60th Annual Meeting of the Divison of Fluid Dynamics
Volume 52, Number 12
Sunday–Tuesday, November 18–20, 2007;
Salt Lake City, Utah
Session KD: Turbulence: Theory I
8:00 AM–10:10 AM,
Tuesday, November 20, 2007
Salt Palace Convention Center
Room: 151 A-C
Chair: Julian Domaradzki, University of Southern California
Abstract ID: BAPS.2007.DFD.KD.10
Abstract: KD.00010 : The Molecular Origin of Turbulence in a Flowing Gas According to James Clerk Maxwell
9:57 AM–10:10 AM
Preview Abstract
Abstract
Author:
Albert de Graffenried
(North American Digital Company)
James Clerk Maxwell was an eminent physicist who operated out of
the University of Edinburgh in the early 1800's. He is
internationally famous for his derivation of the laws governing
the propagation of electro-magnetic waves.
He also derived an equation for the Viscosity of a gas ($\mu )$
in terms of its \textbf{molecular} parameters. This derivation
established clearly and unequivocably that a real (viscous)
flowing gas was a \textbf{molecular fluid}, that is, a flow of
molecules which obeys the Kinetic Theory of Gases.
Maxwell's derivation of the Viscosity of a gas takes place in a
zone of a
flowing gas which (1) is remote from any solid surface, and (2)
is in a
region having a linear velocity-gradient dv$_{x}$/dy .
The derivation which I will present today takes place in a zone
of the
flowing gas which is (1) immediately adjacent a solid surface,
and (2) where
the velocity gradient is unknown.
My analytical approach, the parameters I use, and the theoretical
concepts are all taken from Maxwell's derivation. I have simply
re-arranged
some of his equations in order to solve the 1-dimensional case of
boundary-layer growth over an infinite flat plate, starting with a
step-function of flow velocity, namely: v$_{x}$(y,t) for the initial
condition v$_{x}$(y=0+,t=0+) = U$_{0}$ ,viz: rectilinear flow as
an initial
condition.
Using Maxwell's approach, we write the equation for Net
Stream-Momentum
Flux flowing through an element of area, da$_{y}$ . This quantity
is shown
to be the difference between two Convolution integrals which Laplace
transform readily into an equation in the
s-plane which equation has the same form as a positive-feedback,
single
closed-loop amplifier gain equation, viz:
Output = (input)x(transfer function).
The solution in the Real plane shows v$_{x}$(y,t) equal to the
sum of two exponentials. The coefficients of the two exponents,
r$_{1}$ and r$_{2}$ . are found by using the binomial equation
which contains a square-root radical. If the argument under the
radical (the radicand) is positive, the two roots are real, and
turbulence does not occur.
If the radicand is negative, the two roots are complex conjugates
and turbulence will develop.
The physical reality of the transfer function's feedback-loop
format may be clarified by tracing backwards through the
derivation to the earliest occurrence of v$_{x}$(y,t).
Maxwell's derivation of Viscosity, adapted to solve for the
boundary-layer
growth over an infinite flat plate, is shown to be a nice
application of the
Kinetic Theory of Gases, and is well suited to revealing the
molecular
mechanisms at work in such a flowing pattern.
To cite this abstract, use the following reference: http://meetings.aps.org/link/BAPS.2007.DFD.KD.10