Vortex Core Identification with Applications Markus...

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


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


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!



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



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


λ3+Pλ2+Qλ+R= 0

Q! 0

Q=12

!u2i,i!ui, ju j,i

"=12

!"Ω"2!"S"2

"



kinematic vorticity number NkShows the “Quality of rotation” independent from the magnitude of vorticity

Split ∇u into symm. part S and antisymm. part Ω


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



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


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


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


Identification of vorticesRAE2822 3D profile

Hn not blended Hn, ϑ > -10.0


Identification of vorticesHelicopter with actuator discs

Re = 4.33 Mill.M = 0.2081α = 0.0°

vorticity magnitude = 500



Q = 0.1 Q = 0.01



Nk = 1.0 Nk = 1.1



λ2 = -0.001 λ2 = -0.1



Hn = -0.8/0.8Hn = -0.9/0.9


Identification of vorticesVHBR - generic Very High Bypass Ratio Engine

VHBRRe = 2 Mill.M = 0.22α = 8°

Vorticity magnitude = 1.0



Q = 0 Nk = 1.0



Hn = -0.95/0.95λ2 = -0.001


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


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


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...


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


Adaptation Strategy3D RAE2822 wing

Adaptation of grid with λ2 = -0.01

slices through x = 0.9Initial 1st

2nd 3rd



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


2nd


Vortex downstream - iso-surfaces at λ = -0.01

Initial 1st

3rd



Visibility of vortex with λ2 = -0.01

- Initial- 1st- 2nd- 3rd



Adaptation of grid with normalized Helicity = -0.92 and 0.92

slices through x = 0.9Initial 1st

2nd 3rd



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



Vortex downstream - iso-surfaces at Hn = -0.92 and 0.92

Initial

2nd

1st

3rd



Visibility of vortex with normalized Helicity = -0.92 and 0.92

- Initial- 1st- 2nd- 3rd


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


2nd


Vortex downstream with unsteady effects - iso-surfaces at λ = -0.01

Vortex Core Identification with Applications Markus...

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