Computational Thermodynamics

We test the functionality of FEniCS on the challenge of computational thermodynamics in the form of the EG2 finite element solver of the Euler equations expressing conservation of mass, momentum and energy. We show that computational solutions satisfy a 2nd Law formulated in terms of kinetic energy, internal (heat) energy, work and shock/turbulent dissipation, without reference to entropy. We show that the 2nd Law expresses an irreversible transfer of kinetic energy to heat energy in shock/turbulent dissipation arising because the Euler equations lack pointwise solutions, and thus explains the occurence of irreversibility in formally reversible systems as an effect of instability with blow-up of Euler residuals combined with finite precision computation, without resort to statistical mechanics or ad hoc viscous regularization. We simulate the classical Joule or Joule-Thompson experiment of a gas expanding from rest under temperature drop followed by temperature recovery by turbulent dissipation until rest in the double volume. We present the FEniCS implementation of EG2 including applications to bluff body flow. 1 FEniCS as Computational Science The goal of the FEniCS project is to develop software for automated computational solution of differential equations based on a finite element methodology combining generality with efficiency. Thermodynamics is a basic area of continuum mechanics with many important applications, which however is feared by both teachers, students and engineers as being difficult to understand and to apply, principally because of the apperance of turbulence. In this article we show that turbulent thermodynamics can be made understandable and useful by automated computational solution, as a demonstration of the capability of FEniCS. The biggest mystery of classical thermodynamics is the 2nd Law about entropy and automation cannot harbor any mystery. Expert systems are required for mysteries and FEniCS is not an expert system. Automation requires a continuum mechanics formulation of thermodynamics with a transparent 2nd Law. We present a formulation of thermodynamics based on finite precision computation with a 2nd Law without reference to entropy, which we show can serve as a basis for automated computational simulation of complex turbulent thermodynamics and thus can open to new insight and design, a main goal of FEniCS. In this setting the digital finite element model becomes