Ab initio method of calculating invariant measure for turbulent flow

0557955 - ASÚ 2023 SE eng J - Journal Article
Macháček, Martin
Ab initio method of calculating invariant measure for turbulent flow.
Physica Scripta. Roč. 97, č. 5 (2022), č. článku 055208. ISSN 0031-8949. E-ISSN 1402-4896
Institutional support: RVO:67985815
Keywords : equilibrium statistics * dynamical-systems * fluid turbulence
OECD category: Astronomy (including astrophysics,space science)
Impact factor: 2.9, year: 2022
Method of publishing: Open access

We present a method of calculating the invariant measure (IM) of hydrodynamical systems describing incompressible viscous fluid flow in bounded 3D volume V. Using a basis {w(I)} of zero-divergence vector functions on V satisfying boundary conditions we transform the Navier-Stokes equations (NSE) for the velocity field v(x, t) = Sigma(I) v(I) (t)w(I)(x) into a simple system of ordinary differential equations for v(I) (t). The fact that fluid consists of a finite number of molecules in thermal motion implies that the number N of variables v(I) is finite too and that the system of equations for v(I) (t) contains a white-noise term. We prove that all solutions of this system are global, the IM exists and is unique. Its density psi(v) defined on the phase space R-N satisfies an elliptic partial differential equation (PDE). Expanding psi(v) = Sigma(mu)< He-mu > He-mu(v)phi(epsilon)(v) into Hermite functions He-mu(v)phi(epsilon) (v) (where phi(epsilon) is a Gaussian function on R-N, He (mu) are Hermite polynomials orthonormal with the weight phi(epsilon) and < He-mu > = integral He-mu(v)psi(v)d(N)v are IM mean values of He-mu) we transform this PDE into an infinite system of linear algebraic equations for < He-mu > and suggest somemethods for solving it numerically. From known first- and second-order < He-mu > we could easily calculatemean turbulent velocity v(x) and Reynolds stress at any point x. Cylindrical pipe flow is used to illustrate the method everywhere in the paper. All conclusions are derived from the mathematical model by rigorous mathematics, with no further assumptions. All approximations can be arbitrarily improved.
Permanent Link: https://hdl.handle.net/11104/0337715