Computer Graphics & Geometry

*S.Belotserkovsky, M.Khalenkov, V.Shvilkin
Zhukovsky Air Force Engineering Academy, Russia*

Contents

- Abstract
- 1. Form-Factors
- 1. Basic principles
- 2. Linear steady models
- 3. Linear unsteady models
- 4. Nonlinear steady model
- References

**Abstracts:** the role of computer technologies in aviation is increasing. Their realization and development is in many respects determined by the following:

reliability of calculated data;

availability and practicality of the software;

convenience in the use of calculated information, clearness of the representation of results, computer visualization, serves to the latter.

In Tupolev's aviation firm the activities in the above directions have been fulfilled from the beginning 1960's. Some results are described in the present paper.

**Keywords:** aircraft, aerodynamics, aeroelasticity, numerical methods, method of discrete vortices, visualization.

The prediction of aerodynamic loads acting on the airframe components of an aircraft is a complex multifaceted problem requiring both the usage of all available methods of modern aerodynamics and fundamental understanding of those strength analysis problems whose solution needs these loads.

Computation of loads is generally performed in early stages of an aircraft's design and serves as a chief source of information for determining cross-sectional areas of load-carrying structural elements, levels of acting stresses and providing high overall structural efficiency.

Main techniques of predicting loads are physical simulation using aircraft models instrumented with surface pressure taps and strain gauges, direct measurements in flight tests, and numerical simulation on computers.

Wind tunnel testing of pressure-tapped models allows one to obtain virtually complete information on pressure distribution over airframe components. However, laboriousness and high cost of building and testing such models restrict their usage only to the cases when the aerodynamic outlines of the aircraft under development are completely determined. Test results would be available at best by the time of the flight tests' completion or even after the aircraft's entry into service, and because of this cannot be used for design and serve mainly for checking data obtained with other means. The use of wind tunnel test data obtained for other aircraft is impossible on frequent occasions due to significant effects of peculiarities of one or another configuration on the character of the flow around it.

Loads measured in flight tests are, as a rule, indirect (forces in individual components of the structure) and it is not always possible to restore factors (pressure distribution, inertia forces, deformation) that causes these forces. As for some assembly units of an aircraft, the measurement of loads in flight testing is impossible at all. Nevertheless, nowadays, flight testing is the primary way of checking loads predicted in the phase of structure design and, in particular, for identifying errors significant from the standpoint of flight safety.

Thus, experimental methods feature a number of drawbacks: high cost, long period to get the results available, impossibility of obtaining important data associated with unsteadiness, dynamic effects, etc.

Nowadays, the rise in the cost of energy carriers, materials, equipment and the striving for shortening the time of design and testing have forced major aircraft manufacturing companies to cut the amount of experimental activities, more and more basing on computational methods and their own experience. All this leads to mathematical simulation becoming a chief tool of load prediction.

The development of market relations in Russia has generated a need for the creation of light aviation - light transport airplanes for local airlines as well as business, sport, and private aircraft. The development of such aircraft is often performed by small teams which have not sufficient material resources to carry out necessary full-scale or laboratory tests. To a greater extent, this is true for amateur-built aircraft. It follows from the aforesaid that the problem of assessment of reliability and certification of computational methods being used becomes extremely important.

During the last decades, approaches based on mathematical simulation methods and the use of computers have being enjoyed extremely wide application. The creation of a new branch of mechanics of fluid - computational aerodynamics - has been completed. A fundamental role in this process have been played by the development and introduction into practice of a whole hierarchy of mathematical models for the compressible flow around aircraft based on the method of discrete vortices (MDV) [1 - 4].

This work has been performed and continues to be performed in the following three basic directions:

- creation of general-purpose computational systems of aerodynamics adapted to supercomputers;
- development of separate software modules for most frequently addressed problems, adapted to personal computers;
- establishment of databases for certain types of aircraft.

This work is being carried out on a basis of a constantly refined approach supported by the following principles:

- reasonable sufficiency of models and computational schemes,
- consistent, well-founded transition from simple to more complex simulation of phenomena under study;
- correlation of computed data with the end results of laboratory and flight tests, i. e. the certification of approaches, scheme, and models.

Even if the development and approbation of new methods and computer software remain urgent, it follows from the aforesaid that the work on certification and standardization of already available computational schemes has been becoming first and foremost. It needs a new status, which implies that software, schemes, models must pass through various levels of certification to become officially recognized and legalized.

This process will be rather long, it should be divided into a number of stages with successive updating and expanding areas of computational schemes' application. The most natural and economical course in this direction is expected to be systematic approbation of computational methods in all those areas where there is a wealth of results from laboratory and flight tests. It is important to rationally use for the aims of certification all reliable experimental data obtained during development and test programs of both today's aircraft and those removed from service. All this valuable material could play an important role in the new stage of aerodynamics' evolution. Naturally, for this to be realized, selection and systematization of experimental results are needed.

This work is being performed by both scientific institutes and leading aircraft manufacturing companies. Briefly depicted in the present article is the experience in MDV-based computations for solving the problem on predicting aerodynamic loads at the Tupolev Aviation Scientific-Technology Complex, and an attempt was made to assess a degree of trustworthiness and applicability areas of various computational models. To determine aerodynamic loads acting on various components of an airframe, the strength department of the Tupolev Complex generally uses the following computational schemes:

- linear steady model,
- linear unsteady model,
- nonlinear steady model.

Computer visualization plays an important role in applicatins of the MDV. Various examples are represented below.

Introducing into practice the computer-based numerical solutions of the problem on compressible gas flow about a lifting surface with the help of the linear steady model represents the beginning of employing computational aerodynamics at the Tupolev Scientific-Technological Complex. Later on, this scheme was extended to flow past three-dimensional configurations.

Its underlying concepts are as follows. An aircraft under study is schematized with a set of flat basic elements (Fig. 1), on which gasdynamic singularities are distributed satisfying the continuity equation. All basic elements are arranged in such a way that they are as close as possible to corresponding elements of the aircraft, for example, to the mean surfaces of lifting components. Such components of the aircraft as the fuselage and engine nacelles are schematized with two mutually perpendicular basic planes. All the basic planes are chosen to be parallel to the longitudinal axis of the aircraft.

In the framework of the linear theory, it is assumed that all disturbances - angles of attack and slip, control surfaces' deflections, and angular rates - are small, that is they cause small changes in the freestream velocity. The boundary tangency conditions on the surface of the aircraft are transferred to the basic elements. This is conditioned by the second main assumption of the linear theory, namely the assumption of small distances between the body's surface and the basic planes compared to the length of the body in the freestream direction.

For numerically solving the problem considered, the basic surfaces and the wake behind them are substituted with a continuous vortex layer, which in turn is approximated by a set of discrete oblique horseshoe vortices consisting of a bound vortex segment and free semi-infinite vortices trailing behind the bound vortex. Increasing the number of the discrete vortices, it is possible to represent the continuous vortex layer as accurately as needed. By satisfying the boundary tangency condition at a finite number of points on the basic surfaces, we obtain a system of linear algebraic equations in unknown intensities of discrete vortices. Meeting the Chaplygin-Zhukovsky condition on the finiteness of the flow velocity at the trailing edges of the basic surfaces makes this system closed and its solution unique.

The knowledge of vortex intensities allows one, using the Zhukovsky theorem "in the small", to obtain the local excess pressure differences between the sides of the lifting surface. The basic principles of this computational scheme and its mathematical substantiation are presented in papers [3, 4]. An example of the computational scheme for the Tu-204 aircraft is given by Fig. 2.

Featuring an extreme simplicity with regard to preparation of initial data, this model has turned out to be extraordinarily universal and has allowed one to have solved a great number of problems. It provides obtaining not only global aerodynamic coefficients but also reliable data on pressure distribution over aircraft lifting surfaces with allowance for their mutual interference. Obtaining chordwise and spanwise load distribution over the wing is one of important problems in the strength analysis. A diversity of loading cases for assessment of the wing's static strength [5], a great number of design regimes in typical flights, the need in computations for a wide range of weights and flight speeds, with deflected and undeflected movable surfaces of the wing, the cases of variable sweep, etc., make this task extremely labor-intensive. Continuous searches for optimum values of the wing's parameters, which determine its aerodynamic characteristics (sweep, aspect ratio, shape and thickness-to-chord ratio of the airfoil sections, laws of geometric twist, geometry and deflection angles of high-lift devices) are required due to impossibility of using data related to other aircraft.

It is convenient to divide the problem of determining static loads on the wing into two smaller problems, one of which amounts to obtaining loads at a given deformation with allowance for the wing's fences and winglets, interference with the engine nacelles and fuselage, and another problem comes to determining deformations and their influence on loads. These two problems are addressed at the design stage both for the assessment of the wing's loading state as a whole and for answering specific questions associated with the variation of some parameters (for example, how pressure distribution will change after installation of winglets; how the jig twist of the wing should be changed to decrease bending moments; how efficient the control surfaces are in a load alleviation system, etc.).

In solving these problems, disturbances are considered as a sum of disturbances due to local angles of attack, geometric jig twist of the wing, aerodynamic twist, wing deformation in flight, and kinematic parameters of motion.

For computing general strength of relatively thin, high-aspect-ratio wings, it is often unnecessary to investigate in detail the load distribution over the airfoil contour, but sufficient to know its variation along the span and the position of pressure centers. The linear steady model described above provides a good agreement with experimental data for these parameters at moderate Mach numbers in the area of a linear relationship between local lift coefficients and angle of attack.

Figs. 3 and 4 show a comparison of the computed distribution of the load per unit length with the experimental one for the aircraft whose configuration is shown in Fig. 2. The presented are for two flight regimes with different angles of attack. The experimental data were obtained in wind tunnel testing on a pressure-tapped model. The wing had a supercritical airfoil section of complex shape. Nevertheless, the computational scheme, as can be seen, allows one to obtain reliable data on pressure centers' position.

In Fig. 5, computed bending moments due to aerodynamic loads at a number of cross sections of the Tu-204's wing, which were specified as design data based on the linear steady model, are compared with results of flight tests. The moments are given in a dimensionless form, being nondimensionalized by dynamic pressure and the characteristic length and area.

The linear steady model is indispensable in determining loads on configurations with strong interference between surfaces. An example of such a problem can be asymmetric loading of the horizontal tail on an aircraft's fin in the presence of sideslip. Fig. 6 shows the computed values of the horizontal tail's rolling moment derivative with respect to its position on the fin, and the results of measurements on a pressure-tapped model in wind tunnel testing. The horizontal tail of the model can be positioned at three foxed points along the fin's height. The nondimensional height h is equal to zero when the horizontal tail is on the fuselage with unity corresponding to the tail's position on the top of the fin.

Based on the steady model described above, a computational methodology was developed which has allowed one to determine the influence of the horizontal tail's angle of attack and its elevator's angle on the above-mentioned derivative. A comparison of the results of computation and measurement on a model of the Tu-134 aircraft with the T-shaped tail can be seen in Fig. 7. Just for such a layout, asymmetric loads on the horizontal tail determine the required strength of the fin's upper portion.

To solve problems taking into account the dynamics of structural elastic oscillations, the linear unsteady model of flow about an aircraft is used. The allowance for unsteadiness is straightforward at low Strouhal numbers. Using in this case the assumption that the Strouhal number tends to zero, the above-described steady scheme can be easily extended over the time derivatives of kinematic parameters of motion. In so doing, the dimension of the system of linear algebraic equations having no time derivatives remains independent, which allows one to look for a solution successively [6, 7].

Such an approach yields reliable results in solving principal problems associated with the dynamics of elastic aircraft:

- assessment of degrading the effectiveness of an elastic wing's control surfaces and determination of the critical aileron reversal speed;
- computation of natural frequencies and decrements of airframe's elastic oscillations as a function of flight regimes and determination of the critical flutter speed;
- computation of aerodynamic loads acting on the airframe's components during flight in disturbed air (continuous turbulence and discrete gust);
- assessment of stability margins of the loop "elastic aircraft + automatic flight control system";
- design of automatic load alleviation systems.

It is important for these problems that the MDV provides an accurate prediction of unsteady aerodynamic forces acting on the outer portions of the wing where the oscillation amplitudes are the largest.

Simultaneously updating aerodynamic forces by means of using three-dimensional schemes based on the MDV and mass-inertia forces (three-dimensional beam model on a basis of the polynomial method for low-aspect-ratio wings [8, 9]), allows one to construct adequate mathematical models for aeroelastic problems, increase the amount of parametric and optimization work in early design stages and thus decrease the volume of experiments in wind tunnels on dynamically-similar models.

The adequacy of a computational model to a real object can be demonstrated only by comparing the behavior of the real structure or its dynamically-similar model under some probatory disturbances with corresponding computed results. Impulsive deflection of control surfaces is usually used as probatory disturbances.

Figs. 8 - 10 present examples of the Tu-204 aircraft's response to an impulsive deflection of an aileron, obtained in flight testing, and corresponding calculated transient processes. Aerodynamic loads on all components were determined using the model described above at the Strouhal number approaching zero. The computational scheme is shown in Fig. 2.

The nondimensional roll rate and the bending moment for a span station of the wing (nondimensionalized by the dynamic pressure and characteristic length and area) are presented in Figs. 8 and 9, respectively. The transient alteration of the bending moment at the fuselage plane of symmetry due to an impulsive deflection of the elevator is shown in Fig. 10.

Figs. 11 and 12 compare amplitude and phase characteristics obtained by calculation and frequency wind tunnel testing of an aircraft model.

Fig. 11 shows vertical displacements of the wing tip while lateral oscillations of the rear portion of the fuselage are illustrated by Fig. 12.

It should be noted that for configurations with the engines on elastic pylons under the wing, the modes associated with large amplitudes of engine oscillations have small decrements in the whole range of operational flight speeds and are very sensitive to the smallest changes in the computational model. Dynamic interference between symmetric and antisymmetric oscillation modes begins to perform an important role leading to an asymmetry in dynamic loading of the aircraft. To take into account these factors, further refining and complication are needed in computational models used in practice. The current status of the MDV allows this to be made.

In spite of the fact that linear models provide covering a very wide class of the problems on load determination, there are a number of cases, where the accuracy provided by these models is insufficient and one is forced to reject the assumption about small disturbances. This causes the need in setting out of vortex sheets from the trailing and side edges of airframe components in flow. The location of the vortex sheets is determined by successive approximations simultaneously with updating intensities of both bound and free vortices which simulate the upper vortex layers in this scheme (Fig. 13).

Solving the problem at each step of successive approximations amounts to a system of linear algebraic equations for unknown intensities of discrete vortex segments, which is the result of imposing the boundary tangency conditions.

This nonlinear steady model was best verified in the framework of incompressible fluid [10, 11]. It is extensively applied to investigate aerodynamic characteristics of thin low-aspect-ratio wings at high angles of attack.

Consider the application of such an approach in connection with obtaining aerodynamic loads for designing the wing's structure for a specified longevity and assessing its service life. The service life of a structure is determined in many respects by minimum and maximum values of the load and by the so-called loading cycle "ground-air-ground". In the case of a dominant influence of this cycle on the general damageability, it is very important to determine these extremum values with a high degree of accuracy. For aircraft with swept, high-aspect-ratio wings, the maximums of force factors (bending moment and shear force) at a number of the wing's span stations are quite often realized at the instant of unstick during takeoff. In this case on a highly-cambered wing (due to deflected high-lift devices) at relatively high angles of attack and in the vicinity of the ground. The distance to the ground is different for stations along the wing span.

A comparison of numerical solutions of this problem in the framework of linear and nonlinear schemes with flight test data - in the form of bending moments acting at various span stations of the Tu-154's wing at the instant of unstick - is presented in Fig. 14. As can be seen, the usage of a nonlinear scheme results in a significantly higher accuracy.

The same model allows one to successfully determine loads on deflected high-lift devices and control surfaces of the wing. Fig. 15 shows computed and experimental data for the coefficient of the interceptor's normal force (the panel is positioned on the upper surface of the wing ahead of the flap). The data are given as a function of the interceptor angle. These experimental data were obtained from the actuator's forces measured in flight testing.

However, for complex multi-element flaps, errors associated with numerical methods are large. The problem of refining the computational schemes against reliable experimental data has been the focus of attention of researchers. For example, at the NASA Langley Research Center full-scale experiments are being carried out to provide data for checking numerical methods [12].

A number of problems primarily related to local strength of structural components requires the knowledge not only pressure differences across lifting surfaces but also the pressure distribution over contours of lifting surfaces' cross sections.

Solving this problem in the framework of the considered schemes with meeting the boundary tangency condition results in excessive errors, especially in the areas of the bending and trailing edges. Distribution of discrete singularities directly on the surfaces of airframe components in flow [13] requires databases associated with mathematical description of external contours of the components, high speed and high-capacity memory of computers, that is this operation can be performed only at computation centers having supercomputers.

In such a situation, a more simple and logically successive way (relative to the models considered above) of the solving the problem is seen as a method where discrete singularities are, as before, positioned on the basic surfaces and the boundary conditions imposed on the external surface are carried over, without linearization, to the basic surface normally to it. This method is named the "oblique-normal method". In this case, as opposed to a linear approach, angles of inclination of the surface to the freestream flow are not required to be small. What is more, it was shown that in solving a two-dimensional problem the method turned out to be accurate in the case of elliptical contours [14].

Nowadays, the method is being intensively introduced into computational engineering practice. Results obtained enable one to hope that it will find widespread use. Fig. 16 compares computed results with experimental data for a circular wing with elliptical airfoil sections, while Fig. 17 shows the pressure distribution at various cross sections of a high-aspect-ratio supercritical wing (Tu-204 aircraft) obtained through this computational method, together with results of wind tunnel testing of a pressure-tapped model. As may be seen, this method allows one to have rather good results. The requirements for the capabilities of the computer used are considerably more weak than in the case of using the methods where singularities are positioned directly on the body's surface.

In this article, we have shown the credibility of results obtainable with the use of the MDV when solving a number of problems on determining aerodynamic loads. The work on certification and standardization of computational methods must be continued further. If it is not conducted now, it would be difficult to expect in the future a sufficient level of flight safety for new generation aircraft.

Additional examples of computer visualization for aviation problems on the base of the MDV.

- Wake vortices past an aircraft (Al.S. Belotserkovsky) - ex.1, ex.2
- Flow evolution in open bay of an aircraft (M.I. Kanevsky) - ex.3 (648 Kb- Gif Animation)

References

- Belotserkovsky, S.V., Shipilov, S.D., Pogrebnaya, T.V. Investigation of aerodynamic derivatives of the aircraft on computers. Doklady Academii Nauk USSR, v. 34, No. 1, 1995. In Russian.
- Belotserkovsky, S.M., Lifanov, I.K. (1993). Method of Discrete Vortices. New York: CRC Press.
- Belotserkovsky, S.M. (1967). The Theory of Thin Wings in Subsonic Flow. New York: Plenum Press.
- Belotserkovsky, S.V., Skripach, B.K. (1975). Aerodynamic derivatives of the aircraft and wing at subsonic speeds. Moscow: Nauka. In Russian.
- Airworthiness Regulation for Civil Aircraft, 3d edition (1983). Moscow, Nauka. In Russian.
- Belotserkovsky, S.M., Kotovsky, V.N., Nisht, M.I., Fedorov, R.M. (1993). Two-dimensional Separated Flows. New York: CRC Press.
- Belotserkovsky, S.M., Skripach, B.K., Tabachnikov, V.G. (1971). The wing in unsteady gas flow. Moscow, Nauka. In Russian.
- Belotserkovsky, S.M. (1968) Calculation of flow past a wing of arbitrary planform over a broad range of the angle of attack. Fluid Dyn., No. 4, pp. 38-44.
- Bashkin, V.N., Chudayev B.Ya., Shvilkin, V.A. (1975). Computation of frequencies and modes of natural oscillations of an aircraft with a high-aspect-ratio wing using the initial parameters method. Trudy TsAGI, issue 1662. In Russian.
- Bunkov, V.G. Flutter analysis of a low-aspect-ratio wing with the help of the polynomial method. Trudy TsAGI, issue 905. In Russian.
- Belotserkovsky, S.M., Nisht, M.I. (1978). Separated and attached flow of a perfect fluid over the wing. Moscow, Nauka. In Russian.
- Aviation Week & Space Technology, v. 136, No. 6, February 10, 1992.
- Woodward, F.A. (1973). An improved method for aerodynamic analysis of wing-body-tail configuration in subsonic and supersonic flow. NASA CR-2228.

Shipilov, S.D. (1986). Application of the singular integral equations of the second kind to calculation of pressure on an airfoil section of moderate thickness. Proceedings of the Zhukovsky Air Force Engineering Academy, issue 1313. Moscow. In Russian.

Computer Graphics & Geometry