NASA Contrdctor Report 177994 

lCASE REPORT NO. 85-44 

ICASE 

A FOURTH-ORDER SCHEME FOR THE UNSTEADY 

COMPRESSIBLE NAVIER-STOKES EQUATIONS 

Alvin Bayliss 

Paresh Parikh 

Lucio Maestrello 

Eh Turkel 

NASA Contracts No. NAS1-17070 and No. NAS1-17130 

October 1985 

NASA-CR-177994 

19860004751 

• t I .. ;':' r .. ~ -.~ .. ' 

t .... r I,.i ... ~ ~. ~ • J 

DEC 2 31985 

L, I ,'-"_LY Id:.;:,cAI,C,i cn,TER 

LlCRARY, NASA 

"M'IPTON, VIRGINIA 

INSTITUTE FOR CO~WUTER APPLICATIONS IN SCIENCE AND ENGINEERING 

NASA Langley Research Center. Hampton. Virginia 23665 

Operated by the Universities Space Research Association 

NI\SI\ 

NatIonal Aeronautics and 

Space AdminIstratIon 

Langfey Research Center 

Hampton, VirginIa 23665 

111111111111111111111111111111111111111111111 

NF01019 
A FOURTH-QRDER SCHEME FOR THE 

UNSTEADY COMPRESSIBLE HAVIER-STOKES EQUATIONS 

A. Bayliss 

Exxon Corporate Research Science Laboratories 

P. Parikh 

Vigyan Research Associates, Inc. 

L. Maestrello 

NASA Langley Research Center 

E. Turkel 

ICASE and Tel-Aviv University 

Abstract 

A computational scheme is described which is second-order accurate 1n 

t1me and fourth-order accurate in space (2-4). Th1s method is applied to 

study the stability of compressible boundary layers. The laminar compressible 

Nav1er-Stokes equat10ns are solved with a time harmonic inflow superimposed on 

the steady state solution. This results in spatially unstable modes. It is 

shown that the second-order methods are 1neff1cient for calculat1ng the growth 

rates and phases of the unstable modes. In contrast the fourth-order method 

Y1elds accurate results on relat1vely course meshes. 

Part1al support was provided for the f1rst author under NASA Contract No. 

NASl-17070 and for the second author under NASA Contract No. NASl-17252. 

Research for the fourth author was supported by the National Aeronautics 

and Space Administrat10n under NASA Contracts No. NASl-17070 and NASl-17130 

wh1le he was in res1dence at the Institute for Computer App11cations in 

SC1ence and Engineer1ng, NASA Langley Research Center, Hampton, VA 23665. 

i 
I. Introduction 

This paper is concerned with a fourth-order accurate finite difference 

scheme for the compress~ble, unsteady Navier-Stokes equations. The primary 

~nterest is the computation of spatially unstable disturbances lnto the 

nonl~near regime and the active control of such d~sturbances. Due to the 

wavelike nature of these d~sturbances, this problem has features of both wave 

propagation and flu~d dyna,nics and the numerical scheme must be chosen to 

accurately compute waves propagating in an unstable, high Reynolds number mean 

flow. The applicat~on of this scheme to study the active control of spatially 

growing dlsturbances has been descr~bed previously [1]. Th~s paper describes 

the numerical scheme and the advantages that can be obtained by the use of 

fourth-order accuracy. 

Higher order accurate methods, in particular spectral methods, have been 

successfully used in the computation of incompressible flows. Examples of 

such calculations can be found in [2] - [4]. Generally, spectral methods 

assume that problem is perl0dic in the streamwise direction so that Fourier 

(as opposed to Chebyshev) collocation can be used. (Spatially unstable 

disturbances were considered, however in [4].) The use of hlgher order 

methods for the numerical computation of waves has also been extensively 

considered ~n the literature. Examples can be found in [5] - [7]. 

In section 2, we describe the numerical scheme and discuss certain 

implementation details. In section 3 the performance of this scheme on a 

vanety of problems ~s illustrated. 

results and draw conclusions. 

Finally, ~n section 4 we discuss these 
-2-

II. Numerical Scheme 

The numer1cal scheme is an extension of the second-order MacCormack 

scheme due to Gottlieb and Turkel [8]. For the one-dimensional equation 

Ut = F x where F = F(U), we have (letting the superscripts denote the t1me 

level and the subscripts denote the spatial grid points) 

(2.1a) 

(2.1b) 

There is an obvious symmetric variant of (2.1) starting with a backwards 

predictor and then a forwards corrector. 

If F depends only on U it is shown in [8] that the scheme (2.1) 1S 

second-order accurate in time and fourth-order accurate in space «2-4) 

scheme). Specifically, 1f ~t 1S the time step and ~x the space step, then the 

truncation error is O(~t( (~x)4 + (~t)2»). Thus the scheme is fourth-order 

accurate provided ~t = O(~x)2) as ~x + O. For nonlinear problems this is true 

only if the two variants of (2.1) are alternated. 

Although true fourth-order accuracy is obtained only for ~t = 0(~x)2) it 

has been found that (2.1) is considerably more efficient than second-order 

schemes (see for example [6], [9-10]). For two-d1mensional problems (2.1) can 

be used together with operator splitting [11] to maintain the (2-4) accuracy. 

Specifically, for the equat10n 

we have 

F + G 

x y 
-3-

L L L L Un 

x y y x 

where Lx and Ly are one-dimensional solution operators corresponding to the 

scheme (2.1) applied to the equations Ut = Fx and Ut = Gy respectively. 

Alternatively, one can consider an unsplit version of (2.1) [12]. 

The explicit nature of the scheme (2.1) makes it well suited for current 

vector computers such as the CDC Cyber 205 and the Cray 1-S and XMP series. 

The scheme has been implemented on the CDC VPS 32 at NASA Langley Research 

Center using vector operatlons over a two-dlmensional grid (vector lengths 

- 20,000). The explicit nature of the scheme makes it relatively inefficient 

to compute steady flow, unless acceleration techniques are incorporated 

[ 13] • For the unsteady flows considered here, a time step restriction is 

necessary in order to accurately resolve the fluctuations in time and so 

explicit schemes become efficient. 

The implementatlon of the scheme (2.1) is straightforward if the flux 

functl0n F depends only on the unknown U. For the Navier-Stokes equations we 

have F = F(U,Ux,Uy ) and in the evaluation of Fi lt lS necessary to approximate 

Ux [8]. For a forward sweep, 1.e., (2.1a) Ux is approximated by a two-polnt 

backwards difference, Le., Ux ~ Ui - Ui - 1/f1x and conversely for a backwards 

sweep. It is shown in [8] that for a general, non-constant coefficient 

problem this results in a scheme that is only third-order accurate. However, 

it can be shown that the third-order truncation error is eliminated by 

alternating the sweeps and the resulting scheme is fourth-order accurate for 

both the inviscld and viscous terms. Mixed derivatives, i.e., terms of the 

form Uy in an x sweep are approximated by second-order central dlfferences. 

In many applications these terms are relatively small. We have not 

experlmented wlth fourth-order dlfferences for these terms. 
-4-

In order to complete the description of the numer1cal scheme, 1t is 

necessary to descr1be the implementation of boundary cond1t10ns. We 

distinguish between boundary conditions which must be imposed as part of the 

problem and those boundary conditions which are necessary because the 

straightforward application of the difference formula (2.1) is not valid at 

boundary points. In this section we cons1der only the latter. 

Consider the forward predictor (2.1a). If 1 = N denotes a boundary p01nt 

then the values of F i are not available for i > N and thus the scheme (2.1) 

can not be applied at the p01nts i = N-1 and i N. The most sat1sfactory 

boundary treatment that we have found 1S to def1ne F1 for i > N by third-order 

extrapolation from the interior (see [6] and [8]). Hence, FN+1 and FN+2 are 

defined by 

(2.2) 

In (2.2) the extrapolated fluxes 1nclude both the viscous and inviscid terms. 

The scheme (2.1) has a five-point stencil. Implicit fourth-order schemes 

with a three-point stencil can be constructed by using Pad~ approximations. 

It is well known that compact schemes are more accurate than five-point 

schemes [14], however, they are more expensive because of the impl1cit nature 

of the scheme. One way to increase the accuracy of the scheme (2.1) is to use 

a scheme which is s1xth-order accurate in space. The scheme (2.1) can be 

easily extended to sixth-order. The resulting scheme 1S 
-SIT (2.3a) 1. 

U~+I • i lu~ + u, + 30:~ [37(Fi - Fi_l ) - 8(Fi-1 - F,-2 ) + (F,_2 - F,-3)]I· 

(2.3b) 

The suitab1.l1.ty of (2.3) for the unsteady Navier-Stokes equations is currently 

being invest1.gated. 

III. Numerical Examples 

In th1.s sect1.on we describe some numer1.cal examples illustrat1.ng the 

effects that can be expected from the fourth-order accurate d1.fferencing. The 

present program 1.S pr1.mar1.ly des1.gned to compute unsteady flows and does not 

make use of any acceleration techniques which destory the consistency 1.n t1.me 

[13]. The first numerical example w1.ll, however, illustrate the effect of the 

h1.gher-order d1.fferencing on steady flows. 

Specif1.cally, we consider a supersonic boundary layer over a flat 

plate. The free stream Mach number 1.S 4.S and the unit Reynolds number 1.S 

2.3 x 106 • The inflow data 1.S generated by a boundary layer program [IS] at 

O. S ft. from the lead1.ng edge. The computat1.onal doma1.n is 2.0 ft. in x 

and 11 IS in y where IS is the boundary th1.ckness at 1.nflow. An exponent1.ally 

stretched grid 1.S used normal to the plate w1.th the Jacob1.ans evaluated 

analyt1.cally. F1.gure 1 illustrates the computat1.onal doma1.n. 

In add1.tion to the numer1.cal boundary treatment descr1.bed prev1.ously, 1.t 

is necessary to discuss the boundary condit1.ons which must be 1.mposed. For 
-6-

the superson1c case all quant1t1es are imposed at the inflow while the 

solut10n at the outflow boundary is obtained by zeroth-order extrapolat10n 

from the 1nter1or. We have ver1f1ed that the solution 1S very 1nsens1t1ve to 

the treatment of the outflow boundary as would be expected S1nce the 

11near1zed character1stic var1ables are convected out of the computat10nal 

domain. At the plate, the two veloc1ty components are set to zero and the 

temperature is spec1f1ed. Several d1fferent techn1ques have been used to 

obta1n the pressure. These 1nclude different orders of extrapolat10n 1n the 

normal direction and the use of the normal momentum equation. The accuracy of 

the solut10n 1S 1nsens1t1ve to the boundary treatment (probably because of the 

stretched grid). At present we simply use first-order extrapolation for the 

pressure. 

The treatment of the upper boundary (see F1g. 1) is based on the 

11near1zed characterist1cs 1n the normal direction. 

The Nav1er-Stokes equat10ns can be wr1tten 1n the gener1c form 

F + G x y 

(3.1) 

T where U 1S the vector (p, pu, pv,E) , p is the dens1ty, u and v are the x and 

y veloc1t1es, respect1vely and E is the total energy. The funct10nal forms of 

F and G are standard and are omitted for brevity. In the normal direct10n we 

consider only the system 

(3.2) 

In the applications the vert1cal velocity v 1S small and pos1t1ve at the upper 

boundary. Upon linearizing (3.2) we f1nd that the three character1stic 
-7-

var1ables 

~2 

p - pc (3.3a) 

p + pcv (3.3b) 

u (3.3c) 

are convected to the boundary from the interior. These three var1ables are 

obtained by zeroth-order extrapolat10n from the inter10r. Since we cons1der 

the linearized character1stics, the quant1tl.eS with a - in (3.3), are taken 

from the prevl.ous time step. The final boundary condl.tion is obtained from 

setting 

o (3.3d) 

corresponding to the characteristic variable enterl.ng the computatl.onal 

reg10n. The use of the radl.ation condition (3. 3d) was previously found to 

cons1derably accelerate the convergence to the steady state and to perml.t the 

upper boundary to be brought closer to the plate [16]. 

A typical comparison between the second and fourth-order schemes is shown 

in Fl.g. 2. The streamwise velocity is plotted agal.nst y at a location of 1.0 

ft. from the leadl.ng edge (Re * = 17174), where Re * is the Reynolds number 

o 0 

based on displacement thickness. The results are shown for the fourth-order 

scheme (2.1), the second-order MacCormack scheme and the solutl.on obtal.ned 

from the boundary layer equatl.ons. The f1nite d1fference results were 

obtal.ned from the same gr1d (21 x 31) and the 1mprovement in accuracy of the 

fourth-order scheme is evident. It should be noted that we do not obtain a 

similar improvement for more viscous boundary layers. The reason for this is 

not understood and 1S currently be1ng invest1gated. 
-8-

For our next example, we consider an unsteady d1sturbance in a relat1vely 

low speed subson1c boundary layer. The mean flow is a boundary layer w1th 

free stream Mach number 0.4 and a un1t Reynolds number of 3.0 x 105. At 

1nflow we have Re * 

o 

998 and the computat10nal doma1n 1S chosen so that at 

outflow Re * = 1730. 

o 

A fluctuat1ng d1sturbance 1S 1ntroduced at the 1nflow 

w1th nond1mens10nal frequency F = (2nfv)/U; of .8 x 104 • f 1S the frequency 

1n Hertz, v the kinemat1c Viscos1ty, and U"" the free stream veloc1ty. Th1S 

flow lS nearly 1ncompress1ble. Based on linear, 1ncompress1ble stab1l1ty 

theory it is known that th1S frequency lS unstable at 1nflow but becomes 

stable at Re * = 1450. 

o 

Slnce th1s lS a subson1c flow, three boundary 

cond1t10ns must be 1mposed at the 1nflow. These cond1t10ns are obta1ned by 

computing an eigenfunction of the Orr-Sommerfeld equations (us1ng a program 

developed by R. Dagenhart of NASA Langley Research Center). The real part of 

the Solut10n (t1mes e iFt), lS then used to compute the three llnear1zed 

character1st1c var1ables which enter the computat10nal doma1n from the 

outs1de. Speclfically, 

: :v~~} ~ mean + E (values obtained from Orr-Sommerfeld equatwn) O.4a) 

where € determ1nes the strength of the d1sturbance. The outgolng 

character1stic var1able p - pcv is obta1ned from the 1nter10r by zeroth-order 

extrapolation. 

The outflow boundary cond1tion lS treated slmllarly. The 1ncoming 

character1stic var1able lS set equal to zero by 1mposing the cond1t10n 
-9-

o. (3.5) 

The use of the cond~t~on (3.5) and more accurate rad~at~on cond~t~on to 

compute unsteady d~sturbances ~n subson~c flows ~s d~scussed in [16]. 

Charactenst~c rad~at~on boundary conditions of the form (3.5) are commonly 

used ~n two-dimensional linear wave propagation problems (for example, see 

[17]). The boundary cond~tion (3.5) has been found to be suff~c~ently 

accurate for the present calculat~ons. Th~s ~s based on compar~sons w~th 

linear stab~l~ty theory. 

In F~g. 3 we plot the computed growth rate as a funct~on of Re *. 

o 

growth rates are computed by calculat~ng the RMS =.; (pu)2 - (pu)2 mean 

The 

and 

~ntegrat~ng In y. The Inflow data ~s chosen so that the maximum perturbat~on 

~s 2% of the free stream. In th~s case the (2-4) scheme shows a s~gn~f~cant 

amount of nonl~near growth wh~le the (2-2) scheme, for the same grid, shows 

sign~ficantly less growth. Mesh ref~nements verify the accuracy of the fourthorder code. 

Reduc~ng the ~nflow perturbat~on to 0.2% of the free stream the fourthorder results reproduce the l~near theory results very closely and in 

part~cular, show that the perturbation becomes stable at Re * ~ 1450 as 

o 

pred~cted by linear theory. 

In F~gs. 4a and 4b we plot pu as a function of t~me for Re * = 1263 and 

o 

y = 0.0034 ft. In F~g. 4a the (2-4) scheme is shown for two d~fferent gr~ds 

and shows that a further gr~d ref~nement does not y~eld any add~t~onal 

~nformation. In F~g. 4b the (2-2) scheme is compared w~th the (2-4) scheme. 

At this locat~on the d~sturbance ~s bas~cally l~near and the f~gures ~nd~cate 

sign~f~cant ampl~tude and phase errors for the (2-2) scheme. 
-10-

F1.gures 5a and 5b conta1.n similar results further downstream at the 

locatl.On Re * = 1481 and y = O.OOll ft. 

15 

At th1.s 10cat1.on the d1.sturbance 

exhib1.ts some nonlinear effects wh1.ch are not found by the second-order scheme 

w1.th this mesh. 

In F1.g. 6 the computed growth rates are shown for a d1.sturbance 1.n higher 

speed flow where compress1.b1.lity effects would be expected to be 1.mportant. 

The free stream Mach number 1.S 0.7 and the un1.t Reynolds number is 300,000. 

The computational doma1.n 1.S chosen so that at 1.nflow we have Re * = 900. The 

15 

nond1.mens1.onal frequency F 0.8 x 104• The 1.nflow data was generated from 

the 1.ncompress1.ble stab1.l1.ty program, however, we have not compared growth 

rates with those predicted by the 1.ncompress1.ble stab1.l1.ty theory. The 

results in Fig. 6 show a s1.gnificant reduct1.on 1.n the growth rate using the 

second-order scheme. F1.nally, 1.n F1.g. 7 we plot the mean of pu across the 

boundary layer for Re * = 1400. It is apparent that the second-order scheme 

15 

m1.sses s1.gnif1.cant features of the flow. 

v. Conclusion 

There are several types of errors that appear in the computat1.on of 

unsteady waves and stab1.l1.ty analys1.s. These include both ampl1.tude and phase 

errors. These errors frequently s1.mulate an apparent lower Reynolds number 

and lndicate that the flow is more stable than 1.t physically 1.s. These 

effects are seen 1.n the results presented here. It 1.S shown that second-order, 

1.n space, schemes are not eff1.c1.ent 1.n s1.mulat1.ng nonl1.near stab1.l1.ty of h1.gh 

Reynolds number flows. The eX1.st1.ng program solves the two-d1.mens1.onal 

lam1.nar compressible Nav1.er-Stokes equat1.ons and 1.S currently being extended 
-11-

to three dlmenslons. Both subsonic and transonlC stability regions can be 

calculated uSlng the lmproved fourth-order method. 
-12-

References 

[1] Bayl~ss, A., Maestrello, L., Par~kh, P., and Turkel, E., "Numer~cal 

S~mulat~on of Boundary Layer Exc~tat~on by Surface Heat~ng/Cool~ng," 

AlAA Paper 85-0565, March 1985. 

[2] Mal~k, M., Zang, T. A., and Hussa~n~, M. Y., "A Spect ral Collocat~on 

Method for Navier-Stokes Equat~ons, lCASE Report 84-19, NASA CR-172365, 

June 1984. 

[3] Orszag, S. A. and Kells, L. C., "Trans~t~on to Turbulence for Pla~n 

Po~seu~lle Flow and Pla~n Couette Flow," J. Fluid. Mech., Vol. 96, 1980, 

pp. 159-205. 

[4] Murdock, J. W., "A Numer~cal Study of Nonl~near Effects on Boundary 

Layer Stab~l1ty," AlAA J., Vol. 15, No.8, 1977, pp. 1167-1173. 

[5] Kre~ss, H. o. and Oliger, J., "Compar~son of Accurate Methods for the 

lntegrat~on of Hyperbol~c Systems," Tellus, Vol. 24, 1972, pp. 119-225. 

[6] Turkel, E., "On the Pract~cal Use of Higher-Order Methods for Hyperbol~c 

Systems," J. Comput. Phys., Vol. 35, 1980, pp. 319-340. 

[7] Alford, R. M., Kelly, K. R., and Borre, D. M.," Accuracy of F~n~te 

Difference Model~ng of the Acoust~c Wave Equat~on," Geophys~cs, Vol. 39, 

No.6, 1974, pp. 834-842. 
-13-

[8] Gottll.eb, D. and Turkel, E., "Dl.ssipative Two-Four Methods for Time 

Dependent Problems," Math. Comp., Vol. 30, 1976, pp. 703-723. 

[9] Maestrello, L., Bayliss, A., and Turkel, E., "On the Interaction between 

a Sound Pulse with the Shear Layer of an Axisymmetrl.c Jet," J. Sound and 

Vib., Vol. 74, No.2, January 1981, pp. 281-301. 

[10] Hariharan, S. 1. and Lester, H. C., "Acoustl.C Shocks in a Variable Area 

Duct Contal.ning Near Sonl.c Flows," J. Comput. Phys., Vol. 58,1985, pp. 

134-145. 

[11] Strang, W. G., "On the Constructl.on and Comparison of Dl.fference 

Schemes," SIAM J. Numer. Anal., Vol. 5, 1968, pp. 506-517. 

[12] Rudy, D. H., "Navl.er-Stokes Solutl.ons for Two-Dimensl.onal Subsonl.c Base 

Flow," Southeastern Conference on Theoretl.cal and Appll.ed Mechanics, May 

1984. 

[13] Turkel, E., "Acceleratl.on to a Steady State for the Euler Equatl.on," 

ICASE Report 84-32, NASA CR-172398, July 1984. 

[14] Swartz, Band Wendroff, B., "The Relative Efficiency of Finite 

Difference and Fl.nite Element Methods: 1. Hyperboll.c Problems and 

Splines," SIAM J. Numer. Anal., Vol. 11, 1974, pp. 979-993. 
-14-

[15] Harris, J. E. and Blanchard, D. K., "Computer Program for Solving 

Laminar, Transitional or Turbulent Compressible Boundary Layer Equations 

for Two-Dimensional and AXl.symmetric Flow," NASA TM 83207, 1985. 

[16] Bayliss, A. and Turkel, E., "Far Fl.eld Boundary Condl.tl.ons for 

Compress1ble Flows," J. Comput. Phys., Vol. 48, 1982, pp. 182-199. 

[17] Cohen, M. F., et al., "A Comparl.son of Paraxl.al and Vl.SCOUS Sl.lent 

Boundary Methods l.n Finite Element Analysl.s," Joint ASME-ASCE Mechanl.cs 

Conference, Unl.versl.ty of Colorado, 1981, pp. 67-80. 
-15-

y 

t r--------, 

I ntlow I Computational I Outflow I domain 

Fig. 1. Schematic of computational domain. 
Y xl03 

14 

-16-

----- Boundary layer code 

o 4th order 

12 D. 2 nd order 

10 

8 

6 

4 

2 

o .2 .4 .6 .8 

U!U 00 

1.0 

F~g. 2. Compar~son between the second and the fourth-order 

schemes for a steady state. M = 4.5. 00 
-17-

J (p u )~ I J (p u )~ 0 

7 ---0- linear theory 

--0- Computed, 4th order 

6 --fr- Computed, 2 nd order 

5 

4 

3 

2 

1 

I I I I I I 

1.0 1.1 1.2 1.3 1.4 1.5 

Re6* xl0-3 

Fig. 3. Comparison of amplitude growth for the second 

and the fourth-order schemes. M = 0.4. 00 
pu 

.6 

.5 

.4 

-18-

Grid 

(251 x61) } _____ (313 x8Q) 4th order 

.3 ~ ____ ~ ______ ~ ______ ~ ____ ~ 

4.00 4.25 4.50 

tUea /L 

4.75 5.00 

Fig. 4a. pu vs. time for two grids at Re * = 1263, Y = 0.0034 ft. 

6 
pu 

.6 

.5 

.4 

-19-

Grid 

4 th order} ----- 2 nd order (251 x6U 

.3 ______ ~~ ____ ~ ______ ~ ______ ~ 

4.00 4.25 4.75 5.00 

Fig. 4b. pu vs. time using second and fourth-order schemes. 

Re * = 1263, Y = 0.0034 ft. 

o 
pu 

.3 

.2 

• 1 

4.25 

-20-

Grid 

(251 X 61) } 

(313 x SO) 4th order 

4.50 

tUco/L 

4.75 5.00 

Fig. Sa. pu vs. t1me for two grids at Re * = 1481, Y = 0.0011 ft. 

o 
pu 

.3 

.2 

\ 

• 1 

o 

\ 

4.00 

/ ..... I \ 

I 

I 

I 

I 

4.25 

-21-

Grid 

4 th order} (251 x6l) - - --- 2 nd order 

I 

I 

/ .... I \ 

I 

I \ 

\ 

\ 

4.50 

tUoo/L 

I 

I 

J 

I 

I 

\,. 

/-\ .... I , 

I 

I 

\ I 

\ I 

, I 

\ 

4.75 5.00 

Fig. 5b. pu vs. time uS1ng second and fourth-order schemes. 

Re * = 1481, Y = 0.0011 ft. 

15 
-22-

3 • 4th order, Grid (301 x73) 

4th order} . ____ 2nd order Grid (251 x6U 

2 

1 

I 

9 10 11 14 15 16 

F1g. 6. Comparison of amplitude growth for the second 

and the fourth-order schemes. M = 0.7. 00 
-23-

~ 

.05 I nflow at R = 900 

el)* 

.04, 4 th order } R = 1400 

2 nd order el) * 

.03 

.02 

.01 ,,--- , 

-' ---- o 5 10 15 20 25 x 10-3 

y 

Fig. 7. RMS amplitude d1stribut1on across the boundary layer. 
1 Report No NASA CR-177994 I 2 Government Accession No 3 ReciPIent's Catalog No 

ICASE Report No. 85-44 

4 Title and Subtitle 5 Report Date 

A FOURTH-ORDER SCHEME FOR THE UNSTEADY COMPRESSIBLE October 1985 

NAVIER-STOKES EQUATIONS 6 Performing Organization Code 

7 Author(s) 8 Performing Organization Report No 

A. Bayliss, P. Parikh, L. Maestrello, and E. Turkel 85-44 

10 Work Unit No 

9 Performang Organization Name and Address 

Institute for Computer Applicat10ns in SC1ence 

and Engineering 11 Contract or Grant No 

Ma1l Stop 132C, NASA Langley Research Center NAS1-17070; NAS1-17130 

Hampton, VA 23665 13 Type of Report and Period Covered 

12 Sponsorang Agency Name and Address 

National Aeronautics and Space Admin1strat1on r.nnt-r::art-nr Rp'~QI"t 

Washington, D.C. 20546 14 SponSOring Agency Code 

505-31-83-01 

15 Su pplementary Notes 

Langley Technical Monitor: 

J. C. South Jr. Submitted to AlAA J. 

Final Report 

16 Abstract 

A computational scheme is described which is second-order accurate in 

time and fourth-order accurate in space (2-4). This method is applied to 

study the stabihty of compressible boundary layers. The laminar compress1ble 

Navier-Stokes equations are solved with a time harmonic inflow superimposed on 

the steady state solution. Th1s results in spatially unstable modes. It is 

shown that the second-order methods are 1nefficient for calculat1ng the growth 

rates and phases of the unstable modes. In contrast the fourth-order method 

yields accurate results on relat1vely course meshes. 

17 Key Words (Suggested by Author(s)) 18 Distribution Statement 

fourth-order, stab111ty, 02 - Aerodynamics 

boundary layer, Nav1er-Stokes 34 - Flu1d Mech. and Heat Transfer 

equat10ns 

Unclassif1ed - Unlim1ted 

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lTnrl ::a~'" f, A~ TTn,..l::a",,,,if,An 25 A02 

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

NASA-Langley, 1985 
End of Document