A CONSERVATIVE IMPLICIT FINITE-DIFFERENCE ALGORITHEL FOR THE UNSTEADY TRANSONIC PULL-POTENTIAL EQUATION

J. L. Steger* Flow Simulations, 'Inc. Sunnyvale, California

and

F. X. Caradanna" U.S. Army Aeromechanics Laboratory, AVRADCOM

Ames Research Center, Moffett Field, California

Abstract 11. Governing Equations

An implicit finite-difference procedure is developed to solve the unsteady full-potential equa- tion in conservation-law form. ciency is maintained by using the approximate fac- torization technique. first order in time and second order in space. The circulation model and difference equations are developed for lifting wings in unsteady flow that employ thin-wing body-boundary conditions. Both two- and three-dimensional calculations are carried out.

Computational effi-

The numerical algorithm is

Transformation of the Equations

As a governing equation for three-dimensional, unsteady full-potential flow the conservation of mass equation is used

where density P is determined from the unsteady Bernoulli relation,

I. Introduction

The successful development of an unsteady conservation-law form full-potential finite- difference algorithm offers a practical means to compute unsteady transonic inviscid flow about a large class of complex configurations. Unsteady potential theory can satisfactorily replace the Eder equations if the shack waves are sufficientlv

Here ditions of irrotationality

.$, - u, .$y =,Y, and mZ = w satisfy ehe con-

weak and if the equations employ an equivalent cir- culation model. Yet, the full-potential can have similar computer requirements in time and storage to the simplified transonic small- disturbance theory.

In deriving Eq. (2) the far-field is assumed to be steady. Throughout, the equations have been refer- enced to free-stream quantities,

e = P / P _ , u = u/a,, x = X/i, y = Y/&, ; = z / a ,

I = ta,/a The purpose of this paper is to present an approximate-factorization, implicit, finite- difference algorithm for the unsteady, full-potential equations in conservation-iaw form. ~ ~ ~ ~ ~ ~ ~ ~ ~ i ~ ~ - and the tilda has been suppressed. Here a is the sound speed, and & is a reference length, such as law form is maintained to "capture" shocks with cor- airfoil chord C . rect strength and location. Durinp, the course of this research, two other procedure; have been developed for the unsteady, two-dimensional full- potential equations in conservative form. one cedure, due to Chipman and Jameson,+ solves a system of chree equations for three unknowns, The other, due to Gooriian.5 is a

Boundary conditions are best satisfied by map- Pings which can also be used to cluster grid points so as to enhance numerical solution accuracy. shown by Viviand,6 conservation-law form of Eq. (1) can be maintained under the general transformation

AS

that is suite similar - . to the one developed independently here. E = E(X, Y, 2 , t)

In this report the governing equations are first n = n(x, Y> z > t)

5 = S(X, Y. z i t) ( 4 )

reviewed in Sec. 11. Differencing of the governing equations in conservation-law form and their numeri- cal solution are discussed in Secs. I11 and IV. in Sec. V, results in both two- and three-dimensions are presented to verify the algorithm. Note that small-disturbance boundary conditions are used throughout so as to avoid unduly complicating the development of the numerical method for the full- potential equations.

T = t

giving

aT(p /J ) + aE(Pu/J) + a n m m + ~<(Pw/J) = o ( 5 )

where J is the Jacobian la(s,n,c)/a(x,y,.)l and "Research Scientist. Member AIM.

This pnper is declared a work of the U.S. Government and therefore is in the public domnin.

1

Dow

nloa

ded

by S

tanf

ord

Uni

vers

ity o

n O

ctob

er 3

, 201

2 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.1

980-

1368

v = nt + 'Ixmx + 'Ivmv + nz4=

I4 = ct + 5x4x + 5 m + czmz Y Y

= TX(mx - Xe) + 5 tm - Y,) + cZ(mz - zt) Y Y

The terms mX, my, and mX are expanded by chain rule

mx = 5,m5 + nxm" + 5,m5

my = 5 m + + sym5 (7)

mz = szmc +

Y S

+ 5,4<

Boundary Conditions

As boundary conditions, tangency (i.e., w = O) is imposed on the body surface, and the flow at infinity is required to be uniform and steady. The thin-airfoil body-boundary approximation is utilized since our chief interest here is in verifying the use of an approximately factored finite-difference algorithm for unsteady potential flow. mapping of boundary conditions unduly complicates this task. Surface tangency is replaced by

The complete

(12) dz &lbody slope t so U, V, and W can also be written mz = + z

u = 5, + 05. + mqvn + 45v5)

= 05. (4<85 + m,vn + $'<O5 - rt) which is imposed on the grid lines adjacent to the

+ wing - see Fig. 1.

At distances far from the body u = u- and v = nt + VQ. tm vs + .$"on + m V5) with use of a, as a reference velocity

5 5

+ V c . ( 4 75 + mnVn + $CR - rt) ( 8 c )

where r = (x,y,z)+ and V is the usual Cartesian vector-gradient operator. Subject to the came trans- formation, the Bernoulli equation is written

5 +

For the present purpose af demonstrating an unsteady full-potential algorithm in conservatian- law form the transformation is restricted to the simple Cartesian stretchings

5 - S(X)

'I = ,(Y) (10)

5 = F(Z)

and the thin-airfoil body-boundary approximation is utilized. Because of the transformation Eq. (10). the governing equations then simplify to

+- = M,x (13)

where M, is the free-stream Mach number. Here we chose to cant the wing to obtain angle of attack rather than have the flow vector be specified at an angle, as is often done.

For lifting cases allowance must be made for a jump af potential across the wake-like cut. Ac we intend to partially model an unsteady shear layer leaving the wing, this cut is taken off from the trailing edge (see Fig. 1). On the back boundary then, Eq. (13) does not apply and free-stream pressure (or density in isentropic flaw) must be imposed instead. From Eq. (llb) this implies that

along the back face af Fig. la. Neglecting small perturbations, this simplifies to

m, + SxM-m, = M,'

Equation (13) is imposed along all other far-field boundaries (Fig. la).

(15)

In unsteady flow, vorticity is continuously shed from the wing. this phenomenon can be approximately modeled with a cut aligned with the shear layer. Here we further idealize this flow by keeping the cut in the mean chord line plane, as sketched in Fig. Ib.

In a potential-flow formulation

The jump in potential along the cut, Am = r , will vary with time. By imposing the usual shear layer assumptions, an equation for the cut can be derived. Across an inviscid shear

r(x,y,t) along

2

Dow

nloa

ded

by S

tanf

ord

Uni

vers

ity o

n O

ctob

er 3

, 201

2 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.1

980-

1368

layer, normal velocity and pressure (p) are contin- uous, but tangential velocity and density can be discontinuous. Here, however, the incoming flow is uniform isentropic (i.e., homentropic) and the shocks are weak. that p = py throughout, and density is also taken to be continuous across the cut.

Thus, we have previously assumed

Define r = UmD = mu - +, across the cut as illustrated in Fip. Ib. From the condition that the - velocity normal to the shear layer is continuous, umzn = o

(16) =

The Bernoulli relation together with the continuity of density requires that

(2mt + mx2 + my2 + me2)e - (2mt + mx2 + my? + rnZ2)"

and because normal velocity is continuous

emt + mx2 + m 2, = za ( r n + r ) + [ a x ( + ¶ . + uiz y ¶ . t e

+ [ay(+& + r ) i 2

co that

zrt + zmx/,rx + rx2 + zmylary + ry2 = o (17)

Consistent with thin-airfoil body-boundary condi- tions, Eq. (17) is approximated by

rt + k r x = o (18)

Equation (18) gives an equation to determine r(x,y,t) along the cut.

Although m Z is continuous across the cut, in unsteady or three-dimensional flow m Z z is discon- tinuous. This follows immediately from

uatp + axcPmx) + ay(pmy) + aZ(Pmz)n = o (19)

which on employing density continuity becomes

uax(Pmx) + aycpmy) + paZmZn = o

PU~,,D = -ax(prx) - ay(Pry)

(20)

or alternatively,

(21)

This last equation can be used to develop accurate difference expressions for a,p& or a,Ps%C when Eq. (lla) is differenced to just either side of the CUt.

schemes has always appeared to be straightfoward. Because p = p(+), Eq. (1) can be rewritten as

where

is a differential operator that does not commute. After the indicated operations are performed, Eq. (22) is found to be a second-degree wave equa- tion in nonconservative form

Using variables) gives the familiar textbook form of the potential equation (see , e , Ref. 11). Various finite difference cchemes""have been advanced for simpler forms of this equation and we assume they can be made to work for Eq. ( 2 4 ) .

py-l = a2 (in the chosen nondimensional

The problem arises, however, that Eq. ( 2 4 ) cannot be brought back into conservation-law form with 0 retained as the dependent variable. Never- theless, if one is willing to introduce into the differential equations the same expansions used in deriving the difference formula, conservative form can again be maintained. By Taylor expansion

where o represents a neighboring known state or solution. Substitution of Eq. (25) into Eq. (1) gives

If all of the densitv terms are formed with exactly the same difference operators, we assume that the local linearization term

111. Conservative Differencing and Local Linearizations

Time Derivative Term has only a higher order contribution to numerical

example mn+' - $" or mj - mj-1, where t = nAt Stable conservative difference operators have stability, Moreover, if - is small, for

been successfully developed for the steady-state part of Eqs. (1) or (lid (see Refs. 7-10), so the main task is to develop stable conservative differ-

and = jbx, the error due to expanding second-order accurate and is no greater than that

is

usually made in time differencing. ence operators for the unsteady terms as well. Development of nonconservative time-differencing

3

Dow

nloa

ded

by S

tanf

ord

Uni

vers

ity o

n O

ctob

er 3

, 201

2 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.1

980-

1368

Consequently, Eq. (26) is a conservation-law Consider the spatial derivative term axpa,+ form of Eq. ( i ) for which it is assumed that one of Eq. ( 1 ) . Use of the local linearization, can devise stable difference schemes. Such a scheme Eq. (23) , with this term gives is described in Sec. IV.

r

Space Derivatives

Just as the local linearization approach guides the proper treatment af the time-derivative term, it = ax[P,ax~ - (mxpz-Y),(at + +x103x

can be used to clarify the differencing schemes developed for the spatial terms af Eq. (1) by Jameson,' Holst and Ballhaus,' and Hafez et al." for transonic flow regions. Before proceeding with this analysis, however, some preliminary background information is in order.

Consider first the simple steady two-dimensional

model problem -ax(y)a+ - " Y

(1 - M,2)$,, + myy - o (28 ) (31)

ax(?) a=+ - axfo "

and let the difference operators Ay ox$ - (4j - $j-i)/Ax and It is recognized that the following difference schemes

V,, b x , Vy, and The and mXz terms of Eq. (31) are not crucial to our discussion, and we can justify ignoring them by assuming steady small-perturbation flow. Thus

be defined in a conventional way, for example, Ay$ = ($k+i - +k)/Ay.

(1 - M,z)VxAx4 + V A $ = O EQ 1 (29a)

(1 - L2)V V $ + V A 4 = O (29b)

Y Y

M, > 1 x x Y Y

_. are, respectively, suitable for the model elliptic (M, c i ) and the model hyperbolic (M, > i ) problems. The differencing (29b) is divergent if Eb, < 1 while that of (29a) is convergent for M, , I only if lAx/[(i - M,2)AylI 5 i . an impractical restric- tion for M, ). i . Murman and C01e.l~ aware of these simple constraints, introduced type-dependent dif- ferent operators for the transonic small-disturbance equation. In their approach the streamwise spatial difference operator is switched from central to backward as the local Mach number is less or greater than 1 .

Various refinements to the Murman-Cole differ- encing have been introduced, but usually left unmen- tioned is the fact that Eq. (28) can also be differ- enced as

wichout any need to switch difference operators. An extra boundary condition is needed in the x-direction (which is also true of many higher order difference schemes) and one must ay attention to the physics because replacing M, VxVx+ by &'AXAX+ is also convergent for any EQ. ferencing scheme given by liq. (30) is also less accurate than that of Eq. (29a), but, subject to the usual approximations such as use of periodic bound- ary conditions, the authors can demonstrate the convargence of Eq. (30) .

5 The dif-

Depending on how the operators are transi- tioned. the full-ootential steady-state difference

where we have also replaced ax by the symbolic difference operator 6, and the subtilda is used as a reminder that this operator is the operator used in the Bernoulli equation to form P. Con- sider now the differencing formula of Refs. 7-10. The term aXp$, of Eq. ( i ) is always treated by a central difference formula, although in supersonic flow regions p is evaluated shifted back to the previous point in X. That is, the term axpa,+ is differenced as (see Ref. 8 or 9)

where is a shifted-backward (i.e., upwind) value of density. More cleverly chosen values of Y are used than what is indicated in Eq. (33) . but the crucial paint to be made here is that when den- sity is shifted backwards, the operator 6, of Bq. (32) is also shifted backwards (upwind), for this operator comes from the local linearization of the Bernoulli equation. Comparing Eq. (32) with Eq. (33) leads to

-

central central Y = O backwards v = 1

(34)

which 1s analogous to Eq. (29a) for v = O and Eq. (30) for Y = 1. A rigorous discussion of the actual difference operators is given in Appendix A of Ref. 13. (We remark that the work of Ref. 14 also shows awareness af this concept.)

If the local linearization argument given scheme, af Refs. i - i 0 actually mimic the differenc- ing schemes represented Eq. for < and above Is indeed valid, then the differencing repre-

sented by Eq. (34) with Y - i should be stable in Eq. (30) for M 5 1. The differencing Eq. (30) is not used in subsonic regions because the Pure een- tra1 differencing scheme is more accurate. The switchtng is unnecessary. local linearization process will now be used to (see also Ref. 8) and, with the use of second-order

differencing, the unswitched differencing scheme clarify these remarks.

as well as supersonic regions 5o that This is indeed the case

4

Dow

nloa

ded

by S

tanf

ord

Uni

vers

ity o

n O

ctob

er 3

, 201

2 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.1

980-

1368

gives quite satisfactory results as later demon- strated in the Results section.

IV. Numerical Algorithm

An implicit approximately factored (AF) finite- difference scheme is developed far Es. (11) in conservation-law form by using the local lineariza- tion discussed in the previous section. The advan- tages of an implicit AF algorithm have been dis- cussed before (see, for example, Refs. 1, 2, 8, and 15). The algorithm presented here is option- ally first- or second-order accurate in space and time, but the second order in time option has not proved to be satisfactory.

An implicit finite-difference scheme for Eq. (lla) is given by (here suppressing spatial indices for convenience)

(e"+' - 8") + hS5[ (5x2&)n+165+n+']

E = O for first-order accuracy. The switching parameter v is defined in a way similar to that of Ref. 8 and

Y = [i - (p/p*)2]c Y 5 O if Y c O, i.e., subsonic (37)

1 5 c < 10

Y 2 1 if Y > 1, i.e., supersonic

The parameter u can be set to one throughout, and accuracy will not he impaired if E is also set t<i 2. The operators (Eqs. (36)) assume that the flow will be supersonic only in the positive x-direction (see e.g.. Ref. 9 for extensions). The density is found from the Bernoulli equation with (AS - Aq - AS - 1)

where 6 E (PIJ) , h - [(3 - a)Atll3, and a = O for first-order time accuracy gr 5 = 1 for second-order accuracy. The operators 65. 6,,, and 6 c are defined at each j, k, and t point (x - jAx. y = kAy, z = tAz) by

J -(y). [(l - V j ) P . .' + P.-]

1 - 1 1 2

- a(+" - +n-')]l(2At) = (38d)

The metrics Ex, ny, and S z are obtained from

E , = 2 l ( x . ]+I - x j-1 )

5 , = 2/(zt+1 - (39c)

(39a)

i1 = 2/(Yk+i - Yk-l) (39b) Y

while the term (Cx2/J)j+l/2 either as

(Cx2/J)j+i12 = [ ( S x 2 / J ) j + i + (5x21J)j1/2

of Eq. (36a) is farmed

(404

'k+l + 'k The terms (C,2/J)j-l/2, (ny2/J)k+l/2, etc. receive

essential to add 2 ('k+i - 'k) similar treatment. If Eq. (40a) is used, it is

(36~)

here AC = An = AS = 1 and only the varying indices are indicated. The parameter E - 2 far second- order spatial accuracy in supersonic regions, and

to the right-hand side of Eq. (35) to subtract out a numerical truncation error due to incomplete metric cancellation (see also Refs. 16, 17) . The term, relation (41), should equal O where but this is not obtained if Eq. (40a) is used, and the error can be very appreciable on a highly stretched grid. Bernoulli equation is not needed because of the choice of difference operators and ctretchingc.

m,=N,x,

A similar correction in the

The local linearizations af Sec. I11 are now introduced. Using Eq. (25) to both expand pn+' about pn end to expand p" about pn-' , the terms pn+' - pn of Eq. (35) are approximated

5

Dow

nloa

ded

by S

tanf

ord

Uni

vers

ity o

n O

ctob

er 3

, 201

2 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.1

980-

1368

2 where one must remember that the operators work on any product of terms to their right. 6" = [en - (0"/J)(E, .t 5, $5nE5 + n '+ "6

Y n n e"+' - The opexatOT sr that appears in Eq. (44) is

+ <z20:E5)($n+1 - $">I

- [;"-I - (On-l/J)(E, + Cx2+;-'

given by Eq. (38d). Ac it operates on A+, however, we chose to replace it by the operator

E 5 L (+n+r - +n)/(At) ? (l/At)(I - E-')$"+' + n Y i l 2$n-iEn + cz $< n-'65)(+n -$"-')I (45)

(42) which agrees with Eq. (38d) only if a = O . Equa- where ap/a+ of Eq. (25) is replaced by Eq. (23) tion (45) is now into Eq. (44)'

Bn/(JAt) = (p2-Y)"/(JAt) is divided through, and the left-hand side is approximately factored into E , n, and 5 operators:

[I + AtGZ2$ " E - At(J/0n)h6 ( S 2/J)~"6rl x

[I + Atn '$ "E - At(J/Bn)hán(ny2/J)pn6"l x

[I + AtF,x2+~Eg- At(J/0")h6S(SX2/J)Pn851 <$"+I - $")

= [ i + (On-l/B")](+n- $"-')- (On-'/O")(+n-'-+n-Z)

and 0 =P2-y. Here also the coefficients + , $", and +< while E , , 65, and E,, are the same operators defined by Eq. (38) . Note that both and 6" are linearized, not just

lost. The operators E,, 6 5 . E,,, and E were defined by Eq. (38) because their form is dictated

Bernoulli equation for density.

delta farm and p"+l is eliminated as follows:

are evaluated a5 indicated by Eq. (385,

en+'; otherwise, conservation-law form and second-order accuracy are 5 5 5 2

by the selection of difference operators used in the Y n n

The second derivative terms are rearranged into

+ At(On-i/Bn) (5x2+;-165 n+i- n

+ 65(5x26> S C $

85(5x2/J)pn85($n+i - +n)

+

E,, + 5,2+:-165) í$n - +"-I)

+ At(J/On)I[l - (a/3)](Cn - en-') + h85(5x2/J)[pn + cib" - ~"-')18~+~ (Ex2/J) [p" + a (p" - P"-') ]85#n

(43) + h6,,(ny2/J)[pn + m(pn - P"-~)I~,$"

+ h8 S Z ( 5 '/J)[p" + u(pn - P"-')]~~$"I The i1 and 5 terms are treated in a similar fashion, and for m = 1 second-order time accuracy is maintained. (46)

Applying Eqc. (42) and (43) to Eq. (35) gives the locally linearized form of the implicity differ- enced governing equation:

[ ( B " / J ) ( E , + 5x2+5n65 + ny2$;En + SZ2+,"E5ì

- hs5(5x2/J)pn65 - h6,,(ny2/J)pn8n L 5 A+* = R (48d

This equation has the form

L~L~L~($~+' - 4") = R (47)

and it is implemenced as an algorithm as

L A+** = A$*

n n LL,A$ = A+**

The algorithm Eq. (48) requires only a series of scalar tridiagonal inversions and it is therc- fore very efficiently implemented. equivalent to four levels of $ has to be supplied with p computed from the Bernoulli equation as needed. Alternatively, three levels of 4 and one level of p can be used. although in our test pro- gram two levels of p where stored far convenience in programming a computer code that changed from

nated by using binomial expansions.

x (4" - $n-i) + [ i - (0/3)1(0" - en-') + hIS5(SX2/J)[pn + U(P" - Pn-1)16E+n

+ á,,(ny2/J)lpn + U(P" - P~-')I~,,$~

+ ~C(t,2/J)[Pn + e(Pn - 0n-1)165+n1

Computer storage

( 4 4 ) day to day. All exponential functions were elimi-

6

Dow

nloa

ded

by S

tanf

ord

Uni

vers

ity o

n O

ctob

er 3

, 201

2 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.1

980-

1368

Note that in approximately factoring Eq. (44) into the form Eq. (46) the strict conservative farm is lost in time (only). Although it is possible to subtract off this small error, especially in two dimensions (cf. Goarjian, Ref. 5), we chase not to do so because of the added complexity. We assume that any error in true shock speed due to this added nonconservative term will he small, especially since perfect differences af the streamwise operator are maintained by keeping the 5-operators as the inner- most factor. To within first-order accuracy, the left-hand side variable coefficients can be moved within the operators, and if stable, this would be a simple way to maintain temporal conservation.

The thin-airfoil body-boundary tangency condi- tion, Eq. (121, was directly built into the implicit algorithm. Indeed, it was in order to satisfy this boundary condition that before the 5 and q tridiaganals in Eq. (46) or (47). The difference equation (+/- indicate upper or lower surface)

S-tridiagonals are formed

$, - mik, = [TL(dz/dx) (d~/dt)l/(S~)~ (49)

is implemented by overloading Eq. (46) for points adjacent to the body (see Ref. 13 for more details).

Adjustments to the difference equations and numerical algorithm must also be made for points just above and below the cut indicated in Pig. 1. Por these points, $< is altered as follows

where this change affects the calculation af density as well as coefficients in Eq. (46). The difference formula for aC6rZ2aC$ must also be adjusted for points adjacent to the cut. For points above the Cut

and for points helow the cut

Correction terns can be appended to Eqs. (50)-(52) as a consequence of Eq. (21), but these have been ignored in the present three-dimensional calcula- tions (see Ref. 13 for details).

At the end of each update of the field for $, new values of r arc obtained along the cut by solving Eq. (17) or (18). For example, Eq. (18) is differenced as

with initial data in x supplied from the known value of upstream of the cut.

mu - $ a at the airfoil trailing edge just

V. Results

Calculations in two dimensions were first per- formed to verify the accuracy of the numerical algo- rithm. Unsteady solution accuracy is demonstrated by a comparison with a small-disturbance, low- frequency result obtained by Ballhaus and Steger.’ In this test case, the airfoil is a biconvex pro- file that varies in time from zero thickness to 10% thick and then thins back to zero. A chock initially forms pact midchord, and then moves hack to the trailing edge as the airfoil thickens. As the airfoil then thins, the backward motion of the shock stops, and then the chock propagates upstream. Figure 2 shows a trace of midchord pressure for such a case with a comparison with the low-frequency, small-disturbance result. The small discrepancy between the theories is about what one would expect for this case, and similar results are obtained with the full-potential scheme of Goorjian.

As another test case, the unsteady flow about an impulsively plunged flat plate was computed with M, = 0.8 and a plunge velocity equivalent to a = 1”. good agreement with linear theory is obtained by Chipman and Jameson,’ who solved the full- otential equation, as well as by Beam and Warming,” who solved the Euler equations. Good agreement with linear theory” is also obtained with the present code as the load distributions plotted in Pig. 3 illustrate.

Linear theory is valid for this case and

The steady-state results of Fig. 4 show that essentially the same solution is obtained if the switching parameter u is always set equal to 1 (i.e., no switching), provided the second-order dif- ferencing O = 2 is used. Slight differences in the two solutions for a biconvex airfoil at I-L = 0.857, a = 1‘ edge singularity and at the shock wave. The uncwitched scheme, v = 1, 8 = 2, gives a more pleasing chock-wave result.

are observed at the leading-

Preliminary three-dimensional computations have been carried out for an untwisted thin rectangular AR = 8 wing with a 6% thick biconvex airfoil sec- tion. Steady-state results for a = 1‘ and M, = 0.857 are indicated by the Cp distributions shown over the wing in Fig. 5. This result corre- sponds to 51.4 chords traveled. The wing was then oscillated from this point on such that a = [i + sin(2nt/tp)l at a reduced frequency K = 0,183 where ic = 2a/(&tp) and tp is the non- dimensional time period. Pressure coefficient data, taken during the fifth cycle, are shown in Figs. 6 and 7.

We remark that the above Calculations were carried out over 2,000 time steps on a 65x20~34 grid. This result and similar nonlifting calcula- tions have been used to verify the stability of the numerical algorithm which has otherwise not been analyzed. All of the three-dimensional calculations used the unswitched form of the 5-differencing, Eq. (36a). by setting Y = 1 and O = 1.8. Lower values of O lead to an even more robust scheme, but with loss of accuracy.

,

7

Dow

nloa

ded

by S

tanf

ord

Uni

vers

ity o

n O

ctob

er 3

, 201

2 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.1

980-

1368

On the 65x20~34 grid, approximately 3 sec of CDC 7600 time is needed per time step. noted, however. that the code is currently in an inefficient although very readable state. Finally, we remark that the current unsteady numerical algo- rithm i s a very poor relaxation algorithm - steady states can require over 1,000 time steps.

It should be

VI. Concluding Remarks

An implicit finite-difference procedure was developed to solve the unsteady full-potential equa- tion in conservation-law form. Local linearizations were successfully used to derive a Correct time dif- ferenclng, avoid iterative between-time levels, and correctly analyze the spatial differencing. unsteady circulation model was developed and various test cases were computed to verify the accuracy of the numerical algorithm.

An

References

'Ballhaus, W. F. and Steger, J. L., "Implicit Approximate Factorization Schemes for the Low- Frequency Transonic Equation," NASA TM X-73,082, 1975.

'Ballhaus, W. F. and Goorjian, P. M., "Implicit Finite-Difference Computations of Unsteady Transonic Flows About Airfoils," AIAA Journal, Vol. 15, Dec. 1977.

3Caradonna, F. X. and Isom, M. P., "Numerical Calculation of Unsteady Transonic Potential Flow Over Helicopter Rotor Blades," AIAA Paper 75-168, Jan. 1975.

'Chipman, R. and Jameson, A., "Fully Conserva- tive Numerical Solutions far Unsteady Irrotational Transonic Flow About Airfoils,' AIAA Paper 79-1555, 1979.

'Goorjian, P. M., "Computations of Unsteady Transonic Flow Governed by the Concervatihe Full Potential Equations Using an Alternating Direction Implicit Algorithm," NASA CR-152274, 1979.

'Viviand, H., "Conservative Forms of Gas Dynamic Equations," La Recherche Aerospatiale, No. 1, Jan.-Feb. 1974, pp. 65-68.

'Jameson, A., "Transonic Potential Flow Calcu- lations Using Conservative Form," AIAA Second Compu- tational Fluid Dynamics Conference Proceedings, June 1975, pp. 148-155.

'Holst, T. L. and Ballhaus, W. F., "Fact Con- servative Schemes for the Full Potential Equation Applied to Transonic Flows," NASA TM-78469, 1978. (Also AIAA Journal, Vol. 17, No. 2, Feb. 1979, pp. 145-152.)

9Holrf. T. I , . . "A Fast. Conservative Algorithm ...--. ~ . - for Salving the Transonic Full-Potential Equation," AIAA Fourth Computational Fluid Dynamics Conference Proceedinzc, July 1979, pp. 109-121.

"Hafez, M., South, J., and Murman, E., "Arti- ficial Compressibility Methods for Numerical Solu- tions of Transonic Full Potential Equation," AIAA Journal, vol. 17, Aug. 1979. pp. 838-844.

"Guderley, K. G., The Theory af Transonic Flow, Addison-Wesley Publishing Company, Inc., Reading, Mass., 1962.

"Murman, E. M. and Cole, J. C., "Calculation af Plane Steady Transonic Flow," AIAA Journal, Vol. 9, 1971, pp. 114-121.

Steger, J. L. and Caradonna, F. X., "A Can- servative Implicit Finite Difference Algorithm for the Two Dimensional Unsteady Transonic Full Poten- tial Equation," NASA TM-81211, 1980.

1 3

"South, J. and Jameson, A., "Recent Advances in Transonic Computational Aerodynamics,'' NASA Paper to be published.

"Warming, R. F. and Beam, R. M., "On the Construction and Application of Implicit Factored Schemes for Conservation Laws," SIAM-AMs Proceed- a. Vol. 11, Apr. 1977.

/,

Steger, J. L., "Implicit Finite Difference 16

Solution of Flow About Arbitrary Two-Dimensional Geometries," AIAA Journal, Vol. 16, 1978.

Pulliam, J. H. and Steger, J. L., "On I,

Implicit Finite-Difference Simulations of Three- Dimensional Flow," AIAA paper 78-10, an. 1978.

"Beam, R. M. and Warming, R . F., "Numerical Calculations of Two-Dimensional, Unsteady Transonic Flows with Circulation,'' NASA TN D-7605, 1974.

omax ax, H., Heasiet, M. A., FUILX, F. B., and Sluder, L., "Two- and Three-Dimensional Unsteady Lift Problems in High-speed Flight," NACA Rep. 1077, 1952.

Dow

nloa

ded

by S

tanf

ord

Uni

vers

ity o

n O

ctob

er 3

, 201

2 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.1

980-

1368

3

M

d

y<

Dow

nloa

ded

by S

tanf

ord

Uni

vers

ity o

n O

ctob

er 3

, 201

2 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.1

980-

1368

18 r 0.4 CHORDSTRAVEL M, = 0.8 FLAT PLATE

- LINEAR SOLUTION

- -- CURRENT POTENTIAL SOLUTION 16

O BEAM-WARMING EULER SOLUTION

14 14

FLAT PLATE

- LINEAR SOLUTION NACA REP. 1077,1952 CURRENT POTENTIAL SOLUTION

NASA TN D-76ffi. 1974

12 12

O BEAM-WARMING EULER SOLUTION

10 10

4 8 8

2 a P U

6 6

4 4

2 2

O O X I C X I C

(a) Solution @ 0.2 chords of travel from impulsive plunge.

(b) Solution @ 0 .8 chords of travel.

Fig. 3 Continued.

Fig. 3 Load distributions along plate for plunge velocity equivalent to a = lo

10

Dow

nloa

ded

by S

tanf

ord

Uni

vers

ity o

n O

ctob

er 3

, 201

2 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.1

980-

1368

2.4 CHORDS TRAVEL

- LINEAR SOLUTION

---- POTENTIAL SOLUTION 18

O BEAM-WARMING EULER SOLUTION

xíc

( c ) Solution I.? 2.4 chords of t rave l .

Fig. 3 Concluded.

Dow

nloa

ded

by S

tanf

ord

Uni

vers

ity o

n O

ctob

er 3

, 201

2 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.1

980-

1368

M, = 0.857

BICONVEX PROFILE

BICONVEX PROFILE, a = 1". r=0.06 . M,=0.857

SWITCHED CENTRAL-UPWINO UNSWITCHEO UPWIND

.? O .2 .4 .6 .8 1 .o

X I C

Fig. 4 Comparison of effect of switched and unswitchcd d i f f erenc ing schemes.

-2.0

-1.2 -1.0 -ii' -.8

M, = 0.857

BICONVEX, PROFILE

THICKNESS = 0.06

01 = 1"

::F,', I ~ , , , , ,: .6

O .2 .4 .6 .8 1.0 X I C

Fig. 5 Steady-state Cp d i s t r i b u t i o n over wing.

01 = 1" + 1" sin9

O .2 .4 .6 .8 1.0 XIC

Fig . 6 Pressure dis tr ibut ions during f i f t h o s c i l l a - tory cycle, e = 126'.

M, = 0.857

THICKNESS = 0.06

O .2 .4 .6 .8 1.0 XIC

Fig. 7 Pressure dis tr ibut ions during f i f t h o s c i l l a - tory c y c l e , e = 2880.

Dow

nloa

ded

by S

tanf

ord

Uni

vers

ity o

n O

ctob

er 3

, 201

2 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.1

980-

1368