A Real-Time Method for Estimating Viscous Forebody … A Real-Time Method for Estimating Viscous Forebody Drag Coefficients Stephen A. Whitmore and Marco Hurtado NASA Dryden Flight

NASA/TM-2000-209015

A Real-Time Method for EstimatingViscous Forebody Drag Coefficients

Stephen A. Whitmore

and Marco HurtadoNASA Dryden Flight Research CenterEdwards, California

Jose RiveraUniversity of South FloridaTampa, Florida

Jonathan W. NaughtonUniversity of WyomingLaramie, Wyoming

January 2000

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NASA/TM-2000-209015

A Real-Time Method for EstimatingViscous Forebody Drag Coefficients

Stephen A. Whitmore

and Marco HurtadoNASA Dryden Flight Research CenterEdwards, California

Jose RiveraUniversity of South FloridaTampa, Florida

Jonathan W. NaughtonUniversity of WyomingLaramie, Wyoming

January 2000

National Aeronautics andSpace Administration

Dryden Flight Research CenterEdwards, California 93523-0273

NOTICE

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Available from the following:

NASA Center for AeroSpace Information (CASI) National Technical Information Service (NTIS)7121 Standard Drive 5285 Port Royal RoadHanover, MD 21076-1320 Springfield, VA 22161-2171(301) 621-0390 (703) 487-4650

A REAL-TIME METHOD FOR ESTIMATINGVISCOUS FOREBODY DRAG COEFFICIENTS

Stephen A. Whitmore* and Marco Hurtado†

NASA Dryden Flight Research CenterEdwards, California

Jose Rivera‡

University of South FloridaTampa, Florida

Jonathan W. Naughton§

University of WyomingLaramie, Wyoming

Abstract

This paper develops a real-time method based on thelaw of the wake for estimating forebody skin-frictioncoefficients. The incompressible law-of-the-wakeequations are numerically integrated across theboundary layer depth to develop an engineering modelthat relates longitudinally averaged skin-frictioncoefficients to local boundary layer thickness. Solutionsapplicable to smooth surfaces with pressure gradientsand rough surfaces with negligible pressure gradients arepresented. Model accuracy is evaluated by comparingmodel predictions with previously measured flight data.This integral law procedure is beneficial in that skin-friction coefficients can be indirectly evaluated in real-time using a single boundary layer height measurement.In this concept a reference pitot probe is inserted into theflow, well above the anticipated maximum thickness ofthe local boundary layer. Another probe isservomechanism-driven and floats within the boundarylayer. A controller regulates the position of the floatingprobe. The measured servomechanism position of thissecond probe provides an indirect measurement of bothlocal and longitudinally averaged skin friction.

1American Institute of Aero

*Aerospace Engineer, Associate Fellow, AIAA.†Co-operative Education Engineering Student Intern,

Aerodynamics Branch.‡Engineering Student Intern, Department of Mechanical

Engineering.§Professor, Department of Mechanical Engineering, Member

AIAA.Copyright 2000 by the American Institute of Aeronautics and

Astronautics, Inc. No copyright is asserted in the United States underTitle 17, U.S. Code. The U.S. Government has a royalty-free licenseto exercise all rights under the copyright claimed herein forGovernmental purposes. All other rights are reserved by the copyrightowner.

Simulation results showing the performance of thecontrol law for a noisy boundary layer are thenpresented.

Nomenclature

control equation coefficient

B law of the wake bias parameter

Bk control equation coefficient

base drag coefficient

total drag coefficient

CF longitudinally averaged skin friction

coefficient,

Ck control equation coefficient

cf local skin friction coefficient

local skin friction coefficient calculated by the nonlinear solution method

local skin friction coefficient calculated by the closed-form solution method

estimated skin friction coefficient

C0 curve fit bias coefficient

C1 curve fit slope coefficient

dx longitudinal integration variable

dy normal integration variable

rough surface, skin friction ratio parameter,

F law-of-the-wake velocity defect function, m/sec

Ak

CDbase

CDtotal

CF1x--- c f xd

0

x∫=

c f nonlinear

c f closed form

CF

Gxκ s-----

Gxκ s-----

c f

CF-------=

nautics and Astronautics

K gain parameter

k discrete time index

kPa kilopascals

Medge Mach number at edge of boundary layer

m boundary layer velocity profile exponent

n control gain parameter

local dynamic pressure within boundary layer, kPa

dynamic pressure at edge of boundary layer, kPa

steady-state dynamic pressure ratio

promeasured dynamic pressure, kPa

reference dynamic pressure, kPa

steady dynamic pressure in steady boundary layer

unsteady dynamic pressure in intermittent boundary layer

Rex Reynolds number based on longitudinal coordinate

transition Reynolds number

t time

U time-averaged longitudinal velocity in boundary layer, m/sec

Ue velocity at edge of boundary layer, m/sec

u longitudinal velocity in boundary layer, m/sec

steady-state velocity ratio

u’ unsteady perturbation of longitudinal velocity, m/sec

u+ nondimensional boundary layer velocity,

V time-averaged normal velocity in boundary layer, m/sec

v normal velocity in boundary layer, m/sec

v’ unsteady perturbation of normal velocity, m/sec

W time-averaged lateral velocity in boundary layer, m/sec

w longitudinal lateral in boundary layer, m/sec

w’ unsteady perturbation of lateral velocity, m/sec

x longitudinal coordinate, cm

y normal boundary layer coordinate, cm

yfinal steady-state probe position, cm

yk current probe position, cm

yk+1 commanded probe position, cm

yk–1 old probe position, cm

ylam normal displacement within laminar sublayer, cm

probe position in boundary layer, cm

ytrans point of transition between laminar and turbulent layers

y+ nondimensional normal boundary layer

coordinate,

normal point of transition between laminar and turbulent layers

z lateral boundary layer coordinate, cm

angle of attack, deg.

Clauser nondimensional pressure gradient parameter

intermittency factor

differential pressure, kPa

pressure error feedback, kPa

frame interval, sec

boundary layer thickness, cm

boundary layer thickness, noisy simulation, cm

estimate of boundary layer thickness, cm

boundary layer displacement thickness, cm

laminar contribution to boundary layer displacement thickness, cm

turbulent contribution to boundary layer displacement thickness, cm

controller damping ratio

uniformly distributed random noise function

q

qe

qqe-----

qmeas

qref

qsteady

qunsteady

Rextr

uUe------

final

u+ u y( )

Ue

c f

2-----

-----------------=

yprobe

y+ y

x--Rex

c f

2-----=

y+

trans

α

β

γ

∆p

∆q

∆t

δ

δnoisy

δest

δ*

δlam*

δturb*

ζ

η t( )

2American Institute of Aeronautics and Astronautics

boundary layer momentum thickness, cm

laminar contribution to boundary layer momentum thickness, cm

turbulent contribution to boundary layer momentum thickness, cm

law of the wake velocity distribution slope parameter

surface roughness height, cm

non-dimensional surface roughness,

dummy variable of integration

effective normal height of laminar sub-layer, cm

mean deviation in CF

wake parameter

standard deviation in CF

controller natural frequency, sec–1

longitudinal pressure gradient, kPa/m

longitudinal momentum thickness gradient

Mathematical Operators

first derivative with respect to time

second derivative with respect to time

Introduction

Lifting body and wing body designs currentlyadvocated for transatmospheric reusable launch vehicles(RLV) or for crew return from the space station arederived from variations on the original lifting bodyconcept.1 For a variety of reasons, these designs havelarge base areas relative to the vehicle size whencompared to previous hypersonic designs. For example,on the X-33 and VentureStar™ (Lockheed Martin,Bethesda, Maryland) configurations the large base areasare required to accommodate aerospike engines.Conversely, the X-38—derived from the original X-24Amold lines—has a large blunt base area resulting fromthe upper body flap required to trim the vehicle. In bothof the above cases, the base area flow is highly separatedresulting in large negative base pressure coefficients.Because of the large base-to-wetted-area ratios of thesevehicles, the base drag makes up the majority of theoverall vehicle drag. Thus the base drag has a largeimpact on critical vehicle performance parameters suchas maximum payload and maximum cross-range. Any

decrease in base drag has the potential to significantlyimprove the overall vehicle performance. An increase inthe vehicle lift-to-drag ratio would allow for a much lesssevere glide slope during the terminal phase of thevehicle reentry. Reduction in glide-slope steepnesswould in turn make the terminal area energymanagement task considerably easier.

Early Work Points to a Possible Solution

There is a feasible means for decreasing base drag.For blunt-based objects with base areas that featureheavily separated flows, a clear relationship isdemonstrated between the base drag and the viscousforebody drag.2, 3 Figure 1 shows the base dragcoefficient plotted against the viscous forebody dragcoefficient. The viscous forebody drag coefficient isdefined as the sum of all viscosity-induced dragcomponents on the vehicle forebody. These componentsinclude skin friction, forebody separation, forebodywakes, and parasite drag due to protuberances. Thetrend presented in figure 1 shows that as the forebodydrag is increased the base drag of the vehicle generallytends to decrease. This base-drag reduction is a result ofboundary layer effects at the base of the vehicle. Theshear layer caused by the free-stream flow rubbingagainst the dead, separated air in the base region acts asa “jet pump”2 and serves to reduce the pressurecoefficient in the base areas. The surface boundary layeracts as an insulator between the external flow and thedead air at the base. As the viscous forebody drag isincreased, the boundary layer thickness at the aft end ofthe forebody increases—reducing the effectiveness ofthe pumping mechanism—and the base drag canpotentially be reduced.

Flight Tests on SR-71 LASRE Experiment

Recent work performed on the Linear AerospikeSR-71 Experiment (LASRE) flight program hasdemonstrated that the trend presented in figure 1 can bepractically applied to reduce the base drag of a largeprojectile.4 In the LASRE experiment, a portion of theforebody of the test model was roughened to increasethe forebody drag. A decrease in base drag was observedat all Mach numbers—subsonic, transonic andsupersonic.

Trade-Off Between Forebody Drag and Base Drag

More significantly, figure 1 data show that projectileswith a base drag coefficient greater than 0.30(referenced to the base area) must necessarily lie on therelatively steep segment of the curve presented. In this

θ

θlam*

θturb*

κ

κ s

κ+

λ

λ trans

µ

Π

σ

ωn

∂p∂x------

∂θ∂x------

˙( )˙( )


region a small change in the forebody friction dragshould result in a relatively large change in the basedrag. Conceptually, if the added change in forebody skindrag is optimized with respect to the base drag, then itmay be possible to reduce the overall drag of theconfiguration. That is, based on Hoerner’s data, anoptimal drag bucket should exist. Figure 2 shows thisoptimal drag region. Here the base drag and the totalvehicle drag are plotted against the viscous forebodydrag including measured data for several hypersoniclifting-body and wing-body configurations:3 These arethe X-15, M2-F1, M2-F2, Shuttle, HL-10, X-24A,X-24B, and the SR-71 LASRE. The simple model infigure 2 is presented only as an illustration of the drag-bucket concept. Clearly, the forebody pressure profileand the presence of induced drag and localizedinterference or flow separation drag will likely alter theshape of the optimal curve presented in figure 2. Thechallenge is to modify the forebody drag coefficient toproduce the minimum overall vehicle drag in a real-world configuration.

On the Need for Real-Time Forebody Viscous Drag Measurements

An intriguing concept for implementing the dragoptimization described above utilizes recent advances inmicroelectromechanical systems (MEMS)5 technology.In this concept MEMS technology is used to adaptivelyvary the surface roughness in order to track the optimalskin friction. This design uses a “fish-scale” pattern ofmicroscale actuators distributed along the vehicle skin.The position of these microscale surface panels iscommanded in real-time to modify the local surfaceroughness and increase or decrease the total forebodyviscous drag coefficient, as required, to achieve theoverall drag minimum. This optimization processrequires that the viscous forebody drag coefficient besensed and available for feedback to the MEMScontroller in real-time. To be practical for theoptimization problem, the forebody drag measurementdevice must be self-contained; that is, the measurementsystem must be installed entirely onboard the vehicle.Additionally, the device must be able to adapt to widelyvarying boundary layer thickness.

None of the existing methods for sensing the surfaceskin friction are easily adaptable to the aboveoptimization problem. This paper briefly examines whythere are limitations with the current skin-frictionmeasurement techniques and then proposes a newmethod for sensing the viscous forebody dragcoefficient based on the law of the wake.6 Theengineering model developed for this new method

allows longitudinally averaged skin-frictionmeasurements to be made in a direct manner with asingle measurement of the boundary layer thicknessnear the aft end of the forebody. For simplicity, allanalyses and data presented assume incompressible flowrelationships. It is simply a matter of detail to extendthese discussions to account for the effects ofcompressible flow.

Note that use of trade names or names ofmanufacturers in this document does not constitute anofficial endorsement of such products or manufacturers,either expressed or implied, by the National Aeronauticsand Space Administration.

Background

This section briefly discusses existing methods forsensing surface skin friction and presents the reasonswhy these methods are unsatisfactory for our currentmeasurement problem. Current methods for measuringsurface skin friction can be grouped into two classes:direct measurements and indirect measurements.

Direct Skin Friction Measurements

The direct measurement methods sense surfaceshearing stress by using small floating balances orthrough the use of surface-imaging interferometry. Thebalance methods are generally self-contained and havebeen successfully applied to flight vehicles.7, 8

Unfortunately, experience has demonstrated thatsurface-mount skin-friction gauges are also susceptibleto damage and thermal shifts, and they require frequentre-calibration to achieve accurate results.

Oil-film interferometry has been used to produce veryaccurate localized skin-friction measurements;9

however, these techniques are usually applied only in alaboratory setting. Because they require remote opticalimaging of the flow field, interferometry methods areinherently not self-contained and are not easily adaptedfor use on flying vehicles. The rapid development offiber-optic sensors may provide a practical method toovercome this problem. Developing these opticallysensed skin-shear sensors to be rugged and reliable willbe a challenge for some time yet to come.

Indirect Skin Friction Measurements

Two of the most common indirect methods used tocompute skin-friction coefficients on flight vehicles arePreston tubes and boundary-layer rakes. Both types ofsensors are reasonably easy to fabricate and install, andoffer the advantage of being self-contained, reliable, and


rugged. Unfortunately, both designs have limitationsthat restrict their real-time applicability and make themgenerally unsuitable for the drag optimization problem.

Preston Tubes10, 11 offer a simple hardware solutionwhere a single pitot tube is located as close to thesurface as possible. By assuming a log-law velocityprofile,12 and using measurements of the local wallstatic and surface pitot pressure, the local skin-frictioncoefficient can be calculated. These methods have beensuccessfully applied to a large number of flight andwind tunnel tests. However, the drag optimizationproblem requires that the entire viscous forebody dragcoefficient—which includes the effects of skin friction,forebody separation, forebody wakes, and parasite dragdue to protuberances—must be sensed. Unfortunately,the log-law velocity distribution is not valid in the outerlayers of the boundary layer or within the wake region.Thus it is believed that Preston tubes will not be able tocompletely capture the range of required forebody dragmeasurements. It is possible that the Preston tubeanalyses can be modified to allow for the effects of theouter wake region; however that analysis is beyond thescope of this paper.

Boundary-layer rakes consist of an assembly withmultiple pitot probes placed in a normal line away fromthe skin of the vehicle. Boundary-layer rakes sense thelocal dynamic pressure profile within the boundary layerand use these data to compute the velocity distribution.This velocity distribution is numerically integrated toproduce a measurement of the local momentumthickness.12

In order to obtain accurate momentum thicknessmeasurements using a boundary-layer rake, it is criticalthat the rake be designed to obtain adequate normalresolution within the critical shearing layers near thesurface where the momentum defect distributionreaches a peak value. This “hump” in the momentumthickness curve occurs near the interface of thelaminar sublayer with the outer turbulent log-lawvelocity distribution layer. This resolution requirementmeans that a large number of measurements, typically6–10 probes, must be made in order to sense thevelocity profile with a sufficient amount of resolution.This large amount of data must be sampled andintegrated in real time. Additionally, the frame-by-frameresults must be heavily post-processed to eliminate theeffects of local boundary noise. While these limitationscertainly do not preclude the use of boundary-layerrakes for real-time forebody drag measurements, it isbelieved that these results can be achieved by a morestraightforward measurement technique.

A New Measurement Approach

The real-time measurement technique beingdeveloped in this paper exploits design features of bothPreston tubes and boundary-layer rakes. In thisapproach a reference pitot probe is inserted into theflow, well above the maximum anticipated thickness ofthe local boundary layer. Another pitot probe is mountedon a servomechanism which moves in a directionnormal to the surface. A controller regulates the positionof the servomechanism based on the measured pressuredifference between the moving pitot probe and thereference pitot probe. This design concept is picturedschematically in figure 3. The design will be tailored totypically allow the probe to track the area within theouter wake region near the edge of the boundary layer.A position sensor on the probe gives a measure of thelocal boundary layer thickness. This probe position isrelated to the forebody skin-friction coefficient using alaw-of-wake analysis. This analysis is presented in thenext section. The crux of this analysis is thedevelopment of an integral relationship that describesthe skin-friction coefficient in terms of a localmeasurement of the boundary layer thickness.

Integral Law of the Wake Analysis

The key to the floating-probe concept described in theprevious section is Coles’ law of the wake.6 The law ofthe wake relates the outer (turbulent) boundary-layervariables to the nondimensionalized boundary-layercoordinates

(1)

where,

(2)

is the local boundary-layer thickness, is theslope parameter, B is a bias parameter, is the wakeparameter,13 and is the local viscous forebody dragcoefficient. The currently accepted “best value” for isapproximately 0.41. The bias parameter, B, varies withthe level of surface roughness and for a smooth plate hasa numerical value of approximately 5.0. The wakeparameter is proportional to the local longitudinalpressure gradient, and for a zero-gradient flow conditionthis wake parameter has a numerical value ofapproximately 0.55. The roughness-dependent bias term

u+ 1

κ--- y

+[ ] 2Π 2 π2--- y

δ--sin+ln B+=

u+ u y( )

Ue

c f

2-----

-----------------, y+ y

x--Rex

c f

2-----= =

δ κΠ

CFκ


can be eliminated from equation 1 by expressing the lawof the wake in terms of the local velocity defect.

(3)

General Solution Method for Local Skin-Friction Coefficient

As developed in Appendix A, equation 3 is integratedacross the depth of the boundary layer, including thelaminar sublayer, to relate the momentum anddisplacement thickness to the local boundary thicknessand the local pressure gradient. The resultingrelationships are (1) momentum thickness and(2) displacement thickness.

(1) Momentum thickness:

(4(a))

(2) Displacement thickness:

(4(b))

In equations 4(a) and 4(b), is the Clauserparameter,13 and is related to the local pressure gradientby

(5)

Clearly negative values of correspond to favorablepressure gradients and positive values of correspondto adverse pressure gradients. A pressure gradient with

is considered to be a weak, and a pressure

gradients with is considered to be strong.White14 develops a similar set of integral relationshipsfor the momentum and displacement thickness, but doesnot include the effect of the laminar sublayer.

Closure is added to the problem by applying thecorrelation equation developed by White,14

(6)

Equations 4, 5, and 6 can be used to solve for theentire state of the local boundary in terms of thenormalized boundary-layer thickness, Reynoldsnumber, and the Clauser pressure gradient parameter.Parameters which are estimated include the local skin-friction coefficient, momentum thickness, displacementthickness, and the boundary-layer shape parameter( ). This system of equations is extremelynonlinear but is reasonably well-behaved and isnumerically solved using a simple substitution iterationprocedure. Unfortunately, the correlation of equation 6was developed from smooth surface data only. Thus thissolution procedure should be applied to smooth surfacesonly. For the remainder of this paper, this iterativesolution procedure will be referred to as the nonlinearmethod.

Zero Pressure Gradient Solution for LongitudinallyAveraged Skin-Friction Coefficient

If the local pressure gradient is small or negligible, then the longitudinally averaged forebody

viscous drag coefficient is directly related to the localmomentum thickness using the integral form of theVon Karman momentum equation

(7)

Substituting equation 7 into equation 4(a) andevaluating for , an expression for thelongitudinally averaged viscous drag coefficient interms of the local skin-friction coefficient is derived

(8(a))

1 u y( )Ue

-----------–c f

2-----

1κ--- 2Π 2

π2---

yδ--

yδ--ln–cos ≈

=

c f

2----- 4.878Π 2

π2---

yδ-- 2.439

yδ--ln–cos

δx-- c f

0.16425 2.66116 0.88881 β++[ ] –

10.4874 10.6227 β 2.13276 0.88881 β++ +[ ] c f

+ 55.9194 40.6563 0.88881 β+–

Rex------------------------------------------------------------------------------

+ c f 137.422– 162.4 β 32.6051 0.88881 β++ +[ ]

Rex-------------------------------------------------------------------------------------------------------------------------------

δ*

x-----

δx-- c f 0.16425 2.66116 0.88881 β++[ ][ ]=

55.9194 40.6563 0.88881 β+–Rex

------------------------------------------------------------------------------ ++

β

β δ*

c f-----

∂p∂x------

q------≡

ββ

β 0.1<

β 0.1>

c f0.3e 1.33δ* θ⁄–

log 10Rexθx---

1.74 0.31δ* θ⁄+-----------------------------------------------------------------=

δ* θ⁄

β 0∼( )

θx---

1x---

c f

2----- xd

0x∫

c f

2-----= =

β 0=

CFδx-- c f= 5.3464 24.998 c f–[ ]

1Rex--------- 35.12 213.342 c f–[ ]+


In equation 8(a), is the longitudinally averagedviscous drag coefficient. Evaluating equation 1 at

gives an implicit formula for the local skin-friction coefficient in terms of the and Rex

(8(b))

As mentioned earlier, for a smooth surface .For rough surfaces, White14 has curve-fit thePrandtl-Schlicting sand grain roughness curve to obtainthe following formula

(9)

In equation 9, is the nondimensional surfaceroughness and is defined as

(10)

A surface is defined as “rough” when . For, the surface is said to be “hydraulically

smooth.” Equation 9 is valid for turbulent flowconditions that are smooth, transitionally rough (onlypart of the surface is rough), and fully rough (the entiresurface is rough). Collectively, equations 8, 9, and 10offer a nonlinear set of equations which can be used tosolve for the longitudinally averaged viscous dragcoefficient in terms of the local boundary-layerthickness, Reynolds number, and the surface roughnessratio. This set of equations is well-behaved and can besolved iteratively using a simple substitution method.Since the local skin-friction coefficient is computed byevaluating the law of the wake at the edge of theboundary layer for the remainder of this paper, thissolution method is referred to as the edge method.

Closed-Form Solution Method for Averaged Skin-Friction Coefficient

When the entire surface is rough (a fully rough flowregime), experimental observations show that the ratioof the local and averaged skin-friction coefficients is afunction of only the normalized surface roughness,( ). For a fully rough flat plate Mills,15 et. al. havedeveloped the following relationship for the skin-friction ratio

(11)

Equation 11 is valid over a Reynolds number rangefrom approximately 105 to 108. For large Reynoldsnumbers (Rex > 107) an approximate skin-friction ratiofor fully developed turbulent flow on a hydraulicallysmooth plate is derived by setting the roughness ratio toa very large value. Evaluating equation 11 at

, gives .

By substituting into equation 8(a) andsimplifying, the solution for the averaged viscous dragcoefficient are written in closed form

Equation 12 gives the relationship between the localboundary layer thickness and the averaged surface skin-friction coefficient. This simple result is importantbecause it requires only a single measurement of theboundary layer thickness and an estimate of theReynolds number and surface-roughness ratio todetermine the local momentum defect. When thepressure gradients are small, the local momentum defectis a good approximation of the averaged skin friction.Data to be presented in the Results and Discussion

CF

y δ=δ

2c f----- 2.439

c f

2-----

δx-- Rex 2Π+ln B+=

β 5.0=

B 5.0 2.439 1 0.30κ++[ ]ln–=

κ+

κ+ κ s

x-----Rex

c f

2-----=

κ+5.0>

κ+5.0<

x κ s⁄

c f

CF------- G

xκ s-----

2.635 0.618 xκ s-----ln+

2.57

3.476 0.707 xκ s-----ln+

2.46------------------------------------------------------------------= =

x κ s⁄ 108

= G x κ s⁄[ ] 0.896∼

G x[ ]


(12)CF

12---

5.3464δx-- 213.342

Rex-------------------–

1 24.998G x[ ] δx--+

--------------------------------------------- G x[ ] +

12---

5.3464δx-- 213.342

Rex-------------------–

1 24.998G x[ ] δx--+

--------------------------------------------- G x[ ]

2

35.12

Rex 1 24.99G x[ ] δx--+

------------------------------------------------------+

2

=

section demonstrate that the closed-form model givesresults that are accurate to within 5 percent as long as

. This accuracy estimate holds for fully roughsurfaces with turbulent flow and for hydraulicallysmooth surfaces with turbulent flow occurring at theleading edge. For the remainder of this paperequation 12 is referred to as the closed-form solution.

Calculations from the various solutions methods arecompared in the Results and Discussion section. Thesecomparisons help in determining the regions ofapplicability for the solution methods.

Design of the Boundary Layer Tracking Probe

As mentioned earlier the integral form of the law ofthe wake—where the skin-friction coefficient isexpressed in terms of the local boundary layer thickness—suggests an innovative concept for a skin-frictionmeasurement device. While the actual mechanicaldesign of the probe can vary substantially, the basicconcept is that a reference pitot probe is inserted into theflow, well above the maximum anticipated thickness ofthe local boundary layer. Another pitot probe isservomechanism-driven to move normally to thesurface. A controller regulates the position of thefloating probe. As developed in Appendix B thecontinuous-time control law for this system is

(13)

In equation 13, y is the probe position measuredrelative the to the surface, is the local dynamicpressure at position y within the boundary layer, and

is the difference between the dynamic pressure atthe edge of the boundary layer and the local dynamicpressure. Choosing the natural frequency ( ) anddamping ratio ( ) so that the open-loop systemeigenvalues are stable, causes the unforced probeposition to always decay to zero. The decaying tendencyof the open-loop system insures that the measurement-probe will not remain stuck outside of the boundarylayer where the pressure difference between thereference and measurement-probes is zero. The analogflow diagram for this control equation is depicted infigure 4.

Discrete Control Law

Equation 13 is discretized using the trapezoidal16

rule. After performing some extensive algebra, thediscrete control equation is written as

(14(a))

Where the coefficients of equation 14(a) are

(14(b))

In equation 14 k is the time-frame index and is thetime interval between each frame. These equations canbe implemented recursively starting with the initialposition of the tracking probe. Since the measureddynamic pressure difference is a function of the localposition within the boundary layer, this control law isclearly nonlinear. As such, it is expected that the controllaw behavior will differ somewhat from the response ofa linear second-order system. Clearly though, n is thecontrol gain parameter. An example probe control-lawresponse for an unsteady boundary layer will bepresented in the Results and Discussion section.

Final Probe Position

When the probe is positioned within the boundarylayer so that

(15)

equation 14 is rewritten as

(16)

Since , , and are all positive constants,equation 16 implies that

(17)

Thus the difference equations converge to a steadyvalue. As a result the probe seeks a final position where

(18)

β 0.1<

y 2ζωny ωn2 n2---y+ + ωn

2 n2--- y

q--- ∆q=

q

∆q

ωnζ

yk 1+

Ak

Ck------yk

Bk

Ck------yk 1––=

Ak 2 1 ωn∆t2-----

21

n2--- ∆q[ ]

q-----------

k––

=

Bk 1 2ζωn∆t2-----– ωn

∆t2-----

21

n2--- ∆q[ ]

q-----------

k–+

=

Ck 1 2ζωn∆t2----- ωn

∆t2-----

21

n2--- ∆q[ ]

q-----------

k–+ +

=

∆t

1n2--- ∆q[ ]

q-----------

k– 0=

yk 1+ yk–1 2ζωn

∆t2-----–

1 2ζωn∆t2-----+

----------------------------- yk yk 1––[ ]=

ζ ωn δt

yk 1+ yk– yk yk 1––≤

∆qq

-------final

2n---=


Equation 18 is written in terms of the final velocityratio as

(19)

Equation 19 is used to correct the final probe positionby giving the proper estimate of the boundary layerheight, assuming that the velocity profile in the outerlayers of the boundary layer are approximated by apower-law function. The resulting correction factor is

(20)

In equation 20 the parameter m is the velocity profileexponent and is the function of the local Reynoldsnumber. Values for m, which are required to perform thecorrection of equation 20, are presented in the Resultsand Discussion section. Once the final probe positionhas been achieved; the boundary layer thicknessestimated by equation 20 is used to evaluate the localand averaged skin-friction coefficients using whicheverof the three solutions methods (developed earlier) isappropriate.

Results and Discussion

This section is divided into two parts: a ModelValidation section and a Probe Response section. In theModel Validation section the three solution methods forthe law of the wake model are compared. Thesecomparisons establish the errors that pressure gradientscause in the edge and closed-form solution methods.Next the closed-form solution is compared to the edgesolution for a variety of surface roughness. Finally, skinfriction coefficients and velocity profiles predicted bythe law-of-the-wake model are compared to flight-obtained boundary layer measurements. In the ProbeResponse section an example of dynamic response ofthe probe control law for a noisy boundary layer ispresented.

Model Validation

First an estimate of errors in the edge and closed-form

solution methods caused by longitudinal pressure

gradients are examined. These results are presented in

figure 5. In figure 5 the top graph shows the local skin-

friction coefficient predicted by the nonlinear model as a

function of Reynolds number. The curves correspond to

values of . The

solutions for the edge and closed-form methods (

implicit) are also plotted. For the closed-form solution

the “smooth-plate” value of was

used. For the nonlinear and edge solution

methods give nearly identical results. The closed-form

solution shows approximately a 2-percent error at

. This error diminishes to nearly zero at

. The bottom graph in figure 5 shows the

percent deviation in the closed-form model, defined by

(21)

This parameter is plotted as a function of Reynoldsnumber for values of .The calculated deviations in the edge method aresmaller than those shown in the figure 5 bottom graphby approximately 1 percent. These data are not shown.The deviations shown in this bottom graph of figure 5are also representative of the errors that are expected forthe averaged skin friction values. Clearly, there is errorintroduced by the simplifying assumptions used inderiving the closed-form model. However, if the localpressure gradient is small ( ), the errors areless than 5 percent over a very wide range of Reynoldsnumbers.

The simplifying assumption was introduced into theclosed-form model by the simplifying assumption ofequation 11. However, as shown in figure 5, this error isless than 2 percent over a wide Reynolds number rangefor a hydraulically smooth surface. Figure 6 shows howthis error varies as a function of surface roughness. Thefigure 6 top graph compares the edge solution foraveraged skin-friction coefficient to the closed-formaveraged skin-friction coefficient. The data presented onthis curve were computed for a Reynolds number of5.0 × 107 with roughness levels varying from a verysmooth surface to a very rough surface. Also plotted onthis figure 6 top graph is the transitionally rough skinfriction roughness formula derived by White.14 Thelevel of agreement is very close. The bottom graph offigure 6 shows the percent deviation plots (where theedge solution is assumed to be the “truth set”). Thedeviations are all within 2.5 percent, even at the lowReynolds numbers. As the Reynolds number and surfaceroughness increase, the deviations diminish toapproximately zero.

Comparisons to Flight Data

Figures 5 and 6 show data in the absence ofsignificant longitudinal surface pressure gradients. Inthese two figures the closed-form model predicts the

uUe------

final

qqe-----

final

nn 2+-------------= =

δest y finaln 2+

n-------------

m 2⁄=

β 0.25 0.0 0.25 and1.0, , ,–{ }=

β 0=

G x κ s⁄[ ] 0.896=

β 0=

Rex 106

=

Rex 108

=

percent of deviation = 100c f nonlinear

c f closedform–

c f nonlinear

----------------------------------------------------×

β 0.25 0.0 0.25 and1.0, , ,–{ }=

β 0.10<


averaged skin friction to within 5 percent whencompared to the more complex iterative solutionmethods. This result justifies using the simple model ofequation 12 for the real-time measurement system.Further justification is evident in comparing averagedskin friction predictions of the closed-form modelagainst data obtained from two flight experimentsconducted by Saltzman and Fisher.17, 18 During theseseries of tests, both supersonic and subsonic data wereobtained; however, since this analysis is currentlyapplied only for incompressible flow conditions, onlythe subsonic flight data were analyzed. In these flightexperiments, boundary-layer rake measurements wereobtained on the underside to the fuselage of an A-5Aaircraft, and on the wing of the XB-70-1 aircraft. Forboth series of flight tests a calibrated Preston tube wasalso installed near the rake to supply additionalmeasurements of the local skin-friction coefficients.Figure 7 shows the A-5A instrument locations andfigure 8 shows the XB-70-1 rake and Preston tubelocations. Table 1 summarizes the data points that wereanalyzed.

For the A-5A tests, all of the data points analyzedwere at angles of attack greater than 3.8°, except pointnumber 6 for which the angle of attack was 0.3°.Because the boundary-layer rake for these tests waslocated on the bottom of the vehicle, for the higherangles of attack (points 1–5) the effect of the noseboom

on the Reynolds number was ignored. For these points atotal run length of 508 cm was used. For the low angle-of-attack data point (point 6) the effect of the noseboombecame significant and a run length of 736 cm was used.For the XB-70-1 tests the run length was approximately1557.5 cm.

For each data point the boundary layer thickness wasobtained by fitting the data with a 1/mth power law

(22)

Figure 9 shows an example linear-log plot for datapoint number 2 (A-5A) in table 1. The y-intercept valueis 1.8112 and the boundary layer thickness isapproximately . The linear-logslope represents the power law exponent, hence

. Based on this “measured” boundary layerthickness, the closed-form solution was used to computeboth the averaged and local skin-friction coefficients.These data were then used with equation 3 to computethe velocity profile and momentum distribution withinthe boundary layer. Figure 10 shows the velocity andmomentum profile comparisons for data point 2 (A-5A,Rex = 2.20 × 107). Outside of the laminar sublayer, thecomparisons are excellent. Similar velocity andmomentum distribution profiles are presented infigure 11 for data point 10 (XB-70-1, Rex = 13.0 x 107).The comparisons are very similar to those in figure 10.The characteristics of these selected data points are verysimilar to all of the data points which were examined.

Figure 12 summarizes the flight data analysis.Figure 12 plots the law of the wake (averaged) skin-friction (closed-form solution) coefficients calculatedusing flight-measured boundary layer thickness againstpreviously published17, 18 values determined by theauthors. The published data include local skin frictionestimates determined from the rake data using anadaptation of the Clauser method.13 Flight-measuredPreston tube data are also shown. The Preston tube datawere analyzed using the log-law calibration method ofHopkins and Keener.10 The published data wereexpressed in terms of the local skin-friction coefficientsand must be transformed to averaged values for thecomparisons to be valid. The local skin friction datawere transformed to averaged values using the methodof Mills.15 In this method the skin friction ratio isderived from integrating the local skin friction from theleading edge of the forebody along a flow streamline tothe local position. The effect of laminar flow near theleading edge is accounted for. The result is

Table 1. Summary of subsonic flight data analyzed.

Datapoint Vehicle Rex

1 A5A 1.54e+7 6.2 0.82 508.0

2 A5A 2.20e+7 3.8 0.64 508.0

3 A5A 3.00e+7 6.4 0.60 508.0

4 A5A 3.11e+7 7.0 0.51 508.0

5 A5A 4.94e+7 3.2 0.81 508.0

6 A5A 7.42e+7 0.3 0.81 736.0

7 XB-70-1 1.14e+8 8.7 0.41 1557.5

8 XB-70-1 1.18e+8 8.1 0.49 1557.5

9 XB-70-1 1.24e+8 8.2 0.46 1557.5

10 XB-70-1 1.30e+8 8.9 0.53 1557.5

11 XB-70-1 1.33e+8 8.3 0.51 1557.5

12 XB-70-1 1.41e+8 5.5 0.94 1557.5

13 XB-70-1 1.57e+8 5.2 0.95 1557.5

14 XB-70-1 1.73e+8 5.7 0.91 1557.5

α , deg Medge x, cm

yln δ mu

Ue------ 0.99⁄ln+ln=

δ e1.8112≈ 6.118 cm=

m 7.9613≈


(23)

In equation 23 is the transition Reynoldsnumber and Rex is the local Reynolds number. For thistransformation a transition Reynolds number ofapproximately 5.0 × 105 was assumed.

Figure 12 shows the skin-friction coefficients plottedagainst the normalized boundary layer height. The(adjusted) published data generally scatter about thelaw-of-the-wake curve; however the scatter is large.Table 2 summarizes the mean and standard deviations ofthe adjusted rake and Preston tube CF values away fromthe law of the wake determined values.

The XB-70-1 data show a much better agreement thando the A-5A data. The reasons for the large amount ofscatter in the A-5A data are unclear. Because the testarray was located on the underside of the vehiclefuselage, it is possible that angle-of-attack effectsinduced strong local pressure gradients which had asignificant influence on the local skin frictionmeasurements. The 5- to 10-percent mean deviations arenot, however, unreasonable.

Simulation of the Boundary-Layer Tracking Probe Response

The previous section established that the integral law-of-the-wake analysis is a reasonably accurate predictor(within 5 to 10 percent) of the average skin-frictioncoefficient in the absence of large external pressuregradients. For real-time applications the closed-formsolution given by equation 12 is the preferred solutionmethod. The value of the closed-form solution is that theaveraged skin-friction coefficient can be incorporated in

calculations directly by simply measuring the localboundary layer thickness. This section analyzes thedynamics of the tracking device that is used to obtainthe boundary layer thickness measurement. In thissection the performance of the tracking probe ismodeled using both steady-state and noisy boundarylayer simulations.

Effects of the Control Gain Parameter

In this series of simulations the boundary layerthickness and the velocity profile are assumed to beconstant as a function of time. The probe is released attime with a starting value of . Tomatch the expected transonic Reynolds numbers thatatmospheric reusable launch vehicles encounter, theReynolds number was held constant at

. The simulations were all run at250 samples per second. Figure 13 demonstrates theeffect of the control gain parameter (n) on the systemresponse. Here the simulation is run with the naturalfrequency, damping ratio, and Reynolds number fixedat , , and .The control gain parameter is varied from 10 to 500.The results illustrate an interesting point regarding thetrade-off between final position of the probe and theoscillatory stiffness of the model response. As predictedby equation 19, when the gain is increased, the finalposition of the probe is closer to the boundary layeredge. Unfortunately, high gain values induce anundesirable response in the system, and the probe neversettles down to a stable value. Clearly, it is preferable toallow the probe to track at some fraction of theboundary layer height, and correct the final positionusing equation 20. It is recommended that values of n bekept somewhere within the range from 20 to 50.

Correction for the Final Probe Position

The velocity profile exponent (m), which was requiredto correct the final probe position and estimate the trueboundary layer thickness, was determined by running aseries of simulations in which the Reynolds number andgain parameter were allowed to vary. These results arepresented in figure 14. Figure 14 shows the variation ofm with Reynolds number for control gains of 10, 20, 50,100, and 200. The data of figure 14 were fit with asimple linear model of the form

(24)

Table 2. Deviations of flight measured from law of the wake.

Rake Preston tube

,percent

,percent

,percent

,percent

A-5A 10.68 31.0 1.2 27.0

XB-70-1 –5.9 11.7 –5.3 10.7

CF

c f-------

2.9187

Rextr

---------------- ln2

0.06Rex[ ]Rextr

Rex------------=

+ 1.1495 1Rex

Rextr

------------2 0.06Rex[ ]ln

2 0.06Rextr[ ]ln

-------------------------------------–

Rextr

CF

µ σ µ σ

t 0= y δ⁄ 0.25=

Rex 5.0 107×=

ωn 4π sec1–

= ζ 0.7071= Rex 5.0 107×=

m n Rex,[ ] C0 n[ ] C1 n[ ]10

Rex[ ]log+=


The coefficients of equation 24 are displayed inTable 3.

Linear interpolations of the coefficients in Table 2, asa function of n, are used to obtain a correction functionfor the final probe position.

Tracking an Unsteady Boundary Layer

The steady-state analyses of the previous section wereextremely useful in understanding the dynamics of thetracking probe control law; however, a true boundarylayer is not at all stationary. In reality a turbulentboundary layer has random elements and instantaneousvariations that can only be statistically predicted. Theresult is an intermittent boundary layer with a raggedshape. In the outer regions of the boundary layer thereare turbulent fluctuations which extend beyond thenormally defined boundary layer edge. Conversely,there are intermittent regions inside of the normallydefined boundary layer where the velocity is equal to thefree-stream velocity. The law-of-the-wake analysispresented in this paper deals only with the time-averaged velocity components of the boundary layer.However, the dynamic flow components will effect theperformance of the boundary layer tracking probe andthis flow intermittency must be modeled in the probesimulation. In this paper the random components withinthe boundary layer are modeled using data taken fromKlebanoff.19 This intermittent boundary-layer model isdeveloped in detail in Appendix B.

The response of the probe to an intermittent boundarylayer will now be presented. Reynolds number was setat Rex = 5.0 × 107, and the natural frequency, dampingratio, and control gain parameter were varied todetermine the best combinations required in order toaccurately track the edge of the boundary layer. Usingthe corrected probe position, the skin-friction coefficientis computed using equation 12. Figure 15 presents theresults of a noisy simulation run with a very stiff controllaw. In this simulation the parameters were set at

, , and . Theresponse of the tracking probe and the corrected probeposition are shown relative to the normalized boundarylayer edge in figure 15(a). The estimated skin-frictioncoefficient calculated using equation 12 is showncompared to the known value in figure 15(b). The probeposition correction based on equation 25 works quitewell and the estimated skin-friction coefficient trackswithin 5 percent of the “truth value”–once the initialstartup transient has died out. This simulation ispresented for illustrative purposes only. The particularvalues which are preferred for the control lawparameters must be tuned to the test conditions,hardware characteristics, desired probe response time,and noise rejection characteristics. Clearly more workneeds to be performed in this area.

Summary and Concluding Remarks

Lifting body and wing body designs currentlyadvocated for transatmospheric reusable launch vehiclesare derived from variations on the original lifting bodyconcept. These designs have large base areas relative tothe vehicle wetted area. Thus the base drag has a largeimpact on critical vehicle performance parameters suchas maximum payload and maximum cross-range. Anydecrease in base drag can have the potential tosignificantly improve the overall vehicle performance.Recent work has demonstrated that by increasing theforebody drag of these vehicle shapes it may be feasibleto reduce the base drag so significantly, that the overalldrag of the vehicle can be reduced. Methods foroptimizing this drag reduction process are currentlybeing studied.

Any attempt to implement the drag optimizationrequires a practical method for measuring the viscousforebody drag of the vehicle in real time. To be practicalfor the optimization problem, skin frictionmeasurements must be self-contained, that is, they mustbe installed onboard the vehicle entirely. As discussed,none of the existing methods for sensing the surfaceskin friction are readily adaptable to the aboveoptimization problem.

This paper proposes a new method based on the lawof the wake. In this concept a reference pitot probe isinserted into the flow, well above the anticipatedmaximum thickness of the local boundary layer.Another probe is servomechanism-driven and floatswithin the boundary layer. A controller regulatesposition of the floating probe. The measuredservomechanism-position gives a direct measurement ofthe local boundary layer height. The measured boundary

Table 3. Velocity profile exponent: fit coefficients.

n C0 Cl

10 –0.51785 0.95893

20 0.11626 0.96786

50 0.69694 1.06950

100 1.29530 1.14760

200 1.89040 1.24180

ωn π 2 –1sec⁄= ζ 0.7071= n 25=


layer height is related to the local and longitudinallyaveraged skin friction by an integral form of the law ofthe wake.

In the law-of-the-wake analysis the turbulentboundary layer velocity profile equations werenumerically integrated across the boundary layer depthto develop an engineering model that relateslongitudinally averaged skin-friction coefficients tolocal boundary layer thickness. The law-of-the-wakeanalysis develops three solution methods: 1) an iterativenonlinear solution, 2) an edge solution where the localskin friction is evaluated from the law of the wake at theedge of the boundary layer, and 3) a less-general closed-form solution. The non-linear solution method solvesfor the local skin friction coefficient, momentumthickness, displacement thickness, and the boundarylayer shape parameter. The nonlinear solution method isgenerally applicable to smooth surfaces with arbitrarylongitudinal pressure gradients, but does not allow theaveraged skin-friction coefficient to be evaluated. Theedge method is valid only when the pressure gradientsare small, but allows both the local and averaged skin-friction coefficients to be solved for. The edge method isvalid for smooth, transitionally rough, and fully roughsurfaces. The closed-form solution method is the leastgeneral of the three solution methods and is applicableonly for fully rough surfaces when the pressure gradientis small. The closed-form solution offers the clearadvantage of being an explicit solution; whereas, thefirst two methods require an iterative calculation. Forreal-time applications, as long as the flow constraints arenot violated, the closed-form method is clearly thepreferable choice.

Comparisons of the closed-form solution method toflight data show that the velocity longitudinally

averaged skin friction can be sensed with reasonableaccuracy. The flight data show that it is clearly possibleto achieve mean accuracies on the order of 5 to10 percent. Applying corrections for the longitudinalpressure gradient and the effects of compressibility canlikely reduce these errors further.

The dynamics of the tracking device used to obtainthe boundary layer thickness measurement werepresented. The performance of the tracking probe wasmodeled using both steady-state and noisy boundarylayer simulations. Simulation results showingperformance of the boundary layer tracking probe for anoisy boundary layer were presented. The particularvalues that are preferred for the control law parametersmust be tuned to the test conditions to obtain the desiredprobe response time and noise rejection by the probecontrol law.

To be more generally applicable, additionalcomparisons of the integral law of the wake to transonicand supersonic flight data should be performed. Ifpossible, a compressibility correction must bedeveloped. Additional data comparisons should also beperformed for known pressure gradients, a larger rangeof Reynolds numbers, and for a variety of surfaceroughness patterns. So far, this demonstration isincomplete. Clearly, this tracking device must be builtand tested. A definitive set of criteria for selecting thecontrol law parameters, one that includes the effects ofthe probe hardware, must be developed. However, theresults presented in this paper are very encouraging. Thesimulation results clearly confirm that the trackingprobe can be designed to accurately measure the skinfriction in a real-time environment.


Figure 1. Subsonic correlation of base and viscous forebody drag coefficients.

Figure 2. Visualization of the drag bucket.

.1Viscous forebody drag coefficient

Bas

e d

rag

co

effi

cien

t

1.5

1.0

.5

0

LASRE/X-33

990462

Shuttle

X-15

M2-F3X-24A

X-24B

HL-10

M2-F1

.001 .01 1 10

2-dimensional curve, ref. 23-dimensional curve, ref. 3Full-scale flight vehicle data(Note: Coefficients referenced to base area)

X-15

X24-B

M2-F3

X24-A

M2-F1 HL-10

Shuttle

HL-10

M2-F1

X24-A

M2-F3X24-B

Shuttle

X-15

LASRE

LASRE

Note: Coefficients referenced to base area

.1Viscous forebody drag coefficient

Dra

g c

oef

fici

ent

.8

.6

.4

.2

0

990463

.01 1

CDbase,

model

CDtotal, model


Figure 3. Conceptual schematic of boundary layer tracking probe.

Figure 4. Analog flow diagram of boundary layer probe control law.

qref

yδ

Servomechanism actuator

Reference probe

990464

Moveable measurement probe

Differential pressure sensor

External flow

–

∆q–

Boundary layer

/ / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / /

+

••–

+

•

–

–

∆q

ωn2

ωn2

n/2 X+

–

2 ζ ωn

I (t)

y(0) y(0)

y(t)I (t)

y(t)

990465

Boundarylayer

y(t)Sensordynamics

Feedback control input

q(y)

–

–

qe


Figure 5. Effect of pressure gradient on the law of the wake solution methods.

Note: Skin-friction coefficient referenced to wetted area

Closed-form solution, eqn. 12Edge solution, eqn. 8bNonlinear solution, β = 0Nonlinear solution, β = – 0.25Nonlinear solution, β = 0.25Nonlinear solution, β = 1.0

β = – 0.25

β = – 0.10β = 0

β = 0.10

β = 1.00

β = 0.25

Eqns. 4, 5, and 6

Dev

iati

on

in c

f, p

erce

nt

c f

10

100

0

–10

–20

–30

.001

.002

.003

.004

50

990466

2010

Rex x 106521


Figure 6. Effect of surface roughness on the law of the wake solution methods.

Figure 7. Schematic showing location of boundary layer instrumentation on A-5A aircraft.

Dev

iati

on

in c

f, p

erce

nt

c f

3

100

2

1

0

–150

990467

2010

Rex x 106

Rex = 5 x 107

x/κs

521

.02

.01

010 100 1,000 10,000

cf, White's transitionally rough theory

Smooth surfacex/κs = 5000x/κs = 1000

x/κs = 500x/κs = 100x/κs = 50

508 cm

Rake location 990468


Figure 8. Schematic showing location of boundary layer instrumentation on XB-70-1 aircraft.

Figure 9. Linear-log plot of boundary layer rake data.

990469

56.62 m

5.54 m

9.14 m43.5 m

15.6 mWing sensor complex

32 m

Ln [u/(0.99Ue)]

Ln

[y]

2

1

0

–1

–.5–2

–.4 –.3 –.2 –.1 0

990470

A-5A lower fuselage boundary layer rake data

Ln [y] = 1.8112 + 7.9613 Ln [u/(0.99Ue)]

Rex = 2.22 x 107

Measured profileLinear-log fit


Figure 10. Comparison of law of the wake model against A-5A lower fuselage boundary layer rake.

u/U

e(1

– u

/Ue)

u/U

e

1.2

.0001 .001 .01 .1 1.0 10.0

990471y/δ

1.0

.8

.6

.4

.2

0

.3

.2

.1

0

Law of the wakeA-5A fuselage rake data

Momentum deflect profile

Rex = 2.2 x 107

Velocity profile


Figure 11. Comparison of law of the wake model against XB-70-1 wing surface boundary layer rake data.

1.2

1.0

.8

.6

.4

.2

0

.3

.2

.1

0.0001 .001 .01

y/δ.1 1.0 10.0

990472

u/U

e(1

– u

/Ue)

u/U

e

Rex = 13.0 x 107

Rex = 13.0 x 107

Law of the wakeXB-70-1 wing rake data

Momentum deflect profile

Velocity profile


Figure 12. Summary of flight data, skin-friction coefficient comparisons.

CF – Law of the wake (closed-form solution)CF – XB-70-1 (adjusted rake data)CF – XB-70-1 (adjusted preston probe data)CF – A-5A (adjusted rake data)CF – A-5A (adjusted preston probe data)

Ski

n f

rict

ion

co

effi

cien

t

.003

.02

990473

.01

δ/x

0

.002

.001


Figure 13. Effect of the control gain parameter on probe response.

Figure 14. Velocity profile exponent.

y/δ

2.0

2.0

990474

1.51.0Time, sec

.50

1.8

1.6

1.4

1.2

1.0

.8

.6

.4

.2

n = 500 n = 200 n = 100 n = 50 n = 20n = 10

Rex = 5 x 107, ωn = 4π, ξ = 0.70717

Rex x 106

Bo

un

dar

y la

yer

velo

city

pro

file

exp

on

ent

12

100.010.01.0

990475

.1

10

8

6

4

Curve fit compared to Reynolds number

n = 10n = 20n = 50n = 100n = 200


Figure 15. Response of probe control law to noisy boundary layer.

y/δ

1.5

1.4

1.3

1.2

1.1

1.0

.9

.8

.7

.6

.5

.4

.3

.2

.1

y/δ

.0030

.0028

.0026

.0024

.0022

.0020

.0018

.0016

.0014

.0012

.0010

.0008

.0006

.0004

.00020 10 20 30 40 50

Time, sec990476

Probe position

yprobe/δ

δnoisy/δ

δest/δ

^CF

CF

^

Skin friction coefficient(based on wetted area)


Appendix A: IntegralLaw-of-the-Wake Analysis

This analysis develops an integral relationship thatrelates the longitudinally averaged skin-frictioncoefficient to a local measurement of the boundary layerthickness. The analysis is based on Coles’ empirical“law of the wake.” The law of the wake relates the outer(turbulent) boundary layer variables to thenondimensionalized boundary layer coordinates and canbe written in terms of the velocity defect as

(A-1)

In equation A-1, is the wake parameter and isproportional to the local pressure gradient. Equation A-1is a general expression that is valid outside of thelaminar sublayer for plates, curved surfaces, surfaceswith longitudinal pressure gradients, and roughsurfaces. For a flat plate with zero pressure gradient, thewake parameter has an approximate numerical value of

.

Correction for Laminar Sublayer

Outside of the laminar sublayer, equation A-3 hasbeen found to correlate well with measurements;however, within the laminar sublayer the correlationbreaks down. In the laminar sublayer the correlation isbetter represented by the relationship, u+ = y+, and theresulting velocity defect law in the laminar sublayer is

(A-2)

Equation A-2 strictly holds for and equationA-1 strictly holds for . Between these limitingvalues, the boundary layer velocity profile is

transitional. Spalding20 has offered a transitionalformula for this buffer layer; but with minimal loss inaccuracy a simpler model can be developed by equatingA-1 and A-2

(A-3)

Allowing that the cosine term in equation A-3 isapproximately 1, equation A-3 may be solvednumerically, to produce

(A-4)

Thus for equation A-2 is used as thevelocity defect model and for equation A-1is used as the velocity defect model.

Momentum and Displacement Thickness Calculation

The fundamental definitions for the (incompressible)momentum and displacement thickness are

(A-5)

(A-6)

If the integrals of equations A-5 and A-6 areintegrated across the depth of the boundary layer, usingequation A-2 to evaluate the velocity defect within thelaminar sublayer ( ) and equation A-1 in theturbulent layer ( ), then an integral form ofthe law of the wake can be derived. The laminarcontribution to the momentum and displacementthickness are given as

(A-7)

(A-8)

1 u y( )Ue

-----------– =

c f

2-------- 4.878Π 1 2 π

2--- y

δ--sin–

2.439yδ--ln– =

1 u y( )Ue

-----------– =

c f

2-------- 4.878Π 2 π

2--- y

δ-- 2.439

yδ--ln–cos

Π

Π 0.55∼

1u

Ue------– 1

yδ--δ

x--Rex

c f

2-----– 1

c f

2-----y

+–= =

y+

5<y

+25>

1c f

2-----ytrans

+

–c f

2----- 4.878Π 2

π2---

ytrans

δ--------------cos=

2.439ytrans

δ--------------ln–

ytrans+

10.81≈

y+

10.81<y

+10.81≥

θ u y( )Ue

----------- 1 u y( )Ue

-----------– yd0

δ∫=

δ*1 u y( )

Ue-----------– yd

0

δ∫=

y+

10.81<y

+10.81>

θlam 58.428x

Rex--------- 1 7.207

c f

2-------–

==

δlam*

58.428x

Rex---------=


(A-9)

θturb δ c f 15.288x

Rex---------–

=

1.72463 Π 1+[ ] c f – 4.46154Π29.45545Π 5.9486+ +[ ]

The turbulent contribution is given as

(A-10)

Adding the laminar contributions and turbulentcontributions together the total normalized momentumand displacement thickness equations become

(A-11)

and

(A-12)

Relationship of Wake Parameter to Local Pressure Gradient

As mentioned earlier in this appendix, the wakeparameter ( ) is related to the local pressure gradient.The wake parameter has been correlated to the localpressure gradient by White20 as a polynomial curve fitof the form

(A-13)

In equation A-13 is the Clauser parameter. Solvingfor in terms of and substituting into equationsA-11 and A-12 the momentum thickness anddisplacement thickness equations can be written withthe nondimensional pressure gradient as an explicitparameter in the equations

δturb* δ c f 15.288

xRex---------–

=

1.72463 Π 1+[ ]{ }

θx---

δx-- c f

1.72463 Π 1+[ ]{ }=

c f 4.46154Π29.45545Π 5.9486+ +[ ]–

+ 1

Rex--------- 32.0797 26.3483Π–[ ]

68.208+ c f Π 0.978711–[ ] Π 3.09803+[ ]

δ*

x-----

δx-- c f

1.72463 Π 1+[ ]{ }=

+ 32.0805 26.3483Π–[ ]

Rex------------------------------------------------------

Π

0.42Π20.76Π 0.545–+ β δ*

c f-----

∂p∂x------

q------≡=

βΠ β


(A-14)

and

(A-15)

θx---

δx-- c f

0.16425 2.66116 0.88881 β++[ ]

10.4874 10.6227β 2.13276 0.88881 β++ +[ ] c f –=

+ 55.9194 40.6563 0.88881 β+–

Rex------------------------------------------------------------------------------

+ c f 137.422– 162.4β 32.6051 0.88881 β++ +[ ]

Rex---------------------------------------------------------------------------------------------------------------------------

δ*

x-----

δx-- c f

0.16425 2.66116 0.88881 β++[ ][ ]=

+ 55.9194 40.6563 0.88881 β+–

Rex------------------------------------------------------------------------------ +

Appendix B: Unsteady Boundary Layer Model

In a real-world turbulent boundary layer there arerandom elements whose characteristics can only bestatistically predicted. Mathematically, this randomelement is treated by decomposing the boundary layervelocity components into time-mean and perturbationcomponents.

(B-1)

The point-to-point variations of the perturbationcomponents are inherently intermittent and can never betheoretically predicted. The law-of-the-wake analysispresented in the paper deals only with the time averagedvelocity components of the boundary layer. However,the dynamic flow components will effect theperformance of the boundary layer tracking probe andthis flow intermittency must be modeled in the probesimulation. In this paper the random components withinthe boundary layer are modeled using data taken fromKlebanoff.19 The intermittency factor is the fraction oftime that the flow is turbulent at a given perpendiculardistance away from the wall. The intermittency factor isplotted as a function of the normalized perpendiculardistance in figure B-1. The raw data from Klebanoff’sreport are presented along with a curve-fit model

(B-2)

This intermittency function is used as a probabilitydensity function which describes the time distribution ofa particular velocity component. Assuming that thetime-mean y and z velocity distributions areapproximately zero, then the intermittent boundarylayer model is written as

(B-3)

In equation B-3 is a uniformly distributedrandom number, and the terms , , and are theexpected values for the perturbation magnitudes. Theseparameters are also derived from curve fits of datapresented by Klebanoff. These data are presented inFigure B-2. The unusual shape of the data curves wasdifficult to curve-fit. Instead these data were loaded intothe simulation as tables and their values were computedas a function of using a log-linear interpolation.

The resulting model produces a boundary layer with aragged shape. In the outer regions of the boundary layerthere are turbulent fluctuations which extend beyond thenormally defined boundary layer edge. Conversely,there are intermittent regions inside of the normallydefined boundary layer where the velocity is equal to thefree-stream velocity. At each time point the dynamicpressure profile within the boundary layer is computedby taking the inner produce of the velocity componentsin equation B-3 and using this result to scale theundisturbed dynamic pressure profile.

(B-4)

In equation B-4 u’, v’, and w’ are the random velocityperturbations evaluated using the formulas ofequation B-3, and U is the steady longitudinal velocityin the boundary layer at location y. The intermittentboundary layer thickness is computed as the value of ywhere

(B-5)

u x y z t, , ,[ ] U x y z, ,[ ] , u' x y z t, , ,[ ]+=

v x y z t, , ,[ ] V x y z, ,[ ] v' x y z t, , ,[ ]+=

w x y z t, , ,[ ] W x y z, ,[ ] w' x y z t, , ,[ ]+=

γ 1 5yδ--

6+

1.4–=

u x y z t, , ,[ ] U x y z, ,[ ] u' γη t[ ]+=

v x y z t, , ,[ ] v' γη t[ ]=

w x y z t, , ,[ ] w' γη t[ ]=

η t( )u v w

y δ⁄

qunsteady 1 2Uu' u'2

v'2

w'2

+ + +

U2

----------------------------------------------------+ qsteady=

δ U2

2Uu' u'2

v'2

w'2

+ + + + Ue=→


Figure B-1. Boundary layer intermittency factor.

Figure B-2. Fluctuating velocity components in the turbulent boundary layer.

y/δ

1.2

1.21.0

990477

.8.6.4.2

1.0

.8

.6

.4

.2

0

γ

Curve fit: [1 + 5 [y/δ]6]–1.4

Vel

oci

ty p

ertu

rbat

ion

mag

nit

ud

e

.12

.10

.08

.06

.04

.02

0.0001 .001 .01

y/δ.1 1.0 10.0

990478

|u'/Ue||v'/Ue||w'/Ue|


References

1 Wong, Thomas J., Charles A. Hermach, John O.Reller, Jr., and Bruce E. Tinling, Preliminary Studies ofManned Satellites—Wingless Configurations: TheLifting Body, NACA Conference on High-SpeedAerodynamics: A Compilation of the papers presented,NASA TM X-67369, 1958.

2Hoerner, Sighard F., Fluid-Dynamic Drag, Self-Published, Brick Town, New Jersey, 1965.

3Saltzman, Edwin J., K. Charles Wang, and KennethW. Iliff, “Flight-Determined Subsonic Lift and DragCharacteristics of Seven Lifting-Body and Wing-BodyReentry Vehicle Configurations With Truncated Bases,”AIAA Paper 99-0383, January 1999.

4Whitmore, Stephen A., and Timothy R. Moes, “ABase Drag Reduction Experiment on the X-33 LinearAerospike SR-71 Experiment (LASRE) FlightProgram,” AIAA 99-0277, January 1999.

5Miller, Linda, Michael Hect, Martin Buehler, AminMottiwala, Beverly Eyre, and Indrani Chakraborty,“Ceramic Hybrid Electromechanical Systems(CHEMS),” NASA Tech Briefs, Vol. 23, No. 2, February1999, pp. 61–62.

6Coles, Donald, “The law of the wake in the turbulentboundary layer,” Journal of Fluid Mechanics, Vol. 1,1956, pp. 191–226.

7Garringer, Darwin J., and Edwin J. Saltzman, FlightDemonstration of a Skin-Friction Gage to a Local MachNumber of 4.9, NASA TN D-3830, Feb. 1967.

8Quinn, Robert D., and Frank V. Olinger, Flight-Measured Heat Transfer and Skin Friction at a MachNumber of 5.25 and at Low Wall Temperatures, NASATM X-1921, November 1969.

9Naughton, J. W., and J. L. Brown, “SurfaceInterferometric Skin-Friction Measurement Technique,”AIAA Paper 96-2183, June 1996.

10Hopkins, Edward J., and Earl R. Keener, Study ofSurface Pitots for Measuring Turbulent Skin Friction at

Supersonic Mach Numbers—Adiabatic Wall, NASATN-D-3478, July, 1966.

11Keener, Earl R., and Edward J. Hopkins, Accuracyof Pitot-Pressure Rakes for Turbulent Boundary-LayerMeasurements in Supersonic Flow, NASA TN D-6229,March 1971.

12Mills, Anthony F., Heat and Mass Transfer, IrwinPublishing Co., Homewood, Illinois, 1992.

13Clauser, Francis H., “Turbulent Boundary Layers inAdverse Pressure Gradients,” Journal of AeronauticalSciences, Vol. 21, No. 2, 1954.

14White, Frank M., Viscous Fluid Flow, McGraw-Hill, Inc., New York, 1991.

15Mills, A. F. and Xu Huang, “On the Skin Frictionfor a Fully Rough Flat Plate,” Journal of FluidsEngineering, Technical Briefs, Vol. 105, September1983, pp. 364–365.

16Rade, Lennart, and Bertil Westergren, BetaMathematics Handbook, 2nd ed., CRC Press Inc., BocaRaton, Florida, 1990.

17Fisher, David F., and Edwin J. Saltzman, Local SkinFriction Coefficients and Boundary-Layer ProfilesObtained in Flight from the XB-70-1 Airplane at MachNumbers up to 2.5, NASA TN D-7220, June 1973.

18Saltzman, Edwin J., and David F. Fisher, SomeTurbulent Boundary-Layer Measurements Obtainedfrom the Forebody of an Airplane at Mach Numbers upto 1.72, NASA TN D-5838, June 1970.

19Klebanoff, P. S., Characteristics of Turbulence in aBoundary Layer with Zero Pressure Gradient, NACAReport 1247, 1955.

20Spalding, D. B., and S. W. Chi, “The Drag of aCompressible Turbulent Boundary Layer on a SmoothFlat Plate With and Without Heat Transfer,” Journal ofFluid Mechanics, Vol. 18, 1964, Cambridge UniversityPress, New York, New York, pp. 117–143.


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NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)Prescribed by ANSI Std. Z39-18298-102

A Real-Time Method for Estimating Viscous Forebody Drag Coefficients

WU 529-50-04-T2-RR-00-000

Stephen A. Whitmore, Marco Hurtado, Jose Rivera, and Jonathan W.Naughton

NASA Dryden Flight Research CenterP.O. Box 273Edwards, California 93523-0273

H-2394

National Aeronautics and Space AdministrationWashington, DC 20546-0001 NASA/TM-2000-209015

This paper develops a real-time method based on the law of the wake for estimating forebody skin-frictioncoefficients. The incompressible law-of-the-wake equations are numerically integrated across the boundarylayer depth to develop an engineering model that relates longitudinally averaged skin-friction coefficients tolocal boundary layer thickness. Solutions applicable to smooth surfaces with pressure gradients and roughsurfaces with negligible pressure gradients are presented. Model accuracy is evaluated by comparing modelpredictions with previously measured flight data. This integral law procedure is beneficial in that skin-frictioncoefficients can be indirectly evaluated in real-time using a single boundary layer height measurement. In thisconcept a reference pitot probe is inserted into the flow, well above the anticipated maximum thickness of thelocal boundary layer. Another probe is servomechanism-driven and floats within the boundary layer. Acontroller regulates the position of the floating probe. The measured servomechanism position of this secondprobe provides an indirect measurement of both local and longitudinally averaged skin friction. Simulationresults showing the performance of the control law for a noisy boundary layer are then presented.

Base drag, Boundary layer, Hypersonic vehicles, Skin friction

A03

34

Unclassified Unclassified Unclassified Unlimited

January 2000 Technical Memorandum

Presented as AIAA 2000-0781 at the 38th Aerospace Sciences Meeting and Exhibit, January 10–13, 2000, Reno, Nevada.Stephen A. Whitmore and Marco Hurtado, NASA Dryden Flight Research Center, Edwards, CA; Jose Rivera, Universityof South Florida, Tampa, FL; and Jonathan W. Naughton, University of Wyoming, Laramie, WY.

Unclassified—UnlimitedSubject Category 05

This report is available at http://www.dfrc.nasa.gov/DTRS/

Documents

A Real-Time Method for Estimating Viscous Forebody … A Real-Time Method for Estimating Viscous Forebody Drag Coefficients Stephen A. Whitmore and Marco Hurtado NASA Dryden Flight