dorsal/arxiv
View SchemaTopology and Turbulence
| Authors | R. M. Kiehn |
|---|---|
| Categories | |
| ArXiv ID | physics/0102003 |
| URL | https://arxiv.org/abs/physics/0102003 |
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.
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"abstract": "Over a given regular domain of independent variables {x,y,z,t}, every\ncovariant vector field of flow can be constructed in terms a differential\n1-form of Action. The associated Cartan topology permits the definition of four\nbasic topological equivalence classes of flows based on the Pfaff dimension of\nthe 1-form of Action. Potential flows or streamline processes are generated by\nan Action 1-form of Pfaff dimension 1 and 2, respectively. Chaotic flows must\nbe associated with domains of Pfaff dimension 3 or more. Turbulent flows are\nassociated with domains of Pfaff dimension 4. It will be demonstrated that the\nNavier-Stokes equations are related to Action 1-forms of Pfaff dimension 4. The\nCartan Topology is a disconnected topology if the Pfaff dimension is greater\nthan 2. This fact implies that the creation of turbulence (a state of Pfaff\ndimension 4 and a disconnected Cartan topology) from a streamline flow (a state\nof Pfaff dimension 2 and a connected topology) can take place only by\ndiscontinuous processes which induce shocks and tangential discontinuities. On\nthe otherhand, the decay of turbulence can be described by continuous, but\nirreversible, processes. Numerical procedures that force continuity of slope\nand value cannot in principle describe the creation of turbulence, but such\ntechniques of forced continuity can be used to describe the decay of\nturbulence.",
"arxiv_id": "physics/0102003",
"authors": [
"R. M. Kiehn"
],
"categories": [
"physics.flu-dyn"
],
"title": "Topology and Turbulence",
"url": "https://arxiv.org/abs/physics/0102003"
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