The Physical Foundations of Hydrodynamic Processes

The following sections are included:

• Dedication
• Contents
• Introduction

The following sections are included:

• Euler’s Equations
• Navier–Stokes Equations
• Kinetic Equations and Gas Dynamics
• Summary

The following sections are included:

• Wave Equation and Free Surface Gravity Waves
• Waves in Dispersive Media
• Waves at a Density Interface Between Liquids of Different Densities
• Summary

The following sections are included:

• Equation of State
• Thermodynamical Properties of Fluids and Gases
• Summary

The following sections are included:

• One-Dimensional Fluidodynamics and Rankine–Hugoniot Conditions
• Weak Shocks and the Boltzmann Equation
• Summary

On the example of shock, we saw that nonlinear effects in hydrodynamics may play a very important (critical in the some cases) role. Now, we consider other examples of influence of nonlinearity and special structure of fluidodynamic equations on the real physical properties of liquid and gas. We begin the investigation from the consideration of convection in fluid dynamics, which is the most simple and widely known physical effect in hydrodynamics connected with the critical behavior of the fluid. More complicated types of instability are Saffman, Rayleigh–Taylor, Kelvin–Helmholtz, Richtmyer–Meshkov ones; we will consider these examples too.

The following sections are included:

• Introduction to Classical Turbulence
• The Homogeneous Turbulence and Its Spectrum
• Direct Numerical Simulation of Turbulence
• Summary

In the previous part, we considered the sophisticated mathematical formalism of fluid instabilities which reduce to postlaminar regimes of flows. We need to explain the inner structure of such flows. The nature of turbulence is connected closely with vortices in fluid and multivortex systems which can form structures of essentially varied scales.

To investigate coherent multivortex systems, we must introduce a rigorous definition of coherency in terms of thermodynamics. With this aim, we generalize Hamiltonian dynamical systems.

The possibility of introducing a metric on manifolds associated with hydrodynamic flows various types has been repeatedly discussed in the literature, however, the results to date have either an excessive generality (which makes the development of techniques suitable for describing flows of specific types), or require the introduction of significant restrictions and problem-oriented problems with an a priori rigidly defined additional setting conditions, the physical content of which requires special analysis. If we consider the hydrodynamic flow as a statistical system in a state close to equilibrium, then its geometric properties can be investigated using the general Amari–Weinhold technique, using the possibility of introducing of a Riemannian topology on Gibbs manifolds (these manifolds are determined with the help of the relations between the thermodynamic potentials p = p(ρ, T) and s = s(u, ρ)). In particular, if we turn to the model of a real flow in the form of a set of point vortices of the Onsager type, one can obtain meaningful differential-functional relations that are expressions for the specific heats of the vortex population and the equations of geodesic, binding states equilibrium levels of the system…

The following sections are included:

• Appendix A. Chaplygin’s Gas
• Appendix B. Richtmyer–Meshkov Instability
• Appendix C. Instabilities and Turbulence in Astrophysics

The following sections are included:

• References
• Index