Fully unsteady subsonic and supersonic potential aerodynamics for complex aircraft configurations with applications to flutter

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DEPARTMENT OF AEROSPACE ENGINEERING BOSTON UNIVERSITY COLLEGE OF ENGINEERING BOSTON, MASS. 02213 FULLY UNSTEADY SUBSONIC AND SUPERSONIC POTENTIAL AERODYNAMICS FOR COMPLEX AIRCRAFT CONFIGURATIONS WITH APPLICATIONS TO FLUTTER Kadin Tseng and Luigi Morino Work supported by NASA Grant NGR 22-004-030 Technical Monitor: Dr. E. Carson Yates, Jr.

Fully Unsteady Subsonic and Supersonic Potential Aerodynamics For Complex Aircraft Configurations With Applications To Flutter by Ka Din Tseng and Luigi Morino Boston University Presented here is a new general formulation for the analysis of steady and-unsteady, subsonic and supersonic aerodynamics for complex aircraft configurations. The paper includes the theoretical formulation, the numerical procedure, the description of the program SOUSSA (Steady, Oscillatory and Unsteady, Subsonic and Supersonic Aerodynamics) and numerical results. In particular, generalized forces for fully unsteady (complex frequency) aerodynamics for a wing-body configuration, AGARD wing-tail in- terference in both subsonic and supersonic flows as well as flutter analysis results are included in the paper.

The theoretical formulation is based upon an integral equation presented in Refs. 1 and 2, which includes completely arbitrary motion. Steady and oscillatory aerodynamic flows are considered in Refs. 3 and 4 (enclosed here). A review of the problem is given in Ref. 4 and therefore is not included here.

Here small-amplitude, fully transient response in the time domain is considered. This yields the aerodynamic transfer function (Laplace transform of the fully unsteady operator) for frequency domain analysis (Ref. 5 enclosed here). This is

2.

particularly convenient for the linear systems analysis of

the

whole aircr& Pt. The Formulation briefly outlined in Ref. 5

has now been completed and implemented in the computer program SOUSSA (Ref. 6,for subsonic and supersonic).

The new formulation, program and results will be fully described in the proposed paper.

METHOD.OF SOLUTION The method presented here is based upon a formulation 1,2 For simplicity, only the incompressible developed by Morino steady state is briefly described here. The formulation, by making use of the Green function method applied to the equation of the velocity potential, yields an integral equation relat- ing the unknown potential on the surface of the body to its known normal wash. By making use of the finite-element method, and by the assumption that the potential is constant within each quadrilateral element, the integral equation is approxi- mated by a linear system of N equations relating N (unknown) values of the potential to N (known) values of normal wash at the centroids of N elements.

For the sake of generality and flexibility, in particular, for structural analysis, the downwash is expressed in terms of the generalized coordinates and generalized velocities.

From the potentials at centroids of elements, by an averaging scheme (by which the p otential at a corner is approxi- mated by the average value of potentials at the centroids of the elements in its immediate surroundings), the potentials at

3.

the nodal points are obtained and consequentially the potential at any point on the surface can be expressed by a finite-element interpolating formulation with bi-linear local shape functions.

Finally, the pressure coefficients and generalized forces can be evaluated by a simple finite-element procedure.

ASSESSMENT OF METHOD Next, an assessment of the method is briefly considered.

In particular, new unique features of the methodology (not existing in other methods) are highlighted. Also progress with respect to Ref. 4 is emphasized.

(1) The program can analyze steady, oscillatory as well as fully unsteady potential aerodynamics in both subsonic and super- sonic regimes. To the authors' knowledge this is the only computer program which can handle fully unsteady (complex frequency) aerodynamics for complex configuration (e.g., wing-body-tail combination). No other program can even handle oscillatory supersonic aerodynamics for complex configurations.

(2) Evaluation of the normal wash for complex configurations from prescribed three dimensional mode shapes (Ref. 4 was limited to thin wings with vertical displacements.) is available.

Downwash due to turbalances is also included.

(3) In supersonic flow problems, the present method does not require the use of diaphragms, in which, significantly enough, leads to the unification of the program (i.e., t

the program covers the whole linearized potential flow spectrum - steady, unsteady, subsonic and supersonic).

(Ref. 4 requires the use of diaphragms and hence is limited to simple geometries.)

(4) Finite -element evaluation of pressure. (Ref. 4 used finite-difference and was limited to thin wing wings) Evaluation of the generalized forces for arbitrary geometry (5) and arbitrary three dimensional mode shapes.

The computer code SOUSSA can handle complete wing-body- (6) tail configuration with control surfaces. Results ob- tained for control surfaces are in excellent agreement with existing ones (see next section).

(7) Another unique feature of the present method on unsteady potential flow problems in that the flutter analysis often requires the analysis on a specific geometry for a wide range of frequencies. In the present method, the frequency-dependent coefficients of the aerodynamic transfer matrix, may be expressed as a combination of complex frequency-independent coefficients* with simple frequency- dependent coefficients: the advantage is that every addi- tional frequency analyses other than the first one requires only a minimal amount of CPU time.

optimal design) (8) In iterative procedures (for instance for it is generally required to predict generalized aerodynamic loads due to a variety of vibration modes. In the present method, the aerodynamic coefficient matrix is written as the product of three matrices. The first and the third Did, F i4 , G ij , *B ii , C O W S i3 , coefficients of Ref. 5, i j enclosed here

5.

(for the normal wash and for the evaluation of the generalized forces) are mode dependent but very simple,

while the second one (relating pressure distribution

to normal wash distribution) is mode independent. By the same reasoning as above, the CPU time required for addi- tional modal analysis is reduced to a relatively negli- gible level.

Applications to flutter has been considered. The results (9) (see next section) are in good agreement with existing ones.

NUMERICAL RESULTS Typical numerical results obtained with SOUSSA are prevented in this section. Due to spare limitations, the results are only very briefly outlined.

Figures 1 and 2 are the lift and moment coefficients of a rectangular wing oscillating in pitch with Mach number ranging from 0 to 2.5. Results for the supersonic flow were obtained without the use of diaphragms and have never been presented before. The comparison against Ref. 11 is in general, in excellent agreements. Figures 3, 4 and 5 present the pressure distributions of a rectangular wing in steady subsonic and supersonic flow, and again they are in very good agreements. Figures 6, 7 and 8 are results for a wing-body configuration in both steady and fully un- steady flow, for both subsonic and supersonic speeds.

6.

6 and 7 are presented just to demonstrate ti.e Figures unique feature of the present method over all existing Figures 9 and 10 ones (i.e., fully unsteady flow). in- clude the results for simple wings with control surface In steady and oscillatory flows. Figure 11 presents flutter applications (in excellent agreement with the results of Ref. 17). Tables 1 through 3 are the generalized forces for an AGARD wing-tail configuration In quasi-steady and oscillatory flow in comparison with existing methods.

Further results, such as the fully unsteady aerodynamic analysis of the AGARD wing-tail configuration and other cc:plex configuration (with control surfaces) will be Included in the proposed paper.

In conclusion, whereas only simple configuration results are presented, (in order to assess the accuracy),It is the objective of the proposed paper to emphasize the generality, flexibility, efficiency of the present method. Last, but not least the present method provides a unified approach to cover the whole linearized potential flow spectrum and very

limited human intervention is required in using the computer

code SOUSSA.

7.

CONCLUSIONS There exists several methods to analyze the problem of wing-body, wing-tail interactions. However, it is apparent that the present method, embedded in the computer program SOUSSA, is unique in the following aspects: 1.

It provides a unified approach for steady, oscillatory and fully unsteady, subsonic and supersonic aerodynamic flows.

2. It can be applied to arbitrarily-comp",x configurations.

Wing-body-tail configurations with control surface have been analyzed. (No existing result is available for comparisons. However, simple wing with control surface results shows that the present method is in good agree- ment with existing ones.)

3. It is computationally extremely general, flexible, ef- elimination of ficient and above all, accurate. The diaphragms in supersonic flow improved considerably the simplicity and efficiency of the code.

SOUSSA is the only existing program that can analyze fully 4.

configuration potential aerodynamics in

unsteady complex- subsonic or supersonic regimes. It is also the only oscillatory supersonic aero-

program capable of handling

dynamics for complex configurations.

instances In contrast to existing methods, which in many 5.

requires extensive user's background in aerodynamics and familiarity with the specific method, the present.

8.

.code requires very limited human itervention and is extremely easy to use.

6. Flutter, and optimal design analyses requires evaluation of the aerodynamic influence coefficients for several frequencies and mode shapes. With the unique features mentioned above, the computer time that normally would have been required is dramatically reduced. This is to be added to the fact that preliminary versions of the program already required less computer time than other existing programs (Ref. 4) .

7. ;applications to flutter indicate good agreement with existing results.

References 9.

1. Morino, L. "Unsteady Compressible Potential Flour around Lifting Bodies Having Arbitrary Shapes and Motions", Boston University, College of Engineering, Dept. of Aerospace Engineering, TR-72-01, June 1972.

Superseded by "A General Theory of Unsteady Compressible Potential Aerodynamics", NASA CR-2464. December 1974 2. Morino, L."Unsteady Compressible Potential Flow Around Lifting Bodies - General Theory". AIAA Paper No. 73-196, January 1973.

3. Morino, L. and Kuo, C.C. "Subsonic Potential Aerodynamics for Complex Configurations: A General Theory". AIAA J., Val. 12, No. 2, February 1974, pp. 191-197.

4. Morino, L., Chen, L.T. and Suciu, E.O. "Steady and Oscillatory, Subsonic and Supersonic Aerodynamics Around Complex Configurations" AIAA J., Vol. 13, No. 3, March 1975, pp. 368-374 5. Morino, L. "Subsonic and Supersonic Indicial Aerodynamics and Aerodynamic Transfer Function for Complex Configurations" Boston University, ENG-TN-74-01, September 1974.

Ste ad y ,Oscil- 6. Teeng, K.D., Chen, L.T. and Morino, L. "SOUSSA : latory and Unsteady, Subsonic and Supersonic Aerodynamics for Aerospace Complex Transportation System; A User's Manual" Boston University, ENG-TR-75-03 7. Laschka, B. "Zur Theo , i a der Harmonisch Schwingenden Tragenden Flache bei Unterschallenstromung", Zeitschrift fur Flugwissenschaften, 11 (1963), Heft 7, pp. 265-292.

8. Labrujire, T.E., Loeve, W. and Sloff, J.W. "An Approximate method for the Calculation of the Pressure Distribution on Wing-Body Combinations at Subcritical Speeds", AGARD .specialist Meeting on Aerodynamic Interference, Silver Springs, Md., Sept. 1970, AGARD Conf. Proc. No. 71 9. Leasing, H.C., Troutman, J.C. and Menees,G.P. "Experi- mental Determination of the Pressure Distribution on a Rectangular Wing Oscillating in the First Bending Mode for Mach numbers From 0.24 to 1.30", NASA-TN-D-344, 1960.

10.

I.C. Tijdeman, H. and Zwaan, K.J. "Unsteady Aerodynamics For Wings with Control Surfaces" No. 12, AGARD Symposium on Unsteady Aerodynamics for Aeroelastic Analyses of Interfering Surfaces, AGARD-CP-80-71, 1970 11. Huttsell, L.J., Pollock, S.J. "Unsteady Aerodynamic Loads for the AGARD Interfering Lifting Surfaces" AFFDL paper, 1974.

Rodden, W.P., Giesing, J.P. and Kalman, T.P. "New 12.

Developments and Applications of the Subsonic Doublet- Lattice Method for Nonplanar Configurations" Paper No. 4, AGARD Symposium on Unsteady Aerodynamics for Aeroelastic Analyses of Interfacing Surfaces, AGARD-CP-80-71, 1970.

Schmid, H. and Bechen, J. "Contribution to the AGARD 13.

Program on Unsteady Aerodynamics for Interfering Lifting Surfaces." MBB Paper, 1973.

14. Mykytow, W.J., Olsen, J.J. and Pollock, S.J. "Application of AFFDL Unsteady Cord Prediction Methods to Interfering Surfaces", Paper No. 7, AGARD Symposium in Unsteady Aerodynamics for Aeroelastic Analyses of Interfacing Surfaces, AGARD-CP-80-71, 1970.

15. Davies, D.E. "Applications of Unsteady Airforce Calcula- tion Methods to AGARD Interfering Surfaces.' AGARD-CP-80- 71, 1970.

16. Pollock, S.J. and Huttsell, L.J. "Applications of three Unsteady Aerodynamic Land Prediction methods" AFFDL TR-73-147, May 1974.

17. Appa, K. -'Integrated Potential Formulation. of Unsteady Supersonic Aerodynamics for Interacting Wings" NASA CR-132547, Oct. 1974.

Appa, K. and Jones, W.P. "Into;:eted Potential Formula- 18.

tion of Unsteady Aerodynamic; for Interacting Wings" AIAA paper No. 75-762.

Hammond A.D. and Keffer, i^.M. "The effect at high sub- 19.

sonic speeds cf a flap-type aileron on the chordwise pressure distribution near mid-cemispan of a tapered swept back wir_g of aspect ratio 4 having NACA 35 0 65AO06 airfoil section", NACA RM L53C23.

20, Bisplinghoff, R.L. Ashley, H. and Halfman, R.L.

"Aeroelasticity", Addison-Wesley, Reading, Mass., 1955 21. Davies, D.E. "Calculation of Generalized Airforces on Two Parallel Lifting Surfaces Oscillating Harmonically in Subsonic Flow, RAE Technical Report 72180, 1973.

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