Topology and Turbulence

R. M. Kiehn, "Topology and Turbulence" APS 1992

Abstract: Over a given regular domain of independent variables {x,y,z,t}, every covariant vector field of flow can be constructed in terms of at most four functions and their first derivatives. These functions may be used to construct a differential 1-form of Action. The irreducible number of such functions required to describe the topological properties of the 1-form of Action is defined as the Pfaff dimension of the domain. These functions, their first derivatives, and certain algebraic intersections of these sets, may be used to construct a Cartan topology, and thereby define the four basic topological equivalence classes of Pfaff dimension one, two, three and four on space time. Those vector field solutions to the Navier-Stokes equations which are uniquely integrable in the sense of Frobenious emulate potential flows or streamline processes; they generate an Action 1-form of Pfaff dimension 1 and 2, respectively. Chaotic solutions to the Navier-Stokes equations must be associated with Action domains of Pfaff dimension 3; but such flows are deterministic and reversible in space time. Only domains of Pfaff dimension 4 admit of evolutionary flow processes that can break time reversal and parity symmetry, and therefore may be used to describe the irreversible processes inherent in turbulent flows. When the Pfaff dimension is 2 or less, the Cartan Topology induced by the vector field is connected. When the Pfaff dimension is greater than 2, the induced Cartan topology is disconnected. The topological implication is 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.


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