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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 "~dithout restriction.

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RESEARCH PROPOSAL SUBMITTED TO THE NATIONAL AERONAUTICS AlJD SPACE ADMINISTRATION

BY

NE1v YORK UNIVERSITY COURANT INSTITUTE OF MATHEMATICAL SCIENCES

251 MERCER STREET NEvi YORK, NEW YORK 10012

NUl~RICAL DESIGN OF TRANSONIC SHOCKLESS AIRFOILS

Principal Investigator Name: Paul R. Garabedian Title: Professor Social Security # 565-44-9568

Renewal of NGR-33-016-167 (NYU ACCT. # 0-5743-040)

Starting Date:Sept. 1, 1973 Duration:l2 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 Name: Paul R. Garabedian Title: Professor Signature: ------------------------Telephone: 460-7359 March 26, 1973

Director Peter D. Lax

460-7421 March 26, 1973

.-

Continu·ation of Grant NGR--'33-016-167

Proposed Budget

September 1, 1973 to August 31, 197~

SALARIES:

Professor Paul R. Garabedian (Principal Investigator) 100% 1 Summer Month

·· Associate Professor Samuel Burstein 100% 2 Summer Months

Antony Jameson Sr. Research Scientist 66~2/3% 12 Months

David Korn Research Scientist 30%

0

12 Months

Frances Bauer Research Scientist 50% 12 Months

1 Research Assistant 100% 12 Months

1 Secretary 25% 12 Months

TOTAL SALARIES

OVERHEAD (66% of salaries)

FRINGE BENEFITS (14.5% of salaries)

COMPUTER TIME

TRAVEL

PUBLICATIONS

MISCELLANEOUS (Communications, supplies, etc.)

TOTAL

$ 3,889

!1,020

16,333

6,000

8,750~

5,000

.,.,. ....... -a, 750

$ 46,742 0

30,850*

6,778

12,000

2,000

1,500

•

0 130

$ 100,000

*The above overhead rate does not include 4% of cost sharing or $1,870 representing the university's contribution.

' ·J '·j 0~

NUMERICAL DESIGN OF TRANSONIC SHOCKLESS AIRFOILS

Summary of Progress in 1972-1973 and

_Proposed Research for 1973-1974

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. Compariso~s 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 s'treams 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 wayes 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 conditio&!

is also needed when the tra~ling 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 or 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-

- 2 -

:f.

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

tor assistance in this respect from current testing being

planned at Langley by Whitcomb.

Research will continue on time-dependent transonic flow

~tudies. 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

loo~ing into a simpler formulation of the problem based on

a potential equation in which the effects of entropy varia­

tion are neglected.

- 3 -

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.. BORN:

-EDUCATION

POSITIONS:

MEMBERSHIPS:

HONORS:

Curriculum Vitae Frances Bauer

July 5, 1923 New York City, New York

Brooklyn College

\

A.B. Sc.M. ·pil.D.

1943 19lt5 1948

in Applied Math, Brown University in Applied Math, Brown University

Research Associate, Brown University; 19lt5-lt8 Research Associate, Polytechn~c Institute of Brooklyn,

1949-50 Senior Mathematician, Reeves Instrument Corporation,

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

American University, 1951-52 .· Research Scientist, Courant Institute of Mathematical

Sciences, New York University, 1961-present

_Sigma Xi, AMS, ACM

Rockefeller Foundation Fellowship, 1943-45

·.PUBLICATIONS:

•••••••• , and Reiss, E. L., On the Numerical Determination of Shrinkage Stresses, J. App1. Mech. 37 (1970), pp. 123-127 •

•••••••• , and Reiss, E. L., Stresses in Cantilever Plates, Int~ J. of Computers and Structures 1 (1972).

~ ••••••• , Garabedian, P. R., and Korn, D. G., A Theory of .Supercritical Wing Sections, with Computer Programs and Examp~es, Lecture Note~ in Economics and Mathematical Systems, 66, Springer-Verlag, 1972. -

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· Curriculum vi·t·ae ; · s·amuel· Z. Burstein

\

:BORN: October 4, 1935 .Brooklyn, .New York

.EDUCATION:

POSIT~ONS:

HONORS:

B.M.E. M.M.E. Ph.D.

1957 1958 1962

; .•

Polytechnic Institute of Brooklyn Polytechnic Institute of Brooklyn Polytechnic Ins~itute of Brooklyn

Instructor, Polytechnic Institute of Brooklyn, 1957-62 Research Scientist, Courant Institute of Mathematical ·

Sciences, 1962-65 Assistant·Professor, Courant Institute of Mathe~atical

Sciences, 1965-68 .Associate Professor, Courant Institute of Mathematical

Sciences, 19~8-present

Graduate Cum Laude· PI TAU SIG.[\1A TAU BETA PI SIGMA XI

PROFESSIONAL MEl-IDERSHIPS:

.·SIMi t)

Assoc. Editor, Journal of Comp~tational Physics

. "' PUBLICATIONS : •••••••• , and Youdin, M., "Mathematical analysis ·of model for measuring distribution of XENON-133 in the adult human brain, .. ·Biomedical E~gineering 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 iS, Jan. 1971 .

•••.•••• ,and Mirin, A., "Third o~der difference methods fur hyperbolic equations," Journal of Computational Physics, Vol. 5, No.· 3, June 1970 •

••••••.. , "Difference methods for the inviscid and viscous equations of a compressible gas, .. Journal of Computational Physics· ·2, PP~ 457-4 70 (1968) .

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S. Z. BU~STEIN ·PUBLICATIONS:

.·

•••••••• , "Nonlinear combustion instability in liquid propellant rocket 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

.. · syrilpo_sium ed. by W. F. Ames, Academic Press, pp. 279-290, 1967. .

•••••••• , "High order accurate difference methods in hydro­dynamics," 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, NY0-1480-33, Sept. 1965 ,. Journal of ·computational Physics, pp. 198-222 (1966).

• ••••••• , "Numerical methods in multidimensional shocked flows," AIAA Journal, Vol. 2, No. 12, pp. 21+,1~2117, Dec. 1964 •

• ••• · •••• , "Numerical calculations of multidimensional shocked flO\'lS ," Courant Institute Research and Development Report NY0-10, 433, Nov. 1963 •

•••••••• , 11 Spray combustion model with droplet breakup:

analytical and experimental results," Vol. 6, Progress in astronautics and rocketry, detonation and two-phase flow, Aca~emic Press, New York, 1962 •

•••••••• , "A Spray combustion model with droplet breakup: including· gas dynamic coupling, "P_RL ~1-12, AFOSR 433, June 1961.

"'• ••••••• , 11 A spray combustion model with droplet breakup,

analytical and experimental results," ARS Preprint No. 1{19-61, April 1961 •

• • • • • • • • , "Evaporation and breakup of fuel· droplets in one·­dimensional gas flo\V' in a combustion chamber," PRL No. 60-7,

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

•••••••• , "A spray combustion model with droplet breakup, ProceediPg of the first conferenc~ -kinetics, equilibria, and performance of high temperature systems,rr-Edited by

. G. s. Bahn and E. E. Zukoski, But~erworth and Co., Ltd., Nov. 1959. · ..

•••• ~ ••• , 11Nonlinear theory of combustion instability:

liquid propellant rocket motors," Astia No. 209-491, PRL 58-1, AFOSR Doc. No. 59-60, ·nee. 1958 •

..

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. '

Curriculum Vitae Paul R. Garabedian

BORN: August 2, 1927 Cincinnati, Ohio

EDUCATION: ·A.B. . A.M.

Ph.D;

POSITIONS:

1946 1947

·19!&8.

Brown Harvard Harvard

. . . .. · · ...

National Research Council Fellow, 19~8-49

\

,

Assistant Professor Math, University of California, 19!&9-50

HONORS:

Assistant Professor Math, Stanford University 1950-52

Associate Professor Math, Stanford University, 1952-56

Professor Math, Stanford University, 1956-59 Professor 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.

American Academy of·Arts & Sciences Sloan Foundation Fellowship; 1961-63 Guggenheim Fellowship, 1966

PROFESSIONAL SOCIETIES ·MEMBERSHIPS: ~ American Math Society

Society for Industrial and Applied Math

RECENT PUBLICATIONS:

. -~······~'Bauer, F., and Korn, b. ~., "A theory of supercritical ···wing sections, with computer progran&tJ and examples.," Lecture .. · Notes .~n Economics and Mathematical Systems, 66, Springer-Verlag,

1972 • •

~ ••••••• ,and Korn, D. o., "Numerical desi~n of shockless transonic airfoils," Actes du Cont:rres International des Mathematiciens, Vol. 3, pp. 95-97 (1971) .

. . . II · •••••••• , and Korn, D. 0., Analysis of transonic airfoils, 11

Comm. Pure Appl. r~ath., Vol·. 24, pp. 841:-851 (1971) •

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·P. ·R •. Garabedian RECENT PUBLICATIONS:

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· .••••• ~ •• , Kacprzynski, ·J. ·J,;; Ohman, L. H: 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· Differential Eq~ations - II, ed. by B. Hubbard, Academic Press, 1971.

. .

• ••••••• , and Bloch, Eugene, "Solution of free boundary _problems by the method of steepest descent", p. C5-l and C5-2 in: Proc. of the APS Topical Conference ·of Numerical Simulation of Plasma, ~6s Alamos, 1968. Los Alamos Report LA 3990, 1968 •

· · .•••••••• , 11 Computer experiments with the Bieberbach _conjecture,"

· pp. 627-628 in: Proc. Int'l. Congress of Math., Moscow, 1966~ . Moscow, IZD, "MIR" 1968.

••••••• .", 11The local maximum theorem for the coefficients of

·univalent functions," Archive for Rational Mech. and Anal.· 26 : 1-31 ( 19 6 7) • . . . :

,J •••••••• , 11An extension of Grunsky's inequalities bearin~ on

the Bieberbach conjecture", Journal d'Analyse Math., 18:81-97 ... (1967). . . . .

. ' •.••••••• , ''Conference on Education in Applied Mathematics," Aspen, Colorado, 1966, Proc. of .a conference sponsored by Society for Industrial and Applied Mathematics. SIAM Review 9:289:415 (1967). . . .

•••••• .-., "General solutions in the complex domain as an-answer to problems of mechanics 11 ~ pp. 7-12 in v.2 of: Int'1. Symposium on Applications and the Theory of Functions in Continuum Mechanics, Tbilisi, 1963, P~oceedings, Moscow. 1965.

~.· •••••• , "Asymptotic ~escription of a free boundary at the point pr separation", pp. 111-117 in: Proc. Symposia in App:.Math. App • . or non-1tnear partial_differential_equations in mathematical

·physics Proc., Providence, R.I., American Math. Society, 1964, .P~ 2311. ·.

. . . . . . . . ' ·672 pp. . .

11Partial differential equations", N.Y., Wiley, 196lr,

••• ~~ ••• , "Proof of uniqueness by· ~ymrnetrization", pp. 126-127 in: 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: Int'l. Union of Theoretical and Applied Mechanics, Symposium Transonicum, Aachen, September 3-7, 1962. Ed. by Klaus

.. • Oswatitsch, B~rlin, Springer, 196lt, .490 pp. ·

:~

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·BORN:

EDUCATION:

POSITIONS:

HONORS:

'~

'-:..t . ''\ ' ..

. ·. .. . . ..... .

CUrriculum. Vitae ...

Antony Jameson

November 20, 1934 Gillingham~ Kent, England

: ....... ·

·. . ~ . . . ··.

1955/1963 Cambridge Univ.~rsity, Engl~nd

... :.: .....

B.A. ) M.A. ) ·Ph.D.) ' ..•... . . ... ' ....... ~.i· .. ' ....... ·· ' ~ /. -~'f..l

Research Fellow, Trinity· Hall, Cambridge,.l960-63 . Chief Mathematician and Starr Consultant, Hawker . · Siddeley Dynamics Missile Division, Coventry,

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

Courant Institute of Mathematical Sciences, New Y~rk University, 1972-pre~ent

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

.. :-. :_.,

. SELECTED PUBLICATIONS:

Articles

Generation of Alfven \tlaves, J. Flu·id 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 ~atrix, SIAM J. on Appl. Math., Vol. 16, No. 5, pp.

··1020-1023' 1968

The Analysis of Propeller Hinp; Flo'lr Interaction, Analytic ·_Methods in Aircraft Aerodynamics, NASA Symposium,. Proceedings SP-228, pp. 721-7~Q, October 1969 ·

Comparison of Four Numerical Al~orithms for Salvin~ the Liapunov Matrix Equation, Int. J. of Control, Vol. II, No. 2, pp. 181-198,

· . 1.970 ·(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 Usin~ Output Feedback, IEEE Trans. on Automatic Control, Vol. AC-15, No. 3,

. 1970

'·

;

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. · . . 'Antony Jameson

.· ·sELECTED PUBLICATIO~S!

·Articles . . \

A Direct Aooroach 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 be published, co-author) . . .

Reports

Radiation Patterns of Cylindrical Aerials Hawker Siddeley Dynamics Tech. t'iemorandum R224 ,- 1966

..

· _ · Effect 1 venes s of a Larp:e Sal v.o . .- :

Hawker Siddeley Dynamics Tech. rv1emorandum 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 Hawker Siddeley Dynamics Tech. Memorandum R227, 1966

·: .· Reaction Time and the Seauential Interceotion of rlfultiole Targets, Hawker Siddeley ~Dynamics Tech. i•Iemorandum R22'8, 1966

-.-Optimization of Linear Systems Grumman Aerodynamics Report 393~68-2, 1968· ...

Preliminary Analysis of the Lift of a Win~ in an Elliptic -~lipstream, Grumman Aerodyna~ics Report 393-68-6, 1968

Application of Optimization Techniaues to the Lateral Control of the E-2A, Grumman Report ADN 06~p5-69-l, 1969 (co-author)

Some Ne\'1 Methods oP Solving the Equation AX·+· XB = C \'lith the ... Aid of Similarity Transformation, Grumman Tech. Memo ENG FS

MGR ~0-4, 1970 . - .. . . On tlie Desig;n of a Compensator l·ti-th Dynamic Elements to Realize

- -l.o....-~·--

.the Optimal Control and Reduce Se~sitivity to Parameter~~iar!ations ~rumman Tech Nemo ENG FS MGR 70-5, 1970

On the Use of Cross Product Terms in the Performance Index to Eliminate Specified Feedbacks From a Linear Dptimal Control _Grumman Tech Nemo ENG FS I·~GR 79-6, 1970

Analysis of Wing Slipstream Flow Inte~action NASA C.R. 1632, 1970

··- - .,.

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BOJUI:

EDUCATION:

POSITIONS:

HONORS:

... Curriculum Vitae . . David G. Korn

August 28; 1943 Brooklyn, New York

'.J• ...

B.s... ~965

-Ph.D.. 1969

Mathematics, Rensseiaer Polytechnic Institute

Mathematics, New York University . .

·Research Assistant, Courant Institute of.Mathematical . · .Sciences, 1965-69 Associate Research Scientist, Courant Institute of

Mathematical Sciences, 1969-70; Research Scientist, 1970-present; Adjunct Assis~ant Professor, 1972-present

Graduated Cum Laude Nominated to PI MU EPSILON Math Honorary National Science Foundation Fellowship

PROFESSIONAL MEMBERSHIPS:

SIAM

PUBLICATIONS:

. - " ••••••• :, Bauer, F., and Garabedian, P. R., A Theory of Super-critical Wing Sections, l"ti th Computer Programs and Examples,"

· Lecture ·Notes in Economics and Mathematical Systems, 66, Springer­Verlag, 1972 •

•••••••• , and Garabedian, P. R., "Numerical design of shockless transonic airfoils," Actes du Cone;res International ·des

. Math~maticiens, Vol. 3, pp. 95-97 (1971). · . •••••••• , and Garabedian, P. R., "Analysis of transonic airfoils," Comm~ Pure Appl. Math., Vol. 211, pp.·· 8111-851 (1971) •

. :: •••••• , Ka~przynski, J. J., Ohman, L. H.·and Garabedian, P.-R., ''Analysis of the flow past a shockless lifting airfoil in design and off-design conditioni," N.R.C. of Canada Aeronautical Rep~rt LR-5511, Ottawa, 1971.

.. . •••••••• ,and Garabedian, P. R., "Numerical design of transonic

.airfoils", pp. 253-269 in: Numerical Solution of Partial Differential Equations - II, ed. by B. Hubbai-d, Academic Press, 1971.

~ ••••••• ,·"Computation of shock-free-transonic flows for airfoil design", NYO Report NY0-11180-125. Octoh~,... 1 o~o ·

•.