Variational approach to the volume viscosity of fluids

Abstract

The variational principle of Hamilton is applied to develop an analytical formulation to describe the volume viscosity in fluids. The procedure described here differs from those used in the past in that a dissipative process is represented by the chemical affinity and progress variable ͑sometimes called "order parameter"͒ of a reacting species. These state variables appear in the variational integral in two places: first, in the expression for the internal energy, and second, in a subsidiary condition accounting for the conservation of the reacting species. As a result of the variational procedure, two dissipative terms appear in the Navier-Stokes equation. The first is the traditional volume viscosity term, proportional to the dilatational component of velocity; the second term is proportional to the material time derivative of the pressure gradient. Values of the respective volume viscosity coefficients are determined by applying the resulting volume-viscous Navier-Stokes equation to the case of acoustical propagation and then comparing expressions for the dispersion and absorption of sound. The formulation includes the special case of equilibration of the translational degrees of freedom. As examples, values are tabulated for dry and humid air, argon, and sea water.

Performance of very‐high‐order upwind schemes for DNS of compressible wall‐turbulence

“…There has always been a lot of debate on bulk viscosity [66], and in particular on the way it should be incorporated in the Navier-Stokes equations [67,68]. In the present study, bulk viscosity b ,which is not expected to be important in plane channel flow (at least for the M 1.5 cases studied in the present work), was set to 0 (Stokes hypothesis [67]).…”

Measurement and characterization of infrasound from a tornado producing storm

“…This suggests that latent heat sources are a likely mechanism, and simulations of the liquid-vapor transitions within a cloud were able to produce infrasound between 0.1 and 10 Hz (Akhalkatsi & Gogoberidze, 2009;Schecter & Nicholls, 2010). However, radial vortex oscillations including the non-columnar nature of a tornado (Schecter, 2012) and analysis incorporating nonequilibrium effects (Zuckerwar & Ash, 2006;Ash et al, 2011) are also consistent with observations. As noted by Frazier et al (2014), there are likely multiple acoustic generation mechanisms active, which was based on their datasets exhibiting coherent acoustic energy within two distinct regimes (< 2 Hz and 10-100 Hz with larger tornadoes; EF-2, EF-4, and EF-5).…”

“…The low frequency signal was assumed to be mostly due to wind noise, but there was sufficient coherence for successful beamforming to produce bearings to the tornado producing storms. This lower frequency signature has had several proposed mechanisms including radial oscillations (Abdullah, 1966;Bedard, 2005;Schecter, 2012), electromagnetic sources (Balachandran, 1983;Few, 1985;Pasko, 2009), co-rotating vortices (Powell, 1964;Georges, 1976), vortex-surface-interactions (Tatom et al, 1995), heat-related sources (Nicholls et al, 2004;Akhalkatsi & Gogoberidze, 2009;Schecter & Nicholls, 2010;Markowski & Richardson, 2010;Schecter, 2012), and non-equilibrium effects (Zuckerwar & Ash, 2006;Ash et al, 2011).…”

Variational principle for two-dimensional incompressible inviscid flow

“…Variational-based finite element method has been, and continues to be, a popular numerical tool, however it is very difficult to establish a variational formulation for fluid mechanics [3,4]. For example Zuckerwar and Ash established a variational principle of the dynamics of a fluid with volume viscosity by introducing an additional field variable [9], however, Scholle found the established theory shows inconsistencies [10].…”