Unlike RT instabilities, where the instability can grow only when light fluids push into heavy fluids, RM instabilities can develop both when shockwaves travel from light fluids to heavy fluids and vice versa. We will also discuss the physics associated with the "shock proximity" and introduce nonstandard RM flows.

A more complex initial setup could be constructed for experimental or numerical studies. Many distinct initial interfacial perturbations may be set up: the standard and inverted chevron shapes, enlarged double-bump, V, W, and sawtooth. The so-called "inclined shock tube" method of perturbing the fluid interface is created by inclining the shock tube with respect to the gravitational field. Tilted tank experiments consist of a tank filled with light fluid above heavy, which is then tilted onto one side of the apparatus. These experiments provide two-dimensional data for mix model development.

In this chapter, we focus on some simple cases in which analytical treatments of the Rayleigh–Taylor (RT) and Richtmyer–Meshkov (RM) instabilities can be carried out. This requires neglecting many physical effects and assuming small amplitude perturbations of a single wavelength. The linear stage growth rates are loosely derived and explored. This treatment requires the introduction of the key fluid equations and the concepts of vorticity, species fractions, and diffusion. Comparison is made to experiments.

For analytical simplicity, most research to date on RT and RM instabilities has focused on planar geometries. Such a simplified design is very helpful in easing the diagnostic requirements for laboratory experiments. However, in our limited observations of Chapter 15, we have already witnessed that other geometric configurations may alter the mixing layer growth significantly. In a variety of important applications, one must deal with imploding/exploding flows, the prime examples of which are inertial confinement fusion implosions (convergent geometry) and supernova explosions (divergent geometry). In these configurations, the flows are radially accelerated/decelerated. In contrast to planar geometry, where only RM growth is expected to occur, converging/diverging shock-accelerated interfaces can be RT unstable as they geometrically contract or expand. In the experiments and analytical modeling in this chapter, the amplitude growth depends on the convergence history in a complicated way.

Turbulent flow is a notoriously difficult topic in its own right because it is a truly multi-scale problem with strong nonlinearities. However, in this chapter, I will provide a framework for the key concepts, statistical measurements, and implications for the mixing process, so that the reader can better understand this issue. Both the classic engineering treatment of turbulence as well as the modern statistical closure theories will be introduced and brought together to show the reader how they can be synthesized to describe turbulence mixing induced by hydrodynamic instability driven flows. Some of the key concepts that I will elaborate on include energy transfer and interacting scales. The energy spectrum, and its applicability to RMI and RTI flow, is discussed.

Due to the time-consuming nature of fully 3D simulations of turbulence mixing induced by hydrodynamic instabilities, it is desirable to run computations in 2D when possible. But does 2D turbulence resemble 3D turbulence? The relevance of idealized 2D turbulence to certain aspects of atmospheric motion has been emphasized in many works. Yet molecular mixing occurs at the interfaces of the fluids, and the ratios of area-to-volume in three dimensions are very different than the length-to-area ratios in two dimensions. This has prompted some well-known scientists to claim that "two-dimensional turbulence, ... is a consequence of the construction of large computers." I will investigate this issue in detail and point out that the large-scale structures evolve over a similar time scale in 2D and 3D, indicating that 2D simulations are useful for providing some indication of the amount of instability growth at an interface.

Intense lasers are now being used to probe the physics of fluid dynamics in the high energy density physics (HEDP) regime, a term roughly referring to thermodynamic pressures greater than 1 Mbar. This approach allows us to design dedicated experiments to examine the issue of fluid instabilities in isolation. These laser platforms are also employed to recreate aspects of astrophysical phenomena in the laboratory, a specialized research area frequently referred to as laboratory astrophysics. Studying astrophysical phenomena in the laboratory with intense lasers offers many advantages: Repeatability, advanced diagnostics, controlled initial conditions, etc.

Inertial Confinement Fusion (ICF) recently became the first technology to achieve ignition of hydrogen nuclear fusion fuel in the laboratory. Unlike magnetically confined fusion plasmas such as tokamaks, ICF requires high fuel compression. This implies a high convergence and high velocity implosion, usually driven with laser beams. This allows hydrodynamic instabilities to develop, primarily RTI and RMI. During the initial shock and acceleration phase when the shell is brought up to the peak implosion velocity, RMI instabilities at the various interfaces are followed by ablation front RT growth as the low-density plasma accelerates the dense shell of solid ablator and fuel. The implosion deceleration at the center is also unstable. The resulting spikes and bubbles prevent efficient fuel compression, and can also inject contaminants. I will discuss the measurement and mitigation of this problem. Z-pinch machines, which instead use an electrical current to compress the plasma, will illustrate the role of MHD in the ICF application.

Material strength is important for planetary science and planetary formation dynamics. Inspecting RTI growth in solid-state samples in a high-energy-density setting can be key to determining the strength of a number of materials, such as iron, lead, or tantalum. One of the important applications is the enhanced mixing in the scramjet; I will address this issue as well as detonation in the combustion chamber. Moreover, I will discuss the reactive RMI in detail to address several issues related to turbulence-flame interactions, such as an incident shock wave passing the interface and shock initiation of flow instabilities. Ejecta occurs when small pieces of the material are forced out as a result of stellar explosions or other sharp impacts in the engineering process. RMI is key to understanding the physics processes for the production and distribution of ejecta. Extensive data from numeric simulations and experimental evidence will be offered to provide a comprehensive picture about this topic.

We derive the governing equations for the mean and turbulent kinetic energy and discuss simplifications of the equations for several canonical flows, including channel flow and homogeneous isotropic turbulence. A classical expression for the dissipation rate in isotropic turbulence is provided. In addition, the governing equations for turbulent enstrophy and scalar variance are derived with parallels to the derivation of the turbulent kinetic energy equation. A model for turbulent kinetic energy evolution and dissipation in isotropic turbulence is introduced. Finally, we derive the governing equations for the Reynolds stress tensor components and discuss the roles of the terms in the Reynolds stress budgets in homogeneous shear and channel flows. A crucial link between pressure-strain correlations and the redistribution of turbulent kinetic energy between various velocity components is established. Quantifying how energy is transferred between the mean flow and turbulent fluctuations is crucial to understanding the generation and transport of turbulence and its accompanying Reynolds stresses, and thus properties that phenomenological turbulence models should conform to.

Building on the governing equations and spectral tools introduced in earlier chapters, we analyze the energy cascade, which describes the transfer of turbulent kinetic energy from large to small eddies. This includes an estimate of the energy dissipation rate, as well as the characteristic length and time scales of the smallest-scale motions. Nonlinearity in the Navier-Stokes equations is responsible for triadic interactions between wavenumber triangles that drive energy transfer between scales. Empirical observations suggest that the net transfer of energy occurs from large to small scales. In systems where the large scales are sufficiently separated from the small scales, an inertial subrange emerges in an intermediate range of scales where the dynamics are scale invariant. Kolmogorov’s similarity hypotheses and the ensuing expressions for the inertial-subrange energy spectrum and viscous scales are introduced. The Kolmogorov spectrum for the inertial subrange, which corresponds to a -5/3 power law, is a celebrated result in turbulence theory. We further discuss key characteristic turbulence scales including the Taylor microscale and Batchelor scale.

We discuss properties of numerical methods that are essential for high-fidelity (LES, DNS) simulations of turbulent flows. In choosing a numerical method, one must be cognizant of the broadband nature of the solution spectra and the resolution of turbulent structures. These requirements are substantially different than those in the RANS approach, where the solutions are smooth and agnostic to turbulent structures. We focus on spatial discretization of the governing equations in canonical flows where Fourier analysis is helpful in revealing the effect of discretization on the solution spectra. In high-fidelity numerical simulations of turbulent flows, it is necessary that conservation properties inherent in the governing equations, such as kinetic energy conservation in the inviscid limit, are also satisfied discretely. An important benefit of adhering to conservation principles is the prevention of nonlinear numerical instabilities that may manifest after long-time integration of the governing equations. We end by discussing the appropriate choice of domain size, grid resolution, and boundary conditions in the context of canonical flows with uniform Cartesian mesh spacing.

The spectral description of turbulence allows us to decompose velocity and pressure fields in terms of wavenumbers and frequencies, or length and time scales. We discuss the notion of scale decomposition and introduce several properties of the Fourier transform between physical (spatial/temporal) space and scale (spectral) space in various dimensions, including complex conjugate relations for real functions and Parseval’s theorem. The Fourier transform allows us to develop useful relations between correlations and energy spectra, which are used extensively in the statistical theory of turbulence. The one-dimensional and three-dimensional energy spectra are specifically discussed in conjunction with Taylor’s hypothesis to enable spectra computation from single-point time-resolved measurements. The discrete version of the transform, or the discrete Fourier series, is then introduced, as it is typically encountered in numerical simulations and postprocessing of discrete experimental data. Treatment of periodic data is first considered, followed by nonperiodic data with the help of windowing. The procedure for the computation of various discrete spectra is outlined.

An overview of the three modern categories of methods for numerical prediction of turbulent flows is provided: direct numerical simulation (DNS), solution of the Reynolds-averaged Navier-Stokes (RANS) equations, and large-eddy simulation (LES). We describe zero-equation, one-equation, two-equation, and Reynolds stress transport models for the RANS equations. RANS computations require significantly fewer grid points and lower computational cost since the solutions are smooth and turbulent structures are not captured, but there is a need to tune model parameters for different flows to match experimental data. In LES, only the large-scale motions are resolved, whereas unresolved small scales are modeled. We introduce the notion of filtering, subgrid-scale parameterization, as well as the seminal dynamic Smagorinsky subgrid-scale model. Wall-resolved and wall-modeled LES are briefly discussed. With ever increasing computer power, as well as advances in numerical methods and subgrid-scale models, LES is rapidly becoming a viable tool for practical computations. In selecting a method, one should consider quantities to be predicted, accuracy of the predictions, and the computational cost.