vortex core and its eﬀects on the stability of vortex flow...

Vortex Core and Its Effects on the Stability of

Vortex Flow over Slender Conical Bodies

Jinsheng Cai∗, Her-Mann Tsai†

Temasek Laboratories, National University of Singapore

Kent Ridge Crescent, Singapore, 119260

Shijun Luo‡and Feng Liu§,

Department of Mechanical and Aerospace Engineering

University of California, Irvine, CA 92697-3975

A three-dimensional Euler solver is used to study the vortex core over slender conical

bodies at high angles of attack and low speeds. A three-dimensional conical overset-grid

is established to reduce the computational efforts while accurately resolve the vortex flow.

The numerical results on the vortex core are verified by available experimental data and

theoretical solutions. The line vortex model used in the theoretical stability analyses

made by the present authors for the vortex flow is modified to account for the effects

of the vortex cores. The jetlike flow in the vortex core and inflow at its outer edge are

modeled based on numerical experiments by the Euler methods on slender conical bodies

incorporated with known theoretical and experimental results on vortex cores. Using

the Euler solutions as a benchmark, the modified model yields a better predictions in

the vortex positions than the original, and a favorable shifts of the transition point of

stability in the Sychev similarity parameter.

I. Introduction

The high angle of attack aerodynamics is of interestbecause it is both a result of the intrinsic fascinationof a flow in which a symmetric body under symmetricflight conditions can produce an asymmetric flow pat-tern and hence experience a side force and a practicalresponse to the needs of aircraft and missile designs toimprove maneuverability by extending flight envelopsto higher angles of attack.

A great deal of experimental, theoretical, and com-putational effort has been spent regarding the under-standing, prediction, and control of the vortex asym-metry. The subject has been reviewed by Hunt,1 Er-icsson and Reding,2 and Champigny.3

It is found by numerous experimental observa-tions4–6 and numerical studies7–10 that a micro-asymmetric perturbation close to the nose tip producesa strong flow asymmetry at high angles of attack.There seems little doubt that the vortex asymmetry istriggered, formed, and developed in the apex region,and the after portion of forebody and the after cylin-drical body (if any) have little effect on the asymmetryover the apex region. The evolution of perturbationsat the apex plays an important role in determining theflow pattern over the entire body.

Since the apex portion of any slender pointed bodyis nearly a conical body, high angle-of-attack flow

∗Research Scientist. Member AIAA.†Principal Research Scientist. Member AIAA.‡Researcher.§Professor. Associate Fellow AIAA.

about conical bodies has been studied analytically.Using a separation vortex flow model of Bryson,11

Dyer, Fiddes and Smith12 found that in addition tostationary symmetric vortex flow solutions there ex-ist stationary asymmetric vortex flow solutions overcircular cones when the angle of attack is larger thanabout twice of the semi-vertex angle even though theseparation lines are postulated at symmetric positions.The stability of these stationary vortices over circularcones were later investigated analytically by Pidd andSmith.13 The disturbances which they treated in thestability analysis were spatial rather than temporal.

Using the simplified separation-vortex flow model ofLegendre,14 Huang and Chow15 succeeded in showinganalytically that the vortex pair over a slender flat-plate delta wing at zero sideslip can be stationary andis stable under small temporal, conical perturbations.Using the same flow model, Cai, Liu and Luo16 devel-oped a stability theory for stationary conical vortexpairs over general slender conical bodies under the as-sumption of conical flow and classical slender-bodytheory. The disturbances which they treated in thestability analysis were temporal or transient ratherthan spatial. Small displacements are introduced tothe stationary vortex positions and then removed. Thedisplaced vortices are still ray lines of the conical flow-field. The disturbances are of a global nature ratherthan a localized nature. Cai, Luo, and Liu17–19 ex-tended the method described in Ref. 16 to studythe stability of stationary asymmetric vortex pair overslender conical bodies and wing-body combinationswith and without sideslip.

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In the above analyses, the separation vortices aremodeled by lines of zero diameter with concentratedvortex strength. However, the real scenario is that theshear layer separated from the wing or body curvesupward and inboard and eventually rolls up forminga core of finite diameter in which flow velocity andvorticity are high and pressure is low. Over a slenderconical forebody, the vortex core is roughly conical andhas a diameter as large as one-third of the semi-spanof a delta wing, in which the flowfield is approximatelyaxially symmetric. In fact, the topics of the vortex corehas been studied extensively. For example, Carcail-let, Manie, Pagan, and Solognac20 and Verhaagen andKruisbrink21 by experimental methods, Murman andRizzi,22 Rizzi,23 Powell, Murman, Perez and Baron,24

Rizzetta and Shang25 and Liu, Cai and Luo26 by nu-merical methods, and Hall27 and van Noordenburg andHoeijmakers28 by theoretical methods.

The aim of this paper is to modify the theoreticalmethod in Ref. 16 to account for the effects of the vor-tex core. The features of vortex core are studied by abrief review of a known theoretical solution for vortexcore and a detail investigation of an Euler numericalsolution of the vortex flow over a slender flat-platedelta wing in comparison with known experimentaldata and theoretical solutions. Based on these stud-ies, the modifications to the original vortex line modelare proposed. The modified theoretical method is thenapplied to recalculate the stationary vortex positionsand the vortex flow stability for typical slender coni-cal bodies. The modified theoretical results with theoriginal theoretical predictions are verified by the Eu-ler computations. Lastly a summary and conclusionsare drawn. In this investigation no vortex breakdownis considered.

II. Features of Vortex Core

Examination of measurements of the flow inside theleading-edge vortex of slender wings reveals that theleading-edge vortex core can be described by the dis-tinguishing of the outer inviscid vortex core and aninner viscous subcore. A known theoretical solutionfor the outer part of the vortex core is briefly reviewedand then the detail features of a vortex core over aslender flat-plate delta wing is studied by the Eulermethods. The computed results are then comparedwith the theoretical predictions and known experimen-tal data.

A. A theoretical Outer Solution

Based on the experimental observations, in the pastdecades a large number of theoretical studies have beendone on the vortex core. Almost all of them con-sider an isolated vortex core, that is, in isolation fromthe natural surroundings, by representing the exter-nal influence onto the vortex core through boundaryconditions. A cylindrical coordinates system (a, r, θ) is

defined for the flowfield of the vortex core. The originis located at the apex of the wing (and also the core).The axis a coincides with the vortex-core axis, r is inthe radial direction and θ is in the circumferential di-rection. (ua, ur, uθ) are the three velocity componentsin the coordinates (a, r, θ). The velocity componentsat the outside edge of the vortex core are denoted by(Ua, Ur, Uθ).

Given the flow conditions at the outside edge of thevortex core, Hall27 found a solution of the conical,axisymmetric, incompressible Euler equations for theouter part of the vortex core.

ua

Ua= 1 − ψ lnσ

ur

Ua= −

1

2

R

aψσ

uθ

Ua= [φ2 − ψ2 lnσ]1/2 (1)

where σ = r/R, R is the radius of the vortex core;

ψ = (1+2φ2)1

2 −1; φ = Uθ/Ua is the flow swirl numberat the outside edge of the vortex core.

Form Eq.1, it is seen that inside the vortex core,that is, r < R, ua > Ua, ur < 0, and uθ 6= 0. Theflow of the vortex core is jetlike and swirling with aninflow.

B. Numerical Solution

It is known that the Euler solver can capture au-tomatically the shear layer separated from the sweptsharp leading edge and its spirally rolling up into avortex core over the body leeward side. Although thesecondary features are absent in the Euler solutions,the gross dominant characteristics of the flowfield, i.e.the primary vortex configurations and their interactionwith the body surface are reproduced.

The present Euler solver is based on a multi-block,multigrid, finite-volume method and parallel codefor the three-dimensional, compressible steady andunsteady Euler and Navier-Stokes equations. Themethod uses central differencing with a blend ofsecond- and fourth-order artificial dissipation and ex-plicit Runge-Kutta-type time marching. The coeffi-cients of the artificial dissipation depend on the localpressure gradient. The order of magnitude of theadded artificial dissipation terms is of the order ofthe truncation error of the basic scheme, so that theadded terms have little effect on the solution in smoothparts of the flow. Near the steep gradients the artificialdissipation is activated to mimic the physical dissipa-tion effects. The resulting code preserves symmetry.Unsteady time-accurate computations are achieved byusing a second-order accurate implicit scheme withdual-time stepping. The solver has been validated fora number of steady and unsteady cases.29–32 A newlydeveloped overset-grid techniques33 is implemented to

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z

x

y

o

s

β

α

b

U∝

Ua

Uy

Ux

ar θ

Fig. 1 Slender conical wing-body combination andseparation vortices.

facilitate the grid refinement in the domain of highvorticity.

Figure 1 shows the coordinates system for a conicalbody consisting of a circular cone body and a flat-platedelta wing. The rectilinear coordinates (x, y, z) is abody axes system with the origin located at the wing-body apex point O, where the axis z coincides withthe wing-body axis, y is in the wing spanwise direction,and x is pointed to the leeward side and the axes x, y, zform a right-hand system. The delta-wing has a semi-apex angle ǫ and the semi-spans of the wing and thebody are s and b respectively. The angle of attackis α and the angle of side-slip is β. The freestreamMach number, M∞ is set at 0.1 to approximate anincompressible flow for all computations reported inthe present paper.

The present computational model is a flat-platedelta wing of sweep angle 76◦, or ǫ = 14◦, α = 20.4◦,and β = 0. For the symmetric flow considered, onlya half flow field is to be computed. The wing is aconical body. It is known that a subsonic flow over aconical body cannot be strictly conical. However, ifthe conical body is slender, the flow is nearly conical.This was observed in water tunnel for a triangular thinwing of ǫ = 15◦ at α = 20◦ and the Reynolds numberis 20,000 based on chord by Werle in 1961 as shownin the Reference,34 and also proved by Navier-Stokescomputations of, e.g. Thomas, Kirst and Anderson.35

In principle, a conical flow can be solved in a two-dimensional plane with the appropriate modified equa-tions. However, the present studies maintain the use ofa three-dimensional code on a three-dimensional gridto allow calculation of not perfectly conical flows. Athree-dimesional conical grid for a flat-plate delta wingof ǫ = 8◦ is shown in Fig. 2, part (a) gives the gridon the incidence plane, and part (b) shows the grid onthe exit plane. The upstream boundary is a cone sur-face which shares the same apex with the conical body.This cone surface is 25s distance away from the body

( a ) ( b )

Fig. 2 Conical grid for a flat-plate delta wing,ǫ = 8◦, (a) in the incidence plane, (b) in the exitplane. Only every 8th line is shown in the radialand circumferential directions.

Fig. 3 Close-up view of the conical grid for a flat-plate delta wing in the exit plane, only every 8thline is plotted in the figure for clarity, ǫ = 14◦.

axis at each cross-section normal to the body axis.Zero normal velocity boundary condition is appliedon the body surface. Kutta condition at the sharpleading edges of the wing is satisfied automatically inan Euler code. On the symmetric plan, symmetricboundary condition is applied. Characteristic-basedconditions are used on the upstream boundary of thegrid. On the downstream boundary, all flow variablesare extrapolated. Grids are bunched into one point atthe body apex. No numerical difficulties are encoun-tered at the vertex point since a finite-volume methodis used. Only a few grid lines are needed in the lon-gitudinal direction for conical flow calculations, and5 longitudinal grid lines are chosen for the followingcomputations. However, very fine grids in the radialand circumferential directions in the cross planes mustto be used to resolve the vortical flowfield for the pur-pose of stability studies.

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Table 1 Two grids of different densities

Grid Inner layer Outer layerFine 5 × 481 × 257 5 × 321 × 129Coarse 5 × 385 × 193 5 × 257 × 97

Iteration Step

Max

imal

Res

idua

l

0 10000 20000 30000 40000 50000

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Fig. 4 Convergence history over a flat-plate deltawing, ǫ = 14◦, α = 20.4◦.

For the computation model wing of ǫ = 14◦, an over-set conical grid is designed to match the local flowgradients and to facilitate the parallel processing ofthe computations. A close-up view of the grid on theexit plane is shown in Fig. 3.

It is known that there is a rather large influence ofmesh density and lay-out on the Euler solutions forhigh-angle-of-attack flows. Comparing a given solu-tion with another obtained with the same numericalmethod but on a denser mesh is one of the most certainways to judge how near the given solution is to the ulti-mate accuracy indicated by the converged sequence ofsolutions using successively refined meshes. Two gridsof different densities for a half flowfield shown in Ta-ble 1 are used in this numerical experiments, where thethree grid numbers are given in the longitudinal, radialand circumferential directions. The computational re-sults with the coarse grid turn out to be somewhatdifferent from those with the fine grid in the subcoreinside the vortex core as shown in Fig.11 and Fig.12below. The fine grid is taken under consideration ofour available computing resources, and thus is imple-mented in the following computations.

The total of these two layers of grids constitutes anoverall conical grid of extremely fine density that isneeded to resolve the high vorticity regions and simu-late the vortex interactions. The total number of thefine and coarse grid points for the half space of thesymmetric flow is 5 × 165, 026 and 5 × 99, 234 respec-tively.

With an uniform free stream flow as the initial solu-tion and on the two-layer fine grid, the computationsare run in double precision until the maximum resid-ual is reduced by more than 10 orders of magnitudeas shown by the convergence history in Fig. 4. The

Fig. 5 Pressure contours on a cross flow plane overa flat-plate delta wing, ǫ = 14◦, α = 20.4◦.

y/s

x/s

0.6 0.7 0.8 0.9

0.2

0.3

0.4

0.5

Fig. 6 Local and detail view of the conical grid inthe vortex-core region on a cross-flow plane for aflat-plate delta wing, ǫ = 14◦.

computing time for the 50, 000 iterations in double(64bit) precision is about 12 hours on a 16-processorparallel cluster computer consisting of AMD AthlonXP1600+ CPUs. Such a stringent convergence crite-rion is needed especially for stability studies of highangle-of-attack flows as is pointed out by Siclari andMarconi.36

Figure 5 gives the computed pressure contours on across-flow plane. The center of the core is clearly seenlocated at xc/s = 0.314, yc/s = 0.733.

The local and detail grid configuration used to re-solve the vortex core region is shown in Figure 6. Thereare about 100 × 100 grid points in the radial and cir-cumferential directions respectively. It will be seen atthe end of this section if this grid is fine enough toresolve the vortex core.

Fig.7 shows the contours of the longitudinal veloc-ity component w/U∞ and Figs.8 shows the contoursof the longitudinal vorticity component ωzs/U∞ on across flow plane. In these figures one can find essen-

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Fig. 7 Contours of the longitudinal velocity com-ponent w/U∞ on a cross flow plane over a flat-platedelta wing, ǫ = 14◦, α = 20.4◦.

Fig. 8 Contours of the longitudinal vorticity com-ponent ωzs/U∞ on a cross flow plane over a flat-plate delta wing, ǫ = 14◦, α = 20.4◦.

tially circular contours. Thus the outer edge of thevortex core is nearly a circular cone. This is the pri-mary vortex core formed from the spirally rolling-upof the shear layer separated from the leading edge ofthe wing. It plays an important role in the flow-bodyinteractions.

To further exploration of the vortex-core structureand compare with known experiment and theoreticalresults, the distributions of various flow parametersalong lines passing through the center of the vortexcore are studied. Figure 9 shows the distributions ofthe total pressure loss coefficient Cpt and the staticpressure coefficient, Cp versus (x − xc)/s along theline y = yc, where Cpt = (pt − pt∞)/(ρ∞U2

∞/2). In

the Figure the numerical results are compared withthe experimental data measured by Carcaillet, Manie,Pagan and Solignac20 using a three-dimensional laservelocimeter and a five-hole pressure probe in an 1mresearch low-speed wind tunnel. The tested wing has

Vertical tranverse, (x-x )/s

Sta

tican

dto

talp

ress

ure,

Cp,

Cpt

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

-12

-8

-4

0

c

Cpt

Cp

Experimental

Numerical

Fig. 9 Computed distributions of the total pres-sure loss coefficient and the static pressure coeffi-cient versus (x − xc)/s along y = yc on a cross-flowplane over a flat-plate delta wing, ǫ = 14◦, α = 20.4◦,compared with experimental data.20

Radial distance to vortex axis, r/a

Axi

alvo

rtici

ty,

s/U

-0.05 -0.025 0 0.025 0.05

0.0

20.0

40.0

60.0

80.0

100.0

x-traversey-traverse

ω∝

z

Fig. 10 Computed distributions of the longitudinalvorticity component versus radial distance from thevortex-core axis along the x−traverse, y = yc andy−traverse, x = xc over a flat-plate delta wing, ǫ =14◦, α = 20.4◦.

ǫ = 15◦ at α = 20◦, and the Reynolds number basedon the root chord, c0 is 0.7×106. The data of the mea-surement at the cross-flow plane 58%c0 given in Figure11 of Ref.20 is used here. It is seen that the measureddistributions are in agreement with the correspondingcomputed distributions in Fig.9.

Fig.10 gives the distributions of the longitudinalvorticity component ωzs/U∞ versus r/a along the xtraverse, y = yc and the y traverse, x = xc where(a, r, θ) is the cylindrical coordinates around the vortexaxis a. It is seen that the distribution curves along thetwo perpendicular traverse lines nearly coincide, andthey are almost symmetric with respect to the ordinater/a = 0. Therefore the flow in the vortex core is nearlyaxisymmetric besides conical. Inside the vortex core,

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Axi

alve

loci

ty,u

/U

-0.05 -0.025 0 0.025 0.050.5

1.0

1.5

2.0

2.5

3.0

Numerical (fine grid)Numerical (coarse grid)TheoreticalExperimental

a∝

Fig. 11 Computed distributions of the axial ve-locity component versus radial distance from thevortex-core axis along y = yc over a flat-plate deltawing, ǫ = 14◦, α = 20.4◦, with fine and coarse gridscompared with theoretical solutions27 and experi-mental data.21


Circ

umfe

rent

ialv

eloc

ity,u

/U

-0.05 -0.025 0 0.025 0.05

-1.0

0.0

1.0 Numerical (fine grid)Numerical (coarse grid)TheoreticalExperimental

θ∝

Fig. 12 Computed distributions of the circum-ferential velocity component versus radial distancefrom the vortex-core axis along y = yc over a flat-plate delta wing, ǫ = 14◦, α = 20.4◦, with fine andcoarse grids compared with theoretical solutions27

and experimental data.21

the vorticity increases sharply toward the core centerand reaches a maximum value at the center. From Fig.10 the outside edge of the highly rotational region islocated at R = 0.06a or 0.2s where R is the radiusof the rotational or vortex core. Inside the rotationalregion viscous diffusion has smoothed out completelythe gradients of the velocity distribution, and a shearlayer can no longer be detected. Inside the vortex orrotational core, the static and total pressure decreasetoward the vortex center and reach minimum valuesat the vortex center as shown in Fig. 9.

Fig.11 and Fig.12 give the numerical results of thedistributions of the velocity components, ua/U∞ anduθ/U∞ versus r/a along y = yc. Here the velocity is

Flow Variables Computed Test20 Test21

(uz)max/U∞ 2.8 3.0 3.0(uθ)max/U∞ 1.3 1.4 1.3(ωz)maxs/U∞ 92 152 224

(Cp)min −10.4 −11 −13(Cpt)min −2.8 −2.8 −4.8

R/s 0.2 0.2 0.2Rs/s 0.04 0.08 0.03

Table 2 Comparison of computed vortex-core pa-rameters with experimental data.

decomposed into components along the directions ofcylindrical coordinates, (a, r, θ) where the axis a co-incides with the vortex-core axis. The axial velocitycomponent, ua increases toward the vortex axis, andreaches a maximum value at the vortex axis, while thecircumferential velocity component, uθ first increasestoward the vortex axis, and after reaching a maximumvalue near the vortex axis, decreases abruptly to zeroat the vortex axis. The location of the maximum uθ

defines the edge of a subcore, and inside this subcorelarge gradients of velocity and pressure prevail andnumerical viscous forces dominate. This subcore isa viscous subcore in which an artificial total pressureloss results in. From Figures 11, 12 and 9 the radiusof the subcore, Rs = 0.01a or 0.04s.

Verhaagen and Kruisbrink21 measured the flowproperties of the conical part of a leading-edge vortexusing a five-hole pressure probe in a low-speed and tur-bulence level of about 0.05% wind tunnel. The modelis a sharp leading-edge flat-plate delta wing of ǫ = 14◦

at α = 20.4◦. The Reynolds number is 3.8×106, basedon the model root chord length, c0. The measurementcross-flow plane was at 50%c0. The measured distri-butions21 of the axial velocity component and circum-ferential velocity component along traverses passingthrough the primary vortex core center are also shownin Fig.11 and Fig.12. They agree well with the corre-sponding computed distributions.

From the numerical and experimental results givenin Figures 11 and 12, the swirl number at the outsideedge of the vortex core r/a = 0.06, Ua/U∞ = 1.30and φ = Uθ/Ua = 0.96/1.30. With these boundaryconditions, Eq.1 yields the theoretical results of Hall27

shown in the Figures. The computational results agreewell with the theoretical results in the outer part of thevortex core.

The Figures 11 and Fig.12 also show the numericalresults using the fine and coarse grids given above. Thetwo results are different in the viscous subcore of thevortex core. Hence the coarse grid is not fine enough toresolve the subcore, and the fine grid is implementedas stated before.

The computed characteristic flow parameters aretabulated and compared with the known test data inTable 2.

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As shown in Table 2, most of the important flowparameters are predicted very well by the Euler com-putations. Even the prediction of the level of totalpressure losses in the viscous subcore are quite realis-tic, in consideration of that the total-pressure loss atthe center of the core, just like in the rest of the core,is a purely spurious numerical artifact. Thus, thisagreement between the calculations and experimentsmay be fortuitous. However, similar observations werefound in other free-vortex flow simulations over sharpedge delta wings using Euler methods, e.g. Murmanand Rizzi,22 Rizzi,23 and Powell, Murman, Perez, andBaron.24 They performed systematic studies in whichvarious computational parameters were changed. Inparticular, grid spacing and artificial damping coeffi-cients were changed by an order of magnitude. Theyfound that the magnitude of the total pressure losswas insensitive to all the computational parameters al-though the vortical region was more diffuse on coarsergrid and/or with high damping constants. Rizzi23

claimed that the invariantness of total pressure losswith the grid size appears to result from a singular-ity in the solution. Moreover, Rizzetta and Shang25

reported that total pressure contours from the Eulersolution were virtually identical to those of the Navier-Stokes calculations, except for the zone of secondaryflow not reproduced in the inviscid result. Just asthe separation at sharp leading edge is insensitive toviscosity, the total pressure loss in subcore is insen-sitive to viscosity. Both the sharp edge of the wingand the center of the vortex core are singular pointsof the Euler differential equations. It is the numeri-cal dissipation smoothing out the singularities. Thegeneration of vorticity about sharp edge and the to-tal pressure loss at the center of vortex core are bothinsensitive to the actual magnitude of the numericaldissipation, as long as there is some.

It is noted that the computed maximum axial vor-ticity is lower than the experimental deta. In fact, theexperimental results of the maximum axial vorticityfrom different investigators vary quite substantially aspointed out by Nelson and Visser.37 From an exami-nation of the grid resolution of each investigation, theyfound that the highest derived vorticity values corre-spond to the finest grid resolution and vice versa. Thelower value of the computed maximum axial vorticitymay be also due to the insufficient grid resolution. Thecomputed total pressure loss agrees well with the re-sult of Carcaillet et al.,20 but much lower than that ofVerhaagen et al.21 The discrepancy may be also dueto insufficient grid resolution.

According to Fig. 6, there are approximately 100×100 grid points lying in the cross section of the vortexcore, and about 20 × 20 grid points in the subcore. Itseems that the grid is still not fine enough to resolvethe flow in the subcore.

The computed radial velocity component is one or-

der of magnitude less than the other two velocity com-ponents, and thus is not shown. In the vortex core, theradial velocity component is pointed to the core axis.It first increases toward the vortex axis, and then de-creases abruptly to zero at the core axis.

III. Stability Analysis of Vortex Pair

The reader is referred to Ref. 16 for details of thetheoretical background. Only the derivation of thevortex velocity are reviewed and modified in this sec-tion.

A. The Original Vortex Velocity Expression

Consider the flow past a slender conical wing-bodycombination at an angle of attack α and sideslip angleβ as shown in Fig. 1 with the rectilinear body coor-dinates (x, y, z). The velocity of the free stream flowis U∞. The combination has a slender triangular flat-plate wing passing through the longitudinal axis of thebody. The flow separates from the wing sharp leadingedge and the flow is assumed to be steady, inviscid,incompressible, conical, and slender.

The inviscid incompressible flow considered in theabove model is irrotational except at the lines of theisolated vortices. The governing equation for the veloc-ity potential is the three-dimensional Laplace equationwith zero normal flow velocity on smooth body sur-faces, and Kutta conditions at sharp edges as bound-ary conditions. By the principle of superposition, theflow around the body can be obtained by solving thefollowing two flow problems.

Flow problem 1: The flow due to the normal com-ponents of the freestream velocity, Ux = U∞cosβsinαand Uy = U∞sinβ.

Flow problem 2: The flow due to the axial compo-nent of the freestream velocity, Uz = U∞cosβcosα.

Both problems subject to the boundary conditions.

Under the assumption of conical and slender flow,the three-dimensional flow problem is reduced to atwo-dimensional flow problem. The flow in each crosssection at z may then be regarded as a two-dimensionalflow across the local cross sectional profile governed bythe two-dimensional Laplace equation with the bound-ary conditions. Solution to this two-dimensional ve-locity field is obtained by conformal mapping or otheranalytical or numerical methods for the first flow prob-lem, and by the condition of conical flow in which theflow is invariant along rays emanating from the apexfor the second problem. Let ζ = f(z) is the confor-mal mapping for this profile in the plane Z = x + iyto a circle of radius r0 in a uniform flow of velocity(Ux/2, Uy/2) in the plane ζ = ξ + iη.

The complex velocity at the vortex point Z1 (or ζ1)is obtained by a limiting process (see Rossow38).

u1 − iv1 =

[

1

2

(

Un −Unr2

0

ζ21

)

+iΓ1

2π

(

−1

ζ1 − r20/ζ1

)

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−iΓ2

2π

(

1

ζ1 − ζ2

−1

ζ1 − r20/ζ2

)] (

dζ

dZ

)

1

−iΓ1

4π

(

d2Z

dζ2

)

1

(

dζ

dZ

)2

1

−UzZ1

s/ tan ǫ+

1

2π

N∑

j=1

Qj

Z1 − Zj(2)

where the subscript 1 denotes the values at Z = Z1

(or ζ = ζ1). Un = Ux(1 + iKS); KS = tanβ/ sin α isthe sideslip similarity parameter; K = tanα/ tan ǫ isthe Sychev similarity parameter (Sychev39); ζ1 and ζ2,and Γ1 and Γ2 are the positions and strengths of thevortex 1 and vortex 2, respectively; Qj is the strengthof the point sources at Zj and Qj(j = 1, 2, ...N) areto be determined by N simultaneous equations of theboundary condition on the body contour in the flowproblem 2. A similar expression is obtained for thecomplex velocity, u2 − iv2 at the vortex point Z2 (orζ2).

The stationary positions, Z1 (or ζ1) and Z2 (or ζ2),and strengths of the vortices, Γ1 and Γ2, are deter-mined by solving a set of algebraic equations. Theseare u1 − iv1 = 0 and u2 − iv2 = 0 for the vortexvelocity fields, and two more equations that set theflow velocities to be finite values at the sharp edges ofthe flat plate (Kutta condition). The four algebraicequations are linear in Γ1 and Γ2, and non-linear inZ1 (or ζ1) and Z2 (or ζ2). They are solved by aniteration method. A Newton iteration for the vor-tex locations is constructed for F(X) = 0, whereF = [u1, v1, u2, v2]

T , X = [ξ1, η1, ξ2, η2]T , ζ1 = ξ1+iη1,

and ζ2 = ξ2 + iη2. Given the vortex positions ζ1 andζ2, the vortex strengths Γ1 and Γ2 can be obtained byusing the separation conditions.

Once the stationary positions, Z10 and Z20, of thetwo vortices are determined, the stability of the vor-tices is readily obtained by the Table 1 given in Ref-erence 16.

B. The Modified Vortex Velocity Expression

In consideration of the jetlike inflow of the vortexcore, the theoretical model of Reference 16 is mod-ified by adding a line sink of strength, Qc to eachline vortex of Fig.1, and in the meantime augment-ing the freestream velocity component Uz used in theflow problem 2 by a factor (1 + Kc) where Kc > 0.Evidently, Qc and Kc are related to the strength ofthe vortex considered, Γ.

The strength of the sink Qc and the strength of thevortex Γ for a vortex core are related to the velocitycomponents at the outside edge of the vortex core.

Qc = −2πRUr,Γ = 2πRUθ (3)

Ur and Uθ are related by the the theoretical solutionsof Hall,27 Eq.1 at r = R. Write Qc = −qcΓ. ,

qc =1

2

R

aψ/φ (4)

Using the numerical results of the last Section, R/a =0.06 and φ = 96/130 yields qc ≈ 0.02.

The relation between Kc and Γ is obtained by nu-merical experiments with the Euler methods.

Kc = κ(Γ/(2πsUx))2 (5)

According to the Euler computations of the vortexcores over slender conical bodies of various geometriesand at high angles of attack give κ = 0.3.

Thus, the complex velocity at the vortex point Z1

(or ζ1), Equation (2) is modified.

u1 − iv1 =

[

1

2

(

Un −Unr2

0

ζ21

)

+iΓ1

2π

(

−1

ζ1 − r20/ζ1

)

−iΓ2

2π

(

1

ζ1 − ζ2

−1

ζ1 − r20/ζ2

)

−qcΓ2

2π

(

1

ζ1 − ζ2

+1

ζ1 − r20/ζ2

−1

ζ1

)

−qcΓ1

2π

(

1

ζ1 − r20/ζ1

−1

ζ1

)] (

dζ

dZ

)

1

−(i − qc)Γ1

4π

(

d2Z

dζ2

)

1

(

dζ

dZ

)2

1

−(1 + Kc1)UxZ1

sK+

1

2π

N∑

j=1

Qj

Z1 − Zj(6)

where the subscript 1 denotes the values at Z = Z1

(or ζ = ζ1), and the last term on the right-hand side isdetermined by replacing Uz with (1 + Kc1)Uz as donefor its next preceeding term. A similar expression isobtained for the complex velocity, u2−iv2 at the vortexpoint Z2 (or ζ2).

IV. Analysis of Typical Model

Configurations

To assess the merit of the modified theoretical meth-ods, both original and modified theories are appliedto vortex flows over various typical slender conicalbodies at high angles of attack. Their predictions onthe stationary symmetric and asymmetric vortex pairpositions and stability are compared with the com-putaional results by the Euler methods.

A. Flat-Plate Delta Wing with and withoutSideslip

Stationary vortex configurations and their stabilityover flat-plate delta wing were studied by the origi-nal theoretical methods. Reference 16 discussed thecase without sideslip (KS = 0) in detail. Stationarysymmetric vortex pairs are found over flat-plate deltawings for a range of angles of attack, or in terms of thesimilarity parameter K. Reference 18 further showedthat there are no asymmetric stationary vortices atzero sideslip. Figure 13 plots the maximum real partof the eigenvalues λ1 and λ2 of the vortex motion for

8 of 14

Similarity Parameter , K

Max

{Re(

1),R

e(2)

}

2 4 6 8 10

-0.8

-0.4

0

anti-symmetric

symmetric

Perturbation:

∆

λλ

Fig. 13 Maximum real part of eigenvalues for sym-metric vortex pairs over a flat-plate delta wing vs.K, KS = 0, by original theory.


Max

{Re(

),R

e()

}

2 4 6 8 10-1.2

-0.8

-0.4

0

Windwardanti-symmetric

symmetric Leeward

Perturbation: Asymmetric vortex:∆

λλ

12

Fig. 14 Maximum real part of eigenvalues forasymmetric vortex pairs over a flat-plate delta wingvs. K, KS = 0.5, by original theory.

the stationary symmetric vortex pairs versus the sim-ilarity parameter K. The eigenvalues remain negativefor the whole range of K considered. The original the-ory predicts that the stationary symmetric vortex pairover the flat-plate delta wing is stable for all angles ofattack.

With sideslip, the stationary vortex pair becomesasymmetric.17,18 Figure 14 gives the maximum realpart of the eigenvalues λ1 and λ2 of the vortex mo-tion under small symmetric and anti-symmetric distur-bances against K for the sideslip similarity parameterKS = 0.5. The maximum real part of the eigenvaluesis negative for all the cases considered.

The corresponding results by the modified theoryare given in Figures 15 and 16. It is seen that themaximum real part of the eigenvalues remains negativeafter the modifications.

To verify the above theoretical predictions, the time-accurate three-dimensional Euler code described above

Similarity Parameter, K

Max

{Re(

),R

e()

}

2 4 6 8 10-3

-2

-1

0

1

anti-symmetricsymmetric

Perturbation:

λλ

12

∆

Fig. 15 Maximum real part of eigenvalues for sym-metric vortex pairs over a flat-plate delta wing vs.K, KS = 0, by modified theory.

Similarity Parameter, K

Max

{Re(

),R

e()

}

2 4 6 8 10

-1.5

-1

-0.5

0

0.5


Perturbation:

λλ

12

∆Asymmetric vortex:

LowerUpper

Fig. 16 Maximum real part of eigenvalues of asym-metric vortex pairs over a flat-plate delta wing vs.K, KS = 0.5 by modified theory.

is used. Two particular cases of the flat-plate deltawing of ǫ = 8◦ at an angle of attack of 28◦, orK = 3.783 and β = 0◦ and β = 13◦, or KS =0 and0.4918 respectively were computed in Ref. 40. Thecomputations showed that there exist symmetric andasymmetric stationary vortex pairs for β = 0 and 13◦

respectively and both vortices are stable under smallperturbations. Therefore, the original and modifiedtheoretical predictions for stability features are con-firmed.

Fig.17 and Fig.18 give the stationary vortex posi-tions versus K for the flat-plate delta wing for KS = 0and KS = 0.4918 respectively. They are obtained bythe modified theory and shown by the circle symbols.The pressure contours in a cross-flow plane computedby the Euler code are shown in these figures for thetwo particular cases. It is seen that the centers of thepressure contours computed at K = 3.783 nearly co-incide with the circle symbols of K = 4.0 given by the

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K = 1

K = 9K = 4

Fig. 17 Positions of symmetric vortex pairs overa flat-plate delta wing vs. K, KS = 0, by modifiedtheory; and pressure contours in a cross-flow plane,ǫ = 8◦, α = 28◦ (K = 3.783), by Euler computations.

K = 1

K = 6

K = 4

K = 1

K = 4

Fig. 18 Positions of asymmetric vortex pairs overa flat-plate delta wing vs. K, KS = 0.5 by modifiedtheory; and pressure contours in a cross-flow plane,ǫ = 8◦, α = 28◦ (K = 3.783), β = 13◦ (KS = 0.4918), byEuler computations.

modified theory if the vortex stays close to the wingsurface.

The stationary vortex pair positions predicted bythe two thoeries are compared with the Euler solu-tions in Table 3 and 4. It is seen that the modifiedpredictions on the vortex positions agree better withthe Euler solutions than the original predictions. Theagreement between the theoretical and computationalresults turns into worse when the vortex locates fur-ther away from the body. The agreement worseningmay be attributed to that the slender flow assumptionof the theories is not strictly valid.

B. Flat-Plate Delta Wing and Circular ConeCombination

The vortex flows over a number of wing-body com-binations were studied in Ref. 19 by the originalanalytical method. A typical wing-body combinationis considered in this paper. The flow over a combi-

Coordinates Original Modified Computedxc/s 0.684 0.562 0.544yc/s ±0.774 ±0.771 ±0.713

Table 3 Theoretical vortex center positions over aflat-plate delta wing, ǫ = 8◦, α = 28◦(K = 3.783), β =0◦ (KS = 0.0), compared with Euler computations.

Coordinates Original Modified ComputedLeft xc/s 0.799 0.501 0.507Left yc/s 0.471 0.643 0.554

Right xc/s 1.173 1.150 0.885Right yc/s −1.831 −1.488 −1.063

Table 4 Theoretical vortex center positions overa flat-plate delta wing, ǫ = 8◦, α = 28◦(K = 3.783),β = 13◦ (KS = 0.4918), compared with Euler compu-tations.

y/s

x/s

-3 -2 -1 0 1 2 3

0

1

2

3

K=4.5

K=2.0

K=4.5

K=2.0

∆K=0.5

Asymmetric vortices

Symmetric vortices

K=4.5

Fig. 19 Location of stationary symmetric andasymmetric vortex pairs over a wing-body combi-nation of a flat-plate delta wing and a circular-conebody vs. K, γ = 0.7, by original theory.

nation of a flat-plate delta wing and a circular-conebody with a body-width-to-wing-span ratio γ = 0.7is examined under the condition of no sideslip. Thelocations of the stationary symmetric and asymmetricvortex pairs are shown in Fig.19 by the original the-ory.19 No stationary asymmetric vortex pair is foundat low angles of attack when the Sychev similarity pa-rameter K ≤ 2.0. At higher K, both symmetric andasymmetric stationary vortex pairs exist. As K is in-creased, both stationary symmetric and asymmetricvortex pairs move upward and outboard. The move-ment of the lower vortex of the asymmetric vortex pairis much smaller and that of the upper vortex is muchgreater compared to the movements of the symmetricvortices.

Figures 20 and 21 show the maximum real partof the eigenvalues for the stability of the stationarysymmetric and asymmetric vortex pairs respectively

10 of 14


Max

{Re(

1),R

e(2)

}

2 2.5 3 3.5 4 4.5

-0.4

0

0.4

0.8

anti-symmetric

symmetricPerturbation:

λ

∆

λ

Fig. 20 Maximum real part of eigenvalues of sym-metric vortex pairs over a wing-body combinationof a flat-plate delta wing and a circular-cone bodyvs. K, γ = 0.7, by original theory.


Max

{Re(

1),R

e(2)

}

2.5 3 3.5 4 4.5

-0.4

0

0.4

Upper

anti-symmetric

symmetric

Lower

Perturbation:

Asymmetric vortex:

λ

∆

λ

Fig. 21 Maximum real part of eigenvalues of asym-metric vortex pairs over a wing-body combinationof a flat-plate delta wing and a circular-cone bodyvs. K, γ = 0.7, by original theory.

under small symmetric and anti-symmetric perturba-tions by the original theory. The stationary symmetricvortex pair is stable when K ≤ 2.58 and unstableotherwise. The asymmetric vortex pair is only stablewhen K ≥ 2.62 The upper vortex is the least stableand becomes unstable for K < 2.62. Therefore, wepresume that a stable symmetric vortex pair at lowangles of attack transits into a stable asymmetric pairas the angle of attack is increased beyond somewhereK = 2.58 − 2.62.

The corresponding analytical results by the modi-fied theory are given in Figures 22, 23, and 24. FromFig.22, no stationary asymmetric vortex pair is foundin the modified analyses when K < 3.8. At higherK, both symmetric and asymmetric stationary vortexpairs exist.

Fig.23 shows that the stationary symmetric vortex

Coordinate, y/s

Coo

rdin

ate,

x/s

-2-1012-1

0

1

2

K=6.0

K=0.1

K=1.0

∆

Asymmetric vortex pairs

Symmetric vortex pairs

K=3.8

K=1.0∆

K=6.0

K=3.8

K=4.5

K=6.0

K=0.5∆

Fig. 22 Location of stationary symmetric andasymmetric vortex pairs over a wing-body combi-nation of a flat-plate delta wing and a circular-conebody vs. K, γ = 0.7, by modified theory.


Max

{Re(

),R

e()

}

1 2 3 4 5 6

-1.2

-0.8

-0.4

0

0.4

0.8


Perturbation:

λ

∆

λ 12

Fig. 23 Maximum real part of eigenvalues of sym-metric vortex pairs over a wing-body combinationof a flat-plate delta wing and a circular-cone bodyvs. K, γ = 0.7, by modified theory.

pair is stable when K ≤ 3.55 and unstable otherwise.Fig.24 gives that the stationary asymmetric vortexpair is stable when K ≥ 4.15 and unstable otherwise.There exists no stationary stable conical vortex pairwhen 3.55 < K < 4.15. The general features of stabil-ity predicted by the modified theory remain the sameas those by the original. However, the critical values ofK for the stability transition are somewhat differentbetween the original and modified predictions. Themodified theory yields a larger critical K values thanthe original.

The Euler computations were performed for a cir-cular cone and flat-plate delta wing combination ofǫ = 8◦ and γ = 0.7 in Ref. 40. The computed re-sults for the two angles of attack, α = 18◦, and 30◦, orK = 2.312 and 4.108 respectively are cited here. Atα = 18◦, there exists a stationary and stable symmet-

11 of 14


Max

{Re(

),R

e()

}

4 4.5 5 5.5 6

-0.6

-0.4

-0.2

0

0.2

0.4


Perturbation:

λ

∆

λ 12

Asymetric vortex:LowerUpper

Fig. 24 Maximum real part of eigenvalues of asym-metric vortex pairs over a wing-body combinationof a flat-plate delta wing and a circular-cone bodyvs. K, γ = 0.7, by modified theory.

ric vortex pair. At α = 30◦ both stationary symmetricand asymmetric vortex pairs exist, but the symmetricpair is unstable while in the asymmetric pair the vor-tex located closer to the body is stable and the vortexlocated further away from the body is periodically cir-cling its stationay position with a small radius.

The theoretical stability predictions by both theoriginal and modified methods are assessed by theEuler computations. For the case of α = 18◦ orK = 2.312, the original and modified analyses bothpredict that the stationary symmetric vortex pair ex-ists (see Figures 19 and 22) and is stable (see Figures20 and 23), which agrees with the Euler computations.For the case of α = 30◦ or K = 4.108, the two analy-ses predict the existence of both stationary symmetricand asymmetric vortex pair (see Figures 19 and 22)and that the stationary symmetric vortex pair are un-stable (see figures 20 and 23), which again agrees withthe Euler computations. The stationary asymmetricvortex pair is predicted stable by the original analy-ses (see Fig.21), but unstable by the modified analyses(see Fig.24); while the Euler computations show thatit is neither stable nor unstable but neutrally stable.It is known that the critical value of K for the asym-metric vortex stability transition is predicted as 2.62and 4.15 by the original and modified theories respec-tively. The present case, K = 4.108 is very much closeto the critical K = 4.15 predicted by the modifiedtheory rather than the critical K = 2.62 by the origi-nal. Therefore, the the modified theory improves thestability prediction in this case.

Fig.25 and Fig.26 give the positions of the stationarysymmetric vortex pairs versus K over the wing-bodycombination of γ = 0.7. They are obtained by themodified theory and denoted by the circle symbols.The pressure contours in a cross-flow plane computedby the Euler code for the two particular cases are

K = 1

K = 6

Fig. 25 Positions of symmetric vortex pairs overa wing-body combination vs. K, γ = 0.7, by mod-ified theory; and pressure contours in a cross-flowplane ǫ = 8◦, γ = 0.7, α = 18◦ (K = 2.312), by Eulercomputations.

K = 1

K = 4

K = 6

Fig. 26 Positions of symmetric vortex pairs overa wing-body combination vs. K, γ = 0.7, by mod-ified theory; and pressure contours in a cross-flowplane, ǫ = 8◦, γ = 0.7, α = 30◦ (K = 4.108), by Eulercomputations.


Table 5 Theoretical vortex center positions over acircular cone and a flat-plate delta wing combina-tion, ǫ = 8◦, γ = 0.7, α = 18◦, compared with Eulercomputations.

shown in these figures also. It is seen that the com-puted center of the pressure contours for K = 2.312lies between the two circle symbols of K = 2.0 andK = 3.0; and the computed center for K = 4.108 islocated near to the circle symbol of K = 4.0.

The positions of the stationary symmetric vortexpairs over the wing-body combination of γ = 0.7 andǫ = 8◦ at α = 18◦ and 30◦ predicted by the origi-nal and modified theories and computed by the Euler

12 of 14


Table 6 Theoretical vortex center positions over acircular cone and a flat-plate delta wing combina-tion, ǫ = 8◦, γ = 0.7, α = 30◦, compared with Eulercomputations.

methods are compared in Tables 5 and 6. It is seenthat the modified methods improve the theoretical pre-dictions on the vortex positions.

V. Summary and Conclusions

An Euler solver is implemented to study the essen-tial features of the vortex core on a slender flat-platedelta wing, and to guide to model the effects of thevortex core on the stability of the vortex flow aboutslender conical bodies at high angles of attack and lowspeeds.

The computational method is based on a multi-block parallel three-dimensional finite-volume methodfor the Euler equations on overset grids. The methoduses central differencing with a blend of second- andfourth-order artificial dissipation and explicit Runge-Kutta-type time marching. The resulting code pre-serves symmetry and is run in double precision (64bitarithmetic) on very fine overset grids. Unsteady time-accurate computations are achieved by using a 2nd-order accurate implicit scheme with dual-time step-ping.

The well-known experimental observations that alow speed flow about a slender conical body is nearlyconical justify the implement of a conical grid for theEuler computations. The conical grid has very few gridpoints in the longitudinal direction but very dense gridin the crossflow plane. The conical grid and the Eu-ler solver still are three dimensional. This techniqueis designed to reduce the computational effort whileaccurately resolve the concentrated vortices and theirmutual interactions.

The above computational approach is applied to aflat-plate delta wing of sweep angle 76◦. The totalnumber of the grid points for a half flowfield is 825, 130,including only 5 grid points in the longitudinal direc-tion. There are 100× 100 grid points on the crossflowplane of each vortex core. Stationary symmetric vortexconfigurations are captured by running the Euler codein its steady-state mode until the residual in the con-tinuity equation is reduced by more than 11 orders ofmagnitude starting from the uniform freestream con-dition. To investigate stability of the stationary vortexflow, a small transient asymmetric perturbations is in-troduced to the flow and the Euler code is running inthe time-accurate mode to determine if the flow willreturn to its original undisturbed conditions or envolveinto a different steady or unsteady solution.

The computed Euler solutions automatically satisfythe Kutta condition at the sharp leading edge of theslender delta wing, capture the free shear layer shedfrom the leading edge, and develop it into a compactand coherent rotaional core or vortex core in the lee-side of the wing. The vortex core may have a diameteras large as 40% of the semi-span. Inside the vortexcore the flow is practically axisymmetric, conical andstrongly jetlike swirling. At the outer edge of the core,there exists a small amount of inflow. In comparisonwith available experimental data and theoretical solu-tions, most of the characteristic features of the vortexcore are well modeled by the Euler methods.

In consideration of the jetlike inflow of the vortexcore, the theoretical model used in Reference 16 ismodified by (1) adding a line sink to each line vortex,and (2) augmenting the freestream velocity componentalong the body axis by a factor greater than one. In-corporating a theoretical formula given by Hall27 andknown experimental data, the sink strength is relatedto the associated vortex strength. The axial velocityaugmenting factor is related to the vortex strength bynumerical experiments with the Euler solver.

The modified theory is applied to a flat-plate deltawing with and without sideslip and a combinationbody of a circular cone and a flat-plate delta wing ofspan ratio 0.7. Using the Euler solutions as a bench-mark, the modified theoretical methods improve thepredictions on the vortex positions, and yield favorableshifts of the stability transition points on the Sychevsimilarity parameter. It is noted that the present mod-ifications are semi-emperical and subject to furtherverifications.

References1Hunt, B. L., “Asymmetric vortex forces and wakes on slen-

der bodies,” AIAA Paper 1982–1336, 1982.2Ericsson, L. and Reding, J., “Asymmetric flow separation

and vortex shedding on bodies of revolution,” Tactical MissileAerodynamics: General Topics, Progress in Astronautics andAeronautics, edited by M. Hemsh, Vol. 141, American Instituteof Aeronautics and Astronautics, New York, 1992, pp. 391–452.

3Champigny, P., “Side forces at high angles of atack. Why,when, how?” La Recherche Aerospatiale, , No. 4, 1994, pp. 269–282.

4Dexter, P. and Hunt, B. L., “The effects of roll angle on theflow over a slender body of revolution at high angles of attack,”AIAA Paper 1981–0358, 1981.

5Lamont, P. J., “Pressures Around an Inclined Ogive Cylin-der with Laminar,Transitional,or Turbulent Separation,” AIAAJournal , Vol. 20, No. 11, Nov. 1982, pp. 1492–1499.

6Zilliac, G. G., Degani, D., and Tobak, M., “Asymmet-ric vortices on a slender body of revolution,” AIAA Journal ,Vol. 29, No. 5, May 1991, pp. 667–675.

7Degani, D., “Effect of geometrical disturbance on vortexasymmetry,” AIAA Journal , Vol. 29, No. 4, April 1991, pp. 560–566.

8Degani, D., “Instabilities of flows over bodies at large inci-dence,” AIAA Journal , Vol. 30, No. 1, Jan. 1992, pp. 94–100.

9Levy, Y., Hesselink, L., and Degani, D., “Systematic studyof the correlation between geometrical disturbances and flow

13 of 14

asymmetries,” AIAA Journal , Vol. 34, No. 4, April 1996,pp. 772–777.

10Hartwich, P. M., Hall, R. M., and Hemsch, M. J.,“Navier-Stokes computations of vortex asymmetries controlledby small surface imperfections,” Journal of Spacecraft andRocket , Vol. 28, No. 2, Mar.-Apr. 1991, pp. 258–264.

11Bryson, A., “Symmetrical vortex separation on circularcylinders and cones,” J. Appl. Mech.(ASME), Vol. 26, 1957,pp. 643–648.

12Dyer, D. E., Fiddes, S. P., and Smith, J. H. B., “Asymmet-ric vortex formation from cones at incidence – a simple inviscidmodel,” Aeronautical Quarterly, Vol. 33, Part 4, Nov. 1982,pp. 293–312.

13Pidd, M. and Smith, J. H. B., “Asymmetric vortex flowover circular cones,” Vortex Flow Aerodynamics, AGARD CP-494 , July 1991, pp. 18–1–11.

14Legendre, R., “Ecoulement au Voisinage De La Pointe aA-vant D’une Aile A forte Fleche Aux Incidences Moyennes,”La Recherche Aeronautique, Bulletin Bimestriel, De L’OfficeNational D’Etudes Et De Recherches Aeronautiques, Jan.-Feb.1953.

15Huang, M. K. and Chow, C. Y., “Stability of leading-edgevortex pair on a slender delta wing,” AIAA Journal , Vol. 34,No. 6, June 1996, pp. 1182–1187.

16Cai, J., Liu, F., and Luo, S., “Stability of symmetricvortices in two-dimensions and over three-dimensional slenderconical bodies,” J. Fluid Mech., Vol. 480, April 2003, pp. 65–94.

17Cai, J., Luo, S., and Liu, F., “Stability of Symmetric andAsymmetric Vortex Pairs over Slender Conical Wings and Bod-ies,” AIAA Paper 2003-1101, January 2003.

18Cai, J., Luo, S., and Liu, F., “Stability of Symmetric andAsymmetric Vortex Pairs over Slender Conical Wings and Bod-ies,” Physics of Fluids, Vol. 16, No. 2, Feb. 2004, pp. 424–432.

19Cai, J., Luo, S., and Liu, F., “Stability of Symmetricand Asymmetric Vortex Pairs over Slender Conical Wings-BodyCombinations,” AIAA Paper 2003-3598, June 2003.

20Carcaillet, R., Manie, F., Pagan, D., and Solignac, J.,“Leading edge vortex flow over a 75 degree-swept delta wing –experimental and computational results,” ICAS 86-1.5.1, Sept.1986.

21Verhaagen, N. and Kruisbrink, A., “Entrainment effect of aleading-edge vortex,” AIAA Journal , Vol. 25, No. 8, Aug. 1987,pp. 1025–1032.

22Murman, E. and Rizzi, A., “Applications of Euler equa-tions to sharp edge delta wings with leading edge vortices,”Proceedings of AGARD Conference on Applications of Compu-tational Fluid Dynamics in Aeronautics, Parais des Congress,Aix-en-Provence, France, 7-10, April 1986, pp. 15–1–15–13.

23Rizzi, A., “Euler solutions of transonic vortex flows aroundthe Dillner wing,” Journal of Aircraft , Vol. 22, No. 4, April 1985,pp. 325–328.

24Powell, K., Murman, E., Perez, E., and Baron, J., “Totalpressure loss in vortical solutions of the conical Euler equations,”AIAA Journal , Vol. 25, No. 3, March 1987, pp. 360–368.

25Rizzetta, D. and Shang, J., “Numerical simulation ofleading-edge vortex flows,” AIAA Journal , Vol. 24, No. 2, Feb.1986, pp. 237–245.

26Liu, F., Cai, J., and Luo, S., “Study of stability of vor-tex pairs over a slender conical body by Euler computations,”Proceeding of the conference for d. a. caughey’s birthday, June2004, Ithaca, NY.

27Hall, M. G., “A theory for the core of a leading-edge vor-tex,” J. Fluid Mech., Vol. 11, 1961, pp. 209–224.

28van Noordenburg, M. and Hoeijmakers, H., “Multiple in-viscid solutions for the flow in a leading-edge vortex,” AIAAJournal , Vol. 38, May 2000, pp. 812–824.

29Liu, F. and Jameson, A., “Multigrid Navier-Stokes Calcula-tions for Three-Dimensional Cascades,” AIAA Journal , Vol. 31,No. 10, October 1993, pp. 1785–1791.

30Liu, F. and Zheng, X., “A Strongly-Coupled Time-Marching Method for Solving the Navier-Stokes and k-ω Tur-bulence Model Equations with Multigrid,” J. of ComputationalPhysics, Vol. 128, 1996, pp. 289–300.

31Liu, F. and Ji, S., “Unsteady Flow Calculations witha Multigrid Navier-Stokes Method,” AIAA Journal , Vol. 34,No. 10, Oct. 1996, pp. 2047–2053.

32Sadeghi, M., Yang, S., and Liu, F., “Parallel Computationof Wing Flutter with a Coupled Navier-Stokes/CSD Method,”AIAA Paper 2003-1347, Jan. 2003.

33Cai, J., Tsai, H.-M., and Liu, F., “An overset grid solver forviscous computations with multigrid and parallel computing,”AIAA Paper 2003-4232, June 2003.

34Van Dyke, M., An Album of Fluid Motion, Parabolic Press,1982, photo 90, page 54.

35Thomas, J., Kirst, S., and Anderson, W., “Navier-Stokescomputations of vortical flows over low-aspect-ratio wings,”AIAA Journal , Vol. 28, No. 2, Feb. 1990, pp. 205–212.

36Siclari, M. and Marconi, F., “Computation of Navier-Stokes solutions exhibiting asymmetric vortices,” AIAA Jour-nal , Vol. 29, No. 1, Jan. 1991, pp. 32–42.

37Nelson, R. and Visser, K., “Breaking down the delta wingvortex–The role of vorticity in the breakdown process,” AGARDCP-494, pp. 21-1-21-15, 1990, Vortex Flow Aerodynamics.

38Rossow, V. J., “Lift enhancement by an externally trappedvortex,” Journal of Aircraft , Vol. 15, 1978, pp. 618–625.

39Sychev, V., “Three-dimensional hypersonic gas flow pastslender bodies at high angle of attack,” Journal of Maths andMech. (USSR), Vol. 24, 1960, pp. 296–306.

40Cai, J., Tsai, H.-M., Luo, S., and Liu, F., “Stability ofvortex pairs over slender conical bodies–theory and numericalcomputations,” AIAA Paper 2004-1072, Jan. 2004.

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