Vorticity - FEniCS Project
Vorticity - FEniCS Project
Vorticity - FEniCS Project
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Automation of Turbulence<br />
Simulation<br />
Johan Hoffman<br />
jhoffman@nada.kth.se<br />
Royal Institute of Technology KTH<br />
Johan Hoffman – KTH – p.1
Some References<br />
* J.Hoffman, Computation of mean drag for bluff body problems using Adaptive<br />
DNS/LES, SIAM J. Sci. Comput. Vol.27(1), 2005.<br />
* J.Hoffman and C.Johnson, A new approach to Computational Turbulence Modeling, to<br />
appear in Comput. Methods Appl. Mech. Engrg., 2005.<br />
* J.Hoffman, Simulation of turbulent flow past bluff bodies on coarse meshes using<br />
General Galerkin methods: drag crisis and turbulent Euler solutions, to appear in<br />
Springer Journal of Computational Mechanics, 2005.<br />
* J.Hoffman, Efficient computation of mean drag for the subcritical flow past a circular<br />
cylinder using Adaptive DNS/LES, in review (Int. J. of Numerical Methods in Fluids) .<br />
* J.Hoffman, Adaptive simulation of the turbulent flow due to a cylinder rolling along<br />
ground, in review (Comput. Methods Appl. Mech. Engrg.).<br />
* J.Hoffman, Adaptive simulation of the turbulent flow past a sphere,<br />
in review (J. Fluid Mech.).<br />
* J.Hoffman and C.Johnson, Adaptive FEM for Incompressible Turbulent Flow,<br />
Springer, 2006.<br />
Johan Hoffman – KTH – p.2
Automation of Turbulence Simulation<br />
ACMM: Generality in: Method and Implementation<br />
Method: Adaptive stabilized FEM: General Galerkin G2<br />
Implementation: <strong>FEniCS</strong> project<br />
This talk: Test of Method: G2<br />
Turbulence Simulation; characterized by non-generality:<br />
discretization (geometry specific FD, spectral,...)<br />
turbulence modeling (parameters depend on data,<br />
numerics,...; different filters used in RANS/LES,...)<br />
wall modeling using RANS, boundary layer theory,...<br />
Johan Hoffman – KTH – p.3
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Målilla 88-95: 24-hr mean:<br />
C<br />
30<br />
20<br />
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−10<br />
−20<br />
−30<br />
0 50 100 150 200 250 300 350<br />
Johan Hoffman – KTH – p.4
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Målilla 88-95: weekly mean:<br />
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25<br />
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10<br />
5<br />
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−15<br />
0 5 10 15 20 25 30 35 40 45 50<br />
Johan Hoffman – KTH – p.5
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Målilla 88-95: monthly mean:<br />
C<br />
20<br />
15<br />
10<br />
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2 4 6 8 10 12<br />
Johan Hoffman – KTH – p.6
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Yearly mean 1860-2003:<br />
C<br />
Johan Hoffman – KTH – p.7
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Cube:<br />
, mean<br />
1.9<br />
1.8<br />
1.7<br />
1.6<br />
1.5<br />
1.4<br />
1.3<br />
1.2<br />
2 2.5 3 3.5 4 4.5 5 5.5<br />
cube: xy-plane<br />
cube: xz-plane<br />
Johan Hoffman – KTH – p.8
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Turbulence: Chaotic Dyn. System<br />
Pointwise quantities strongly sensitive to perturbations<br />
and thus unpredictable (to any tolerance of interest)<br />
Mean values in space/time moderately sensitive to<br />
perturbations and thus predictable up to a tolerance<br />
Ex: Weather prediction (Målilla):<br />
24-hr mean temperature unpredictable (<br />
monthly mean temperature predictable (<br />
C)<br />
C)<br />
Ex: Bluff body flow (experiments/computations):<br />
pointwise drag<br />
mean value drag<br />
unpredictable (<br />
predictable (<br />
)<br />
)<br />
Johan Hoffman – KTH – p.9
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viscosity,<br />
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Pointwise existence & uniqueness unknown:<br />
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Existence (but not uniq.) of weak solution (Leray 1934):<br />
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Johan Hoffman – KTH – p.10
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Navier-Stokes Equations (NSE)<br />
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velocity,<br />
pressure,<br />
viscosity,<br />
Computational turbulence: Mean values computable<br />
(<br />
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Computational cost (#dofs):<br />
(resolving all scales in Direct Num. Simulation DNS)<br />
).<br />
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Limit today:<br />
(many industrial appl.<br />
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Johan Hoffman – KTH – p.11
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Analysis & Computation of NSE<br />
Scientific goals today:<br />
Prove exist & uniq of pointwise NSE:<br />
Push limit of DNS (wrt<br />
constraint)<br />
Turbulence modeling: find model for unresolved scales<br />
(filtering of NSE: Reynolds stresses, closure problem)<br />
Alternative scientific goals:<br />
Approximate weak solution<br />
(mean value) output<br />
: weak uniqueness in<br />
: stability of<br />
wrt<br />
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Adaptive algorithm: min #dof :<br />
Johan Hoffman – KTH – p.12
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Existence of -Weak Solutions<br />
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Johan Hoffman – KTH – p.13
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Johan Hoffman – KTH – p.14<br />
Weak Uniqueness: Duality<br />
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Johan Hoffman – KTH – p.15<br />
Weak Uniqueness: Duality<br />
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Johan Hoffman – KTH – p.16<br />
G2 for NSE: Adaptive DNS/LES<br />
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G2 for NSE: Adaptive DNS/LES<br />
Galerkin discretization error + stabilization modeling error<br />
Adaptive algorithm: From coarse mesh<br />
do<br />
(1) compute primal and dual problem on<br />
(2) if<br />
TOL then STOP, else<br />
(3) refine elements<br />
with largest<br />
(4) set<br />
, then goto (1)<br />
Johan Hoffman – KTH – p.17
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Bluff Body Benchmark Problems<br />
Drag Coeff.<br />
= normalized mean drag force<br />
circular cylinder Re=3900<br />
dual solution<br />
sphere Re=10 000<br />
dual solution<br />
square cylinder Re=22 000<br />
dual solution<br />
cube Re=40 0000: xy<br />
dual sol. xy<br />
dual sol. xz<br />
cube xz<br />
Johan Hoffman – KTH – p.18
Ref. Mesh wrt<br />
: circular cylinder<br />
Johan Hoffman – KTH – p.19
Ref. Mesh wrt<br />
: sphere<br />
Johan Hoffman – KTH – p.20
Ref. Mesh wrt<br />
: square cylinder<br />
Johan Hoffman – KTH – p.21
Ref. Mesh wrt<br />
: surf mount cube<br />
Johan Hoffman – KTH – p.22
Circular cylinder:<br />
2<br />
1.8<br />
1.6<br />
1.4<br />
1.2<br />
1<br />
0.8<br />
4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5<br />
Johan Hoffman – KTH – p.23
Sphere:<br />
2<br />
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1.6<br />
1.4<br />
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0.6<br />
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0.2<br />
3.8 4 4.2 4.4 4.6 4.8 5<br />
Johan Hoffman – KTH – p.24
Square cylinder:<br />
2.5<br />
2<br />
1.5<br />
1<br />
4.2 4.4 4.6 4.8 5 5.2<br />
Johan Hoffman – KTH – p.25
Surface mounted cube:<br />
4.5<br />
4<br />
3.5<br />
3<br />
2.5<br />
2<br />
1.5<br />
1<br />
0.5<br />
2.5 3 3.5 4 4.5 5 5.5<br />
Johan Hoffman – KTH – p.26
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Summary Benchmark Problems<br />
G2 for NSE: No filtering. No Reynolds stresses. G2<br />
automatic “turbulence model”.<br />
Adaptive algorithm captures separation points, and<br />
“correct” (finite limit) dissipation in the turbulent wake.<br />
Mean value output (drag, lift, frequencies, separation<br />
points, pressure coeff,...) computable up to a tolerance<br />
corresponding to experimental accuracy (<br />
1-5%).<br />
About 10-100 times less mesh points needed to<br />
compute drag than in non-adaptive LES.<br />
All computations on a standard PC (2 GHz, 0.5Gb)<br />
Johan Hoffman – KTH – p.27
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Ex: Rotating Cylinder<br />
Rotating cylinder, ground moving at free stream speed<br />
Model of rotating cyl on ground: moving coordinate frame<br />
(Wheels of F1 racing car, airplane at take-off or landing,...)<br />
Compare with stationary wheel (simple wind tunnel testing)<br />
velocity xz velocity xy transversal velocities<br />
Johan Hoffman – KTH – p.28
G2: rot vs stat: velocity<br />
Johan Hoffman – KTH – p.29
G2: rot vs stat: pressure<br />
Johan Hoffman – KTH – p.30
G2: rot vs stat: vorticity<br />
Johan Hoffman – KTH – p.31
G2: rot vs stat: vorticity<br />
Johan Hoffman – KTH – p.32
G2: rot vs stat: dual velocity<br />
Johan Hoffman – KTH – p.33
G2: rot vs stat: dual pressure<br />
Johan Hoffman – KTH – p.34
G2: rot vs stat: mesh<br />
Johan Hoffman – KTH – p.35
G2: rot vs stat: mesh<br />
Johan Hoffman – KTH – p.36
G2: rot vs stat: mesh<br />
Johan Hoffman – KTH – p.37
G2: rot vs stat: drag : 1.3 vs 0.8<br />
4.5<br />
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3<br />
2.5<br />
2<br />
1.5<br />
1<br />
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4 4.2 4.4 4.6 4.8 5<br />
Johan Hoffman – KTH – p.38
G2: rot vs stat: forces<br />
2<br />
1.5<br />
1<br />
0.5<br />
0<br />
−0.5<br />
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Johan Hoffman – KTH – p.39
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Turbulent boundary layers<br />
So far: No slip boundary conditions seems ok for modeling<br />
laminar boundary layers (separation and skin friction)<br />
For high<br />
boundary layer undergoes transition<br />
Extremely expensive to resolve turbulent boundary layer:<br />
We need wall-modeling<br />
Johan Hoffman – KTH – p.40
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Turbulent boundary layers<br />
Separation given by:<br />
Retarding flow near the boundary:<br />
Turbulent boundary layer<br />
boundary<br />
delayed separation<br />
higher momentum near the<br />
Johan Hoffman – KTH – p.41
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Friction boundary condition<br />
Slip with friction boundary condition [Maxwell, Navier,....]<br />
: no slip b.c.<br />
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: slip b.c.,<br />
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Friction coefficient<br />
[LES + Boundary Layer theory: Layton, John, Illiescu,....]<br />
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Johan Hoffman – KTH – p.42
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drag crisis;<br />
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Johan Hoffman – KTH – p.43
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drag crisis;<br />
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Johan Hoffman – KTH – p.44
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drag crisis:<br />
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Johan Hoffman – KTH – p.45
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drag crisis;<br />
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Johan Hoffman – KTH – p.46
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EG2 and Turbulent Euler solutions<br />
;<br />
(<br />
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Euler/G2 + slip b.c. (EG2)<br />
EG2: no empirical parameters; only<br />
(very general...)<br />
What happens in the limit? The potential solution (<br />
)?<br />
Johan Hoffman – KTH – p.47
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Velocity & pressure: t=1.0:<br />
Johan Hoffman – KTH – p.48
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Velocity & pressure: t=1.25:<br />
Johan Hoffman – KTH – p.49
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Velocity & pressure: t=1.5:<br />
Johan Hoffman – KTH – p.50
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Velocity & pressure: t=1.75:<br />
Johan Hoffman – KTH – p.51
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Velocity & pressure: t=2.0:<br />
Johan Hoffman – KTH – p.52
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Velocity & pressure: t=2.25:<br />
Johan Hoffman – KTH – p.53
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Velocity & pressure: t=2.5:<br />
Johan Hoffman – KTH – p.54
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Velocity & pressure: t=2.75:<br />
Johan Hoffman – KTH – p.55
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Velocity & pressure: t=3.0:<br />
Johan Hoffman – KTH – p.56
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Velocity & pressure: t=4.5:<br />
Johan Hoffman – KTH – p.57
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Velocity & pressure: t=5.0:<br />
Johan Hoffman – KTH – p.58
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Velocity & pressure: t=5.5:<br />
Johan Hoffman – KTH – p.59
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Velocity & pressure: t=5.75:<br />
Johan Hoffman – KTH – p.60
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Velocity & pressure: t=11.0:<br />
Johan Hoffman – KTH – p.61
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<strong>Vorticity</strong>:<br />
,<br />
,<br />
: t=1.0:<br />
Johan Hoffman – KTH – p.62
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<strong>Vorticity</strong>:<br />
,<br />
,<br />
: t=1.25:<br />
Johan Hoffman – KTH – p.63
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<strong>Vorticity</strong>:<br />
,<br />
,<br />
: t=1.5:<br />
Johan Hoffman – KTH – p.64
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<strong>Vorticity</strong>:<br />
,<br />
,<br />
: t=1.75:<br />
Johan Hoffman – KTH – p.65
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<strong>Vorticity</strong>:<br />
,<br />
,<br />
: t=2.25:<br />
Johan Hoffman – KTH – p.67
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<strong>Vorticity</strong>:<br />
,<br />
,<br />
: t=2.5:<br />
Johan Hoffman – KTH – p.68
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<strong>Vorticity</strong>:<br />
,<br />
,<br />
: t=2.75:<br />
Johan Hoffman – KTH – p.69
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<strong>Vorticity</strong>:<br />
,<br />
,<br />
: t=1.0:<br />
Johan Hoffman – KTH – p.70
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<strong>Vorticity</strong>:<br />
,<br />
,<br />
: t=1.25:<br />
Johan Hoffman – KTH – p.71
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<strong>Vorticity</strong>:<br />
,<br />
,<br />
: t=1.5:<br />
Johan Hoffman – KTH – p.72
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<strong>Vorticity</strong>:<br />
,<br />
,<br />
: t=1.75:<br />
Johan Hoffman – KTH – p.73
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<strong>Vorticity</strong>:<br />
,<br />
,<br />
: t=2.0:<br />
Johan Hoffman – KTH – p.74
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<strong>Vorticity</strong>:<br />
,<br />
,<br />
: t=2.25:<br />
Johan Hoffman – KTH – p.75
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<strong>Vorticity</strong>:<br />
,<br />
,<br />
: t=2.5:<br />
Johan Hoffman – KTH – p.76
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¡<br />
<strong>Vorticity</strong>:<br />
,<br />
,<br />
: t=2.75:<br />
Johan Hoffman – KTH – p.77
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<strong>Vorticity</strong>:<br />
,<br />
,<br />
: t=1.0:<br />
Johan Hoffman – KTH – p.78
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<strong>Vorticity</strong>:<br />
,<br />
,<br />
: t=1.25:<br />
Johan Hoffman – KTH – p.79
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<strong>Vorticity</strong>:<br />
,<br />
,<br />
: t=1.5:<br />
Johan Hoffman – KTH – p.80
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<strong>Vorticity</strong>:<br />
,<br />
,<br />
: t=1.75:<br />
Johan Hoffman – KTH – p.81
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<strong>Vorticity</strong>:<br />
,<br />
,<br />
: t=2.0:<br />
Johan Hoffman – KTH – p.82
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¡<br />
<strong>Vorticity</strong>:<br />
,<br />
,<br />
: t=2.25:<br />
Johan Hoffman – KTH – p.83
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¡<br />
<strong>Vorticity</strong>:<br />
,<br />
,<br />
: t=2.5:<br />
Johan Hoffman – KTH – p.84
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<strong>Vorticity</strong>:<br />
,<br />
,<br />
: t=2.75:<br />
Johan Hoffman – KTH – p.85
Oscillating EG2 solution:<br />
2.1<br />
2<br />
1.9<br />
1.8<br />
1.7<br />
1.6<br />
1.5<br />
1.4<br />
1.3<br />
1.2<br />
1.1<br />
1 2 3 4 5 6 7<br />
Johan Hoffman – KTH – p.86
Oscillating EG2 solution<br />
Note: there is only one separation point (in each section)<br />
Johan Hoffman – KTH – p.87
Guadalupe, Canaries:<br />
Note: only one separation point (consistent with EG2)<br />
Johan Hoffman – KTH – p.88
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EG2 and Turbulent Euler solutions<br />
No experimental results for cylinder at<br />
Therefore not much is known about this flow regime.<br />
For geophysical flow (<br />
) we observe “von<br />
Karman vortex shedding”; but with separation in one<br />
point only (no wake!) consistent with the EG2 solution.<br />
EG2: no empirical parameters (only parameter is<br />
)<br />
“In a reasonable theory there are no dimensionless<br />
numbers whose values are only empirically determinable.”<br />
(Einstein)<br />
Johan Hoffman – KTH – p.89
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Summary: NSE & G2<br />
Approximate weak solutions: existence by G2<br />
Weak uniqueness: duality: computation of<br />
Computational method: G2 (Adaptive DNS/LES)<br />
Adaptivity<br />
Generality<br />
Quantitative error control<br />
very low computational cost<br />
quick & easy to adjust to new problem<br />
reliable results<br />
No filtering<br />
no Reynolds stresses<br />
Turbulent boundary layer by simple friction b.c.<br />
EG2: parameter-free model of high<br />
flow<br />
Johan Hoffman – KTH – p.90
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