AD-A104 M3 ARM4Y ARMAMENT RESEARCH AND DEVELOPMENT COMMAND A8ER--ETC F/S 20/14

A FLOW MODEL FOR THE EFFECT OF A SLANTED BASE ON DRAG.(U)JL 81 R SEDNEY

LWCLASSIFIED ARRL-TR-02341 NLED ,IIIII EIIEEIoEEEEEEEE.EllEEEElis

TECHNICAL REPORT ARBRL-TR-02341 '

t A FLOW MODEL FOR THE EFFECT OF ASLANTED BASE ON DRAG

Raymond Sedney

DTICJuly1981 ELECTE

SEP10 1981

B

US ARMY ARMAMENT RESEARCH AND DEVELOPMENT COMMANDBALLISTIC RESEARCH LABORATORY

ABERDEEN PROVING GROUND, MARYLAND

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TECHNICAL REPORTi ARBRL-TR-02341 L4D-,A /0 V4. TITLE (and Subtitle) S. TYPE Of REPORT & PERIOD COVERED

A FLOW MODEL FOR THE EFFECT OF A SLANTED BASE ON FinalDR AG ,6. PERFORMING ORG. REPORT NUMBER

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Raymond/Sedney

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' U.S. Army Ballistic Research Laboratory(ATTN: DRDAR-BLL) RDTF-,1L161102AH43Aberdeen Proving Ground, Maryland 21005R / 1 A

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Il. SUPPLEMENTARY NOTES

IS. KEY WORDS (Continue on reveree aide if nece y mid Identify by block number)

DragBase DragVehicle DragVorticesVortex Breakdown

2(L A6TRACT rcenmtu em rver.., iaW swy mE Identify by block number)

Experiments by Morel have shown that slanting the base of a bluff bodycauses large variations in the drag, as the slant angle changes. For aparticular, critical slant angle the drag changes discontinuously. Thisphenomenon was correlated with a drastic change in the type of base flow. Amechanism to explain this change, and therefore, the discontinuity in drag, is

osed.. Th basis pf the echanism is breakdown of the side edge vortices.nroStjmate o theolui 1 angi weje vortices is obtainedusing a swe t-as utio s wir ngle and the other elements of the

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21. ABSTRACT (Continued)

-theory provide an estimate for the critical slant angle that is entirely con-sistent with the observed value.

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TABLE OF CONTENTS

LIST OF ILLUSTRATIONS ...... .. .. .. .. .. .. . ... .. . ... . ... .. .. .. .. . .. 5

IT. THE EXPERIMENTAL EVIDENCE .......... ..... ...... .. . ...... .. 8

III. THE SIDE EDGE FLOW............... ... *....**......*..... *.. 9

IV. THE VORTEX BREAKDOWN HYPOTHESIS................................ 11

REFERENCES ..... *******........*...............*......... . ..... 16

LIST OF SYMBOLS ............ .. .. ... . .. . ... . .. ... .. .. . .. .. ... . .... 23

DISTRIBUTION LIST ......... . .................. ....... ... 25

: r

r -.S

LIST OF ILLUSTRATIONS

Figure Page

1 Models Tested by Morel (Reference 2). Dimensions in mm...... 17

2 Drag Coefficient of the Vehicle-Like Model in the FreeStream Location, From Reference 2 .............. 17

3 The Two Types of Base Flows for a Body with a Slanted Base ... 18

4 Rear View of a Slanted Base Showing Part of the SurfaceFlow Pattern. The Flow Along the Side (Out of the Planeof the Paper) Separates at the Side Edge and Reattachesat the Dashed Line. Sketched from a Photograph in

5 A Wedge, Semi-Infinite in the y-Direction, as an IdealizedConfiguration for Side Edge Flow. The Free StreamVelocity, U, is Parallel to the Top and the Side. TheSlanted Base is z = 0, x > 0, y > 0.................. 19

6 Flow Over an Infinite, Yawed Side Edge at y =0, z = 0 .... 20

7 Vortex Breakdown in a Duct Flow, Made Visible with DyeIntroduced into the Flow, from Reference 6. B is theBreakdown Point..................... ..... ..... .. .. . .. 21

I/

I. INTRODUCTION

In the search to reduce drag of road vehicles, it has become clear that amore basic understanding of the complex flows about such bodies is required.Many investigations of drag and flow phenomena exist; see Reference 1 forsurveys of recent work.

A particularly interesting phenomenon is the effect that slanting thebase has on the drag of a bluff body. Following up on work by Janssen andHucho, partially reported in Reference 1, Morel did a comprehensive study ofthat effect, see References 1 and 2. Janssen and Hucho observed an overshootin drag and a change in separation pattern in tests on a model of a hatch-backcar, when the angle of the slanted portion of the roof was varied over a smallrange. In order to examine this effect more closely and gain some understand-ing of it, Morel1'2 made extensive tests on two models (see Figure 1): (i) anogive cylinder with a slanted base, mounted in the center of a wind tunnel tominimize wall effects; and (ii) a vehicle-like model simulating a hatch-backcar, mounted in the center of the tunnel and close to the tunnel wall.Because the wind tunnel models are simpler to discuss and there are moreexperimental details, the results of Morel will be used in this paper.

The most striking result from the tests of Morel was the extremely rapidchange in drag coefficient, CD, as the slant angle, 8, is varied. In fact the

data show, essentially, a discontinuity in CD for a certain 8 = Oc . Also,

for 8 < $c , the variation of CD with 8 is much greater than for B > Oc. The

results of Janssen and Hucho are qualitatively the same, but the variation ofCD with a is smooth, i.e., no discontinuity. (The term discontinuity is usedhere for convenience and its descriptive accuracy, even though a mathematicaldiscontinuity does not exist.)

Visualizing the flow with smoke, Morel showed that there are two distinct

types of base flow. For a > ac a closed base flow, typical of blunt-based

axisymmetric bodies, was found. For a < $c streamwise vortices were formed at

the side edges with a resultant 3-D separation pattern. It was concluded thatswitching from one separation pattern to the other caused the discontinuity inCD-

1. Sovran, G., Morel, T., and Mason, Jr., W.T., Aerodynamic Drag Mechanismsof Bluff Bodies and Road Vehiles, Symposiun held at the General MotorsResearch Laboratories, Plenum Press, New York, 1978.

2. Morel, T., "Aerodynamic Drag on Bluff Body Shapes Characteristic of Hatch-Back Cars", Research Publication GMR-2581, General Motors ResearchLaboratories, November 1977. See also S.A.E. Technical Paper 78026?,1978.

7

Additional evidence for the existence of the streamwise vortices spring-ing from the side edges is given by Carr 3. He used the surface indicator (oilflow) method to visualize the flow and found clear and distinct edge vorticesfor 0 = 250 but not for 0 = 35* . Carr also discusses the downwash producedby, and the effect on the rear lift force of, the side edge vortices.

The primary purpose of this paper is to propose a flow mechanism toexplain the change in separation pattern and the discontinuity in CD. The

mechanism involves breakdown of the side-edge vortices. Vortex breakdown hasbeen relatively well studied and is partially understood; the work on thissubject through 1972 has been reviewed by Hall 4 , and later work byLeibovich5 . To make use of the empirical knowledge on vortex breakdown, amodel for the flow over a side edge is required. For an idealized side edgeconfiguration, a flow model is constructed which allows a simple estimate ofthe swirl in the vortex. Empirical evidence shows that breakdown occurs for avalue of swirl corresponding to 8 = %c, approximately. The consistency of

this theoretical result and the experimental data is sufficient to warrantfurther examination of this mechanism and to test it in additional experi-ments.

II. THE EXPERIMENTAL EVIDENCE

For the two models tested by Morel, see Figure 1, the ac were

different. It is simpler to discuss the vehicle-like model because it has astraight side edge. Some discussion of the ogive-cylinder model is givenlater.

The drag coefficient, based on projected frontal area, as a functionof 0 is shown in Figure 2. The base flows corresponding to Regimes I and IIare shown in Figure 3. In Figure 3a, for Regime I, the base flow is essenti-ally like that of an axisymmetric body; in the mean the base flow is a closedregion. In Figure 3b, for Regime II, a vortex springs from each side edge;between them there is attached longitudinal flow. In Reference 3 Carr quotesthe work of Potthoff, 1969, who pointed out the existence of the side edgevortices and the fact that they can prevent the slanted base flow from separ-ating for a as large as 30', or even greater with suitable shaping of the bodysides.

3. Carr, G.W., "Influence of Rear Body Shape on the AerodynamicCharacteristics of Saloon Cars" Motor Industry Research Association,Report No. 1974/2, Nuneaton, Warwickshire, February 1974.

4. Hall, M.G., 'Vortex Breakdown". Annual Review of Fluid Mechanics, Vol. 4,1972, Annual Review, Inc.

5. Leibovich, S., "The Structure of Vortex Breakdown", Annual Review of FluidMechanics Vol. 10, 1978, Annual Reviews, Inc.

For Carr's model, which had a short "boot length" at the end of the slantbase, 8 = 25. The surface flow pattern clearly showed the trace of the sideedge vortices. A part of this pattern is sketched in Figure 4, which showsthe rear view of the slanted part of the roof, i.e., slanted base. The flowalong the side separates at the side edge and reattaches along the dashedline. The surface streamlines, easily visible in the oil film used for thevisualization, which emanate from the dashed line, are typical of reattach-ment, forming what is often called a herringbone pattern close to the reat-tachment line. Using the reattachment line to indicate the inboard boundaryof the vortex, this result shows that the vortices extend over about one-halfof the width of the slant base, for this case.

There is enough evidence to conclude that side edge vortices exist fora < ac. For smaller 8, say B < 100, they probably exist but would be diffi-

cult to detect because they are weak. As pointed out by Morel 1,2, theinitial decrease in CD, for 0 4 8 ' 10', may be a boattail type of effect,

familiar in the design of projectiles, rather than a side edge vortexeffect. The latter probably begins to dominate for 8 > 100. For a = sc thediscontinuity in CD occurs and the side edge vortices disappear. This massive

change in the slanted base flow field occurs suddenly, for a small changein 8.

Before discussing the flow mechanism for this sudden change, some esti-mates of the side edge flow are necesary.

III. THE SIDE EDGE FLOW

A description of the complete flow field over either of the models shownin Figure 1 by analytical methods would be very difficult. To obtain a tract-able idealization the local flow at the side edge should be examined. Onepossibility is shown in Figure 5 where the side edge is along the x-axis andthe leading edge of the slanted base, or corner, is along the y-axis. The z-axis is normal to the base and forms a right-handed, rectangular, Cartesiancoordinate system. The external flow, U, is parallel to the top and sidesurfaces. The boundary layer would be neglected until the flow reaches theside edge and corner. This idealization, however, will not yield the simpleestimates needed here.

As a further simplification, appropriate to the side edge flow away fromthe corner, consider an infinite, slanted, or yawed, side edge. That is,there is no corner. The side view, y = 0 and the back view, x = constant, areshown in Figures 6a and 6b, respectively. The solution to this problem isindependent of x. This kind of idealization is basic in the study of sweptwings. As an additional approximation the incoming flow, U, is taken to bethe free stream velocity over the model in Figure lb. The thickness of theboundary layer, before it separates from the edge, is neglected. This is thesame assumption made in the classical Gortler, solution for the 2-D, free,turbulent, shear layer, often called the mixing zone problem. We obtain thisproblem, discussed in Reference 6, if 8 = 900. This idealization of the sideedge flow can be called a combination of the 2-D mixing zone solution with the

sweep-back principle.- 9

However, an additional caution must be discussed. In most applicationsof sweep-back theory the independence principle holds, i.e., the flow in the(y,z) plane can be computed, and then the flow in x-direction determined. Forlaminar flow this is always possible but, because of the Reynolds stress termsin the momentum equations, the independence principle is not strictly validfor turbulent flow. It will be assumed here that the independence principleholds; this assumption should cause a small error in the swept-back mixingzone problem.

Therefore, from Figure 6, the mixing zone flow is determined for theexternal velocity U sin 8. The velocity, w, in the z-direction, in the mixingzone is given by the well known solution6 ,

w = (1/2) U sin 8 [1 + erf (-ay/z)] (1)

where a =12 is the mixing coefficient. The vorticity, Dw/ay, is solely inthe x-direction. The velocity parallel to the edge, U cos 8, is constant forall z. This is as far as the idealization can be carried. Specifically, inthe absence of pressure gradients, the free shear layer cannot reattach to thewall z = 0.

On the actual model, the separated flow must reattach on the slanted basein order to form the vortex. Only empirical criteria for reattachment areavailable and these require knowledge of the pressure. The criterion given by

Crabtree 7 , for leading edge bubbles on airfoils is S < 0.35 where

S = (Cpr- C ps)/(1 - cps) (2)

and Cpr and Cps are the pressure coefficients at reattachment and separation,

respectively. Using Morel's pressure datal, reattachment is indicated byCrabtree's criterion. At this stage this is only a consistency statement. Itwould be useful deductively if estimates of the pressure could be obtained.

The most important quantity that can be estimated from the solution tothe swept-back mixing zone problem is the swirl in the side edge vortex.Assume that the flow over the actual model reattaches on the slanted base.The swirl angle is defined as

* : tan- 1 (ve/u)

6. White, F.M., Viscous Fluid Flow, p. 509, McGvazw-Hill Book Company, NewYork, 1974.

7. Crabtree, L.F., "Effects of Leading-Edge Separations on Thin Wings in Too-

Dimensional Incompressible Flow" I.A.S. Preprint 659, January 1957.

10

where v, is the azimuthal velocity in the vortex and uw is the velocity in the

x-direction, i.e., axial velocity in the vortex. *varies with position inthe vortex because yv and u vary. The estimate for v0 is obtained from (1)and that for u from U cos 8. Thus,

v= 0 (U sin 0)

U =0 (U Cos 0)

so that

This result for will be used as an approximate relation

. (3)

The velocities ye and u could be measured using a laser Doppler velocimeter to

obtain a check on this result.

An estimate of 0 is needed to relate breakdown of the side edge vortex tothe discontinuity in CD. The error in this estimate may be considerable; but

it should decrease as x increases. Since only the ratio, ye/u, is used to

estimate 0 and it may have smaller error than v6 and u.

IV. THE VORTEX BREAKDOWN HYPOTHESIS

Vortex breakdown has been observed in a number of flows; e.g., flow overhighly swept wings and flow in ducts. Breakdown is one of the more remarkableaspects of vortex cores. Although it has been studied vigorously, there is nocompletely satisfactory theory for it. The description of it given by Hall4

will be quoted here; he refers to flow in a duct with swirl imparted by vanesat the duct entrance. "If we follow the fluid as it spirals along the duct wefind, typically, that the structure of the vortex, as indicated for example bythe velocity distribution over a cross section of the duct, varies only slowlyin the axial direction and then, suddenly and, at first sight, unexpectedly,'there is an abrupt change in the structure with a very pronounced retardationof the flow along the axis and a corresponding divergence of the stream sur-faces near the axis." This abrupt change is called vortex breakdown. SinceHall's review many papers on the subject have appeared, too numerous tomention here. Leibovichs reviews the recent work. The important facts forpresent purposes are the abruptness of the change in the vortex core flow andits sensitivity to small changes in flow conditions. Vortex breakdown occursin two forms, mainly, Reference 5. One is called the bubble type and isnearly axisyninetric, at least close to the breakdown region; the second iscalled the spiral type and is highly asyrmetric. These are illustrated in

Figures 7a and 7b, respectively. In the duct flow a filament of dye is intro-duced along the axis which then gives a visual record of the breakdown.

Most of the quantitative information on vortex breakdown is obtained fromduct flows because it is more difficult to run a controlled experiment inother flows. It seems that the phenomenon was first discovered in flows overhighly swept wings, see Reference 4 for references and a smoke flow photo-graph. This photograph shows a bubble type breakdown on one side of a deltawing and a spiral type on the other. Clearly the flow over a wing, withvortices on each side of the plane of symmetry, is more representative of theslanted base flows discussed here than duct flows. For typical wings, thespanwise separation of the vortices is much greater than in slanted baseflows; the interaction between the vortices is then quite different.

According to HalI' the necessary conditions for breakdown are: (1) themaximum * > 40% (2) an adverse pressure gradient; (3) stream tube divergencein the vortex core. Conditions (2) and (3) are satisfied for the side edgevortices. The estimate obtained from the swept-back mixing zone problemgave * = so that condition (1) would give a8 40' for breakdown.

Condition (1) is clearly approximate. It was deduced by Hall from thelimited experimental data available at the time. Since then laser-Doppleranemometry has come into wide use and is ideal for velocity measurements invortex breakdown flow fields because no probe is introduced into the flow; aprobe in the breakdown region can introduce large perturbations in the flowfield. Results from many experiments on duct flows, using laser-Doppleranemometry, are presented by Leibovich5. In particular, his Table 1 sum-marizes data for the velocity in the approach flow, i.e., the flow to withinabout 1.5 vortex core diameters upstream of the breakdown point. Both bubbleand spiral types of breakdown were observed and measured at three Reynoldsnumbers, which is based on the axial velocity far from the axis and the vortexcore diameter. From that data, Table 1 was constructed.

Table 1.*

CoreType of m Expansion

Re Breakdown (deg) Ratio

1920 Spiral 31.7 1.64Bubble 30.4 2.54

2812 Spiral 30.6 1.78Bubble 30.8 1.97

3348 Spiral 33.5 2.10Bubble 29.9 2.76

*Data from Reference 5.

The swirl angle, *,presented in Table is defined as12

tan' (v /u

It is calculated using certain functional forms for v. and u; the parameters

in those forms are determined from a best fit to the data. The value of 0 m isa measure orl the swirl angle for vortex breakdown; it is more reliable thanthe criterion given in condition (1).

In Table 1, *m varies between 29.90 and 33.5%* No trend in the variation

Of om with Reynolds number can be detected; the average value of *m is 31.90

for the spiral type, 3Q.* for the bubble type, and 31.20 if both types areconsidered. For purposes of estimation, *m =30' can be used. Assuming

that om approximates the estimate of * in (3), breakdown of the side edgevortex is estimated to occur at a = 300.

Another feature of vortex breakdown that enters the proposed mechanism toexplain the discontinuity in CD is the increase in size of the vortex coreafter breakdown. Downstream of the breakdown region a new vortex is formed,in the wake region. The ratio of the wake-core radius to that in the flowupstream of breakdown is called the core expansion ratio, given in Table 1.The core radius is defined as the radial coordinate of the maximum in theazimuthal velocity. The smallest entry in the table gives a 64% increase inthe size of the core.

One of the major results of the above theory is that breakdown of theside edge vortices is possible for the range of parameters covered in Morel'sexperiments. Assuming it does occur, its relation to the sudden change in CDcan be described as follows. Before breakdown the two vortices occupy asubstantial part of the slanted base, see the discusssion of Figure 4,above. The sudden billowing of these vortices, as measured by the coreexpansion ratio, will cause the mutual interaction between them and theirinteraction with the outer stream to increase. The adverse pressure gradientacting on the vortices will increase and cause the breakdown region to movefurther upstream, etc. Rapid movement of the breakdown region is observed induct flow experiments. The 3-D separation pattern, or open base flow, thencollapses into the quasi -axi symmnetric base flow. The final stage of thiscollapse cannot be described by the model.

8The estimates of swirl, the relationship between swirl and slant angle,a , and the model proposed here can now be combined. They give the criti-

cal slant angle for the discontinuity in CD, 0 c =30% Morel's experimentsgive, Figure 2, Bc = 30'. Considering the idealizations made in arriving atthe theoretical result, its agreement with the experimental value must beconsidered fortuitous. The proper conclusion is that the theory is consistentwith the available experimental findings.

For the ogive-cylinder wind tunnel model tested by Morel2, Figure la, theCD versus 8 variation was generally the same as that in Figure 2. However,

13

iithe discontinuous decrease in CD, from 0.62 to 0.3 in this case, occurs at

Bc = 430 rather than ac = 300 as for the straight side-edge. Development of a

flow model for the ogive-cylinder slanted base, where the edge is an ellipse,will not be attempted. Some understanding of the flow for 8 < 0c can be

obtained by isolating three modules of the base flow field, assuming steadyflow. (1) At the top of the base the flow separates. The scale of thisseparated region, measured by the location of reattachment line, is a smallfraction of the major axis. (2) In the neighborhood of the side of the basethe flow separates and side-edge vortices are formed. The flow pattern wouldbe topologically the same as that in Figure 4. The scale of the separationregion is a significant fraction of the minor axis. (3) At the bottom of thebase the flow has, at most, a small scale separated region.

It appears reasonable to neglect first order interaction between thesethree modules; then some portion of the base surface streamlines can besketched. For small enough a the side-edge vortices must depart from the baseeither by reaching the edge or by lifting off. It is conjectured that, for8 = 300, either the swirl is decreased below the critical value before theydepart the surface or they break down off the surface. In the former, largera is required for breakdown on the surface. In the latter, their mutualinteraction and that with the outer stream, which is necessary to explain thesudden change in base flow and CD, is relatively weak. Stronger vortices,

i.e., larger 8, would be required to have breakdown on the surface and therequired interaction. In either case the conclusion is consistent with theobserved larger 0c.

V. DISCUSSION

More refined estimates for the swirl in the side edge vortices are pos-sible, e.g., using the configuration of Figure 5, but would require much moreanalysis. It would be preferable to test the central hypothesis of thistheory experimentally. Introduction of a blunt body along the center of thevortex core, which is known to promote breakdown, could be useful as a diag-nostic tool.

An explanation of the discontinuity in CD at 8 = c requires a process

with the abruptness (almost explosive nature) of vortex breakdown. Theoriesbased on some other hypothesis are possible, however. One of these could bebased on the inability of the free shear layer to reattach for 8 > 8c.

In addition to an understanding of the phenomena that lead to the discon-tinuity in CD, it would be desirable to be able to predict drag, especially

for 8 < 8c. Linearized theory might be useful for the case of small o.

Otherwise, only numerical methods could be successful.

Although the discussion in this report is limited to the effect on dragof slanting the base, the phenomenon of side edge vortices and the flow modelproposed for their breakdown has wider applicability. Other parts of a roadvehicle have side edges and they occur in other applications. One of these is

14

in ballistics: the non-conical boattail proposed by Platou8. No studies ofvarying $ have been made, however. A more complete understanding of thegeneration and breakdown of side edge vortices will be useful in many fluiddynamic problems.

7

8. Platou, A.S., "An Improved ProjectiZe Design", U.S. A y BallZieticResearch Laboratory/ARRADCOM Memorandum Report No. 2395, Aberdeen ProvingGTound, Mamland, July 1974. AD A785520

15

REFERENCES

1. Sovran, G., Morel, T., and Mason, W.T., Jr., Aerodynamic Drag Mechanismsof Bluff Bodies and Road Vehicles, Symposium held at the General MotorsResearch Laboratories, Plenum Press, New York, 1978.

2. Morel, T., "Aerodynamic Drag of Bluff Body Shapes Characteristic of Hatch-Back Cars," Research Publication GMR-2581, General Motors Research Labor-atories, November 1977. See also S.A.E. Technical Paper 780267, 1978.

3. Carr, G.W., "Influence of Rear Body Shape on the Aerodynamic Character-istics of Saloon Cars," Motor Industry Research Association, Report No.1974/2, Nuneaton, Warwickshire, February 1974.

4. Hall, M.G., "Vortex Breakdown," Annual Review of Fluid Mechanics, Vol. 4,1972, Annual Review, Inc.

5. Leibovich, S., "The Structure of Vortex Breakdown," Annual Review of FluidMechanics, Vol. 10, 1978, Annual Reviews, Inc.

6. White, F.M., Viscous Fluid Flow, p. 509, McGraw-Hill Book Company, NewYork, 1974.

7. Crabtree, L.F., "Effects of Leading-Edge Separations on Thin Wings in Two-Dimensional Incompressible Flow," I.A.S. Preprint 659, January 1957.

8. Platou, A.S., "An Improved Projectile Design", U.S. Army BallisticResearch Laboratory/ARRADCOM Memorandum Report No. 2395, Aberdeen ProvingGround, MD, January 1974. AD A785520

16

L351 --38

(a)

U900 J 270 -

nominal length

(b)Figure 1. Models Tested by Morel (2). Dimensions in mm.

(a) Ogive-Cylinder Model(b) Vehicle-Like Model

0.5 I

'U

0.4-

CD ,

0.3

Regime II 1 Regime I

0.1 1 ,1

0 30 60 90

Figure 2. Drag Coefficient of the Vehicle-Like Model in the

Free Stream Location, From (2)

17

IIPPWa Quasi-axis mmtric Separation Pattern

(b) 3-D Separation Pattern

Figure 3. The Two Types of Base Flows for a Body with a Slanted Base

(a) The Closed Type Exists in Regime I(b) The Open Type with Side Edge Vortices Exists in

Regime II

CENTERLINE

SID

REATTACHMENTLINE

Figure 4. Rear View of a Slanted Base Showing Part of the Surface FlowPattern. The Flow along the Side (Out of the Plane of thePaper) Separates at the Side Edge and Reattaches at theDashed Line. Sketched from a Photograph in(3

18

z* TOP

SIDE ,

SIDE /EDGE

Figure 5. A Wedge, Semi-Infinite in the Y-Direction, as an IdealizedConfiguration for Side Edge Flow. The Free Stream Velocity,U, is Parallel to the Top and the Side. The Slanted Baseis z = 0, x > 0, y > 0

19

z

.1 SIDE

U U o

U sin 3 Uo/

(a) x

U sin /w w7I77z;777 z

MIXINGZONE

(b)

Figure 6. Flow Over an Infinite, Yawed Side Edge at y = 0, z = 0

Sa) The View in the Plane y = 0.b) The View in the Plane x = constant. The Flow

Separates from the Side Edge, Forming a FreeShear Layer or Mixing Zone.

20

I

(a)

Im

(b)

Figure 7. Vortex Breakdown in a Duct Flow, Made Visible with DyeIntroduced into the Flow, From (6). B is the BreakdownPoint

(a) Bubble Type(b) Spiral Type

21/

LIST OF SYMBOLS

CD - drag coefficient based on frontal area

Cp - pressure coefficient

0 - order of magnitude symbol

S - see (2)

u - axial velocity in vortex and velocity in x-direction

U - free stream velocity

ve - azimuthal velocity in vortex

w - velocity in z-direction

x,y,z - Cartesian coordinates, see Figure 5

a - angle of slanted base

0c - angle of slanted base dividing Regimes I and II

r - swirl angle in vortex

o - mixing coefficient

Subscripts

r,s - reattachment and separation

23/

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