view structure data, is consistent with the mixing width growth and the RAGE simulations of sTKE.

Simulation of shock–turbulence interaction in non-reactive flow and in turbulent deflagration and detonation regimes using one-dimensional turbulence

The one-dimensional turbulence (ODT) methodology is extended to include an efficient compressible implementation and a model for capturing shock-induced turbulence is presented. Lignell et al. recently introduced a Lagrangian ODT implementation using an adaptive mesh. As the code operates in the incompressible regime (apart from constant-pressure dilatation) it cannot handle compressibility effects and their interactions with turbulence and chemistry. The necessary algorithmic changes to include compressibility effects are highlighted and our model for capturing shock–turbulence interaction is presented. To validate our compressible solver, we compare results for the Sod shock tube problem against a finite volume Riemann solver. To validate our model for shock–turbulence interaction, we present comparisons for a non-reactive and a reactive case. First, results of a shock traveling from light (air) to heavy (SF6) with reshock have been simulated to match mixing width growth data of experiments and turbulent kinetic energy results from LES. Then, for one-step chemistry calibrated to represent an acetylene/air mixture we simulate the interaction of a shock wave with an expanding flame front, and compare results with 2D simulation (2D-sim) data for flame brush formation and ensuing deflagration-to-detonation transitions (DDT). Results for the Sod shock tube comparison show that the shock speed and profile are captured accurately. Results for the non-reactive shock–reshock problem show that interface growth at all simulated Mach numbers is captured accurately and that the turbulent kinetic energy agrees in order of magnitude with LES data. The reactive shock tube results show that the flame brush thickness compares well to 2D-sim data and that the approximate location and timing of the DDT can be captured. The known sensitivity of DDT characteristics to details of individual flow realizations, seen also in ODT, implies that model agreement can be quantified only by comparing flow ensembles, which are presently unavailable other than in an ODT run-to-run sensitivity study that is reported herein.

Investigation of the initial perturbation amplitude for the inclined interface Richtmyer–Meshkov instability

A simulation studying the effects of inclination angle and incident shock Mach number on the inclined interface Richtmyer–Meshkov instability is presented. Interface inclination angle is varied from 30° to 85°, with incident shock Mach numbers of 1.5, 2.0 and 2.5 for an air over SF6 interface. The simulations were performed in support of experiments to be performed in the Texas A&M shock tube facility, and were created with the ARES code developed at Lawrence Livermore National Laboratory. The parametric cases are separated by inclination angle into nonlinear and linear initial perturbation cases. A linear initial perturbation is defined as when the interface amplitude over wavelength is less than 0.1. Density, pressure gradient and vorticity plots are presented for a nonlinear and a linear case to highlight the differences in the flow field evolution. It is shown that the nonlinear case contains strong secondary compressible effects which reverberate through the interface until late times, while in the linear case these waves are almost completely absent. The inclined interface scaling method presented in previous work (McFarland et al 2011 Phys. Rev. E 84 026303) is tested for its ability to scale the mixing width growth rate for linear initial perturbation cases. This model was shown in the previous work to collapse data well for varying Mach numbers and nonlinear inclination angles. The scaled data is presented to show that a regime change occurs in the mixing width growth rate near an inclination angle of 80° which corresponds to the transition from a linear to nonlinear initial perturbation.

Effect of shear on Rayleigh-Taylor mixing at small Atwood number

The effect of shear on the development of Rayleigh-Taylor instability (RTI) is studied at an Atwood number of 0.035 using the gas tunnel at Texas A University. Two types of diagnostics, imaging and simultaneous hot wire and cold wire anemometry, are used to measure mix widths, pointwise instantaneous velocities, and density. Image analysis has shown that the superposition of shear on RTI development increases the mixing width and growth rate at early times ($\ensuremath{\tau}=\frac{x}{U}\sqrt{\frac{{A}_{t}g}{H}}\ensuremath{\ll}1$). In particular, the mixing region shows distinct characteristics of shear (Kelvin-Helmholtz instability) and buoyancy (RTI). The Kelvin-Helmholtz (KHI) instability is observed to be dominant at early times, and the RTI at late times ($\ensuremath{\tau}1$). In the late-time self-similar regime ($\ensuremath{\tau}1$), the mix width growth rate coefficient obtained using digital image analysis converges to a value between 0.06 and 0.07 for the compound buoyancy and shear (KH+RT) driven flows. Vertical velocity fluctuation rms values at the mixing layer centerline are measured using a hot-wire technique. These rms values are correlated to the centerline mixing width growth rate, and this growth rate coefficient is found to lie between 0.06 and 0.07 at $\ensuremath{\tau}1$ for the KH+RT flows. The transition in flow dominance from shear instability to RTI is observed to correspond with Richardson numbers ($\mathrm{Ri}=\frac{\ensuremath{-}2hg\ensuremath{\Delta}\ensuremath{\rho}}{\ensuremath{\rho}\ensuremath{\Delta}{U}^{2}}$) of \ensuremath{-}1.5 to \ensuremath{-}2.5. Molecular mixing between the fluids is examined by looking at the probability density function distribution of the density fluctuations. A different type of mixing behavior is observed over time for the compound cases compared with the transient development of Rayleigh-Taylor driven mixing.

Optical Density Measurements and Analysis for Single-Mode Initial-Condition Buoyancy-Driven Mixing

The Texas A&M water channel experiment is modified to examine the effect of single-mode initial conditions on the development of buoyancy-driven mixing (Rayleigh-Taylor) with small density differences (low-Atwood number). Two separated stratified streams of ~5°C difference are convected and unified at the end of a splitter plate outfitted with a servo-controlled flapper. The top (cold) stream is dyed with Nigrosine and density is measured optically through the Beer-Lambert law. Quantification of the subtle differences between different initial conditions required the optical measurement uncertainties to be significantly reduced. Modifications include a near-uniform backlighting provided through quality, repeatable, professional studio flashes impinging on a white-diffusive surface. Also, a black, absorptive shroud isolates the experiment and the optical path from reflections. Furthermore, only the red channel is used in the Nikon D90 CCD camera where Nigrosine optical scatterring is lower. This new optical setup results in less than 1% uncertainty in density measurements, and 2.5% uncertainty in convective velocity. With the Atwood uncertainty reduced to 4% using a densitometer, the overall mixing height and time uncertainty was reduced to 5% and 3.5%, respectively. Initial single-mode wavelengths of 2, 3, 4, 6, and 8 cm were examined as well as the baseline case where no perturbations were imposed. All non-baseline cases commence with a constant velocity that then slows, eventually approaching the baseline case. Larger wavelengths grow faster, as well as homogenize the flow at a faster rate. The mixing width growth rates were shown to be dependent on initial conditions, slightly outside of experimental uncertainty.

Computational parametric study of a Richtmyer-Meshkov instability for an inclined interface

A computational study of the Richtmyer-Meshkov instability for an inclined interface is presented. The study covers experiments to be performed in the Texas A&M University inclined shock tube facility. Incident shock wave Mach numbers from 1.2 to 2.5, inclination angles from 30° to 60°, and gas pair Atwood numbers of ∼0.67 and ∼0.95 are used in this parametric study containing 15 unique combinations of these parameters. Qualitative results are examined through a time series of density plots for multiple combinations of these parameters, and the qualitative effects of each of the parameters are discussed. Pressure, density, and vorticity fields are presented in animations available online to supplement the discussion of the qualitative results. These density plots show the evolution of two main regions in the flow field: a mixing region containing driver and test gas that is dominated by large vortical structures, and a more homogeneous region of unmixed fluid which can separate away from the mixing region in some cases. The interface mixing width is determined for various combinations of the parameters listed at the beginning of the Abstract. A scaling method for the mixing width is proposed using the interface geometry and wave velocities calculated using one-dimensional gas dynamic equations. This model uses the transmitted wave velocity for the characteristic velocity and an initial offset time based on the travel time of strong reflected waves. It is compared to an adapted Richtmyer impulsive model scaling and shown to scale the initial mixing width growth rate more effectively for fixed Atwood number.