The nonlinear stage starts when the amplitude of the unstable flow feature becomes significant. This chapter first studies the nonlinear growth of the interface amplitude and its associated terminal velocity with potential flow models, both for RM and RT. Next, one describes several models intended to predict the evolution of the bubble and spike heights, and the corresponding velocities, for the nonlinear stage. The success and limitations of each model are assessed with comparison to experiments and numerical simulations. The sensitivities to viscosity, density ratio and Mach number are discussed.

I will describe how certain external factors, such as rotation and time-dependent acceleration/deceleration, could suppress the evolution of the hydrodynamic instabilities.

By necessity, experimental studies have been the key to advancement in fluid dynamics for centuries. However, with the rapid increase of computational capabilities, numerical approaches have become an acceptable surrogate for experiments. Calculations must resolve the Navier–Stokes equations or approximate methods constructed from them. I will discuss the pros and cons of various types of approaches used, including direct numerical simulations, subgrid models, and implicit grid-discretization-based large-eddy simulation.

This chapter contains a discussion of the coupling of a magnetic field, through the framework of magnetohydrodynamics (MHD), to the hydrodynamic body forces. This leads to an additional body force, namely the Lorentz force on electrical currents in the fluid. Due to their conductivity, this effect is especially important for ionized plasmas. The intuitive result is that the magnetic field lines follow the flow, and they have an effective tension that can stabilize the RTI. As with the RTI, the RMI can be suppressed by a magnetic field.

The challenge confronting researchers is significant in many ways. One can start by noting that multiple instabilities might exist simultaneously and interact with each other. As an example, oblique shocks generate all three instabilities: RT, RM, and KH. In this chapter, several combined instabilities are discussed: RTI and RMI, RTI and/or RMI with KHI.

In this chapter, we will focus on the statistical spectral dynamics which are paramount to understanding the development of the integrated mixing quantities described in Chapter 5. Reynolds flow averaging and the turbulent kinetic energy are introduced. In addition, I will discuss how the energy of the flows is transferred from large scale to small scale modes, as well as the impact of the shockwave and gravity on the isotropy of the flows. The flow spectra allow several important length scales to be defined. Numeric simulations and experimental data will be offered to provide insights on the mixing processes.

This chapter will provide a detailed presentation of the basic structure of the supernova and its core collapse process to illustrate the roles that RMI, RTI, and KHI play in the different stages of these processes. During the explosions, the shockwave passing through the onion-like supernova core will generate both RMI and RTI. The RTI is the key physical process creating the filament structures observed in the Crab Nebula. MHD RT instabilities will be presented to show how they can further improve the comparison between simulations and observations. Several additional applications where hydrodynamic instability plays an important role will also be examined. Geophysics and solar physics also present effective lenses to view the importance of hydrodynamic instabilities. In the case of solar physics, I will describe how RTI’s impact can be viewed through various phenomena, such as the plumes that rise from low density bubbles as well as eruptions that occur as material returns to the solar surface. Once again, MHD RT instabilities are relevant.

After the RM instability grows from a first shock, it can be hit by a second shock. These reshock scenarios have been found in the key applications of inertial confinement fusion implosions or supernova explosions. In this chapter, I will introduce the efforts to model the growth of the mixing layer induced by the first shock and subsequent reshock and describe how the turbulence kinetic energy and anisotropy might be affected by the reshock events. Data from shock tube experiments and numeric simulations will also be introduced to provide insight into the reshock RM induced flows.

There are a number of microphysics and transport processes that can be extremely important to suppress or enhance the growth of these instabilities. I will provide a detailed description of how the hydrodynamic instability evolutions can be modified by incorporating the viscosity, surface tension, diffuse interface, and compressibility of the flows into the governing equations and growth rates.

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.