NASA Technical Reports Server (NTRS) 19910021148: Verification of the proteus two-dimensional Navier-Stokes code for flat plate and pipe flows

10 Pages·0.52 MB·English

by  NASA Technical Reports Server (NTRS)

t/d y
i
NASA Technical Memorandum 105160 ._
..... 29
AIAA-9!-20!3 ...................................
.'w Verification of the Proteus Two-Dimensional
. =
Navier'Stokes Code for Flat Plate and
Pipe Flows
Julianne M. Conley
Lewis Research Center
Cleveland, Ohio
and
Patrick L. Zeman
Arnold Engineering and Development Center
Arnold AFB, Tennessee
Prepared for the
27th Joint Propulsion Conference
" cosponsored by AIAA, SAE, ASME, _d ASEE
Sacramento, California, June 24-27,1991 -
Ngi-30462
THE
_(_!AS A-T'-4-105 L60) V_I_ICATION _
NAV I_R- ST('IKES C F_LE_
PROTEIIS TWO -oIM_NSInNAL
FOR FLAT PLATE AN r) PIPE FLOWS (NASA) g P uncl as
CSCL 20D
0036902

VERIFICATION OF THE PROTEUS TWO-
DIMENSIONAL NAVIER-STOKES CODE FOR FLAT
PLATE AND PIPE FLOWS
Julianne M. Conley'
National Aeronautics and Space Administration
Lewis Research Center
Cleveland, Ohio
Patrick L. Zeman
Arnold Engineering and Development Center
Arnold AFB, Tennesee
In order to assess the code's validity for calculating funda-
mental fluid flows encountered in most aerospace propulsion
The Proteus Navier-Stokes Code is evaluated for two- applications, aseries of validation cases have been rim, using the
dimensional/axisymmetric, viscous, incompressible, internal two-dimensional planar/axisymmetric version of the code.
and external flows. The particular cases to be discussed are These cases are for both internal and external incompressible
laminar and turbulent flows over aflat plate, laminar and turbu- flows. This paper describes validation studies for laminar and
lent developing pipe flows and turbulent pipe flow with swirl. turbulent flat plate boundary layers with zero pressure gradient,
Results are compared with exact solutions,empirical and for laminar and turbulent developing pipe flows and
correlations and experimental data. Adetailed description of the turbulent pipe flow with swirl. Incompressible cases in Proteus
code set-up, including boundary conditions, initial conditions, were simulated by running ataMach number between 0.1 and
grid size and grid packing is given for each case. 0.3. In the results for both flat plate and pipe flow to be
presented, constant total enthalpy was assumed, and the energy
Introduction equation was not solved.
An effort is underway atthe NASA Lewis Research Center Test Cases
to develop atwo and three-dimensional Navier-Stokes code,
called Proteus, for aerospace propulsion applications.(1) The Laminar Flat Plate Flow
emphasis in this effort is not algorithm development or research
on numerical methods, but on the development of the code Incompressible laminar flow over aflat plate with zero pres-
itself. The objective is to develop acode that is user-oriented, sure gradient can be compared with the exact solution of
easily modified, and well documented. Code readability, Blasius.(8) The results of one such comparison are shown in
modularity, and both internal and external documentation have Figures 2-6, plotted with the results of Blasius. For this test
been emphasized. case, the freestream Mach number was 0.2 and Rex, the
Reynolds number based on x, ranged from 20,000 at the
Proteus solves the Reynolds-averaged, unsteady, upstream computational boundary to 100,000 at the downstream
compressible Navier-Stokes equations in strong conservation computational boundary. A 201x101 grid was used, with
law form. Turbulence is modeled using aBaldwin-Lomax(2) packing in the vertical direction near the plate surface such that
based algebraic eddy viscosity model. The governing equations the ratio of the minimum to maximum cell height, defined as the
are written in Cartesian coordinates and transformed into packing ratio, was 0.05; the grid was uniform in the x-direction.
generalized nonorthogonal body-fitted coordinates. They are The grid extended horizontally from x/L = 0.25 to x/L = 1.25
solved by marching in time using afully-coupled alternating and vertically from y/L = 0.0 to y/L = 0.05, where L is a
direction implicit solution procedure with generalized first or reference length used by Proteus to normalize input values. For
second order time differencing.(3-4) The boundary conditions this test case, L = 52_nax, where _nax is the maximum
,arealso treated implicitly, and may be steady or unsteady. All boundary layer thickness. Aportion of the grid extending from
terms, including the diffusion terms, are linearized using second x/L = 0.25 to x/L = 0.29 is illustrated in Figure I. For the
order Taylor series expansions. initial conditions, u, the horizontal x-velocity, and v, the vertical
y-velocity, were computed using the Blasius solution. The static
Two versions of the Proteus code exist: one for two- pressure, p, was set to p**, the freestream static pressure,
dimensional planar and axisymmetric flow, and one for three- everywhere. For the boundary conditions, at the upstream
dimensional flow. In addition to solving the full time-averaged boundary, p, u and v were held at the initial condition values. At
Navier-Stokes equations, Proteus includes options to solve the the downstream boundary, p = p,,,, and o_2u/o_x2= o')2v/o_x2 = 0.
thin-layer or Euler equations, and to eliminate the energy At the surface, bp/Oy = 0, and u = v = 0. At the freestream
cquation by assuming constant stagnation enthalpy. Artificial boundary, p = p_, u = u_, and _v/c_y = 0.
viscosity is used to minimize the odd-even decoupling resulting
from the use of central spatial differencing for the convective The results shown in Figures 2-6 were obtained after 4100
terms, and to control pre- and post-shock oscillations in super- iterations. Figure 2shows the x-velocity profile plotted against
sonic flow. Two artificial viscosity models are available -- a the Blasius similarity coordinate, rl, where
combination implicit/explicit constant coefficient model (5), and
an explicit nonlinear coefficient model designed specifically for
r/= yu._'-./vx
flows with shock waves.(6-7). At the NASA Lewis Research
Center, the code is typically run either on the CRAY X-MP or with v a_ the kinematic viscosity. Here, the Proteus results are
the CRAY Y-MP computer, and is highly vectorized.
indistinguishable from the Blasius profile, indicating excellent
*Aerospace Engineer performance by Proteus. In the y-velocity profile of Figure 3,
Member AIAA the Blasius results are also indistinguishable from the Proteus
10
0.05
8
....... t-
1 Oo 6
I ...................... 0
i__
"e-
"_ 4
---!-- --I
____, t .......
y/L _ 2
___ PBrlaosteiuuss SRoelustuiolnts
i 1 1 1
0
o.0o0 0.002 0.004 0.006 0.008
v/u** Velocity
Fig.3. Y-velocityprofileforlaminar flat plate flow.
0.00750 L
0.00625
Proteus Results
0.0
0.29 ---HD--- Blasius Solution
0.25 x/L
Fig.1 Aportion ofthegrid usedforlaminarflat plate
calculations.
"_ 0.00375
10
G 0.00500
L 0.00250
¢.-
8
Proteus, Rex=40,000
t'-' Proteus. Rex=60,O00
_S
Proteus, Rex=80,000 0.00125 t
Oo 6
-- _ Blasius Solution
O o.ooooo I 1 I I I I I I 1 I I
j 90 110 130 150 170 190 210
Ree
E 4
Fig.4. Local skinfrictioncoefficientforlaminar flatplate
flow.
03
0.0025
_ 2
0.0020
0.0 0.2 0.4 0.6 0.8 1.0 1.2
u/u_ Velocity 0.0015
Fig.2. X.velocity profiles for laminar flat plate flow.
results. Figure 4shows the local skin friction coefficient plotted 0.0010
Proteus Results
against Re0, the Reynolds number based on0, the momentum
Blaslus Solution
thickness. Figure 5shows 0versus xand Figure 6shows the
displacement thickness, 5",versus x. Figures 4through 6all 0.0005
exhibit excellent agreement between the Proteus results and the
Blasius solution. Thus, Proteus iscapable of accurately
calculating incompressible laminar flow over a flat plate.
o.oooo I .1 I
0.2 0.4 0.6 0.8 1.0
X
Fig.5. Momentumthicknessfor laminar flatplate flow.
0.006
A study was done to minimize the number of grid points in
the streamwise and normal directions required to accurately
compute the above described laminar flow. The results of this
study are shown in Figures 7 and 8. In Figures 7a-7d, the
0.005 number of streamwise grid points is decreased from 101 to 13
points, while the number of normal grid points is held constant
at 101. The results begin to deviate from the Blasius solution
below 26 points, as shown in the deviation of the 13xl0] cu_'e
of Figure 7d. This shows that aminimum of 26 streamwise
"_o 0.004
grid points are required for an accurate solution. In Figures 8a-
8c, 26 grid points are used in the su'eamwise direction, and the
number of grid points in the normal direction is decreased from
101 to 26 points. The results begin to disagree with the Blasius
0.003 curve when fewer than 51 points are used, as seen by the
deviation in the 26x26 curve of Figure 8c. Thus, the smallest
!O _ BlasiusSolution
grid needed to accurately calculate this laminar flow over aflat
plate is a26x51 grid.
o.oo2 I I I
0.2 0.4 0.6 0.8 1.0
X
Fig. 6. Displacement lhickness for laminar flat plate flow.
0.00750 7a, 0.00750 7b.
0.00625 BlasiusSolution 0.00625
Blasius Solution
+ ProteuslOlxlOl Grid t _ Proteus 51x101 Grid
0.00500 l 0.00500
c7 cT
0.00375 "_ 0.00375
O _°9
..J
0.00250 0.00250
0.00125 0.00125
I I] I I I I I I I I I I I I I I I t I I I
0.00000 0.00000
90 110 130 150 170 190 210 9O 110 130 150 170 190 210
Re e Ree
0.00750 7¢. 0.00750 / 7d.
/ _ Proteus13xt01 Grid
0.00625 BlasiusSolution 0.00625
+ Proteus26x101Grid
0.00500 [ 0.00500
c.7
0.00375 "_ 0,00375
O
O q
J
0.00250 - _ 0.00250
0.00125 0.00125
0.00000 t I 1 1 I I I 1 1 I I o.ooooo I t 1 1 I I 1 I I 1 l I
90 110 130 150 170 190 210 90 110 130 150 170 190 210
Re o Reo
Fig. 7a-d. Variation inthe number of streamwise grid points.
3 "
8a, Turbulent Flat Phtte Flow
Results for incompressible turbulent flow over afiat plate
0.00625 are shown in Figures 9-11. For the cases shown, the freestream
0.00750 t "---O---- Proteus 26x101 Grid Math number was 0.2 and Rex ranged from 4,000,000 at the
Blasiu¢ Solution upstream computational boundary to 16,000,000 at the
0.00500 t downstream boundary. A lOlx191 grid was used with packing
in the vertical direction atthe plate surface such that the packing
G
ratio was 0.005. Grid packing was also used in the x-direction
0.00375 at the upstream boundary such that the packing ratio was 0.05.
O
3 The grid extended from x/L = 0.33 to x/L = 1.33 and from y/L =
0.0 to yfb --0.048, where L _,58_nax. For the initial
0.00250 Conditions, u was determined from an expression developed by
Musker(9), with v = 0, and p = poo. The boundary conditions
were identical to those for the laminar flat plate case.
0.00125
1.20
1 I I I I I I IJ LL
0,00000
1.08
90 110 130 150 170 190 210
Ree 0.96 Proteus, Rex-7x t06
Proteus, Rex- 10x10e
0.84
-- * -- Proteus. Rex.13x10 e
0.72 O Klebanoff Data
00..0000765205 _ 8b.
---O--- Proteus26x51 Grid 0,60 i
(
0.48
G 0.36
0.00375
O 0.24
0.00500 " "_
0.12
0.00250
I
0.00
0.0 0.2 0.4 0.6 0.8 1.0 1.2
U/U_ Velocity
0.00125
I 1 1 I 1 I I I I I Fig. 9. X-Velocity Follies for turbulent flat plate flow.
0.00000 1
90 110 130 150 170 190 210
Figure 9 shows the x-velocity plotted with y/8, where 5 is
Re_
the boundary layer thickness. Note that the Proteus results at the
8C.
three different Reynolds numbers shown are each represented
by acurve and agree so closely that it is difficult to distinguish
them on the plot. This is to be expected since the profiles are
0.00625
plotted with similarity coordinates. The results also show good
0.00750 L -----O---- Proteus 26x26 Grid
agreement with the experimental data of Klebanoff.(10) Figure
----O--- Blaslus Solulion 10 shows the same Proteus results plotted on asemi-log graph
0.00500 t
with u+ and y+ coordinates, where u+ = u/ux, with u,_equal to
G the shear velocity, and y+ = yux/v. These results show good
0.00375 agreement with the law of the wall correlation.(11) Figure 11
3 shows aplot of the local skin friction coefficient, cf, versus Re0,
compared with the Karman-Schoenherr correlation(12) and the
0.00250 experimental data of Weighardt.(13) Notice that the Proteus
results exhibit adrop in cf at the upstream boundary, where Rex
= 4,000,000 or Re0 = 6,500 with the remaining portion of the
0.00125 -- curve in agreement with the data of References 14 and 15. This
drop at the upstream boundary is most likely aresult of using an
I 1 l 1 1 I 1 I I .1_[ inexact boundary condition at this boundary. Recall that at the
0.00000 upstream boundary, a u-profile was approximated and v was set
90 110 130 150 170 190 210
to zero. Other upstream boundary conditions were also
Ree
Fig. 8a-c. Variation In the number of normal grid points.
40
Laminar Developinm_
The Proteus calculations for incompressible laminar
ProteusRex=7x106
developing pipe flow are shown in Figures 12and 13compared
.... ProteusRex=lOxlO6
with the experimental data of Reshotko.(14) For this test case,
3O
-- - -- Proteus Rex=13xtO 6 an average Mach number of approximately 0.1 was used, and
ReD, the Reynolds number based on the pipe diameter, was 100.
The pipe length was set to 10diameters, and a51 axial by 21
radial grid was used. For the initial flow field, u =v = 0 and p =
b
2O Pr, where Pr is the reference pressure which was set to standard
sea level pressure. For the boundary conditions, the inlet and
exit pressure were chosen to achieve apressure drop calculated
by pipe design formulas. For the remaining inlet boundary
conditions, 632uf_x2 =0 and 3v/3x = 0. The remaining exit
10
conditions were 3u/3x = avf0x = 0. At the pipe wall, 3p/Or = 0
and u = v = 0. The centerline boundary conditions were
standard symmetry conditions such that Op/Or = 3u/3r = 0 and v
-"_'IIIIIIII I llJJllll I IIIIIIII I lllllll =0.
0
10-0 10"1 10"'2 10"3 10"4
y÷
Fig. I0. X-Velocity profiles for turbulent flat plate flow.
o1,
0.0060
0 Proteus Results
_o.5._
0.0050 -- Karman-Schoenherr Correlatior
0 Weighardt Data
o
0.0040 -o
o 0.2
qb
0.1
0.0030 -- Oo O ,
0 O0 - • , - : . , • , ' , • ,
.J 0.00 0.25 0.50 0.75 t.O0 1,25 1.50 1.75 2.00
Axial Velocity U/Uo
0.0020
Fig. 12. Axial velocity profiles for laminar developing plpe
flow.
0.0010
In Figure 12, the nondimensionalized axial velocity, u/u0, is
o.oooo I I I I I
plotted against the nondimensionalized radial position, r/R, with
0 4000 8000 12000 16000 20000 24000
u0 equal to the average velocity, requal to the radial position in
Re 0 the pipe and R equal to the pipe radius. Results are plotted at
various axial locations in the pipe, represented nondimensionally
Fig. 11. Localskin friction coefficientforturbulentflat as x/[(R)(ReR)], with ReR equal to the Reynolds number based
plate flow. on R. The Proteus results are shown as curves and the
experimental data as symbols. As can be seen, the Proteus
velocity profiles coincide fairly well with the experimental data
and exhibit the distinctive bullet or Poisueille profile. Figure 13
considered, such as using au-profile computed from the
represents the axial velocity in the pipe at selected radial
Musket expression and v either computed 'from the continuity
positions. Again, the Proteus results are shown as curves and
equation or extrapolated, or moving the upstream boundary to the experimental data as symbols. Here, the Proteus agreement
the leading edge of the plate where u --u=, and v=0. These is also good, but note that in both Figures 12and 13, there is a
boundary conditions, however, were not as effective as the
slight deviance in the near-wall region where r/R = 0.9.
chosen conditions. Overall, Figures 9-11 show that the Proteus
Previous work has shown that this might be improved by
performance is very good for incompressible, turbulent fiat plate packing more grid points near the wall to better resolve the
flow.
steep gradients imposed by the no-slip wall boundary
conditions. Overall, Proteus performs well for laminar
developing pipe flows.
1.25 '
201
1,20"
161
1.15'
1.61 • t/R.O 4 t.I0 '
o
14" 1.05'
>,12" • r/R.O.6 1.00
ooJ
-_ t0"
_o •
r/R=07
• o t • • qp • •
08- - 0 0 r/Fl.O.75
m 0.85 -
0.6" l<
t/R=O,8 <
0.80-
0,4'
0.75 -
0.2ooo• o.o5•--T--o.,o- oi.,_• o._o 0.25 " 0,;0 " 0.35 0.40
0.70
xI(R'Re)
0,65 " 10 " _ " 30 ' 40 ' 50 ' 60 70 80 90 100
Fig. 13. Axialvelocityprofilesfor laminar developingplpe Axial Location x/R
flow.
Fig.15. Axialvelocityprofiles for turbulent developing plpe
flow.
Tt_rbulent Developing Pipe Flow
Swirlinl_ Deveioped PiPeFlow
The results for incompressible turbulent developing pipe
flow are shown in Figures 14and 15compared with the In this validation case, the Proteus results for swirling
experimental data of Barbin. 15 This case had an average Mach incompressible turbulent pipe flow are compared with the
number of approximately 0.09 and aRED=388,000. The pipe experimental data of Weske. 16 The Mach number was 0.1 and
length was set to 50diameters, and a 101axial by51radial grid RED=30,000. The pipe length was set to50 diameters with a
was used with apacking ratio of 0.05 near the wall. The initial 400 axial by50 radial grid, and apacking ratio of 0.1 near the
and boundary conditions used were identical tothose of the wall. The initial conditions for uwere calculated using the I/Tth
laminar developing pipe flow case. power law, with the boundary layer thickness approximated as
10% of the pipe radius. The initial swirl velocity profile was
Figures 14and 15show the Proteus results inamanner linear with the swirl velocity w = 0at the centerline and
analogous toFigures 12and 13 for laminar developing pipe increasing toa maximum of w = u0near the wall, where u0is
flow. The value of uat the pipe inlet isapproximately equal to the centerline axial velocity for this case. This gives the swirl
the average velocity, u0, which would be expected for turbulent number of o = 1.0, where o =Wmax]U0.The remaining initial
developing pipe flow. Also, the Proteus results closely agree conditions were p= Pr,and v=0. For the boundary conditions,
with the experimental data, with aslight deviation inthe near- the inlet and exit pressure were chosen so that the pressure drop
wall region. Thus, Proteus iscapable of calculating turbulent coincided with the design pipe calculation value, ignoring the
developing pipe flows• unknown effects of the swirling velocity component. The inlet
velocities were held at the initial condition values and at the exit,
au/'dx = av/ax = aw/ax =o. At the pipe wall, ap/'0r =0and u=
v= w =0. Atthe centerline, ap/_ = au/ar =0and v=w =0.
1.1
• x,.,R,,.3
1.0- _"" " --" +• #xR/R,-.303 1.0
0,9" • #R-81 oo
0,9-
0,8 _ O 4- 'I-
0.7 _ 0.8" 0 + o
o.6:
0.7" 0 4- 0
._0.5-
0.4- o + o
_" 0.3: 0.6" 13 4, 0
02 _ 0.5"
_" 0.4" I:i + 0
0.1"
0.0 --" o:_ 0.3" 0 .i, o
0,5 0,7 0.8 0.9 1.0 1.1 1.2
Axial Vel_¢lty U/Uo s
0.2'
Fig.14. Axial velocity profiles for turbulent developing pipe
0.1'
flow. o 50Diarr_lers
0.0 • i • i _ i • i • i • i • i , i • i
0.0 0.1 0.2 0.3 0.4 0,5 0.6 0.7 0.8 09 1.0
Swirl Velocity W/Uo
Fig.16. Swirl velocity profiles in apipe. (Inlet swirl number
o:1)
Figure 16 shows the Proteus swirl velocity profiles as 10. Klebanoff, P.S., "Characteristics of Turbulence in a
curves and the experimental data as symbols. The plot shows a Boundary Layer with Zero Pressure Gradient," NACA Report
general agreement of the profile shape and swirl decay as the 1247, 1953.
flow works its way down the pipe; however, the Proteus curves I1.White, F. M., Viscous Fluid Flow, McGraw-Hill, Inc.,
do not agree well with the data. The disagreement can be 1974.
attributed to the inability of the algebraic eddy viscosity 12. Hopkins, E. J., and Inouye, M, "An Evaluation of
turbulence model to handle the anisotropies of this complex Theories for Predicting Turbulent Skin Friction and Heat
flow. Yoo et al.(17) describe the problems of computing the Transfer on Flat Plates at Supersonic and Hypersonic Mach
turbulence field for asimilar flow. Numbers," AIAA J.Vol. 9, No. 6, June, 1971, pp. 993-1003.
13. Weighardt, K., "Flat Plate Flow. u_,,= 33 nVsec",
AFOSR-IFP-Stanford Conference on Computation of
Turbulent Boundary Layers-1968", Vol. II, Ed. by Coles, D.E.
Concluding Remarks
and Hirst, E.A., 1968.
Validation cases for both laminar and turbulent 14. Reshotko, E., "Experimental Study of the Stability of
Pipe Flow, I. Establishment of an Axially Symmetric Poisueille
incompressible flow over a flat plate atzero pressure gradient
Flow," Progress Report No. 20-354, Department of the Army
showed excellent agreement with exact solutions, empirical
Ordinance Corps, October, 1958.
correlations and experinaental data. It was also shown that a
15. Barbin, A. R., "Development of Turbulence in the Inlet
26x51 grid with packing near the wall gives sufficient resolution
of aSmooth Pipe," Ph.D. Thesis, Purdue Univ., 1961.
to calculate laminar flat plate flow. The velocity profiles of both
16. Weske, D. R., "Experimental Study of Turbulent
laminar and turbulent developing pipe flow agreed with
Swirled Flows in a Cylindrical Tube," Fhdd Mechanics --Soviet
experimental data, with slight deviations near the pipe wall. Pipe
Research, Vol. 3, No. 1,Jan.-Feb., 1974, pp. 77-82.
flow with a swirl number of 1.0 showed the expected profile
17. Yoo, G. J., So, R. M. C., and Hwang, B. C.,
shape and swirl decay; however, the swirl velocity profiles did
"Calculation of Developing Turbulent Flows in aRotating
not coincide with experimental data. This is ashortcoming of
Pipe," ASME J. ofTurbomachinary, Vol. 113, January 1991,
the algebraic eddy viscosity model used in Proteus for
pp. 34-41.
computing swirling pipe flows. With this exception, Proteus, is
proven to be effective for calculating simple internal and
external, incompressible, viscous flows.
Validation of Proteus is ongoing. Future plans include
verification of higher Mach number flows'and flows with heat
transfer.
References
1. Towne, C. E., Schwab, J. R., Benson, T. J., and Suresh,
A., "PROTEUS Two-Dimensional Navier-Stokes Computer
Code -- Version 1.0, Vols. 1-3," NASA TM 102551-102553,
March, 1990.
2. Baldwin, B. S., and Lomax, H., "Thin Layer
Approximation and Algebraic Model for Separated Turbulent
Flows," AIAA Paper 78-257, Jan. 1978.
3. Briley, W. R., and McDonald, H., "Solution of the
Multidimensional Compressible Navier-Stokes Equations by a
Generalized Implicit Methods," J. Comp. Phys., Vol. 24, Aug.
1977, pp. 373-397.
4. Beam, R. M., and Wanning, R. F., "An Implicit Factored
Scheme for the Compressible Navier-Stokes Equations," AIAA
J., Vol. 16, April 1978, pp. 393-402.
5. Steger, J. L., "Implicit Finite-Difference Simulation of
Flow about Arbitrary Two-Dimensional Geometries,"AIAA J.,
VoI. 16, July 1978, pp. 679-686.
6. PuIliam, T.H., "Artificial Dissipation Models for the
Euler Equations," AIAA J., Vol. 24, Dec. 1986, pp. 1931-1940.
7. Jameson, A., Schmidt, W., and Turkel, E., "Numerical
Solutions of the Euler Equations by Finite Volume Methods
Using Runge-Kutta Time-Stepping Schemes," AIAA Paper 81-
1259, 1981.
8. Schlicting, H., Boundary Layer Theory, McGraw-Hill
Book Company, New York, 1968.
9. Musker, A. J.,"Explicit Expression for the Smooth Wall
Velocity Distribution in aTurbulent Boundary Layer,"AIAA J.
Vol. 17, No. 6, June 1979, pp. 655-657.
7
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1. AGENCY USEONLY (Leaveblank) 2. REPORT DATE 3. REPORT TYPE ANDDATES COVERED
Technical Memorandum
4. TITLE AND SUBTITLE
S. FUNDING NUMBERS
Verification of the Proteus Two-Dimensional Navier-Stokes Code
for Flat Plate and Pipe Flows
6. AUTHOR(S) WU-505- 62-21
Julianne M. Conley and Patrick L. Zeman
7. PERFORMING ORGANIZATIC)N NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION
REPORT NUMBER
National Aeronautics and Space Administration
Lewis Research Center
E-6449
Cleveland, Ohio 44135 - 3191
9. SPONSORING/MONITORING AGENCY NAMES(S) ANDADDRESS(ES) 10. SPONSORING/MONrrORING
AGENCY REPORT NUMBER
National Aeronautics and Space Administration
NASA TM- 105160
Washington, D.C. 20546- 0001
AIAA-91-2013
11. SUPPLEMENTARY NOTES
Prepared for the 27th Joint Propulsion Conference cosponsored by AIAA, SAE, ASME, and ASEE, Sacramento,
California, June 24-27, 1991. Julianne M. Conley, NASA Lewis Research Center; Patrick L. Zeman, Arnold Engi-
neering and Development Center, Arnold AFB, Tennessee. Responsible person, Julianne M. Conley, [phone].
12s. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE
Unclassified -Unlimited
Subject Category 34
13. ABSTRACT (Maximum 200 words)
The Proteus Navier-Stokes Code is evaluated for two-dimensional/axisymmetric, viscous, incompressible, internal
and external flows. The particular cases to be discussed are laminar and turbulent flows over a flat plate, laminar
and turbulent developing pipe flows and turbulent pipe flow with swirl. Results are compared with exact solutions,
empirical correlations and experimental data. A detailed description of the code set-up, including boundary
conditions, initial conditions, grid size and grid packing is given for each case.
14. SUBJECT TERMS lS. NUMBER OFPAGES
8
Navier-stokes equation; Two-dimensional flow; Axisymmetric flow; Laminar flow;
Turbulent flow; Flat plates; Boundary layer flow; Pipe flow; Swirling 16. PRICE CODE
A02
17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT
OF REPORT OFTHIS PAGE OF ABSTRACT
Unclassified Unclassified Unclassified
NaN 7540-01-280-5500 Standard Form 298 (Roy. 2-89)