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24th Symposium on Naval Hydrodynamics Fukuoka, JAPAN, 8-13 July 2002 Tracking Vortices Over Large Distances Using Vorticity Confinement RainalLd LJohner i, Chi Yang ~ and Robert Roger2 (i George Mason University, 2 Johns Hopkins University) ABSTRACT This paper discusses the use of the vorticity confine- ment method on unstructured grids to simulate vortex dominated flows. A general vorticity confinement term has been derived using dimensional analysis. The re- sulting vorticity confinement is a function of the lo- cal vorticity-based Reynolds-number, the local element size, the vorticity and the gradient of the absolute value of the vorticity. The vorticity confinement term disap- pears for vanishing mesh size, and is applicable to un- structured grids with large element size disparity. The new term has been found to be successful for a num- ber of test cases, allowing better definition of vortices without any deleterious effects on the flow field. INTRODUCTION The requirement to accurately track vortices over large distances is common to many areas of engineering, e.g. rotating helicopter blades (Caradonna, 2000; Strawn 2000; Wenren et al 2001) and vortices shed by sub- marines. Due to the inherent dissipation built into nu- merical flux functions in order to avoid numerical in- stabilities and unphysical solutions, any current Euler or RANS field solver will tend to dissipate these vor- tices too fast. In order to obtain a rough estimate for the grid sizes required to capture accurately typical trailing edges vortices, consider a helicopter blade with radius r = 5m and a vortex of size ~ = 10cm. The element size required to describe such a vortex will be smaller than h = lcm (10 elements across the vortex zone). The volume occupied by the vortex per blade rotation Is approximately Vie, = 1r 102cm2 Or 5 102cm ~ 106cm3 Given that the vortex is moving, and possibly interact- ing with blades and other vortices, an isotropic grid seems prudent. Such a grid would require at least 106 points per blade rotation. This should be viewed as a lower limit, as the only free parameter is the mesh size, which, in all likelihood, was estimated too large. Simi- larly, estimates for the number of grid points required to track accurately one or more vortices for one submarine length easily approach billions of points, making such calculations impractical for the next decade (if Moore's law continues). In order to avoid the rapid dissipation of vortices, Steinhoff and co-workers have introduced the concept of vorticity confinement (Steinhoff et al, 1992; Steinhoff et al,1994; Steinhoff et al, 1999; Hu et al, 2000), refining it over the last decade and applying it successfully to a number of relevant flows (Moulton et al, 2000; Wenren et al, 2001; Dietz et al, 20011. The basic technique consists of adding a force-term to the momentum equations, resulting in: pa, + pvVv + Vp = V,uVvcon x A, (1) where p, v, p, ,u and ~ denote, respectively, the density, velocity, pressure, viscosity and vorticity of the fluid, ~ is a user-defined number and n is a 'normal' vec- tor. Note that this additional force acts in the direction normal to the vorticity and n, thus convecting vortic- ity back towards the centroid as it diffuses away. It can also be noted that this term is limited to the vertical re- gions and does not affect nonvertical regions. A typical choice for n is: Vows ~V~ ~ (2) Steinhoff and co-workers (Steinhoff et al, 1992; Stein- hoff et al,1994; Steinhoff et al, 1999; Hu et al, 2000; Moulton et al, 2000; Wenren et al, 2001; Dietz et al, 2001) were able to demonstrate that vortices could be captured and maintained for long distances without dis- sipation. Steinhoff and co-workers have worked mainly on uniform Cartesian grids, for which ~ could be kept constant. Murayama (Murayama et al, 2001 ) attempted to use this type of vorticity confinement on an unstruc- tured grid, i.e. leaving ~ constant. The results were
mixed. For some values of c, an improvement of re- sults was observed. Other values of ~ lead to unphysi- cal results, e.g. premature vortex burst on a delta wing. These results make it clear that for non-uniform grids a general solution has to be found. Vorticity confinement has also been used recently for the visual simulation of smoke (Fedkiw et al, 2001~. These animations were performed on Cartesian grids using the incompress- ible Navier-Stokes equations, and included (for the first time, to the author's knowledge) an explicit, linear de- pendence on the mesh size h. DIMENSIONAL ANALYSIS From dimensional analysis, one can see that ~ must have the dimension of a velocity. One could either use Eve, hack or h2 ~V~ I. Considering Eqn.~2), the last form is particularly appealing, leading to: pv,~ + pvVv + Vp = V]UVVCV ph2 V ~w ~ x ~ , (3) where cv is now a true constant, regardless of the grid. One can see immediately that the vorticity confinement term is of the form of an anti-diffusion, and that it will disappear as the grid gets finer and finer (h ~ O). PROPER LENGTH SCALE h A crucial ingredient in the vorticity confinement given by Eqn.~3) is the length scale h. It was found that for isotropic grids, most of the possible forms: average of edge-lengths surrounding a point, volume to surface ra- tio of elements surrounding a point, etc., gave similar results. As expected, the situation is markedly differ- ent for highly stretched grids. Here, is was found that taking h as the characteristic length in the direction of V ~w ~ was the proper choice. The determination of char- acteristic lengths in the x, y, z directions is performed by observing that the gradient, computed as: Ok = Mi · ~ Ck (Ui + uj) , (4' . . for point i, direction k and edges i, j surrounding point i, will be of dimension ffu]/:h]~. Therefore, an approx- imate estimate for the characteristic element length in direction k may be obtained from: (ink) = 2M~: · ~ ark ~ . (5) Denoting by h the characteristic element lengths com- puted, the final form of h is given by: h=h IVI 11 . (6) TREATMENT OF BOUNDARY LAYERS The primary function of vorticity confinement is to en- hance the capture of relevant physics in regions where mesh density is insufficient. This is not the case in well resolved boundary layers close to solid surfaces. In fact, it was found that switching on vorticity con- finement in these highly resolved regions could lead to numerical instabilities. Therefore, an explicit switch, based on the local Reynolds-number Reh, was at- tempted: he = min(`l'Reh\) h; Reh = ~ · (~7) This form did not prove satisfactory. A more universal form, that is tied to the terms used for vorticity con- finement, is a local Reynolds-number defined with the vorticity. From dimensional analysis, one may observe that the following three candidates could be used: p~Av~h Rew,h= .. ; Red h = Path 7 ~ plVlwI ~h3 Reich = I,,. (8) The final form of the vorticity confinement force then takes the form: f = g(Re~,7h)cvph2VI~l x ~ g=max O,min 1, R 1 Re0 (9) [ [ Cash Lo he ] NUMERICAL RESULTS The vorticity confinement described above was imple- mented into FEFLO, a general adaptive unstructured fi- nite element flow solver (Lohner, 2002~. The results shown were all obtained by using a projection-type in- compressible flow solver (Lohner et al, 1998) that em- ploys a second-order upwind advection operator and a fourth-order pressure damping for the divergence con- straint (Lohner et al, 19991.
NACA0012 (Euler) The first case considered is that of a finite width NACA0012 wing, characteristic of control surfaces. The upstream boundary is located at 5 chord lengths ahead of the leading edge and the downstream bound- ary is located at 10 chord lengths behind the trailing edge. The incompressible Euler equations are solved for an angle of attack of or = 15°. Figure la shows the surface mesh employed, as well as a cut normal to the x-direction at 15% chord length downstream of the trailing edge. Note that a line-source was specified in the approximate position of the vortex in order to ob- tain a finer grid. The mesh has approximately 120,000 points, with 12,000 points on the boundary. Figure lb shows the results of the surface pressure contours and vorticity contours in four cut planes with the vorticity confinement terms switched on. These four cut planes are normal to the x-direction and are located at 5%, 1, 4 and 8 chord lengths downstream of the trailing edge. The vorticity confinement coefficient is cat, = 0.1 . The vortex core visualization in this fig- ure shows how well the vortex is captured till 8 chord lengths downstream of the trailing edge. Figures lc,d compare the vorticity and helicity (v w) in the same four cut planes downstream of the wing. The effect of vorticity confinement is clearly visible. Without vorticity confinement, the vortex is dissipated after only one chord length of the airfoil. With vorticity confinement, the vortex is still maintained very well up to eight chord lengths of the airfoil. Delta Wing (Laminar NS) The second case considered is that of the delta-wing measured by Hummel (Hummer et al, 1967) and com- puted by Murayama (Murayama et al, 2001~. This is a laminar case, making it ideally suited for bench- marking. The angle of attack is al = 20.5°, and the Reynolds-number based on the length of the wing is Re = 106. The grid, shown in Figure 2a, is typical of RANS calculations. In the proximity of the wall, the elements are highly anisotropic with extremely fine spacing normal to the wall. Away from the wall the mesh coarsens rapidly and becomes isotropic. The pri- mary vortex generated by the delta wing rapidly enters regions of low mesh density. Figure 2b shows the vor- ticity and pressure contours for planes located at 30%, 50% and 70% root chord length for the case with vor- ticity confinement terms switched on and the vorticity confinement coefficient cat,. = 0.1 . The vortex strength for the 50% chord plane are compared in Figure 2c. As before, vorticity confinement has a marked effect on the strength of the detached vortex. Figure 2d compares the measured and computed cp distribution at the plane x = 0.7 . One can see that the results with vorticity confinement deviate from the experimental measurements at the wing tip. 2-D Cylinder (Laminar NS) The third case considered is a 2-D circular cylinder. Although the case is two-dimensional, it was modeled as three dimensional, with two parallel walls in the z-direction. The simulations are performed for two Reynolds numbers, Re = 110 and Re = 190, re- spectively. The Reynolds number is based on the di- ameter of the circular cylinder. The incompressible laminar Navier-Stokes equations are solved since the flow is laminar for the Reynolds numbers considered here. The mesh consists of 306,160 tetrahedral ele- ments, 60,291 points and 17,332 boundary points. The grid, shown in Figure 3a, is typical of RANS calcula- tions. In the proximity of the cylinder, the elements are highly anisotropic with extremely fine spacing nor- mal to the cylinder. Away from the cylinder, the mesh coarsens rapidly and becomes isotropic. The vortex generated by the circular cylinder rapidly enters regions of low mesh density. Figure 3b shows comparison of the time history of the lift coefficient for Re = 110 between present results obtained without vorticity confinement and the numer- ical results predicted by Walhorn (Walhorn, 2001) us- ing a space finite element method. It can be seen from Figure 3b that both predictions agrees fairly well ex- cept slightly different frequency with which vortices are shed in a Karman vortex street behind a circular cylin- der. The dimensional frequency of vortex shedding ob- tained from the present numerical prediction is com- pared with the measurements performed by Roshko (Roshko, 1954) in the table 1. Our numerical predic- lion shows slightly high Strouhal number. The slight high Strouhal number in our prediction in comparison with both Roshko (Roshko, 1954) and Walhorn (Wal- horn, 2001) is due to constrain of channel in our simu- lation model. Table 1: Strouhal Number I 11 Re=110 1 Re=190 1 l Roshko 0.171 0.188 Present results cv0 0.185 0.205 Present results cv 76 0 0.188 0.207
NACA 0012: Surface Grid (nboun= 12,087, npoin=121,314) NACA 0012: Cut for Plane x=1.15 Figure la Finite NACA0012 Wing: Surface Mesh (left) and Cut Plane x = 1.15 (right) A.. / Figure lb Finite NACA0012 Wing: Surface Pressure, Vorticity in 4 Cut Planes and Vortex Core (4 Cut Planes: x = 1.05,2,5,9; cat. = 0.1 )
Figure to Finite NACA0012 Wing: Comparison of Vorticity in 4 Cut Planes 4 Cut Planes: x = 1.05, 2, 5, 9 (left: cv = 0; right: cv = 0.1) ~1 - Figure ld Finite NACA0012 Wing: Comparison of Helicity in 4 Cut Planes (4CutPlanes: x = 1.05,2,5,9) (left: cv = 0; right: cat. = 0.1)
Figure 2a Delta Wing: Surface Mesh (left), Detailed Surface Mesh and Mesh in Cut Plane (right) Figure 2b Delta Wing: Vorticity (left) and Pressure (right) in Planes x = 0.3, 0.5, 0.7 Figure 2c Delta Wing: Comparison of Vorticity for Plane x = 0.5
0.8 0.6 Q O 0.4 0.2 O -0.2 0.8 0.6 0.4 J 0.2 c' o -0.2 -0.4 -0.6 -0.8 Experiment Cv=O. x Cv=0.05 0 Cv=0.1 ~ 6< - 11 ~ ,.~.- _. _ ~ .,_.~ ~-'e' 0 0.2 0.4 0.6 0.8 1 X Figure 2d Delta Wing: Comparison of Cp for Plane x = 0.7 0.6 0.4 c' 0.2 o -0.2 -0.4 -0.e Present Results, Cv=O.O Walhorn's Results --~ 135 140 1 4t5 150 160 Figure 3b 2-D Cylinder: Comparison of C~, for Re = 110 Figure 3a 2-D Cylinder: Surface Mesh 0.8 0.6 0.4 0.2 o -0.2 -0.4 -0.6 -0.8 -1 - 150 155 160 90 . . · . · . Re=110, Cv=O.O Re=1 10. Cv=0.25 --- ----- - 130 135 140 145 Re=190, Cv=O.O - Re=190, Cv=0.2 Figure 3c 2-D Cylinder: Comparison of CL, Computed with and without Vorticity Confinement (left: Re = 110; right: Re = 190)
Re= 110 ,t= 146 Re=llO,t=146 Re= 110 ,t= 148 Re = 110 , t = 148 Re= 110 ,t= 150 Re = 110, t = 150 Re=llO,t=152 Re = 110, t = 152 Re = 110 , t = 154 Figure 3d 2-D Cylinder: Pressure Contours without Vorticity Confinement (cv = 0) Re = 110 , t = 154 Figure Be 2-D Cylinder: Pressure Contours with Vorticity Confinement (cv = 0.25)
Re = 190, t = 108 Re = 190, t = 108 Re= 190 ,t= 110 Re=l90,t=110 Re= 190 ,t= 112 Re=l90,t=112 Re = 190 , t = 114 Re = 190, t = 114 Re= 190 ,t= 116 Figure 3f 2-D Cylinder: Pressure Contours without Vorticity Confinement act, = 0) Re=l90,t=116 Figure 3g 2-D Cylinder: Pressure Contours with Vorticity Confinement act, = 0.2)
The time histories of the lift coefficient computed with and without vorticity confinement are compared one another in Figure 3c for both Reynolds numbers. The vorticity confinement coefficients are taken as cv = 0.25 for Re = 110 and cat, = 0.2 for Re = 190. It can be seen from Figure 3c that the vorticity confinement has a relatively minor effect on the lift coefficient and the frequency with which vortices are shed in a Karman vortex street behind a circular cylinder. The pressure contours at four different times are shown in Figures 3d-g for the Re = 110 and Re = 190. It can be seen from Figure 3d and Figure 3f that the Karman vortex street is dissipated quickly for both Reynolds numbers without vorticity confinement. It can also be seen from Figure 3e and Figure 3g that the Karman vortex street has been preserved very well for both Reynolds numbers with the vorticity confinement terms switched on. Submarine (RANS + Baldwin-Lomax turbulence model) The fourth case considered is a submarine with a 42- foot diameter. The speed is 7 knots and the Reynolds number per foot is Re = 1. l X 106. The front half of the submarine is modeled to study the vortex shed by the sail. The mesh has approximately 6,000,000 tetrahedral elements. The incompressible RANS equations with Baldwin-Lomax turbulence model are solved. Fig- ures 4a,b show the RANS grid used in the simulation. Figure 4a Submarine: Surface Mesh Figure 4b Submarine: Detailed Surface Mesh Figure 4c shows the preliminary results of the pres- sure contours on the submarine and the absolute ve- locity contours in two cut planes. A tip vortex can be seen from this figure. Note that the present preliminary results are computed without vorticity confinement. Figure 4c Submarine: Surface Pressure Contours and Absolute Velocity Contours at Two Cut Planes CONCLUSIONS AND OUTLOOK A general vorticity confinement term for unstructured grids has been derived, implemented and found to be successful for some cases. The vorticity confinement terms are of the form: f = g(Re`~,h~c~,ph2V~ x
where cat, = 0~1), h is a characteristic element size and 9 depends on the local vorticity-based Reynolds- number Rew,h. We are currently investigating in more depth the theoretical aspects associated with vorticity confine- ment. In particular: - One can see that the vorticity confinement as given by Eqn.~3) is in the form of a body force. As such, these terms may add or subtract axial and/or tan- gential moment from the surrounding flow. We are attempting to derive hard estimates/ proofs for the axial and tangential moment attributed to vor- ticity confinement. Our present conjecture of that such estimates can be obtained by making use of Stokes' theorem for vorticity. It is possible that the vorticity confinement terms introduce errors in the flow field. For benchmark problems (e.g. delta wing), more tests are required to determine the source of errors and to quantify them. Finally, other formulations for vorticity are cer- tainly possible. We will also continue the submarine run with vor- ticity confinement terms switched on to track the vor- tices shed by the sail over a large distance. ACKNOWLEDGEMENTS This research was partially supported by ONR, with Dr. Patrick Purtell as the technical monitor. REFERENCES Caradonna, F., "Developments and Challenges in Ro- torcraft Aerodynamics," AIAA-00-0109, 2000. Dietz, W., Fan, M., Steinhoff, J. and Wenren, Y., "Ap- plication of Vorticity Confinement to the Prediction of the Flow Over Complex Bodies," AIAA-01-2642, 2001. Fedkiw, R., Stam, J. and Jensen, H. W., "Visual Simula- tion of Smoke," Proceedings of SIGGRAPH, Los An- geles, CA, 2001. Hummel, D. and Srinavasan, P. S., "Vortex Break- down Effects on the Low-Speed Aerodynamic Charac- teristics of Slender Delta Wings in Symmetrical Flow," Royal Aeronautical Society Journal, vol. 71, 1967, pp. 319-322. Hu, G., Grossman, B. and Steinhoff, J., "A Numer- ical Method for Vortex Confinement in Compressible Flow," AIAA-00-0281, 2000. Lohner, R., Yang, C. and Onate, E., "Viscous Free Surface Hydrodynamics Using Unstructured Grids," Proceedings of the 22nd Symposium on Naval Hydrodynamics, Washington, D.C., 1998. Lohner, R., "FEFLO User's Manual," GMU-CSVCFD- 01-01, 2002. Moulton, M., and Steinhoff, J., "A Technique for the Simulation of Stall with Coarse-Grid CFD," AIAA-00- 0277, 2000. Murayama, M., Nakahashi K. and Obayashi S., "Nu- merical Simulation of Vortical Flows Using Vorticity Confinement Coupled With Unstructured Grid," AIAA- 01-0606, 2001. Roshko, A., "On the Development of Turbulent Wakes from Vortex Streets," NACA Report 1191, 1954. Steinhoff, J., Yonghu, W., Mersch, T. and Senge, H., "Computational Vorticity Capturing: Application to Helicopter Rotor Flow," AMA-92, 1992. Steinhoff, J., "Vorticity Confinement: A New Tech- nique for Computing Vortex Dominated Flows," Frontiers of Computational Fluid Dynamics, D.A. Caughey and M.M. Hafez eds., J. Wiley & Sons, 1994. Steinhoff, J., Yonghu, W. and Lesong, W., "Efficient Computation of Separating High Reynolds Number Incompressible Flows Using Vorticity Confinement," AIAA-99-3316-CP, 1999. Strawn, R., "Computational Modeling of Hovering Ro- tors and Wakes," AIAA-00-0110, 2000. Walhorn, E., "Ein ganzheitliches Berechnungsmodell fur Fluid-Struktur-Wechselwirkungen," Ph.D. thesis, Der Technischen Universitat Carolo-Wilhelmina zu Braunschweig, 2001. Wenren, Y., Fan, M., Dietz, W., Hu, G., Braun, C., Steinhoff, J. and Grossman, B., "Efficient Eulerian Computation of Realistic Rotorcraft Flows Using Vor- ticity Confinement," AIAA-01-0996, 2001.
DISCUSSION John Steinhoff University of Tennessee Space Institute, USA I believe that Dr. Loehner et al. have done excellent, much needed research in method for computing vertical flows. The authors have added an important capability to their unstructured grid methodology. Two points should be made, however: First, earlier work has been done on varying the Vorticity Confinement parameter so that the effects vanish when the grid is sufficiently refined. Second, we believe that it is important that a "biased" finite difference method should be used so that no Vorticity Confinement corrections are made outside the vortex core. Otherwise, they could erroneously effect the computed velocities. This could be the reason for the deviation from experiment of the reported results for computed delta wing surface pressures. Comments, paraphrased from Ref. t1], are "Vortices convecting past airfoils and wings (blade - vortex interactions) were treated in Ref. A. In this early study, unlike in our current studies, near the surface a surface- fitted grid was used for the wing with surface grid refinement to resolve the actual Navier-Stokes equations, since only a low Reynolds number, laminar, case was treated. To accommodate this grid refinement with Vorticity Confinement, the parameter specifying the strength of the Vorticity Confinement term ~ ~ ~ was made to be proportional to grid size so that it automatically vanished in the fine-grid boundary layer region, but was able to confine the convecting vortex in the external, coarse-grid region. When using unstructured grids, which have rapidly changing cell sizes, care must be taken not only that £ varies properly with cell size, but also that the confinement correction does not extend beyond the vortex core due to numerical artifacts of the implementation. This property is true in the continuum limit, and should be preserved in the discretization. If the correction does extend beyond the vortex, then it could erroneously affect surface pressure if a vortex is passing near a surface. This could be important, for example, for delta wings and similar cases, where vortices convect near surfaces. In fact, for delta wings, it is well known that there is a feeding sheet from the leading edge causing the vortex to grow in strength as it convects and causing the characteristics to point towards it. In such cases, for a reasonable grid, confinement is not really needed (until the vortex convects past the trailing edge). If confinement is used correctly, however, it should not change the nearby pressure on the surface even in these cases, for high Reynolds number flow." References 1. Fan, M., Wenren, Y., Dietz, W., Xiao, M., Steinhoff, J., "Computing Blunt Body Flows On Coarse Grids Using Vorticity Confinement", to appear in Journal of Fluids Engineering, December, 2002. 2. Steinhoff, J., and Raviprakash, G., "Navier- Stokes Computation of Blade-Vortex Interaction Using Vorticity Confinement", AIAA-95-0161. AUTHORS' REPLY Thank you very much for your interest in our paper and for providing information about your own work on the vorticity confinement. We would like to address your comments as follows. Point 1: The effect of Vorticity confinement should vanish as a) Antidiffusion: One can develop V|CA X ~ with CO = V X v . One of the terms appearing in the double vector product is of the form V2v, i.e. an antidiffusion. The problem is that one cannot get rid of the other terms, making the analysis murky. Clearly, the antidiffusion is conservative. b) Different Limiting: One can also submit that the diffusion one is trying to avoid is the one that is trying to spread the vortex. If one decomposes locally the velocity vector with the CO, n, v system, one can apply different limiters in the different directions. For example, one could apply a steepening limiter like super-B in the direction of n. We tried this. The results of the vorticity confinement were still far superior to those obtained with this procedure.
DISCUSSION Luigi Martinelli Princeton University, USA I would like to congratulate the authors for their interesting contribution and for a well written paper. Tracking vortices over large distances using CFD methods based on the Euler and RANS equations is indeed a challenging and important task. The authors have chosen to implement a vorticity confinement method, which is shown to improve the resolution of vortices propagating in the far field. In the paper, the authors correctly note that the vorticity confinement term acts as a body force. Thus, it alters the conservation of momentum, and therefore the overall dynamics of the flow. In light of this, I was surprised by the little theoretical work done to justify this approach. I believe that a theoretical investigation of the vorticity confinement method should precede the application of it to the very complex flows discussed in the paper. So I am wondering why the authors have decided to compute very difficult flows first, and eventually carry out the theoretical analysis at a later time. Also, I expect that a more accurate formulation of their method will require a confinement term consistent with the discretization error of the numerical scheme. Have the authors looked at this alternative approach? AUTHORS' REPLY Thank you very much for your interest in our paper and for the good comment. The point of carrying out first some theoretical analysis before proceeding to large-scale problems is well taken. We have tried to bridge the gap between theory and practice, but have not been successful to date. As anecdotal note, among the things we tried the following two seemed the most . . promising: a) Antidiffusion: One can develop V|Cd XCO with ~ = V X v . One of the terms appearing in the double vector product is of the form V2v, i.e. an antidiffusion. The problem is that one cannot get rid of the other terms, making the analysis murky. Clearly, the antidiffusion is conservative. b) Different Limitina: One can also submit that the diffusion one is trying to avoid is the one that is trying to spread the vortex. If one decomposes locally the velocity vector with the do, n, v system, one can apply different limiters in the different directions. For example, one could apply a steepening limiter like super-B in the direction of n. We tried this. The results of the vorticity confinement were still far superior to those obtained with this procedure. DISCUSSION Ernie O. Tuck University of Adelaide, Australia This is a worrying procedure, and I am pleased to see (in the 1St two data points of the written "conclusions and outlook" section) that the authors are hoping to address these worries in future work. For those of us who doubt methods like RANS, precisely because they sometimes seem to over- dissipate, the "black magic" of adding a non- physical additional body force that somehow reverses that over-dissipation is specifically attractive. However, there is a worry that what it might be doing is allowing generation of vortices by the code, but then propagating them into the far field via a non-physical and perhaps even incorrect mechanism. The quality of the present results suggests otherwise, and I wish the authors luck in further study of this interesting technique. AUTHORS' REPLY Thank you very much for your interest in our paper. We agree that there is the worry of generating vorticity due to too much antidiffusion. We, like so many others, also wish for the perfect non-dissipative scheme. But after waiting for 50 years, we decided we could wait no longer. If we were ever going to translate vortices in our lifetime, this technique looked very good. We furthermore believe that although not perfect, it has certainly improved the quality of the results obtained. Its biggest contribution may be a renewed thinking along the antidiffusive theme.
