The Hopf fde for the Spatial Characteristic Functional: Cartesian Case

Variational derivatives according to the definitions of Gateaux and Fréchet are applied to derive the Hopf fde governing the evolution of the characteristic functional for turbulent flows. Several versions of the Hopf fde for the characteristic functional \(\varTheta \lbrack \mathbf {y};t\rbrack ,\mathbf {y}\in \mathcal {N}\) derived from the dimensionless Navier-Stokes equations plus appropriate initial and boundary conditions are considered. The terms representing convection, viscous effects, and the pressure gradient are analyzed in the spatial description and Cartesian coordinates. The pressure gradient functional is formulated for Cartesian coordinates in the flow domain; the computation of the Green’s function for the periodic pipe flow domain \(\mathcal {D}\) is presented in detail in Chap. 28 . The effects of external forces on the momentum pdes are discussed for non-random and random forces. The resulting Hopf fdes given for non-random and random external forces, for homogeneous Dirichlet conditions set for velocity and solenoidal basis, is shown to be free of pressure terms due to orthogonality of scalar gradients and argument fields. The spectral version of the Hopf fde is derived to show that convection appears as diffusion in argument space.