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SAE TECHNICALPAPER SERIES 2000-01-0354

Selecting Automotive Diffusers toMaximise Underbody Downforce

Kevin R. Cooper and J. SymsNational Research Council of Canada

G. SovranGM Research, retired

Reprinted From: Vehicle Aerodynamics(SP1524)

SAE 2000 World CongressDetroit, Michigan

March 6-9, 2000

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ISSN 0148-7191Copyright 2000 Society of Automotive Engineers, Inc.

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1 2000-01-0354

Selecting Automotive Diffusers toMaximise Underbody Downforce

Kevin R. Cooper and J. SymsNational Research Council of Canada

G. SovranGM Research, retired

Copyright 2000 Society of Automotive Engineers, Inc.

ABSTRACT

Underbody diffusers are used on racing cars to generatelarge downforce that will permit them to achieve reducedlap times through aerodynamically-enhanced traction.Both the configuration of these cars and their underbodyflows are complex, so the design of optimum underbodygeometries is a formidable task. The objective of thepresent study is to generate data and understanding thatwill facilitate design through knowledge of the relevantphysics and the application of a numerical analysis thatprovides generalised design guidance.The approach taken is one that is traditional in the studyof complex problems: to identify a less-complex but stillrelevant sub-problem that has the key elements and flowphysics of the main one, and study it to generate a firstphase of cause-and-effect relationships. In addition tohaving immediate utility, it can serve as the foundationupon which future research activity can be built.The result of the present study is an analytical model thatwill facilitate the selection of optimum length and arearatio for the underbody diffusers of flat-bottomed racingcars. While not a universally-applicable design tool, theguidelines developed should reduce the effort required todevelop underbody configurations that produce largedownforce.

INTRODUCTION

In an initial paper [1], an extensive set of wind tunnel testresults was reported for a simple rectangular-block modelwith circular-arc front corners that was fitted with plane-

walled underbody diffusers of two lengths. Both fixedand moving-ground test conditions were utilised. Theobjective was to measure the influence of underbody dif-fusers on lift and drag and to resolve the associated flowphysics.At the relatively large ground clearance typical of passen-ger cars, drag reduction that is important for fuel econ-omy was achieved under some circumstances. Largedownforce production that is important for race cars wasachieved under many circumstances, particularly at thesmall ground clearances typical of that application.The focus of this paper is on racing-car applications of adiffuser, and specifically on the effect of diffuser lengthand area ratio on downforce generation. An analysis isperformed that permits diffuser measurements from [1] tobe generalised and applied to the identification of opti-mum underbody geometries. The previous paperexplored the physics of downforce generation. Thispaper extends the understanding gained and applies it toprimarily flat-bottomed, diffuser-equipped racing cars likecontemporary sports-racing prototypes.

BACKGROUND

The simple, wheel-less model of this investigation isshown in Figures 1 and 2. It was fitted with diffusers ofeither 25% or 75% of its length. The angle of the diverg-ing wall was adjustable from 0 to more than 16. Thediffusers were fitted with partial side plates, and nearly allthe measurements were made at zero pitch angle. Fig-ure 1 presents a photograph of the model on its overheadsting.

2Figure 1. View of Model with Long Diffuser Showing Maximum Diffuser Angle and Partial Side Plates

A circumferential distribution of surface-pressure tapswas positioned on or near the longitudinal centre plane ofthe model at the locations shown in Figure 2. The pres-sure taps on the vertical rear face were connectedtogether and averaged pneumatically to give the averagepressure over the full base area. Two other sets of tapson the roof of the model, marked by the curly parenthe-ses in Figure 2, were averaged in the same fashion.The model was mounted above a moving-belt systemthat was 4.24 times wider and 3.85 times longer than themodel. It was tested over a full matrix of diffuser configu-rations (2 diffuser lengths, 9 diffuser angles, 22 rideheights), for both moving-ground and fixed-ground simu-lations.

In [1], CFD was applied to three of the test configurationsto provide assistance in understanding the measureddata, and for comparison with experiment. The numeri-cal model did not have the partial side plates of the phys-ical model. The code was fully turbulent, and thereforeincapable of computing laminar separation if it were tooccur.

DOWNFORCE MECHANISMS

Lift coefficient data were used in [1] to identify three dif-ferent downforce mechanisms for the test body: under-body upsweep (body camber), ground interaction anddiffuser pumping. In the present paper, surface-pressuredistributions will be used for the same purpose. Theseoffer a more detailed picture of the physical mechanisms.As developed in Appendix A, equation (A5), the lift coef-ficient of the test body can be expressed as,

(1)

Figure 2. Model Geometry (dimensions in mm)

where the streamwise-distance-averaged, mean-effectivepressure coefficients (equation (A4)) are defined as,

(2)and the subscripts l and u denote the lower and theupper surfaces, respectively. The overall length andheight of the body are L and H, respectively (see FigureB2). Managing lift is a matter of managing the differencebetween and . Since the primary interest of thisstudy is the underbody flow, a test body of simple overallconfiguration was chosen so that the flow over its uppersurface would be relatively insensitive to changes inunderbody geometry and ground clearance. That this

= puplL CCH

LC

iX0 pipi dx)x(Cx1C

plC puC

3was the case is demonstrated by Figure 3 for several dif-fuser wall angles, , and ride heights, (h1/H).

With this relative constancy of , equation (1) indi-cates that the behaviour of CL is essentially determined

by the underbody through . Since downforce is nega-tive lift, it is maximised by making as negative aspossible.

Figure 3. Example of Invariance of Upper Surface Pressure Distribution with Underbody Variations

As also shown in Appendix A, equation (A6), can beresolved into two major components,

(3)

where the component coefficients are defined accordingto equation (2). The subscript f designates the under-body surface (L-N) upstream of the diffuser (including thefront radius, but referred to as flat for convenience) andd the diffuser of length N. The relative weighting of thetwo component pressure coefficients is determined by(N/L), the length of the diffuser relative to that of the body. Two different diffuser lengths were evaluated in the testprogram, a short one of (N/L) = 0.247 and a long one of(N/L) = 0.752. Figure 4 shows their underbody pressuredistributions for the non-lifting symmetric configurationshaving =0, at the relatively large ground clearance of(h1/H) = 0.338 that is the largest value tested for the longdiffuser.

Figure 4. Comparison of the Zero-Diffuser-Angle Pressure Distributions from the Short and the Long Diffusers; Ground Moving

The general form of the nearly identical pressure distribu-tions is characterised by a strong suction peak at thefront corner, produced by strong streamline curvature inthe flow around its circular-arc geometry. This is followedby a subsequent pressure relaxation that asymptotes tothe base pressure and is essentially completed by mid-body length. The relaxation process has a local reversalimmediately downstream of the front radius that isthought to be the consequence of a laminar separationbubble resulting from a strong adverse pressure gradientin the flow around the second half of the corner. While the two configurations are nominally identical ingeometry, they have a small difference. The hinge at thediffuser inlet that permits variation of the wall angle pro-trudes slightly from the otherwise flat underbody surfaceand is at a different streamwise location for the twocases. For the short diffuser, this hinge produces a localdistortion in the pressure immediately downstream ofitself that is clearly evident in Figure 4. It is not presentfor the long diffuser which does not have a surface dis-continuity at that streamwise location.There is a small difference in the local pressure reversaljust downstream of the front corner. This may also behinge related since the long diffuser has a hinge near thatregion while the short one does not. The generation of downforce by a symmetric, non-liftingbody as it is brought close to ground is illustrated in Fig-ure 5 for the short-diffuser configuration. Only data forthe short diffuser are used in the analysis of this paper. Itwas tested at 22 ride heights over the range 0.031(h1/H)0.646. The curves in the Figure are for different ride

puC

plC

plC

-2 .0

-1 .0

0 .0

1 .0

0 1 00 20 0 30 0 4 00

D ista n ce fro m F ro n t F a ce , m m

Pre

ss

ure

C

oe

ffic

ien

t

solid lin e, =0.00, (h1/ H)=0.338cha in -dashed line , =9.6 4, (h 1/ H)=0.338das hed line, =9.64, (h1/ H)=0.062

plC

pdpfpl CLNC

LN1C

+

=

-2.0

-1.0

0.0

1.0

0 10 0 2 00 3 00 4 00

D ista n ce fro m F ro n t Fa c e , m m

Pre

ss

ure

C

oe

ffic

ien

t

=0.0 0, (h 1/ H)=0.338so lid lin e, s hor t dif f u ser ; CL= -0.004 4dashed line, long dif f user; C

L= -0.058

star t of longdif f us er

end of radiu s

s ta rt of sh or td if f us er

bas epress ure

4heights. The lift coefficient at the largest ride height, (h1/H)=0.646, is close to zero, and essentially the same as infree air. Consequently, the corresponding pressure distri-bution is the baseline from which to assess downforceproduction as ride height, (h1/H), is reduced to very smallvalues (downforce due to ground proximity).

Figure 5. Effect of Ground Proximity on the Underbody Pressure Distribution with Zero Diffuser Angle; Ground Moving

The three curves in Figure 5 have similar characteristics,but noticeable and important differences. The magnitudeof the suction peak at the front corner is only slightlyaffected by ride height but the local pressure-reversalimmediately downstream of the leading-edge radius issignificantly reduced as the local underflow is con-strained by the approaching impervious ground. How-ever, the major effect is an increase in the distancerequired for the pressure to asymptote to the essentiallyinvariant base pressure. The relaxation process is one ofdeceleration, requiring the streamtube to expand withdownstream distance. This increase in area is con-strained by the ground as (h1/H) decreases. The area between curves at low ride height and that at(h1/H) =0.646, sensibly out of ground effect, representsthe downforce produced by ground proximity. This identi-fies the increased distance required for pressure recov-ery from the suction peak as the major mechanism ofdownforce production. The ground-proximity downforceis a maximum at (h1/H) = 0.062.The generation of downforce by an underbody diffuser asa body is brought close to ground is illustrated for theshort-diffuser configuration in Figure 6.

Figure 6. Effect of Ground Proximity with an Underbody Diffuser; Ground Moving

There are four curves for different ride heights, all forrear-underbody upsweep of 9.64, all having the samegeneral shape. As already discussed, at (h1/H)=0.646the body is essentially in free air. The upsweep producesa pressure recovery over its length. Since the base pres-sure is only slightly increased from its value for the sym-metric body, producing a small reduction in drag, theconsequence of this pressure-recovery process is areduction in the pressure at the beginning of theupsweep. The pressure relaxation from the front-cornersuction peak must decrease to this level and so a localmaximum is produced in the flat-underbody pressure,resulting in a downward-concave profile.As ride height is reduced, the combination of the upsweptunderbody and the flat ground plane forms a diffuser ofasymmetric geometry whose ratio of outlet area to inletarea (area ratio, AR) becomes increasingly greater thanunity. This generates greater and greater diffuser pres-sure recovery that, since the base pressure remainsnearly constant, increasingly depresses the pressure atthe diffuser inlet. This results in higher flow velocity overthe upstream flat underbody, lowering the pressurethroughout that region and contributing to increasingdownforce. The diffuser effectively "pumps down" the flatunderbody, producing the large downforce measured atlow ride heights. At very low ride heights the underbodyflow resistance increases due to viscous effects, causingless flow to enter the underbody so that the downforcemechanism created by the diffuser is curtailed.The area between any two (h1/H) curves in Figure 6 rep-resents the change in downforce produced with the asso-ciated change in ride height with a diffuser. However, notall of the downforce is due to diffuser pumping. As wasseen in Figure 5, changes in ride height alone, even with-out a diffuser, also produce changes in ground-proximitydownforce. The separation of these two mechanisms isillustrated in Figure 7.

-2 .0

-1 .0

0 .0

1 .0

0 10 0 20 0 30 0 4 00

D ista n ce fro m F ro n t F a ce , m m

Pre

ss

ure

C

oe

ffic

ien

t (h 1/H)=0.646(h1 /H)=0.100

(h 1/H)=0.062

=0 .0 0

-2.0

-1.0

0.0

1.0

0 1 00 20 0 30 0 4 00

D ista nc e from F ro nt F a ce , m m

Pre

ss

ure

C

oe

ffic

ien

t

(h 1 /H)=0.646(h1/H)=0 .1 00

(h1 /H)=0 .0 62

=9.64

(h1/H)=0.192

5Figure 7. Mechanisms of Downforce Generation; Ground Moving

The top two curves in Figure 7 are for the symmetricbody. The upper curve of this pair is for the free-air con-dition at (h1/H)=0.646, and the lower one for the value of(h1/H)=0.062 that produces maximum ground-proximity-induced downforce. The area between the curves repre-sents the downforce due to ground proximity. If the dif-fuser angle of 9.64 that produces maximum downforce isintroduced at the lower ground clearance, the bottomcurve results. The area between this curve and the oneimmediately above it represents the downforce due to dif-fuser pumping.The downforce mechanism called underbody upsweep(body camber) that was identified in [1] is not an indepen-dent mechanism when a body is close to ground.

ANALYTICAL MODEL OF VEHICLE DOWNFORCE

An analytical model of vehicle downforce that permitsexamination of the dependence of downforce on diffuserdesign will be developed. This requires a diffuser pres-sure-recovery map in the dominant variables of area-ratioparameter and non-dimensional length, and a method forits application. The stated application of this analysis isto the diffusers of flat-underbody racing cars, where thediffusers are typically less than one-half of the underbodylength. The analysis will be performed for only the mov-ing-ground boundary condition, since this is the on-trackstate.

The combination of equation (1) and Figure 3 indicatesthat changes in the lift coefficient of the simple bodyunder consideration are determined by changes in themean-effective underbody pressure coefficient. Refer-ring to equation (3), it is seen that this coefficient is com-prised of two components that are,

(3)

The more negative this mean-effective pressure coeffi-cient, the greater the downforce. Its evaluation requiresinformation on the component mean-effective pressurecoefficients for the diffuser, , and for the flat upstream

underbody, , in turn requiring a diffuser pressure-recovery map.

DIFFUSER MEAN-EFFECTIVE PRESSURE COEFFICIENT

The driving force behind the downforce generation pro-cess is the diffuser pressure-recovery performance, sothis is the first element of the analysis to be developed.The axial pressure distribution in a subsonic diffuser hasa characteristic non-linear shape that must be estab-lished so that its mean-effective pressure coefficient canbe determined. This distribution and an equation for themean-effective coefficient are developed in Appendix B(equation (B18)) for asymmetric, plane-walled, under-body diffusers in inviscid, incompressible, one-dimen-sional flow. The result is,

(4)

where is the pressure coefficient at diffuser exit (seeFigure B2 for geometry),

(5)

and is the overall pressure-recovery coefficient(equation (B16)),

(6)

For real flows in which the values of and areknown, it will be assumed that equation (4) suitablyreflects the non-linear behaviour of the pressure distribu-tion in underbody diffusers and, for the purposes of this

paper, provides an adequate approximation of .

It is assumed that can be best represented by thearea-averaged, base-pressure coefficient that was mea-sured. This was essentially constant at a value of for all the configurations tested. That is,

(7)

The diffuser pressure-recovery coefficient, , is a func-tion of diffuser area-ratio parameter, (AR-1), and non-dimensional length, (N/h1). Its evaluation for particularcombinations of underbody-diffuser geometry andground boundary condition (moving ground, fixed

-2 .0

-1 .0

0 .0

1 .0

0 1 00 20 0 30 0 4 00

D ista n ce fro m F ro n t F a ce , m m

Pre

ss

ure

C

oe

ffic

ien

t=0.00, (h 1 /H)=0.646

=0.0 0, (h1 /H)=0.062=9.64 , (h1/H)=0.062

dif fus e r p um ping

g r ou n d pr o xim ity

pdpfpl CLNC

LN1C

+

=

pdC

pfC

( )p

2ppd

C1

C11C

=

2pC

qppC 22p

pC

( )( )1p

1p2pp C1

CCC

=

2pC pC

pdC

2pC

19.0

19.0CC pb2p ==

pC

6ground) requires a performance map with these vari-ables. In the next section, two such maps will beextracted from the experimental data.

FLAT-UNDERBODY MEAN-EFFECTIVE PRESSURE COEFFICIENT

Once diffuser performance has been established, anexpression is required for the mean-effective, flat-under-body pressure coefficient that it induces. The axial pres-sure distribution of the flat underbody is morecomplicated than that of the diffuser. As has been seenin Figure 6, it is characterised by a strong suction peak asthe underflow curves and accelerates around the frontcorner of the body, followed by a downward-concavepressure-recovery profile that terminates at the diffuserinlet. As diffuser length increases, the length of the con-cave distribution decreases, as seen in Figure 8. Thecorresponding change in diffuser affects the magni-tude of the front-corner suction peak and, more signifi-cantly, the pressure level of the concavity.An analytical description of this pressure distribution andits integration to directly produce the flat-underbodymean-effective pressure coefficient is not feasible.Instead, the approach taken is to seek a correlation

between and , thereby relating the flat under-body pressures to the diffuser pressure recovery. Theexperimental program used only two diffuser lengths, andthis is insufficient for establishing such a correlation.Recourse has therefore been made to CFD for supplyingsuitable information. Computations with moving groundat (h1/H)=0.062 and (AR-1)=2.67 were made for eight dif-ferent diffuser lengths over 0.25(N/L)0.75. The resultsare shown in Figure 8.

Figure 8. CFD Predictions of Underbody Pressure Distributions; Ground Moving

For each diffuser length in Figure 8, the underbody pres-sure distribution was numerically integrated to provide

and . The ratio of these quantities for each dif-fuser length is shown in Figure 9.

Figure 9. CFD-Predicted Relationship Between

and as a Function of Diffuser Length; Ground Moving

This ratio was also calculated for the experimental datafrom the two diffuser lengths studied. For each length,the ratio of mean-effective pressures was insensitive toboth ride height (h1/H) and area-ratio parameter

. Consequently, although the data in Figure 9were derived for a single combination of ride height andarea-ratio parameter, they will be taken as representativeof all combinations of ride height and area-ratio parame-ter.

The data variation with length in Figure 9 is essentiallylinear, and adequately represented by,

(8)This correlation indicates that at any given area ratio, as(N/L) is increased from a small value to improve diffuser

and make its corresponding mean-effective pressurecoefficient more negative, the mean-effective pres-sure coefficient of the flat underbody becomes morenegative even faster. This is the consequence of thelength and pressure level of the downward-concave partof the pressure distribution along the flat componentbeing reduced (see Figure 8). Since the downward-con-cave pressure profile contains the most-positive portionof the flat-underbody pressures, these changes make thenet value of the flat-underbody component more nega-tive.

pC

pfC pdC

-3.0

-2.0

-1.0

0.0

1.0

0 10 0 2 00 30 0 4 00

Dista n ce fro m F ro n t F a ce , m m

Un

de

rbo

dy

Pre

ss

ure

C

oe

ffc

ien

t

0 .2 50.28

N/L = 0.75 0.59

0 .5 4

0.48

0.42

0.37

(h 1 /H)=0.06 2, ( A R- 1)=2.67

pfC pdC

y = 3 .61 x + 1.02R 2 = 0.9 9

1.0

2.0

3.0

4.0

0.0 0 .2 0 .4 0 .6 0 .8 1 .0

Diffu se r L e ng th F ra ction , (N /L )

(Fla

t-U

nd

erb

od

y/D

iffu

se

r) M

ea

n-

Effe

cti

ve

P

res

su

re C

oe

ffic

ien

ts

(h 1 /H)=0 .0 62, (AR-1)=2 .6 7

pfC

pdC

)1AR(

)L/N(61.302.1C/C pdpf +=

pC

pdC

pfC

7UNDERBODY MEAN-EFFECTIVE PRESSURE COEFFICIENT

Using the correlation of equation (8) in equation (3), themean-effective pressure coefficient of the whole under-body becomes,

(9)

The term in square brackets in this equation is positiveand has a local maximum of 1.91 at (N/L)=0.50.Introducing equations (4) and (7) into equation (9) resultsin the final expression for the total underbody mean-effective pressure coefficient,

(10)

As will be shown, if a diffuser map of is available, thisequation will permit an optimum underbody geometry tobe determined. For example, the best geometry irre-spective of diffuser length could be found. Alternatively,the best area ratio and ride height for a given diffuserlength, or the best area ratio for a given ride height anddiffuser length could be found.

DIFFUSER PRESSURE-RECOVERY PERFORMANCE

In [1], a performance map of diffuser-based downforcewas generated with coordinates of area-ratio parameter,

, and non-dimensional length, (N/h1), the so-called diffuser plane. This downforce map mimicked thediffuser pressure-recovery coefficient, . In the presentstudy, a pressure-recovery map is evaluated directly. It isderived from the pressure distributions measured on thecentre plane of the underbody. Only the short-diffuserdata is used in the development of this map. The long-diffuser data are excluded because the diffuser inlet flowinteracts with the front-corner flow.

The pressure-recovery coefficient can be expressed inthe following form,

(11)

where the subscripts 1 and 2 designate the diffuser inletand outlet, respectively, and the infinity symbol desig-nates a station far upstream of the body.

Numerical values for the three required inputs to equation(11) will be considered one at a time.As already discussed in the analytical model for the dif-fuser mean-effective pressure coefficient and stated inequation (7), is best represented by the area-aver-aged base-pressure coefficient, which is essentially con-stant at a value of 0.19.

In conventional diffuser research, a test diffuser is pre-ceded by a constant-area inlet duct along which the pres-sure gradient is very small. The inlet pressure to thediffuser is taken as the wall static pressure a short dis-tance upstream of the inlet, where the influence of flowcurvature resulting from the diverging flow at the inlet isnegligible. Even though the underbody flow path preced-ing the diffuser is of constant height in the present study,the pressure along it varies significantly. As was seen inFigure 6, there is a downward-concave pressure profileimmediately upstream of the diffuser. In view of this, itwas decided that the best measure of the inlet staticpressure coefficient, , would be obtained by extrapo-lating the downstream end of this profile to the diffuserinlet using a quadratic function. The inlet dynamic head of the diffuser, q1, was not mea-sured, and so has to be inferred. Considering the flowfrom far upstream of the body to the diffuser inlet,

giving,

(12)

where is the loss coefficient of this flow. Unfortu-nately, the value of is not known. In order to pro-ceed, it is assumed to be zero. With , theresulting inlet dynamic pressure coefficient, , isover-estimated and its usage in equation (11) yields,

(13)

which produces an under-estimate of . This conces-sion to practicality does not limit the use of the resultingdiffuser map in the search for maximum vehicle down-force. The major objective of the present study is thedetermination of optimum diffuser geometry, not thevalue of the corresponding maximum downforce coeffi-cient. The procedure that has been described was usedwith the measured underbody pressure distributions togenerate the results of Figure 10 for the short diffuser.For clarity, only five of the nine available diffuser anglesare shown.

pdpd

pd

pfpl CL

NCC

CLN1C

+

=

pd2

pl CLN61.3

LN59.302.1C

+=

+=

p

2pl

C1

19.11LN61.3

LN59.302.1C

pC

)1AR(

pC

( )

=

11p2p

1

12p qqCC

qppC

2pC

1pC

oloss

oloss

oo111 PqpPPPqp +=+

)CC1(qq

loss1p1

=

lossClossC

0Closs =)q/q( 1

( )( )1p

1p2pp C1

CCC

=

pC

8Figure 10. Diffuser Pressure-Recovery Coefficients for Fixed-Ground and Moving-Ground Simulations

The pressure-recovery coefficients for all diffuser angleswere converted to the diffuser plane using the transfor-mations of equations (14) and (15).

(14)

(15)

The results are plotted in the diffuser pressure-recoverycontour maps of Figure 11.

Figure 11. Diffuser Pressure-Recovery Coefficient Maps (Based on Data from Figure 10)

The maps for the two ground simulations are similar.There are, however, significant differences betweenthem, which can be summarised as: 1. the pressure-recovery contours with the moving

ground are open at the higher non-dimensionallengths, not closed as with the fixed ground,

2. the location of the point of maximum pressure recov-ery is different, with the maximum occurring at lowerarea-ratio parameter and higher non-dimensionallength with the ground moving.

Constant-pressure-recovery-coefficient contours nearmaximum pressure recovery for both ground simulationsare compared in Figure 12.

0 .0

0 .2

0 .4

0 .6

0 .0 0.2 0 .4 0.6

N o n -dim e n sio na l R id e He ig ht, (h 1/H )

Pre

ss

ure

-R

ec

ov

ery

Co

eff

icie

nt

2.87 deg 6.82 deg9.64 deg 13.50 de g15.5 9 deg

Ground f ixed

0.0

0.2

0.4

0.6

0.0 0 .2 0.4 0.6

N o n -d im e nsion a l R ide He igh t, (h 1 /H )

Pre

ss

ure

-R

ec

ov

ery

Co

eff

icie

nt 2.87 deg 6.82 deg

9.64 deg 13.5 0 deg15.59 deg

Ground mo ving

( )

=

111 h

HLN4.2

hH

HL

LNh/N

( )

= tanhN1AR1

0 5 10 15 20Non-Dimensional Length, (N/h1)

0

2

4

6

Area

Rat

ioP

ara

me

ter,

(AR

-1)

0.300

0.350

0.3750.4000.410

0.420

0.375

0.350

0.3000.250 0.200

0.100

Short-Diffuser Pressure RecoveryFixed Ground

0 5 10 15 20Non-Dimensional Length, (N/h1)

0

2

4

6

Area

Rat

ioP

ara

me

ter,

(AR

-1)

0.300

0.350

0.375

0.400 0.410

0.415

0.3750.350

0.3000.250 0.200 0.100

Short-Diffuser Pressure RecoveryMoving Ground

9Figure 12. Comparison of Pressure-Recovery Contours with Fixed and Moving Ground

The effect of ground simulation on optimum diffuser per-formance is primarily caused by differences in diffuserblockage. Velocity non-uniformity in cross sections of aninternal flow stream represents blockage. Such non-uni-formity in the flat-underbody region upstream of the dif-fuser, due to viscous flow effects, is increased by thesubsequent diffusion process, as described in [1]. Thedistortion in the moving-ground velocity profiles is smallerthan with fixed ground because the ground boundarylayer is reduced by ground motion. Consequently, withmoving ground the effective area ratio at any given geo-metric area ratio is always greater than that with fixedground. Thus, a given pressure recovery can beachieved at a smaller geometric area ratio with the mov-ing ground, explaining why the pressure-recovery contourin Figure 12 with the ground moving falls below that withthe ground fixed.Furthermore, at any given geometric area ratio, greatervalues of diffuser length can be tolerated with the groundmoving before blockage in the diffuser reduces the effec-tive area ratio to values for which the pressure recovery isdiminished, closing the contours. Such values of diffuserlength are not reached in the present data with theground moving, so the pressure-recovery contours areopen to the right.The pressure distributions on which the diffuser maps arebased were measured only on the centre plane of theunderbody. How well they represent the whole width ofthe body depends on the nature of the local flow at theside edges of the diverging wall and on the ratio of thewidth, W, to the height of the diffuser inlet, h1, that is, onthe aspect ratio of the diffuser inlet, where aspect ratio isdefined as,

(16)

For the maps of Figure 11,

(17)

In diffuser research, aspect ratios of about 10 are used inorder to have the internal flow approach two-dimensional-ity. For the current study, equation (17) indicates that thisvalue is achieved at (N/h1) =4.61. Most of the measure-ments in the maps of Figure 11 are in the region of (N/h1)4.61, where the diffuser aspect ratios are well over10. However, the presence of only partial side plates per-mits lateral flow into the diffuser, introducing flow three-dimensionality. In spite of this, the analyses of [1]showed that integration of the centre-line pressure distri-butions gave sectional lift forces that closely matched thebalance lift measurements. It would appear, perhaps for-tuitously, that the centreline pressures are representativeof the average distribution over the full width of themodel.

MAXIMUM VEHICLE DOWNFORCE

The analysis and rationale that have been presentedsuggest a strong correlation between diffuser pressure-recovery coefficient and the lift on the body. That this isthe case is shown in Figure 13.

Figure 13. Correlation of Lift Coefficient with Diffuser Pressure-Recovery Coefficient; Moving Ground

Here, the pressure-recovery coefficient and the lift coeffi-cient for the short-diffuser are plotted against ride heightfor the moving-ground simulation. When increaseslocally, becomes locally more negative (more down-force), and when decreases, becomes less nega-tive (less downforce). Each change in pressure-recoverycoefficient is mirrored by a reverse change in lift coeffi-cient.

0.0

1.0

2.0

3.0

4.0

0.0 5 .0 1 0 .0 1 5.0 2 0.0

N o n -Dim e n sio na l L e ng th , (N /h 1)

Are

a R

ati

o P

ar

am

ete

r, (A

R-1) be lt f ixed

be lt mov ing

40.0Cp ====

11 hN

NL

LW

hWA

=

1hN17.2A

-1 .00

-0 .75

-0 .50

-0 .25

0 .00

0 .25

0 .50

0 .00 0 .20 0 .40 0.60 0.80

Rid e He ig h t, (h1/H )

Lift

a

nd

Pre

ss

ure

-R

ec

ov

ery

C

oe

ffic

ien

ts

p r e s s u r e r e co ve r y

lift

(N/L) = 0.24 7 = 15.59 d e g.

pCLC

pC LC

10

A more detailed correlation is provided by the constant-lift-coefficient contours on the diffuser plane ((AR-1) vs.(N/h1)) presented in Figure 14. The lift contour plots canbe compared to the pressure-recovery plots of Figure 11.It is seen that the lift contours are similar to the pressure-recovery contours, but have their maximum magnitudesat lower values of area ratio and length, and are closed atlonger lengths for both ground simulations.

Figure 14. Contours of Constant Lift Coefficient; Short Diffuser

APPLICATION TO UNDERBODY DIFFUSER DESIGN

The preceding analytical model, summarised in equa-tions (6) and (10), offers the opportunity for examiningoptimum diffuser design. This requires a diffuser pres-sure-recovery map as a function of the dominant vari-ables, area-ratio parameter and non-dimensional length.It will be assumed that the variation of diffuser inflow con-ditions with decreasing length of the upstream flat under-body (increasing N/L) is sufficiently small that the diffusermaps of Figure 11 can be applied over a range of diffuserlengths. A requirement for this assumption to be valid isthat the diffuser entry flow not interact with the flow enter-ing the underbody region. If there is interaction, the entryconditions to the diffuser are particularly uncertain andthere can be an effect on diffuser pressure recovery.Consequently, at some diffuser length it is to be expectedthat this assumption will fail, although at what length isuncertain. The long, three-quarter-body-length diffuserfalls into the interaction category, so the upper-lengthlimit for general applicability of the analytical model isless than this. For the flat-underbody racing cars that arethe stated application of the analysis, diffusers are typi-cally less than one-half the underbody length, and it willbe assumed that the existing maps of Figure 11 are ade-quate to at least this length.Referring to equation (3), it is seen that the mean-effec-tive underbody pressure coefficient, proportional tounderbody downforce coefficient, is comprised of twocomponents.

(3)

When (N/L) is small, the flat component is the greatercontributor, both because is significantly more nega-

tive than and because (1-N/L) is greater than (N/L).As (N/L) increases at constant (AR-1) in order to increase

and make more negative, there is a twofold ben-efit to the diffuser component. The flat component, how-ever, has opposing effects because while becomesmore negative with (see equation (8)), its (1-N/L)weighting decreases. At some diffuser length, even-tually reaches a maximum and then decreases with fur-ther lengthening of the diffuser, and the opposing effectsin the flat component continue. It is likely that thesephysics will produce a minimum in at some value of(N/L).

0 5 10 15 20Non-Dimensional Length, (N/h1)

0

2

4

6

Are

aR

atio

Par

ame

ter,

(AR

-1)

-0.400

-0.500

-0.600

-0.700

-0.800

-0.60

0

-0.500

-0.400-0.300 -0.200

-0.750

-0.8

20

Short-Diffuser Lift CoefficientFixed Ground

0 5 10 15 20Non-Dimensional Length, (N/h1)

0

2

4

6

Are

aR

atio

Par

ame

ter,

(AR

-1)

-0.60

0

-0.700

-0.750

-0.8

00

-0.

820

-0.8

40

-0.7

50

-0.700

-0.600-0.500

-0.400-0.300

Short-Diffuser Lift CoefficientMoving Ground

pdpfpl CLNC

LN1C

+

=

pfC

pdC

pC pdC

pfC

pdCpC

plC

11

The preceding behaviour is captured in equation (10) ofthe analytical model.

(10)

The analytical model is more useful if the moving-ground,diffuser map of Figure 11 can be represented by a sur-face fit. This is outlined in the following, where the varia-tion of in the map has been fitted by,

(18)where the coefficients (a,b,c) are functions of (N/h1) inthe form,

(19)

The numerical values of the coefficients for theground-moving case are:

With this input, equations (18) and (19) can be used tocalculate the variation of with (N/h1) at various

, and this then used in equation (10) to deter-mine the variation of with (N/L). A typical result for(h1/H)=0.062 and (AR-1)=2.67 is plotted in Figure 15.The inclusion of (h1/H) is required for equation (14) whichtransforms (N/h1) in the diffuser pressure-recovery mapto (N/L).The diffuser map of Figure 11 only extends to a non-dimensional length, (N/h1), of approximately 20. For theride height of Figure 15, this transforms to (N/L)=0.52.Consequently, the curves in Figure 15 calculated fromthe analytical model are dotted for (N/L) greater than thisvalue to indicate that the diffuser map is being extrapo-lated.The diffuser pressure-recovery coefficient increasesmonotonically with increasing (N/L) up to (N/L)=0.40,after which it decreases slightly. As suggested to belikely, the analytical model for , equation (10), has alocal minimum corresponding to maximum downforce,and it occurs at (N/L)=0.46. The predictions of diffuser-length effects from equation (10) are compared with CFDresults in this Figure. The CFD computations were madefor eight different diffuser lengths over 0.25(N/L)0.75.The two curves are not the same, but are close at small

diffuser lengths, and both show a minimum. The CFDcurve is flatter, and has its minimum at (N/L)=0.37 (theshaded circle) instead of at (N/L)=0.46 from the analysis(unshaded circle). The reasons for the differences arenot known, but the lack of side plates and the absence oflaminar flow in the CFD simulation may be contributors.

In any event, the differences in the curves do notdetract from the utility of equation (10). Its major contri-bution is its delineation of the physics of downforce gen-eration.

Figure 15. Typical Dependence of Underbody Mean-Effective Pressure Coefficient on Diffuser Length; Moving Ground

Before proceeding further, it is instructive to analyse thecomponent effects in equation (10) that determine theminimum in the underbody mean-effective pressure coef-ficient . Using the case of and (h1/H)=0.062 for illustration, Figure 16 shows the variation ofthe three contributing factors, and their net effect, with (N/L).

The diffuser pressure-recovery coefficient, , is thedriver in the downforce generation process, and it has amaximum at (N/L)=0.31. The corresponding diffusermean-effective pressure coefficient, , is negative andhas a minimum at the same diffuser length. This is thesecond term of the product in equation (10). The qua-dratic pre-multiplier of that equation has a maximum at(N/L)=0.50 and it is not explicitly dependent on diffuserperformance. The product of the two terms is and ithas a minimum at (N/L)=0.41, which is between theextrema of its two contributors. This diffuser length formaximum downforce (most-negative ) is not coinci-dent with that for maximum diffuser pressure recovery,but is greater than it in this example. The latter seemscontrary to intuition and therefore warrants elaboration.

Table 1. Coefficients for Variation of with (AR-1) and (N/h1); Moving Ground

x(N/h1) x1 x2 x3 x4a 0.27 1.40 0.75 1.00b -0.20 -45.00 1.50 3.40c 1.40 3.00 0.03 2.50

+=

p

2pl

C1

19.11LN61.3

LN59.302.1C

pC

( )( ) ( )c1p )1AR(bexp)1AR(ah/N),1AR(C =

4x13

211 )h/N(x1

xx)h/N(x

++=

ix

pC

pC)1AR(

plC

plC

plC

-1.5

-1.0

-0.5

0.0

0.5

0 .00 0 .25 0 .50 0.7 5

Diffu se r L e n gth F ra ctio n, (N /L )

Un

de

rbo

dy P

res

su

re C

oe

ffic

ien

ts

(h 1 /H)=0 .0 62, (A R-1)=2.67

me

an

-e

ffec

tive

u

nde

rbo

dypr

es

su

re c

oe

ffic

ien

t

d if f us er p ress urerecov ery, eqn. (18)

f rom CFD

analy tica l mode l eqn . (10)

plC 0.2)1AR( =

pC

pdC

plC

plC

12

Figure 16. Breakdown of Component Effects in Underbody, Mean-Effective Pressure Coefficient; (AR=1)=2.0, (h1/H)=0.062, Moving Ground

When (N/L) is increased just past the value for which is a maximum, decreases and becomes lessnegative, which is unfavourable for downforce production.However, the correlation of equation (8) indicates that thisis offset by the mean-effective pressure coefficient, ,of the flat underbody continuing to become more nega-tive. Initially, this happens at a faster rate than becomes less negative, and so the quadratic pre-multi-plier continues to increase, which is favourable for down-force production. The net result is that the mean-effective pressure coefficient for the whole underbody,

, continues to become more negative past the lengthfor maximum diffuser pressure recovery. Eventually,however, with further increase in (N/L), the unfavourablereducing negativity of dominates and passesthrough its minimum value and begins to increase (lessdownforce).The results presented in Figure 15 are for only one arearatio and for only one ride height. The analytical modelcan also examine the effect of changes in area ratio andride height, as is shown in Figures 17 and 18.Figure 17 presents the variation of underbody mean-effective pressure coefficient with area ratio and lengthfor two ride heights. The area-ratio curve of maximumdownforce is shown in bold in each graph. This area ratiois different for each ride height, but is near in both cases, and the optimum lengths are only slightlydifferent.

Calculations like those in Figure 17 were made at twoadditional ride heights. For each of them, the area ratiogiving the minimum mean-effective underbody pressurecoefficient (maximum downforce) was found and plottedin Figure 18.The optimum diffuser length changes as ride height andarea ratio vary, but is generally near (N/L)=0.5. The opti-mum area ratio increases with decreasing ride height.The magnitude of the maximum downforce (minimummean-effective pressure coefficient) increases monotoni-cally with increasing area ratio up to andthen decreases for larger area ratios.

Figure 17. Effect of Area Ratio on Optimum Diffuser Geometry for Two Ride Heights; Moving Ground

-2.0

-1.0

0.0

1.0

2.0

0.0 0 .2 0 .4 0.6Diffu se r L e ng th F ra ctio n, (N /L )

An

aly

tica

l-M

od

el

Co

mp

on

en

ts

qu adra tic pre-multiplier , eqn . (10)

p ress ure- rec ove ry c oef ., e qn. (1 8)

dif f use r mea n-ef f ec tivepre ssu re coe f., e qn. (4)

u nde rbod y me an-e ff e ctivep ress ure coe f ., eqn . (10 )

(h 1 /H)=0.06 2, (A R-1 )=2.00

pC

pC pdC

pfC

pdC

plC

pdC plC

0.2)1AR( =

02.2)1AR( =

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0 .00 0 .25 0 .50 0 .75

D iffu se r L e n gth F ra ctio n, (N /L )

Un

de

rbo

dy M

ea

n-E

ffe

cti

ve

Pre

ss

ure

C

oe

ffic

ien

t(h 1 /H)=0.062

(AR-1) = 7.50

3.50

2.67

op t2.20

1.30

5.00

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0 .00 0 .25 0 .50 0 .75

D iffu se r L e n g th F ra c tio n , (N /L )

Un

de

rbo

dy M

ea

n-E

ffe

cti

ve

Pre

ss

ure

C

oe

ffic

ien

t

(h 1 /H)=0.100

(A R-1 ) = 7.50

5.00

3.50

2.67

o pt.2.02

1.30

13

Figure 18. Effect of Ride Height on Optimum Diffuser Geometry; Moving Ground

DISCUSSION

The analysis and the analytical model that have beenpresented capture the underlying physics of the down-force generated by underbody diffusers. The under-standing that they provide should be generally useful inguiding vehicle design. However, the assumptions andconstraints involved need to be clearly appreciated, andare collected here for easy reference.The caveats for the analytical model are the following:1. It is assumed that the streamwise pressure distribu-

tion p(x) in a real diffuser flow between prescribedinlet and outlet pressures is approximately the sameas for an ideal flow between the same pressures.

2. It is assumed that the underbody front-corner suctionpeak of the simple test model is reasonably repre-sentative of actual racing cars and that it does notinteract directly with the diffuser flow.

3. It is assumed that the correlation for the ratio of themean-effective pressure coefficient of the flat under-body to that of the diffuser, equation (8), derived fromCFD results is adequately representative of experi-ment.

The analytical model requires a diffuser pressure-recov-ery map. The moving-ground map that has beenextracted from the experimental data is, to the authorsknowledge, the first for a diffuser with one wall movingrelative to the other. Although based on data for only thediffuser length (N/L) = 0.25, it is assumed to be adequatefor determining optimum performance even though:4. it only extends to a non-dimensional diffuser length

(N/h1) of approximately 20.

5. the diffuser pressure-recovery coefficients, , areunder-predicted because of the manner in whichtheir normalising inlet dynamic head q1 is deter-mined.

6. there are viscous effects included in the region (N/h1)>10.

While it is too optimistic to expect the model to exactlypredict the optimum underbody performance for arbitrarycombinations of diffuser length, area ratio and rideheight, it should give values that are nearly correct.Therefore, it can serve to narrow the range of diffusergeometries that need to be evaluated experimentally. Italso offers a technique for generalising diffuser perfor-mance measurements so that they can be used in thevicinity of the experimental domain to evaluate diffuserdesign variations that were not tested.

CLOSING REMARKS

Underbody diffusers are used on racing cars to generatelarge downforce that will permit them to achieve reducedlap times through aerodynamically-enhanced traction.Both the configuration of these cars and their underbodyflows are complex, so the design of optimum underbodygeometries is a formidable task. The objective of thepresent study is to generate data and an improved under-standing of the relevant physics of underbody-diffuser-induced flows. The application of this knowledge throughthe numerical analysis presented can provide genera-lised design guidance that should facilitate effectiveunderbody design.The approach taken is one that is traditional in the studyof complex problems: to identify a less-complex but stillrelevant sub-problem that has the key elements and flowphysics of the main one, and study it to generate a firstphase of cause-and-effect relationships. In addition tohaving immediate utility, it can serve as the foundationupon which future research activity can be built.The general objective of the present study is to generateinformation that will facilitate the selection of optimumlength and area ratio for the underbody diffusers of flat-bottomed racing cars, reducing the experimental effortrequired to develop underbody configurations that pro-duce large downforce. Specific objectives are to demonstrate that optimum dif-fuser geometries exist, with the details of the geometriesbeing a function of regulatory and physical constraints,and to develop a semi-empirical, mathematical model ofthe underbody flow that will permit such optimae to bepredicted. The design problem for any flat underbodycould be to identify the best diffuser subject to no restric-tions, or to do so subject to diffuser-length and/or ride-height constraints imposed by a particular set of racingrules. The approach taken is to model the downforce

-1 .2

-1 .0

-0 .8

-0 .6

-0 .4

-0 .2

0.00 0.25 0.50 0.75

Di ffu se r L e ng th Fr a c tion , (N /L)

Me

an

-E

ffe

cti

ve

Un

der

bo

dy

Pre

ss

ure

C

oe

ffic

ien

t

0.062, 2 .2 0

0.250, 1.13

0.150, 1 .7 0

0.100 ,2.02

(h1/H), (A R-1 )

pC

14

generation of a simple diffuser/body combination andthen, based on measurement, to generalise the experi-mental diffuser pressure-recovery characteristics as aninput to this model.The analytical model has predicted the best diffuserlength to be of order one-half the length of the underbody,or less, thereby demonstrating that effective underbodiescan be produced with relatively short diffusers. Also, itprovides insight into the underbody geometries requiredfor best downforce production. For example, it showsthat optimum diffuser length reduces as area ratiodecreases and that optimum area ratio increases as rideheight decreases.The analytical model offers an improved understanding ofthe role of the diffuser in downforce generation and canbe applied by other researchers. If the appropriate forceand pressure measurements are made during diffuserdevelopment, they may then be used to search the nearvicinity of the test domain for a better configuration thanthose tested.This first iteration of underbody downforce prediction pro-vides a preview of what may be possible with more infor-mation. It was shown in [1] that body pitch angle has a

strong effect on downforce generation, and it is alsoknown that the introduction of a vortex into a diffuser canenhance its performance. These techniques are utilisedin racing car design but have not yet been studied in sim-ple geometries like the current one. Their study couldprovide topics for future research.

REFERENCES

1. K. R. Cooper, T. Bertenyi, G. Dutil, J. Syms, G. Sov-ran - The Aerodynamic Performance of AutomotiveUnderbody Diffusers, SAE 980030, InternationalCongress and Exposition, Detroit, MI, USA, Feb.1998.

CONTACT

The first author can be contacted at:The National Research CouncilBuilding M-2, Montreal Rd.Ottawa, Ont.Canada K1A 0R6Telephone (613) 993-1141, Fax (613) 957-4309e-mail Kevin.Cooper@nrc.ca.

APPENDIX A: LIFT COEFFICIENT

Assuming that the centre-line pressures on the upperand lower surfaces of the model adequately representthe average pressures across the width of the body, W, atall of its cross sections, as was shown in [1], then theaerodynamic lift on a vehicle equipped with an under-body diffuser is,

(A1)

where pl and pu are the local static pressures on thelower and upper surfaces, respectively. Reducing the liftto coefficient form,

So,

(A2)

where pressure coefficient is defined as,

(A3)

The distance-averaged, mean-effective pressure coeffi-cient over a streamwise length xi is defined as,

(A4)

Introducing this into equation (A2) gives the final liftexpression,

(A5)

The mean-effective pressure coefficient on the lower sur-face can be divided into two components, one for themostly flat under-body and one for the diffuser,

(A6)

where the subscript f denotes the total underbody sur-face of length (L-N) upstream of the diffuser and ddenotes the diffuser of length N.

===L

0u

L

0l dx)x(pdx)x(pW)Wdx(ppdaL

==

L

0u

L

0lL dx)pp(dx)pp()WH(q

W)WH(qC

L

= Lo puLo plL dx)x(Cdx)x(CH1C

q

ppC ipi

iX0 pipi dx)x(Cx1C

= puplL CCH

LC

+

L

NLpl

NL

0plpl dx)x(Cdx)x(CL

1C

pdpfpl CLNC

LN1C

+

=

15

APPENDIX B: DIFFUSER MEAN-EFFECTIVE PRESSURE COEFFICIENT

Consider an inviscid, incompressible, one-dimensionalflow in a rectilinear diffuser of any wall geometry, as inFigure B1.

Figure B1. Two-Dimensional Diffuser

Ignoring any losses, the local conditions at any stream-wise station x within the diffuser are related to those atinlet by,

Thus,

(B1)

Applying continuity,

(B2)

Substituting this into equation (B1) gives the ideal pres-sure distribution,

(B3)

Applying equation (B3) at the outlet, station 2, providesthe ideal, diffuser pressure-recovery coefficient,

(B4)

The distance-averaged, mean-effective pressure over adiffuser of length N is defined as,

(B5)

This mean-effective pressure can be obtained from themean-effective value of the pressure coefficient on theleft-hand side of equation (B3) in the following fashion,

(B6)

The diffuser in the present study is at the rear under-bodyof the simple test model and is plane-walled and asym-metric, as sketched in Figure B2.For this diffuser,

(B7)

and its area ratio is,

(B8)

Figure B2. Geometry of an Underbody Diffuser

The integral in equation (B6) can be evaluated usingequations (B3), (B7) and (B8),

resulting in

(B9)

Using equation (B9) to replace the left-hand side of equa-tion (B6),

(B10)

111 q,p,A222 q,p,A

x

N21

)x(q),x(p),x(A

11o1

o qpPP)x(q)x(p +=+

=

11

1q

)x(q1q

p)x(p

21

2

11 )x(AA

V)x(V

q)x(q

=

=

( )2111

A/)x(A11

qp)x(p

=

( )

==

22121

12pAR

11AA11

qppC

No dx)x(pN1p

dxq

p)x(pN1

qpp N

0 1

1

1

1

=

=

1

11

1 qpp

NNpp

q1

+=

=

tanhx1

h)x(h

A)x(A

111

+=

=

= tanhN1

hh

AAAR

11

2

1

2

111 q,p,hground plane

q,p

222 q,p,h

)x(q),x(p),x(hx

N21

L

H

dx)A/)x(A(

11N1

qpp N

0 211

1 = [ ] dxtan)h/x(1

11N1 N

0 21 +=

=

AR11

qpp1

1

( )

=

=

1

1

1

1qq

qpp)pp(

qpp

AR11

16

Define the coefficient of mean-effective pressure over thelength of the diffuser as,

(B11)

Using this in (B10),

(B12)

Relating conditions at the diffuser inlet, station 1, to thosefar upstream of the model,

so

(B13)

Using this equation in equation (B12) and solving for ,

(B14)

This is the desired result. However, for the purposes ofthis study, an alternative expression in terms of the vari-

ables and is more useful because these are theavailable experimentally-measured quantities.Area ratio can be eliminated from equation (B14), usingequation (B4),

(B15)

while can be eliminated by recourse to equations(B4) and (B13),

(B16)

leading to,

(B17)

Substituting equations (B15) and (B16) into equation(B14) provides the final form of the expression for themean-effective diffuser pressure coefficient, where thesubscript d has been added to denote a diffuser value.

(B18)

qppCp

=

11pp q

qCCAR11

=

qq1

qpp 11

( )1p1 C1qq

=

pC

( )1pp C1AR11C

=

pC 2pC

pC1AR1

=

1pC

( ) ( )( )1p

1p2p

1

12p C1CC

qq

qpp)pp(C

=

( ) ( )( )p2p

1p C1C1

C1

=

( )p

2ppd

C1

C11C

=