% see also numbers.tex
Fully developed turbulent flow is attained when the distance downstream
is x>100d.
(d is depth)
page 32 Sleith
A Reynolds number is $\bar{U} d/ \nu$.
% 2 m/s * 100 m / (viscosity of water)
If this is bigger than 5000 then laminar flow is unstable and turbulence
dominates.
For tides I reckon R is something like 200,000,000.
Turbulent flow is chopped into 3 layers:
Inner layer
Overlap layer
Outer layer
Inner layer assuming a smooth bed
Mean velocity at height y is
\bar{u} = y \bar{u}^2_* / \nu
where
\bar{u}_* = ( \bar{\tau}_0 / \rho )^{1/2}
[$\tau$ is mean shear stress.]
Thickness of this viscous sublayer
\delta = 11.6 \frac{ \nu }{ \bar{u}_* }
Overlap layer - overlap between wall layer and `defect' layer.
Possible form of velocity (Prandtl--Von Karman)
\frac{ \bar{u} }
{ \bar{u}_* } = \frac{1}{K} \ln \frac{ y}{y_0}
where K is 0.4. (Magic Karman constant.)
And $y_0$ must be determined by experiment. It depends on the
size $k_s$ of the bed roughness compared with the
thickness $\delta$ of the viscous sublayer. Which depends on
a Reynolds number ubar* k_s / nu.
Outer layer
From sanjoy@mrao.cam.ac.uk Fri Dec 15 03:05:09 2006
To: "David J.C. MacKay"
Subject: Re: bed
Date: Fri, 15 Dec 2006 03:05:07 +0000
From: Sanjoy Mahajan
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Status: RO
X-Status: A
This relates to one of the two sole results in the whole theory of
turbulence. One result is the Kolmogorov scaling, which gives the
power spectrum in isotropic turbulence (so no walls). The other
result is the 'law of the wall' also called the 'law of the
logarithm', first guessed by Prandtl and then by his student von
Karman.
I don't remember the whole story -- need to do it for my book though.
But here are half-remembered pieces of it.
The friction is caused by the eddies (like the eddies that the
Kolmogorov spectrum gives you the power spectrum for) that turn into
smaller eddies that turn into small-enough eddies for viscosity to
gobble them (the last step being the dissipative one).
One result in general is that eddies turn into smaller eddies in one
turnover time. So an eddy of size l with velocity v(l) has a turnover
time l/v(l), and in that time it turns into smaller eddies. So the
energy in the eddy (per mass) is v^2, and the power dissipation is
v^2 * 1/time = v^3/l.
(The Kolmogorov spectrum comes from saying that v^3/l is a constant,
because all the energy from one size goes into the next smaller size,
and cannot build up.)
Another question then is how big the eddies are as a function of
height above the bottom of the river. That's where dimensions comes
in. There's no length scale except the height z, so l = constant*z.
The constant is called the von Karman constant and has value,
emprically, of \kappa \approx 0.4. There are lots of hokey
theoretical derivations of it (using the renormalization group, and
what not) but none of them are convincing. So for now it's a number
that is somewhere in the Navier-Stokes equations but our mathematics
is not strong enough to find it.
The eventual result is that the flow velocity at height z is
v \propto (1/\kappa) * log(z/z0),
where z0 is the "roughness scale". You'd think the roughness scale
would be the boulder or pebble size, but emprically it's pretty random
but has a weak relation. Taking all that into account, the fit with
experimental data is good.
The whole result is the subject of a huge war in the fluids world.
G. I. Barenblatt (of Caius at times), who Tadashi knows, says that
it's not a logarithmic law at all but a set of power laws. He gets it
by a different argument which I don't know at all. But as you can
imagine, it's hard to distinguish a high-order root (like a 1/7th
power) from a logarithm, so the wars rage. [The old tables, e.g. for
pipe flow, all used various high-order roots, which worked well
emprically, and the order of the root depended on the Reynolds number.
Tritton's fluid book is a good one for the von Karman law, and
Barenblatt's book on asymptotics has his view of how it works, which
as I say I don't understand at all.
-Sanjoy
`Never underestimate the evil of which men of power are capable.'
--Bertrand Russell, _War Crimes in Vietnam_, chapter 1.
Further references found in
http://www.kirj.ee/public/Engineering/2007/issue_3/eng-2007-3-4.pdf
39. Shin, S. Determination of the shear velocities, the bottom roughness and friction factors.
Publications of the Oregon State University, Swash Zone Turbulence Group, Report 2,
2004, 15 p.
http://www.google.com/search?hl=en&safe=off&sa=X&oi=spell&resnum=1&ct=result&cd=1&q=kajiura+brevik+jensen&spell=1
trying to find a figure of stephen s's
which shows many coefficients versus
something.
kajiura brevik jenssen sleath
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 96, NO. C8, PAGES 15,237%Gâ%@15,244, 1991
Velocities and Shear Stresses in Wave-Current Flows
John F. A. Sleath
Engineering Department, Cambridge University, Cambridge, England
Abstract
A model is presented for the velocity distributions in combined wave-current flows over rough beds. The model is based on a new analysis of turbulence measurements in oscillatory flow. The resulting formulas for the velocity distribution are remarkably simple and show good agreement with experiment. The formula derived for friction factor also shows good agreement with experiment.
Received 10 October 1990; accepted 23 April 1991.
Sleath defines friction factor f_w
to include the 0.5:
horizontal stress
tau_0
= f_w rho U_0^2 / 2
U0 is the amplitude of the
free stream velocity (oscillatory).
Variation of friction factor with
a/k_s
what's a, k_s?
a is the semiorbital amplitude of the
grid.
k_s is the roughness length of the
bed of sediment