Computer Visualization of Vortex Wake Systems
3
AIAA JOURNAL, VOL. 37, NO. 4: TECHNICAL NOTES 511 Computer Visualization of Vortex Wake Systems Roger C. Strawn ¤ U.S. Army Aviation and Missile Command, Moffett Field, California 94035-1000 David N. Kenwright † MRJ Technology Solutions, Moffett Field, California 94035-1000 and Jasim Ahmad ‡ MCAT Institute, Moffett Field, California 94035-1000 Introduction T HE general problem of identifying and visualizing free-shear vortex centers in large transient three-dimensional computa- tional uid dynamics(CFD)datasetsis addressed.This vortexiden- ti cationis particularlyimportantforrotorcraftproblems,wherethe interactions between rotor blades and their vortical wake systems play an important role in determining rotor airloads, noise, and vi- brations.These rotorcraftvorticestypicallyconsistof an inner core of rotational ow coupled to an irrotational outer ow as modeled by Eq. (1) from Ref. 1: V μ D r 0 2¼.r 2 C a 2 / (1) The radius from the vortex center is given by r , with a rotational core radius given by a. V μ is the vortex tangential velocity, and 0 is the circulationon the blade, which ultimately determines the vortex strength.Beyond the core radius, the velocity eld rapidly becomes irrotational. The vortex center coincides with a local maximum in vorticity magnitude. References 2– 4 describe a variety of methods for vortex identi - cation in three dimensions based on different vortex de nitions. In particular, Jeong and Hussain 4 develop a general scheme for iden- tifying vortex core regions for both wall-bounded and free-shear ows. This scheme works well for most applications; however, for the rotorcraft problem, we wish to identify the vortex centers, or skeletons, in addition to the outer surfaces of the vortex cores. The vortex centers are important because these regions have the high- est velocity gradients relative to the local ow and, hence, have the greatest in uence on the rotor aerodynamics. Also, the traces of vortex centers show where the vortices originate on the rotor blades. Vortices as Centers of Swirling Flow There is really no formal de nition for a vortex in three dimen- sions. However, a number of practicaltechniqueshave been used to helpinvestigatorsidentifyvortices,includingthecharacterizationof a vortex as the center of a swirling ow. This model has been used by Sujudi and Haimes 5 as a visualization tool for vortex centers. They tested the eigenvalues and eigenvectorsof the velocity rate of deformation tensor [@ u i =.@ x j /] to determine the direction and the center of any swirling ow. If a swirl center lies inside the cell under examination, then this cell contains a vortex center. This identi cation method works well for many practical cases; however, confusion arises when multiple vortices have overlapping rotational cores. Figure 1 shows cross-sectionalvelocity vector and Received May 26, 1998; revision received Nov. 9, 1998; accepted for pub- lication Dec. 3, 1998. This paper is declared a work of the U.S. Government and is not subject to copyright protection in the United States. ¤ Research Scientist, Army/NASA Rotorcraft Division, Aero ightdynam- ics Directorate. Senior Member AIAA. † Senior Research Scientist. ‡ Research Scientist. a) b) Fig. 1 Velocity vectors and vorticity contours for two line vortices with overlapping rotational cores. vorticity plots for a model problem with two identical line vortices inside a unit cube grid. These vortices have the same counterclock- wise sense of rotationand overlappingrotationalcores. Each vortex has a velocity eld de ned by Eq. (1) with a rotationalcoreradiusof 0.35. The overall velocity eld is obtained by superposition of the individualvortexvelocities,andalthoughthisis an idealizedcase, it demonstrates what happens in typical CFD data set problems when rotational vortex cores overlap. The velocity vector eld in Fig. 1b shows only a single rotational center. A vortex identi cation tech- nique that searchesfor centers of swirl will identifyonly one vortex for this case, and its location will be at the center of the square. Sim- ilarly, a vortex identi cation technique that identi es vortex core surfaces 4 will not delineate two vortex centers. Instead, it will show only a combined vortex core surface. The basic problemis that the combinationof rotationalvortexve- locities does not preserve the center of rotation for each individual vortex.This observationhas importantimplicationsfor rotary-wing calculations, which typically do not have suf cient grid resolution to completely resolve the rotor wake system. This lack of grid reso- lution causes the computed vortices to arti cially diffuse and leads to very large rotational vortex cores, which can easily overlap. Vortices as Local Maxima of Vorticity Magnitude Our scheme de nes a vortex center as a local maximum of vor- ticity in a plane that is normal to the vorticity vector. This rules out shear layers, which have high vorticity values but no local vortic- ity maxima. This idea is similar to that of Banks and Singer, 3 who identi ed vortex centers as local pressure minima in a plane that is normal to the vorticity vector. For practical simplicity, our imple- mentation searches for local vorticity maxima on the faces of each interior element in a structured hexahedral grid rather than on the exact planes that are normal to the vorticity vector. We derive nodal values for the uid vorticity vector ! from the velocity eld u, using central differences to compute the velocity derivatives.The subsequentsearch for local vorticity maxima takes place in uniform computational space to simplify the analysis. The vortex detection scheme extracts blocks of 3 £ 3 hexahedral cells, as shown in Fig. 2. We analyze each face of this central cell and its surrounding nodes in turn. The rst step is a candidate test to determine whether a local maximum exists on or near the central face.If thevorticitymagnitudeis highestatoneof thecentralvertices of the 4 £ 4 nodal patch, then the central face is a candidate for a vorticity maximum. The second step determines the exact location of the maximum viathesolutionof two nonlinearsimultaneousequations.The center is located where the gradients of the vorticity magnitude are zero. Constructionofthesenonlinearequationsrequiresthatwe alsocom- pute values for the gradientsof vorticityat each vertex of the central face. We use centraldifferencesto compute these gradients,thereby drawing on data only from the 4 £ 4 patch of nodal values. Bilinear functions describe the variation of these gradients over the central face, as shown in Eq. (2). The coef cient values in Eq. (2) can be determinedby matchingthe gradientequationsto the known values at the four corners of the face: j!j Ƈ D a 1 C b 1 Ƈ C c 1 ´ C d 1 Ƈ´ (2) j!j ´ D a 2 C b 2 Ƈ C c 2 ´ C d 2 Ƈ´ Downloaded by CARLETON UNIVERSITY LIBRARY on June 28, 2014 | http://arc.aiaa.org | DOI: 10.2514/2.744
Transcript of Computer Visualization of Vortex Wake Systems
AIAA JOURNAL, VOL. 37, NO. 4: TECHNICAL NOTES 511
Computer Visualizationof Vortex Wake Systems
Roger C. Strawn¤
U.S. Army Aviation and Missile Command,Moffett Field, California 94035-1000
David N. Kenwright†
MRJ Technology Solutions,Moffett Field, California 94035-1000
andJasim Ahmad‡
MCAT Institute, Moffett Field, California 94035-1000
Introduction
T HE general problem of identifying and visualizing free-shearvortex centers in large transient three-dimensional computa-
tional � uid dynamics (CFD)data sets is addressed.This vortex iden-ti� cation is particularlyimportant for rotorcraftproblems,where theinteractions between rotor blades and their vortical wake systemsplay an important role in determining rotor airloads, noise, and vi-brations. These rotorcraft vortices typically consist of an inner coreof rotational � ow coupled to an irrotational outer � ow as modeledby Eq. (1) from Ref. 1:
Vµ Dr0
2¼.r 2 C a2/(1)
The radius from the vortex center is given by r , with a rotationalcore radius given by a. Vµ is the vortex tangential velocity, and 0 isthe circulationon the blade, which ultimately determines the vortexstrength.Beyond the core radius, the velocity � eld rapidly becomesirrotational. The vortex center coincides with a local maximum invorticity magnitude.
References 2–4 describe a variety of methods for vortex identi� -cation in three dimensions based on different vortex de� nitions. Inparticular, Jeong and Hussain4 develop a general scheme for iden-tifying vortex core regions for both wall-bounded and free-shear� ows. This scheme works well for most applications; however, forthe rotorcraft problem, we wish to identify the vortex centers, orskeletons, in addition to the outer surfaces of the vortex cores. Thevortex centers are important because these regions have the high-est velocity gradients relative to the local � ow and, hence, havethe greatest in� uence on the rotor aerodynamics. Also, the tracesof vortex centers show where the vortices originate on the rotorblades.
Vortices as Centers of Swirling FlowThere is really no formal de� nition for a vortex in three dimen-
sions. However, a number of practical techniqueshave been used tohelp investigatorsidentifyvortices, includingthe characterizationofa vortex as the center of a swirling � ow. This model has been usedby Sujudi and Haimes5 as a visualization tool for vortex centers.They tested the eigenvalues and eigenvectorsof the velocity rate ofdeformation tensor [@u i=.@ x j /] to determine the direction and thecenter of any swirling � ow. If a swirl center lies inside the cell underexamination, then this cell contains a vortex center.
This identi� cation method works well for many practical cases;however, confusion arises when multiple vortices have overlappingrotational cores. Figure 1 shows cross-sectionalvelocity vector and
Received May 26, 1998; revision received Nov. 9, 1998;accepted for pub-lication Dec. 3, 1998. This paper is declared a work of the U.S. Governmentand is not subject to copyright protection in the United States.
¤Research Scientist, Army/NASA Rotorcraft Division, Aero� ightdynam-ics Directorate. Senior Member AIAA.
†Senior Research Scientist.‡Research Scientist.
a) b)
Fig. 1 Velocity vectors and vorticity contours for two line vortices withoverlapping rotational cores.
vorticity plots for a model problem with two identical line vorticesinside a unit cube grid. These vortices have the same counterclock-wise sense of rotationand overlappingrotationalcores. Each vortexhas a velocity � eld de� ned by Eq. (1) with a rotationalcore radiusof0.35. The overall velocity � eld is obtained by superposition of theindividualvortex velocities,and although this is an idealizedcase, itdemonstrateswhat happens in typical CFD data set problems whenrotational vortex cores overlap. The velocity vector � eld in Fig. 1bshows only a single rotational center. A vortex identi� cation tech-nique that searches for centers of swirl will identifyonly one vortexfor this case, and its locationwill be at the centerof the square. Sim-ilarly, a vortex identi� cation technique that identi� es vortex coresurfaces4 will not delineate two vortex centers. Instead, it will showonly a combined vortex core surface.
The basic problemis that the combinationof rotationalvortexve-locities does not preserve the center of rotation for each individualvortex. This observationhas important implicationsfor rotary-wingcalculations, which typically do not have suf� cient grid resolutionto completely resolve the rotor wake system. This lack of grid reso-lution causes the computed vortices to arti� cially diffuse and leadsto very large rotational vortex cores, which can easily overlap.
Vortices as Local Maxima of Vorticity MagnitudeOur scheme de� nes a vortex center as a local maximum of vor-
ticity in a plane that is normal to the vorticity vector. This rules outshear layers, which have high vorticity values but no local vortic-ity maxima. This idea is similar to that of Banks and Singer,3 whoidenti� ed vortex centers as local pressure minima in a plane that isnormal to the vorticity vector. For practical simplicity, our imple-mentation searches for local vorticity maxima on the faces of eachinterior element in a structured hexahedral grid rather than on theexact planes that are normal to the vorticity vector.
We derive nodal values for the � uid vorticity vector ! from thevelocity � eld u, using central differences to compute the velocityderivatives.The subsequent search for local vorticity maxima takesplace in uniform computational space to simplify the analysis. Thevortex detection scheme extracts blocks of 3 £ 3 hexahedral cells,as shown in Fig. 2. We analyze each face of this central cell andits surrounding nodes in turn. The � rst step is a candidate test todetermine whether a local maximum exists on or near the centralface. If thevorticitymagnitudeis highestatoneof thecentralverticesof the 4 £ 4 nodal patch, then the central face is a candidate for avorticity maximum.
The second step determines the exact location of the maximumvia the solutionof two nonlinearsimultaneousequations.The centeris located where the gradients of the vorticity magnitude are zero.Constructionof thesenonlinearequationsrequires that we also com-pute values for the gradientsof vorticityat each vertex of the centralface. We use centraldifferencesto compute these gradients,therebydrawing on data only from the 4 £ 4 patch of nodal values. Bilinearfunctions describe the variation of these gradients over the centralface, as shown in Eq. (2). The coef� cient values in Eq. (2) can bedeterminedby matching the gradient equations to the known valuesat the four corners of the face:
j!j» D a1 C b1» C c1´ C d1»´(2)
j!j´ D a2 C b2» C c2´ C d2»´
Dow
nloa
ded
by C
AR
LE
TO
N U
NIV
ER
SIT
Y L
IBR
AR
Y o
n Ju
ne 2
8, 2
014
| http
://ar
c.ai
aa.o
rg |
DO
I: 1
0.25
14/2
.744
512 AIAA JOURNAL, VOL. 37, NO. 4: TECHNICAL NOTES
Fig. 2 Tests for local vorticity maxima take place on each face of thecentral cell in a 3 £ £ 3 £ £ 3 block.
The maximumin vorticitymagnitudeoccurswhere j!j» D j!j´ D0. Setting Eqs. (2) to zero and solving them simultaneously givesrise to two quadratic equations for » and ´. Only one of the rootsof each quadratic equation will, in general, lie within the face ofthe center cell, and we discard the other root. If the computed co-ordinate (»; ´) lies inside the central face, then we have found apoint in the vortex core. The corresponding coordinates in three-dimensional physical space are obtained from a Jacobian transfor-mation.The procedure� nisheswhen all six faceshavebeen checkedfor every cell in the mesh. The end result consists of a collectionofpoints that identify the local maxima of vorticity magnitude on cellfaces.
In addition to the procedure just described, we use two thresh-olds to cut down on the number of vortex centers that marginally� t the criteria already described. The � rst threshold requires thateach cell face must have a vorticity magnitude that is larger than auser-determined value. This procedure removes weak vortex cen-ters from the solution.The second threshold addresses the potentialfor misalignment between a cell-face normal and the vorticity vec-tor on that face. Ideally, the vorticity vector should be aligned withthe cell-facenormal.Any misalignmentcan causea small projectionerror in the computedvortex center.To avoid these errors,we do notconsider a cell face for a vortex center if the angle between the facenormal n and the vorticity vector ! is larger than a user-prescribedthreshold of approximately60 deg. Any vortex center that fails thisangle threshold test on a particular cell face will be picked up byother cell faces with better alignment.
We have tested the new vortex detection scheme for the caseshown in Fig. 1. Computed vortex traces show two vortex centersthat coincide with the two local maximums in vorticity magnitude.
Results and DiscussionWe demonstrate our new vortex visualization scheme using two
computed solutions. The � rst case models the steady vortex rollupfrom a delta wing (Fig. 3). This symmetric case was computed byChaderjian6 on a 67 £ 49 £ 107 grid with a freestream Mach num-ber of 0.27 and an angle of attack of 15 deg. The vorticity-maximascheme clearly distinguishes what Chaderjian describes as the pri-mary, secondary,and tertiary vorticeson each side of the deltawing.
The second test case simulates a wind-tunnel test of the V-22tiltrotor blades. The solutioncontains11 separateoverset grids witha total of 1.02 £ 106 grid points. The vortex tracing algorithm isapplied only in the off-body rotor-wake grids, not in the viscousboundary-layergrids near the rotor blade surfaces.The V-22 bladeshave a hover-tip Mach number of 0.64, an advance ratio of 0.1,and a shaft tilt of 5 deg. Periodic solutions for this case requiredapproximately 55 CPU hours on a Cray C-90 computer for threerevolutions. Figure 4 shows tip and root vortices from each rotor
Fig. 3 Vorticity contours and vortex traces on a delta wing.
Fig. 4 Vortex core traces for the V-22 rotor blades.
blade that convect below and to the rear of the rotor disk. Figure 4also shows the overall vortex rollup around the sides of the rotordisk. The vortex identi� cation computations require approximately6:7 £ 10¡4 s per hexahedral cell on a Silicon Graphics Onyx IIworkstation with one 195-MHz R10000 CPU.
ConclusionThis new vortex detection scheme can identify vortex centers in
overlapping regions of rotational � ow, and it is easy to implementfor structured-griddata sets. In addition to vortex identi� cation, thisscheme can be used to guide adaptive-grid CFD methods that seekto locally improve the grid resolution for vortex wake systems. Forrotorcraft applications, we expect that visualization tools such asthis will become more and more important as solutions on largercomputationalgrids begin to capture more of the � ne details of therotor blade–vortex interactions.
References1Scully, M. P., “Computation of Helicopter Rotor Wake Geometry and Its
In� uence on Rotor Harmonic Airloads,” Massachusetts Inst. of Technology,ASRL TR 178-1, Cambridge, MA, March 1975.
2Roth, M., and Peikert, R., “Flow Visualization for Turbomachinery De-sign,”ProceedingsofVisualization’96 (SanFrancisco,CA), IEEE ComputerSociety Press, Los Alamitos, CA, 1996, pp. 381–384.
3Banks, D. C., and Singer, B. A, “Vortex Tubes in Turbulent Flows: Iden-ti� cation, Representation, Reconstruction,”Proceedingsof Visualization’94(Washington,DC), IEEE Computer Society Press, Los Alamitos, CA, 1994,pp. 132–139.
4Jeong, J., and Hussain, F., “On the Identi� cation of a Vortex,” Journalof Fluid Mechanics, Vol. 285, 1995, p. 69.
5Sujudi, D., and Haimes, R., “Identi� cation of Swirling Flow in 3-DVector Fields,” AIAA Paper 95-1715, June 1995.
6Chaderjian, N. M., “Navier–Stokes Prediction of Large-AmplitudeDelta-Wing Roll Oscillations,” Journal of Aircraft, Vol. 31, No. 6, 1994,pp. 1333–1340.
A. PlotkinAssociate Editor
Dow
nloa
ded
by C
AR
LE
TO
N U
NIV
ER
SIT
Y L
IBR
AR
Y o
n Ju
ne 2
8, 2
014
| http
://ar
c.ai
aa.o
rg |
DO
I: 1
0.25
14/2
.744
This article has been cited by:
1. K Nakayama. 2014. Physical properties corresponding to vortical flow geometry. Fluid Dynamics Research 46:5,055502. [CrossRef]
2. L. Zhang, Q. Deng, R. Machiraju, A. Rangarajan, D. Thompson, D. K. Walters, H.-W. Shen. 2014. BoostingTechniques for Physics-Based Vortex Detection. Computer Graphics Forum 33:1, 282-293. [CrossRef]
3. Václav Kolář, Jakub Šístek, Fehmi Cirak, Pavel Moses. 2013. Average Corotation of Line Segments Near a Pointand Vortex Identification. AIAA Journal 51:11, 2678-2694. [Abstract] [Full Text] [PDF] [PDF Plus]
4. Alexander Brian Phillips, Stephen Richard Turnock. 2013. Application of the VORTFIND algorithm for theidentification of vortical flow features around complex three-dimensional geometries. International Journal forNumerical Methods in Fluids 71:11, 1461-1474. [CrossRef]
5. Wei Liu, Eraldo Ribeiro. 2012. Detecting singular patterns in 2D vector fields using weighted Laurent polynomial.Pattern Recognition 45:11, 3912-3925. [CrossRef]
6. Kondwani Kanjere, David Angland, Xin ZhangA Numerical Study to Optimise Blowing Flow Control for FlapSide-Edge Noise Reduction . [Citation] [PDF] [PDF Plus]
7. J. Kitzhofer, T. Nonn, C. Brücker. 2011. Generation and visualization of volumetric PIV data fields. Experimentsin Fluids 51:6, 1471-1492. [CrossRef]
8. N. Kasmai, D. Thompson, E. Luke, M. Jankun-Kelly, R. Machiraju. 2011. Feature-based adaptive mesh refinementfor wingtip vortices. International Journal for Numerical Methods in Fluids 66:10, 1274-1294. [CrossRef]
9. Václav Kolár. 2009. Compressibility Effect in Vortex Identification. AIAA Journal 47:2, 473-475. [Citation] [PDF][PDF Plus]
10. J. W. Lim, R. C. Strawn. 2008. Computational Modeling of HART II Blade-Vortex Interaction Loading and WakeSystem. Journal of Aircraft 45:3, 923-933. [Citation] [PDF] [PDF Plus]
11. Roger C. Strawn, Francis X. Caradonna, Earl P. N. Duque. 2006. 30 Years of Rotorcraft Computational FluidDynamics Research and Development. Journal of the American Helicopter Society 51:1, 5. [CrossRef]
12. Ming Jiang, Raghu Machiraju, David ThompsonDetection and Visualization of Vortices 295-XXVIII. [CrossRef]13. Hee Jung Kang, Oh Joon Kwon. 2001. Effect of Wake Adaptation on Rotor Hover Simulations Using Unstructured
Meshes. Journal of Aircraft 38:5, 868-877. [Citation] [PDF] [PDF Plus]
Dow
nloa
ded
by C
AR
LE
TO
N U
NIV
ER
SIT
Y L
IBR
AR
Y o
n Ju
ne 2
8, 2
014
| http
://ar
c.ai
aa.o
rg |
DO
I: 1
0.25
14/2
.744
