NASA TECHNICAL. NOTE

CMCO

NASA TN D-7732

fp)f (EI I1WIFTHE NUMERICAL CALCULATION OF

LAMINAR BOUNDARY-LAYER SEPARATION

by John M. Klineberg and Joseph L. Sieger

Ames Research Center

Moffett Field, Calif. 94035

1974

^

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION • WASHINGTON, D. C. • JULY 1974

1. Report No.

TN D-77322. Government Accession No. 3. Recipient's Catalog No.

4. Title and Subtitle

THE NUMERICAL CALCULATION OF LAMINAR BOUNDARY-LAYER SEPARATION

5. Report Date

JULY

6. Performing Organization Code

7. Author(s)

John M. Klineberg and Joseph L. Steger

8. Performing Organization Report No.

A-5281

9. Performing Organization Name and Address

NASA-Ames Research CenterMoffett Field, Calif. 94035

10. Work Unit No.

501-06-01

11. Contract or Grant No.

12. Sponsoring Agency Name and Address

National Aeronautics and Space AdministrationWashington, D . C . 20546

13. Type of Report and Period Covered

Technical Note

14. Sponsoring Agency Code

15. Supplementary Notes

Presented at the AIAA 12th Aerospace Sciences Meeting, Washington, D. C., Jan. 30-Feb. 1, 1974.

16. Abstract

Iterative finite-difference techniques are developed for integrating the boundary-layerequations, without approximation, through a region of reversed flow. The numerical proceduresare used to calculate incompressible laminar separated flows and to investigate the conditionsfor regular behavior at the point of separation. Regular flows are shown to be characterizedby an integrable saddle-type singularity that makes it difficult to obtain numerical solutionswhich pass continuously into the separated region. The singularity is removed and continuoussolutions ensured by specifying the wall shear distribution and computing the pressure gradientas part of the solution. Calculated results are presented for several separated flows and theaccuracy of the method is verified. A computer program listing and complete solution case areincluded.

17. Key Words (Suggested by Author (si 1

Separated FlowBoundary-Layer FlowFinite-Difference Methods

18. Distribution Statement

Unclassified - Unlimited

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Unclassified

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Unclassified

21. No. of Pages

101

22. Price

$4.50

'For sale by the National Technical Information Service, Springfield, Virginia 22151

NOMENCLATURE

A matrix formed by difference equations

a local speed of sound

b(x) function of x, equation (9)

c constant

skin friction coefficient, — — -p *u *2CO OO

/

_ _

u dy

f1? f2, fs, . . . coefficients in Taylor series expansions, seeequation (38)

»-gH conditioning matrix, also Hu, Hv

h parameter for relaxation scheme

I identity matrix

J maximum j index

j discrete index in streamwise direction

K maximum k index

k ^ discrete index in normal direction

£ typical length scale

M , Mach numberdu

X 6m normalized velocity gradient, — -\ —

U dX

eHM

— X 6m normalized Mach number gradient, rr- -j- —

C

p fluid pressureu *lOO

R Reynolds number, -

R residual vector, also Ru, Rv\

111

p u xe eRex Reynolds number,

u*u normalized component, —^00

u* streamwise velocity component, physical variable

u transformed component, —

v* c;v normalized component, —5- /Roo

v* normal velocity component, physical variable

/x~ 1v transformed component, v /— + y (m - l)yu/ e

x*x normalized streamwise coordinate, —JO

x* streamwise coordinate

v* i—y normalized normal coordinate, -r- /Rx>

y* normal coordinate

/Vy transformed normal coordinate, y/ —• A

z u 2 - u2e

a arbitrary parameter

88 2

Ay22 r— , also ratio of specific heats

truncation error term or a small parameter

X eigenvalue of iteration matrix (I + hHA); also X = I (1 - u2)dyJo

y coefficient of viscosity

v kinematic viscosity, —

p fluid density

iv

a eigenvalue of HA

_ 9u

fy stream function, J u dy

a) relaxation parameter or vorticity

Subscripts

B backward difference operator

C central difference operator

e condition at edge of viscous layer

F forward difference operator

j,k location at a grid point or an index

max with J or K, maximum number of grid points j or k in the field

s condition at separation

x partial derivative with respect to x

0 constant value of u or v; also a quantity evaluated at y = 0

1,2 conditions on either side of a plane in physical space

°= far upstream condition

|| • || Euclidean vector norm or induced matrix norm

Superscripts ' !

* physical variable

transformed variable, see equation (5)

~ ' perturbation term

(n) iteration level

->- vector quantity

3 , d. g— , also -j— with equations (39)-(42)

THE NUMERICAL CALCULATION OF LAMINAR

BOUNDARY-LAYER SEPARATION*

John M. Klineberg and Joseph L. Steger

Ames Research Center

SUMMARY

Iterative finite-difference techniques are developed for integrating theboundary-layer equations, without approximation, through a region of reversedflow. The numerical procedures are used to calculate incompressible laminarseparated flows and to investigate the conditions for regular behavior at thepoint of separation. Regular flows are shown to be characterized by an inte-grable saddle-type singularity that makes it difficult to obtain numericalsolutions which pass continuously into the separated region. The singularityis'removed and continuous solutions ensured by specifying the wall sheardistribution and computing the pressure gradient as part of the solution.Calculated results are presented for several separated flows and the ccuracyof the method is verified. A computer program listing and complete ..olutioncase are included.

INTRODUCTION

During the past decade, various approximate methods have been developedto calculate separated flows by using the boundary-layer equations. The mostpopular schemes have been integral, or moment, methods based on the early workof Abbott, Holt, and Nielsen (refs. 1-3) or Lees and Reeves (refs. 4-8). Inthe integral approach, the boundary-layer equations are multiplied by a powerof u and converted into a system of ordinary differential equations by inte-grating across the viscous layer. Regions of attached and separated flow aretreated similarly because the average convection in the boundary layer isalways in the downstream direction.

A second type of approximate method, first proposed by Reyhner andFlugge-Lotz (ref. 9) uses finite-difference techniques (refs. 10 and 11).This approach uses a forward-marching procedure, with all convective deriva-tives set to zero in regions of reversed flow for numerical stability. Theconservation of momentum and energy is therefore violated in the portion ofthe separated flow bounded by the zero-velocity line, although the errorsintroduced by this approximation are not expected to be significant for smalllaminar separation bubbles. Both the finite-difference and integral methodshave produced good agreement with experimental data, particularly forcompression-corner flows and shock-wave/boundary-layer interactions (see thereview in ref. 12). ^

*Presented at the AIAA 12th Aerospace Sciences Meeting, Washington, D. C.,Jan. 30-Feb. 1, 1974.

The first finite-difference integration of the complete boundary-layerequations through a region of reversed flow was performed by Catherall andMangier (ref. 13). This report provides the best previous numerical evidenceof flows that are regular at separation. A continuous solution was obtainedby specifying the displacement thickness downstream of an appropriate pointnear separation and determining the pressure gradient by streamwise integra-tion. The numerical procedure developed instabilities in the reversed-flowregion, however, and the integration was continued only by decreasing the con-vergence criterion at each station. As the authors point out, this difficultyis to be expected because the region of separated flow should actually beintegrated in the upstream direction, with boundary conditions provided fromdownstream.

There have also been several numerical studies of nonlinear parabolicequations of mixed type, where the direction of increasing "time" reverses insome region of the flow field. One of these investigations, by Klemp andAcrivos (ref. 14), considers the flow over a finite, stationary flat platewhose surface moves at a constant velocity opposite that of the free stream(i.e., a. rotating belt). The boundary layer is divided into two regions alongthe unknown zero-velocity line and the equations are integrated in the appro-priate flow directions, with the final solution obtained by iterating for thelocation of the common boundary. It is not evident that this technique wouldprove effective for calculating boundary-layer separation because the regionof reversed flow results only from the upstream motion of the surface of theplate. Also, the pressure gradient is assumed to be zero and the shearstresses remain positive throughout the flow field. The singularities atseparation and reattachment are therefore caused by discontinuities in theboundary conditions and are not associated with the vanishing of the surfaceskin friction.

A more useful numerical procedure for calculating the flow past animpulsively started flat plate has recently been developed by Dennis (ref. 15).For this problem, the motion at short times is given by Rayleigh's errorfunction solution, while the final steady-state condition is given by theBlasius profile. Although the transition from the initial to-the final statecan be calculated directly in the three independent variables (ref. 16),Dennis formulated the problem in similarity coordinates where the governingequation is parabolic and of mixed type. The convective derivatives wereapproximated by backward or forward differences where appropriate, and thesolution was obtained through a successive overrelaxation procedure. Thisnumerical technique with certain modifications can also be applied to theequations that describe boundary-layer separation. The two problems are, ofcourse, different in many important respects. In particular, there is nothingcorresponding to reattachment for the impulsively started flat plate, and thedownstream (large time) boundary conditions are given. One of the moreinteresting features of boundary-layer separation is that although there is anembedded region of reversed flow and of upstream influence, the overallproblem remains parabolic in the downstream direction.

The present investigation develops a numerical procedure for integratingthe laminar, incompressible boundary-layer equations, without approximation,through a region of reversed flow. Under Development of Numerical Method, a

model problem is examined to determine convergence and stability criteria, anditerative finite-difference schemes are developed to solve the nonlinearequations. Under Results and Discussion, the numerical procedures are used toinvestigate the conditions for regular behavior at the point of separation.The separation C^nd reattachment) points are shown to be saddle-type singu-larities in the physical plane, which make it difficult to obtain numericalsolutions that pass continuously from the attached region to the separatedregion. The singularities are effectively removed, however, by specifying thewall shear distribution and determining the pressure as part of the solution.These inverse calculations are used to infer the type of pressure distributionrequired for the boundary layer to pass smoothly into a region of reversedflow. Where possible, results are compared to relevant analytical or similar-ity solutions to verify the accuracy of the calculations. The extension ofthe method to compressible flows and to the solution of complete viscous-inviscid interactions is indicated in a separate section.

DEVELOPMENT OF NUMERICAL METHOD

The Differential Equations

The boundary-layer equations for two-dimensional, laminar, incompressibleflow are

U1H+ v I" u ^£ + uU 3x + V 8y - Ue dx 9y2

where the Reynolds number has been explicitly removed by introducing theusual scaling x = x*/£, y = (y*/£)A, u = u*/uco, v = (

v*/Uco) T> R = uco*£/v.Here superscript (*) indicates the physical or untransformed variable.Boundary conditions are u = v = 0 and u -»• ue as y -> °°. In a directproblem, ue is specified as a function of x, while, in an inverse problem,an alternate condition such as (9u/8y)0 or ve is given as a function of x.In this case, ue must be determined as part of the solution process.

The parabolic nature of the equations is evident in von Mises coordinates:

37 a27Q L Q L /•"»•*-^7 = - —~ (2a)

with ue2 - u2 = z and v = -3ijj/3x. Equation (2a) is clearly a heat equation

in which the coefficient u changes sign in regions of reversed flow. Becausethere is no downstream boundary condition, the solution is not unique unlessthe separated zone is entirely confined within the domain of integration.

The equations can also be written as a system of nonlinear first-orderequations in conservative form, for example,

9u 9v _ ,_ ,^ + 97 = ° (3a)

- uv + g = 0 ' (3c)

Because the equations are nonlinear, discontinuities may occur in the flowfield even though continuous boundary conditions are .specified. Equations(3a) , (3b), and (3c) possess the following weak solutions:

[u2 - ujjsin 9 = [v2 - VI]CQS 9 (4a)

[u22 - U!2]sin 9 = [g2 - gjcos 9 (4b)

0 = [u2 - ujcos 9 (4c)

where ue is assumed to be continuous and 9 is the angle between the axisand a plane separating conditions 1 and 2. If 9 < ir/2, equation (4c) ensuresthat u2 = ui and, consequently, all the variables are continuous. When9 = ir/2, the weak solutions are indeterminate. In particular, v may be dis-continuous with a jump of indeterminate strength even with u continuous.Furthermore, if u is discontinuous, then, from equation (4a), [v2 - vj] -> °°.

Preliminary Numerical Considerations

As equation (2a) in particular shows, in the separated-flow region,information must be allowed to propagate upstream with the reversed flowvelocity. A natural way to fulfill this requirement consistent with restric-tions of numerical stability is to treat the x-derivatives with backward(upwind) finite-difference formulas in attached flow regions and with forward(downwind) finite-difference formulas in the reversed flow region. However,this means that at least a portion of the difference equations will requiresimultaneous solution. Furthermore, the extent of the separated region isunknown and, because the equations are nonlinear, an iterative finite-differencemethod appears to be the most efficient way to find a solution. Here, ofcourse, one can rely on experience obtained with type-dependent relaxationmethods employed for transonic flow fields (refs. 17 and 18).

As an alternative to a type-dependent differencing scheme, interpolative(elliptic) finite-difference formulas such as central differencing can be usedover the entire flow region. In fact, in the absence of discontinuities,

parabolic and hyperbolic problems can be solved with interpolativedifferencing, provided the boundary conditions are properly satisfied. Ofcourse, for a simple initial-value problem, a marching process that uses back-ward differencing is generally far more efficient than a simultaneous solutionprocess.

The choice of whether to use backward-forward differencing, centraldifferencing, or some hybrid of these will depend on the efficiency andaccuracy obtainable in the iterative finite-difference method. In any case,no downstream boundary conditions can be supplied for the boundary-layerequations, so the last computed profile must be attached to allow the use ofbackward differencing for the x-derivatives.

The success of a numerical method also depends on the choice ofvariables into which the equations are cast. Equation (2), for example, isnot suitable because the variable ty is multivalued in the separated region.For the most part, equations (la) and (Ib) appear to be the most appropriateto difference with a high probability of being readily extended to morecomplex (e.g., three-dimensional) flows.

Because the boundary-layer exhibits extensive growth in the. x-direction,it is essential for numerical efficiency that this growth be scaled out.This can be accomplished by introducing a variable, growing grid system orby using a transformation that keeps the viscous layer of nearly uniformthickness. The following transformation is used:

/uey = y / —' / x

(5)

so that equations (la) and (Ib) become

x H. + 3v + 2L±_i G = 0 (6a)

xu H. + v = m(l - u2) + (6b)

8X 3y 3y2

Boundary conditions are indicated in figure 1. These equations can also bewritten as a single equation for the stream function

fin + 5L±_! ff» + mCi . f ' 2 ) = x(f ' f ' - f f") (7)^ A. A

ATTACHED FLOW SEPARATED FLOW BLENDING

•_ a x

H4~ ~hH

.005<

-.01 <u«.or

12

5 8ocOOO

2 45ccO

UPSTREAM/BOUNDARY:

EQUATIONS

u = /(?)

= m(1 -u^) + ur/c

NO CONDITIONON DOWNSTREAM

BOUNDARY

SURFACE: u = v = 0, m orlpinwJ = /(x)

Figure 1.- Difference operators and boundary conditions forrelaxation calculation.

Iterative Finte-Difference Method

In the first stages of developing a finite-difference method, it isuseful to begin with the study of a model problem. A model equation isobtained here by linearizing equations (la) and (Ib); the iterative convergencecriteria are reviewed and an appropriate choice of difference formulas is madeso that the simple model equation is iteratively stable. In the followingsection, the convergence of the difference equations to the differential equa-tion is considered, and iteratively convergent differencing schemes for thenonlinear boundary-layer equations are subsequently given without analysis.

Model problem- Equations (la") and (Ib) are simplified with

u = UQ + u |

v = VQ + v )

so that the model equation becomes

8u 3u

(8)

(9)

In any local domain, UQ and v0 are treated as constants. Equation C9) alsorepresents the transformed equations, equations (6a) and (6b), if an averagevalue for xu0 is substituted for UQ.

If convergent difference algorithms and convergent iterative procedurescan be selected for equation (9), subject to all reasonable choices of UQand v0, it is assumed that such schemes can be successfully adapted to equa-tions (la) and (Ib) . While explicit and implicit marching procedures havebeen developed and extensively studied for parabolic equations of standardtype, a comparable development does not exist for relaxation schemes. Thedevelopment of such a scheme is undertaken below where the primary concern is

to ensure that the relaxation procedure is valid for both positive andnegative values of UQ.

Iterative convergence criteria- Once equation (9) is differenced over adiscrete network of grid points, one is left with the task of inverting thelinear system of equations

Au - c = 0 (10)

where the components of u consist of the dependent variables at each gridpoint. Then the most general first-degree iteration scheme for equation (10)is

_ = hR[Au- (n)

where H is a conditioning matrix usually implicitly built into the iterativesolution algorithm; here we chose to extract a parameter h from H. Itshould be understood that this type of iterative solution algorithm can treatnonlinear equations with the same ease as linear equations.

Equation (11) has the recursive solution:

n-1

= (I + hHA)nu(0;) m(I + hHA)hHc (12)

m=o

so if the matrix (I + hHA) has a spectral radius (i.e., largest eigenvalue inabsolute magnitude) less than 1, then (I + hHA)n -> 0 for n sufficientlylarge. Furthermore, from the Neumann lemma (ref. 19, p. 26, or ref. 20,p. 82), it is evident that

n-1

(I "1 (13)

m=0

"" i""or u -*- A-1c as required.convergence is that all

Thus the sufficient condition for iterative

. + ho.. < 1 (14)

where ct are the eigenvalues of HA.possibly complex cfj arc of the same sign, h can e cosen

This is an asymptotic convergence criterion for

Hence, if all the real parts of theo assure con-

vergence. Ths s an asymptotc convergence crteron or n sufficientlylarge. For the scheme defined by equation (11) to be efficient, the matrixHA must not have a large condition number (refs. 19 and 21) nor should theimaginary parts of aj be large compared to the real parts. The eigenvalue-convergence criterion does not guarantee that the norm of (I + hHA)nuC°) willnot grow appreciably during intermediate iterations - a situation likely tooccur if the eigenvectors of HA are linearly dependent or almost so.

Convergence of the model problem- The advantage of studying the modelproblem is that analytic expressions are obtained to describe its behavior forvarious choices of differencing. It is assumed that the nonlinear problemwill share at least some common features. Here, let

92u

jkAy

- 2u.,jk u. 0(Ay2)3 (15a)

'"]* 0(Ay2) (15b)

and

9u3x

0(Ax) (16)

where a = 1 is first-order backward, a = 0 is second-order central, anda = -1 is first-order forward. Using these approximations in equation (9)with g = Ay/2 and y = (Ay)2/2Ax, one obtains

2u.k

- (1 - a = b.

(j = 2, 3, 4 . . ., J ; k = 2, 3, 4 . . ., K - 1)^J ' ' ' max max (17)

-V.

If u is the vector whose components are the u-j^ over the ordered gridpoints, equation (17) can be written as the linear system of equations,equation (10). The eigenvalues of A are given by

jk = -2\l + /(I + v06)(l - v0g) cos (jr

-2u0y[a + /-(I + a)(l - a) cos

(k = 1, 2 . . K; j = 1, 2 . . ., J; K = K - 2; J = J - 1)' J ' > ' max ' max

(18)

where u is assumed to be given on a boundary as needed. If a is 0 or 1when UQ > 0 or if a is 0 or -1 when UQ < 0, the a roots always havenegative real parts and A is a stable matrix. Thus the point-iterationscheme with H = I and h = o)/(2 + 2u0ya) is proven to be convergent for anappropriate u) $ 1. As another example, the point successive overrelaxation(SOR) method has the roots

+ ayu°)= A ~ ajkl~ / c i + v°3)(1 " v°3) COS(FTT)

/ (1 + a) (-1 + a) cos

(j = 1, 2 . . . K; j = 1, 2 . . . J) (19)

and is also iteratively convergent with h = -1 and a proper choice of therelaxation parameter u>.

Equations (18) and (19) show that the roots will be complex if a = 0 orif |vo&| > 1. This can be detrimental to the convergence rate of a first-degree iteration scheme if the imaginary parts become large enough, so thecentral differencing should be restricted to regions where UQY is small.The product Vg3 is normally expected to be less, than 1 in absolute value andthus has the beneficial effect of reducing the term cos kir/(K + 1).

In place of the complex roots that occur for a = 0, when a - 1 or -1,the eigenvectors of HA appear in multiples of the number of J grid points.Under these conditions, the norm of an iteration matrix can be expected togrow before it decays; however, study of £2 and £„, induced matrix norms(ref. 19) for the point iteration scheme shows that residual growth cannotoccur if the spectral radius is kept less than 1. Conversely, numericalexperimentation with the heat equation demonstrates that the SOR forward-differenced scheme (a = -1) swept from left to right can experience appreciableresidual growth if Ax « (Ay)2. If swept from right to left, the residualsdecay rapidly.

Convergence to the Differential Equations

Although the previous analysis shows that iteration algorithms can beused to find a solution to the system of difference equations, it does notprove that the solution of the difference equations will converge to the solu-tion of the differential equations as the grid is refined. However, with theexception of the central differencing scheme, all the schemes to be introducedare known to be stable and consistent for the heat equation (cf. ref. 22).

If one assumes periodic boundary conditions in x and end conditions iny, then sufficient conditions for convergence of the centrally differencedheat equation

are Ax - 0(Ay2) and .Ay - 0(Ax2). (This is not an explicit leap-frog scheme.)Here convergence implies that the difference between the exact solution tothe differential equation and the exact solution to the difference equationwill vanish as the grid is uniformly refined over a fixed domain. That is,the summation of truncation errors given by A~^e -*• 0 as Ax, Ay -*• 0 where_^ Ais the matrix formed by the difference equations over both y and x, and eis the vector of truncation errors. While the complete convergence proof istoo lengthy for this report, note that A is a normal matrix and hence isunitary similar to a diagonal matrix of its eigenvalues (ref. 21). Theeigenvalues are

ajk(2AxA) = 4Ax

Ay2- cos 2i(±l)sin

(j = 1, 2 . . . J; k = 1, 2 . . ., K) (20)

and 11 ~11|2 = (min l^kl)"1 so II A"1 II 2 11 e 11 2 is simply determined.

Finite-Difference Equations and Solution

Two second-order-accurate differencing schemes were developed to solvethe boundary-layer equations (6a) and (6b). The first of these provedsuperior for the separated flows computed in this investigation. The secondmore conventional method is described because it may prove efficient forcertain extensions of the present approach.

The first method employs the central-differencing schemes for Uyy and uygiven by equations (15a) and (15b). The term xuux in equation (6b) isregrouped as 0.5x(u2)x and backward-differenced:

2 3x

- 4u? 7 'Ui-9 kJ 2,K

2Ax 0(Ax2) (21)

(B)

for u > 0.015 or j = Jmax. When u < 0.005 or if j = 2, centraldifferencing is used:

_2 9x

0(Ax2) (22)

(C)

10

In the intermediate zone, 0.005 * u * 0.015, the backward and central formulasare combined according to the relation

9u2

9x jkU + a)

9u2

9x9u2

jk(B)

9xjk(C).

C23)

with a E 1 + 200(u - 0.015).figure 1.

The difference stencils are indicated in

We emphasize that the blending defined by equation (23) is used solely toenhance the iteration process and is not otherwise fundamental. It is obviousthat when the difference equations are switched at a given value of u', a dif-ferent set of data points is sampled and slightly different truncation errorsresult. The change in the residual error vectors at this point can be largeenough to drive u(n+1) back across the value at which switching occurs.This can then start an oscillatory mode with little decay. The blending sim-ply modifies the differencing relations in a continuous fashion so that theresiduals vary smoothly. In the present scheme, the blending is completed atu = 0.005 to avoid a special operation at separation and reattachment. Theblending can also be used between 0 < u < 0.01 without changing the results.

The continuity equation is differenced with the modified Euler scheme(i.e., trapezoidal rule or Crank-Nicholson differencing):

vjk-vjk-i

9v

jk

9vay jk - i /

0(Ay2)

with

9v3y

jk

r /u. , . -vI j + i,k-[xj\ ^T 0(Ax2)

(24)

(25a)

for j = 2, J - 1, andJ ' max '

9v9y 2Ax Ujk 0(Ax2) (25b)

at j = Note that xux is central -differenced at all times (except atJmax) in both the attached and reversed flow regions. While equation (24) isgenerally recommended, two schemes implicit in the y direction are presentedas alternatives. Either the second-order-accurate "shifted" scheme

f-i , + 4v., - v.,jk- l jk jk+ i + 0(Ay2) (26)

(where point jk is updated in the relaxation) or the third-order accurate-in-y "abated Hermite" scheme:

11

+ 8v., - 3v. 3vjk+l jk-1

4 3v6 3y

5_ 3y_6 9y jk+l

0(Ay3) (27)

can be used with Sv/Sy^ again defined by equation (25). Both alternativeschemes generate diagonally dominant tridiagonal blocks if a backward two-point differencing is used at the edge where v varies linearly. Effectively,equations (24), (26), and (27) give the same results.

An additional difference algorithm must be introduced if an inverseproblem is solved. To impose a specified shear distribution, the momentumequation is evaluated at the surface:

m = - (28)

y=0

The second derivative is differenced as a function of T(X) to generate thesecond-order-accurate relation:

32u -?V8u.^ - u.032 ]3

2(Ay):

Wake flow is treated in the same fashion withvelocity UQ specified:

(29)

T = 0 and the centerline

du0

(30)

With the choice of differencing established, the solution procedure isstraightforward. An approximate solution is input, usually by assuming aBlasius profile with m = 0 everywhere. For an inverse problem, a new distri-bution of m is then predicted for the specified boundary condition usingeither equation (28) or (30), with m updated by the relaxation (herewritten for T(X) specified):

(n+1) (n)m. •* = m.3 3

ii-..)2(Ay)2 /

(31)

For a poor guess of the initial solution, a) is initially kept small,to = 0(0.05). New values of u are then found from relaxation of the momentumequation, while new estimates of v follow from continuity. This iterationsequence continues (with to increased as the initial guess is improved) untilan equilibrium or converged state is reached.

Solutions are found by both point and line successive underrelaxation(SUR) by using the iterative correspondence:

12

=u u '(32a)

-(n+l) ^v = v oiH Rv v (32b)

The residual vectors Ru and Rv represent the differenced momentum anddifferenced continuity equations, Hu and Hv are the conditioning matrices ofthe SUR algorithm, and <D is the relaxation factor. The line method (notused in eq. (32b)), in general, converges faster than the point scheme, but itis more sensitive to changes in m, making it more difficult to control in acomputer batch mode. For moderate reversed flows and grid spacings with Axapproximately equal to Ay, the optimum relaxation parameter is slightly lessthan 1 for point SUR with ui = 0(0.5) for equation (31). For line SUR, theoptimum relaxation parameter is 0(0.4) and o> is the 0(0.15). The point SURmethod fully converges in 400 to 800 iterations for a grid of 80 j-points and50 k-points. Highly separated cases with rapid variations in the flowquantities require the higher iteration counts.

Note that, when u is negative, it is possible to blend from thecentral into a three-point forward difference and that this variant of therelaxation procedure is iteratively convergent. For very large reversed-flowregions, it may be advantageous to program this additional logic. Experiencealso shows that switching at u = 0 from a three-point .backward differencinginto a three-point forward differencing without blending first into thecentral differencing is not convergent.

The second method developed is patterned after the Crank-Nicholson scheme.Equations (6a) and (6b) are first put into conservative form

3v 8xu (m - (33a)

-2 _ (33b)

The continuity equation is treated as before, and the y derivatives in themomentum equation are again centrally differenced by use of relations (15a)and (15b). The x-differencing is Crank-Nicholson

9(xu2) 9xu2 1(u > 0.01 or j = Jmx)

with

3(xu2)

jk

3uv

(34)

(35)

13

where the appropriate central-difference formulas are substituted for the yderivatives. For reversed flow, forward differencing is used

(xu2).j + lk 3x+ 3(xu

2)

j + lk ,J(u < -0.01) (36)

and the two schemes are linearly blended in the interval -0.01 < u < 0.01 (seefig. 1). As before, the blending is used solely to enhance the iterationprocess.

The Crank-Nicholson scheme has been solved by both point and line SUR, andfor either process the relaxation parameters are approximately those describedfor the previous point method. This second method requires slightly morealgebra per step and, in general, has a slower rate of convergence than thefirst method.

The conservation-law form of the Crank-Nicholson method is notconsidered to be an advantage, and the procedure generally predicts m dis-tributions that are slightly oscillatory. The oscillations decay as Ax/Aydecreases, and they are.confined to the relatively uninteresting attached flowregions. Of course, m is a sensitive function of the solution and the u andv distributions are much smoother. A nonconservative version of the Crank-Nicholson scheme was also programmed. In this case, the oscillations in mwere negligible in attached-flow regions but observable in the separated zone.

Finally, we remark that a very stable first-order-accurate method can bedeveloped by replacing the x differencing by

x 3u2

2 ^\ -.oX

x 3u2

2 3x

jk(B)

jk(F)

x u2 V

x. /u2 V

*• -u*-jk j-1Ax

J2+l.k-u

Ax

,k

i

2 >

jk

i

Cu > 0.01)

(u < -0.01)

(37a)

(37b)

and

x 8u2 9T

x.

jkcO 9x

jk(B)

(1 -3u2

jk(F)

(-0.01 5 u 5 0.01; a = lOOu)

14

This scheme, with xux of continuity also first-order-accurate and switched inan identical fashion, will generally give "computational results" for thefirst problem, m specified. This first-order method is not recommendedbecause a much finer x-grid spacing is required to maintain accuracy. Thismethod proved useful for the numerical experiments described in the nextsection.

RESULTS AND DISCUSSION

In this section, the iterative finite-difference procedure is used tointegrate the boundary-layer equations through a region of reversed flow. Theseparation-point singularity is investigated and conditions for regularbehavior are determined. Calculated results are presented for a number ofseparated flows and the accuracy of the method is verified. Possible indica-tions of the breakdown of the boundary-layer assumptions are also examined.

Direct Solutions

One of the most extensively studied problems in separating boundary-layerflows is the response of a flat-plate boundary layer to a linearly retardedexternal stream. This problem has been investigated by Howarth (ref. 23),Hartree (ref. 24), and many others; recent solutions have also been obtainedby Briley (ref. 25) and Leal (ref. 26) using the full Navier-Stokes equations.

A sequence of calculations for different mesh spacings is shown infigure 2. The external velocity was specified to decrease linearly from the

origin and the first-order-accuratedifference scheme was used becauseof stability considerations. As themesh is refined, the separationpoint moves upstream, with the lastcalculation in exact agreement withthe accurate results of Smith andClutter (ref. 27) for the flowupstream of separation. In the limitof zero-mesh spacing, it is evidentthat the solution is singular, withthe wall shear approaching zero asthe square root of the distanceupstream of the separation point andthe normal velocity unbounded. Thistype of behavior has been discussed

1.2 1.6

Figure 2.- Calculation for linearlyretarded flow.0

in detail by Goldstein (ref. 28)by Landau and Lifshitz (ref. 29)among others.

and

The interesting result .here is that the use of an iterative finite-difference scheme which contains type-dependent operators allows the solution

15

to be "continued" in the downstream direction. As the mesh is refined, itbecomes evident that the flow fields upstream and downstream of separation areessentially independent, and the solution is therefore not meaningful. Thewall shear jumps discontinuously to a negative value at separation and thenormal velocity v becomes unbounded; all flow quantities subsequently remaincontinuous downstream of the jump and through reattachment. The magnitude ofthe discontinuity is determined by the specified pressure distribution in theseparated zone. In a set of simple numerical experiments, a constant externalvelocity distribution was smoothly joined to the linearly retarded flow atdifferent values of x. As the joining point was moved downstream,.the magni-tude of the jump and the extent of the reversed-flow region increasedmonotonically, with separation remaining at x = 0.96.

Singular behavior at the point of separation is thus related to the factthat the wall shear T = (8u/9y)o i-s nonanalytic; in particular,T ~ (xs - x) V

2 and 8i/8x -»- °° as x -»• xs, the separation point. Therefore, themost obvious means of ensuring regular solutions at separation is to specify acontinuous wall-shear distribution. The pressure distribution can then bedetermined as part of the solution by satisfying the momentum equation at thesurface. Note that because the equations are nonlinear, it is not possible toguarantee that discontinuities will not occur in the flow field even withanalytic boundary conditions prescribed (see ref. 30 for hyperbolic equations,or the weak solutions, eqs. (4a), (4b), and (4c)).

Inverse Solutions

With the wall-shear distribution specified, m can be determined fromequation (31) and the second-order-accurate differencing scheme generates con-tinuous solutions that give no indication of singular behavior at eitherseparation or reattachment. These solutions are demonstrated to be regularunder Accuracy Check. An inverse calculation cannot be duplicated by thedirect method, however. Starting with a fully converged inverse solution, thecalculation diverges if the iteration is continued with m fixed, that is,the relaxation parameter to is set to zero in equation (31). Two examples ofthis type of inverse (T specified) and direct (m given) calculation sequencesare shown in figure 3. After as many as 500 iterations (less if the solutionis initially perturbed), the residuals begin to grow and the relaxation pro-cedure either becomes unstable or converges to a different "solution" of thedifference equations. As the mesh is refined, the second-order scheme failsto converge while the first-order method, for moderate grid spacing, generatescomputational results containing a discontinuity.

The fact that the direct calculation fails to duplicate a convergedinverse solution-cannot be ascribed to instabilities in the numerical scheme.The only difference between the two calculations is the value of the relaxa-tion parameter u in equation (31), and the solution processes are essentiallyidentical. The numerical evidence therefore strongly suggests the existenceof a saddle-type singularity at the separation point. Because of this criticalpoint, roundoff and residual errors are sufficient to cause a completelyconverged solution to diverge when the pressure-gradient parameter is held

16

COMPUTED mFOR T SPECIFIE

(INVERSE)

COMPUTED T FOR-. m SPECIFIED

VDIRECT)

-.04 -

-.08

Figure 3.- Inverse/direct calculationsthat indicate existence of saddlepoint.

u(x,y) = f!y + f 2

where f3 = 0 and the notation is used

fl = T,

fixed. There are no other possiblesources of error in the calculations:the variation of m is determined toarbitrarily high accuracy by theinverse solution, and no interpolationor differentiation is required as forcomputations with experimentallydetermined pressure distributions.With the pressure gradient correspond-ing to a completely regular flow fieldprescribed, the equations contain asaddle-type singularity at separationthat makes a continuous numericalsolution difficult to obtain. Thesaddle point is removed from thedomain of integration, however, byspecifying the wall shear rather thanthe pressure gradient as a boundarycondition. A discussion of theessential differences between the twotypes of calculations is presentedbelow. In the following section, theconditions for regular behavior atthe point of separation are examined.

Saddle Point

The difference between thedirect- and inverse-calculation pro-cedures can best be illustrated byexamining the boundary-layer equationsnear the surface. Expanding thevelocity profile in a Taylor seriesin y yields

y2 ll3! (38)

2 = P

Either fi or f^ (but not both) is prescribed and all otherdetermined as functions of x by the differential equations,must satisfy the following set of relations:

[ areThe coefficients

17

£7 -

£8 -

fl

fs

fefi£1

f - fl f{

- 2f!f •

- 2f2q

+ Sf.fJ

+ 9fcf '

= 0

= 0

= 0

= 0

= 0

(39)

NUMERICALSOLUTION

ORDER OFEXPANSION:

.4 .6 .8 1.0

Figure 4.- Series expansion forsimilar separation profile.

where the prime denotes differentia-tion with respect to x. One of thefi is given by the outer boundarycondition that u -> ue as y ->- °° (seeref. 28). The validity of the expan-sion procedure near the separationpoint is demonstrated in figure 4 forthe particular case of similar flowwith m = -0.09044, corresponding tozero shear (see eq. (7)). For thiscase, the only nonzero coefficientsmultiply terms of the ordern1"1-2 (n = 1, 2, . . .), and theexpansion has been continued throughthe twenty-second power of the normalcoordinate.

Direct calculations- For thepressure gradient specified, thecoefficients in equations (39) mustbe determined by integrating the fol-lowing system of first-orderdifferential equations:

(40a)

f!f7 + 5f42

2f i 2 [ f i o * 8!

+ 33£1fit£7 - (40b)

The remaining coefficients are given by the algebraic relations:

18

= 2p,

xx

9px(£l£7 + Sf^2) + 4f!2(10pxxxfi2 -

= SfjCSp, * '

(41)

If one arbitarily terminates the expansions at this point and assumes thatflO can be correctly specified, then equations (40) and (41) provide rela-tions for all coefficients of the lower-order terms. Given the velocity pro-file at a particular station, standard numerical techniques can be used tointegrate equations (40a) and (40b) to determine the adjacent profile providedfl is nonzero. As f^ -»- 0, however, the solution becomes increasingly sensi-tive to the calculated value of fj, and numerical errors are propagated inthe direction of integration. The equations are highly nonlinear, with thecoefficient f^ of the derivatives determined by f^, which in turn dependson f7, f10) etc., and on the outer boundary condition. Even for a pressuredistribution corresponding to a regular solution at separation, the numericalintegration of equations (40a) and (40b) is unlikely to result in values offl and fi^ that vanish simultaneously. In that event, either f{ will beinfinite, leading to a square-root singularity, or fi will remain positiveand the calculation will fail to show boundary-layer separation.

We emphasize that with the pressure gradient specified, the nonlinearequation for the wall shear (eq. (40a)) is inherent to the system of differ-tial equations. Even with special procedures that would guarantee that f^vanishes at T = 0, the saddle point would remain to confound the numericalsolution process. The behavior shown in figure 3 is to be expected because aconverged solution perturbed by small roundoff and residual errors cannotremain converged in the presence of the saddle-point singularity.

Inverse calculations- For the wall shear specified and the pressuredistribution determined as part of the solution, a different system ofordinary differential equations results:

2 = £ (42a)

= 2£8

+ 93T £8 + 3f5(160TT - 201x 2)A AA A ,

27f,(19T 3 - 16TT T - 20T2T )^^ X X XX XXX (42b)

including the algebraic relations

19

f =

£7 = T(4TT - T 2)' XX X

Tfg = 4(f2f8 - f52) + 26T £2f5 - 36f2

2(TT + T 2)- A AA A

(43)

T2f10 = 4f2(£2£8 - fs2) + 26t £2

2£5 -36f23(TT + T 2)

A . AA A

+ T3(27T 3 - 24TT T + 28T2T )X X XX XXX

These equations are linear, with the coefficient of the derivatives Tspecified as a function of x. The system is therefore less susceptible tonumerical error, and although the matrix of coefficients still vanishes atT = 0, the saddle-point singularity has been effectively removed. If thenumerical integration is accurate enough to ensure that f5 = 0 when Tvanishes, the solution will pass smoothly through the separation point.

The basic difference between the inverse and direct problems is that,for the pressure gradient prescribed, the unknown shear distribution isdetermined by a nonlinear equation that contains a saddle-type singularity atseparation. For the wall shear specified, on the other hand, the pressuregradient is given by a linear equation that is much less sensitive to numeri-cal error. This is probably also the case when the displacement thickness isprescribed (see ref. 13). The fact that most numerical evidence indicates asingularity at separation is therefore misleading because of the difficulty innumerically integrating through the saddle point. Of course, not all pressuredistributions admit a regular solution (as discussed in the following section).

An interesting point is that, provided the correct numerical proceduresare used, no difficulties are encountered at reattachment (see fig. 2 or 3).The reason for this is that any numerical errors made at the reattachmentpoint are either integrated out of the downstream boundary or upstream towardseparation. The direction of the flow, and therefore the differencing scheme,results in a solution process that allows integration away from the saddlepoint at reattachment but that requires integration into the singularity atseparation.

Several numerical experiments were performed to verify theseconclusions. In one set of computations, the velocity profiles at separationand in the immediate vicinity of that point were held fixed after convergingthe inverse calculation. For these cases, the inverse and direct proceduresgave identical solutions. Similar results were obtained when an artificial-viscosity term equal to euxx

was introduced into the difference equations.As the coefficient e was decreased, however, the direct calculation wouldagain diverge from the inverse solution.

20

Pressure Gradient at Separation

As shown in the previous section, the existence of a regular solutionrequires that fi+ = 0 at the point of separation (see also refs. 31 and 32) .The coefficients f5, fy, and fg must also vanish at the point of zero shear,and the pressure gradient must therefore satisfy certain specific conditionsto permit the flow to pass smoothly through separation. The constraints onthe pressure distribution cannot be determined directly because of the saddlepoint, but must be obtained from the inverse, or shear-specified, calculations.

It is reasonable to expect that only certain pressure distributions willadmit regular solutions. The separation profile, for example, is determinedby both the upstream and downstream flows so that some compatibility relationmust be satisfied at this station. Also, from kinematic considerations, theboundary -layer approximation to the vorticity transport equation is

(44)

where to = 3u/9y and px = 3a>/9y at y = 0. The restriction on the pressuregradient at separation can thus be interpreted as a constraint on allowableboundary conditions: - the normal gradient of vorticity at the surface isrequired to satisfy some local condition for the vorticity to remain continuousat the singular point.

From physical considerations, a constraint on the allowable pressuregradient implies that the interaction between the inner viscous layer and theouter fluid essentially determines the conditions at separation. Prandtl(ref . 33) recognized this in 1938 when he stated that the pressure fieldcould not be chosen arbitrarily for the flow downstream of separation "to agreewith observation." Most numerical solutions of the Navier-Stokes equations,including the recent investigation by Leal (ref. 26) in particular, alsoindicate that, when the interaction with the outer flow is included, there isno evidence of singular behavior at separation.

Because of the nonlinearity of the boundary- layer equations, it is notpossible to determine the precise pressure-gradient condition that permits aregular solution. Certain restrictions on the pressure distribution can beinferred from the Taylor series expansion and from the numerical solutions,however. The acceleration of a fluid particle near the surface, for example,can be approximated as follows:

du du y i o y y ,. ?•> y I r * r ^u a T + v s ^ = V T T v + 2rPw t + PvPw fT + Tf4TTw ~ Tv ) fiiT + • • ' f45^ox oy ^ i x xx o x xx D • xx x t>u I

Immediately upstream of separation, T and px are positive and TX is nega-tive. As T •> 0, the fluid in a stream tube near the surface continues todecelerate, and the streamlines continue to move away from the wall provided

21

y]D ^ + 2r ^- 1 < I TT I + I TT 2 I 2—- + f461Px 6 ^ 3 I '"x1 |T x I 60 ' ' ' . (-40J

For the flow to separate smoothly, then, a restriction on the pressure fieldis that

p < 0 as T -* 0 (47)rxx

There will therefore be an inflection on the pressure distribution upstream ofthe separation point. This requirement is consistent with experimental evi-dence, and the existence of a "knee" in the pressure curve is often taken toindicate boundary-layer separation.

The numerical evidence suggests that this condition is not sufficient,however. All regular solutions, in fact, satisfy the requirement:

$L > o at T = 0 (48)dx ^ J

This is a more restrictive condition than that given by equation (47) becausemx can be negative for pxx negative. The linearly retarded flow consideredunder Direct Solutions, for example, satisfies equation (47) but not equation(48). In a series of papers, Meksyn (refs. 34 and 35) has contended that theexistence of a minimum in mx was a necessary condition for regular separa-tion. He cited Schubauer's (ref. 36) measurements of the flow over an ellipticcylinder as experimental verification of this requirement. Similar argumentshave also been advanced as a result of the use of approximate methods to cal-culate supersonic viscous-inviscid interactions (see, e.g., ref. 37).

The most useful means of examining the numerical results is in theT - m phase space (fig. 5). Several typical computations are presented,including the locus of solutions for similar flow. In these coordinates, xis a parameter that varies along the curves, with Ax -»• °° for the similaritysolutions. For this limiting curve, dm/dr, and therefore dm/dx, is zero atthe point of zero shear. All nonsimilar trajectories, on the other hand, havepositive mx at both the separation and reattachment points. This conditionwas never violated in approximately 30 different calculations using variousspecified shear distributions. Note that the locus of similar flows is some-times taken to indicate singular behavior at separation becauseT ~ (mo - m)1/2 and di/dm ->• °° at T = 0. The similarity solutions are obtainedfor mx = 0, however, and the limiting value of df/dx (= mx dr/dm) must becarefully determined if an actual flow is replaced by a sequence of similarflows. In any event, the condition for regular separation, that mx - 0 atthe point of zero shear, is satisfied by both the similar and the nonsimilarflows.

22

.24

.20

.16

.12

.08

I

° .04

0

-.04

-.08

-.12

SIMILARSOLUTIONS

ARROWS SHOWDIRECTION OF

- INCREASING X

-.20 -.16 • -.08m

-.04 .04

Figure 5.- Phase space representation.

10

0.0826-0.06630.0835 - 0.0660

m T0.0783-0.07760.0767 - 0.0769

POINT A POINT B>

SIMILAR SOLUTIONSPRESENT METHOD

1.0 -.5 .5 1.0

Accuracy Check

The phase-space representationof solutions presents an opportunityto verify the accuracy of the numeri-cal procedure. The points labeled Aand B in figure 5, for example, havethe same value of T and m as a cor-responding similarity solution. Theleft-hand side of equation (7), whichis completely determined byT = f"(0) and m, is therefore zero.The local x variation vanishes andthe similar and nonsimilar profilesmust be identical at those points.The velocity profiles calculated bythe present scheme are compared toadjacent solutions of the similarityequation (obtained by fourth-orderRunge-Kutta integration in ref. 38)in figure 6. There are essentiallyno differences in the resultsobtained by the two methods.

With a continuous sheardistribution specified, the solutionis constrained to be regular at bothseparation and reattachment. Thisresult can be verified by comparingthe calculated streamline patternwith the local solution of theNavier-Stokes equations obtained byOswatitsch (ref. 39) (see also Dean(ref. 40) and Legendre (ref. 41)).At the point of zero shear, a regularsolution of the Navier-Stokes equa-tions requires that the angle of thedividing streamline be proportionalto the ratio of the x derivativeof the shear and the pressuregradient. In the transformed vari-ables, the precise condition is

figure o.- <_0iupanson with similarsolutions.

JFT *„„r i\ uaii (49)

where 9 is the angle of the dividing streamline. For a prescribed sheardistribution, the calculated values of m can be integrated in x to obtainue. The flow in the vicinity of separation and reattachment for a refined-mesh calculation (Ax = Ay = 0.1) is compared with equation (49) in figure 7.The calculated results agree exactly with the local Navier-Stokes solution at

23

the point of zero shear, again demonstrating that the boundary-layer solutionis regular.

\ - 0.005

J I \ I I 0.001 I

-1.4X-XR

Figure 7.- Detailed flow field in the vicinity of T = 0.

Flow-Field Solutions

As previously mentioned, a number of different shear distributions werespecified in an effort to determine the behavior of the boundary-layer equa-tions in separated flow. Some of those results are presented in this sectionand the following one. Figure 8, for example, shows the streamlines and skin-friction variation, in physical coordinates, for a typical parabolic shear

distribution. The relation between\ the physical and transformed\ / variables is

6 r

4 -

Vv/R

Figure 8.- Streamlines for specifiedshear distribution.

= 2u r?-"e _TX

and

= /xu

f'•'ody

(50)

In figure 9, the skin friction andstreamline patterns for a differentshear distribution are shown. Forthis case, the maximum reversed flow

24

C,/R

Figure 9.- Streamlines for specifiedshear distribution.

3.0BOUNDARY LAYER—I WAKE FLOW

Figure 10.- Velocity profiles fortrailing-edge flow.

occurs toward the reattachment sideof the separation bubble. The divid-ing streamline has several rapidchanges in slope, and this solutionwould be difficult to obtain if itwere necessary to explicitly iteratefor the location of the u = 0 line.Note that in all cases the normalcoordinate is multiplied by thesquare root of the Reynolds numberand that these solutions representshallow separated regions confinedto the interior of the viscous layer.

The present method can also beused to calculate flows wherereattachment occurs in a wake ratherthan on a solid surface. The detailsof this type of flow field in theimmediate vicinity of the trailingedge are shown in figure 10. Here,the transition from boundary-layerflow to wake flow is assumed to occuron a scale that is small compared tothe thickness of the viscous layer(see ref. 42). The prescribedboundary conditions of zero velocityand negative wall shear were thusdiscontinuously changed to zeroshear and specified reversed-flowvelocity at the trailing edge. Basedon order-of-magnitude considerations,the initial reversed-flow velocitywas taken to be equal to the valueof the wall shear at the joining

point. No attempt was made to ensure continuity of the dividing streamlineor displacement thickness, although mass and momentum are conserved in thesolution to the differential equations.

Indications of Breakdown

In the previous sections, it was demonstrated that the boundary-layerequations have regular solutions at separation and reattachment. The flowstructure at the separation point agrees with the limiting form of theNavier-Stokes equations, and the Goldstein solution does not appear to berelevant for real flows. The square-root singularity in the boundary-layerequations is a consequence of specifying an external pressure distributionbased on an inviscid solution determined as though there were no separation.In practice, the pressure gradient is locally modified near the separationpoint such that the boundary-layer solution remains regular. The questionthat arises then concerns the manner in which the boundary-layer equations

25

eventually break down. Real flows tend to separate toward the rear of a closedbody and vorticity is transported into the outer fluid. In some cases, thevorticity is confined to a relatively narrow region, or wake, downstream ofthe body. In other situations, behind a circular cylinder, for example, alarge region of the fluid becomes rotational. The vorticity is no longerrestricted to a thin viscous layer and the normal component of velocityceases to be small compared with the"tangential component. In the presentinvestigation, the region of separated flow is, of course, constrained toremain close to the surface, inside a layer of order 1/i/R. The numericalsolutions may, however, suggest when this approximation is no longer realistic.

An indication of the possiblebreakdown in the boundary-1ayerequations is shown in figure 11 fora highly separated flow. As themesh is refined, the computed valuesof m appear to become discontinuousat a point downstream of separation.Apparently, there are two solutions,one associated with separation andthe other with reattachment, thatare joined in the reversed-flowregion.

The distribution of ve, thetransformed normal velocity, isshown on an expanded scale in fig-ure 12. The normal velocitiesincrease rapidly downstream of theseparation point, and the viscouslayer begins to break away from thesurface. Because of constraintsimposed by the boundary conditions,however, a discontinuity in v (andin 3u/9x) occurs at the maximumvalue of ve, and the remainingsolution is continuous. Althoughthere is a certain degree of smooth-ing in the numerical results, thediscontinuity in ve is evident infigure 12. A jump in v is anallowable weak solution of the dif-ferential equations and is apparentlyrequired for certain boundary condi-tions (e.g., large negative shears).If strong discontinuities occur whenthe shear distribution correspondingto a real flow is prescribed, how-ever, this can be taken to indicatethe breakdown of the boundary-layerassumptions.

Figure 11.- Evidence of weak solutionsfor highly separated flow.

40

AX10.05

1.0

Figure 12.- Normal velocity distribu-tion for highly separated flow.

26

-.04 -

TBL(X-2)(X-6)/l2 =

= T0[l+a(X-2)(X-6)

2<x<6

Figure 13.- Effect of shear variationin separated region.

-.08 L

4X

Figure 14.- Effect of shear variationin vicinity of reattachment.

The rapid variation of m, ve,and of the other flow quantitiesdepends on the amount of reversedflow. This is illustrated in fig-ure 13 for a sequence of solutionswhere the specified shear distribu-tion was modified in the separatedregion. As the values of the shearbecome less negative, the solutionsbecome increasingly smooth and con-tinuous. The streamlines correspond-ing to a = 0.1 were previously shownin figure 8. Even for this relativelymild case, the separating flowappears to undergo a rapid transitionto the reattaching portion of theflow field at x = 2.7 approximately.

The results of an additionalnumerical experiment are shown infigure 14. For this case, the wallshear was varied only in the down-stream portion of the separatedzone and kept constant elsewhere.The nonlinearity and upstream influ-ence of the boundary-layer equationsis evident in the computed distribu-tions of m and ve. Note also, how-ever, that the flow in the immediatevicinity of separation (x < 2.5) isnot significantly affected byrelatively large changes nearreattachment.

Upstream Influence

Part of the success ofapproximate methods that use forward-marching schemes (e.g., refs. 9 and13) may be related to the limitedupstream influence discussed above,particularly for flows with smallseparated zones. For the cases shownin figure 14, of course, it wouldnot be possible to obtain accuratesolutions downstream of x = 2.5without including the boundary con-ditions at reattachment. To investi-gate this question, calculationswere made with the convective term

27

uux set to zero for u ^ 0 with both the first- and second-order-accuratedifference schemes used in a marching mode. Only backward differencing wasemployed for both, momentum and continuity, and the equations were completelyrelaxed at each x station before proceeding.

-.04

-.16

"EXACT"SECOND-ORDERRELAXATION SOLUTION

AX = O.I MARCHING

Figure 15.- Comparison with forward-marching procedure.

The three point backwardsecond-order scheme could be marchedaccurately a short distance into theseparated zone. It always divergedrapidly, however, at approximatelythe location where mx became nega-tive. The first-order scheme, withmoderate grid spacing, could be used

j for small bubbles but diverged for8 more separated flows. A typical

calculation for a mildly separatedflow is compared in figure 15 withan "exact" solution obtained usingthe correctly differenced second-

As the grid spacing was refined in x,The

order scheme with smaller step size.the first-order marching began to diverge from the correct solution.instability could be delayed by keeping Ay < Ax and by accepting a lessstringent iterative convergence criterion at each x station, but overall,the difference equations failed to converge to a solution as the grid wasrefined.

This experiment indicates that backward differencing, even with uux = 0for u < 0, is always unstable. For mild separation, the eigenvalues in theunstable range are small and dominate the numerical calculation only after asufficient number of steps is taken. It is probable that the schemes of ref-erences 9 through 11 and 13 are also divergent, although they are useful forcertain applications.

To determine the effect of neglecting the upstream convection of momentum,additional calculations were performed with the term uux set to zero foru 5 0, but with the term ux in the continuity equation centrally differenced.In this manner, upstream influence is retained and the solution must again beobtained by relaxation methods. The results were essentially identical to theexact second-order solution, verifying that the upstream convection ofmomentum is not significant for laminar flows with limited separated regions.

POSSIBLE EXTENSIONS

An important extension of the present method is to match an inner,boundary-layer solution to an outer inviscid flow to calculate completeviscous-inviscid interactions. It would also be useful to compare results ofthe present method to experimental measurements of laminar separating andreattaching flows. Because low-speed boundary layers rarely remain laminarthrough reattachment, the computations must be extended to supersonic flows.

28

There are, for example, a number of reliable experiments for compression-corner interactions at supersonic speeds, as well as several different approxi-mate solutions and Navier-Stokes calculations available for comparison (e.g.,refs. 43 and 44). It is indicated below how the method can be adapted tocompressible flows, and an integral relation is proposed that offers promiseof allowing the treatment of complete viscous-inviscid interactions.

Compressible Flows

To apply the method to compressible boundary layers, the followingtransformation can be used:

X =

v = p y ue e epv + xu Tr-

(51)

If it is assumed that the density-viscosity product is constant through thelayer, the following equations result:

xu _ 3uv — 32u-9y2

im = 03y [(Y -

(52)

where

x ^ein = —. and Me =Me dx e

These equations can then be solved in exactly the same fashion as equations(6a) and (6b), with Me calculated by integrating m.

Viscous-inviscid Interations

The solution for a complete interaction is complicated by the fact thatT is specified. The following integral relation can, however, be used:

29

1 ., 2V dX X\- Me2J(* dx" + 2 j

M

[(Y - D/2]Me2

"o

u2 dy m (53)

For an assumed T distribution, the solution of equations (52) givescalculated values of m and hence of Me and pe. Using an inverse inviscidprocedure, the distribution Me can be specified to obtain a new effectivebody shape, that is, the streamline slopes ve/ue. Then, from equation (53),a new estimate for T can be determined and the procedure continued untilconvergence is achieved. Based on recent experience with an integral scheme(ref. 8), it will probably not be advantageous to precisely match theintermediate iterations for ve/ue.

It would, of course, be easier to specify ve directly for the viscoussolution. For similar flows, an efficient scheme was developed by differen-tiating the continuity equation with respect to y and using standard second-order central differencing for v. The value of m was then updated byevaluating the continuity equation at the edge of the layer. This approachfailed, however, for the complete boundary-layer equations with separatedregions and was much slower for attached flows than the T specified schemes.An alternate approach, perhaps using the vorticity equation, may be required.All analytical and numerical evidence indicates, however, that the wall shearis the optimum boundary condition for calculating separated flows.

CONCLUDING REMARKS

The numerical procedures developed in the investigation provide an exactmeans for integrating the boundary-layer equations through separation andreattachment. The approach appears to be adaptable to the treatment of com-plete viscous-inviscid interactions for flow fields where the boundary layerremains confined to a narrow region: compression-corner flows or separationat the trailing edge of a streamlined body, for example. The method may alsoprove useful in evaluating different turbulence models for separated flows.As compared to complete Navier-Stokes solutions, the present approach allowsan order-of-magnitude better resolution of the viscous region and requiresconsiderably less computation time. Finally, a method based on the boundary-layer equations provides the most promising means for investigating theimportant problem of three-dimensional flow separation.

Ames Research CenterNational Aeronautics and Space Administration

Moffett Field, Calif., 94035, March 8, 1974

30

REFERENCES

1. Abbott, D. E.; Holt, M.; and Nielsen, J. N.: Investigation ofHypersonic Flow Separation and Its Effects on Aerodynamic ControlCharacteristics. Vidya Rept. 81, 1962; also Studies of SeparatedLaminar Boundary Layers at Hypersonic Speed With Some Low ReynoldsNumber Data, AIAA Paper 63-172, 1963.

2. Nielsen, J. N.; Lynes, L. L.; and Goodwin, F. K.: Calculation of LaminarSeparation With Free Interaction by the Method of Integral Relations:Part I - Two-Dimensional Supersonic Adiabatic Flows. AFFDL-TR-65-107,Air Force Flight Dynamics Lab., Wright-Patterson AFB, Ohio, Oct. 1965.

3. Nielsen, J. N.; Lynes, L. L.; and Goodwin, F. K.: Calculation ofLaminar Separation With Free Interaction by the Method of IntegralRelations: Part II - Two-Dimensional Supersonic Nonadiabatic Flow andAxisymmetric Supersonic Adiabatic and Nonadiabatic Flows. AFFDL-TR-65-107, Air Force Flight Dynamics Lab., Wright-Patterson AFB, Ohio,Jan. 1966.

4. Lees, L.; and Reeves, B. L.: Supersonic Separated and ReattachingLaminar Flows: I. General Theory and Application to AdiabaticBoundary-Layer/Shock-Wave Interactions. AIAA J., vol. 2, no. 11, Nov.1964, pp. 1907-1920.

5. Reeves, B. L.; and Lees, L.: Theory of Laminar Near Wake of BluntBodies in Hypersonic Flow. AIAA J., vol. 3, no. 11, Nov. 1965, pp.2061-2074.

6. Holden, M. S.: An Analytical Study of Separated Flows Induced by ShockWave - Boundary-Layer Interaction. Rept. AI-1972-A-3, CornellAeronautical Lab., 1965.

7. Klineberg, J. M.; and Lees, L.: Theory of Laminar Viscous - InviscidInteractions in Supersonic Flow. AIAA J., vol. 7, no. 12, Dec. 1969,pp. 2211-2221.

8. Klineberg, J. M.; and Steger, J. L.: Calculation of Separated Flows atSubsonic and Transonic Speeds. Proceedings of the Third InternationalConference on Numerical Methods in Fluid Mechanics, Springer-Verlag,vol. II, 1973, pp. 161-168.

9. Reyhner, T. A,; and Flugge-Lotz, I.: The Interaction of a Shock WaveWith a Laminar Boundary Layer. Tech. Rept. 163, Div. of EngineeringMechanics, Stanford Univ., Nov. 1966; see also Internatl. J. NonlinearMech., vol. 3, June 1966, pp. 173-199.

10. Werle, M. J.; Polak, A.; and Bertke, S. D.: Supersonic Boundary LayerSeparation and Reattachment Finite-Difference Solutions. Rept. AFL 72-12-1, Dept. of Aeronautical Engineering, Univ. of Cincinnati, Jan.1973.

31

11. Dwoyer, D.-L.: Supersonic and Hypersonic Two-Dimensional Laminar FlowOver a Compression Corner. Proceedings of AIM Computational FluidDynamics Conference, Palm Springs, Calif., July 19-20, 1973, pp. 69-83.

12. Murphy, J. D.: A Critical Evaluation of Analytic Methods for PredictingLaminar Boundary-Layer, Shock-Wave Interaction. NASA TN D-7044, 1971.

13. Catherall, D.; and Mangier, K. W.: The Integration of the Two-DimensionalLaminar Boundary-Layer Equations Past the Point of Vanishing SkinFriction. J. Fluid Mech., vol. 26, pt. 1, 1966, pp. 163-182.

14. Klemp, J. B.; and Acrivos, A.: A Method for Integrating the Boundary-Layer Equations Through a Region of Reverse Flow. J. Fluid Mech., vol.53, pt. 1, 1972, pp. 177-191.

15. Dennis, S. C. R.: The Motion of a Viscous Fluid Past an ImpulsivelyStarted Semi-Infinite Flat Plate. J. Inst. Math. Applications, vol. 10,Aug. 1972, pp. 105-117.

16. Hall, M. G.: The Boundary Layer Over an Impulsively Started Flat Plate.Proc. Roy. Soc. A, vol. 310, 1969, pp. 401-414.

17. Murman, E. M.; and Cole, J. D.: Calculation of Plane Steady TransonicFlows. AIAA J., vol. 9, no. 1, Jan. 1971, pp. 114-121.

18. Steger, J. L.; and Lomax, H.: Generalized Relaxation Methods Applied toProblems in Transonic Flow. Proceedings of the Second InternationalConference on Numerical Methods in Fluid Dynamics, Springer-Verlag,1971, pp. 193-198.

19. Ortega, J.: Numerical Analysis, Academic Press, Inc., 1972.

20. Varga, R.: Matrix Iterative Analysis, Prentice-Hall, Inc., 1962.

21. Franklin, J. : Matrix Theory, Prentice-Hall, Inc., 1968.

22. Richtmyer, R. D.; and Morton, K. W.: Difference Methods for Initial-Value Problems, 2nd ed., John Wiley § Sons, New York, 1967, Ch. 8.

23. Howarth, L.: On the Solution of the Laminar Boundary Layer Equations.Proc. Roy. Soc. A, vol. 164, Feb. 15, 1938, pp. 547-579.

24. Hartree, D. R.: A Solution of the Laminar Boundary-Layer Equation forRetarded Flow. ARC Reports and Memoranda 2426, England, March 1939,pp. 156-182.

25. Briley, W. R.: A Numerical Study of Laminar Separation Bubbles Using theNavier-Stokes Equations. J. Fluid Mech., vol. 47, pt. 4, 1971,pp... 713-736.

32

26. Leal, L. G.: Steady Separated Flow in a Linearly Decelerated FreeStream. J. Fluid Mech., vol. 59, pt. 3, 1973, pp. 513-535.

27. Smith, A. M. 0.; and Clutter, D. W.: Solution of the IncompressibleLaminar Boundary-Layer Equations. AIAA J., vol. 1, no. 9, Sept. 1963,pp. 2062-2071.

28. Goldstein, S.: On Laminar Boundary-Layer Flow Near a Position ofSeparation. Quart. J. Mech. Appl. Math., vol. 1, March 1948, pp. 43-69.

29. Landau, L. D.; and Lifshitz, E. M.: Fluid Mechanics, Pergamon Press Ltd.,Oxford, 1959, pp. 151-156.

30. Courant, R.; and Hilbert, D.: Methods of Mathematical Physics, vol. II,Interscience Publishers, New York, 1962, pp. 486-490.

31. Curie, N.: The Laminar Boundary Layer Equations, Oxford MathematicalMonographs, Oxford at the Claredon Press, 1962, pp. 34.

32. Brown, S. N.; and Stewartson, K.: Laminar Separation. Ann. Rev. FluidMech., Annual Reviews Inc., 1969, pp. 45-72.

33. Prandtl, L.: On the Calculation of the Boundary Layer. From Z. angew.Math. Mech., vol. 18, no. 1, 1938, pp. 77-82; translated as NACA TM959, 1940.

34. Meksyn, D.: Integration of the Laminar Boundary Layer Equation. I -Motion of an Elliptic Cylinder. Separation. II - Retarded Flow Alonga Semi-Infinite Plane. Proc. Roy. Soc. A, vol. 201, March 1950, pp.268-283.

35. Meksyn, D.: Integration of the Boundary Layer Equations. Proc. Roy.Soc. A., vol. 237, Nov. 1956, pp. 543-559.

36. Schubauer, G. B.: Air Flow in a Separating Laminar Boundary Layer.NACA TR 527, 1935.

37. Lees, L.: Viscous-Inviscid Flow Interactions at Supersonic Speeds.First Annual Dean's Lecture, Dept. of Mechanical Engineering, Univ. ofNotre Dame, April 10-11, 1967.

38. Klineberg, J. M.: Theory of Laminar Viscous-Inviscid Interactions inSupersonic Flow. Ph.D. Thesis, Calif. Inst. of Technology, Pasadena,Calif., 1968.

39. Oswatitsch, K.: Die Ablosungsbedingung von Grenzschichten. BoundaryLayer Research, edited by H. Gortler, Springer-Verlag, 1958, pp. 357-367.

40. Dean, N. L.: Note on the Motion of a Liquid Near a Position of Separa-tion. Proc. Camb. Phil. Soc., vol. 46, 1950, pp. 293-306.

33

41. Legendre, R.: Decollement Laminaire Regulier. Comptes Rendus, vol. 241,Sept. 1955, pp. 732-734.

42. Riley, N.; and Stewartson, K.: Trailing Edge Flows. J. Fluid Mech.,vol. 39, pt. 1, 1969, pp. 193-207.

43. Lewis, J. E.; Kubota, T.; and Lees, L.: Experimental Investigation ofSupersonic Laminar Two-Dimensional Boundary-Layer Separation in a Com-pression Corner With and Without Cooling. AIM J., vol. 6, no. 1, Jan.1968, pp. 7-14.

44. Carter, J. E.: Numerical Solutions of the Navier-Stokes Equations for theSupersonic, Laminar Flow Over a Two-Dimensional Compression Corner.NASA TR R-385, 1972.

34

APPENDIX

A program listing for the point-relaxation version of the first methodis included. Only a description of the input and output is given; however,program variable names are the same as used in the text and should be self-explanatory. No effort was made to optimize the code or even to use a veryefficient procedure for solving the attached region of the flow. A solutioncase corresponding to a = 0.1 in figure 13 is included.

INPUT PARAMETERS

(Subroutine INIT)

JMAX = maximum number of points in x, 3 < JMAX < 120KMAX = maximum number of points in y, 4 - KMAX 5 100DX = AxDY = AyXO = x-location of initial profileUEXO = ue at XOSMO = m at XOALPHU, ALPHV, ALPHM - relaxation parameters to update u, v, m

(Subroutine PROFL)

DYO = Ay in which initial profile is givenKMAXO = number of data points to specify initial profileU(K),V(K) = u and v of initial profile

(Main)

ITERM = maximum number of iterations permittedRMAX = calculation is terminated if the maximum residual exceeds RMAXRMIN = residual at which iteration ceases and the converged solution is

printedALPHM2 = after an initial number of iterations, ALPHM is reset to this valueADDAL = increment to ALPHU and ALPHV after an initial number of iterations

Wall Shear is analytically input in the present program.

OUTPUT

- The input parameters and the initial profile are printed.

- Minimum output from a marching routine that calculates the attached flowregion is printed.

35

- Maximum residuals and their locations are printed every 10 iterations.

- The basic solution as a function of x is printed; data include j, x, m,ue, ve, and T.

- The solution profiles are printed at each x station. Data include k, y,u, v, y, u, v, and \\>/-/R and interpolated values of y at constant valuesof \l>/R.

36

PROGRAM LISTING

AND

CASE RUN

37

PROGRAM FLOSEP (INPUT,OUTPUT,TAPE5=1NPUT,TAPE6=GUTPUT)CC MAIN PROGRAMC AN ITERATIVE FINITE-DIFFERENCE METHOD FOR INTEGRATING THEC LAMINAR INCOMPRESSIBLE BOUNDARY-LAYER EQUATIONS THROUGHC SEPARATION AND REATTACHMENT

COMMON SMC120) , SMC U20) ,U (120, 100 )., V {120 , 100) » XB (120)COMMON /PARAM/ JMAX , KMAX , DX , DY , UEXO ,

1 ALPHU , ALPHV , ALPHM , RMAX , RMINCOMMON /RESID/ ITER > JREST i KREST , RESU i RESV,JU, KU

CDIMENSION TAUU20)

CC CONSTANTS

ITERM = 1000RMAX a 10,RMIN = o.oooosALPHM2 = 0,5ADDAL - 0.06

CC INITIALIZATION

CALL INITSY2 s 1,/(DY*OY)

CC INPUT WALL SHEAR INTO TAU(J) AR R A YC EXAMPLE CASE

ALF =0.1TO = 0,33238/12.DO 10 J=1,JMAXX2 = XBCJJ-2.X6 s X2-<*.IF(X2) 9,9,5

5 IFCX6) 7,9,97 TAU(J) = TO*X2*X6*(1.+ALF*X2*X6)

GO TO 109 TAU(J) = TO*X2*X6

10 CONTINUE

C INITIALIZATION COMPLETECC MARCHING IN ATTACHED FLOW REGIONS

CALL MARCH (Jl, J2, TAU3

1FU1-J2) 12.50,5012 C O N T I N U E

Cc RELAXATION PARTc

WRITEC6,500)C

ITER * 015 CONTINUE

RMTST s 0.0JRM a 1

C UPDATE M, EQ. 31

00 25 JsJl,JZRM = SM(J)*SY2*(U.*U(J»2)-0.5*U(J,3)-3.*DY*TAU(J))SM(J) s SM(J)-4LPHM*RMSMC(J) s 0.5*(SMCJ)+1.0)RMP = ABS(RH)IFCRMP-RMTST) S5,2«,2"

24 RESM a RHRMTST * RMPJHM = J .

25 CONTINUE30 CONTINUE

C CAUL RELAXATION ROUTINE. METHOD ONE

ITER s ITER +1CALL RELAX (Jl, J2)IF (ITER - (XTER/10)*10) 32,31,32

c . PRINT MAXIMUM RESIDUALS AND LOCATIONC EVERY 10 ITERATIONS..

31 CONTINUEWRITE (6,501) ITER,RESV,JREST,KREST,RESU,JU,KU, RESM,JRM

3£ REST s ABS(HESV)IF (ITER -. 200} 38,3«,38

C CHANGE RELAXATION PARAMETER AFTER 200 ITER

31 ALPHM s ALPHM2ALPHU : ALPHU + ADDALALPHV » ALPHV + ADDALIF (ALPHV - 1.0) 36,35,35

35 CONTINUEALPHU s 0,98ALPHV s 0.98

36 CONTINUEWRITE(6,508) ALPHV,ALPHU,ALPHM

C TEST WHETHER TO TERMINATE CALCULATION

36 IF(REST-RMAX) <iO,«0,60

39

40 IF (ITER • ITERM) 45,45,4242 CONTINUE

WRITE (6,502)GO TO 50

05 If (REST - RMIN) 46,46,1546 CONTINUE

WRITE C6,507)

50 CONTINUE

TERMINATION WITH PRINT OUT

CALL PRNT

60 CONTINUESTOP

500 FORMAT(1HO,35X22HRELAXATION CALCULATION //1 7X,4HITER,5X,5HHES V,7X,7HJV Ky,6X,5HRES U»7X,2 '7HJU KU,6X,5hRES M,7X,2HJM)

501 FORMAT(5X,I5,3(E15.5,2I5))502 FORMATC34HO MAXIMUM ITERATIONS COMPUTED...,)507 FORMATC25HO CONVERGED SOLUTION,,..)508 FORMATdHO, 17HALPHV,ALPHU,ALPHM ,3F13.5)

END

SUBROUTINE INIT

INPUT DATA SUBROUTINE

COMMON SM(120) , SMC(120).,U(120,100),V(120.100),XB(120)COMMON /PARAM/ JMAX , KM A X , ox , or , UEXO ,

1 ALPHU / ALPHV , ALPHM , RMAX , RMIN

DIMENSION UKIOO) , VK100) , •

READ (5,500) JMAX,KMAX,DX,DY,XO,UEXO,SMOREAD (5,501) ALPHU, ALPHV, ALPHMWRITE (6,505) JMAX, DX, XO, ALPHV, KMAX, Of, UEXO,ALPHU,SMO,ALPHM

INTERPOLATION OF INITIAL PROFILE IF NEEDED

CALL PROFL (KMAX, DY, Ul, VI)

DO 30 Jcl,jMAXXB(J) = XO+(J»1)*DXSM(J) = SMOSMC(J) > 0.5«(SHO+1.)DO 20 K=1,KMAXU(J,K) B Ul(K)V(J,K) s VKK)

20 CONTINUE30 CONTINUE

RETURN500 FORMAT(2I5,5F10.0)501 FORMAT(BFIO.O)505 FOPMATC1H1,35X,12HINPUT VALUES//

1«X,6HJMAX s,!5,4X,«HDX s,fJ0.5,6X,«HXO =,F10.5,«X,7HALPMV s,F10.S/2UX.6MKMAX s,I5,aX,t|HOY s,F10,5,«X,6HUEXfl =,F10.5,«X,7HALPHU »,3F10.5/ 39X,«HMO = ,F10.5,«X,7HALPHM =.F10.5)END -

SUBROUTINE PROFL CKMAX1, OYl» Ul» VI)

INTERPOLATION OF INITIAL PROFILE.

DIMENSION Y(00),U(60),V(60), C(5), S(5), T(5) ,1 YK100), Ul(lOO) , V1C100)

INT B 2

INPUT INITIAL PROFILE

READC5,501) OYO,KMAXOMRITE(6,510)00 2 K=1,KMAXOREAD(5,500) U(K3,V(K)YCK) = (K-1)*DYOWRITE (6,511) K, YtK), U(K), V(K)

2 CONTINUEIF( KMAX1 - KMAXO) 30,30,3

3 CONTINUEKSAVE s 1DO 20 Klsl,KMAXlYl(Kl) = (K1-1)*DY1

1 DO 5 KSKSAVE,KMAXO • •KK s KIFCY(K)-YiCKl)) 5,5,6

5 CONTINUE6 IFCKSAVE-1) 9,9,77 IFtY(KK-l)-YHKl)) 9,9,88 KSAVE s 1

60 TO H9 KK • KK-UNT + D/2

IFCKK) 10,10,1110 KK = 1

GO TO 1311 M = KK+1NT

IF(M-KMAXO) 13,13,1212 KK = KK-1

GO TO H13 INT1 s INT+1

KSAVE a KKDO 11 L=1,INT1C(L) » YKKl)-Y(KK)SCL) = U(KK)T(L) = V(KK)

14 KK - KKtl

00 16 KKsl,INT1 = KK+1

15 D = C(KK)-CCI)SCI) s CCCKK)*SCI)-C(1)*SCKK))/DTCI) e CCCKK)*TCI)-C(I)*TCKK))/DI = 1 + 1IFCI-INT1) 15,15,16

16 CONTINUEU1CK1) = SCINT1)V1CK1) s TCINT1)

20 CONTINUERETURN

C NO INTERPOLATION

30 DO 31 K*1,KMAX1U1CK) s UCK)VI (K) = VCK)

31 CONTINUE

RETUKN

500 FORMATC8F10.0)501 FORMATCFIO.0,15)510 FORMATC1 HO,15X,1«HINITIAL VALUES//4X,1HK,7X,IKY,1IX,lHU» 11 X, 1HV)511 FORMATC2X,I3,3F12.6)

ENDC

SUBROUTINE MARCH CJ1. J2» TAU)

C MARCHING IN ATTACHED REGION

COMMON SMC120) , SMC(120),U(120,100),V(120,100).XB(120)COMMON /PARAM/ JMAX , KMAX , DX , DY , UEXO ,

1 ALPHU , AUPHV , ALPHM ,'RMAX , RMIN

DIMENSION TAUC120) , UX(IOO)

TAUWT = 0.02ALPHM2 B .5SY = ,5*DYSYY = DY*DYSX2 a ,5/DXSY2 = l./CDY*DY)

CN a 2JINT » -1KM B KMAX-1IFCTAUC3) - TAUWT) 50,50,5

C5 N = N +1

J s NJl B J t JINTJ2 = JJINT = 0

IFC J - JHAX) 6,6,50

C TEST TO SEE IF PROFILE IS ATTACHED

6 IFCTAU(J) - TAUWT) 50,50,77 CONTINUE

C OBTAIN GOOD GUESS BY USING EXTRAPOLATION OF LAST COMPUTED PROFILES

DO 8 K=2,KMUCJ,K) B 2.*UCJ-1,K) -U(J-2,K)

6 V(J,K) s 2,*V(J.1,K) - V(J-2,K)SM(J) = SMCJ-1)

CITER = 0UXC1) = 0.

C10 ITER = ITER + I

DO 20 J=J1,J2JR = J-lJRR = J-2REST = 0.RM = SM(J)+SY2*U.*UCJ»2)-Ot5*UCJ,3)-3.*DY*TAU(J))SM(J) = SM(J)-ALPHM*RMSMC(J) = 0.5*CSM(J)+l,0)00 18 K=2,KMKR a K-JKP s K +1

CIF( J-2) 12.12,13

12 UX(K) a SX2*t U(J+1,K) - UCJRrKnU2X = UX(K) *( UCJ+l.K) + UCJR.K))DIAX a 0.GO TO 14

13 UX(K) = SX2* C 3.*U(J/K) -«.*U(JR,K) +UURR.K))U2X s SX2*( 3.*U(J,K)**2 -4.*U(JR,K)**2 » U(JRR,K)**2)DIAX = 3,*XB(J) *SX2

14 CONTINUEUY = SY*(UCJ,KP)-UCJ,KR))FU = UY*V(j>K)tSYY*C0.5*XBtJ)*U2X-SM(J)*(lt-U(J/K)**2))RU = U(J,KR)-2.*UCJfK)+U(J,KP)-FURV s VCJ,Kj-V(J,KR)tSY*(XB(J)*CUXCK)»UXCKR»+SMCCj)*(UCJ»K)+

1 U(J,KR)))OT s 2. + SYY*U(J,KJ«CIAXou = RU/DTDV = - RVUCJ,K) = U(J,K) + DUVCJ,K) s V(J,K) + DVRT s ABS(RV)IFCRT - REST) 18,18,15

15 REST s RTIB CONTINUE20 CONTINUE

CIFC ITER - 20) 26,26,25

as ALPHM sas CONTINUE

IF( REST -.RMAX) 27,100,100Z7 IF( REST - RMIN) 30,JO,28ae IF( ITER - 600) 10,100,10030 CONTINUE

IFCN-3) 35,35,«035 WRITE(6,501) . -

IZ = 0RZ = 0.DO 36 J = 1,2V(J,KMAX) s VCJ,KM) - DY*SMCCJ)TAUW = ,5*( -3,*UCJ»1) »1.*U(J»a) -UtJ,3))/DYWRITE(6,500) J,IZ,RZ,V(J,KMAX),TAUW,SMCJ)

36 CONTINUE(10 J a N

TAUW - .5*( -3.*U(J,1) +«.*UCJ,a) -UCJ,3))/DYVCJ.KMAX) « V(J,KM) - OY*SMC(J)WRITE(6,500) J,ITER,REST,VCJ,KMAX),TAUW,SM(J)GO TO 5

"50 Jl = Nja s JMAXRETURN

100 J s NTAUW = ,5*( -3.*UCJ,D +«.*UtJ,a) -U(J,3))/OYV(J,KMAX) = V(J,KM) - DY*SMC(J)WRITE (6,500) J,ITER,RtST,VCJ,KMAX),TAUW,SM(J)STOP

500 FORMATC1H ,ai5,«F13.5)501 FORMATfiHl, 35X18HMARCHING PROCEDURE // 5X1HJ, aX«HlT£R,

1 7X5HRES V, SXaHVMAX, 1OX3HTAU,10X1HM )

END

SUBROUTINE RELAX CJ1, J2)

RELAXATION FOR REVERSED FLOW

COMMON sM(iao) , sMC(iao),u(iao,ioo),vciao,ioo),xB(iao)COMMON /PARAM/ JMAX , KMAX , ox ., DY , UEXO ,

. 1 ALPHU , ALPHV , ALPHM , RMAX , RMINCOMMON /RESID/ ITER , JREST , KRtST , HESU , RE5V,JU, KU

DIMENSION UX(IOO)

EPS1 = .015EPSa = 0.005KM u KMAX "I

EPS = EPSl-EPSaRESU s o.

RESV = 0.0RTEST a 0,REST « 0.0SY a 0,5*DY8YY = DY*DYSX2 = 0.5/OX.00 50 JaJl,J2JR a J-lJP a J+lUX(l) a 0.000 30 KB 2,KMKR • K.IKP = K*l

Cc

IF(J-JMAX) 10,13,1310 UX(K) a SX2*tU(JP,K)-U(JW,K))

IFCJ-2) 11,11/1211 U2X = SX2*(U(JP,K)**2-U(JR,K)**2)

DIAX * 0.GO TO 20

12 T = U(J,K)IF(T-EP81) 16,16,la

13 UX(K) a SX2*(3.*U(J,K)-U.*U(JR,K)+U(JR-1,K))tC ATTACHED FLOWC

14 JQ a J-2U2X = SX2*(U(JQ,K)**2-«.*UCJH,K)**2»3.*U(J,K)**2)OIAX a 3.*XB(J)*SX2GO TO 20

CC SEPARATED FLOWC

16 USX a SX2*(UCJP,K)**2-U(JR,K)**2)DIAX = 0.IF(T»EPS2) 20,20,18

CC SEPARATION POINTC REATTACHMENT POINTC

18 J(J - J-2U2P - U2XU2X » SX2*(U(JQ,K)**2-U.*UCJR,KJ**2+3,*U(J,K)**2)TA = (T-EPS2)/EPSU2X = TA*U2X+C1.-TA)*U2POIAX e 3.*TA*XB(J)*SX2

C20 CONTINUE

UY a SY*(U(J,KP)-UU»KR))FU a UY*V(J,K)fSYY*CO.S*XB(J)*U2X"SM(J)»(l.-UCJ,K)**2))RU a U(J,KR)-2,*UCJ,K)+U(P,KP)-FURV - VCJ»K)-V(J,KR)»SY*(XB(J)*(UX(K)+UX(KR))»SMCCJ)*(UCJ»K)+

1 UCJ,KR)))DT a 2. » 8YY*U(J»K)*DIAX

DU = HU/DTDV = • RVUCJ,K) = UCJ,K)tALPHU*DUVCJ,K) s V(J,K)*ALPHV*DV

RT s ABSCRU)IF(RT-RTEST) 22,22,21

21 RESU a RURTEST s RTJU « JKU f K

22 CONTINUERTT = ABSCRV)IF(RTT-REST) 27,27,26RESV = RVREST = RTTJREST a JKREST = KIFCREST-RMAX) 27,27,100CONTINUE

30 CONTINUEK s KMAXRV = V(J,KMAX)-V(J,KM)+SY*(XBCJ)*UXCKM)+SMC(J)*C1.+UCJ,KM)))V(J,KMAX) s V(J,KMAX)-ALPHV*RVRTT = ABSCRV)If-CRTT-REST) 35,35,34

34 RESV s RVREST = RTTJREST = JKREST s KCONTINUECONTINUERETURNEND

SUBROUTINE PRNT

26

27

3550100

OUTPUT SUBROUTINE

COMMONCOMMON /PARAM/

1COMMON /RESID/COMMON /STRM/

SMC120)JMAXALPMUITER

PSI(IOO)KIM

SHC(120),U(120,100),VC120,100),XB(120)KMAX , DX , DY , UEXO ,ALPHV , ALPHM , RMAX , RMIN, JREST , KREST , RESU > RESV,JU, KUYST(IOO) , POUT(60) , YIC60) , Y2(60) ,K2M

DIMENSION UY(120) , UEC120) , F(120) , FX(120) , UST(IOO) ,1 VST(IOO), ETA(IOO)

HY s 0.5/DYKEND = KMAXSX - 0.5*DXDO 10 J=1,JMAXUUJ) = HY*(-3.*UCJ,m«.*U(J.2)-UCJ«3))

10 CONTINUE

REST = RESVWRI TEC 6,500) ITER,JREST,KREST,REST

CC INTEGRATION FOR UECX)C • ' .

UEU) a UEXOF(l) a ALOG(UEXO)IFC XBC1) -.00001) 12,13,13

12 XB(1) a ,0000113 CONTINUE

FX(1) a SMCD/XBCl)00 15 Ja2,JMAXFX(J) a SMCJ)/X6CJ)F(J) B F(J-1)+SX*CFX(J)+FX(J-1))UE(J) a EXPCF(J))

15 CONTINUE

WRITE (6,SOS)WKITE (6,503) CJ,XB(J)»SMCJ),UE(J),V(J,KMAX)•UY£J),Jel,JMAX)DO 20 Kal,KMAXETACK) a (K-1)«DY

20 CONTINUEYST(l) * 0,VST(l) = 0,PSK1) s 0.

DO 50 JcltJMAXCl » SQRTCX8(J)/UE(J)>C2 = 1./C1Ci = SQRTCXBCJ)*UE(J))51 a 0.5*(SM(J)-1.)52 = 0.5*DY*C3USTCl) s UE(J)*U(J»1)

DO 30 K=2,KHAXY s ETACK)YSTCK) = C1*YUSTCK) a UCJ,K)*UECJ)VSTCK) s CVCJ,K)-S1*Y*UCJ,K))*C2PSlCt<) a PSI(K-1)+S2*(UCJ»K)+UCJ,K-1))

30 CONTINUE

DSTR B YSTCKMAX).PSICKMAX)/UECJ)WRITEC6,510) XBCJ).DSTR

CCALL STREAM CKEND)

CWRITE (6,511)IF CK1M ,EQ. 0) GO TO UOWRITE (6,512) (K,ETACK),U(J,K),V(J,K),YSTCK),UST(K), VST(K),

1 PSlCK), POUTCK), Y1CK), Y2CK), KB!,KIM)

C KIM - NUMBER OF POINTS WITH SEPARATION

40 CONTINUE

K2 s KIM + 1WRITE C6.51<1) (K,ETACK).UCJ,K),V(J,K),YSTCK),UST(K), VST(K),

1 PSICK), POUTCK), YKK), KsK2,K2M)

C . K2M - TOTAL NUMBER OF INTERPOLATED POINTS,C (SEPARATED PLUS SINGLE VALUED) .

K3 = K2M t 1WHITE (6,M3) (K,ETA(k),U(J,K),V(J,K),YST(K),UST(K), VST(K),

1 PSICK) , KsK3,KMAX)50 CONTINUE

RETURN

500 FORMATC9H1 ITER'=,I5,9H JRESV =,Ib,9H KRESV s,l5,8H RESV «»1 H3.5)

502 FORMATCSHO J, 6X, UHX C J} , 6X, 2MSM,9X,6HU EDGE,7X,6HV EDG£,6X,1 10HDU/DY WALL)

503 FOHMATC2X,I3,F10.3,4F13,6)510 F O R M A l t l H l f b H X t J ) = F12.5, 2X8HDELSTR = F12.6 )511 F O R M A T ( 1 0 2 X , 1 2 H I M T E R P O L A T E D /

1 3 X , 1 H K , « X , 3 H E T A , 9 X , 1 H U , 1 0 X , 1 H V , 1 5 X , 1 H Y , 1 1 X , 3 H U S T , 9 X , 3 H V S T , 9 X ,2 3 H P S I , 1 5 X . 3 M P S I , 7 X , 1 H Y , 9 X , 1 H Y ) •

512 FONHAT( ia ,F6 ,3 ,2F12 .6 ,4X ,aF12 .6 ,10X,F7 .4 ,2F10 .5 )513 FURHAT(I<1,F8.3 ,2F12.6 ,«X,UF12.6)51« FORMAT( I« ,Fe .3 ,2F12.6 ,«XrUF12.6 ,10X,F7 .« ,F l0 .b )

ENDC

SUBROUTINE STREAM (KMAX)

c INTERPOLATION FOR STREAM FUNCTION

COMMON /STRM/ PSIClOO) • YST(IOO) > POUT(60) > Yl(60) > Y2C60) »1 KIM , K2M

DIMENSION CCa) , SC«)

CPSMIN s -0.10PSMAX = 10,0

CNO » *200.*PSMINNO s NlO + 1M B NO+9N2 = NltlON3 = N2+10DO 8 N=l,60IFCN-N3) 2,1,1

1 POUT(N) s 6*N-N3NMAX s N

2 IFCN-N2) 4,3,33 POUT(N) = 0.5*(N-N2+1)

GO TO 8

0 IF(N-Nl) 6,6,55 POUT(N) s O.Ob*(N-Nl)

GO TO 86 POUTCN) s 0.005*(N«NO)6 CONTINUE9 CONTINUE

CC**»**FIND MINIMUM PSI

INT = 2DO 10 Ks2,KMAXKK = KIF(PSI(KK)-PSICKK-in 10,10,11

10 CONTINUE11 KPMIN s KK-1

IFCKPMIN-INT) 12,12,1512 NMIN s NO

KPMIN a 1GO TO 40

15 PMIN s PSI(KPMIN)CC*****FIND INITIAL PRINTOUT VALUE

DO 17 Nsl,NMAXNN = NIF(POUTCN)-PMIN) 17,17»18

17 CONTINUE16 NMIN = NN

NI = NO-NMINcC*****INTERPOLATION.,.PSI FROM HALL TO U = 0

00 30 L=1,N1NN = NO-LDO 20 K=l,KPMINKK B KIFCPOUTCNN)-PSI(KK)) 20,20,21

20 CONTINUE21 KK » KK-(INT+l)/2

IFCKK) 22,22,2322 KK = 1

GO TO 2523 M s KK+INT

IF(M-KPMIN) 25,25,2«24 KK B KK-1

GO TO 2325 INT1 = INT+1

DO 26 J=1,INT1 • •CCJ) = POUTCNN)-PSI(KK)SCJ) = YST(KK)

26 KK s KK+1DO 28 J B 1,INT1 B J + l

27 S£I) = (C{J)*SfI)-C(I)*5(J))/(C(J)-C(I))I s 1 + 1

IFCI-INT1) 27,27,2826 CONTINUE

Y2CNN) * SCINTl)30 CONTINUE

KPMIN s KPMIN+i40 CONTINUE

KSAVE s KPMINCC*****INTERPOL*TIUN,..PSI FROM U = 0 TO EDGE

DO 60 NBNMIN,NMAXNN • NDO 45 KsKSAVE,KMAXKK B KIFCPSHKK)-POUT(NN)) 45,45,46

45 CONTINUE46 KK = KK-1

IFCKK-KPMIN) 47,47,4847 KK = KPMIN

GO TO 4948 M a KK+2

IFCM-KMAX) 49,49,4649 KSAVE s KK

DO 50 jsl,3C(J) = POUT(NN)-PSKKK)S(J) a YST(KK)

50 KK = KK+1S(2) = CC(l)*S(a)-C(Z)*S<l))/(C(l)-C<2))S(3) = (CC1)*3(3)"C(3)*8(1))/(C(1)-C(3))Yl(NN) B CC(2)*3(3)-C(3)*S(2))/(C(2)-C(3))

60 CONTINUEKl B 0IFCNO-NMIN) 75,75,65

65 NO! = NO-1DO 70 L=NMJN,NOlKl s Kl + 1POUTCK1) « POUT(L)Y l C K l ) B YKL)Y2CK1) B Y2CL)

70 CONTINUE75 CONTINUE

KIM B KlK2 B KIMDO 80 L B NO,NMAXK2 = K2 + 1POUTCK2) B POUT(L)YKK2) s YKL)

80 CONTINUEK2M = K2RETURN

END

52 52.125 .24 0, 1. 0.0.9 0.9 0.05

,250,000000,0830«0.165970,246480.330030,109910.487260.561110.630470.694420,752160.603130,647040,883900,914000.937900.956320,970100,980100,987120,991910,995060,997080,998320,999060,999490,999730.999860,999930,999960,999980,999990,999990,999990,999990,999990,999990,999990,999990,999990,999990,999991.000001.000001,000001,000001,000001,000001,000001,00000XENDDS

500.00000-0.00519-0.02075-0.04665-0,08281-0.12906-0.18513-0.25065-0.32513•0.40793-0.49834-0,59555•0.69869-0.80687-0,91924-1,03496-1.15337-1.27377-1.39566-1.51861•1.64230-1.76649-1.69100•2.01571-2.14055-2,26546-2.39041-2.51538-2.64037-2.76537•2.89036-3.01536-3.14036-3,26536-3.39036-3.51536-3.64036•3.76536-3.89036-4,01536-4.14036-4.26536-4.39036-4.51535-4.64035•4.76535-4.89035•5.01535-5.14035-5.26535

INPUT VALUES

JMAX = 52UMAX a 52

DX s .12500 XO BOY s .24000 UEXO =

MO =

INITIAL VALUES

K123u<367e9

101112131415161716192021222324252627282930313233343536373639uo41424344454647464950

y0.000000.250000.500000.750000

l.QOQOOO1.2500001,5000001.7500002.00QOOO2.2500002.5000002.7500003.0000003,2500003.5000003.7500004.0000004.2500004,5000004,7500005.0000005,2500005.5000005,7500006.0000006,2500006.5000006.7500007.0000007.2500007.5000007.7500006,0000006.2500008.50QOOO8,7500009.0000009.2£00009,5000009.7SOOOO10.00000010.25000010.50000010.75000011.00000011.25000011.50000011.75000012.00000012,250000

U.0.000000,063040.165970.248460,330030.409910,487260.561110.630470.69U420.752160.603130.647040.883900.91UOOO.937900.956320.970100.980100.987120.991910.995060.997080.998320.999060,999490.999730,999860.999930,999960,999980,999990.999990.999990.999990,999990,999990.999990.999990.999990.999990.999990.000000.000000.000000.000000.000000.000000.0000001.000000

V0.000000-.005190-.020750-.046650-.082810-. 129060-.185130-.250650-.325130-.407930-,498340-.595550-, 698690-.606870-.919240•1.034980-1.153370-1.273770-1.395660-1.518610-1.642300-1.766490-1.891000-2.015710-2. 140550-2.265460-2.390410-2.515380-2.640370-2.765370-2.890360-3.015360-3,140360-3.265360-3.390360-1.515360-3.640360-3.765360-3.890360-4.015360-4.140360-4.265360-4.390360-4.515350-4,640350-4.765350-4.890350-5.015350-5.140350-5.265350

0.000001,000000.00000

ALPHV *ALPHU =ALPHM *

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