Despite the intensive efforts to develop increased computational capabilities, mix models remain the most viable approach for the solution of many applications. These are an approximation to the true solution of the Navier-Stokes equations. The reason for this state of affairs becomes abundantly clear when one considers the difficulties of achieving the desired turnaround time for applied fluid dynamics calculations. In this chapter, we focus on some of the methodologies currently utilized to tackle the practical problem of simulating hydrodynamic instabilities in engineering designs.

The multi-mode instability is the simultaneous growth across many wavelengths. This is closer to the reality of many applications. We provide a detailed treatment of the various stages of development. It is widely believed that many turbulent flows, such as RTI, RMI, and KHI mixing layers, evolve toward self-similarity. Here, the RTI grows quadratically with time, and the suitable proportionality constant is the subject of ongoing research. The growth exponent for RMI is also the subject of ongoing research. I also discuss measurements of these parameters in experiments and simulations arising from multimodal initial perturbations.

I will describe the earlier efforts in both hydrodynamic instability and turbulence mixing research and provide a broad non-mathematical overview of the significance of turbulence mixing on scientific and engineering applications. I will briefly explain several varied applications in which hydrodynamic instability plays a critical role, namely, inertial confinement fusion (ICF), supernova explosions, solar prominences, paintings, and combustions and detonations, among others, to provide the reader with an idea of what will be discussed later in the book.

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.

Two selected Exercises are solved in full: a question relevant to the twin paradox and a method using special relativity for deriving the Biot–Savart equation.

The Lorentz transformation between frames of reference is determined and presented in a matrix form. The transformation of velocities is derived with an example presented in the form of the relativistic rocket equation. The Doppler effect is discussed. Relativistic momentum and energy are presented. The quantum nature of particles and light is discussed. The four-vector and tensor notation are introduced.