THE EQUATIONS OF NEARLY INCOMPRESSIBLE FLUIDS .1. HYDRODYNAMICS, TURBULENCE, AND WAVES

A unified analysis delineating the conditions under which the equations of classical incompressible and compressible hydrodynamics are related in the absence of large-scale thermal, gravitational, and field gradients is presented. By means of singular expansion techniques, a method is developed to derive modified systems of fluid equations in which the effects of compressibility are admitted only weakly in terms of the incompressible hydrodynamic solutions (hence "nearly incompressible hydrodynamics"). Besides including molecular viscosity self-consistently, the role of thermal conduction in an ideal fluid is also considered. With the inclusion of heat conduction, it is found that two distinct routes to incompressibility are possible, distinguished according to the relative magnitudes of the temperature, density, and pressure fluctuations. This leads to two distinct models for thermally conducting, nearly incompressible hydrodynamics-heat-fluctuation-dominated hydrodynamics (HFDH's) and heat-fluctuation-modified hydrodynamics (HFMD's). For the HFD case, the well-known classical passive scalar equation for temperature is derived as one of the nearly incompressible fluid equations and temperature and density fluctuations are predicted to be anticorrelated. For HFM fluids, a new thermal transport equation, in which compressible acoustic effects are present, is obtained together with a more-complicated "correlation" between temperature, density, and pressure fluctuations. Although the equations of nearly incompressible hydrodynamics are envisaged principally as being applicable to homogeneous turbulence and wave propagation in low Mach number flow, it is anticipated that their applicability is likely to be far greater.