From the Navier-Stokes equations
via the Reynolds decomposition
to a working turbulence closure model
for the shallow water equations:
The compromise between complexity and pragmatism.
Hans Burchard
Leibniz Institute for
Baltic Sea Research Warnemünde
hans.burchard@io-warnemuende.de
Why are we stirring our cup of coffee?
Milk foam: light, because of foam and fat
Coffee: relatively light, because hot
Milk: less light, because colder than coffee
Why the spoon?
…OK, and why the coocky?
little mixing
strong mixing
little stirring strong stirring
From stirring to mixing …
Tea mixing (analytical solution)
Put 50% of milk into tea.
z
Let m(z) be the milk fraction with m=1 at the bottom
and m=0 at the surface.
With a constant mixing coefficient,
the m-equation is this:
10cm
Let us take the spoon
and stir the milk-tea mix
n-times such that we
get a sinosodial milk-tea
variation in the vertical
and then see the
resulting mixing after 1 min:
Conclusion: stirring leads to increased mixing.
Set of equations that describes turbulent mixing
Navier-Stokes equations (for velocity vector u1, u2, u3):
tendency
advection
Incompressibility
stress divergence
condition:
Earth rotation
pressure
gradient
gravitational
force
 6 equations for 6 unknowns (u1, u2, u3, p, ,  )
Temperature equation:
stirring
Equation of state:
mixing
Example for solution of Navier-Stokes equations (KH-instability)
Direct Numerical Simulation (DNS) by William D. Smyth, Oregon State University
Reynolds decomposition
To reproduce system-wide mixing, the smallest dissipative scales
must be resolved by numerical models (DNS).
This does not work in models for natural waters due to limited
capacities of computers.
Therefore, the effects of turbulence needs to be partially
(= Large Eddy Simulation, LES) or fully (Reynolds-averaged
Navier-Stokes, RANS) parametersised.
Here, we go for the RANS method, which means that small-scale
fluctuations are „averaged away“, i.e., it is only the expected
value of the state variables considered and not the actual value.
Reynolds decomposition (with synthetic tidal flow data)
Any turbulent flow can be decomposed
into mean and fluctuating components:
Reynolds decomposition
There are many ways to
define the mean flow, e.g.
time averaging (upper panel)
or ensemble averaging (lower
panel).
For the ensemble averaging,
a high number N of
macroscopically identical
experiments is carried out
and then the mean of those
results is taken. The limit
for N   is then the
ensemble average (which is
the physically correct one).
Time averaging
Ensemble averaging
Reynolds decomposition
For the ensemble average 4 basic rules apply:
Linearity
Differentiation
Double averaging
Product averaging
The Reynolds equations
These rules can be applied to derive a balance equation
for the ensemble averaged momentum.
This is demonstrated here for a simplified (one-dimensional)
momentum equation:
The Reynolds stress constitutes a new unknown
which needs to be parameterised.
The eddy viscosity assumption
Reynolds stress and
mean shear are
assumed to
be proportional
to each others:
eddy viscosity
The eddy viscosity assumption
The eddy viscosity is typically orders of magnitude larger
than the molecular viscosity. The eddy viscosity is however
unknown as well and highly variable in time and space.
Parameterisation of the eddy viscosity
Like in the theory of ideal gases, the eddy viscosity can be assumed
to be proportional to a characteristic length scale l and a velocity scale v:
In simple cases, the length scale l could be taken from geometric arguments
(such as being proportional to the distance from the wall). The velocity scale
v can be taken as proportional to the square root of the turbulent
kinetic energy (TKE) which is defined as:
such that
(cl = const)
Dynamic equation for the TKE
A dynamic equation for the turbulent kinetic energy (TKE) can be derived:
with
P: shear production
B: buoyancy production
e: viscous dissipation
Dynamic equation for the length scale (here: e eq.)
A dynamic equation for the dissipation rate of the TKE) is constructed:
with the adjustable empirical parameters c1, c2, c3, se.
With this, it can be calculated
with simple
stability functions cm and cm‘.
All parameters can be calibrated to characteristic properties of the flow.
Example on next slide: how to calibrate c3.
Layers with homogeneous stratification and shear
For stationary & homogeneous stratified shear flow,
Osborn (1980) proposed the following relation:
which is equivalent to
(N is the buoyancy frequency),
a relation which is intensively used to derive the
eddy diffusivity from micro-structure observations.
For stationary homogeneous shear layers, the k-e model reduces to
which can be combined to
.
Thus, after having calibrated c1 and c2,
c3 adjusts the effect of stratification on mixing.
Umlauf (2009), Burchard and Hetland (2010)
Mixing = micro-structure variance decay
Example: temperature mixing
Temperature equation:
Temperature variance equation:
Mixing
Second-moment closures in a nut shell
Instead of directly imposing the eddy viscosity assumption
With
one could also derive a transport equation for
and the turbulent heat flux (second moments). These second-moment
equations would contain unknown third moments, for which also equations could
be derived, etc. The second-moments are closed by assuming local
equilibrium (stationarity, homogeneity) for the second moments. Together
with further emipirical closure assumptions, a closed linear system of equations
will then be found for the second moments. Interestingly, the result may be
, where now cm and cm‘ are
formulations as follows:
functions of
and
with the shear squared, M2.
Such two-equation second moment-closures are now the workhorses in
coastal ocean modelling (and should be it in lake models) and have been
consistently implemented in the one-dimensional
General Ocean Turbulence Model (GOTM)
which has been released in 1999 by Hans Burchard and Karsten Bolding
under the Gnu Public Licence. Since then, it had been steadily
developed and is now coupled to many ocean models.
GOTM application: Kato-Phillips experiment
Stress-induced
entrainment
into linearly
stratified fluid
Dm(t)
Empirical
G<0.2
G>0.2
Empirical:
GOTM application: Baltic Sea surface dynamics
unstable
Reissmann et al., 2009
Take home:
Due to stirring, turbulence leads to an increase of effective
mixing and dissipation by several orders of magnitude.
For simulating natural systems, the Reynolds decomposition into
mean (=expected) and fluctuating parts is necessary.
Higher statistical moments are parameterised by means of
turbulence closure models.
Algebraic second-moment closures provide a good compromise
between efficiency and accuracy. Therefore such models are
ideal for lakes and coastal waters.
Question: Will we be able to construct a robost and more
accurate closure model which resolves the second moments
( inclusion of budget equations for momentum and heat flux)?