Topology and Turbulence

Abstract

Over a given regular domain of independent variables {x,y,z,t}, every covariant vector field of flow can be constructed in terms a differential 1-form of Action. The associated Cartan topology permits the definition of four basic topological equivalence classes of flows based on the Pfaff dimension of the 1-form of Action. Potential flows or streamline processes are generated by an Action 1-form of Pfaff dimension 1 and 2, respectively. Chaotic flows must be associated with domains of Pfaff dimension 3 or more. Turbulent flows are associated with domains of Pfaff dimension 4. It will be demonstrated that the Navier-Stokes equations are related to Action 1-forms of Pfaff dimension 4. The Cartan Topology is a disconnected topology if the Pfaff dimension is greater than 2. This fact implies that the creation of turbulence (a state of Pfaff dimension 4 and a disconnected Cartan topology) from a streamline flow (a state of Pfaff dimension 2 and a connected topology) can take place only by discontinuous processes which induce shocks and tangential discontinuities. On the otherhand, the decay of turbulence can be described by continuous, but irreversible, processes. Numerical procedures that force continuity of slope and value cannot in principle describe the creation of turbulence, but such techniques of forced continuity can be used to describe the decay of turbulence.