Morphing continuum theory for turbulence: Theory, computation, and visualization

Abstract

A high order morphing continuum theory (MCT) is introduced to model highly compressible turbulence. The theory is formulated under the rigorous framework of rational continuum mechanics. A set of linear constitutive equations and balance laws are deduced and presented from the ColemanNoll procedure and Onsager's reciprocal relations. The governing equations are then arranged in conservation form and solved through the finite volume method with a second order Lax-Friedrichs scheme for shock preservation. A numerical example of transonic flow over a three-dimensional bump is presented using MCT and the finite volume method. The comparison shows that MCTbased DNS provides a better prediction than NS-based DNS with less than 10% of the mesh number when compared with experiments. A MCT-based and frame-indifferent Q-criterion is also derived to show the coherent eddy structure of the downstream turbulence in the numerical example. It should be emphasized that unlike the NS-based G-criterion, the MCT-based Q-criterion is objective without the limitation of Galilean-invariance.

First-order approximation to the Boltzmann–Curtiss equation for flows with local spin

“…In the meantime, a high order continuum theory considering local spin, e.g. morphing continuum theory [15], can be derived under rational continuum thermomechanics (RCT) or constructed from a high order distribution function, e.g. Boltzmann-Curtiss distribution [14].…”

“…With the definition of the heat flux in equation 21, and the stresses in equations 22, 23, and 24, along with the definition of the average in equation 6, the zeroth and first-order approximations to the missing terms in the balance laws can be calculated. Beginning with the zeroth order approximations, the averaging is carried out with f 0 [15]:…”

“…This challenge is eliminated if rotation is treated as an explicit, independent variable, allowing for a single distribution function to describe the small-scale rotation in the system. Recently, an additional theory derived from the perspective of rational contin-num thermomechanics (RCT) has provided governing equations for the mass, momenta, and energy of a fluid composed of spherical particles [15,16,24,25,26]. These equations start from a description of the fluid that deviates from any classical mechanical fluid.…”

“…When the particles are given a finite size and allowed to rotate, additional motions bring additional degrees of freedom to the system. If the angular motion of the particles is independent from the translational motion and is dependent on its orientation, then the transport equation has the form [15,20]:…”

“…Still, the physical picture from which equations 69, 70, 71, and 72 are derived differs from Boltzmann's classical picture of a monatomic gas only through the introduction of the variable of gyration. When the gyration of a particle is distinct from macroscopic rotation, as defined by the angular velocity, 1 2 sab ∂U b ∂xa , the new form of the linear momentum equation 70 and the independent angular momentum equation 71 can provide an alternative description to the classical Navier-Stokes picture.The difference of vorticity and gyration forms an objective (frame-indifferent) description of rotational motion, absolute rotation, as[15] The disappearance of absolute rotation indicates that vorticity is solely responsible for the local rotation and the dependence of gyration vanishes. Thus, such relation is called as angular equivalence.…”