On Gauge Fields In Fluid Turbulence
Author(s)
T. H. Moulden
Abstract
A standard gauge velocity field is introduced into the constant density Navier–
Stokes equations and extended to treat fluid turbulence. Some of the elementary
properties of such gauge fields are explored and relationships between them
established. In particular, the gauge fields separate out the irrotational motion
from the vortical motion. An example that describes the inactive component of
a turbulent flow is presented.
Keywords: gauge fields, turbulence models, energy estimates.
1 Background
Write the field equations for the flow of a constant density viscous fluid in the
standard form:
∂vi
∂xi
= 0; ∂vi
∂t
+ ∂
∂xj
(vivj) + ∂P
∂xi
= ν
∂2vi
∂xj ∂xj
+ fi (1a, b)
where P(x, t) is the pressure field normalized by the constant fluid density.
Equations (1a,b) define the field (v, P) in which v(x, t) is the instantaneous
velocity field at location x and time t. f(x) denotes a time independent body
force in equation (1b). Herein, the kinematic viscosity coefficient, ν, is taken to
be constant. To be useful these equations must possess a unique regular solution
for meaningful boundary conditions; and the existence of such a solution is
assumed in the following discussion. Certain additional assumptions must be made
before equations (1a,b) can be adopted to describe the turbulent flow of a viscous
fluid. These assumptions include the need for the turbulent velocity and pressure
fluctuations to be continuous and have bounded energy. It is also required that all
velocity-related correlation functions be bounded.As is standard, the divergence of
equation (1b) produces a Poisson equation to compute the pressure field directly:
∂2P
∂xi ∂xi
= ∂fi
∂xi
− ∂2
∂xi ∂xj
(vivj) (1c)
Keywords
gauge fields, turbulence models, energy estimates.
Related Book
Other papers in this volume
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