Accurate predictions with quantifiable uncertainty are essential to many practical turbulent flows in engineering, geophysics, and astrophysics typically comprising extreme geometrical complexity and broad ranges of length and timescales. Dominating effects of the flow instabilities can be captured with coarse-graining (CG) modeling based on the primary conservation equations and effectively codesigned physics and algorithms. The collaborative computational and laboratory experiments unavoidably involve inherently intrusive coarse-grained observations – intimately linked to their subgrid scale and supergrid (initial and boundary conditions) specifics. We discuss turbulence fundamentals and predictability aspects and introduce the CG modified equation analysis. Modeling and predictability issues for underresolved flow and mixing driven by underresolved velocity fields and underresolved initial and boundary conditions are revisited in this context. CG simulations modeling prototypical shock-tube experiments are used to exemplify relevant actual issues, challenges, and strategies.

Originating from irreversible statistical mechanics, the Mori–Zwanzig (M–Z) formalism provides a mathematical procedure for the development of coarse-grained models of complex systems, such as turbulence, that lack scale separation. The M–Z formalism begins with the application of a specialized class of projectors to the governing equations. By leveraging these projectors, the M–Z procedure results in a reduced system, commonly referred to as the generalized Langevin equation (GLE). The GLE encapsulates the system’s behavior on a macroscopic (resolved) scale. The influence of the microscopic (unresolved) scales on resolved scales appears as a convolution integral – often referred to as memory – and an additional noise term. In essence, fully resolved Markovian dynamics is transformed into coarse grained non-Markovian dynamics. The appearance of the memory term in the GLE demonstrates that the coarse-graining procedure leads to nonlocal memory effects, which have to be modeled. This chapter introduces the mathematics behind the projection approach and the derivation of the GLE. Beyond the theoretical developments, the practical application of the M–Z procedure in the construction of subgrid-scale models for large eddy simulations is also presented.

It is critical to evaluate whether the flow has transitioned into turbulence because most of the impact of large-scale mixing occurs when the flow becomes fully developed turbulence. Hydrodynamic instability flows are even more complex because of their time-dependent nature; therefore, both spatial and temporal criteria will be introduced in great detail to demonstrate the necessary and sufficient conditions for the flow to transition to turbulence. These criteria will be extremely helpful for designing experiments and numeric simulations with the goal to study large-scale turbulence mixing. One spatial criterion is that the Reynolds number must achieve a critical minimum value of 160,000. In addition, the temporal criteria suggest that flows need to be given approximately four eddy-turnover-times. This chapter will expand on these issues.

We focus on three integrated measures of the mixing: the mixed-width, mixedness, and mixed mass. I will also examine the dependence of these mixing parameters on density disparities, Mach numbers, and other flow properties. It is shown that the mixed mass is nondecreasing. The asymmetry of the bubble and spike is also discussed.

There is significant simulation and experimental evidence suggesting that hydrodynamic instability induced flows may be dependent on how the initial conditions are set up. The initial surface perturbations, density disparity, and the strength of the shockwaves could all be factors that lead to a completely different flow field in later stages.

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.