Energy exchange in two-dimensional compressible Taylor-Green vortex flows

The internal-kinetic energy exchange and the effect of the Mach number in a 2D compressible Taylor-Green vortex flow are investigated, using direct numerical simulation with initial Mach numbers (Ma₀) ranging from 0.2 to 1.0 and a Reynolds number of 1600. Kinetic energy is decomposed into dilatational and solenoidal components using Helmholtz decomposition. The flow evolution is described in three stages, which are the generation of compressibility, transition, and oscillatory decay stages, respectively. In the generation stage, the dilatational kinetic energy is analytically derived to increase quadratically with time, which is generated by advection from solenoidal kinetic energy and partially transformed into internal energy via pressure-dilatation. In the transition stage, pressure-dilatation reaches a minimum and scales linearly with initial Mach number. Shock waves emerge at high initial Mach numbers, enhancing both the dilatational and solenoidal dissipations. In the oscillatory decay stage, the kinetic energy is dominated by the solenoidal component, which decays exponentially with superimposed oscillations. These oscillations are explained as acoustic energy exchange between internal energy and the negligible dilatational kinetic energy. The oscillation frequency and initial Mach number are found to follow an inverse proportionality for Ma₀ < 0.6. As a first simplified example, this research sheds light on the understanding of internal-kinetic energy exchange in compressible flows.