NASA Technical Paper 1208

Inverse Boundary-Layer Theory

and Comparison With Experiment

James E. Carter

SEPTEMBER 1978 G@P¥

Ill/_A

NASA Technical Paper 1208

Inverse Boundary-Layer Theory

and Comparison With Experiment

James E. Carter

Langley Research Center

Hampton, Virginia

NI A

National Aeronautics

and Space Administration

Scientific and Technical

Information Office

1978

CONTENTS

SUMMARY ......................................... 1

INTRODUCTION ...................................... 1

SYMBOLS ......................................... 2

ANALYSIS ......................................... 6

Incompressible Boundary-Layer Fornmlation ..................... 6

Numerical Procedure .................................. 9

Finite-difference equations ............................. 9

Newton linearization of viscous term ........................ 10

Solution of linearized difference equations ..................... 12

Compressible Boundary-Layer Formulation ...................... 14

RESULTS AND DISCUSSION ............................... 20

Comparison With Experimental Data .......................... 20

Results Obtained With _-Method ............................ 21

Similar solutions ................................... 21

Laminar separation ................................. 23

Turbulent separation ................................. 23

Viscous interaction .................................. 24

CONCLUDING REMARKS ................................. 25

APPENDIX A - COEFFICIENTS OF FINITE-DIFFERENCE EQUATIONS

FOR 5*-METHOD .................................... 27

APPENDIX B - FINITE-DIFFERENCE EQUATIONS AND SOLUTION

TECHNIQUE FOR _-METHOD ............................. 30

RE FERENCES ....................................... 36

FIGURES ......................................... 39

iii

SUMMARY

Two newinverse boundary-layer computationalprocedures, which permit nonsingular solutions at separation and reattachment, are presented. In the first technique,which

is for incompressible flow, the displacementthickness is prescribed and the pressure is

deducedimplicitly alongwith the solution. Comparisonsare madeby using this procedure with the turbulent experimental data of ChuandYoung,andexcellent agreement

is obtained. Thepurpose of the secondtechniqueis to obtain aninverse computational

procedure for compressible separatedflows that is nonsingular at separation. In this

formulation, a perturbation massflow is prescribed for the compressible boundary-layer

equationsandthe pressure is deducedimplicitly alongwith the solution. Laminar and

turbulent computationsmadeby using this techniqueare presented andare typical of separated flow. In both inverse procedures, finite-difference techniquesare usedalongwith

Newtoniteration. The present boundary-layer procedures canbeused either in the

inverse or direct (pressure prescribed) mode,dependingonthe boundary conditions; in

either modethe solution procedure is no more complicated than conventionalboundarylayer computations. Both inverse techniquespresentedherein appearto be well suited

for completeviscous-inviscid interaction computations.

INTRODUCTION

In recent years, significant progress hasbeen madein the solution of flow fields

which contain regions of separatedflow. Despite the diversity of approaches,which have

beenreported, there seems to be a clear distinction that most techniquesare either based

on the Navier-Stokes equationswith relatively large computer costs or more approximate

procedures in which the simultaneoussolution of the boundary-layer and inviscid-flow

equations is obtained. This latter approachis often referred to as interacting boundarylayer theory. In cases in which massive separation occurs, suchas a stalled airfoil, it

seemsapparentthat the solution of the Navier-Stokes equationsmust be usedto obtain a

detailed description of the flow field. In this case, the separatedviscous region occupies

a significant portion of theflow andthereby makes the inherent assumptionsof interacting

boundary-layer theory highly questionable. Alternately, there are manyflows of interest

in which, despite the presenceof separation,the viscous flow is confinedto a relatively

thin layer adjacentto the body surface, thereby rendering the use of the Navier-Stokes

equationsunnecessaryfor most, if not all, of the flow field. In the present paper, a
technique is presented for obtaining approximate solutions for boundary layers in which

the separated region is of a limited extent. Despite this simplifying feature, conventional or direct boundary-layer theory (pressure distribution prescribed) cannot be used

without significant modification, such as that used in reference I, because the boundarylayer solution is singular at separation.

In a recently developed boundary-layer theory, the displacement thickness is prescribed in the streamwise direction, thereby resulting in the calculation passing smoothly

through the separation point. The surface pressure distribution is then deduced from the

boundary-layer calculation. This idea of an inverse boundary-layer procedure to avoid

the separation singularity was first presented in reference 2 and later was reformulated

in reference 3. An alternate inverse boundary-layer procedure is to prescribe the skinfriction distribution as demonstrated in references 3 to 5 and others. However, the prescribed displacement-thickness technique is preferable when the simultaneous interaction

of the boundary layer and inviscid flow is considered.

The purpose of the present paper is to give a new prescribed displacement-thickness

procedure, which is considerably simpler than that given previously in reference 3. The

new procedure is no more complicated than a direct boundary-layer calculation. This

procedure is presented for incompressible flow, and comparisons are made with the turbulent experimental data presented in reference 6. With a few modifications, this method

is then extended to compressible flow, and some sample calculations for laminar and turbulent separated flow are presented.

An axisymmetric version of the incompressible procedure has been successfully

interacted with an inviscid program described in reference 7 in which the relaxation solution of the potential equation is obtained. Calculations for the low-speed separated flow

over an axisymmetric boattail-sting juncture are discussed in reference 8.

SYMBOLS

An,Bn, Cn,Dn, _

En,Hn, Pn,QnJ

coefficients in finite-difference equations

A' E' H' "_

n' n' n'_

_' R' S'l

coefficients in recurrence equations

if

cf

skin-friction coefficient

skin-friction coefficient multiplied by
f

F

h

k

transformed stream function

perturbation stream function in N-method

nondimensional velocity ratio, u/u e

= So(0 - 1) dg

;0 for two-dimensional flow

\1for axisymmetric flow

constant of proportionality used in similar form of equations for K_-method

K ratio of two adjacent q-mesh increments

L reference length

m perturbation mass flow

M Mach number

N maximum number of points across boundary layer

Npr

Npr,turb

q

Prandtl number

turbulent Prandtl number

column iteration counter

r B

R

body radius

free-stream Reynolds number,

p_UL

/'LOG
T

Coo

V

V

x

o/

]

¥

6*

_x

Aft

4

nondimensional static temperature, To, _

velocity component parallel to surface divided by

free-stream velocity

velocity component normal to surface divided by

transformed normal-velocity component

coordinate along surface divided by L

V_

coordinate along surface divided by L and multiplied by

= (V - 1)Me2

pressure-gradient parameter for Levy-Lees formulation,

pressure-gradient parameter for iS-method, d(ln Ue)

d_

du e

pressure-gradient parameter for 5*-method, 5*2Ue

dx

ratio of specific-heat coefficient

transformed displacement thickness multiplied by

displacement thickness, _7 (l - ---P-_ dy

PeP e,]

dimensional displacement thickness

U and multiplied by R_

2 _ dUe

u e dE

increments in dependent variable between two successive column iterations

grid spacing in x-coordinate

grid spacing in _-coordinate
grid spacingin y-coordinate

grid spacingin f-coordinate

eddy-viscosity coefficient

=i+ _

E Npr =1+

/_ Npr,tur b

transformed normal coordinate for 6*-method

transformed normal coordinate for _-method

transformed normal coordinate for Levy-Lees formulation

static-temperature ratio, T/T e

molecular-viscosity coefficient

transformed tangential coordinate

density divided by Po_

nondimensional stream function

perturbation stream function in 5*-method

OA vorticity

Subscripts:

edge _ boundary layer

m_n indices for 4- and q-coordinates, respectively
w wall

oo free stream

ANALYSIS

Incompressible Boundary-Layer Formulation

The boundary-layer forms of the nondimensional continuity and x-momentum equations for incompressible flow are

+ = 0 (1)

_x 0y

and

dUe u--+v--=u e-+ ayk 0y} (2) ax _y dx

The normal coordinate y and velocity

R_. For laminar flows, the quantity

unity, whereas for turbulent flow

v have been scaled in the usual manner by

_, which appears in equation (2), is set equal to

(3)

where e/p is the ratio of the eddy-viscosity coefficient to the molecular-viscosity

coefficient. The eddy-viscosity coefficient is used to relate the Reynolds stress to the

velocity gradient in the usual manner; thus

-pu'v' = e _u (4)

_y

In the present calculations, a two-layer eddy-viscosity formulation has been used and is

discussed by Cebeci and Smith in reference 9. In the inner region the formulation is

based on Prandtl's mixing length model along with the van Driest damping factor. In the

outer region Clauser's velocity defect model is used along with Cebeci's low Reynolds

number correction. This two-layer eddy-viscosity model has been successfully used for

many attached-flow calculations with mild pressure gradients; for separated flow it

serves only as an approximate model. The purpose of the present paper is to develop a
computational techniquefor separatedturbulent flows; at a later time this method, when

used to make computations for comparison with separated turbulent data, could conveniently be used to test new and improved turbulence models.

A key feature of the present inverse boundary-layer method as well as that given

previously (ref. 3) is the elimination of the continuity equation by introducing the stream

function

u = _ (5)

_y

v = ----_ (6) 8x

The combination of equation (5) and the definition of displacement thickness

asymptotic value of the stream function at the boundary-layer edge

-u e(y- 5") as y_ o_

As a result, a perturbation stream function _ can be introduced as

6" gives the

(7)

u(y- (8)

such that t_ - 0 as y - _. In this manner a boundary condition is imposed on the

stream function at the outer edge provided that the displacement thickness is known.

This condition replaces the usual outer boundary condition u - u e as y -- _¢. If the

perturbation stream function is introduced into the governing equations (1) and (2) and the

normal coordinate is conveniently scaled by 6" to give _ = y/6*, then the governing

equations become

8__ 6" [ rl) 8u (9)

_*2u_xaU-_, _ + (_ - l_uoj _ = Z + (10)
where

dUe (11)

=-5*2Ued--xTheseequationscanbe solvedeither in the usual direct modewith ue given or in the

inverse modewith 5" prescribed and ue and fi unknown. In the direct mode 5",

which appears in equations (9) to (11), is not the displacement thickness but is some given

function of x, such as 5" = 2_. In the direct and inverse modes the boundary conditions

are

u=_=0 at rT=0

Direct: u-u e as r7 -

Inverse: _- 0 as _ -

(12)

In the inverse mode it is assumed that the y-derivatives vanish at the outer boundary;

consequently, the x-momentum equation becomes

5.2u 0u = _ (13)

0x

The discussion on the numerical scheme shows that the unknown pressure gradient

is deduced implicitly along with the solution for u and _. Hereafter this formulation,

combined with the numerical procedure described in the following section, is referred to

as the 5*-method.

In the previous 5*-method presented in reference 3, the m_known pressure gradient

was eliminated by differentiating the x-momentum equation with respect to y, thus introducing the vorticity transport equation. The present formulation is preferable because,

not only can it be solved in either the direct or inverse mode, but it avoids the complication of having to solve for the unknown surface vorticity as was required in the previous

formulation. Furthermore, the vorticity formulation introduces the second y-derivative

of the eddy-viscosity coefficient, as shown in reference 10. This formulation was sometimes found to cause oscillations in the outer part of the boundary layer when the twolayer model described previously was used.
Numerical Procedure

The governing equationsare solvedby using the Crank-Nicolson finite-difference

schemewith Newtonlinearization. In the reversed flow region the Reyhnerand F1LiggeLotz (ref. 11)approximation

_u

u--=0 for u <0 (14)

ax

is used to prevent instability while preserving the usual rapid forward-marching scheme

used for the boundary-layer equations. In a previous paper, Carter and Wornom (ref. 12)

showed that a similar approximation

u--=0 for u <0 (15)

ax

introduces negligible errors in the reversed flow region if the absolute value of the

reversed flow velocity is less than approximately 0.10u . Cebeci (ref. 13) repeated some

of the calculations made by Carter and Wornom (ref. 12) and showed that the approximations given in equations (14) and (15) yield about the same result.

Finite-difference equations.- A variable grid is used across the boundary layer to

account adequately for the two-layer structure of a turbulent velocity profile with as few

grid points as possible. With n used to denote the grid-point index in the normal direction, the first and second derivatives are approximated by

0o_ = Un+l - Un-i (16)

n (i + K) ArTn_I

and

__-(_/I _? n - K(1 + K)2 Z_77n_ 2 1 I n+l Un+l - (_n+l+K_n_l/Un +Ken_lUn_l 1

(17)

where AT?n-1 = _?n - _?n-1 and the constant K = A_n/AT?n_l, which is a variable grid that

has been used by a number of investigators of turbulent boundary layers. The finitedifference forms of the governing boundary-layer equations were obtained by using the

difference expressions given in equations (16) and (17) in conjunction with the CrankNicolson scheme. Newton linearization was used on the nonlinear terms to accelerate the

convergence of the iterative solution made at each streamwise location. The linearized
finite-difference form of the stream function andx-momentum equations,given in equations (9)and (10), are written as

5_n-5_n-1 + Pn(bUn- SUn-l) =Qn (18)

and

AnbUn_ 1 + Bn6U n + CnbUn+ 1 + Dn6tk n = E n + HnSfl (19)

where 5u, 5_, and 5fi (inverse mode) denote the change in the dependent variable at a

given point between two successive column iterations; that is

5Un = uq+ln - uqn (20)

where q is the iteration index. The coefficients in equations (18) and (19), which are

assumed known from the previous iteration, are given in appendix A.

The truncation error in equations (16)and (17)is 0ffl- K)Ar/n_l_; thus, the scheme

is second-order accurate provided that K = 1 + 0(A_). Blottner (ref. 14) used the same

finite-difference expressions and demonstrated numerically that they are second-order

accurate. Blottner reports to have made these calculations for a fixed value of K = 1.82;

however, it can be shown that this statement is in error, and the actual K used by

Blottner depends on the number of points N used across the boundary layer and is given

by

K = 1.8210/N (21)

Equation (21) reduces to

K= 1 + 10 In 1.82

N

(22)

as N - _ and thus shows that K = 1 + 0(A_), thereby resulting in a second-order

accurate scheme.

Newton linearization of viscous term.- The number of column iterations in the turbulent calculations could be halved by using Newton linearization on the viscous term in

the inner region. When the shear is positive in that region (the following procedure was

used only in that case), the viscous term is written as

10
,23,

where c I is assumed known. With the difference expression given in equation (17) and

Newton linearization, the viscous term can be expressed as

X6u n+K(2_ n_12 - 1) 5Un_ 1 + _n+_1 Un+l n+_ n-- )Un

+K_n__21 Un-I 1

J

(24)

Comparison of equation (24) with the usual formulation, where Newton linearization is not

used on the eddy-viscosity coefficient

n

K(1 +K)_0n_l n+lbUn+l 1 +K_ 5u n

+ K_ 1 SUn-1 + g 1 Un+l -

n --2 n+g _n 1 + K_ 1/U n+2 n-

-1

+ K_ 1 Un-ll

n-_ J

(25)

shows that Newton linearization of the viscous term adds only a slight complication.

Keller and Cebeci (ref. 15) were the first to use Newton linearization on the boundarylayer equations; however, they did not use it on the turbulent viscous term. Nevertheless

they claimed that their calculations converged quadratically; that is, the residuals decrease

as

11
5uq+ 1 -- (5uq)2 (26)

n \ n

with each subsequent iteration.

This convergence rate was observed only in the laminar calculations in the present

study; even with the Newton linearization on the inner eddy-viscosity law, the present

turbulent calculations did not converge quadratically. On the basis of the observations

in the present paper, it is not clear how Keller and Cebeci (ref. 15) obtained quadratic

convergence for their turbulent calculations.

Solution of linearized difference equations.- Repeated application of equations (18)

and (19) across the boundary layer results in a system of block tridiagonal linear equations, which are solved by the recurrence relations

bUn = E'n - H'n5_ - Ai15Un+ I (27)

and

where

!

5_n = Qn- S'nbfl- R'nSUn+l

En - AnE'n-i - Dn Qn + Qn-IT + PnEn-I, N

E'n=

ql

Hn +An n-1 + Dn S -1 + Pn H -

ql

A' - Cn

n ql

(28)

(29)

12
and

' = ' -E' Qn Qn + Qn-1 + PnE'n-i nq2

' :S' Sn n-I + PnH'n-i - H'nq2

' = A'

R n - nq2

q2--n-i +Pn (1 + A,) n-I

(30)

The recurrence constants given in equations (29) and (30) are computed from the wall to

the edge of the boundary layer. Since the surface boundary conditions are all homogeneous (i.e., 5Ul= 5_1=0), the recurrence constants are initialized at n=l by

setting

(31)

In the inverse case the perturbation in the edge velocity 6u N (where N denotes the

grid point at the boundary-layer edge) and the perturbation in the pressure gradient 5/_

are deduced at the edge of the boundary layer. These quantities are found from the

simultaneous solution of the recurrence relations, which results in

? ^ T

SUN_ 1 = EN_ 1 - HN_16 fl - AN_15u N (32)

A T T _ t

5_N-1 = QN-1 - SN-15fl - RN-15UN (33)

and the finite-difference forms of equation (13) and equation (9), where the latter is evaluated at n=N- 1/2, gives

BNSU N = E N + HNSfl (34)

and

13
-5_N-l+ PN(SUN- SUN-l) =QN

in which the boundary condition 5_N = 0 has been imposed.

The solution of equations (32) to (35) results in

(35)

5uN =

QN +QN-1'+ PN E -1 +EN ¢'SN-1 + PNHN-1')

RN-1 ' + PN (AN-1) 1 + + BN(SN- 1 + PNHN - 1)

(36)

and then SUN_l, 5_N_l , and 5_ are deduced from equations (32) to (34). Now the

back-substitution process is initiated, and the process continues down to the wall by using

equations (27) and (28). Note that in this procedure the pressure gradient fl is deduced

simultaneously along with the boundary-layer solution u and _. This procedure is

similar to that developed previously by Reyhner (ref. 16). In a direct calculation, since

u e and fl are known, 5u N = 5_ = 0, and the back-substitution process begins by using

equations (32) and (33). In either the direct or inverse calculation, the column iterative

procedure is continued until

m i?Un 

/

where e c is set equal to 10 -4 in the present calculations.

tions typically required is 5.

(37)

The number of column iteraCompressible Boundary-Layer Formulation

The compressible formulation is obtained by using a suitable compressibility transformation followed by an additional transformation to obtain an inverse formulation.

These transformations could be considered together, but it is .¢impler to treat them separately. In the present study the Levy-Lees compressibility transformation (ref. 17), is

used, although an analogous inverse formulation would have resulted if the Stewartson

(ref. 18) transformation had been used. The nondimensional boundary-layer equations

expressed after the Levy-Lees transformation (ref. 19) are given as follows:

Continuity:

__8F+F+__=08V (38)

14
Momentum:

Energy:

+ ^

Npr

(39)

(4O)

where the coordinates are transformed by

= PeUe_er B dx

and

(41)

- epeuerjB 2_ _0Y _eP dy

and the dependent variables are given by

(42)

F- u

u e

__

V __

T

Te

2j ax

PeUetterB

(43)

Additional quantitiesin equations (38) to (40) are given by

-%

l- P_

PeP-e

= (y - 1)M 2 (44)

15
The boundary conditions are given as follows:

Wall boundary:

F(_,0) = V(_,0) = 0

=

or

(45)

Edge conditions:

F and 6- 1 as q -

The eddy-viscosity formulation used is the same as that in the 5*-method.

bulent Prandtl number enters the analysis in the compressible case in the term

(46)

The turwhere

e Npr (47) }=I+

tt Npr,turb

_ u'v'(0T/0y) (48)

Npr'turb v'T'(au/ay)

Harris (ref. 19) presents a discussion of the turbulent Prandtl number; in the present

analysis Npr,tur b = 0.95.

The development of the inverse compressible formulation proceeds in an analogous

manner to that developed previously for incompressible flow. A stream function f is

introduced as

af _ F (49)

and the continuity equation is integrated to give

16
where the boundary condition V(},O) = 0 has been imposed. Introduction of a perturbation stream function is desired so that an outer boundary condition can be imposed from

the prescribed displacement thickness. This boundary condition replaces the usual condition of fi, the pressure gradient, being prescribed. The edge value of the stream function is

f_ O0

= '_0 F d_ (51) fe

The compressible displacement thickness is given by

which becomes, after the Levy-Lees transformation

6'= _. _f0_ (0 - F) d_/ (53)

OeUer _

Combining equations (51) and (53) results in

fe = ,fO 0 dfl PeUer]BS*

A perturbation stream function is defined as

_0 _ OeUerJBS" 1

such that the stream-function boundary conditions are

i(_,0) = _(_,o_) = 0

It is convenient to prescribe the perturbation mass flow

(54)

(55)

(56)

17
-=PeUerJBS* (57)

instead of the displacement thickness because this grouping of quantities appears in the

definition of the perturbation stream function. An inverse boundary-layer procedure

based on mass flow instead of displacement thickness should be useful in a complete

viscous-inviscid interaction since the effect of the boundary layer on the inviscid flow may

be represented by imposing an injection (or suction) velocity at the solid surface given by

1 d_

V n -

Per_ dx

(58)

It is also convenient to simplify the normal coordinate by introducing

= (59)

m

which in terms of physical variables gives

0_P dy

5"

Note that _ is the usual Dorodnitsyn variable (ref. 20) scaled by the displacement thickness. Use of this variable helps to reduce the effect of boundary-layer growth in the

computation. With these definitions, the perturbation stream function becomes

F_ (_ _ 1 + h) (60)

T=f

where

h(_) -- (_)- 1) d_

The inverse compressible formulation is deduced from equations (38) to (40) by

transforming from _ to _ and from V to f. The resulting system of equations is

referred to as the prescribed perturbation mass-flow formulation and is given as

18
Of'_ _ (1 -_ - h) aF

_2F _F-m-_f'_+_tF(F/- a-_- l+h)_

(61)

(62)

and

(63)

These equations are solved for F, 0, _, and _- d\ln[ Ue/

d_

distribution _ subject to the following boundary conditions:

for a prescribed streamwise

Wall boundary:

F(_,0) :'f(_,0) : 0

: ew( )

(64)

or

Edge conditions:

F=O-1 and _--0 as @_.oo (65)

Hereafter this formulation, combined with the numerical technique, which is now described,

is referred to as the _-method.

The Crank-Nicolson finite-difference scheme with Newton iteration is used to solve

equations (61) to (63). The resulting finite-difference equations and solution technique are

presented in appendix B since the technique is the same as that described previously. The

energy equation is uncoupled from the combined stream function and x-momentum equations; as a result, quadratic convergence is not observed in the iterative convergence. At

19
a later time, whenthe present schemeis embodiedin a completeviscous-inviscid interaction problem, it is plannedto treat all three governingequationsin a coupledmanner

by using Newtoniteration. The resulting finite-difference equationswill be a block tridiagonal system with eachblock being a 3 x 3 matrix provided that h(_) is assumedto

beknownfrom the previous boundary-layer sweep.

RESULTSANDDISCUSSION

ComparisonWith Experimental Data

ChuandYoung(ref. 6) reported anexperiment in which measurementswere made

on a flat plate approximately 21⁄2m long, which was first subjectedto a favorable pressure

gradient andthen to an adversepressure gradient. This pressure distribution was

obtainedby placing a 7.5-cm-diameter circular cylinder 11cm abovethe flat plate with

its axis parallel to the plate andnormal to the windstream.

Calculationswere madefor this low-speedflow (U = 18.7 m/s) with the 5*

deduced by Chu and Young from velocity profile measurements prescribed in the streamwise direction. The 5*-method is used instead of the incompressible version of the

l-_-method since, with the latter fornmlation, the computation is done in terms of a transformed x-coordinate

= _0 _ u e dx (66)

and requires m = UeS* to be prescribed as a function of _. Since u e is unknown, it is

preferable in the present application of inverse boundary-layer theory to use the 5*-method

which requires only 5*(x) to be specified. Note that in a complete interaction calculation,

in which an appropriate inviscid scheme is combined with the _-method, repeated streamwise computations are made, and therefore the most recent distribution of u e can be

used to transform from the x-coordinate to the _-coordinate and conversely.

The upstream profile of u was obtained by using Coles' (ref. 21) wall-wake velocity

distribution. This profile formulation contains three constants, which are determined by

matching with experimental data. In the present case four quantities, 5", cf, _, and Ue,

were known from Chu and Young's measurements. In order to use all the data to deduce

the constants in Coles' formulation, a least-squares error calculation was used. Also,

the wall-wake profile was modified near the surface by Kleinstein's formulation (ref. 22),

thereby accounting for the laminar sublayer. At the outer edge of the boundary layer an

exponential function was used to force the wall-wake profile for u to approach asymptotically ue. After the u-profile was known, the perturbation stream function _ was

2O
deducedby integrating equation(9) from _?= 0 where _(},0) = 0. Ninety-three grid

points were used across the boundary layer with K = 1.09.

A cubic spline fit to the displacement-thickness distribution measured by Chu and

Young is shown in figure 1. This curve fit to the data was obtained by using a computer

program developed by Smith and others (ref. 23) to represent the desired function by a

cubic polynomial over prescribed intervals. Note that the displacement thickness shown

in figure 1 is dimensional and is denoted as 5". In order to demonstrate the capabilities

of the present technique in separated flow, the prescribed 5*-distribution extends in figure 1 beyond the data of Chu and Young in a manner typical of a turbulent boundary layer

incurring both separation and reattachment.

Figure 2 shows the comparison of the computed skin-friction coefficient on the flat

plate with the experimental distribution obtained by Chu and Young. The agreement is

good despite the use of an algebraic eddy-viscosity model, which is generally considered

to be inappropriate for turbulent boundary layers subjected to large pressure gradients.

Chu and Young also applied several direct boundary-layer procedures to these data and

found that only the Bradshaw (ref. 24) and Kuhn and Nielsen (ref. 25) procedures gave

good agreement with the data. None of these direct procedures can be used downstream

of the _eparation point as exhibited by the present inverse calculation shown in figure 2.

Figure 3 shows good agreement between the computed and measured velocity distribution

at the edge of the boundary layer on the flat plate. Although more data are needed to

assess the present method in a separated region, the good agreement shown in figures 2

and 3 between theory and experiment is encouraging and motivates the further use of

inverse boundary-layer theory. The present comparison was simplified by the use of

the experimental 5*-distribution, thereby requiring only one streamwise boundary-layer

calculation to be made. In a complete interaction calculation it is necessary to combine

a suitable inviscid calculation with an inverse boundary-layer calculation and to solve the

two simultaneously until convergence. A study demonstrating this technique for incompressible flow is reported in reference 8.

Results Obtained With _-Method

Similar solutions.- A similar form of the mass-flow prescribed formulation is

obtained by setting _ = k 2_-, where k is a prescribed constant, and by assuming that

! = 0. The similar form of these equations for incompressible laminar flow is given by

f' = k(1 - _) F' (67)

and

21
(68)

with the boundary conditions

_(0) = F(0) = 0 (69)

and

-0 and F- 1 as _- _ (70)

where the prime denotes differentiationwith respect to _. A general, compressible,

similar formulation cannot be obtained from the _-method because the edge Mach number M e appears explicitlyin the energy equation. If 13_ 0, then M e = Me(1) , and thus

the energy equation is not independent of 4. A compressible similar formulation can be

obtained by using the total-enthalpyform of the energy equation as was done by Cohen

and Reshotko (ref.26), thereby eliminating M e from the equation when the Prandtl number is unity.

Incompressible similar solutions were obtained by solving equations (67) and (68)

for different values of k subject to the boundary conditions given in equations (69)

and (70). The prescribed constant k is easily shown to be equal to 8, where 8 is the

transformed displacement thickness expressed in terms of the Levy-Lees coordinate _;

thus

8 = (1 - F) d_ (71)

Results of these solutions are compared in figure 4 with so-called exact solutions, which

previous investigators (e.g., see ref. 27) have obtained by numerical means. Excellent

agreement is obtained. It is observed in figure 4 that for each value of fl, the pressuregradient parameter, there are two values for both the transformed displacement thickness 5 and the wall shear F w. Thus, in a direct calculation, that is when fl is

prescribed, it is not clear what values of

unless a shooting technique was used, the

sponding value of 8 always resulted for

presented a double iterative procedure to

Fw and 8 will be obtained. Previously,

forward flow solution F w > 0 and correa given value of ft. Werle and Bertke (ref. 28)

obtain the reverse flow solutions Fw < 0, which

were originally discovered by Stewartson (ref. 29). In the present approach there was no

difficulty in obtaining either the forward or reversed profile because figure 4(a) indicates

that, given a value of 8, a single value of _ results.

22
Laminar separation.- As a demonstration of the present procedure for computing

separated boundary layers, the _-distributions shown in figure 5(a) as a function of

were prescribed for both an incompressible and a compressible calculation. In the compressible case, Moo = 2 and the wall temperature was set equal to the free-stream

stagnation temperature with Npr = 1. The two _-distributions in figure 5(a) differ by a

constant so that both _-distributions match that of a flat-plate solution at _ = x = 1.

The initial boundary-layer profiles at this point were flat-plate similar solutions.

The resulting calculations for both cases are shown in figures 5(b), 5(c), and 5(d)

for the edge velocity, skin friction, and displacement thickness, respectively. These

quantities are plotted against the physical coordinate x, which is obtained after the calculation is completed from

_ d_ (72) x = =0 PeUe_te

which for isentropic flow and a linear temperature-viscosity law becomes

x=_}_0 d_ (73)

Ue +_--_ M2(1-u 2 7/(7-1)

Although figures 5(b) to 5(d) show that the calculations for Moo = 0 and Moo = 2 differ

in detail, the resulting solutions show the same basic features. This result is expected

because equation (73) shows that if u e remains close to unity (as shown in fig. 5(b)),

then the physical and transformed streamwise coordinates are nearly the same, independent of the free-stream Mach number. Both solutions are smooth, and no difficulty was

encountered near the separation or reattachment point in either solution. As a check on

the computer program for the _-method, the displacement thickness shown in figure 5(d)

for the incompressible case was prescribed for the 6*-method. The resulting solutions

for u e and cf were identical to those shown in figures 5(b) and 5(c), respectively,

for M = 0.

oc

Turbulent separation.- A turbulent computation, similar to that shown for laminar

flow, was made to demonstrate the present procedure for both an incompressible and

compressible separated turbulent boundary layer. The prescribed _-distributions are

shown in figure 6(a) for the calculations with Moo = 0 and Moo = 2. As was the case for

the laminar calculation, the prescribed _-distributions match a turbulent flat-plate solution at the initial streamwise station x = } = 0.95 ,(Roo,x = 0.95 × 106).. When this calculation was made the computer program using the _-method had not been written so that

23
a direct calculation could be made. Hence, the initial boundary-layer profiles had to be

obtained elsewhere. In the incompressible case the program for the 6*-method was run

in the direct mode with _ -- 0, and the flow was assumed to be turbulent from the leading

edge of the flat plate. For the compressible case the upstream profiles were obtained

with the computer program reported in reference 30.

The resulting distributions of edge velocity, skin friction, and displacement thickness are plotted against the physical coordinate x in figures 6(b), 6(c), and 6(d), respectively. In both cases, flow separation and reattachment occurred, and no difficulty was

encountered in obtaining a smooth solution. In contrast with the laminar case, the distributions of u e shown in figure 6(b) differ significantly from unity, thereby resulting in

different inverse transformations from the f-coordinate to the x-coordinate using

equation (73).

In view of the different distributions of Ue, it is not surprising that large differences

occur between the distributions of skin friction and displacement thickness for Moo = 0

and M = 2 shown in figures 6(c) and 6(d), respectively. A common feature is that both

solutions show that the points of reattachment and maximum displacement thickness occur

close to the point of minimum-edge velocity (maximum pressure). This trend has been

observed in separated turbulent flow measurements, such as those of Alber and others in

reference 31.

Viscous interaction.- With the development of the _-:-method described in this paper,

efforts should now be directed at the complete problem of viscous interaction. This

inverse boundary-layer procedure, which to the author's knowledge is the only compressible, inverse, finite-difference procedure based essentially on displacement thickness,

appears to be well suited for viscous interaction as shown by equation (58). As mentioned

previously, a study has been completed using the 5*-method on axisymmetric boattails and

is reported in reference 8. The results and interaction scheme seem promising. Several

compressible interaction calculations using inverse integral procedures have been

reported in the literature, such as those by Kuhn and Nielsen (ref. 4) and Thiede (ref. 32).

Integral techniques have the disadvantage that the shape of the velocity profile must be

specified by the user, whereas the finite-difference calculation does not. Also, the present

finite-difference inverse boundary-layer procedure should provide a good means to test

various turbulence models for separated flow much more economically than can be done

with solutions of the Navier-Stokes equations. Most of the current Navier-Stokes computations are forced to use relatively coarse grid spacing in the viscous region adjacent to

the body region. In constrast, boundary-layer procedures provide complete resolution of

the viscous region because the computation is confined entirely to the viscous region.

The present m-method should be useful for separated flows with supersonic as well

as subsonic edge conditions. The main motivation for the development of inverse

24
boundary-layer theory has been directed at subsonic separated flows to eliminate the

singularity encountered at the separation point. Werle and Vatsa (ref. 33) developed a

new supersonic viscous interaction technique, which was a major achievement because

it converted the calculation from an ill-posed initial value problem to a well-posed

boundary-value problem. The disadvantage of their procedure was that, unless a great

deal of care was taken, a singularity was obtained at the point of separation. If the present f-I-method were used, while retaining the interaction model of Werle and Vatsa, it is

anticipated that a well-posed boundary-value problem would result with no singularity at

the separation point.

CONCLUDING RE MARKS

Two inverse boundary-layer formulations and solution techniques are presented for

separated boundary-layer calculations. In the first technique, which is only for incompressible flow, the displacement thickness is prescribed; comparisons made with this

method show good agreement with experimental low-speed turbulent data up to the point

of separation. Further comparison with experimental data is needed to assess the

accuracy of the present boundary-layer method in a region of separated flow. It is anticipated that the boundary-layer approximations are still valid provided that the separated

region is not large but, based on the current literature, significant improvements are

needed in the turbulence model for a separated region.

The second inverse technique presented in this paper is applicable to compressible

flows and is based on a prescribed perturbation mass-flow distribution. A solution technique is presented for this formulation, and sample calculations are presented for laminar

and turbulent incompressible and compressible flows. In all these calculations the solution passes smoothly through the points of separation and reattachment.

Both inverse procedures appear to be well suited for use in complete viscousinviscid interactions because they are based primarily on the displacement thickness.

The next step in the development of a complete viscous-inviscid interaction theory is to

combine the compressible inverse procedure with an appropriately modified inviscid technique for subsonic or supersonic flows. This viscous-inviscid iteration technique can

then be compared with experiment to test its accuracy as well as the simultaneous

development of more sophisticated turbulence models. Interacting boundary-layer theory

provides a more economical means by which different turbulence models for separated

flows can be tested than current Navier-Stokes computations. Finite-difference procedures

25
shouldprove superior to integral techniquesbecausethe latter requires the specification

of the shapeof the velocity profile.

Langley ResearchCenter

NationalAeronautics andSpaceAdministration

Hampton,VA 23665

July 19, 1978

26
APPENDIXA

COEFFICIENTS OF FINITE-DIFFERENCE EQUATIONS FOR 6*-METHOD

The stream function and x-momentum equations, given by equations (9) and (10), are

differenced about the points (m,n-1⁄2) and (m-l,n), respectively, where 2n denotes

the x-grid-point index and n denotes the T/-grid-point index. The resulting finitedifference equations are .given by equations (18) and (19) in which the following coefficients

appear:

1

m_n -

A n = e - (AI)

(1 +K) 2 A77n- 1

1 +K_ 1

m,n+ m,n

Bn = 2aUm,n -fcl + (A2)

K(1 + K) A 2

_?n-1

C n = -e -

m,n+-

2

K(1 + K) 2 AUn_ 1

(A3)

E n

Dn = -blf (A4)

= fl 1 +a u m-l,n- Um,n 2 )

m --

2

+ ef

+

K(1 + K) 2 m,n + m,n+l 1

A_?n_1 m,n +_ m,n -

U

m,n

+ Kg lUm,n-1 + g 1 Um-l,n+l -

m,n -_ m-l,n +_-

+Kg m-l,n _ Um_l,n_ _

- -

(g m-l,n +_1 +K_ m-l,n - 1) Um-l,n

(A5)

27
APPENDIXA

Hn= 1 (A6)

P

n = r/ lnm

(A7)

Qn = _m,n-1 - _m,n- Fn(Um,n - urn,n-l) (A8)

5*2 1

m --

a- 2

2Ax (A9)

6* 1

bl = 2(1 + K)A_n_ 1 \A-_-'x 1

1]l - --2

m 1

b2 = 2(1 + K)At/n_ 1 \A-xx 2_2-@' l

11"I--- 2

5* i(77n-I)

m-_- (5* 1 dS* 1

el = 2(1 + K) k,Tn_ 1 _ + 2 -d-x-x] 1

2

c2= 2 ld6"l _x 2 _-x '/ 1

2(1 + K) At/n_ 1 m --

2

(A10)

(All)

(A12)

(A13)

and

e = bl_m, n - b2_'m_l, n + ClUm, n - C2Um_l, n

(A14)

f = urn,n+ 1 - urn,n_ 1 + Um_l,n+ 1 - Um_l,n_ 1

(A15)

28
APPENDIX A

Note that

1

n+_

is given by

1 =

n+_

_n + _n+l

(A16)

29
APPENDIX B

FINITE-DIFFERENCE EQUATIONS AND SOLUTION TECHNIQUE

FOR _-_-ME THOD

Application of the Crank-Nicolson scheme with a variable grid and with Newton

iteration to equations (61) and (62) about the points (m,n-1) and (m-l,n),resp eetively, results in the following system of linearized finite-difference equations where the

notation is the same as that used previously:

AnSFn_ 1 + BnSF n + CnSFn+ 1 + Dn6f n = E n + HnS_

(B1)

_n- 5_n-1 + Pn(6Fn - 6Fn-1) =Qn (B2)

where

An= e I -

(1 + K)A_2_1

(B3)

Bn = 2aFm,n - cle2 + (ffl2fl) 1 IFm'n +$ m-l'n )

m-_

1 L7 ) + K(/_) (B4) + K(1 + K) -2 1A_n_l m,n+_ m,n -

C n = -e 1 -

(Zg) 1

m,n+g

K(1 + K) _-2_n-1

(B5)

D n = -ble 2

(B6)

30
E n =a(F_m_X,n- F2D

APPENDIX B

+ ele2 + Hnfi 1

Irl---

2

+

K(I + K) A_2_I Qlg) m,n+2 1 Fm,n+l - _lF) m,n 1 + K(/g) _Fm, n +2 m,n--

+ K(/g) 1 Fm,n-1 + (/g) 1 Fm-l,n+l

re,n-2 m-l,n+_

- IlT) + K(/7) m-l,n+2 I- m-l,n- Fm-l,n

(B7)

H n

m-_ i_

Im,n+ m-X,n 2 - 2(BS)

=

m

_l-l+h

2 m) (B9)

Qn =Tm,n-1-Tm,n + Pn(Fm,n_l - Fm,n) (BI0)

The additional coefficients which appear in equations (B3) to (B10) are given by

m2

1 m --

2

a2A_ (Bll)

31
APPENDIX B

bl=

m 1

2(1 + K) ASn_ 1 1

Ill--

2

(B12)

m

m 1

b2 = 2(1 + K)_Tjn_ 1 2 1

Ill - --

2

(m3)

1 _1 1 ram(On - 1 + hm) (B14)

Cl = 2(K + 1) A_n_IA} m-_

1 m 1 ff]m-l(_n - 1 + hm_l) (BI5)

c2 = 2(K + I) A_n_IA _ m -_

el = bl_m,n _ b2_m_l,n + ClFm,n _ C2Fm_l,n (B16)

e2 = Fro,n+ I - Fro,n_ 1 + Fm_l,n+ 1 - Fm_l,n_ 1

(B17)

In the present calculations for simplicity l - PP - 1, which is an approximation frePete

quently used by others. Also, Newton linearization of the eddy viscosity, in the

x-momentum equation was used in the same manner as described in the main text for the

6*-method.

As in the 5*-method equations (B1) and (B2) can be solved either in the inverse mode

where K_ is prescribed and fi is deduced or in the direct mode where _ is prescribed.

In the direct mode ff m = V_, then _ = _, _ = t3, and the resulting formulation is the

usual Levy-Lees boundary-layer formulation where the transformed normal-velocity component has been replaced by

v : -2_-_ _ +F(_ - 1 +h

32
APPENDIX B

Equations (B1) and (B2) are solved by the same solution technique presented for the

incompressible formulation in the main text. In the inverse case, 6fl, 5FN_I, and

5{N_ 1 are deduced from the simultaneous solution of

i i --

5FN_ 1 = EN_ 1 - HN_15 fl (B19)

t i --

6fN-1 = QN-1 - SN-lbfl (B20)

5_N-I + PN6FN-I = -QN (B21)

where the boundary conditions 6F N = 5_N = 0 have been applied. In the direct case equation (B21) is replaced by

_N - 5_N-1 - PNSFN-1 = QN (B22)

and equations (B19), (B20), and (B22) are solved for 5FN_I, 5_N_l, and 6_N with

6fl = 0. The back-substitution procedure is then carried out from the boundary-layer

edge to the surface.

After F and "f are updated by solving the x-momentum and stream-function

equation, the linearized form of the energy equation is solved. The finite-difference form

of this equation is given by

An60n_ 1 + Bn50 n + Cn60n+ 1 = D n (B23)

where

1

m,n - --

2

A n = e I - (B24)

Npr(l + K) A_n_2 1

(l'_) 1 + K(/_) 1

m,n + _ m,n - -

Bn = a(Fm,n + Fm_l,n) + 2 (B25)

NprK(I + K) AUn_2 1

33
APPENDIXB

C n = -el -

(l?) 1

m,n +_

NprK(1 + K) Arln_ 2 1

(B26)

Dn=a(Fm,n+Fm-l,n)(0m_l,n-0m,n) +el(0m,n+l-0m,n_l

+Om-l'n+l-Om-l'n-1) +(l_)m-l,n (2 (1 + K)Zl_/n_ 1 ;) 2

X

Fm,n+ - Fm,n_l) 2 + (Fm_ 1 ,n+l- Fm-l,n-1)_

1

+

NprK(1 + K) A _n-12

X

_Z_) m,n +_10m,n+l-l/_) m,n +_-1 +K(/_) m,n - ___Om,n

+ K(/?) 1 0m,n-1

m,n -

+ (l?) 1 0m-l,n+l

m-l,n+g -

Il e)m-l,n +_1 + K(/}) m-l,n - _0m_l, n

h

+ K(/?) 1 0m-l,n-l_

m-l,n -_ -J (B27)

The solution to the energy equation is given by

50 n = D'n - An' 50n-1 (B28)

where

D' = Dn - CnD'n+l

n Bn CnA,n+ 1

(B29)

A' = An

n Bn _ CnA'n+ 1

(B30)

34
APPENDIX B

The constants D'n and A n' are evaluated from the outer boundary n = N with

t t

D N = A N = 0 to the wall. If the wall is isothermal, then 5_ 1 = 0 and the solution for

5_ n is deduced from equation (B28). If the heat transfer _/_fT(_,0) is specified, then

the finite-difference form of this Neumann condition is solved along with equation (B28)

to deduce 5_ 1. Then the back-substitution process given in equation (B28) begins.

35
REFERENCES

1. Bauer, Frances; Garabedian,Paul; Korn, David; and Jameson,Antony: Supercritical

Wing SectionsII. Volume 108of Lecture Notes in Economicsand Mathematical

Systems,Springer-Verlag, 1975.

2. Catherall, D.; and Mangler, K.W.: The Integration of the Two-Dimensional Laminar

Boundary-Layer Equations Past the Point of Vanishing Skin Friction. J. Fluid

Mech., vol. 26, pt. 1, Sept. 1966, pp. 163-182.

3. Carter, James E.: Inverse Solutions for Laminar Boundary-Layer Flows With

Separation and Reattachment. NASA TR R-447, 1975.

4. Kuhn, Gary D.; and Nielsen, Jack N.: Prediction of Turbulent Separated Boundary

Layers. AIAA Paper No. 73-663, July 1973.

5. Klineberg, John M.; and Steger, Joseph L.: On Laminar Boundary-layer Separation.

AIAA Paper No. 74-94, Jam-Feb. 1974.

6. Chu, J.; and Young, A.D.: Measurements in Separating Two Dimensional Turbulent

Boundary Layers. Flow Separation, AGARD-CP-168, Nov. 1975, pp. 13-1 - 13-12.

7. Keller, James D.; and South, Jerry C., Jr.: RAXBOD: A Fortran Program for

Inviscid Transonic Flow Over Axisymmetric Bodies. NASA TM X-72831, 1976.

8. Carter, James E.: A New Boundary-Layer Interaction Technique for Separated

Flows. NASA TM-78690, 1978.

9. Cebeci, Tuncer; and Smith, A. M.O.: Analysis of Turbulent Boundary Layers.

Academic Press, Inc., 1974.

10. Carter, James E.; and Wornom, Stephen F.: Solutions for Incompressible Separated

Boundary Layers Including Viscous-Inviscid Interaction. Aerodynamic Analyses

Requiring Advanced Computers - Part I. NASA SP-347, 1975, pp. 125-150.

11. Reyhner, T. A.; and Fliigge-Lotz, I.: The Interaction of a Shock Wave With a

Laminar Boundary Layer. Int. J. Non-Linear Mech., vol. 3, no. 2, June 1968,

pp. 173-199.

12. Carter, James E.; and Wornom, Stephen F.: Forward Marching Procedure for

Separated Boundary-Layer Flows. AIAA J., vol. 13, no. 8, Aug. 1975, pp. 1101-1103.

13. Cebeci, Tuncer: Separated Flows and Their Representation by Boundary-Layer

Equations. Report ONR-CR215-234-2, U.S. Navy, Sept. 1976. (Available from

DDC as AD A035 693.)

36
14. Blottner, F.G.: Variable Grid Scheme Applied to Turbulent Boundary Layers.

Comput. Methods Appl. Mech. & Eng., vol. 4, no. 2, Sept. 1974, pp. 179-194.

15. Keller, Herbert B.; and Cebeci, Tuncer: Accurate Numerical Methods for BoundaryLayer Flows. II: Two-Dimensional Turbulent Flows. AIAA J., vol. 10, no. 9,

Sept. 1972, pp. 1193-1199.

16. Reyhner, Theodore Alison: The Interaction of a Shock Wave With a Laminar

Boundary Layer. Ph.D. Diss., Stanford Univ., 1967.

17. Lees, Lester: Laminar Heat Transfer Over Blunt-Nosed Bodies at Hypersonic Flight

Speeds. Jet Propul., vol. 26, no. 4, Apr. 1956, pp. 259-269, 274.

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Proc. R. Soc. (London), ser. A, vol. 200, no. A1060, Dec. 22, 1949, pp. 84-100.

19. Harris, Julius E.: Numerical Solution of the Equations for Compressible Laminar,

Transitional, and Turbulent Boundary Layers and Comparisons With Experimental

Data. NASA TR R-368, 1971.

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Oxford Univ. Press, Inc., 1964.

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vol. 1, pt. 2, July 1956, pp. 191-226.

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Boundary Layers. AIAA J., vol. 5, no. 8, Aug. 1967, pp. 1402-1407.

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May 26, 1967, pp. 593-616.

25. Kuhn, Gary D.; and Nielsen, Jack N.: An Analytical Method for Calculating Turbulent

Separated Flows Due to Adverse Pressure Gradients. Tech. Rep. NEAR-I-PU,

U.S. Navy Project Squid, Oct. 1971. (Available from DDC as AD 731 744.)

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Laminar Boundary Layer With Heat Transfer and Pressure Gradient. NACA

Rep. 1293, 1956. (Supersedes NACA TN 3325.)

27. Christian, James W.; Hankey, Wilbur L.; and Petty, James S.: Similar Solutions of

the Attached and Separated Compressible Laminar Boundary Layer With Heat

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(Available from DDC as AD 705 581.)

37
28. Werle, M. J.; and Bertke, S. D.: A Finite-Difference Method for Boundary Layers

With Reverse Flow. AIAA J., vol. 10, no. 9, Sept. 1972, pp. 1250-1252.

29. Stewartson, K.: Further Solutions of the Falkner-Skan Equation. Proc. Cambridge

Philos. Soc., vol. 50, pt. 3, July 1954, pp. 454-465.

30. Anderson, E. C.; and Lewis, C.H.: Laminar or Turbulent Boundary-Layer Flows of

Perfect Gases or Reacting Gas Mixtures in Chemical Equilibrium. NASA

CR-1893, 1971.

31. Alber, Irwin E.; Bacon, John W.; Masson, Bruce S.; and Collins, Donald J.: An

Experimental Investigation of Turbulent Transonic Viscous-Inviscid Interactions.

AIAA J., vol. 11, no. 5, May 1973, pp. 620-627.

32. Thiede, P.G.: Prediction Method for Steady Aerodynamic Loading on Airfoils With

Separated Transonic Flow. Prediction of Aerodynamic Loading, AGARD CP-204,

Feb. 1977, pp. 16-1 - 16-12.

33. Werle, M. J.; and Vatsa, V.N.: New Method for Supersonic Boundary-Layer

Separation. AIAA J., vol. 12, no. 11, Nov. 1974, pp. 1491-1497.

38
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42
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43
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(a) Prescribed perturbation mass-flow distributions.

Figure 5.- Laminar-separation computations for T w = T

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x

x M_ = 0

x × M_ = 2

I I I I I I

1.2 1.4 1.6 1.8 2.0 2.2

x

(d) Computed displacement-thickness distributions.

Figure 5.- Concluded.

I

2.4

49
m

25 --

20--

15--

10--

5--

0

.8

Moo=0

M =2

/ \

I I I I J

1.0 1.2 1.4 1.6 1.8

(a) Prescribed perturbation mass-flow distributions.

Figure 6.- Turbulent separation computations for T w = To,o_ and Ro_

= 106.

48
u

e

1.1

1.0

.9

.8

.7u

.6

.8

X

X

X

x

X

X

X

X

X

X

X

X

X

X

X

X

=0

=2

I I I I I I

1.0 1.2 1.4 1.6 1.8 2.0

x

(b) Computed edge-velocity distributions.

Figure 6.- Continued.

49
cf

6 --

5--

4--

3--

2--

I--

0

--I --

-2

.8

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

X

M_ _ O

x M : 2

xx,?_,rx x

x x

x x

/

I 1 I I I I

1.0 1.2 1.4 1.6 1.8 2.0

x

(c) Computed skin-friction distributions.

Figure 6.- Continued.

5O
30--

25 m

20 m

15

10  ̧

0

.8

I I I I

1.0 1,2 1.4 1.6

X

(d) Computed displacement-thickness distributions.

Figure 6.- Concluded.

I

1,8

I

2.0

51
1. Report No. 2. Government Accession No.

NASA TP-1208

4. Title and Subtitle

INVERSE BOUNDARY-LAYER THEORY AND COMPARISON

WITH EXPERIMENT

7. Author(s)

James E. Carter

9. Performing Organization Name and Address

NASA Langley Research Center

Hampton, VA 23665

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration

Washington, DC 20546

3, Recipient's Catalog No.

5. Report Date

September 1978

6. Performing Organization Code

8. Performing Organization Report No,

L-12190

10. Work Unit No.

505-06 -33-08

'11. Contract or Grant No.

13. Type of Report and Period Covered

Technical Paper

14 Sponsoring Agency Code

15. Supplementary Notes

16. Abstract

Two new inverse boundary-layer computational procedures, which permit nonsingular

solutions at separation and reattachment, are presented. In the first technique, which

is for incompressible flow, the displacement thickness is prescribed; in the second technique, for compressible flow, a perturbation mass flow is the prescribed condition. The

pressure is deduced implicitly along with the solution in each f_f these techniques. Laminar

and turbulent computations, which are typical of separated flow, are presented and comparisons are made with experimental data. In both inverse procedures, finite-difference

techniques are used along with Newton iteration. The resulting procedure is no more complicated than conventional boundary-layer computations. These separated boundary-layer

techniques appear to be well suited for complete viscous-inviscid interaction computations.

17 Key Words (Suggested by Author(s))

Ae r odynamic s

Bomldary layers

Separated flow

Finite difference

18. Distribution Statement

Unclassified - Unlimited

19 Security Classif. (of this report) I 20. Security Classif.(of this paget 21. No. of Pages

Uncla ss if ted Unc la s si,f led 53

Subject Category 34

I 22. Price" $5.25

For sale by the National Technical Information Service, Springfield, Virginia 22161

NASA-Langley, 1978