A Coherent Structure Dynamics Model for Non-Equilibrium Turbulence

This paper introduces a computational model that computes the time-dependent evolution of a non-equilibrium turbulence spectrum from the system size down to the viscous dissipation scale. Turbulence models for use with Computational Fluid Dynamics (CFD) have at best treated the inertial-range below the CFD resolution as if obeying a renormalizable or scale-similar equilibrium described by the Kolmogorov spectrum with a spectral energy density that scales as k–5/3. The “Coherent Structure Dynamics” (CSD) model introduced here addresses situations where the timescale for changes in the macroscopic fluid dynamics is short and thus the resulting turbulence is far from an equilibrium cascade because the turbulent small scales that drive the dissipation will not have had time to equilibrate. Such circumstances can be caused, for example, by strong shocks passing through passive density gradients or fuel injection into supersonic flows. Mixing on the molecular scale and thus chemical reactions will be delayed until the short scales in the velocity spectrum are energized. The CSD model is not derived from the Navier-Stokes equations. It is constructed to satisfy the important physical conditions of the problem including scale consistency of the inviscid, nonlinear, fluid-dynamic interactions between the coherent structures that actually comprise turbulence. In addition to treating the kinetic energy density as a function of scale size down to the Kolmogorov dissipation scale, a number density of coherent structures at each scale is introduced to account for the fact that the relative spacing of the structures comprising turbulence, particularly away from equilibrium, may not be the same at all scales. This dynamic system relaxes to the Kolmogorov spectrum with a definite pre-dissipative bump (the bottleneck). Two scaleindependent parameters in the model are calibrated using the Taylor-Green vortex problem. Examples are presented and tests of the model are discussed. _______________ Manuscript approved September 14, 2018. 2 Glossary: ρ mass density of the fluid (gm/cc). ν Kinematic viscosity of the fluid (gm/(sec cm) ). L%&% System scale length, 10 m in examples following. R( Length scale (cm) of the rotors in scale size bin k. The following variables change in time due to the stiff evolution equations . . . E( Energy density (ergs/cc) of rotors of size Rk (k = 0, kmax). Nk Number density of rotors (#/cc) of scale size Rk. ε( Energy in a single rotor of size Rk. ε( = 3πρR(V( = E(/N( The following derived quantities are also used . . . P( 4 Packing fraction for rotors of size Rk (dimensionless). P( 4 ≡ 3πR( N(. Thus E( ≡ 3πρR(V(N( = ρP( V( P( Packing fraction temporary. P( ≡ (P( )0/. = (3πR(N() V( Characteristic average velocity (energy weighted) of rotors of size R( .