NASA NOTICE FOR HANDLING PROPOSALS

(October 1969)

This proposal shall be used, or duplicated, only for NASA

evaluation purposes and this notice shall be applied to any

reproduction or abstract thereof*

Disclosure of this proposal outside the Government for NASA

evaluation purposes shall not be made unless the policy and

procedures prescribed by NASA Procurement Regulation 1.304-2(f),

including the requirements for approval and for an arrangement

with the outside evaluator prior to disclosure, are followed.

The restrictions contained in this notice do not apply to

technical data or business information obtained from another

source without restriction.

RESEARCH PROPOSAL SUBMITTED TO THE

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

BY

NEW YORK UNIVERSITY

COURANT INSTITUTE OF MATHEMATICAL SCIENCES

251 MERCER STREET

NEW YORK, NEW YORK 10012

NUMERICAL DESIGN OF TRANSONIC SHOCKLESS AIRFOILS

Principal Investigator

Name: Paul R. Garabedian

Title: Professor

Social Security # 565-^4-9568

Renewal of NGR-33-016-167 (NYU ACCT. # 0-57^3-0^0)

Starting Date:Sept. 1, 1973 Duration:12 Mos. Amount Requested:$100,000

Residual funds expected to be on hand -0- as of August 31» 1973

Other support - None

Approval Signatures:

Principal Investigator Director

Name: Paul R. Garabedian Peter D. Lax

Title: Professor

Signat ure:

Telephone: 460-7359 460-7421

March 26, 1973 March 26, 1973

Continuation of Grant NGR-33-016-167

Proposed Budget

September 1, 1973 to August 31. 1971*

SALARIES:

Professor Paul R. Garabedlan

(Principal Investigator)10058 1 Summer Month $ 3,889

Associate Professor Samuel Burstein

10035 2 Summer Months it,020

Antony Jameson

Sr. Research Scientist

66-2/358 12 Months 16,333

David Korn

Research Scientist

30$ 12 Months 6,000

Prances Bauer

Research Scientist ^

5056 12 Months 8,750<

1 Research Assistant

10058 12 Months . " - 5,000

1 Secretary

2558 12 Months —2,750

TOTAL SALARIES $ 46,7*2

OVERHEAD (6656 of salaries) 30,850*

FRINGE BENEFITS (1M.5* of salaries) 6,778

COMPUTER TIME 12,000

TRAVEL 2,000

PUBLICATIONS . 1,500

MISCELLANEOUS (Communications, supplies, etc.) 130

TOTAL " $ 100,000

*The above overhead rate does not include k% of cost sharingor $1,870 representing the university's contribution.

NUMERICAL DESIGN OP TRANSONIC SHOCKLESS AIRFOILS

Summary of Progress in 1972-1973

and

Proposed Research for 1973-1971*

A numerical method for the analysis of three-dimensional

transonic flow past a skew supercritical wing has been devel

oped and programmed. It is based on a parabolic coordinate

system in plane sections perpendicular to the span and on a

rotated finite difference scheme that generates appropriate

artificial viscosity. Comparisons with experiment Indicate

that lift-drag ratios computed by this method are physically

realistic.

During the next year we plan to make improvements in the

accuracy and effectiveness of our three-dimensional program.

For calculations with supersonic free streams the present

first order accurate difference scheme can only give good

accuracy with a fine mesh. We wish to investigate the possi

bility of constructing difference schemes with second order

accuracy, or at least reduced truncation errors. It is also

desirable to achieve a sharper representation of the weak

shock waves on the wing, perhaps by the use of special differ

encing procedures near the shock waves. Better resolution is

needed at t*?e tips. Better resolution of the Kutta condition

is also needed when the trailing edge does not coincide with

a mesh point, so that the effects of twist can be reliably

assessed. It should be possible to use the program to make

comparisons of the merits of various shockless airfoils

- 1 -

developed by our design method, to be discussed below.

It would be desirable to move forward from the model of

a skew wing to that of a swept wing or of a wing-body combina

tion. As a preliminary step in that direction, a method for

the analysis of yawed flow past an axially symmetric body

would be desirable. In this connection there has been colla

boration with South at Langley on computations of axially

symmetric transonic flow. In the current proposal a signi- •

ficant request for purchase of computer time has been included

because of the long runs required for the study of the three-

dimensional scheme.

We are preparing a report describing a number of shockless

airfoils of interest to industry which will include extensive

off-design analyses. We have found that drag creep presents

a significant problem with some of the airfoils we developed

in the past. We intend to make available a new improved

series of profiles that will give better evidence of what

our hodograph method is able to provide. In this connection

we hope to receive visitors from industry interested in learn

ing how to make effective use of the design method. Finally,

in this report we will give an updated version of the design

and analysis programs which include a number of improvements

that have arisen in practice. An effort will be made to speed

up the program and make it more understandable.

We have developed a version of the two-dimensional

analysis program that exploits the best features of two

versions prepared earlier by Korn and Jameson. The new

version incorporates a straightforward boundary layer correc-

tion. This seems to handle boundary layer-shock wave inter

action with some success. However, there is a distinct

difficulty in formulating a correct model of the flow at the

tail for a heavily aft-loaded supercritical wing section*

While progress on this problem has already been made, we

anticipate that a significant amount of further work will

be necessary to bring the program to a satisfactory final

state. In particular, it will have a semi-empirical feature

that requires comparison with experimental data. We hope

for assistance in this respect from current testing being

planned at Langley by Whitcomb.

Research will continue on time-dependent transonic flow

studies. Progress has been made on a numerical method in

corporating the effect of entropy change across a shock

wave. However, the work has been slowed down by difficulties

with the boundary conditions, especially in the region near

the tail. We plan now to use new coordinates based on the

parabolic system that has proved to be successful in our

three-dimensional work. Also, there is the possibility of

looking into a simpler formulation of the problem based on

a potential equation in which the effects of entropy varia

tion are neglected.

- 3 -

Curriculum Vitae

Prances Bauer

BORN: July 5, 1923 .New York City, New York * . * .

EDUCATION - • .. , '

A.B. 19^3 Brooklyn CollegeSc.M. 19^5 in Applied Math, Brown UniversityPh.D. 1948 in Applied Math, Brown University

POSITIONS: . • •

Research Associate, Brown University, 19^5-48

Research Associate, Polytechnic Institute of Brooklyn,

19^9-50

Senior Mathematician, Reeves Instrument Corporation,

1950-51, 1952-61Mathematician, Bureau of Standards, contract with

American University, 1951-52.

. Research Scientist, Courant Institute of Mathematical

Sciences, New York University, 196l-present

MEMBERSHIPS: '

Sigma Xi, AMS, ACM

HONORS:

Rockefeller Foundation Fellowship, 1943-45

PUBLICATIONS: -.. •

• •• , and Reiss, E. L., On the Numerical Determination of

Shrinkage Stresses, J. Appl.-Mech. 37 (1970), pp. 123-127.

., and Reiss, E. L., Stresses in Cantilever Plates, Int.

J, of Computers and Structures 3^ (1972). '

• , Garabedian, P. R., and K6rn5 D. G., A Theory of

Supercritical Wing Sections, with Computer Programs and Examples,

Lecture Notes in Economics and Mathematical Systems, 66, Springer-

Verlag, 1972. ~

Curriculum Vitae

' Samuel Z. Burstein

BORN: October 4, 1935

Brooklyn, New York < . .

EDUCATION:

B.M.E. 1957 Polytechnic Institute of Brooklyn

M.M.E. 1958 Polytechnic Institute of Brooklyn

Ph.D. 1962 Polytechnic Institute of Brooklyn

POSITIONS:

Instructor, Polytechnic Institute of Brooklyn, 1957-62

Research Scientist, Courant Institute of Mathematical

Sciences, 1962-65

Assistant Professor, Courant Institute of Mathematical

Sciences, 1965-68

Associate Professor, Courant Institute of Mathematical

Sciences, 1968-present

HONORS:

Graduate Cum Laude •

PI TAU SIGMA

TAU BETA PI

SIGMA XI •

PROFESSIONAL ' • .

MEMBERSHIPS:

SIAM o

Assoc. Editor, Journal of Computational Physics

PUBLICATIONS: • .

, and Youdin, M., "Mathematical analysis of model for

measuring distribution of XENON-133 in the adult human brain,"

Biomedical Engineering Society, 1972.

, and Mirin, A., "Difference methods for hyperbolic

equations using space and time split difference operators of

third order accuracy," Springer-Verlag Lecture Notes in Physics

#8, Jan. 1971.

, and Mirin, A., "Third older difference methods fur

hyperbolic equations," Journal of Computational Physics, Vol. 5,

No.'3, June 19 70. > ^

, "Difference methods for the inviscid and viscous

equations of a compressible gas," Journal of Computational

Physics- 2, pp. 457-470 (1968),

S. Z. BURSTEIN

PUBLICATIONS:

""Nonlinear combustion instability in liquid propellantrocket engines - the transverse mode," National Aeronautic

and Space Administration TR 32 - 1111, Sept. 1967.

•...•..., "High order accurate difference methods in hydro

dynamics, nonlinear partial differential equations," a

symposium ed. by W. F. Ames, Academic Press, pp. 279-290,

1967.

, "High order accurate difference methods in hydro

dynamics," symposium on nonlinear partial differential

equations, University of Delaware, Dec. 1965, published by

Academic Press, 1967.

• •••.•••, 'Finite difference calculations for hydrodynamic

flows containing discontinuities,*" AEC Research and Development

Report, NYO-1480-33, Sept. 1965, Journal of Computational

Physics, pp. 198-222 (1966).

, "Numerical methods in multidimensional shocked

flows," AIAA Journal, Vol. 2, No. 12, pp. 2111-2117, Dec. 1964.

........, "Numerical calculations of multidimensional shocked

flows," Courant Institute Research and Development Report

NYO-10, 433, Nov. 1963.

, "Spray combustion model with droplet breakup:

analytical and experimental results," Vol. 6, Progress in

astronautics and rocketry, detonation and two-phase flow,

Academic Press, New York, 1962.

, "A Spray combustion model with droplet breakup:

including gas dynamic coupling,"PRL 61-12, AFOSR 433, June

1961. m '

^ .., "A spray combustion model with droplet breakup,

analytical and experimental results," ARS Preprint No.

1719-61, April 1961.

■

, "Evaporation and breakup of fuel droplets in one-

dimensional gas flow in a combustion chamber," PRL No. 60-7,

AFOSR Doc. No. Tn 60-229, Jan. 1960.

, "A spray combustion model with droplet breakup,

Proceeding of the first conference - kinetics, equilibria,

and performance of high temperature systems,""Edited by

G. S. Bahn and E. E. Zukoski, Butterworth and Co., Ltd.,

Nov. 1959. ' •■• *

, "Nonlinear theory of combustion instability:

liquid propellant rocket motors," Astia No. 209-491,

PRL 58-1, AFOSR Doc. No. 59-60, Dae. 1958.

■y

Curriculum Vitae

. ■ •

BORN:

EDUCATION:

POSITIONS:

Paul R.

August 2, 1927

Cincinnati, Ohio

A.B.

A.M.

Ph.D: • •1948

Garabedian

•

Brown

Harvard

Harvard

National Research Council Fellow, 1948-49Assistant Professor - Math, University of California,

1949-50Assistant Professor - Math, Stanford University

1950-52

Associate Professor - Math, Stanford University,

1952-56Professor - Math, Stanford University, 1956-59Professor - Math, Courant Institute of Mathematical

Sciences, New York University, 1959-present;

Director, AEC Computing & Applied Mathematics

Center, Courant Institute of Mathematical Sciences,

New York University, 1972-present.

HONORS:

American Academy of Arts & Sciences

Sloan Foundation Fellowship; 1961-63Guggenheim Fellowship, 1966

PROFESSIONAL SOCIETIES ' : ■MEMBERSHIPS:

American Math Society

Society for Industrial and Applied Math

RECENT PUBLICATIONS: ...4

......... Bauer, P., and Korn, D. G., ffA theory of supercritical

wing sections, with computer programs and examples," Lecture

Notes \n Economics and Mathematical Systems, 66, Springer-Verlag,

1972,• .

v , and Korn, D. G., "Numerical design of shockless

transonic airfoils," Actes du Congres International des

Mathematiciens, Vol. 3, pp. 95-97 (1971).

,'and Korn, D. G., "Analysis of transonic airfoils,"

Comm. Pure Appl. Math., Vol. 24, pp. 811-851 (1971).

•P. R. Garabedian ' * ' . • • " .'.

RECENT PUBLICATIONS:

•......., Kacprzynski, J. J., Ohman, L. HI and Korn, D. G.,

"Analysis of the flow past a shockless lifting airfoil in design

and off-design conditions," N.R.C. of Canada Aeronautical Report

LR-554, Ottawa, 1971.

•......., and Korn, D. G., "Numerical design of transonic airfoils",

pp. 253-269 in: Numerical Solution of Partial DifferentialEquations - II, ed. by B. Hubbard, Academic Press, 1971. .

, and Bloch, Eugene, "Solution of free boundary problems

by the method of steepest descent", p. C5-1 and C5-2 in: Proc.

of the APS Topical Conference of Numerical Simulation of Plasma,

Los Alamos, 1968. Los Alamos Report LA 3990, 1968.

, "Computer experiments with the Bieberbach conjecture,"

pp, 627-628 in: Proc. Intfl. Congress of Math., Moscow, 1966-Moscow, IZD, "MIR" 1968. . . . . •

, "The local maximum theorem for the coefficients of

univalent functions," Archive for Rational Mech. and Anal.

26:1-31 (1967).

, "An extension of Grunskyfs inequalities bearing on

the Bieberbach conjecture", Journal dfAnalyse Math., 18:81-97

(1967).

..•••••.., "Conference on Education in Applied Mathematics,"

Aspen, Colorado, 1966, Proc. of a conference sponsored bySociety for Industrial and Applied Mathematics. SIAM Review

9:289:415 (1967). . -

• • , "General solutions in the complex domain as an answer

to problems of mechanics", pp. 7-12 in v.2 of: Int'l. Symposium

on Applications and the Theory of Functions in Continuum Mechanics,

Tbilisi, 1963, Proceedings, Moscow 1965.

.. , "Asymptotic description of a free boundary at the point

of separation", pp. 111-117 in: Proc. Symposia in App..Math. App.

of non-linear partial,differential equations in mathematicalphysics Proc, Providence, R.I., American Math. Society, 1964,

p. 234. •

, "Partial differential equations", N.Y., Wiley, 1964",

;672 pp.

I........ "Proof of uniqueness by symmetrization", pp. 126-127in: Studies in Mathematical Analysis and Related Topics. Essays

in honor of George P'olya Stanford University Press, 1962, 447 pp.

, "Transonic flow behind a detached shock wave", pp. 110-

111 in: Intfl. Union of Theoretical and Applied Mechanics,

Symposium Transonicum, Aachen, September 3-7, 1962. Ed. by Klaus

Oswatitsch, Berlin, Springer, 1964, .490 pp. •

Curriculum Vitae

Antony Jameson

BORN:

POSITIONS:

HONORS:

November 20, 193^Gillingham, Kent, England

EDUCATION: B.A. )

M.A. )Ph.D.)

1955/1963 Cambridge University, England

Research Fellow, Trinity Hall, Cambridge," 1960-63Chief Mathematician and Staff Consultant, Hawker

Siddeley Dynamics Missile Division, Coventry,

1963-66Project Leader/Engineer, Grumman Aerospace, 1966-72Adjunct Associate Professor; Sr. Research Scientist,

Courant Institute of Mathematical Sciences, New

York University, 1972-present

M.A. in Engineering (first class honors)Alfred P. Sloan Foreign Post Doctoral Fellow, 1962

SELECTED PUBLICATIONS: . ■ .' . ' ■'

Articles • •" >

. Generation of Alfven Waves, J. Fluid Mechanics, Vol. 19, Part

4, pp. 513-527, 1964

Solution of the Equation AX + XB = C by Inversion of an M x M

or N x N Katrix, SIAM J. on Appl. Math., Vol. lb, No. 5, PP.

'1020-1023, 196« '■ .

The Analysis of Propeller Wing Flow Interaction, Analytic

Methods in Aircraft Aerodynamics, NASA Symposium, Proceedings

SP-228, pp. 721-71*0, October 1969 •

Comparison of Four Numerical Algorithms for Solving the Liapunoy

Matrix Equation, Int. J. of Control, Vol. II, No. 2, pp. ltil-19ti

1970 '(co-author)

*

Optimization of Linear Systems of Constrained Configuration;

Int. J, of Control, Vol." II, .No. 3, pp. 409-421, 1970

Design of a Single Input System for Specified Roots Using: Output

Feedback, IEEE Trans, on Automatic Control, Vol. AC-15, No. 3,

1970

Antony Jameson " T

, 'SELECTED PUBLICATIONS! . •qS ... ^

Articles •

' A Direct Approach to the Design of Asymptotically Optimal

Controllers, Int. J. Control, Vol. 13, No. 6, pp. 1041-1050,

1971 (co-author)

On Criteria for Closed Loop Sensitivity Reduction, J. Math.

* >Analysis and Applications (to be published, co-author)

Inverse Problem of Optimal Control, SIAM J. Control (to bepublished, co-author) " ~ . • .

Reports . . " ■ •

Radiation Patterns of Cylindrical Aerials

Hawker Siddeley Dynamics Tech. Memorandum R22il, 1966

. -Effectiveness of a Large Salvo

Hawker Siddeley Dynamics Tech. Memorandum R225, 1966

. Shocks and Kinks In Strings

Hawker Siddeley Dynamics Tech. Memorandum R226, 1966

Lift of a Body of Circular Section with Three or More Wings

tiawker Siddeley Dynamics Tech. Memorandum R227, 1966

•- • Reaction Time and the Sequential Interception of Multiple

targets, Hawker Siddeley Dynamics Tech. Memorandum R22b, 1966

. -...Optimization of Linear Systems • •

Grumman Aerodynamics Report 393-68-2, 1968

Preliminary Analysis of the Lift of a Wing; In an Elliptic

Slipstream, Grumman Aerodynamics Report 393-68-6,

Application of Optimization Techniques to the Lateral Control

of the E-2A, Grumman Report ADN 06-05-69-1, 1969 (co-author)

Some New Methods of Solving the Equation AX > XB = C With the

Aid of Similarity Transformation, Grumman Tech. Memo ENG FS

MGR 70-4, 1970 " .

On the Design of a Compensator with Dynamic Elements to Realize

»the Optimal Control and Reduce Sensitivity to Parameter Variations

Grumman Tech Memo ENG FS MGR 70-5, 1970

On the Use of Cross Product Terms in the Performance Index to

Eliminate Specified Feedbacks From a Linear Optimal Control

Grumman Tech Memo ENG FS MGR 70-6, 1970

Analysis of Wing Slipstream Flow Interaction

NASA* C.R. 1632, 1970* \

Curriculum Vltae

David G. Korn

BORN: August 28, 19*13 * •Brooklyn, New York '

EDUCATION:

B.S.. 1965 Mathematics, Rensseiaer PolytechnicInstitute

Ph.D.. 1969 Mathematics, New York University

POSITIONS:

Research Assistant, Courant Institute of. Mathematical

Sciences, 1965-69Associate Research Scientist, Courant Institute of

Mathematical Sciences, 1969-70; Research Scientist,

1970-present; Adjunct Assistant Professor, 1972-

present

HONORS: . . ; .Graduated Cum Laude * ■

Nominated to PI MU EPSILON Math Honorary

National Science Foundation Fellowship

PROFESSIONAL • •

MEMBERSHIPS:

SIAM ' •

PUBLICATIONS: . . . " ' . "

., Bauer, F., and Garabedian, P. R., ffA Theory of Super

critical Wing Sections, with Computer Programs and Examples,"

Lecture Notes in Economics and Mathematical Systems, 66, Springer-

Verlag, 1972.

, and Garabedian, P. R., "Numerical design of shockless

transonic airfoils," Actes du Congres International des

Mathfematiciens, Vol. 3, pp. 95-97 (1971). \ .

, and Garabedian, P. R., "Analysis of transonic airfoils,"

Comm. Pure Appl. Math., Vol. 21, pp.-! 841-851 (1971).

.'•/• , Kacprzynski, J. J., Ohman, L. H. and Garabedian, P. R.,ffAnalysis of the flow past a shockless lifting airfoil indesign and off-design conditions," N.R.C. of Canada Aeronautical

Report LR-554, Ottawa, 1971.

••• , and Garabedian, P. R., "Numerical design of transonicairfoils", pp. 253-269 in: Numerical Solution of PartialDifferential Equations - II, ed. by B. Hubbard, Academic Press, 1971,

• ,-"Computation of shock-free-transonic flows for airfoil

design", NYO Report NYO-1480-125.