AlAA 87-0205 Vortical Flow Aerodynamics - Physical Aspects and Numerical Simulation

R. W. Newsome NASA Langley Research Center Hampton, Virginia 0. A. Kandil Old Dominion University Norfolk, Virginia

AIM 25th Aerospace Sciences Meeting January 12-15, 19871Ren0, Nevada

VORTICAL PLOW AERODYNAMICS--PHYSICAL ASPECTS AND NUMERICAL SIUULATION

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Richard W. Newsome* Flight Dynamics Laboratory

Air Force Wright Aeronautical Laboratories AFSC Liaison Office, NASA Langley Research Center

Hamp ton, Virginia

and

Osama A. Kandil** Old Dominion University

Norfolk, Virginia

Abstract

Progress in the numerical simulation of vortical flow due to three-dimensional flow separation about flight vehicles at high angles of attack and quasi-steady fliqht conditions is surveyed. Primary emphasis is placed on Euler and Reynolds- averaqed Navier-Stokes methods where the vortices are "captured" as a solution to the qoverninq equations. A discussion of the relevant flow physics provides a perspective from which to assess numerical solutions. Current numerical prediction capabilities and their evolutionary development are surveyed. Future trends and challenges are identified and discussed.

Introduction

The flow about aircraft and missiles at moderate to high angles of attack is characterized by the presence of large- scale vortices on the leeside of the body due to three-dimensional boundary-layer separation. In many instances, the vortices are the predominant aerodynamic feature, and greatly affect the resulting forces and moments on the vehicle. The "non-linear lift" due to the low pressure on the leeward surface of a delta wing beneath the vortex core is perhaps the best known example.

Most flight vehicles are designed for attached flow at cruise conditions. However, for modern fighter aircraft and missiles, the high angle-of-attack flight regime is of primary importance under combat and maneuverinq conditions. Slender bodies and highly swept, sharp leading-edge wings, common to both fighter aircraft and missiles, lead to extensive regions of vortical flow at high angles of attack. Where the vortices are both stable and symmetric, the large increment in normal force due to the vortices may be exploited to aerodynamic advantage. This

*Major, USAF. Member AIAA. **Professor of Mechanical Engineering and Mechanics. Member AIAA. Thi7 onper i s declared s work 01 fhr U.S. Gorrmnmt and / I

rial rvbjrel 10 cop)rlght pr0lrctlon In the United Smtrs.

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region of favorable influence is terminated by the onset of vortex breakdown or the occurrence of asymmetric vortices. Such phenomena produce large, abrupt changes in force and moment coefficients which may jeopardize flight safety. Asymmetric vortex development, for example, may lead to aircraft spin. Other less catastrophic vortex-induced phenomena include structural problems due to hiqh rates of heat transfer along reattachment lines in hvoersonic flow and the onset of buffeting h;e to vortex breakdown. Controllability problems may result from vortex impingement on control surfaces.

Vortical flow may also be used to improve the cruise performance of aircraft. Near the design point, regions of vortical flow then coexist with regions of attached flow. For example, the strakes on the F-16 and F-18 aircraft are also effective in producing lift- augmentation at low angles of attack. Recently, considerable effort has been directed toward the development of leading-edqe vortex flaps which attempt to recover a thrust component as well as a lift component from the vortex-induced normal force on the flap.

In addition to its critical importance to the hiqh anqle-of-attack flight regime, vortical flow has been studied because of its intrinsic interest to aerodynamicists. In many cases, very simple bodies produce complex patterns of three-dimensional (3-D) boundary-layer separation and reattachment. Our understandinq of 3-D flow separation, primarily based upon flow visualization, is incomplete. For example, a universally accepted definition of boundary-layer separation in three- dimensions or of sufficient conditions fOK its occurrence does not yet exist. In many instances, vortical flow is further complicated by transitional or turbulent boundary layers or by shocks which may interact with either the boundary layer or the vortices. Vortex breakdown and asymmetric vortex formation are outstanding problems whose physics are poorly understood.

Numerical methods for the prediction of vortical flow can be classified as: ( 1 ) methods which model the vortex in an approximate manner or ( 2 ) methods which capture the vortex as a solution to the fundamental governing equations. The first category of methods has been principally applied to the prediction of leading-edge vortex flow about sharp-edged delta wings in which the location of the separation line and the topology of the vortical flow are known a priori. Perhaps the simplest method is the Polhamus suction analogy' in which the normal force due to the vortex is assumed to be equivalent to the leading-edge suction force as predicted by linear. attached flow theory. Numerous methods exist in which vorticity is introduced as a singular solution to the linearized potential equation. Either point, line, or panel singular solutions are distributed on the body and also in the flow-field at the assumed location of the vortex. Typically, the free-shear layer and the initial part of the vortex roll-up are approximated by a series of discrete panels and terminated by a line vortex in the vortex core. The satisfaction of velocity boundary conditions on the surface and the fit vortex is sufficient to determine the strength of the singularities and the location of the fit vortex. Such models are computationally inexpensive and have been widely used for preliminary analysis and design. A qood review of this category of methods has been given by Hoeijmakers2 and by Smith.3

The present survey is principally concerned with the second class of methods: those methods which capture the vortex through solution of either the Euler or Navier-Stokes equations. While the Euler equations describe vortical flow under a more restrictive set of conditions, the Reynolds-averaged Navier- Stokes equations correctly model the relevant flow physics and provide a uniformly valid description of vortical flow about arbitrary geomtries throughout the range of flight speeds and Reynolds numbers. In particular, the Navier-Stokes equations are required to describe flows with strong viscous-inviscid interaction in the form of massive boundary-layer separation or shock-boundary layer interaction. Both sets of equations are computationally more expensive to solve than the first category of methods.

The objective of this paper is to survey progress in the numerical prediction of vortical flow due to three- dimensional flow separation about flight vehicles (wings, bodies, and complete configurations) at high angles of attack and quasi-steady flight conditions. While full potential equation solutions with discrete vorticity are briefly reviewed, the predominant emphasis of the paper is on solutions to the Euler and Reynolds-

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averaged Navier-Stokes equations. The fluid physics relevant to vortical flow about wings and bodies, the governing equations, and solution algorithms are first reviewed. Progress in the numerical simulation of high angle-of-attack vortical flows by inviscid and viscous methods is then surveyed. An assessment is made of current capabilities and areas for future work are identified. Although the review is, no doubt, less than comprehensive, it is hoped that a balanced perspective of the rather impressive Progress to date will emerge.

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Physical Characteristics--High Angle-of-Attack Vortex Dominated Flows

The literature dealing with experimental and theoretical aspects of high angle-of-attack vortex-dominated flows is extensive. No attempt has been made to cover this material in a comprehensive manner. However, in order to provide a perspective from which to assess numerical simulation of this flow regime, the relevant physical characteristics of high incidence, vortical flows are briefly sketched. The interested reader is referred to the many excellent comprehensive reviews of the subject. 4 - 1

The flow pattern about a given confiquration depends upon numerous qeometric and flight parameters. However, a useful and generally valid classification can be made in which the L1

flow remains attached and vortex-free. At higher angles of attack, Region 11, (moderate to high angles of attack) three- dimensional boundary-layer separation results in the formation of large vortices on the lee-side of delta wings, swept, low aspect ratio wings. and slender bodies. In this range of angle of attack, the vortices are both stable and symmetric, and the large increments in lift, due to the low pressure induced on the leeward surface by the vortices, may be used to aerodynamic advantage. The great majority of research in vortex-dominated flows has been directed towards understanding and exploiting this region. Region I11 (at still higher anqles of attack) is characterized by the loss of either stability or symmetry in the vortex structure. In particular, the vortex may burst or breakdown, or the vortices about a symmetric body may become quite asymmetric. Either phenomenon may occur in a quasi-steady or unsteady fashion. Both vortex breakdown and asymmetric vortices produce large, unfavorable changes in force and moment characteristics. Due to its proximity to the region of stable and symmetric flow, ,d

Region 11, and an inability to accurately predict the onset of either vortex breakdown or vortex asymmetry, a better understanding of this region is

w/ essential. Finally, at extremely large angles of attack, Region IV (up to g o a l , the leeside flow is characterized by an unsteady diffuse wake frequently accompanied by periodic vortex shedding (depending primarily upon the Reynolds number, and Mach number). Since flight vehicles are unlikely to operate in this flow region, it is of less importance than Regions I1 and 111.

Historically, sharp-edged delta wings and slender bodies have been the two geometries most studied both theoretically and experimentally. They serve as generic models for the principal components of real aircraft and missiles. A brief review of their relevant characteristics in the regions of interest, Regions I1 and 111, serves to highlight the challenges involved in numerically simulating the often complex vortical flows which occur for even simple geometries.

Region 11. Stable and Symetric Flow

Shary-Edge Delta Wing.- Stanbrook and Squire' demonstrated that the flow about a sharp-edqed delta wing could he conveniently classified by decomposing the Mach number, M_, and anqle of attack, a*, into components normal to the winq leadinq edqe MN, a N ,

M = M _

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N

-1 tana (a) aN = tan

where A is the wing leading-edge sweep angle. The Stanbrook-Squire boundary delineates leading-edge separated and attached flows according to whether the leading-edge normal Mach number. MN, is less than or greater than one. For subsonic flight conditions, M _ , < 1, the flow underqoes a strong expansion at the leading edge from the windward to leeward side. If the flight condition is supersonic, M- > 1, hut the normal component is subsonic, MN < 1, the leading edge is swept behind the Mach cone, and a similar leading-edge expansion is observed. In this case, a curved bow shock of variable strength is also present. For sharp leading edges, the boundary layer is unable to negotiate the strong adverse pressure gradient of the subsequent recompression on the leeward edqe just inboard of the expansion, and the flow separates at the leading edge producing a free-shear layer of distributed vorticity. The separation is fixed since separation always occurs at the leading edge irrespective of the

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overall flight condition, boundary-layer condition, or leading-edge geometry. The shear layer then rolls up in a spiral fashion to form a well-defined region of rotational flow - the leading-edge vortex (Figure 11. These leeward vortices, which occur in counter-rotating pairs as the flow is shed about opposite leading edges, are the dominant features of the flow. Most noticeably, the vortices induce very low pressure levels on the wing surface directly below the vortex core. The resultant normal force is then much larger than what would occur if the flow were to remain attached and the vortices were not present. For moderate to high angles of attack, the vortices are both stable and symmetric. As can he seen in Figure 1, the vortex lift occurs as a nonlinear function of angle of attack. Frequently, a secondary, counter-rotating vortex will form under the primary vortex due to the adverse pressure gradient in moving from the low pressure under the primary vortex core to the higher pressure further outboard. Here, the location of the secondary separation line depends critically upon the condition of the boundary layer. The separation line is later defined to he a particular skin friction line onto which adjacent skin friction lines converge as the boundary layer leaves the surface in the form of a free shear layer. Since the laminar boundary layer separates before its turbulent counterpart, the secondary vortex in a laminar flow lies inboard of its position for a turbulent flow. The most obvious effect of turbulence is to decrease the size and strength of the secondary vortex. It has been experimentally observedL3 that the suction pressure peak under the primary vortex is then stronger than that found in the laminar flow.

The previously described vortex flow is termed "classic leading-edge separation." For normal Mach numbers to the right of the Stanbrook-Squire boundary, M,,> 1, a variety of flow patterns are possible. Ideally, since the wing leading edge lies outside of the Mach cone, the leading-edge flow will he attached with a Prandtl-Meyer expansion on the leeward edge and an attached how shock on the windward edge. Although either the boundary layer or leading-edge bluntness may serve to detach the bow shock and cause leading-edge separation, there is a general tendency for the flow to he attached with increasing values of the normal Mach number. Szodruch and Peake14 identified six distinct flow types for thick delta wings, principally to the right of the Stanbrook-Squire boundary. Miller and Wood,15 considering the flow about thin wings at supersonic speeds, refined the classification of Szodruch to include the seven flow types shown in Figure 2. For supersonic normal Mach numbers at small angles of attack, Region 4 . the flow

remains attached, and the windward crossflow expansion is terminated by a crossflow shock. At hiqher angles of attack, the strength of the crossflow shock is sufficient to separate the flow producing a shock-induced bubble, Reqion 5. Still higher angles of attack result in leading-edge separation with a very shallow separation bubble contained within the boundary layer and a crossflow shock on top on the bubble. Region 6 . At very large angles of attack, whether the normal Mach number is subsonic or supersonic, leading-edge separation occurs with very large leeside vortices and strong crossflow shocks on top of the vortices, Region 7. At hypersonic speeds, the flow types, when classified according to leading-edge normal Mach number and anole of attack are similar to those found at supersonic speeds. Szodruch and Peake'" identify one additional flow type, "attached leading-edqe flow with vortices" at low angles of attack and low values of the hypersonic viscous interaction

3 parameter, x = M - / q . by Rao and Whitehead,16 a shear layer forms within the boundary layer due to lateral pressure qradients and rolls u p to form embedded vortices.

As described rn

The flow about a conical delta winq is three-dimensional due to the nonlinear growth of the viscous layer. However, for flow which is either entirely laminar or turbulent, most flow Properties are essentially conical and the separation and reattachment lines are straight, radial lines originating near the wing apex.

the separation line on a body is determined by the three-dimensional boundary layer. Perhaps the simplest h o d y shape is the circular cone. For flows that are either all laminar OK all turbulent, the boundary layer separates along conical rays, and the free-shear layer rolls up to form the primary separation vortex. At sufficiently hiqh angles of attack, secondary counter- rotatinq vortices are formed also alonq conical separation lines.

Slender Bodies.- Unlike the delta wing,

For non-conical bodies, the boundary layer no longer separates along straiqht, radial lines. Indeed, the three- dimensional, vortical flow patterns of separation and reattachment about such bodies are among the most interesting and complex in all of fluid dynamics. Experimentally, oil-streak techniques have been used to visualize flow separation and reattachment on the surface of the body. The convergence of oil-streak lines toward a particular line indicates a line of separation and conversely, the divergence of oil-streak lines from a particular line indicates a line of reattachment. The most successful attempt to interpret the

oil-streak lines and relate them to the overall vortical flow pattern surrounding the body has been a mathematical approach based upon the properties of a continuous vector field and topological rules. A continuous vector field has the property that at any regular point there is only one field line which passes through that particular point. A finite number of singular points may occur within the field. The types of singular points, their number, and the topological rules governing relations between them provide a "flow grammer"17 which may be used to describe and categorize patterns of three- dimensional flow separation.

Legendrela and Maske11lg assumed that limiting streamlines (streamlines arbitrarily close to the body surface) form a continuous vector field on the body surface. However, this assumption breaks down near lines of separation where limiting streamlines rapidly leave the body surface. LighthillLa showed that the skin friction lines (Loci of the surface shear stress vectors) are uniquely defined everywhere on the surface and in particular in the vicinity of the separation lines. Skin friction lines thus form a continuous vector field.

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At the isolated singular points, the skin friction is identically zero. Singular points may either be nodes or saddles. Nodes may be further classified as nodal points or foci of either

number of skin friction lines are tangent to a particular skin friction line at a nodal point and are directed inward toward a node (of separation) or away from a node (of attachment). A focus or spiral node has no common tangent and an infinite number of skin friction lines spiral inward or outward around the singular point. A saddle point has two common skin friction lines. The direction of the lines is inward on either side of a sinqular point for one of the lines and outward for the other. The two skin friction lines separate the adjacent skin friction lines into four quadrants. A node of attachment is typically a forward staqnation point from which skin friction lines emerge and a node of separation is a rearward stagnation point on a closed body into which skin friction lines disappear.

separation or reattachment. An infinite LI

A line of separation is a particular skin friction line toward which adjacent skin friction lines converge as the boundary-layer flow leaves the body surface in the form of a free-shear layer which rolls u p to form the vortex. A line of separation may be classified as either a global or local separation line. A

saddle point of separation. A local separation line is a particular skin friction line onto which adjacent skin

separation line originates from a

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friction lines converge where both the separation line and adjacent skin friction lines all originate at a common, upstream, node of attachment. The separation

v' resulting from a global se aration line is variouslv termed a alobalI7 or closedz'

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separation while the separation resulting from a local separation line is termed either as local," open,2' or crossflowzz separation. The converse of a line of separation is a line of reattachment from which adjacent skin friction diverge. Permissible separation patterns must obey topoloqical rules - the most basic being that the number of nodes must exceed by two the number of saddle points on a closed three-dimensional body. Additional rules17 have been devised for symmetry planes, crossflow planes, etc. The topological rules provide a rational framework for understandinq complex three- dimensional flow separations based upon limited surface information. However, the pattern of skin friction lines on the body surface does not uniquely determine the structure of the flow field away from the body.

The hemisohere-cvlinder ~roblern dramatically' I1lust;ates maLy of the concepts previously discussed. Although the geometry is extremely simple, the three-dimensional separation patterns are quite complex. Numerous e~perimental~'-~~ and computational studies have been performed to fully understand the detailed flow structure. A photoqraph of the experimental oil-streak flow

v visualizationz5 at Mach = 1.2 and a = 19' is shown in Figure 3a. The primary and secondary separation lines are clearly seen on the cylinder downstream of the hemispherical cap. Figure 3b ~ h o w s conceptual drawings of the pattern of skin friction lines on body near the hemispherical nose and the correspondinq 3-D streamline pattern.27 Two sets of symmetrically placed foci are evident. The foci qive rise to so called "nose vortices" which lift off the body and are swept into spiral nodes of attachment in the symmetry plane. The primary separation line originates from the windward node of attachment (not shown) while the second separation line originates at the saddle point downstream from the second EOCUS. When the windward node of attachment and a postulated node of separation downstream are accounted for, the flow satisfies the basic topological rule (ZN = 9 , ES = 7 ) . At other angles of attack, the topological patterns are different but vary from that shown in a continuous manner. The complexity illustrated in the present example is not unique but rather should he viewed as representative of that found in qeneral three-dimensional flow separations.

The vortical flow about more complex qeometric configurations (real aircraft,

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missiles) involves no new flow phenomena. However, additional complexity is introduced by the interaction of the vortices generated by one surface with adjacent body surfaces. Examples include canard/wing, and strake/wing interactions which are generally favorable and forebody/tail or wing/tail interactions which may cause a degradation of control surface effectiveness or premature vortex breakdown.

Region 111. Vortex Asymmetry, Vortex Breakdown

Asymmetric Vortices.- At sufficiently hiqh anqles of attack, the vortical flow about symmetric wings and slender bodies at zero deqrees side slip transitions to a bi-stable asymmetric condition, Figure 4 , 2 8 in response to the small perturbations in the geometry or flight condition. The phenomenon has attracted research interest because the forces resulting from the asymmetry are, in many instances, sufficiently large to trigger aircraft soin. At anales of attack near

11 as asymmetric. The mechanisms which lead to asymmetric vortices are not well understood. The first of two hypothesesz5 suggests that the asymmetry results from an asymmetric boundary-layer transition. On smooth, slender bodies, the onset of the asymmetry is accommpanied by a rapid asymmetric movement of the secondary separation and then the primary separation lines, precipitated by an asymmetric transition reqion. However, asymmetric vortices have been observed in laminar as well as transitional and turbulent flows. Moreover, asymmetric vortices have been documented for sharp- edged delta wings where the primary separation is fixed at the leading edge. Generally, the sharp leading edge. which enforces a symmetric separation, does increase the angle of attack at which asvmmetrv is first observed. A second hypothesis relates the asymmetry to the stability of the singular saddle point in the crossflow planes above the body. Keener and Chapmanz9 experimentally examined the onset of asymmetry on slender forebodies at subsonic speeds and correlated their data with the delta wing data of Shanks.30 Asymmetry was found only at fineness ratios ( a = length/diameter or length/span) above 2.5 and at angles of attack greater than 20 for forebodies and higher values for delta wings. Its occurrence was attributed to the inviscid instability mechanism due to vortex crowding at high fineness ratios.

Ericsson and Redinq3' have surveyed experimental and analytical studies of 3-D vortex asymmetry. For an arbitrary configuration, current knowledqe cannot predict either the onset of the asymmetry

or its magnitude

Vortex Breakdown- The phenomenon of vortex breakdown or bursting in flows over delta wings was first observed by Peckham and AtkinsonSZ in the water vapor condensation trails along the leading-edge vortex cores of a gothic wing. The vortex breakdown location was shown to depend upon the angle of attack and the wing sweep angle. For small angles of attack and larqe sweep angles, vortex breakdown first occurred downstream of the wing and moved upstream to a location above the wing surface with increasing angle of attack or decreasinq sweep angle. Lambourne and BryerS3 were the first to document the two forms of leadinq-edqe vortex breakdown through their well known photograph, Figure 5 . The lower vortex shows an almost axisymmetric sudden swellinq of the core into a bubble that is open at the downstream end (bubble form) and a trailinq spiral turbulent flow. The upper vortex shows an asymmetric spiral filament followed by a rapidly spreading turbulent flow (spiral form). Both forms are characterized by an axial stagnation point and a limited region of reversed axial flow. The spiral form of breakdown was described in three sucessive stages: a sudden deceleration of the fluid moving along the core, an abrupt kink of the core which then turns with a whirling motion that persists for a few turns and a breakdown to large scale turbulence. Although the bubble form occasionally occurred for short periods, the spiral form was the predominant type observed. Particularly at flight Reynolds numbers, leading-edge vortex breakdown is an unsteady phenomenon characterized by the whirlinq motion of the spiral form or the oscillation of the bubble about a mean position in addition to the downstream coherent unsteadiness.

Hummel and Srinivasan3' showed that lift and pitching moment are substantially affected by vortex breakdown once its location moves upstream of the wing trailing edge. There is an abrupt decrease in lift due to the loss of suction pressure under the vortex and a corresponding nose-up pitching moment. Breakdown may also produce buffeting, unsteadiness, and loss of control surface effectiveness.

Much of our knowledge of vortex breakdown has been obtained from experimental studies of vortex breakdown in tubes where the free-vortex can be generated under controlled conditions. Investigations have been reported b

Orloff and Bossel,38 Faler and Leibovich,39 Escudier and Zehnder,'O and Escudier and Keller,'' among others. In References 35-39, the vortex was qenerated by quide vanes located at the upstream end of the tube while, i n References 40-41 ,

Kirkpatrick,36 Sarpkaya, 3 Y

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the vortex was generated by a jet tangent to the wall of a cylindrical cavity with a side slot entry which was then exhausted

vortex generated by the latter process better resembles the leading-edge vortex, neither process reproduces exactly the vortex generated by a swept delta wing.

to a cylindrical tube. Although the d

Both bubble and spiral forms of vortex breakdown have been generated in tube experiments. By varying the Reynolds number and the circulation number (a measure of swirl), SarpkayaS7 generated both the bubble and spiral forms and a third type of vortex disruption, termed a double helix. Faler and Leibovich39 identified seven modes of disruptions where the four additional modes were combinations of the previously mentioned three. Since the double helix does not show a stagnation point neither it nor its variants are termed vortex breakdown. A useful map of the location and type of breakdown as a function of Reynolds number (based on diameter) and circulation number was presented in Reference 39. For a fixed Reynolds number above 3000, the spiral type of breakdown first occurs and moves upstream with increasing values of the circulation number until a certain value of the circulation number is reached after which the breakdown transforms to the bubble type at a more forward location. There is controversy as to whether the bubble and spiral forms are distinct types. Esc~dier'~ has argued that the bubble form is basic and that the spiral type of breakdown is a result of the instability of the bubble type while L e i b ~ v i c h ~ ~ Suggests that the two forms are Separate and distinct phenomenon.

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There is no clear understanding of the mechanisms leadinq to vortex breakdown. Essential elements seem to be a reqion of hiqh total pressure loss within the vortex core and an externally imposed axial pressure qradient. Hall'4 has shown that increasing values of swirl amplify the pressure gradient along the vortex axis. Despite numerous attempts, a generally accepted theoretical description of vortex breakdown does not exist. As outlined by Leibovich,43 Hall,"' Escudier,45 and Wedeme~er'~ in their review papers, existing theories for vortex breakdown may be classified into three groups:

1. Hydrodynamic Instability theories: based "Don either the inflection point instability or the Rayleigh-Taylor instability subject to axiymmetric or spiral disturbances.

2. Stagnation point theories: based on the failure of the numerical inteqration scheme for approximate forms of the qoverning equations (slender cores, quasi-cylindrical flow). 4

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Similar to the usual boundary- layer equations, breakdown is predicted by the failure of the scheme to march beyond the stagnation point (assumed location of vortex breakdown) due to extreme velocity gradients in the vortex core.

3 . Wave-motion theories: vortex breakdown is related to occurrence of a "critical state" where axisymmetric waves in a swirling cylindrical flow propaqate upstream and build up to large values at a location corresponding to the critical state resulting in vortex breakdown. The flow (upstream of the critical state) is termed supercritical where waves can only propagate downstream while the flow (downstream of the critical state) where waves can also propagate upstream is termed subcritical.

Governing Equations

The Reynolds-averaged Navier-Stokes equations describe the conservation of mass, momentum, and enerqy for a turbulent, viscous fluid or qas. For a compressible qas with no external heat addition or body forces, the three- dimensional equations, in conservative form, are qiven as 4

where

L J and the inviscid and viscous flux vectors are

1 F,G,H = - J ~~~

d

( P E + p)Uc

0 a T t a r + a 7

a T t 0 T + o r

C I T + c a r + a T

axbx t ayby + azbz

x xx y xy 7. xz

x XY Y YY YZ

x xz Y YZ z 7.7.

( 3 )

( 4 )

( 5 )

In the above, (u,v,w) are Cartesian velocity vectors, P is the density, p is the static pressure, and E is the total energy. The generalized coordinates ( s , n , i ) are related to the Cartesian coordinates (x,y,z) through the transformation 5 = S(x,y,z), II = (x,y,z), i = (x,y,z) and J is the Jacobian of the transformation, J = a(C,",i)/a(x,y.z) . The form of the equations may also be interpreted in a finite volume sense." For example, the cell volume is 1/J, the surface area of the cell interface in the 5 direction (for example) is (gradS(/J, and the direction cosines are ( k X . ~ y . ~ z ) = ( F x r S )/[grad<(. Y Z

The shear stress tensor, T and the X . X . ' 1 1

heat flux vector, qxi, are defined in

tensor notation (summation convention implied) as:

and the chain rule is used to evaluate derivatives with respect to xi(x,y,z). For most aerodynamic applications, the gas is assumed to be perfect with constant specific heats, C and Cv, (calorically perfect) giving tRe perfect gas equation of state, p = oRT. For turbulent flow, the mass averaginq procedure of Rubesin and produces a form of the qoverning equations ( 1 ) in terms of the mass averaqed variables which is identical to the laminar equations when the shear stress and heat flux terms are redefined to include the Reynolds stress terms. The Boussinesq assumption further relates the Reynolds stress to the rate of mean strain through the eddy viscosity coefficient, E . For a constant turbulent Prandtl number, Prt = 0.9, the shear stress and heat flux are then given by equations (6)

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and ( 8 ) . Laminar viscosity is specified by Sutherland's law and Stokes hypothesis. For air, a Prandtl number, P r = 0.72, is used. The eddy viscosity coefficient is determined by the turbulence model.

Several simplified forms of the full 3-D Navier-Stokes equations have been employed for various vortical flow calculations. The Euler equations neglect viscosity and heat conduction (Fv, G v ,

HV = 0 ) and are valid for non-interacting, inviscid flow outside of the boundary layer. For a body oriented coordinate system in which one of the coordinates (0 is approximately normal to the body surface, the thin layer approximation is often used. All viscous derivatives except those in the normal direction are dropped since these terms are small and are not adequately resolved in most qrids. Thus, the thin-layer equations are obtained from equation (1) by setting Fy. GV = 0 and retaininq only derivatives with respect to 5 in H v . The parabolized Navier-Stokes equations arc valid for stead". suoersonic flow without

~~

. . . ~~~~

streamwise separation. They are obtained from equation ( 1 ) by discarding the time dependent and streamwise diffusion terms aQ/at = 0. Fy = 0 land GV thin layer assumption is made). When the streamwise pressure gradient is modified to suppress negative eigenvalues in the subsonic part of the boundary layer, the equations may be spatially marched and substantial reductions in computation effort and storage requirements are achieved. The less general conical flow equations are valid for supersonic flows about conical bodies only. For inviscid flow, the abscence of a characteristic lenqth scale in the radial direction results in a solution which is invariant along radial lines originating at the apex of the body. In terms of the conical variables. Y. Z,

= 0. if the

Y = y / x 2 = z / x

the conical flow requirement is a 6 / a E = 0. then simplified since a(F - Fv)/a, is replaced by a source term. and the number of independent spatial variables drops from three to two. For viscous flow, a length scale dependence is contained in the Reynolds number and the flow is no longer radially invariant. However, the

The governing equations are " "

flow may be thought of as "locally conical" with the Reynolds number determining the location of the crossflow

solution is obtained. plane (or spherical surface) at which the u

In the limit of infinite Reynolds number, zero thickness of the free-shear layers (free-vortex sheets) and zero radius of the vortical core (vortex line), the flow field becomes irrotational exclusive of the free-vortex sheets and lines. The velocity field is derivable from a scalar potential function, ?. The continuity and the Bernoulli equations are then sufficient to solve for the flow field--including surfaces of both tanqential velocity discontinuity (fit vortex sheets) and normal velocity discontinuity (isentropic shocks). For steady flow, the qoverninq equations are

The non-conservative form of the full potential equation ( 9 )

111) v, 2 v m = - ( V + - V P / P )

has proven most convenient for integral equation solutions.

Solution Algorithms

Vortical flow solutions have been obtained using central and upwind differencinq by finite-difference and finite-volume methods. The evolution of methods has been from explicit, finite- difference methods using central differencing to implicit, finite-volume methods using upwind differencing.

The majority of the reported solutions have been obtained with the central difference alqorithms developed by MacCormack,49 Beam and Warming,50 and Jameson, et a1.51 Although implicit extensions52 as well as finite-volume implementation^^^ have been introduced by MacCormack, the basic explicit, predictor- corrector finite-difference scheme has been the most widelv used. The Beam and

algorithm as generalized by ~ ~ ~ ~ ~ ~ 2 4 and Pulliam and Steger55 and embodied in the ARC2D and ARC3D codes is an implicit finite-difference approximate factorization scheme. The Jameson alsorithm is an explicit multi-stage Runse-Kutta time discretization with a

a

finite-volume spatial discretization. The three-dimensional version developed by R i ~ 7 . i ~ ~ has been widely used to compute Euler solutions for the flow about delta wings. Extensions for viscous flows7 and converqence accelerations8 have been reported.

V'

Quite recently, upwind methodsr7, 59-6' have matured to the point where general two and three-dimensional algorithms for inviscid and viscous flow about arbitrary geometries have been developed. The essential idea is to better match the numerics to the physics of the flow by identifying the characteristic directions of signal propagation and upwind differencinq along those directions. M o r e t t i ' ~ ~ ~ ' characteristic based h scheme for the non-conservative Euler equations with shock fitting has been used to compute the vortical flow about cones at high angles of attack. The predominant interest, however, has been conservative schemes where the natural dissipation due to the upwind differencing combined with proper treatment of signal propagation enable stronq shocks to be captured sharply without oscillation. The increased diaqonal dominance due to upwinding allows the implementation of alternate solution strateqies such as relaxation. The implicit, upwind, finite volume, thin layer Navier-Stokes code developed by Thomas,63 CFL3D, has been applied to the flow about delta wings. The implicit solution is obtained either by streamwise relaxation and approximate factorization in the crossflow planes or entirely by approximate factorization. A hybrid upwind/central difference code, F 3 D . was reported by Ying, Steger, Schiff, and Baganoff.27 H a r t ~ i c h ~ ~ has developed an implicit flux vector split scheme for the incompressible Navier-Stokes equations similar to the scheme of Thomas.

-4,

Algorithms for the parabolized Navier- Stokes equations which have been applied to vortical flows include those due to Lubard and Helliwe11,65 Vigneron, et a1.,66 and Schiff and Steger.67 The latter two codes both employ a non- iterative, implicit, approximate- factorization alqorithm with different techniques to suppress the occurrence of departure solutions.

The Baldwin-Lomax turbulence has been widely used for turbulent, vortical flows. The use of this relatively simple alqebraic model reflects a hesitancy to introduce additional uncertainty into the computation of complex, three-dimensional, massively separated flows. Solutions of enqineering accuracy are frequently achieved. Degani and Schiff69 introduced modifications to the basic model to ensure that the length and velocity scales used to determine the outer eddy viscosity are correctly scaled relative to the boundary layer and not the

4'

9

vortex core. These modifications siqnificantly improve the agreement with experimental results. Nearly a l l computations to date have been for fully turbulent flows. The accurate modeling of transitional flow regions for complex flows would appear to be a difficult prohlem.

The emphasis of the present survey is on the Euler and Navier-Stokes equations which admit solutions with distributed vorticity. Alternatively, a large class of methods have developed based upon solutions to the linearized potential (Prandtl-Glauret) equation with discrete vorticity. For completeness, it should he noted that solutions to the full potential equation ( 1 1 ) with the full nonlinear compressibility have also been developed and applied to vortical flows. Murman and Streme170 used the 'cloud-in-cell" algorithm to track and compute the vortex roll-up behind a larqe aspect-ratio wing. Kandil and Yates" developed an integral equation technique in which the compressibility term (R.H.S. of equation (11)) was modeled as a volume inteqral source term to the linear L.H.S. of equation (11). The method was applied to the transonic leading-edge vortex flow about a hiqhly swept delta winq.

SURVEY OF APPLICATIONS

Inviscid Flow

The development of methods which solve the Euler equations and the application of these methods to the prediction of vortical flow about delta wings and slender bodies at angle of attack has been an area of active research. It has also been the source of considerable controversy. The controversy centers upon the question of how separation occurs in the numerical simulation of an inviscid flow and the degree of realism that the inviscid model provides in describing the actual flow physics.

The Euler equations are midway in complexity between the full potential equation and the Navier-Stokes equations. In contrast to the potential equation, the Euler equations provide the correct Rankine-Hugoniot shock jump conditions. They also allow for the transport (and for three-dimensional flow, stretchinq, and tilting) of vorticity. The more complex Navier-Stokes equations describe the qeneration of vorticity through the no-slip boundary condition in a viscous flow. If geometric singularities (sharp edges) in the body geometry are excluded, then there is only one valid mechanism for vorticity generation in an inviscid flow. In accord with Crocco's theorem

where is the velocity, the vorticity, s the entropy, and ht the total enthalpy, the Euler equations allow for the generation of vorticity through non- constant shock strength (shock curvature, shock intersection, etc.) For an inviscid flow, entropy is constant along streamlines except in crossing a shock where the entropy jump is a function of the local shock strength. The gradient in entropy normal to streamlines results in the production of vorticity. The significance of vorticity generation is clear when it is realized that the presence of vorticity in the flow coupled with an adverse pressure qradient are necessary €or flow separation.

From the discussion above, it is clear that theoretically valid solutions to the Euler equations with flow separation do exist. S a l a ~ ~ ~ first demonstrated shock- induced inviscid separation for the transonic flow about a circular cylinder. Marconi used the lambda scheme of Moretti to produce similar results for the supersonic flow about circular cones73 and, more recently, elliptic cones.74 Although such inviscid solutions are theoretically valid, they do not necessarily provide an accurate description of the separation that occur s in the actual viscous flow.

In the past several years, numerous solutions to the conservative Euler equations for the conical and three- dimensional flow about delta wings with both sharp and rounded subsonic leadinq edges have been reported. The computations invariably reveal the characteristic leading-edge separation vortex. The computed surface pressure coefficient frequently shows qualitative aqreement with experimental measurements. Rizzi, a strong proponent of Euler methods for leading-edge vortex flows, has presented numerous solutions €or flow about medium and low aspect ratio wings using a three-dimensional finite-volume Runge-Kutta scheme. In Reference 56, the ONERA MK wing (rounded edge) revealed the existence of a tip vortex due to flow separation at the tip. The Dillner wing, a 7 0 ° swept delta wing with chordwise sections composed of 6% thick circular arcs, was later ~ o n s i d e r e d . ~ ~ Solutions were given at 15’ angle of attack for Mach numbers of 0 . 7 and 1.5, both of which correspond to the classic leading-edge separation vortex when correlated in terms of leadinq-edge normal Mach number and anole of attack. The Dillner wino

~ ~

calculations were later repeated with mesh densities exceeding 1 million (193 x 57 x 9 7 ) grid points.76 In both the coarse and fine grid transonic and supersonic solutions, the flow separates at the leadinq edge forming the primary

vortex. Both the fine grid and coarse grid solutions exhibit large total pressure losses of similar magnitude in the vortex core. However, the total pressure losses are confined to a much Smaller region centered about the vortex core in the finer grid solutions. Isobars of static pressure on the leeward wing surfaces are shown in Figure 6 for the transonic case. Rizzi found that the fine grid solutions predicted crossflow shocks between the vortex core and the wing surface which were not apparent in the coarse grid solution. A comparison of the predicted surface pressure coefficients with experiment is shown in Figure 7. As the qrid is refined, the suction pressure peak becomes sharper and reaches rather unrealistic values in the transonic flow case. This tendency is less pronounced in the supersonic case where the peak suction pressure is much closer to the vacuum pressure limit. Noteably absent is any indication of a secondary vortex.

Similar leading-edge separation vortices have heen reported €or both rounded and sharp leading edges using a variety of numerical schemes. A partial list includes the works of Raj and Sikora,77 Murman and Powell78 also with the finite-volume Runge-Kutta algorithm; Fujii and Obayashi” using an LU factored scheme whose right-hand side is identical to the Beam and Warming scheme; and Manie, et al.@O and Newsome@’ using a MacCormack scheme. Two rather basic questions have risen out of these and similar solutions: in the absence of a clearly defined physical mechanism, such as shock curvature, how does separation occur in an inviscid flow and what is the mechanism leadinq to large total pressure losses? Two hypotheses have been sugqested. The first hypothesis82 suggests that the vorticity necessary for flow separation may he generated by the transient appearance of a shock as the flow expands around the leading edge. The shock may then disappear in the steady state. The second hypothesis suqqests that separation occurs due to numerical dissipation in the algorithm. The same two explanations have been qiven for the related question of why Euler equation solutions for lifting airfoils produce the correct lift without an explicitly imposed Kutta condition,56 as is necessary for the potential equation. Subsequent evidence has supported the second hypothesis. Raj77 reported leading-edge separation for a low subsonic Mach number (Mm = 0.3) condition in which supersonic flow did not occur throughout the transient. Rizzia3 noted a sensitivity to mesh refinement for the ONEKA M6 wing.

u

4

Any numerical algorithm must be dissipative for stability. Purely central difference schemes are not naturally dissipative and dissipative terms must be added to the discretized equations. The

...2 10

added dissipative terms are generally of two types: a third-order fourth- difference term to provide dissipation in

difference term to control shock oscillations. The later term reverts to first order at a shock, but it is also significant in regions of rapid expansion. Several studies have examined the effect of dissipation and other numerical issues in solutions to the Euler equations for flow about delta wings.

It is necessary to differentiate between sharp and rounded leading edges and, as will be seen later, between central and upwind difference schemes. At present, interest is focused on rounded leading edges and central difference schemes. For any finite tip radius, the flow at the leading edge is resolvable and must approach the limit of vacuum pressure for compressible, inviscid flow as the tip radius approaches zero. Newsome8' considered the flow about a thin elliptic cone with half angles of 2 0 ° and 1 . 5 O in the lateral and vertical directions at a Mach number of 2 . 0 and 10' angle of attack. by Siclaris4 and Squire subsequently published experimental data for the same elliptic cone with a circular centerbody at similar flight conditions. Newsome presented solutions to the conical Euler equations on a fine, viscous grid (151 x 6 5 pts) and a coarse inviscid grid (75 x 5 5 pts) and, for comparison,

d solutions to the conical Navier-Stokes equations on the fine arid at Reynolds numbers of 0.1 x l o 6 and 0.5 x l o 6 .

vi smooth regions and an adaptive second

This case was greviously computed

The Navier-Stokes solutions resulted in a leading-edge separation with primary and secondary vortices. On the coarse grid, the Euler solution, Figure 8a-c, resulted in a similar vortex but no secondary votex. A comparison of the surface pressure coefficient revealed similar pressure levels under the vortex core but an over expansion, relative to the viscous solution, near the tip. Referring to Figure 8, the separation is recognized as spurious since there is no valid entropy generating mechanism at the leading edge. Variation of the damping coefficient over the limited range of values investigated produced minor changes in the point of separation but not in the overall character of the flow. On the finer viscous grid, an entirely different Euler solution is seen. Most significantly, the flow remains attached and there is no entropy generated at the leading edge, Figure 8d-f. Entropy is, however, generated by the crossflow shock which is present in the absence of leading-edge separation. Entropy variation normal to streamlines produces vorticity in accord with Crocco's theorem. This leads to the development of a small separation bubble downstream of

d

the shock which is a valid Euler solution. The two solutions demonstrate quite clearly that the leading-edge separation for the rounded leading edge is entirely a numerical phenomenon. Similar results were published by Barton and Pulliama6 for airfoil at high angles of attack and Fujii and O b a y a ~ h i ~ ~ for conical flow about elliptic cones.

numerical origin of the leading-edge separation, it does not identify a precise cause since boundary condition error, truncation error, and artificial dissipation all go to zero in the limit as the grid is refined. Kandil and Chuang8' showed conclusively that numerical dissipation was responsible for inviscid separation about rounded edges. Solutions were given for an elliptic cone with half angles of 20' and 2'. again at Mach 2 , loo angle of attack on very coarse ( 2 9 x 3 9 ) , coarse ( 6 5 x 6 5 ) , and fine (100 x 100) grids. Solutions were obtained with the finite-volume Runge-Kutta scheme. Figure 9 shows a comparison of two solutions on the very coarse ( 2 9 x 3 9 ) grid which differ only in the value of the prescribed damping coefficients. At lower coefficient values, separation did not occur: a factor of five increase in the coefficient values produced leading-edge separation. On the fine grid, Kandil found attached flow at the leading edge irrespective of the magnitude of damping coefficient with a shock-induced separation bubble downstream of the crossflow shock. In all cases, the separated or attached flow solutions were found to be insensitive to whether a l oca l

While grid refinement demonstrates the

or global minimum (time accurate) time step was used.

All reported inviscid, central difference, solutions for delta wings with sharp. subsonic, leading edges have resulted in separation at the leading edqe. Powell, et a1.88 argued that a Kutta condition exists at the leading edge. Just as the point of separation is fixed at the edge irrespective of the Reynolds number in the real flow, it is similarly fixed at the edge in the computation, irrespective of the level of numerical viscosity. Numerical solutions to the conventional Euler equations, equation (1). (Fv, Gv, HV = 0 ) for the conservative variables, equation ( 2 ) , with leading-edge separation exhibit large total pressure losses in the vortex core. For constant total enthalpy, the total pressure loss may also be expressed as an equivalent entropy increase

~"~

P.

Figure 10 compares the computation of Murman and with experimental

11

pitot measurements of Monnerie and Werle.89 Althouqh the Euler solution does not model the secondary vortex and instead predicts a crossflow shock underneath the primary vortex, both the location and the maximum total pressure loss are well predicted. Based upon extensive computations for conical inviscid flows with the finite-volume Runge-Kutta method, Powell and Murman,a8>90 concluded that the magnitude of the total pressure loss in the vortex core was essentially insensitive to all computation parameters, grid topologies and boundary condition implementations. The distribution was found to be sensitive to computational parameters and grid densities with increased dissipation leading to more difEuse vortices. The maqnitude of total pressure losses was determined to depend upon the relevant aerodynamic parameters (Mach number, angle of attack, sweep anqle). An explanation for the total pressure loss was given by Powell, et a1.88 In essence, the argument states that any vortical region (vortex sheet) with finite thickness must have a total pressure loss. The magnitude of the total pressure is set by the jump in the tangential velocity (strength of the sheet) since for any finite thickness, the tangential velocity must transition throuqh a minimum (zero) value between opposite sides of the sheet. Central difference schemes with added artificial viscosity produce weak solutions (shocks, contact discontinuities) to the conservative Euler equations with finit-e thickness and thus give rise to total pressure losses. Powell and Murman also presented conical flow solutions for delta wings with yaw and angle of a t t a ~ k ’ ~ and for delta wings with vortex flaps.78,91 In the latter case, it appears that a viscous model is necessary to accurately capture hinge-line separations and secondary vortices.

~

12

Several recent papers have Eurther addressed the issue of total pressure loss in Euler equation solutions. Marconig2 has qiven solutions using the lambda scheme of Moretti for leading-edge separated flows about sharp-edqed delta wings with no total pressure loss. The nonconservative lambda scheme solves the entropy equation, the momentum equations in terms of flow anqles, and the continuity equation in terms of pressure. When used with shock fittinq, the scheme is a general Euler solver. Entropy is constant along streamlines and can chanqe only when crossing shocks. However, for a flat plate delta wing, leading-edqe Separation with a shed layer of distributed vorticity was predicted. For steady flow, total enthalpy is constant and Crocco‘s theorem, equation ( 1 2 ) , is satisified only when the velocity vector is parallel to the vorticity vector. The parallel alignment OE these two vectors has, in fact, been veri€ied numerically.

Powell and Murman” modified the Einite volume Runqe-Kutta scheme for the conservative Euler equations to produce similar solutions with no total pressure loss. The streamwise velocity is computed not Erom the streamwise momentum equation but from the requirement that total pressure remain constant. Kandil, et a 1 . 9 4 investigated several sets of non- standard Euler equations in which one OK more of the conservation equations is replaced by the isentropic gas equation and the steady energy equation for constant total enthalpy. The isentropic assumption, as used by Powell and Murman and Kandil, et al., is not valid for strong shocks. Both approaches produce leading-edge separated flows with no total Pressure loss which are quite similar to the results of Marconi. A typical solution for the leading-edge vortex flow about a zero thickness delta wingg4 is shown in Figure 11. Crocco‘s relation is satisified since the velocity vector is aligned with the vorticity vector. Solutions with and without total pressure loss produce essentially identical crossflow Mach numbers and pressures. The difference is reflected in the streamwise velocity component.92-94

L,

Upwind-difference methods have only recently been applied to the computation of €low about delta wings. The behavior of the upwind schemes is different from that of the previously described central- difference schemes. Newsome and Thomas95 investigated Euler and thin-layer Navier-Stokes solutions for the flow about both rounded and sharp leading-edge delta wings using the finite-volume code developed by Thomas based upon the Van Leer €lux vector splitting. Since the upwind discretization is naturally dissipative, no added artificial viscosity was necessary. By varying the accuracy of the interpolations of the conservative variables to the cell interfaces where the fluxes are Eormed, schemes of first or second order are possible. The first- order method is the most dissipative. Both first- and second-order solutions were obtained for the flow about the previously described elliptic cone on coarse and Eine grids at Mach 2 and l o o anqle of attack. Quite suprisingly, even the first-order accurate coarse grid solution showed no evidence of leading- edqe separation. The second-order . solution on the coarse arid correctly resol.ved the shock-induced separation bubble. On the finer grid, the second- order solution was virtually identical to the MacCormack solution. A thinner version of the elliptic cone (half angles of 20’ and 0.75‘) with a sharp ( l o o half angle) leading edge was taken as a model for a sharp-edged delta wing. First- and second-order accurate solutions were again obtained on coarse ( 7 5 x 5 5 ) and fine grids (151 x 75). In contrast to central difference methods, both the converged,

V

I v

steady state, first- and second-order solutions predict attached flow at the leading edge. Crossflow velocity, Mach

order solution are shown in Figure 12. Significant levels of entropy are generated both by the crossflow shock and the leading edge; and consequently, the flow downstream of the crossflow shock is separated. On the much finer (viscous) grid, both the first- and second-order upwind-difference solutions predict leading-edge separation.

Chakravarthy and Otag6 applied a

V' number and entropy plots fo r the second-

finite-volume upwind-difference code based upon Roe's approximate Riemann solver to the elliptic cone problem. On very coarse grids ( 3 8 x 27 pts), even the highly dissipative first-order scheme did not separate at the leadinqe ed e. In contrast, Kandil and Chuangg7 maintained attached flow at the leadinq edge on similarly coarse grids only with minimal damping. Chakravarthy was able to induce leading-edge separation only by the use of a spatially variable (local) timestep. His suggestion that the use of a variable timestep leads to separation in central difference solutions is contrary to the findings of Kandil and Chuang8' and Newsome.gs In addition, the attached flow upwind solutions of Newsome and Thomasg5 were obtained with a spatially variable timestep.

Spatial marching solutions to the 4 steady-state Euler equations for

supersonic inviscid flow have been presented by several authors as an alternative to the time-dependent approach. Kopfler and N i e l ~ e n $ ~ - ~ ~ applied a space marching finite-difference MacCormack scheme to predict the vortical flow about missile winqs, forebodies and forebody/wing flows (where wing-body interference is a major factor). Weilandloo qave similar results for sharp-edged delta wings with a different space marchinq algorithm. In both cases, a Kutta condition was applied at the sharp leading edge of the wing. The purpose of Kutta condition was not to force separation where it otherwise would not occur hut to specify conditions at the singular leading-edge point as required in the fini te-difference procedure. Numerically induced inviscid separation occurs in the spatial marching solutions just a s it does in time marchin solutions. Kopfler and NielsenY9 reported flow separation about a cone at angle of attack and A l l e n L o L reported Separation about an elliptical-body missile with the SWINTlo2 code. Kopfler and Nielsen98199 also investigated the use of a Kutta condition to force separation on smooth bodies. M a r ~ o n i ~ ~ used the separation model of Smithlo3 for force separation on cones. AS noted previously for sharp leading-edge separations, no total pressure loss occurs when the lambda 4

scheme is used with the forced separation model. In the absence of a sharp leading edge, the Kutta condition approach has the obvious problem that the separation line must he known a priori either through experiment, correlation, or by a boundary- layer solution coupled with the inviscid solution.

Large-scale Euler solutions for geometrically complex configurations have been presented by Rizzi104 (cranked delta winos). Rai, Sikora. and Keenlo5 (Wino- - ~. .~ body-strake), and Karman, Steinbrenner, and KisielewskiLo6 (entire F-16 aircraft). Rizzi reported an unsteady vortical flow on fine grids which he attributed to the early stages of turbulence transition due to vortex stretching/tilting. Raj, et al. found unsteady vortical flow at higher angles of attack and suggested vortex breakdown as a possible cause. Vortex breakdown is further discussed in a later section. The multi-block grid for the complete F-16 aircraft as reported by Karman, et al. is particularly impressive. However, at the time of publication, a converged solution was not presented.

Viscous Solutions

When viscosity is included in the numerical model, all relevant flow ohvsics . . are, in principle, correctly modeled. In practice, the fidelity of numerical solutions is limited by grid resolution due to speed and memory constraints of available computers and the lack of adequate turbulence models. The current generation oE supercomputers (Cyber 205, Cray 2. Fujitsu VP400) with fast vector processors and large in-core memories (up to 256 MWORDS, Cray 2) now provide the tools for accurate simulation of realistic three-dimensional flows. Recent proqress in the numerical simulation of vortical flows by conical, parabolized, and thin- layer or full Navier-Stokes equation methods is surveyed. The scope of reported applications spans the range from simple cones and delta wings to complex configurations such as the X-24C and F- 16. For convenience, the survey is organized according to the following categories:

1. Wings in isolation (delta, and low and medium aspect ratio finite wings)

cylinders, forebodies) 2. Bodies in isolation (cones, ogive-

3 . Complex configurations (wing-bodies, complete configurations)

Wings.- Vigneron, Rakich, and Tannehilllo7 provided early solutions to supersonic flow about the sharp edged delta winq considered experimentally by Monnerie and WerleS9 (M_ = 1.95, a = l o o , A s = 75"). Solutions were obtained by the

1 3

conical flow approximation previously used by McRaelQB and by a widely used parabolized procedure introduced in Reference 107. Although the primary vortex was correctly captured, resolution was not sufficient to capture the secondary vortex or produce good agreement with the experimental pitot pressure measurements.

Venkatapathy, Tannehill, and RakichloY later generalized the computer code of Reference 107 to compute parabolized solutions for cases in which the grid surface was aligned normal to the body and not necessarily aligned with the axial coordinate direction. Calculations were made for the hypersonic flow about blunt delta wings at M = 6.8 and 9.6, and a = 20' and 41.5' corresponding to the experiment of Bertram and Everhart."O The authors reported agreement with experimental pressures and shock shapes and a clearly defined primary vortex.

Blufordlll computed conical Navier- Stokes solutions for the hypersonic flow about thin, planar delta wings corresponding to the experimental data of Cross.112 At the computed test condition (M- = 10.17, ReL = 0.334 x l o 6 , A = 750), the Mach number normal to the leadinq edge was supersonic and the flow was attached at the leadinq edge. However, at anqles of attack of 11' and 15', the embedded vortex in the leeside boundary layer due to shock/boundary-layer interaction was correctly predicted.

Rizzetta and Shang"3 gave 3-0 Navier- Stokes solutions for the Monnerie and Werle delta wing ( A = 75OI at l o o angle of attack at Mach numbers of 1.95, 4 . 0 , 7.0 corresponding to subsonic (0.5), sonic ( 1 . 0 2 ) . and supersonic (1 .78) Mach numbers normal to the leading edge. Leading-edqe separation was reported at the two lower Mach numbers with secondary separation also present at the lowest Mach number ( 1 . 9 5 ) . At Mach 7.0, the leading-edge flow was attached and a smaller primary vortex was found with separation apparently due to shear generated within the boundary layer in response to the lateral pressure gradient as described by Rao and Whitehead.l6 Buter and Rizzetta"' recomputed the 1.95 Mach number case with greater resolution and found better aqreement with experimental pitot measurements. A comparison of the computed and experimental total press re

shown in Figure 13. The result may also be compared with the conical Navier-Stokes solution of Thomas and NewsomeG3 and with the inviscid result of Murman and

contours (x/L = 0.8. Rel = 0.965 x 10 Y , . 1 1s

Figure 10.

Thomas and NewsomeG3 produced solutions to the three-dimensional and conical thin- layer Navier-Stokes equations with an

upwind/relaxation algorithm. The three- dimensional solutions produced little streamwise variation (with the exception of the Viscous boundary-layer growth) and thus serve to validate the conical solutions which were obtained at much less computational expense. Comparisons were made with the extensive experimental data of Miller and Wood.15 Good agreement with experimental surface pressure measurements and vaporscreen photographs was demonstrated in all seven of the flow reqimes identified by Miller and Wood. Figure 14 shows a comparison of leeside surface pressure prediction with experiment at two different Mach numbers with variation in angles of attack and sweep. Figures 15-16 show a comparison of vaporscreen and computation at two different conditions corresponding to subsonic and supersonic leading edges. Figure 15 (Mw = 1.7, a = E o , A = 75OI corresponds to region 1 in Figure 2 and Figure 16 (M- = 2.8, a = e o . A = 67.5OI corresponds to region 6 in Figure 2.

Fujii and Kutler1I5 reported three- dimensional laminar compressible thin layer Navier-Stokes solutions for a delta winq and a strake delta wing with rounded leadinq edges at Mach 0.5. Corresponding experimental data were not available however. A second was made with a qeometry which resembled the experimental qeometry of Hummell" ( A = 7b0, AR = 1, M = 0 .1 , ReL= 0.95~10 ] at 20.5' angle of artack. Because of the grid topology IC-grid) and the finite- difference scheme used (ARC3D1, the leading edge was rounded so that the sharp edqe of the experiment was not simulated. To avoid excessive computation time with the compressible code, the simulation was made at Mach 0.5 rather than at the lower experimental value. Both primary and secondary separations were predicted. However, due to the differences between the computation and experiment, the agreement in the surface pressure was only fair.

6

Good agreement with the data of Hummel (A=760, AR=l, M =0.1, a=20.5Q, ReL = 0.95~10~1 was recently given by Thomas; et al.lL8 Multigrid acceleration was used to speed convergence of the approximate factorization, finite volume, upwind, thin-layer Navier-Stokes code on qrids of up to 550,000 points. A laminar solution was obtained at Mach 0.3, and the wing qeometry was accurately represented. Total pressure contours are shown in Fiqure 1 7 . Primary. secondary, and a small tertiary vortex are indicated and are in agreement with the experiment. A comparison of the surface pressure coefficient at several streamwise stations is given in Figure 18. At 40° angle of attack. the computations indicated a region of reverse streamwise velocity in the primary vortex core which is

-

1 4

consistent with the experimental onset of vortex breakdown between 30' and 3 5 O angle of attack. The discussion of vortex breakdown is deferred until a later section. v'

Solutions for incompressible flow about sharp-edged delta wings at low Reynolds numbers were presented by Krause, Shi, and H a r t w i ~ h . ~ ' ~ Hartwich and H s u ~ ~ solved the artificial compressibility form of the incompressible Navier-Stokes equations with an upwind flux-vector-split algorithm for the Hummel wing. The first-order accuracy of the scheme did not provide sufficient resolution for good agreement with experimental data. The algorithm was subsequently extended to second-order accuracy120 and good agreement with Hummel's experimental surface pressure measurements was reported. Calculations were also presented for the double-delta winqs examined experimentally in References 121 and 122.

Delta winqs. by design, produce controlled vortical flow at positive incidence. In contrast, medium aspect ratio, finite wings are designed for attached flow. However, at supercritical Mach numbers and moderate angles of attack, shock-boundary-layer interaction leads to massive boundary-layer separation and complex three-dimensional vortical flow over the upper wing surface. VatsaIz3 gave solutions for the Onera M6 wing at Mm = 0.84, a = 6.060, Re = 11 x lo6 using a viscous version of the finite-volume Runqe-Kutta alqorithm with a grid of 400,000 points. Vatsa noticed that a relaxation turbulence modellZ4 significantly improved the surface pressure correlation with the experiment. Fujii and ObayashilZ5 puhlished solutions for the "W-14" (AR = 9.4) wing using a hybrid scheme which combines the Beam and Warming right- hand side with a different implicit factorization. 2.0 x lo6, increasing regions of separated vortical flow were predicted at 2.46O. 4.66", and 7.5' angle of attack. In the latter two cases, a swept separation line extended from the wing root to the tip.

The low aspect ratio wing, "Wing C," has been the sub'ect of several experimental j I z 7 and computational studies.128-'3n Wing C characteristics include an aspect ratio AR = 0.83, taper ratio TR = 0 . 3 , sweep angle = 45O, and twist angle of 5.17*. The nominal design condition Mm = 0.85, 5' angle of attack resulted in shock-induced three- dimensional boundary-layer separation. MansourIZ8 and Srinivasan, et a1.1z9 were primarily interested in the tip vortex. The subject of present interest, vortical flow resulting from large-scale, three- dimensional boundary-layer separation was carefully investiqated by Kaynak, et

-!

At Mach = 0.82 and ReL =

i

Kaynak, as well as Srinivasan, et al. used the previously developed transonic Navier-Stokes (TNS) code of Holst, et a1.13' The transonic Navier- Stokes code is a zonal, multiple block grid implementation of the ARC-3D algorithm. The present application used four blocks. The two inner, fine grid, blocks adjacent to the wing solved the Navier-Stokes equations while the two outer coarse grids solved the Euler equations. Outer boundary conditions corresponded to either wind-tunnel walls or free air. A comparision of computed skin friction lines with the postulated skin friction lines as determined from the oilflow is given in Figure 19. Although the location and extent of the separation line is well predicted, the critical points (foci and saddle point) are not evident in the calculation. At the higher Mach number, Mach 0.9, the computation failed to predict the large spiral node (focus) and the smaller counter-rotating focus further outboard that dominate the surface oilflow. The earlier results of Holst, et a1.I3l for a simpler finite wing based upon the NACA 0012 airfoil (AR = 3.0, TR = 1.0, n = 200) at Mach = 0.82, a = 20, Re = 8.0 x lo6 a similar defficiency in resolving the skin friction line topology. Kaynak, et demonstrated improved agreement with the skin friction lines as postulated from the experimental oilflow by decreasing the effect of the nonlinear dissipation model within the boundary layer and implementing a relaxation correction in the Baldwin-Lomax turbulence model. Computations at higher grid densities are in progress. Solutions for the NACA 0012 wing at higher Mach numbers (0.85, 0.9) and angle of attack ( a = 5O) did qive rise to prominent spiral foci in the corn uted skin friction

massively separated flow solutions (beyond C ) for subcritical Mach numbers

(M = 0.5). Unfortuantely, these latter soyutions did not have corresponding experimental data. In all of the above solutions, although the computation did not fully reproduce the experimental oilElow, the computed skin friction lines never-the-less obey the usual topological rules.

LE show

lines, and Chaderjian' B has qiven

Lmax

Bodies.- The sharp circular cone is perhaps the simplest body which exhibits a well developed leeside vortical flow at incidence due to crossflow boundary-layer separation. The 3-D boundary-layer solutions of Lin and Rubin,' the parabolized Navier-Stokes solutions of Lubard and Helliwell,bs and the conical Navier-Stokes solutions of McRael were among the first viscous vortical flow solutions and the earliest application of the respective solution methods. All three methods were applied to laminar,

15

hypersonic flow about cones in the weak interaction region and each computed (among others) the 10' sharp cone tested experimentally by Tracy135 (M- = 1.95, Reft = 1.3 x lo6, oc = IOo) at 1 2 ' angle of attack. General agreement was reported in the surface pressure measurements, bow shock location, and the location of the separation line. At 2 4 - anqle of attack, McRae captured the leeside crossflow shock which occurred in the experiment at the hi her angle of attack. Lin and RubinI3? later published laminar solutions to the parabolized Navier-Stokes equations at very large angles of attack (up to 5 4 ' ) for the IO" circular cone of Tracy.

predicted leeside vortical flow depends critically upon the computed eddy viscosity in this region of massively separated flow. M ~ R a e l ~ ~ considered the turbulent, high Reynolds number, supersonic flow about the cone correspondinq to the experiment of Rainbird138 (M =1.8, 0 =5O, Re = 29x10 , a = 12.5, turbElent). 'This ge6metry was further tested in the wind tunnel and a l s o in flight and compared with computation in Reference 139. McRae found that the

For turbulent flow, the accuracy of the

~~ ~~ ~~

alqehraic cddy viscoqity :nodel o f I l ~ c r i s : - ~ over>redicr.cd rhe eddy viscosity in the leesidc vortical f l9 .1 . As a consequence, the experimentally observed secondary vortex was not found computationally. Modifications to the turbulence model to reduce the predicted eddy viscosity in the separated region by (1) freezing the eddy viscosity at computed minimum pressure location and ( 2 1 implementing a relaxation model in the circumferential direction qave much better agreement.

Degani and Schiff69 computed parabolized Navier-Stokes solutions for the flow about supersonic cones and ogive- cylinders using the Steger-Schiff algorithm. The previously described over- prediction of the turbulent eddy viscosity in the leeside crossflow separation was also noted with the unmodified Baldwin- Lomax model. Degani and Schiff introduced appropriate modieications in the outer eddy viscosity velocity and length scales to correctly predict the leeside eddy viscosity. Figure 20 demonstrates the improved agreement with the experimental cone data of Rainbird"' (M_=1.8, 0 = 1 2 . S o , Re = 2 1 x 1 0 , a = 2 2 . 5 1 in terms of surface pressure and surface angle. In particular, the surface streamline angle, w , is a sensitive indicator of the accuracy of the computation. In Figure 20b. w is positive for circumferential flow directed toward the symmetry plane. Thus primary and secondary separation lines are indicated where w crosses the zero point from the

6 C

16

positive direction. In contrast to the model used by McRae, a-priori knowledge of the flow field is not required.

Paraholized Navier-Stokes solutions for turbulent flow over ogive-cylinders with crossflow separation include the works of Rakich, et al.,142 Schiff and StUrek,143 and Degani and Schiff.69 Laminar, 3-D Navier-Stokes solutions for the ogive- cylinder were given by Graham, et al.144 with the explicit MacCormack algorithm. Computations for the 3 caliber ogive- cy inder at M m = 1.98, ReD = 0 . 4 4 x 10 , CI = 20° corresponding to the data of Jorgensen and perk in^"^ showed good agreement in the predicted separation lines, vortex locations, and wall pressure distributions. In a subsequent paper,'46 a pioneering attempt was made to compute the highly asymmetric vortex patterns observed by Jorgenson and Perkins at 30° anqle of attack and Mach 1.6. Although a modest degree of asymmetry was observed, it did not correlate closely with the experimental pitot pressure surveys. Graham and Hankey found that the computed asymmetry was numerically induced by the MacCormack algorithm due to noncentered spatial differencing. By switching the o r d e r of spatial differencing in the predictor-corrector sweeps, the asymmetric vortex pattern could be reversed. The computations were made on a rather coarse qrid ( 2 6 x 30 x 60) corresponding to the capahilities of the CRAY 1-S computer and no other attempt was made to introduce an asymmetric perturbation into the flow either through the geometry or initial or boundary conditions. To date, this work is the only published attempt to numerically capture vortex shedding through solution of the fundamental qoverninq equations--the 3-D Navier-Stokes equations.

L 4

k '

'L/

Newsome and computed 3-D Navier-Stokes solutions for the Mach 2.5, turbulent flow (Ref 7 4 : 6 6 x l o 6 ) about the sharp-tipped e liptlcal missile body considered experimentally by Allen, et

The cross sectional area corresponded to the Adams minimum drag distribution with a 3:l elliptic cross section. Solutions were obtained for symmetric conditions: IO", 20- angle of attack, 0" roll angle, and asymmetric conditions: loo, 20° angle of attack, 4 5 O r o l l angle. The turbulence model was modified as suggested by Degani and Schiff.69 In all cases, excellent agreement between computed and experimental surface pressures in addition to a close correspondence between the experimental vapor screen photos, and a color qraphics representation of the computed density flow field was demonstrated. Figure 21 shows the surface pressure coefficient comparison at nine crossflow stations for 2 O 0 angle of attack, 4 5 O r o l l anqle corresponding to a grid of (35 x 101 x 65) points. The // " \

/'

comparison of vapor screen and the computed color graphics (black and white here) density representation is shown in Figure 22 at three crossflow stations. The predicted stronger windward vortex, the weaker leeward vortex, and the crossflow shock above the windward vortex agree in size, shape, and location with the vapor screen.

4

Numerical solutions for incompressible (Mach 0.03) three-dimensional separated flow about a 6 : l prolate spheroid at I O o angle of attack have been given by Rosenfeld, et a1.149 (incompressible PNS equations) and Pan and P ~ l l i a m ' ~ ~ (compressible three-dimensional Navier- Stokes equations) corresponding to the experimental data of Meier et a1.151 Both methods sucessfully computed the leeside vortical flow where orevious three-dimensional boundary-liyer methods failed. However, since the lees'de flow was transitional (Re, = 1 . 6 x 10 6 ), significant differenges with the experiment in this reqion were noted in both the laminar and turbulent solutions The separation pattern in this case was topologically simple (as was the case with previously discussed sharp-tipped bodies). The primary and secondary separation lines are of the "local" type. They originate at a forward stagnation point (node of attachment) and disappear at a rearward stagnation point (node of separation).

v In contrast, the separation pattern for the flow about the hemisphere-cylinder is much more topologically complex. Three- dimensional Navier-stokes solutions have been qiven b several researcher^.^^, I s 2 - l s 5 The earlier calculations152 3 l S 3 were severely limited in resolution due to the computers then available. However, several recent ~ a l c u l a t i o n s ~ ~ ~ ~ ~ ~ l ~ ~ ~ utilizing grids of 200,000 points have successfully resolved more of the detailed critical point behavior on the hemisphere nose suggested in the experimental oilflow. In addition, qood agreement with experiment was found in the predicted surface pressures, downstream separation lines, and crossflow vortices. Although the flow was transitional in the region of interest, all calculations have been for laminar flow. Yinq. et a1.27 presented solutions at O D , loo, 19' anqles of attack, at Mach 1.2, ReD = 445,000, and 0", 1 9 O anqles of attack, Mach 0.9, ReD = 425,000. At O' and 10' angles df attack, the computed solutions were topologically simpler than what was postulated in Reference 25 based upon surface oilflow visualization. The diEferences were not due to insufficient numerical resolution. Instead, the numerical solutions provided sufficient information to reinterpret the somewhat ambiquous oilflow results in a manner consistent with the numerical results and topological rules. The computed skin

d

friction lines for 19' angle of attack, Mach 1 . 2 are shown in Figure 23. The corresponding topological interpretation was previously given in Figure 3. The downstream primary and secondary separation lines are clearly evident, as are the two prominent foci which give rise to the "nose vortices" which are swept into the node of attachment in the symmetry plane. The two smaller foci shown in Figure 3 are not clearly seen in Figure 23. The leeside separation occurs along a separation line which originates from a saddle point and is thus of the "global type." Figures 3 and 23 both differ substantially from the earlier postulated skin Eriction topology given by PeakeZ5 based upon surface oilflow. At present, neither the computation nor the oilflow provide a definitive picture of the surface topology. Fuji and Obaya~hi'~' also gave similar results for this case. Solutions for 19' angle oE attack, Mach 0.9 were given by Ying, et a1.27 and Kordulla, et a1.Is5 Laminar flow calculations using different algorithms (Ying-streamwise flux split, crossflow central difference; Kordulla- implicit/explicit MacCormack) both resulted in an unsteady flow. The unsteadiness produced only minor variation in the observed skin friction lines, Figure 24.155 The results of Ying and Kordulla are quite similar. In contrast to the Mach 1.2 case, the upstream focus is more developed and has moved to a more windward pas it ion.

Complex Configurations.- Fujii and Obayashils6 computed Navier-Stokes solutions for the transonic flow over the W-18 wing-fuselage on a grid of 700,000 ( 6 2 x 74 x 151) points. The solutions were obtained with the previously mentioned LU-AD1 central-difference scheme and a Baldwin-Lomax turbulence model. At a Mach number oE 0.82, solutions were computed at 2". 4', 6' angles of attack. In each case, shock-induced boundary-layer separation occurred and increased in size with increasing angle of attack. The computed skin-friction lines are shown in Figure 25a for 69 angle of attack. A well developed spiral node can be seen in the wing-fuselage juncture on both the winq and fuselaqe. In the correspondinq partical path traces, Figure 25b, the vortex which occurs in this region resembles a coiled spring which is hent so that the vortex axis is perpendicular to hoth the wing and fuselage surfaces. Further outboard on the winq, a separation line appears. Corresponding oilflow results were not available for comparison. Good agreement with experimental surface pressure measurements was demonstrated.

Several PNS s ~ l u t i o n s ~ ~ ~ - ' ~ ~ have been given for a maneuvering re-entry vehicle at a Mach number of 10 at angles of attack up to 14O. The geometry consists of a

spherical nose tip, a biconic central section, and after body with cuts and flats. A 3-0 Navier-Stokes solution for this configuration was also given by Shang.160 A primary leeside vortex resulting from a local separation line was predicted and generally good agreement was demonstrated in comparison with experimental surface pressure and heat transfer measurements.

References 1 5 9 , 161-162 present PNS solutions for the turbulent hypersonic flow about the X-24C lifting body at conditions(M_ = 5.95; ReL = 1 1 . 5 x lo6; a = 6 ” , 2 0 ” ) corresponding to available experimental data.164!165 The solutions were started from a 3-D Navier- Stokes blunt body solution and marched down the body until the beginning of the strake, x/L = 0.75, where subsonic axial flow prevented further marching. Srinivasan, et al.162 found that excessive damoina. reauired €or stable marchino.

~ ~. degraded soiution accuracy at the hiqher angle of attack.

Parabolized Navier-Stokes solutions For the turbulent, hypersonic flow about the space shuttle orbiter at large angles of attack have been given by Srinivasan, et al.162 and by Chaussee, et al.’63 The marchinq solution was again started from the blunt body solution and marched hack to a streamwise location, x/L = 0.66, corresponding to the strake wing intersection. At this point, the wing- shock/bow shock interaction produced a region of streamwise subsonic flow which prevented further marching. Figure 26 shows computed skin friction lines at a fliqht condition of Mach 7.9, 2 5 * angle of attack, Re Chaussee, ik ,l.163 be seen at the strake/fuselage juncture and on the leeside of the fuselage.

- 7 x l o 5 , as computed by Separation lines can

Shang166 published the first Navier- Stokes solution €or a complete aircraft. The flow ahout the X24C-1OD was solved with the explicit MacCormack algorithm on a sinqle grid of 4 7 5 , 0 0 0 points. A turbulent flow solution was obtained for a Mach 5.95, 6O angle of attack fliqht condition at a Reynolds number ReL =

11.7 x l o 6 correspondinq to experimental data16r,165 and previous PNS solutions.’59,’61,162 In contrast to earlier PNS solutions, the computation included both the strakes and fins on the rear of the configuration. The computed surface pressure was shown to he in excellent agreement with experimental measurements while heat transfer was less accurately predicted. The skin friction lines for the left half of the vehicle are shown in Figure 2 7 . The large separation line on the leeward surface corresponds to the primary vortex. Smaller separation lines are indicated upstream of the canopy

and at the strake/fuselage and fin/fuselage intersections. An integration of a l l pressure and shear stresses over the body (excluding the base) was within 4.7% and 6.7% of the experimentally measured lift and drag coefficients.

Two papers reported elsewhere in this meeting are devoted to Navier-Stokes solutions for the F-16A. The previously described transonic Navier-Stokes code was applied to each of 16 different zonal grids to describe the flow about the complete coneiguration with the exception of the tail assembly and the inlet, which was faired over. A total of 300,000 grid points were used between the finer viscous grids near the body and the coarser inviscid grids away from the body. In Reference 1 6 7 . Flores, et al. describe the algorithm and results in general while Reference 168, Reznick and Flores, is devoted to the problem of strake-generated vortex interactions. Since specific results were not available for review, the reader is directed to the references for details. At the outset, however, the present calculations represent the most ambitious attempt yet to compute the flow over a realistically complex geometry. The utility of the zonal schemes in such calculations is demonstrated. Further, the development of this capability enables one t.o consider the simulation of the various vortex interactions which occur in r e a l flight vehicles.

Vortex Breakdown

The first attempts to numerically simulate vortex breakdown were given for an isolated axisymmetric vortex. Attention here is focused on methods which predict not only the onset of the breakdown. but describe the flow within and downstream of the breakdown as well. As pointed out by Liu, et a1.,169 the solutions diEfer from the leading-edge vortex breakdown over a delta wing principally in the assumption of axisymmetry and the restriction to low Reynolds number. Several numerical solutions have been presented for an isolated vortex with a Reynolds number based uoon core radius. Re. = 200. and . ~~

~~ ~ .~~~ ~~~~ ~~~~~ ~. identical inflow profilesi6 Somewhat different results were obtained. Unfortunately, experimental data at the corresponding Reynolds number does not exist.

Grabowski and Bergerl’O solved the incompressible, steady, axisymmetric Navier-Stokes for an unconfined viscous vortex with a Reynolds number based upon core radius of 100 and 200 and velocity profiles corresponding to experimentally measured values. Converged solutions were not obtained for either highly subcritical or supercritical initial flows due to numerical difficulty. For moderate

18

subcritical and supercritical conditions, steady-state solutions for the bubble form of breakdown were predicted. The

subcritical initial flows is at odds with "wave-motion" or "critical-state" theories which predict breakdown to occur only for flows with a supercritical initial state as the critical state is reached.

4 prediction of vortex breakdown for

Shi119,171 integrated the incompressible, unsteady Navier-Stokes equations using an AD1 scheme for an isolated axisymmetric vortex for a Reynolds number of 2 0 0 . A periodic bubble form of breakdown was observed involving the formation, disappearance, and reappearance of two internal vortices within the bubble. The unsteady, periodic behavior of the two-cell vortex structure within the bubble is similar to the experimentally observed behavior of a confined vortex at a Reynolds number, ReD = 2 5 0 0 , studied by Faler and Lelbovich.17* Figure 2 8 shows a particular instantaneous streamline pattern which agrees well with the time- averaged streamline Battern measured by Faler and Leibovich' and the flow visualization of Escudier and Keller.41 Figure 2 9 shows the more general periodic flow behavior in the bubble as computed by Shi.17'

Hafez, et a1.173 solved the incompressible, steady, axisymmetric Euler and Navier-Stokes equations for an

calculations of Grabowski and Berger and Shi (for viscous flow at Reynolds numbers of 100 and 200). The Euler equation solutions agreed with the inviscid calculations of Ta'assan174 in which a bubble type structure was predicted. The inviscid results were quite different, however, from the viscous results and from experimental results at higher Reynolds number. Viscous result were in substantial agreement with Grabowski and Berger (and disagreement with Shi) in which a steady solution was obtained with a single cell vortex structure within the bubble.

d isolated vortex corresponding to the

Several inviscid calculations for leading-edge vortex flows about delta wings have been interpreted as simulations of vortex breakdown. Hitzel and Schmidta2 reported a chaotic vortical flow in coarse grid calculations for the Mach 0 . 5 flow about an arrow wing at 35' angle of attack. Rizzi and Purce11104 examined the Mach 0 . 3 flow about a rounded leading-edge cranked delta wing with mesh densities of up to 600,000 points. While calculations at much lower densities were steady, the hiqh resolution solutions were unsteady with a fine-scale chaotic structure downstream of the crank. Rizzi interpreted the unsteady Structure as the initial staqes of the transition to turbulent flow in which vorticity is 'd

transferred to successivly smaller scales to the smallest scale supported by the mesh. In a subsequent paper,175 calculations were given for the same wing at 1 2 . 5 O and 20' angle of attack. At 12.5O angle of attack, the calculation was unsteady while at 2 0 ° angle of attack the calculation was steady hut with reduced values of the lift and drag coefficients (CL f 0.516, Cp = 0 . 1 8 2 ) . condition was interpreted to be vortex bursting. Significantly, how'ever, experimental results did not indicate vortex burstinq and the measured lift and drag coefficients were much hi her (CL = 0 . 8 8 . CD,= 0 . 3 2 ) . Raj, et al.'05 noted a similar inability to obtain a converged solution at 30° angle of attack for a strake-wing-body configuration while solutions at lower angles of attack did converge. The unsteadiness was attributed to vortex bursting over the aft part of the wing and the lift coefficient was reduced compared to the maximum computed value at a lower angle of attack. Again, experimental measurements indicated a substantially higher lift coefficient in which the vortex had not yet burst.

The latter

The interaction and merger of the two vortices originating from the wing apex and the wing kink at higher angles OE attack is an important feature of strake- wing and cranked wing geometries. Vortex breakdown may occur within the joined vortices. The interaction of tangled vortex lines, essential to the development of full scale turbulence. is modeled by the Euler equations. Physically, this process is dissipated by viscosity on the smallest turbulent length scales while numerically it is dissipated by a grid dependent artificial viscosity. How well the numerical simulation reproduces the physical process is an open question.

AS previously mentioned, Thomas, et a1.118 computed laminar, thin-layer Navier-Stokes solutions for the delta wing of H ~ m m e l ' ~ ~ for angles of attack up to 4 O 0 in 5' increments. At 4 O 0 angle of attack, a steady-state solution was obtained in which a bubble form of breakdown was observed beginning at x/L = 0 . 6 . In Figure 3 0 , a limited region of reverse axial flow can be seen within the vortex core beginning at the breakdown point and extending to the trailing edge. From Figure 3 1 , the computed C corresponds to an angle of attack of 35O while experimentally it lies between 30O and 3 5 * . At 40" angle of attack, the lift coefficient is modestly overpredicted compared with experiment.

Lmax

Conclusions

Progress in the numerical simulation of vortical flows due to 3-D boundary-layer separation about flight vehicles at high

19

angles of attack by inviscid and viscous methods has been examined. The emphasis of the survey has been on methods which "capture" vortical flow as a solution to fundamental conservation laws of fluid dynamics--the Euler and Navier-Stokes equations.

for the prediction of separated flows remains a matter of some controversy. Separated flow solutions may, in fact, be valid Euler solutions for flow where there is an inviscid vorticity generatinq mechanism--such as shock curvature. However, the predicted solution may be quite different from what occurs in the actual viscous flow. Euler methods have principally been applied to the prediction of vortical flow about highly swept leading-edge delta wings. In the case of delta wings, with subsonic. rounded leading edges, spurious separation, as predicted by conventional central difference schemes, was shown to occur due to numerical dissipation. In contrast, upwind-difference schemes appear to he much less susceptible to spurious inviscid separation and thus provide a better description of inviscid flow about rounded leading-edqe wings. For subsonic, sharp leadinq edqes, different methods give different answers. Central difference methods consistently predict leadinq-edqe separation while upwind difference methods, in some instances, predict attached flow. Where leading-edge separation was predicted, the solution either contained significant total pressure loss or no total pressure loss depending upon which set of equations was used. Further study is required to establish the correct Euler solution near the singularity and to ensure that this behavior is accurately reflected in the numerical solutions.

The value of Euler equation solutions

The Euler equations do not model the secondary separation of the boundary layer on the leeside of the swept delta wing, which, for laminar flow, may substantially affect the surface pressure distribution. However, due to compensating errors in the predicted distribution, lift and drag coefficients are often in reasonable agreement with experimental measurements. The practicinq engineer, concerned principally with force and moment coefficient prediction, will likely continue to use Euler equation solutions for this purpose.

The Reynolds-averaged Navier-Stokes equations, unlike the Euler equations, provide a complete description of flow separation about arbitrary bodies. They are essential to accurately describe massive 3-0 boundary-layer separation and the resultinq vortical flow which is the subject of present interest. Paced by the rapid evolution in speed and memory of available computers, the numerical

simulation of vortical flow over complete confiqurations is now possible. The development of this capability, as reflected in earlier calculations about simple wing and body geomtries, has been surveyed. Detailed quantitative as well as qualitative agreement with experiment has been demonstrated for the laminar and turbulent flow about a variety of qeometries at subsonic through hypersonic speeds.

The increased computational capability now afforded by current computers will he reflected in two key areas of vortical flow simulation. First, further efforts with greater mesh resolution will be made to fully resolve the detailed critical point behavior in the computed skin friction lines of highly complex vortical flows. Such efforts will assess and minimize numerical influence (truncation

W

~~ ~ ~ . ~~~~~~~ ~ ~~~~

error, artificial dissipation) on the computed solutions and allow an unbiased judgement of the adequacy of the rather simple turbulence models used to date. Where detailed agreement between the computed skin frFction lines and the oilflow is demonstrated, confidence is established in the overall 3-D flow-field prediction. The oilflow provides an often less-than definitive picture of the skin friction lines from which the 3-D flow structure must be inferred. Computational results have already substantially contributed to our understanding of 3-D flow separation, particurlarly in relating the limited information provided by surface oilflow to the overall 3-D flow structure.

I/

The numerical simulation of more geometrically complex bodies is a trend which is readily apparent in recent calculations. This effort is especially imoortant for vortical flows where the vortices generated by one component interact with and greatly alter the flow about other components. Calculations for vortex interaction about a strake wing, a double delta winq, and a wing-body have recently been reported. It is expected that future calculations will address forebody/wing, canard/wing, forebody/tail and wing/tail vortex interactions. The development of zonal solution methods for ComnleX geometries appears essential. The zonal methodology simplifies the grid qeneration process, allows local grid refinement, and decreases computational expense.

Given the rather remarkable progress to date, it is difficult to speculate as to what will, in the future, be considered a reasonable calculation for design purposes. Today, however, the previously described calculations may require many CPU hours on the largest supercomputers and significant effort to post-process and understand the large solution data base. This burden will, of course, he reduced by d

20

newer computers, and more efficient alqorithms. The reader, interested in a general discussion of pacing items in the

dynamics and its application in the design and analysis process, is referred to the excellent surveys given by K ~ t l e r l ~ ~ and Shang. 7 7

4 development of computational fluid

In the near term, numerical simulation of complex 3-D vortical flows will greatly contribute to our knowledge of such flows. Our understanding of 3-D separation and reattachment has profited from numerical simulation. The first solutions for the breakdown of the leading-edge vortex over a delta wing have recently been reported. It is quite likely that our eventual understandinq of both vortex breakdown and vortex asymmetry will come from complimentary numerical simulation and detailed experimental investiqation.

Althouqh the present survey was intentionally restricted to quasi-steady fliqht conditions, numerous examples exist in which the dynamics of the vehicle motion are important and the fluid dynamics equations must be solved in a time accurate manner coupled with the flight mechanics equations. Examples include the dynamic lift overshoot resulting from delayed development of the leeside vortical flow with ra id vehicle pitch-upZ7 and the wing-rocklY8 phenomenon due to asymmetric vortex movement. Such problems are an important area of future

and exploit the vortical flow about flight vehicles at hiqh anqles of attack.

4 emphasis in our quest to Sully understand

References

1. Polhamus, E. C.: A Concept of the Vortex Lift of Sharp-Edqe Delta Wings Based on a Leadinq-Edge Suction Analogy. NASA TN D-3767, 1966.

2. Hoeiimakers. H. W. M.: Comoutational

Paper No. 18, July

3 . Smith, J. H. B.: Theoretical Modeling of Three-Dimensional Vortex Flows in Aerodynamics. Aerodynamics of Vortical Flows in Three Dimensions, AGARD CP-342, Paper No. 17, July 1983.

4. Peake, D. J.; and Tobak, M.: Three- Dimensional Interactions and Vortical FLOWS With Emphasis on High Speeds. AGARD AG-252, July 1980.

5 . High Angle-of-Attack Aerodynamics, AGARD CP-247, Jan. 1979.

6 . High Angle-of-Attack Aerodynamics, AGARD LS-121, Dec. 1982. .-'

7. Aerodynamics of Vortical Type Flows in Three Dimensions, AGARD CP-342, July 1983.

8. Salas, M. D.: ed.: Studies of Vortex Dominated Flows. Springer-Verlag, New York, NOV. 1986.

9. Campbell, J. F.: Osborn, R. F.: and Foughner, J. T.; eds.: Vortex Flow Aerodynamics. Vol. 1, NASA CP-2416, vol. 2, NASA CP-2417, Oct. 1985.

10. Introduction to Vortex Dynamics, van Karman Institute for Fluid Oynamics Lecture Series 1986-08, May 1986.

11. Nielsen, J. N.: Nonlinearities in Missile Aerodynamics. AIAA Paper No. 78-20, Jan. 1978.

12. Stanbrook, A.: and Squire, L. C.: Possible Types of Flow at Swept Leading Edqes. Aeronautical Quarterly, vol. XV, pt. 1, Feb. 1964, pp. 72-82.

13.

1 4 .

15.

16.

17.

18.

19.

20.

21.

Carcaillet, R.: Manie, F.: Pagan, D.: and Soliqnac, J. L.: Leading Edqe Vortex Flow Over a 75 Degree-Swept Delta Wing - Experimental and Computational Results. ICAS-86- 1.5.1, Sept. 1986.

Szodruch, J. G.; and Peake, D. J.: Leeward Flow Over Delta Wings at Supersonic Speeds. NASA TM-1187, April 1980.

Miller, D. S . ; and Wood, R. M.: Leeside Flow Over Delta Wings at Supersonic Speeds. NASA TP-2430, June 1985.

Rao, D. M.: and Whitehead, A. H.: Leeside Vortices on Delta Wings at Hypersonic Speeds. AIAA Journal, vol. 10, NOV. 1972, pp. 1458-1465.

Peake, D. J.: and Tobak. M.: Three- Dimensional Separation and Reattachment, High Angle of Attack Aerodynamics. AGARD LS-121, Paper No. 1, Dec. 1982.

Leqendre. R.: Separation de l'ecoulement laminare tridimensional. Rech. Aero. no. 54, 1956, pp. 3-8.

Maskell, E. C.: Flow Separation in Three Dimensions. RAE Aero Rept. 2565, 1955.

Lighthill, M. J.: Attachment and Separation in Three-Dimensional Flow. In Laminar boundary Layers, Chap. 11, Sec. 2.6, ed. L. Rosenhead, Oxford Univ. Press, 1963, pp. 72-82.

Wanq, K. C.: Separation of Three- Dimensional Flow. Reviews in Viscous

21

Flow, Proc. Lockheed-Georgia Co. Symp. LG-77ER004, 1976, pp. 33-35.

Aeronautical Research Council, R&M 3282, 1961.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

Chapman, G. T.: Topological Classification of Flow Separation on Three-Dimensional Bodies. AIAA Paper No. 86-0485.

Dallmann, U.: To~oloaical Structures of Three-Dimensional cortex Flow Separation. AIAA Paper NO. 83-1735, July 1983.

Hsieh, T.: An Investigation of Separated Flow About a Hemisphere- Cylinder at O' to 19' Incidence i n the Mach Number Range from 0.6 to 1 . 5 . AEDC-TR-76-112, 1976.

Peake, D. J.: and Tobak, M.: Three- Dimensional Flows About Simple Components at Angle of Attack. AGARD LS-121, High Angle of Attack Aerodynamics.

Bippes, H.: and Turk, M.: Oil Flow Patterns and Separated Flow on a Hemisphere Cylinder at Incidence. DFVLR-FB 84-20, 1984.

Ying, S. X . : Steger, J. L.: Schiff, L. B . : and Baganoff, D.: Numerical Simulation of Unsteady, Viscous, Hiqh Angle of Attack Flows Using a Partially Flux-Split Alqorithm. AIAA Paper No. 86-2179, 1986,

Skow, A.: and Peake, D. J.: Control of the Forehody Vortex Orientation by Asymmetric Air Injection, (Part 8 ) - Details of the Flow Structure. Hiqh Angle-of-Attack Aerodynamic, AGARD LS-121, Paper No. 10, Dec. 1982.

Keener, E. R.: and Chapman, G. R.: Similarity in Vortex Asymmetries Over Slender Bodies and Wings. AIAA Journal, vol. 15, Sept. 1977, pp. 1370-1372.

Shanks, R. E.: Low-Subsonic Measurements of Static and Dynamic Stability Derivatives of Six Flat- Plate Wings Having Leading-Edge Sweep Angles of 70-84 Deg. NASA TN D-1822, 1963.

Ericsson, L. E . : and Reding, J. P.: Vortex-Induced Asymmetric Loads in 2-D and 3-D Flows, AIAA Paper No. 80- 0181. Jan. 1980.

Peckham, D. H.: and Atkinson, S. A.: Preliminary Results of Low-Speed Wind-Tunnel Tests on a Gothic Wing of Aspect Ratio 1.0. Aeronautical Research Council, CP-508, 1957.

Lambourne, N. C.: and Bryer, D. W.: The Bursting of Leading-Edge Vortices: Some Observations and Discussion of the Phenomenon.

34. Hummel, D . ; and Srinivasan, P. S . : Vortex Breakdown Effects on the Low- Speed Aerodynamic Characteristics of Slender Delta Wings in Symmetrical Flow. Journal of Royal Aeronautical Society, vol. 71, April 1966, pp. 319-322.

35. Harvey, J. K.: Some Observations of the Vortex Breakdown Phenomenon. Journal of Fluid Mechanics, vol. 14, no. 4, Dec. 1962, pp. 585-592.

Investigation of the Breakdown of a Vortex in a Tube. Royal Aircraft Establishment Technical Note No. Aero 2963, 1965.

36. Kirkpatrick, D. L. I.: Experimental

37. Sarpkaya, T.: Vortex Breakdown in Swirling Conical Flows. AIAA Journal, vol. 9, no. 9, Sept. 1971, pp. 1791-1799.

38. Orloff, K. L.: and Bossel, H. H.: Laser-Doppler Velocity Measurements of Swirling Flows with Upstream Influence. Bullentin of American Physical Society, vol. 2, no. 16, 1971, pp. 1331.

39. Faler, J. H.: and Leibovitch, S.: Disrupted States of Vortex Flow and Vortex Breakdown. Physics of Fluids, "01. 20, no. 9, 1977. pp. 1385-1400.

vortex Flow Regimes. Journal of Fluid Mechanics, "01. 115, 1982, pp.

L/

40. Escudier, M. P.: and Zehnder, N.:

105-121.

41. Escudier, M. P.; and Keller, J. J.: Vortex Breakdown: a Two State

883, Paper P

42. Escudier, M. P.: Vortex Breakdown in Technology and Nature. von Karman Institute for Fluid Dynamics Lecture Series 1986-08, May 1986.

43. Leibovich. S.: Vortex Stability and Breakdown. Survey and Extension. AIAA Journal, "01. 22, Sept. 1984, pp. 1192-1206.

4 4 . Hall, M. G.: Vortex Breakdown. Annual Review of Fluid Mechanics, "01. 4, 1972, pp. 195-218.

45. Escudier, M. P.: Vortex Breakdown Theories, VIK Institute for Fluid Dynamics Lecture Series 1986-08, May 1986.

46. Wedemeyer, E . : Vortex Breakdown. Hiqh Angle of Attack Aerodynamics, AGARD LS-121, Dec. 1982, Paper No. 9. d

22

47.

Y'

48.

49.

50.

51.

52.

53.

4

54.

55.

56.

57.

58.

Thomas, J. L.; van Leer, B.; and Walters, R. W.: Implicit Flux-Split Schemes for the Euler Equations. AIAA Paper No. 85-1680, July 1985.

Rubesin, M. W.; and Rose, W. C.: The Turbulent Mean Flow Reynolds Stress and Heat Flux Equations in Mass Averaged Dependent Varibles. NASA TMX-62, 248, March 1973.

MacCormack, R. W.: The Effect of Viscosity in Hypervelocity Impact Cratering. AIAA Paper No. 69-354, Jan. 1969.

Beam, R. M.; and Warming R. F.: An Implicit Factored Scheme for the Compressible Navier-Stokes Equations. AIAA Journal, "01. 16, April 1978, pp. 393-402.

Jameson, A.: Schmidt, W.; and Turkel, E.: Numerical Solutions of the Euler Equations by Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes. AIAA Paper No. 81-1259, June 1981.

MacCormack, 8 . W.: A Numerical Method for Solving the Equations of Compressible Viscous Flow. AIAA Journal, "01. 20, Sept. 1982, pp. 1275-1281.

MacCormack, R. W.; and Paulley, A. J.: The Influence of the Computational Mesh on Accuracy for Initial Value Problems With Discontinuous or Non-Uunique Solutions. Computers and Fluids, vol. 2, pp. 339-361.

Steqer, J. L.: Implicit Finite Difference Simulation of Flow About Arbitrary Two-Dimensional Geomtries. AIAA Journal, vol. 16, July 1978, pp. 679-686.

Pulliam, T. H.; and Steger, J. L.: Implicit Finite-Difference Simulations of Three-Dimensional Compressible Flow. AIAA Journal, "01. 18, Feb. 1980, pp. 159-167.

Rizzi. A.: Damped Euler-Equation Methods to Compute Transonic Flow Around Winq-Body Combinations. AIAA Journal, vol. 20, Oct. 1982, pp. 1321-1328.

Swanson, R:C.; and Turkel, E.: A Multistage Time-Stepping Scheme for the Navier-Stokes Equations. AIAA Paper No. 85-0035 . Jan. 1985.

Jameson, A.: The Evolution of Computational Methods in Aerodynamics. Journal of Applied Mechanics, vol. 50, 1983.

59.

60.

61.

62.

6 3 .

64.

65.

6 6 .

67.

68.

69.

70.

71.

Thomas, J. L.; and Walters, R. W.: Upwind Relaxation Algorithms for the Navier-Stokes Equations. AIAA Paper No. 85-1501CP, July 1985.

Chakravarthy, S . R.; and Osher, S . : A New Class of High Accuracy TVD Schemes for Hyperbolic Conservation Laws. AIAA Paper No. 85-0363, Jan. 1985.

Chakravarthy, S . R.; Szema, K-Y.; Goldberg, U. C.; Gorski, T. J.; and Osher, S . : Application of a New Class of High Accuracy TVD Schemes to the Navier-Stokes Equations. AIAA Paper No. 85-0165, Jan. 1985.

Moretti, G.: The Lambda-Scheme. Computers and Fluids, vol. 7, pp. 191-205, 1979.

Thomas J. L.; and Newsome, R. W.: Navier-Stokes Computations of Leeside Flows Over Delta Wings. AIAA Paper No. 86-1049, May 1986.

Hartwich, P. M.; and HSU, C. H.: An Implicit Flux-Difference Splitting Scheme €or Three-Dimensional Incompressible Navier-Stokes Solutions to Leading Edge Vortex Flows. AIAA Paper No. 86-1839CP, June 1986.

Lubard, S. C.; and Helliwell, W. S.: Calculation of the Flow on a Cone at High Angle of Attack. AIAA Journal, vol. 12, July 1974, pp. 965- 974.

Vigneron, Y. C.; Rakich, J. V.; and Tannehill, J. C.: Calculation of Supersonic Viscous Flow Over Delta Wings with Sharp Subsonic Leading Edges. AIAA Paper No. 78-1137, 1978.

Schiff, L. 8.; and Steger, J. L.:

30, Jan. 1979

Baldwin, B. S . ; and Lomax, H . : Thin Layer Approximation and Algebraic Model for Separated Turbulent Flows. AIAA Paper No. 78-257, Jan. 1978.

Degani, D.: and Schiff, L. B.: Computation of Supersonic Viscous Flows Around Pointed Bodies at Large Incidence. AIAA Paper No. 83-0034, Jan. 1983.

Murman, E. M.; and Stremel, P. M.: A Vortex Wake Capturing Method for Potential Flow Calculations. AIAA Paper No. 82-0947, June 1982.

Kandil, 0. A.; and Yates, E. C.: Computation of Transonic Vortex Flows Past Delta Wings - Integral Equation

23

72.

73.

74.

75.

76.

77.

78.

79.

80.

81.

82.

83.

Approach. AIAA Journal, no. 11, NOV. 1986, pp. 1729-1736.

Salas, M. D.: Recent Developments in 84. Transonic Euler Flow Over a Circular Cylinder. Mathematics and Computers in Simulation, vol. XXV, 1983, pp. 232-236.

Marconi, F.: The Sprial Singularity in the Supersonic Inviscid Flow Over a Cone. AIAA Paper No. 83-1665, July 1983.

Marconi, F.: Shock Induced Vorticities on Elliptic Cones in Supersonic Flow. AIAA Paper No. 85- 0433, Jan. 1985.

Rizzi, A.; Erickson, L. E.; Schmidt, W.: and Hitzel, S. M.: Simulatinq Vortex Flow Around Wings. Aerodynamics of Vortical Type Flows in Three Dimensions, AGARD CP-342, Paper No. 21, July 1983.

Rizzi, A.: Euler Solutions of Transonic Vortex Flow Around the Dillner Wing-Compared and Analyzed. AIAA Paper No. 84-2142, Ruq. 1984.

Raj, P.; and Sikora, J.: Free Vortex Flows: Recent Encounters With an Euler Code. AIAA Paper No. 84-0135, Jan. 1984.

Murman, E. M.: Powell, K. G . ; Miller, D. S.; and Wood, R. M.: Comparison of Computational and Experimental Data for Leading Edqe Vortices- Effects of Yaw and Vortex Flaps. AIAA Paper No. 86-0439, Jan. 1986.

Fujii, K.; and Obayashi, S.: Evaulation of Euler and Navier-Stokes Solutions f o r Leading-Edge and Shock- Induced Separations. AIAA Paper No. 85-1563, July 1985.

Manie, F.; Neron, M.; and Schmitt, V.: Experimental and Computational Investigation of the Vortex Flow Over a Swept wing. 14th International Council of the Aeronautical Sciences. ICAS-14-2.8.1. Sept. 1984.

Newsome, R. W.: Euler and Navier- Stokes Solutions for Flow Over a Conical Delta Winq. AIAA Journal, April 1986, pp. 552-561.

Hitzel, S. M.: and Schmidt, W.: Slender Wings With Leading-Edge Vortex Separation-A Challenge for Panel Methods and Euler Solvers. Journal of Aircraft, vol. 21, Oct. 1984, pp. 751-759.

Rizzi, A.: Mesh Influence on Vortex Shedding in Inviscid Flow Computations. Recent Contributions

85.

86.

87.

88.

89.

90.

91.

92.

93.

94.

95.

to Fluid Mechanics, W. Hasse, ed., Sprinqer-Verlaq, Berlin, 1982.

Siclari, M. 3.: Investigation of Cross Flow Shocks on Delta Wings in Supersonic Flow. AIAA Journal, vol. 18, Jan. 1980, pp. 85-93

Squire, L. C.: Leading Edge Separation and Crossflow Shocks on Delta wings. AIAA Journal, vol. 23, March 1985, pp. 321-325.

W

Barton, J. T.; and Pulliam, T. H.: Airfoil Computations at High Angles of Attack, Inviscid and Viscous Phenomena. AIAA Paper No. 84-0524, Jan. 1984.

Kandil, 0. A.; and Chuang, A.: Influence of Numerical Dissipation in Computing Supersonic Vortex-Dominated Flows. AIAA Paper No. 86-1073, May 1986.

Powell, K.; Murman, E . ; Perez, E.; and Baron, J.: Total Pressure Loss in Vortical Solutions of the Conical Euler Equations. AIAA Paper No. 85- 1701, July 1985.

Monnerie, B.; and Werle, H.: Etude de 1'Ecoulement Supersonique and Hypersonique Atour d'une Aile Elancee en Incidence. AGARD CP-30. Paper No. 23, 1968.

Murman, E. M . : Goodsell, A.; Powell, LI K.; and Landahl, M.: Leading Edge Vortex Solutions with Large Total Pressure Loss. AIAA Paper No. 87- 0039, Jan. 1987.

Powell, K.; Murman, E. M.; Wood, R. M.; and Miller, D. S . : A Comparison of Experimental and Numerical Results for Delta Wings With Vortex Flaps. AIAA Paper No. 86-1840-CP, June, 1986.

Marconi, F.: Flat Plate Delta Wing Separated Flows with Zero Total Pressure Loss. AIAA Paper No. 87- 0038, Jan. 1987.

Powell, K.; and Murman, E. M.: Vortical Solutions of Conical Euler Equations. Vortex Flow Minisymposium SIAM National Meeting, Boston, Mass., July 1986, (oral presentation).

Kandil, 0. A.; Chuang, A.: and Shiffette, 3. M.: Finite Volume Euler and Navier-Stokes Solvers for Three Dimensional and Conical Vortex Flows Over Delta Wings. AIAA Paper No. 87-0041, Jna. 1987.

Newsome, R. W.; and Thomas, 3. L.: Computation of Leading-Edge Vortex Flows. NASA CP-2416, OCt. 1985, pp. 305-330. il

2 4

96. Chakravarthy, S . R.; and Ota, D. K.: Numerical Issues in Computing Inviscid Supersonic Flow Over Conical Delta Wings. AIAA Paper No. 86-0440, Jan. 1986. d

97. Klopfer, G. H.; Nielsen, J. N.: Euler Solutions for Wing and Wing- Body Combination at Supersonic Speed with Leading-Edge Separation. AIAA Paper No. 80-0126, Jan. 1980.

98. Klopfer, G. H.; and Nielsen, J. N.: Euler Solutions of the Body Vortices of Tangent Ogive Cylinders at High Angles of Attack and Supersonic Speeds. AfAA Paper No. 81-0361, Jan. 1981.

99. Klopfer, G. H.; Kuhn, G. D.; and Nielsen, J. N.: Euler Solutions of Supersonic Wing-Body Interference at High Incidence Including Vortex Effects. AIAA Paper No. 83-0460, J a n . 1983.

108. McRae. D. S . : A Numerical Studv of Supersonic Viscous Cone Flow at' High Angle of Attack. AIAA Paper No. 76- 97, 1976.

109. Venkatapathy, E.; Rakich, J. V.; and Tannehill, J. C.: Numerical Solution of Supersonic Viscous Flow Over Blunt Delta Winqs. AIAA Paper No. 82-0028, Jan. 1982.

110. Bertram, M. H.; and Everhard, P. E.: An Experimental Study of the Pressure and Heat Transfer Distribution on a 70° Sweep Slab Delta Wing in Hypersonic Flow. NASA TR R-153, Dec. 1963.

111. Bluford, G.: A Numerical Solution of Supersonic and Hypersonic Viscous Flow Fields Around Thin Planar Delta Winqs. A I M Journal, vol. 17, no. 9, Sept. 1979, pp. 942-949.

112. Cross. E. J.: Exnerimental and ~~ ~~.~ ~ ~ ~~~~

Analytical Investigation of the 100. Weiland, C.: Vortex Flow Simulations Expansion Flow Field Over a Delta

Past Wings Using the Euler Wing at Hypersonic Speeds. ARL 68- Equations. Aerodynamics of Vortical 0027, Wright-Patterson AFB, Ohio, Type Flow in Three Dimensions, AGARD Aerospace Research Laboratories, CP-342, Paper No. 19, July 1983. August 1971.

101. Allen, J. M . ; and Townsend, J. C.: Application of the SWINT Code to Wing-Body-Tail Geometries. AIAA Paper No. 85-1811, Aug. 1985.

102. Wardlaw, A. B.; Hackerman, L. B.; and Baltakis, F. T.: An Inviscid Computational Method for Supersonic Missile Type Bodies. NSWC TR-81-459, Dec. 1981.

4

103. Smith, J. H. B.: Behavior of a Vortex Sheet Separating from a Smooth Surface. R.A.E. TR-77058, 1977.

104. Rizzi, A.; and Purcell, C. J . : Numerical Experiment with Inviscid Vortex-Stretched Flow Around a Cranked Delta Wing: Subsonic Flow. AIAA Paper No. 85-4080, Oct. 1985.

105. Raj, P.: Sikora, J. S . ; and Keen, J. M . : Free-Vortex Simulation Using a Three-Dimensional Euler Aerodynamic Method. ICAS-86-1.5.2, Sept. 1986.

J

106. Karma", S . L.; Steinbrenner, J. P.; and Kisielewski, K. M.: Analysis of the F-16 Flow Field by a Block Grid Euler Approach. AGARD Symposium on Applications of Computational Fluid Dynamics in Aeronautics, Aux-En- Province, April 1986.

107. Viqneron. Y. C.: Rakich, J. V.; and Tannehill, J. C.: Calculation of Supersonic Viscous Flow Over Delta Wings With Sharp Subsonic Leadinq Edqes. A I M Paper No. 78-1137, July 1978.

113. Rizzetta, D. P.; and Shang, J . S . : Numerical Simulation of Leadinq-Edge Vortex Flows. AIAA Journal, vol. 24, Feb. 1986, pp. 237-245.

Steady Supersonic Navier-Stokes Solutions of a 75' Delta Wing. NASA CP-2416, Oct. 1985, pp. 331-347.

115. Fujii, K.; and Kutler, P.: Numerical Simulation of the Leadinq-Edqe Vortex f o r a Wing and Strake-Wing Configuration. AIAA Paper No. 83- 1908, July 1983.

114. Buter, T. A.; and Rizzeta, D. P.:

116. Fujii. K.; and Kutler. P.: Numerical Simulation of the Viscous Flow Over Three-Dimensional Complicated Geometries. AIAA Paper No. 84-1550, June 1984.

117. Hummel, D.: On the Vortex Formation Over a Slender Wing at Large Angles of Incidence. High Angle of Attack Aerodynamics, AGARD CP-247, Paper No. 15, Jan. 1979.

118. Thomas, J. L.; Taylor, S . L.; and Anderson, K.: Navier-Stokes Computations of Vortical Flows Over Low Aspect Wings. AIAA Paper No. 87- 0207, Jan. 1987.

119. Krause, E.; Shi, X. G.; and Hartwich, P. M.: Computations of Leading Edge Vortices. AIAA Paper No. 83-1907, July 1983.

25

120. Liu, C. H.: Hartwich, P. M.; and HSU, C. H.: Incompressible Navier-Stokes Solutions for a Sharp Edged Double- Delta Wing. AIAA Paper No. 87-0206. Jan. 1987.

121. Brennenstuhl, U.; and Hummel, D.: Vortex Formulation Over Double Delta Wings. ICAS-82-6.6.3, pp. 1133-1146.

122. Verhagqen, N. G . : An Experimental Investigation of the Vortex Flow Over Delta and Double Delta Wings. Aerodynamic of Vortical Type Flows in Three Dimensions. AGARD CP-342, Paper No. 7, July 1983.

123. Vatsa, V.: Accurate Solutions for Transonic Viscous Flow Over Finite Wings. AIAA Paper No. 86-1052, May 1986.

124. Shang, J. S.; and Hankey, W. L.: Numerical Solution for Supersonic Turbulent Flow Over a Compression Ramp. AIAA Journal, "01. 13, Oct. 1975, pp. 1368-1374.

125. Fujii, K.: Practical Applications of New LU-AD1 Scheme for the Three- Dimensional Navier-Stokes Computation of Transonic Viscous Flows. AIAA Paper No. 86-0513, Jan. 1986.

126. Hinson, 0. L.; and Hurdges, K. P.: Acquisition and Application of Transonic Wing and Far-Field Test Data for Three-Dimensional Computational Method Evaluation, Vol. 11. Appendix. B, Experimental Data. Lockheed Georgia Company, AFOSR-TR-80-0422, 1980.

127. Keener, E. R.: Computational Experimental Pressure Distributions on a Transonic, Low-Aspect-Ratio Wing. AIAA Paper 84-2092, Aug. 1984.

128. Mansour, N. N.: Computations of the Tip Vortex Off a Low-Aspect Ratio Wing. AIAA Journal, vol. 23, Aug. 1985, pp. 1143-1149.

129. Srinivasan, G . R.: McCroskey, W. J.: Baeder, J . D.; and Edwards, T. A.: Numerical Simulation of Tip Vortices on Wings in Subsonic and Transonic Flows. AIAA Paper No. 86-1095, 1986.

130. Kaynak, U.; Cantwell, H. J . : Holst, T. L.: Numerical Simulation of Transonic Separated Flows Over Low- Aspect Ratio Wings. AIAA Paper No. 86-0508, Jan. 1986.

131. Holst, T. L.: Kaynak, U., Gundy, K.: and Thomas, S. D.: Numerical Solution of Transonic Wing Flows Using an Euler/Navier-Stokes Zonal Approach. AIAA Paper No. 85-1640. 1986.

132. Kaynak. U.: Holst, T. L.: and Cantwell, B. J . : Computation of Transonic Separated Wing Flows Using

Approach. NASA TM-88311, July 1986.

133. Chaderjian, N. M.: Transonic Navier- Stokes Wing Solutions Using a Zonal Approach: Part 2. High Angle of Attack Simulation. NASA TM-88248, April 1986.

an Euler/Navier-Stokes Zonal L/

134. Lin, T. C.: and Rubin, S. G . : Viscous Flow Over a Cone at Incidence, Part 2. Boundary Layer. Journal of Fluid Mechanics, vol. 1, July 1973, pp. 593-620.

135. Tracy, R. R.: Hypersonic Flow Over a Yawed Circular Cone. Ph.D. Thesis, Augs. 1963, Aeronautical Labs, Firestone Flight Sciences Lab., California Institute of Technology, Pasadena, California.

136. Lin, A.; and Rubin, S. G . : Three- Dimensional Supersonic Viscous Flow Over a Cone at Incidence. AIAA Journal, vol. 20, NOV. 1982, pp. 1500-1507.

137. McRae, D. S . : and Hussaini, M. Y.: Numerical Simulation of Supersonic Cone Flow at High Angle of Attack. High Angle of Attack Aerodynamics, AGARD-CP-247, Oct. 1978, Paper No. 23.

W 138. Rainbird, W. J.: The External Flow

Field About Yawed Circular Cones. Hypersonic Boundary Layers and Flow Fields. AGARD-CP-30, May 1968. Paper No. 19.

139. Peake, D. J . ; Fisher, D. F.; and McRae, D. S.: Flight, Wind-Tunnel, and Numerical Experiments With a Slender Cone at Incidence. AIAA Journal, v01. 20, Oct. 1982, pp. 1338-1345.

140. Harris, J . E.: and Morris, D. L.: Solution of Three-Dimensional Compressible, Laminar and Turbulent Boundary-Layer Equations with Comparison to Experimental Data. Proceedings of the Fourth International Conference on Numerical Methods in Fluid Dynamics, June 1974, pp. 204-211.

141. Rainbird, W. J . : Turbulent Boundary- Layer Growth and Separation on a Yawed Cone. AIAA Journal, "01. 6 , Dec. 1968, pp. 2410-2416.

142. Rakich, J. V.; Vigneron, Y. C.; and Agarwal. R.: Computation of Supersonic Viscous Flows Over Ogive- Cylinders at Angle of Attack. AIAA Paper No. 79-0131, Jan. 1979.

26

143. Schiff, L. B.: and Sturek, W. 0.: Numerical Simulation of Steady Supersonic Flow Over an Ogive-

No. 80-0066, Jan. 1980. ' W Cylinder-Boattail Body. AIAA Paper

144. Graham, J. E.; Hankey, W. L.; and Shang, J. S.: Navier-Stokes Solution of a Slender Body of Revolution at Large Incidence. AIAA Journal, vol. 20, June 1982, pp. 776-781.

145. Jorgensen, L. H.; and Perkins, E. W.: Investigations of Some Wake and Vortex Characteristcs of an Inclined Ogive-Cylinder Body at Mach Number 2. NACA Rept. 1371, 1958.

146. Graham. J. E.: and Hankev. W. L.: Computation of the Asymmetric Vortex Pattern for Bodies of Revolution. AIAA Journal, vol. 23, NOV. 1983, pp. 1500-1504.

147. Newsome, R. W.: and Adams, M. S.: Numerical Simulation of Vortical Flow Over an Elliptical Body Missile at High Angles of Attack. AIAA Paper No. 86-0559, Jan. 1986.

148. Allen, J. M.; Hernandez, G.; and Lamb, M.: Body-Surface Pressure Data on Two Monoplane-Wing Missile Configurations With Elliptical Cross Sections at Mach 2.5. NASA TM-85645.

149. Rosenfeld, M.; Israeli, M.; and

Separation on a Spheroid at Incidence. ICASE Report No. 86-49, July 1986.

u Wolfshtein, M.: A Numerical Study of

150. Pan, D.; and Pulliam, T. H.: The Computation of Steady 3-D Separated Flows Over Aerodynamic Bodies at Incidence and Yaw. AIAA Paper No. 86-0109.

1 5 1 . Meier, H. U.: Kreplin, H. P.; Vollmers, H.: Development of Boundary Layers and Separation Patterns on a Body of Revolution at Incidence. 2nd Symposium on Numerical and Physical Aspects of Aerodynamic Flows, Jan. 1983.

152. Pulliam, T. H.; and Steqer, J. L.: Implicit Finite-Difference Simulations of Three-Dimensional Compressible Flow. AIAA 3.. vol. 18, Feb. 1980, pp. 159-167.

1 5 3 . Hsieh, T.: Calculation of Viscous, Sonic Flow Over Hemisphere-Cylinder at 19O Incidence: The Capturing of Nose Vortices. AIAA Paper No. 81- 0189, 1981.

154. Fujii, K.; and Obayashi, S.: Computation of Three-Dimensional Viscous Transonic Flows Usinq the LU-

AD1 Factored Scheme, National Aerospace Laboratory, TR-889T.

155. Kordulla, W.: Vollmers, H.: and Dallmann, U.: Simulation of Three- Dimensional Transonic Flow With Separation Past a Hemisphere-Cylinder Configuration. AGARD CPP-412, Applications of Computational Fluid Dynamics in Aeronautics, Paper 31, April 1986.

156. Fujii, K.; and Obayashi, S.: Navier- Stokes Simulation of Transonic Flow Over Wing-Fuselage Combinations. AIAA Paper No. 86-1831, June 1986.

157. Helliwell, W. S.; Dickinson, R. P.: and L u b a r d . S. C.: Viscous Flow Over ~~. ~~.~~~ ~. ~~

Arbitrary Geomtries at High Angle of Attack. AIAA Paper No. 80-0064, Jan. 1980.

158. Rizk, Y. M.: Chaussee, D. S . ; and McRae, D. S.: Computation of Hypersonic Flow Around Three- Dimensional Bodies at Hiqh Angles of Attack. AIAA Paper No. 81-1261, 1981.

159. Chaussee, D. S.; Patterson, J. L.; Kutler, P.; Pulliam, T. H.; and Steger, J. L.: A Numerical Simulation of Hypersonic Viscous Flows Over Arbitrary Geometries at High Angle of Attack. AIAA Paper No. 81-0050, Jan. 1981.

160. Shang, J. S.; and MacCormack, R. W.: Flow Over a Biconic Configuration with an Afterbody Compression Flap - A Comparative Numerical Study. AIAA Paper No. 83-1668, July 1983.

161. Kaul, U . ; and Chaussee, D. S.: A Comparative Study of Parabolized Navier-Stokes (PNS) Code Usinq Various Grid Generation Techniques. AIAA Paper No. 84-0459. Jan. 1984.

162. Srinivasan, G. R.; and Nicolet, W. E.: Viscous Hypersonic Flow Over Complex Bodies at High Angles of Attack. AIAA Paper No. 84-0015, Jan. 1984.

163. Chaussee, D. S.; Rizk, Y. M.; and Buninq. P. G.: Viscous Computation of a Space Shuttle Flow Field. NASA TM-85977.

164. Wannernwetsch, G. D.: Pressure Tests of the AFFDL X-24C-1OD Model at Mach Numbers of 1.5, 3.0, 5.0 and 6.0. AEDC-DR-76-92, "on Karman Institute Gas Dynamics Facility, Arnold Engineering Development Center, TN, NOV. 1976.

165. Carver, D. 0.: AFFDL X24C Flowfield Survey. von Karman Gas Dynamics J

27

Facility, Arnold Engineering Development Center, TN, Project No. V41B-47, June 1979.

166. Shang, J. S . : Scherr. S . T.: Navier- Stokes Solution of the Flow Field Around a Complete Aircraft. AIAA Paper No. 85-1509CP. July 1985.

167. Flores, J.; Reznick, S. G.: Holst, T. L.: and Gundy, K.: Transonic Navier- Stokes Solutions for a Figher-Like Configuration. AIAA Paper No. 87- 0032, Jan. 1987.

168. Reznick, S . G.; Flores, J.: Strake- Generated Vortex Interactions €or a Fiqhter-Like Configuration. AIAA Paper No. 87-0589, Jan. 1987.

169. Liu, C. H.; Krause, E.; and Ting. L.: Vortex-Dominated Flow with Viscous Core Structure. AIAA Paper No. 85-1556, July 1985.

170. Grabowski, W. J.: and Berger, S. A.: Solutions of the Navier-Stokes Equations for Vortex Breakdown. Journal of Fluid Mechanics, v01. 75, part 3, 1976, pp. 525-544.

171. Shi, X. G.: Numerische Simulation des Aufplatzens von Wirbeln. Dissertation, RWTH Aachen, West Germany, 1983.

Region Angle oE Attack ( a )

I low

I1 moderate to high

I11 very high

IV extreme (up to 90')

172. Faler, J. H . ; and Leibovich, S . : An Experimental Map of the Internal Structure of a Vortex Breakdown. J.

313-335. of Fluid Mech., vol. 86, 1978, pp. Y'

173. Hafez, M.; Kuruvila, G.: and Salas, M. D.: Numerical Study of Vortex Breakdown. AIAA Paper No. 86-0558, Jan. 1986.

174. Ta'asan, S.: Multigrid Method for a Vortex Breakdown Simulation. ICASE Report No. 86-29, May 1986.

175. Rizzi, A.; and Purcell, C. J.: Disordered Vortex Flow Computed Around a Cranked Delta Wing at Subsonic Speed and High Incidence. ICAS-86-1.4.1, Sept. 1.986.

176. Kutler, P.: A Perspective of Theoretical and Applied Computational Fluid Dynamics. AIAA Journal, vol. 23, no. 3, March 1985, pp. 328-341.

177. Shang, J. S.: An Assessment of Numerical Solutions of the Compressible Navier-Stokes Equations. Journal of Aircraft, May 1985, pp. 353-370.

Flow Characteristics

Attached flow

Stable and symmetric vortical flow

Onset of vortex asymmetry and/or vortex breakdown. Nominally steady (lower a), unsteady (higher a )

Unsteady diffuse wake possibly with periodic vortex shedding depending upon Re, Me, geometry. etc.

178. Ericsson, L. E.: The Fluid Mechanics of Slender Wing Rock. Journal of Aircraft, May 1984, pp. 323-328.

2 8

d

. I I

I Y

I

'I 0

-2 Fig. 1.- Flow over a slender, sharp-edged wing (schematic) (Humel, Ref. 1 1 7 ) (a) vortex formation, (b) pressure distribution, (c) lift characteristic.

Fig. 3 a . - Oil-streak pattern about hemisphere-cylinder, M_ = 1.2, a = 1 9 O . Re,, = 653,000 (Peake and Tobak, Ref. 26)

1

2

Classical Vortex Separation Bubble with No Shock No Shock/ No Separation Shock with no Separation Shock-Induced Separation Separation Bubhle with Shock

. ,r, THREE DIMENSIONAIL ,iN SPACE,

Fig. 3b.- Conceptual drawing of the flow Pattern for a hemisphere-cylinder, M_ = 1.2, 01 = 1 9 O (Ying, et al., Ref. 2 7 ) .

Fig. 2 . - Flow classification for sharp leading-edqe delta winqs according to angle of attack and Mach number normal to leadinq edqe (typical variation with anqle of attack, Mach number, and wing sweep anqle indicated) (Miller and Wood, Ref. 1 5 ) . .&

2 9

Fiq. 4 . - Vortex asymmetry--right circular cone, u/Bc = 3 . 2 (a) laser vapor screen, (b) crossflow streamlines (Skow and Peake, Ref. 2 8 ) .

I

Fiq. 6.- (1 - p/p ) contours, Dillner wing, M* = 0 . 7 , am= 15'. (a) coarse qrid ( 6 5 x 21 x 29), ( b ) fine qrid (161 x 49 x 81) (Rizzi, Ref. 76).

Fig. 5 . - Vortex bursting over sharp-edged delta wing, water tunnel, (Lambourne and Rryer, Ref. 3 3 ) .

DILLNER WING b . 0 ' L1.lid.G

Fig. 7.- Spanwise pressure coefficient, Dillner wing, 8 0 % chord, effect of qrid refinement for Euler equations in comparison with experimental and potential flow results ( a ) M- = 0 . 7 , c( = 15* ( b ) M _ = 1.5, o. = 1 5 O (Rizzi, Ref. 7 6 ) .

30

Z l X

YIX

V I X

0 .2 . 3 V l X

0 . 2 . 3 V l X

YIX

Fig. 8.- Elliptic cone, y / x = tan(20'). z/x = tan(l.5O). M _ = 2.0, a = lo', coarse grid conical Euler (a) crossflow Mach number, ( b ) entropy, (c) crossflow velocity fine grid conical Euler I d ) crossflow Mach number, (e) entropy, If) crossflow velocity, (Newsome, Ref. 81).

31

Fig. 9.- Elliptic cone, y / x = tan ( 2 0 ' ) . z /x = tan (2.0-), M_ = 2.0. 01 = loo, large damping coefficient, conical Euler, (a) crossflow velocity, ( b ) crossflow Mach number, small damping coefficient. conical Euler, (c) crossflow velocity. (d) crossflow Mach number (Kandil and Chuang, Ref. 8 7 )

Fiq. 10.- Pitot pressure, sharp-edged delta wing, A = 7 5 O , M_ = 1.95, 01 = loo, (a) conical E u l e r (Murman, et al., Ref. 7 8 ) , ( b ) experiment (Monnerie and Werle, Ref. 89).

3 2

W

-.05 I I I I

0. . 2 -4 ,6 .8 1 . 0 I 05

.15 00-000.004.0 000 000

I 25 0

Fig. 11.- Isentropic separation, zero- thickness, delta wing, A = 70'. M = 2.0, 01 = l o o , conical Euler, l a ) c?ossflow velocity, (b) surface pressure coefficient IKandil and Chuang, Ref. 94).

VI x I

. 2 Y l X

L . 3

YIX Fig. 12.- Sharp-edged delta wing, A = 70', M _ = 2.0, a = l o o , coarse grid upwind Euler, la) CroSSfloW Mach number, lb) entropy, IC) crossflow velocity (Newsome and Thomas, Ref. 95).

33

Fig. 13.- Pitot pressure, sharp-edged delta wing, A = 75', M_ = 1.95, a = IOo, (a) 3-D Navier-Stokes (Buter and Rizzetta. Ref. 114). ( b l experiment (Monnerie and Werle, Ref. 891, (c) conical Navier-Stokes (Thomas and Newsome, Ref. 63).

-.i IIIIIIIIII/ -.3

YIM NO.nt.70 : sWEEpr;7S.O DEG. (a)

I I I I I I I I I

0

-. 1

0- I

0 7 I

.I c SWEEP. OEG. = 75.0

-.2c I

Fig. 14.- Upper surface pressure coefficient. sharp-edged delta wing, conical Navier- Stokes, (a) effect of angle of attack, ( b ) effect of sweep angle (Thomas and Newsome, Ref. 6 3 ) .

(a) vapor-screen photograph.

~~~ ~~

\

'.. . -. ~. ( a ) vapor-screen photograph.

/ ~... ',.\ ',. , . / *

4 L . A b

(b) T o t a l pressure Contours.

Fig. 15.- Flow-field comparison, A = 7 5 " ,

M- = 1 . 1 , a = E o , ReL = 3.5 x l o 6 , ( a )

vapor-screen photo (Miller and Wood, Ref. Fiq. 16.- Flow-field comparison, 1 5 ) , ( b ) total pressure contours, conical Navier-Stokes (Thomas and Newsome, Ref. 63). Wood, Ref. 151, ( b ) density contours,

Ib) Density contours .

A = 67.5'. M_ = 2 . 8 , a= 8 " . ReL = 2.3 x l o 6 , ( a ) vapor-screen photo (Miller and

Conical Navier-Stokes (Thomas and NewsOme, Ref. 6 3 ) .

34

Fig. 17.- Total pressure conto~rs, AR=1.0 delta wing, M- = 0.3, m = 20.5 , ReL = - 0.95 x lo6 (Thomas, et al., Ref. 118).

Semispan distance

Fig. 18.- Surface pressure coefficient, AR = 1.0 delta wing, M_ = 0.3, ReL = 0.95 x lo6, (Thomas, et al., Ref. 118).

a = 20.5O,

'.,,

Fig. 19.- Skin friction lines, Wing C, 6 M_ = 0.85. a = 5.9', ReM.A.C. = 6.8 x 10 I

(a) postulated Erom o i l flow, (b) 3-0 Navier-Stokes (Kaynak, et al., Ref. 130).

35

0 EXP.. REF. 17 ~

MODIF!ED MODEL

0 30 Bo 00 120 150 180 WIND LEE

0. d.p

~ i g . 20.- Circular cone, oC = 1 2 . 5 ' . M- = 1 . 8 , a = 22.750, Re, = 2 1 x l o 6 , l a ) surface pressure coefficient, ( b ) surface flow direction anqle (Deqani and Schiff. Ref. 69).

Fig. 21.- Surface pressure coefficient, elliptical-body missile, Mm = 2.5, a = 20" . 6 = 4 5 ' . ReL = 4.6 x lo6 (Newsome and Adams, Ref. 1 4 7 ) .

36

\

X/L = 0.64

X/L = 1.0

Fig. 22.- Comparison of experimental vapor-screen and computed density fields, elliptical body missile. M _ = 2.5,

6 a = 20°, B = 45’, ReL = 4.66 x 10

(Newsome and Adams. Ref. 147).

TOP VIEW

SIDE VIEW

Fig. 23.- Computed skin friction lines, hemisphere-cylinder, M = 1.2, m = 19’. ReD = 445,000 (Ying, er al., Ref. 27).

~ i q . 24.- Computed skin friction lines, hemisphere-cylinder, M _ = 0.9, Re = 425,000 (Kordulla, et al., Ref. 15g).

a = 19’.

Fig. 25.- W-18 wing-fuselage, M _ = 0.82, 01 = 6 O , Rer = 1.6 x lo6, (a) computed skin friction lines, ( b ) particle path traces (Fujii and Obayashi, Ref. 156).

3 7

61 x 45 GRID

Fig. 26.- Computed skin friction lines, space shuttle orhiter, M_ = 1.9, CL = 2 5 0 ,

ReL = 2.4 x 106/m, (Chaussee, et al. Ref. 1 6 3 ) .

R\ M-= 5.95 Ruy = 1.64 x lO'/m a = 6 degrees

Fig. 21.- Computed skin friction lines, X24C-lOD. M_ = 5.95, a = 6". ReL = 1.64 x

107/m, (Shang and Scherr, Ref. 166).

L,

3 8

Fig. 28.- Comparison of experimental and computed bubble structure, ( a ) time- averaged streamline pattern from experimental measurements, ReD = 2500, (Faler and Leibovich, Ref. 1 7 2 ) , (b) flow visualization (Escudier and Keller, Ref. 4 1 ) . (c) instantaneous streamline pattern, Re6 = 200 (Krause, et al., Ref. 119).

piq. 2 9 . - computed periodic flow behavior in bubble-type breakdown, Re6 = 200 (Shi, Ref. 1 7 1 ) .

o Experiment (Hummell M,=O.l - Thin-layer Navier-Stoker LCFUDl M,=Q.3 1.4

Hummel Delta wing

CL

Fig. of attack for AR = 1 delta wing, M - = 0.3,

ReL = 0.95 x lo6 (Thomas, et al., Ref.

31.- Lift curve variation with angle

118).

~ i g , 30.- Streamwise velocity contours for AR = 1 delta wing, M- s 0.3, ReL = 0.95 x l o 6 (Thomas, et al., Ref. 118).

a = 40°,

\.J

39