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1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22..June.2006
Vortex Core Identification with ApplicationsMarkus Widhalm
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Content
MotivationIdentification of vortices
vorticity magnitude
λ2 Method
second invariant Qkinematik vorticity number Nk
normalized helicity HnAdaptation strategy with applicationsConclusion & Outlook
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Motivation
Prior developments made from Markus Ruetten and Thomas Alrutz with successful applicationsIdentifiaction of vortices
Based on point methodsVisualize flow phenomena, especially fighter aircraftsVisibility of rotation of vorticesVortex break down
Wake turbulence behind transport aircraftsAdaptation
Adapt the grid only at specific areas of interestSafes time and memory required
Usage of adaptation with the user defined section
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Identification of vortices
What is a vortex?Up to now many many different definitions for a vortex exists.
But: One characteristic value, the vorticity, describes the formation, the magnitude and distortion of a vortex in a flow field.
Identification of vortices with point-methods
Due to the appearance of flow variables at discrete grid points and neighboring point information - the computation of properties will show a vortex or not.the magnitude of vorticity is at most cases not sufficientSolution: Search for coherent vortex structures or pressure minima from the velocity gradient tensor ∇u
!ω= rot!v
!!ω!
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Identification of vortices
Method
compute eigenvalues from with decomposition of ∇u
symm. part S (rotation) and antisymm. part Ω (rate-of-strain) of ∇u
Eigenvalues λ1, λ2 and λ3 of matrix A
Vortex identification with
λ2S2+Ω2
∇u=
!
"#
∂u∂x
∂u∂y
∂u∂z
∂v∂x
∂v∂y
∂v∂z
∂w∂x
∂w∂y
∂w∂z
$
%&
S2+Ω2 = A
det | A!λI |= 0
λ3+Pλ2+Qλ+R= 0
λ1,λ2,λ3
λ1 ! λ2 ! λ3 and λ2 < 0
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Identification of vortices
second invariant Qderived from the characteristic equation
P,Q and R are the three invariants of ∇u
Q represent the local balance between the shear strain and vorticity magnitude
Vortex identification with
λ3+Pλ2+Qλ+R= 0
Q! 0
Q=12
!u2i,i!ui, ju j,i
"=12
!"Ω"2!"S"2
"
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Identification of vortices
kinematic vorticity number NkShows the “Quality of rotation” independent from the magnitude of vorticity
Split ∇u into symm. part S and antisymm. part Ω
Vortex identification with
Nk =!Ω!!S! =
!| ω |2
2Si jSi j
"
!Ω!=| tr | ΩΩT ||2,Ωi j =12
| ui j"u ji | !S! =| tr | SST ||2,Si j =12
| ui j+u ji |
Nk ! 1
and
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Identification of vortices
normalized Helicity Hndescribes the angle between the velocity and the vorticity vector
values derived between -1 and 1visualization of interacting vortices - primary and secondary vortices
Vortex identification withvalues between -1 and ~ -0.9 and
between ~0.9 and + 1.0
Hn =!v!ω
|!v ||!ω | = cosα
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Identification of vorticesRAE2822 3D profileRe = 6.5 Mill.
M = 0.7α = 10.0°
Nk = 1.0 λ2 = -0.001 Hn = -0.9/0.9
Q = 0
vorticity magnitude = 500
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Identification of vorticesRAE2822 3D profile
Blending of valuesfast computation provided in comparison to main value
even small parallel vorticity and velocity vectors are countedshould overcome most of the “noise” introduced by the velocity gradient computationBlending derived from characteristic equation
vortex core is a region with complex eigenvalues - discriminant ϑ gets positiv
λ3+Pλ2+Qλ+R= 0
with R= det(ui, j)ϑ=!Q3
"3
+!R2
"2
> 0
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Identification of vorticesRAE2822 3D profile
Hn not blended Hn, ϑ > -10.0
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Identification of vorticesHelicopter with actuator discs
Re = 4.33 Mill.M = 0.2081α = 0.0°
vorticity magnitude = 500
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Identification of vorticesHelicopter with actuator discs
Q = 0.1 Q = 0.01
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Identification of vorticesHelicopter with actuator discs
Nk = 1.0 Nk = 1.1
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Identification of vorticesHelicopter with actuator discs
λ2 = -0.001 λ2 = -0.1
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Identification of vorticesHelicopter with actuator discs
Hn = -0.8/0.8Hn = -0.9/0.9
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Identification of vorticesVHBR - generic Very High Bypass Ratio Engine
VHBRRe = 2 Mill.M = 0.22α = 8°
Vorticity magnitude = 1.0
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Identification of vorticesVHBR - generic Very High Bypass Ratio Engine
Q = 0 Nk = 1.0
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Identification of vorticesVHBR - generic Very High Bypass Ratio Engine
Hn = -0.95/0.95λ2 = -0.001
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Adaptation Strategy
Absolute values are not feasible - a lot of post-processing neededmagnitude of vorticity
total pressure losspreferable are values which points out the behavior of the flow
second invariant Qkinematic vorticity number Nk
Or, looking for the pressure minimum and derive the eigenvalues of ∇u
λ2 method
at least, leave classical methods - looking directly on flow conditionnormalized Helicity Hn
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Adaptation Strategy
vortex core adaptation can now be handled more straight forwardsecond invariant Q
kinematic vorticity number Nk
λ2 method
normalized Helicity Hnvalues between -1 and ~ -0.9 andbetween ~0.9 and + 1.0
Q! 0
λ1 ! λ2 ! λ3
Nk ! 1
λ2 < 0
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Adaptation Strategy
adaptation needs now new input parameterIndicator type (4) - User defined value
handling adaptation for values greater thanvalue < limit considered
handling adaptation for values greater thanvalue > limit considered
and a range around a valuelimit - eps < limit < limit + eps considered
all main features are untouchedmaximum point number
percentage of new points...
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Adaptation Strategy
Does this approach work as desired?Values with one limit
greater then or less than for Q, Nk and λ2
Values with two limitsgreater than and less than for Hn
Approach should refine sparsely in boundary layer regionEffect of vortex downstream should last longer
minimize dissipation through flow solver in core region
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Adaptation Strategy3D RAE2822 wing
Adaptation of grid with λ2 = -0.01
slices through x = 0.9Initial 1st
2nd 3rd
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Adaptation Strategy3D RAE2822 wing
0
625.000
1.250.000
1.875.000
2.500.000
initial 1st 2nd 3rd
npnts tetras prisms
Adaptation of grid with λ2 = -0.01
0
38
75
113
150
initial 1st 2nd 3rd
% new points
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
2nd
Adaptation Strategy3D RAE2822 wing
Vortex downstream - iso-surfaces at λ = -0.01
Initial 1st
3rd
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Adaptation Strategy3D RAE2822 wing
Visibility of vortex with λ2 = -0.01
- Initial- 1st- 2nd- 3rd
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Adaptation Strategy3D RAE2822 wing
Adaptation of grid with normalized Helicity = -0.92 and 0.92
slices through x = 0.9Initial 1st
2nd 3rd
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Adaptation Strategy3D RAE2822 wing
0
500.000
1.000.000
1.500.000
2.000.000
initial 1st 2nd 3rd
npnts tetras prisms
Adaptation of grid - normalized Helicity = -0.92 and 0.92
0
21
43
64
85
initial 1st 2nd 3rd
% new points
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Adaptation Strategy3D RAE2822 wing
Vortex downstream - iso-surfaces at Hn = -0.92 and 0.92
Initial
2nd
1st
3rd
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Adaptation Strategy3D RAE2822 wing
Visibility of vortex with normalized Helicity = -0.92 and 0.92
- Initial- 1st- 2nd- 3rd
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
Conclusions & Outlook
Vorticity magnitude is not suitable for adaptation - vortex core boundaries are ambiguousAdaptation process still requires pre-knowledge of vortex generation
Normalized Helicity shows most promising results for wake turbulence and separation casesAdaptation preferable done outside boundary layerAfter to many adaptations point number increases exponentialBe aware of unsteady effects in a steady case
Predefined boxes for refinementImplementation with whole adaptation functionalities soon available
1st Adaptation Workshop Göttingen > Markus WidhalmDokumentname > 21.-22. June 2006
2nd
Adaptation Strategy3D RAE2822 wing
Vortex downstream with unsteady effects - iso-surfaces at λ = -0.01