MODELLING AND COMPUTATIONAL SIMULATION OF EULERIAN FLOW

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MODELLING AND COMPUTATIONAL SIMULATION OF
EULERIAN FLOW

• Wayne Lawton
• Department of Mathematics
• National University of Singapore
• 2 Science Drive 2
• Singapore 117543

Email matwml_at_nus.edu.sg Tel [phone]
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RIGID BODIES
Eulers equation
for their inertial motion
angular velocity in the body
inertia operator (from mass distribution)
Theoria et ad motus corporum solidorum seu
rigodorum ex primiis nostrae cognitionis
principiis stbilita onmes motus qui inhuiusmodi
corpora cadere possunt accommodata, Memoirs de
l'Acad'emie des Sciences Berlin, 1765.
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IDEAL FLUIDS
Eulers equation
for their inertial motion
pressure
velocity in space
outward normal
of domain
Commentationes mechanicae ad theoriam corporum
fluidorum pertinentes, M'emoirs de l'Acad'emie
des Sciences Berlin, 1765.
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GEODESICS
Moreau observed that these classical equations
describe geodesics, on the Lie groups that
parameterize their configurations, with respect
to the left, right invariant Riemannian metric
determined by the inertia operator (determined
from kinetic energy) on the associated Lie algebra
Une method de cinematique fonctionnelle en
hydrodynamique, C. R. Acad. Sci. Paris 249(1959),
2156-2158
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LIE GROUPS, LIE ALGEBRAS, AND THEIR
REPRESENTATIONS
Lie group
Lie algebra
linear dual
define the adjoint
For
and coadjoint representations
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CARTAN-KILLING OPERATOR
Theorem 1.The Cartan-Killing operator
defined by
is self-adjoint and satisfies
B is nonsingular iff G is semisimple (Cartan), B
is positive semidefinite iff G is compact (Weyl)
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TRAJECTORIES AND VELOCITIES
Definition If
is a smooth trajectory
and
in a Lie group G then
are trajectories in its Lie algebra
These trajectories are called the (angular)
velocities in the body and in space.
and define
In the sequel we will let
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INERTIA OPERATORS AND KINETIC ENERGY
is an inertial operator
Definition
if it is self-adjoint and positive definite. Then
A defines an inner product
and a left, right invariant Riemannian metric on G
Kinetic energy of rigid bodies and fluids defines
an
inertial operator
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EULERS EQUATION ON LIE GROUPS
Theorem 2. The motion of a physical system whose
configuration space is a Lie group G is a
trajectory in G. If the kinetic energy E is left,
right invariant then the trajectory is a geodesic
(shortest path between any two points) with
respect to the left, right invariant Riemannian
metric induced by E
Theorem 3. A trajectory is a geodesic with respect
to the left
right
invariant
Riemannian metric on G iff u satisfies Eulers
eqn.
Arnold, V. I., Mathematical Methods of
ClassicalMechanics, Springer, New York, 1978
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GLOBAL ANALYSIS
based on this geometric formulation provides a
powerful tool for studying fluid dynamics
Arnold used it to explain sensitivity to initial
conditions in terms of curvature
Ebin, Marsden, and Shkoller used it to derive
existence, uniqueness and regularity results for
both Eulers and Navier-Stokes equations
These ideas are fundamental for the study of a
large class of nonlinear partial differential
equations and have developed into the extensive
field of topological hydrodynamics
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VORTICITY FORMULATION
Theorem 4. Let
be an inertial operator,
be the Cartan-Killing operator, and
with
If
satisfies the vorticity equation
then
satisfies the equation
If
is nonsingular then the converse holds
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VORTICITY FORMULATION
Corollary. Let
be an inertial operator
defined on a semisimple Lie group
is a geodesic for the left
Then
invariant Riemannian metric defined
right
by
iff
where
and
for
is the vorticity
Under these conditions the enstrophy
is constant
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LAGRANGIAN FORMULATIONS
and
Theorem 5. Let
and
for
then
satisfies
iff
Corollary A trajectory
is a geodesic
with respect to the left
right
invariant Riemannian metric iff
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LAGRANGIAN FORMULATIONS
and
Theorem 6. Let
and
for
then
satisfies
iff
Corollary If G is semisimple then
is a geodesic with respect to the left
invariant Riemannian metric
right
induced by an inertial operator A iff the
vorticity
satisfies
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EULER SYMPLECTIC INTEGRATOR
Preserves
Error
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IMPROVED SYMPLECTIC INTEGRATOR
Preserves
Error
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SPECTRAL BASIS
for
There exist a basis
and
such that
and
If
then
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SPECTRAL REPRESENTATION
The vorticity equation has the spectral
representation
where the structure constants
are defined by
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STREAM FUNCTION
Then
(orthogonal coordinates
The operator
is convolution
with the Greens function
is constant along particles in the flow,
therefore
the moments
are invariant
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EULER FLOW ON
Identified with ideal flows on
that are
periodic with respect to the subgroup
with average value zero, for the spectral basis
of the complexified Poisson Bracket Lie algebra
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FAIRLIER, FLETCHER AND ZACHOS
for odd
defined the map
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ZEITLIN
used the approximation
to approximate flow on
by flows on
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NUMERICAL EXPERIMENTS
Simplest model is the 8 dim Lie group
Vorticity is represented as a real valued
function on
the group
Rotations, translations, and reflections (later
must be combined with multiplying the vorticity
by 1) are Lie group automorphisms that also
commute with the inertial operator A and
therefore with Euler flow
Vorticities that are invariant under these
operations are either fixed points or periodic
points of Euler flow
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WAVELET BASES
Neither the canonical Fourier basis nor the
canonical sparse matrix basis provides a
sparse representation of Eulers equation on SU(n)

• Wavelet vorticity bases provide nearly sparse
• representations for Eulers equations because
• Greens operator is Calderon-Zygmund
• (ii) Poisson bracket is exponentially localized

Wavelet bases provide simple approximations for
invariant moments and energy
We are using wavelet bases to study
Okubo-Weiss criteria for two-dimensional
turbulence
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FUTURE STUDIES
Determine if Euler flow on SU(3) is integrable
Analyze the effects of curvature
Investigate vorticity cascade starting with SU(5)
Study the role of symmetry
Explore gauge theoretic issues including instanton
s