Numerical solutions of the Navier-Stokes equations, with an algebraic turbulence model, for time-dependent two dimensional flow about multi-element airfoils were developed. Fundamental to these solutions was the use of numerically-generated boundary-conforming curvilinear coordinate systems to allow bodies of arbitrary shape to be treated. A general two dimensional grid generation code for multiple-body configuration was written as a part of this project and made available through the COSMIC code library

Thompson, J. F.

English

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N84-i6876
0
Final Report
for the Period
January 1, 1974 through December 31, 1983
(NASA-CR-173251) NUMERICAL SOLUTION OF THE
NAQI ER-STOKES EQUATIONS FOR ARBITRARY
2-DIMENSIONAL !MULTI-ELEMENT AIRFOILS Final
Report, 1 Jan. 1974 - 31 Dec. 1983 	 Unclas
(MiSSiSSiMi State Uni v .. !lississi DDi 	 G3/64 18243
t Submitted Under
NASA Grant NCR-25-001-55
Entitied
NUMERICAL SOLUTION OF THE NAVIER-STOKES EQUATIONS
FOR ARBITRARY TWO-DIMENSIONAL MULTI-ELEMENT AIRFOILS
by
Dr. Joe F. Thompson
Mississippi State University
Department of Aerospace Engineering
Mississippi State, MS 39762
January 23, 1983
4	 !r ^ I
IV
	
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Table of Contents
Pag,2
Introduction	 1
Numerical Grid Generation 	 1
Incompressible Navier-Stokes Codes 	 3
Compressible Navier-Stokes Codes
	 5
Incompressible Potential Code
	 6
Conclusion
	 6
Publications	 7
L;
	
List of Graduates Supported	 11
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Introduction
This research project covered a period of ten years, from 1 January
1974 through 31 December 1983. The overall purpose was to develop numerical
solutions of the Navier-Stokes equations, with an algebraic turbulence
model, for time-dependent two-dimensional flow about multi-element airfoils.
Fundamental to these solutions was the use of numerically-generated
boundary-conforming curvilinear coordinate systems to allow bodies of
arbitrary shape to be treated. This project, in fact, provided the initial
support for the study of elliptic grid generation ; .^;tems in two dimensions,
and much of the development of numerical grid generation based on this
technique has been done under this project. A general two-dimensional
grid generation code for multiple-body configuration was written as
a part of this project and made available thrcugh the COSMIC code library.
Numerical Grid Generation
	 }
The developments under this project in the area of numerical grid
generation are summarized as follows:
(1) Elliptic grid generation with coordinate line control. This initial
work was covered in the Ph.D. dissertation of Thames (1975) and was reported
in Thompson, Thames and Mastin, Journal of Computational Physics (1974),
and in Thames, Thompson and Mastin, NASA SP-347 (1975). Results were also
reported at the AIAA 2nd Computational Fluid Dynamics Conference in
Hartford, Conn. (1975). The basic techniques of elliptic generation
were developed and coded, including in particular the means of user-
controlled attraction of coordinate lines and/or points to other coordinate
1
lines and/or points to provide a priori concentration of coordinate lines
in regions of expected high gradients, e.g.,boundary layers,
(2) Extension to multiple-body fields:the TOMCAT code. This work culminated
in the development of the TOMCAT code which generates two-dimensional
boundary-conforming coordinate systems for fields containing any number
of arbitrarily-shaped bodies, with a priori control of coordinate line
spacing as mentioned above. This code was made available through the
COSMIC code library, as reported in the NASA Tech Briefs (Spring 1978,
p. 150). The code was described fully in a contractor report, CR-2729
(1977), and also was reported in Thompson, Thames and Mastin, Journal of
Computational Physics (1977).
(3) Automatic coordinate line concentration in boundary layers. This
allowed a prescribed number of lines to be placed within a specified
boundary layer thickness on an airfoil, requiring only the specification
of the Reynolds number. This technique was discussed in Thompson, et.al .,
NASA CP-2045 (1978); in Thompson, Computational Fluid Dynamics (1980);
and in Thompson, Numerical Grid Generation (1982).
(4) Evaluation of grid-induced truncation. error. Studies were made of
the truncation error induced by rapid changes in coordinate line spacing,
and several distribution functions were compared in this regard. Some
of this work was involved in the MS thesis of D. Thompson (1980).
(5) Grids with infinity transformed to the orgin. This type of grid
is particulary appropriate to double-airfoil configurations. The work
was discussed in the MS thesis of Long (1977) and was also reported
in Thompson, et.al ., NASA CP-2045 (1978).
p
2
i).
e(6) Grid generation surveys. This project provided support for extensive
coverage of the literature on grid generation over the years, contributing to
the surveys given by Thompson, Warsi and Mastin, Journal of Computational
Physics (1982), and that presented by Thompson at the AIAA 21st Aerospace
Sciences: meeting in Reno (1983).
(7) Exposition on grid generation. In the course of this long project
the principles of numerical grid generation from elliptic generation systems,
and the use thereof in the numerical solution of partial differential
equations, were formulated and refined to the point of consistent presentation
for general use. This is reflected in the expositive presentations by
Thompson in Computational Fluid Cynamics (1980), and in Numerical Grid
Generation (1982).
Incompressible Navier-Stokes Codes
The codes developed for the incompressible Navier-Stokes equations
are summarized below. All these, except the first, were based on the
velocity-pressure formulation, using the Poisson equation for the pressure.
The pressure boundary condition on the body surface was obtained from
the normal momentum equation, although in later work a zero normal pressure
gradient on the surface was found to be sufficient because of the very
close spacing of the first coordinate line off the surfacee typically
set at about 1% of the expected boundary laver thickness.
In all these codes, provision was made for the use of artificial viscosity
proportional to the divergence of the velocity, and hence analyticaly
zero, to control the solution during the initial transients. The start
was through continuous acceleration, instituted through the inclusion
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of ficticious body force terms during the period of acceleration. The
Baldwin-Lomax algebraic turbulence model was used in all cases.
(1) Stream function-vorticity. This code used point SOR iteration
for the vorticity equation and the stream function Poisson equation.
This work was discussed in the Ph.D. dissertation of Thames (1975) and
was reported in Thames, Thompson and Mastin, NASA SP-347 (1975).and in
Thames, et.al . Journal of Computational Physics (1977). Results for
application to boundary layers were discussed in the MS thesis of Walker
-	 (1974), also being reported in Thompson, et.al ., Symposium on Unsteady
Aerodynamics (1975),and in Thames, et.al .,(1977) cited above. Applications
to airfoils at higher Reynolds numbers were discussed in the MS thesis
of Bearden (1977) and were reported in Thompson, et.al ., NASA CP-2045
(1978).
(2) Point SOR. This code was based on point SOR iteration with under-
relaxation,using a variable acceleration parameter field :alculated from
the local velocity. This code was discussed in Thompson, et.al ., NASA
CP-2045 (1978); in Thompson, Computational Fluid Dynamics (1980); and in
the MS thesis of D. Thompson (1980).
(3) Approximate factorization. This code used approximate factorization
on the two momentum equations and line SOR iteration on the pressure
Poisson equation, thus requiring only one pass of the momentum equation
at each time step, although the pressure equation still was iterated.
This work was discussed in the Ph.D. thesis of 3ernard (1981), and results 	 !
were presented at the AIAA/ASME 3rd Joint Thermophysics, Fluids, Plasma
and Heat Transfer Conference in St. Louis (1982) and also in Mueller,
4
et.al ., Computers in Flow Predictions and Fluid Dynamics Experiments
(1981).
(4) Vectorized checkerboard SOR. This code was based on checkerboard
SOR iteration in all equations, and was fully vectorized on the CYBER 203.
The vectorization was explicit using ,.he vector FORTRAN with bit vectors
to control' the manipulation of the arrays. This code was discussed in
the Ph.D. dissertation of Patel (1983), and a paper has been submitted
for presentation at the AIAA 17th Fluid Dynamics, Plazmadynami^-s, and Lasers
Conference in Snowmass,Colorado (1984), This was the most promising
_	 of the incompressible codes, showing considerable speed advantage because
of the vectorization.
Compressible Navier-Stokes Codes
The compressible Navier-Stokes codes are summarized as follows.
All used backward-time and central-space differences and were second-
order in time and space. Again the Baldwin-Lomax turbulence model was
used.
(1) Point SOP.. This code used point SOR iteration of all equations at
	 =
each time step, with a variable acceleration parameter field calculated
from the local velocity. Since these parameters proved to be very small
for the continuity equation,this code was unreasonably slow. The code
was discussed in the Ph.D. dissertation of Turner (1979).
(2) Density Gradients as dependent variables. A small investigation
of the use of density gradients as dependent variables was made, as
reported in the MS thesis of Kwon (1977), but little promise of advantage
was found.
5
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(3) Approximate factorize*ion. This code was based on approximate factorization
in the delta form for all equations and was discussed in the Ph.D. dissertation
of Cooper (1980). Results were presented at the AIAA 14th Fluid and Plasma
Dynamics Conference in Palo Alto (1981).
Incompressible Potential Code
In addition to the Navier-Stokes codes, an incompressible potential flow
code for multiple-body fields was developed in the early stages of the
project. This solution was also discussed in the Ph.D. dissertation of
Thames (1975). Some results were reported in Thames, et.ai ., Journal of
Computational Physics (1977), and more extensive results were collected later
in the MSU report of Thompson and Thames (1982).
Conclusion
The major accomplishments of this long project were the fundamental
development of elliptic grid generation and the establishment thereof
as a basic component in the numerical solution of partial differential
equations, particularity in computational fluid dynamics_ A significant
additional accomplishment was the development of a vectorized code for
the incompressible Navier-Stokes equations. Finally, perhaps the real
significance of this project was the basic support it provided over t`,e
years for the development, ^ori-elation and dissemination of ideas on
t;umerical grid generation and the graduates it produced.
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6
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PUBLICATIONS
Journal Articles
J. F. Thompson, F. C. Thames, and C. W. Mastin, "Automatic Numerical Generation
of Body-Fitted Curvilinear Coordinate System for Fields Containing any
Number of Arbitrary Two-Dimensional Bodies," Journal of Computational Physics,
15, 299, (1974).
F. C. Thames, J. F. Thompson, C. W. Nastin, and R. L. Walker, "Numerical
Solutions for Viscous and Potential Flow about Arbitrary Two-Dimensional
Bodies Using Body-Fitted Coordinate Systems," Journal of Computational
Physics, 24, 245 (1977).
J. F. Thompson, F. C. Thames, C. W. Mastin, "TOMCAT" - A Code for Numerical
Generation of Boundary-Fitted Curvilinear Coordinate Systems on Fields
Containing any Number of Arbitrary Two-Dimensional Bodies," Journal of
Computational_ Physics, 24, 274 (1977).
Joe F. Thompson, Z. U. A. Warsi, and C. W. Mastin, "Boundary-Fitted Coordinate
System for Numerical Solution of Partial Differential Equations - A Review,"
Journal of Computational Physics, 47, 1 (1982).
Jae F. Thompson, "Elliptic Grid Generation", Applied Mathematics and Computation,
10-11, 79, (1982}.
J. F. Thompson, "General Curvilinear Coordinate Systems", Applied Mathematics
and Computation, 10-11, 1, (1982).
J. F. Thompson, "A Survey of Grid Generation Techniques in Computational
Fluid Dynamics," accepted for publication in AIAA Journal.
Conference Presentations
F. C. Thames, J. F. Thompson, and C. W. Mastin, "Numerical Solution of
the Navier-Stokes Equations for Arbitrary Two-Dimensional Airfoils," Pro-
ceedings of NASA Conference on Aerodynamic Analyses Re5uirinq Advanced
Computers, Langley Researc F.. Center, NASA P-34	 (1975).
J. F. Thompson, F. C. Thames, R. L. Walker, and S. P. Shanks, "Numerical
Solutions of the Unsteady Navier-Stokes Equations for Arbitrary Bodies Using
Boundary-Fitted Curvilinear Coordinates, Proceedings of Arizona AFOSR
Symposium on Unsteady Aerodynamics, UniversitT y of Arizona, (1975).
7
J. F. Thompson, F. C. Thames, C. W. Martin, and S. P. Shanks, "Use of
Numerically Generated Body-Fitted Coordinate Systems for Solution cf
the Navier-Stokes Equations," Proceedings of the AIAA 2nd Computational
Fluid Dynamics Conference_, Hartford, Connecticut,	 .
J. F. Thompson, F. C. Thames, S. P. Shanks, R. N. Reddy, and C. W. Mastin,
"Solutions of the Naviar-Stokes Equations in Various Flow Regimes on
Fields Containing any Number of Arbitrary Bodies Using Boundary-Fitted
Coordinate Systems," Proceedings of V International Conference on Numerical
Methods in Fluid Dynamics, Enschede, the Netherlands, Lecture Notes in
Physics, Springer-Verlag, (1976), 'ol. 59, p. 421.
Joe F. Thompson, Louie Turner, W. Serrill Long, and John H. Bearden,
"Numerical Solution of the Navier-Stokes Equations for Arbitrary Two-
Dimensional Multi-Element Airfoils," Advanced Technology airfoil Research,
NASA Conference Publication 2045 (1978).
Joe F. Thompson, "Numerical Solution of Flow Problems Using Body-Fitted
Coordinate Systems," Lecture Series in Computational Fluid Dynamics,
von Kaman Inst. for Mid Dynamics, Belgium, 1978. Published in
Computational Fluid Dynamics, W. Kollmann (ed), Hemisphere, 1980.
Joe F. Thompson and C. Wayne Mastin, "Pro gress in the Use of Boundary-
Fitted Coordinates for the Numerical Solution of Partial Equations",
17th Society of Engineering Science Meeting, Atlanta, 1980.
G. Kyle Cooper and Joe F. Thompson, "A Panasonic Navier-Stokes Solver",
AIAA Paper 81-1195, AIAA 14th Fluid and Plasma Dynamics Conference, Palo
Alto, 1981.
T. J. Mueller, B. J. Jansen, Jr., R. S. Bernard, and Joe F. Thompson,
"Experimental and Numerical Studies of the Incompressible Viscous Flow
Over a Two-Dimensional Airfoil", in Computers in Flow Predictions and
Fluid Dynamics Experiments, ASME, 1981.
Joe F. Thompson, "Elliptic Grid Generation", in Numerical Grid Generation
Joe F. Thompson (ed.) Elsevier, 1982.
J. F. Thompson, "General Curvilinear Coordinate Systems", in Numerical
Grid Generation, J. F. Thompson (ed.) Elsevier, 1982.
R. S. Bernard and J. F. Thompson, "Approximate Factorization with an
Elliptic Pressure Solver for Incompressible Flow," AIAA -82-0978, AIAA/ASME
3rd Joint Thermophysics, Fluids, Plasma and Heat Transfer Conference,
St. Louis, Missouri, 1982.
Joe F. Thompson, "A Survey of Grid Generation Techniques in Computational
Fluid Dynamics," AIAA Paper No. 83-0447, AIAA 21st Aerospace Sciences
Meeting, Reno, Nevada, January 1983.
8
f®r\
Reports
J. F. Thompson, F. C. Thames, and C. W. Mastin, "Boundary-Fitted Curvilinear
Coordinate System for So'iution of Partial Differential Equations on Fields
Containing any Number of Arbitrary Two-Dimensional Bodies," NASA CR-2729
(1977).
Joe F. Thompson and F. C. Thames, "Numerical Solution of Potential Flow
About Arbitrary Two-Dimensional Multiple Bodies", MSSU-EIRS-ASE-82-2,
Mississippi State University, 1982.
Thesis
R. L. Walker, "Numerical Solution of the Navier -Stokes Equations for
Incompressible Viscous Laminar Flow Past a Semi-Infinite Flat Plate,"
M.S. Thests, Mississippi State University, (1974).
John H. Bearden, "A High Reynolds Number Numerical Solution of the Navier-
Stokes Equations in Stream Function -Vorticity Form," M.S. Thesis, Mississippi
State University, August 1977.
W. Serrill Long, "Two-Body Coordinate System Generation Using Body - Fitted
Coordinate System and Complex Variable Transforma^ion," M.S. Thesis,
Mississippi State University, August 1977.
Jang-Hyuk Kwon, "Numerical Solution of the Compressible Navier -Stokes
Equations Using Density Gradients as Additional Dependent Variables,"
M.S. Thesis, Mississippi state University, December 1977.
David S. Thompson, "Numerical Solution of the Navier-Stokes Equations for
High Reynolds Number Incompressible Turbulent Flow", M.S. Thesis, Missis ippi
State University May, 1980.
Dissertations
F. C. Thames, "Numerical Solution of the Incompressible Navier-Stokes
Equations about Arbitrary Two-Dimensional Bodies," Ph.D. Dissertation,
Mississippi State University, (1975).
Louie Turner, III, "Numerical Solution of the Navier-Stokes Equations for
Laminar, Transonic Flows," Ph.D. Dissertation, Mississippi State " - iversity,
May 1979.
9
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(TY
G. Kyle Cooper, "An Approximate Factorization Solution of the Navier-
Stokes Equations for Transonic Flow Using Body-Fitted Coordinates with
Application to NACA 64A010 Airfoils", Ph.D. Dissertation, Mississippi
State University, August 1980.
Robert S. Bernard, "Approximate Factorization for Incompressible Flow,"
Ph.D. Dissertation, Mississippi State University, August 1981.
Nisheeth Patel, "A Fully Vectorized Numerical Solution of the Incompressible
Navier-Stokes Equations,` Ph.D. Dissertation, Mississippi State University;
December 1983.
10
D 
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List of Graduates Supported
De ree	 Date
	
Present Affilation
Ii .
Rav L. Walker M.S. 1974 CIA
Frank C. lhames Ph.D. 1975 NASA Langley Research Center
John H. Bearden M.S. 1977 Cabot Corp. (entering K.D. program
at Georgia Tech)
W,	 Serrill Long M.S. 1977 United Airlines
J. H.	 Kwon M.S, 197? unknown (last known as Ph.D.
student at Cornell)
Louie Turrer Ph.D. 1979 Boeing
David S. Tiompson M.S. 1980 Ph.D. student at Iowa State
G. Kyle Cooper Ph.D. 1980 Srerdrj.p Corp.
Robert S. Bernard Ph.D. 1981 U.S. Army Engineer Waterways
[Aperiment Station
Nisheeth Patel Ph.D. 1983 U.S. Army Ballistic Research
Laboratory
Of these ten graduates, all but two wr;re U.S. citizens.
11


Numerical solution of the Navier-Stokes equations for arbitrary 2-dimensional multi-element airfoils

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