Rotational transformations of the sum of the squares of the flux Jacobians of the Navier–Stokes equations

Abstract

We report the discovery of a new mathematical identity for the Navier–Stokes equations, which model the flow of a viscous compressible gas. The identity relates the sum of the squares of the viscous flux Jacobians in rotated coordinate systems, and takes the form of a remarkably compact similarity transformation: ∑ i=1 3 A * i d =G{ ∑ i=1 3 A i d A i d } G T . The proof of an analogous relationship for the sum of the squares of the Euler flux Jacobians appeared in Venkatasubban (Proc. R. Soc. Lond. (2001) A457, 1111–1114). At that time, the extension to the Navier–Stokes equations was not evident, given that this requires a separate and detailed derivation. It may be remarked that these relationships possess an elegant simplicity that the flux Jacobians themselves do not have. In developing numerical algorithms for flow solvers based on the Navier–Stokes equations, it is often necessary or advantageous to work in a rotated coordinate frame. This relation thus has potential applications in the development of new and improved algorithms for Navier–Stokes computer codes used to solve industrial problems in aerodynamics and fluid dynamics.