COMPUTATlON OF STEADY AND UNSTEADY CONTROL SURFACE LOADS

IN TRANSONIC FLOW

Bala K. Bharadvaj*

Douglas Aircraft Company McDonnell Douglas Corporation

Long Beach, California

Abstract

A computational procedure based on the transonic full potential equation for the analy- sis of loads due to steady and oscillatory control surfaces is presented here. The con- trol surface deflection is modelled using an equivalent body velocity approach without modifying the grid. Viscous effects, including rnild separation, are modelled using an inter- active inverse boundary layer and the transpiration velocity approach. R e s ~ ~ l t s are presented for a fighter wing and a transport aircraft wing configuration, for control sur- faces located at the trailing edge and the leading edge, for steady as well as oscillatory deflections.

Introduction

The accurate prediction of unsteady con- trol surface loads is becoming increasingly important because of recent developments in the technology for flutter suppression, load alleviation, and improving aircraft perforni- ance and stability. Efficient analysis methods are needed for the accurate prediction of steady and unsteady aerodynamic loads due to control surfaces. In transonic flow, this is a challenging problem because of the strong influence of viscosity and possible shock- boundary layer interactions including sepa- ration.

In principle, this problem can be ad- dressed by computational methods based on the Navier-Stokes equations. However, Navier-Stokes methods available at the pres- ent time require computers with very large storage and use up a great deal of cornputa- tional time. Moreover, these methods are not always robust and require expert users mak-

* Principal Engineer Scientist, Analysis Technology Group, Aircraft Structure Center of Excellence, Member AlAA

ing them inappropriate for routine analysis of this important problem. Simpler techniques based on the lifting surface formulation are available but are restricted to purely subsonic flows and often yield results that have large '

errors in the control surface hinge moments since effects of viscosity and separation are not modelled. Transonic small-disturbance analysis has been used in the past (Ref. 1) but the accuracy of this approach is questionable for thick supercritical wings and at leading edges of thin wings because of the inherent limitations of the theory used.

For many practical problems, inviscid methods that account for viscous effects through interactive boundary layers provide a very efficient alternative to solution of the full Navier-Stokes equations. For two dimen- sional problems, it has been shown that inviscid flow analysis with an interactive boundary layer yields results comparable in accuracy to those obtained from Thin-Layer Navier-Stokes analysis (Ref. 2) even for cases with laminar separation bubbles, provided the regions of flow separation are not very large. For steady three dimensional flows, the use of viscous corrections to inviscid analysis based on interactive boundary layer approach has been shown to improve the results signif- icalilly (see e.g., Refs. 3, 4). Similar results have been obtained by the present author for wings oscillating in pitch.

This paper presents an efficient method for the analysis of steady and unsteady control surface loads using a three dimensional com- pr~tational. rnethod based on solution of the transonic full-potential equation using the strongly irriplicit procedure used previously by Malorie arid Sarikar (Ref. 5). Corrections for viscous effects, including mild separation,

1349 Copyright 0 1990 by McDonnell Douglas Corporation Published by the American Institule of Aeror;autics and Astronautics, Inc with permission

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are included through an interactive inverse boundary layer formulation due to Cebeci, et. al. (Refs. 6, 7). Correlations have been per- formed for a thin wing fighter configuration (the F5 wing) as well as for a high aspect ratio wing (HARW) with thick supercritical wing sections. The present paper is believed to be the first effort to correlate the test data ob- tained 'for the HARW using transonic compu- tational methods. Results have been obtained both for trailing edge arid leading edge control surfaces by the present method and are com- pared with experimental data as well as with a lifting surface analysis based on Ref. 8.

Formulation

The forniulation used in the analysis is briefly explained below.

Governing Equations: The fundamental governing equation used here is that of mass conservation. _&The velocity potential m, is de- fined (V@ = V) and a body fitted curvilinear coordinate system introduced through the transformatiori 5 = [ ( x , y , z ) ; y = I ? ( x , ~ , z ) ; = [ (x ,y ,z ) and T = t. When combined with

the principle of conservation of energy and the isentropic relations, the governing

In the standard solution procedure, the coefficients of the matrix [ A ] and the residue are evaluated at a given time step based on values at the previoi~s time step, and the equations solved for A& using a strongly irn- plicit approximate factorization scheme due to Stone (Ref. 9). As the solution marches in tirne, the computed A+ is used to update the velocity potential at every time step. In the analysisvsed here, this standard procedure is modified to improve the time accuracy by using with Newton iterations: several iter- ations are performed at each time step and successively more accurate estimates of A$ are obtained by modifying the matrix [A ] as well as the residue [R] based on data at the previous iteration of the current time step.

Boundary Conditions: At the far field upstream boundary, the perturbation potential d) is set to zero. At the downstream boundary,

is zero at subsonic points; at supersonic points, the potential is obtained by propa- gation of d) frorn the nearest non-boundary point. At the body surface, the normal com- ponerit of the relative velocity between the body and fluid is set to zero for the inviscid case.

equation for the velocity potential may be pressed as

where, p is the density, U, V and W are the contravariant velocity components and J is the Jacobian of the transformation. In the analysis, a perturbation potential q!! = @ - x cos a, - z sin a. (where a, is the an- gle of attack of the free stream) is introduced and the equatiori for the velocity potential ex- pressed in conservation form. Then the equation is time linearized and discretized in space using central differences along with appropriate ~ ~ p w i n d bias in regions of super- sonic flow. Backward differences are used for the time derivatives. This leads to a set of equations for Ad) at each time step which can be expressed in matrix form as [A ] {Ad)] = { R ) where [ A ] is a matrix with seven di-

agonals and consists of terms containing p , U, V, W, etc. and ( R ) is the residue which is Ihe discretized form of the continuity eq~~at io l i .

ex- The location of the wake is prescribed in the analysis. The wake surface is tangential to the wing surface at the trailing edge, and extends to the dowristrearn boundary where it becomes parallel to the free stream. A po- tential jump is imposed across the wake; its value i's related to the discontinuity at the trailing edge and is convected along the wake.

Control Surface Modelling: Strictly speaking, when a control surface is deflected, it results in a modification of the geometry of the original wing surface. For two dimen- sional steady flow problems, this can be ac- commodated easily by modifying the grid and applying the standard boundary conditions as has been done in Ref. 10. In real life three dimensional problems, the control surfaces typically extend only over part of the wing span. A deflection will not only cause changes in the section geometry at span lo- cations including the control surface, but also create discontinuities in the leading edge and/or the trailing edge at the inboard and outboard sections of the control surface. This introduces additional complexity in the grid generation needed to model the wing wit11 a deflected control surface.

Moreover, when the control surface de- flection is a function of time, the section ge- ometry will also change with time and require a time-dependent grid system in order to rnodel the effects accurately. This increases the complexity level of the problem consider- ably and further magnifies the computational effort necessary to solve the problem.

In the interest of keeping the analysis simple and inexpensive, these issues of grid generation and motion have been circum- vented by modeling the effect of the control surface approximately. This is done by gen- erating the C-grid used in the potential flow solution for the wing with no control surface deflection and including the effect of the con- trol surface deflection through a modification of the boundary conditions, as explained be- low.

For the case of the control surface with a deflection 6 (positive for trailing edge down) the tangency boundary condition is linearized and replaced by an equivalent downward body velocity (plunge) of 6 U, for the region of the control surface.

For non-steady control surface deflection with angular velocity h(t) the motion of the control surface is linearized to yield an effec- tive downward velocity of (x - xhi,,,) d(t).

Combining the contributions of the aileron deflection arid motion yields a total equivalent downward velocity of

Viscous Effects: The effects of viscosity are included in the potential flow analysis by the transpiration or blowing velocity concept. The boundary layer over the wing, includirig any region of separation, is computed using a quasi three dimensional approach, i.e., treat- ing each of the computational span stations as being locally two dimensional. An equiv- alent blowing velocity v, is obtained and used as a correction to the normal boundary condi-

the cornbination of the potential flow and the boundary layer converges to a final steady flow solution.

For sinusoidal motion of the control sur- face, three levels of approximation can be im- plemented to include the viscous effects. The most elementary option is to use the 'frozen boundary layer' approach in which the boundary layer computed for the converged steady flow problem at the mean control sur- face deflection is kept frozen for the unsteady analysis. This option has the least computa- tional cost, however, it completely ignores any changes in the viscous effects due to the oscillation of the control surface. The next level of complexity is to compute the con- verged steady flow boundary layer at several deflected positions of the control surface that would be realized during the oscillation and use this data-base of viscous corrections to estimate the viscous effects during the analy- sis of the unsteady problem by interpolation. This quasi-steady analysis using the 'interpo- lated boundary layer' accounts for changes in the viscous effects due to variations in the geometry of the problem, but ignores any phase effects. In the next higher level of complexity, the boundary layer is computed at every time step used to compute the inviscid solution. This includes the correct viscous effects due to geometry changes as well as the phase effects. All the three tech- niques discussed above have been investi- gated numerically in this study and are discussed below.

Numerical Implementation

The formulation described above has '

been incorporated into a computer code TUFPAL (Transonic Unsteady Full Potential Aeroelastic Loads program) and results ob- tained for a variety of steady and unsteady flow problems. Some of the specific issues relating to the numerical implementation of the above formulation to the analysis of con- trol surface loads are discussed below.

tion to account for the addition or subtraction of mass flux through the surface to simulate Grid Generation: The grid used for the

the growth or decay of the displacement analysis is a sheared parabolic C-grid system.

thickness on the wing surface; any viscous I"rder to improve resolution of the compu-

effects on the wake are not accounted for. tation over the control surface, the grid is clustered near the leading edge or the trailing

For steady flows, the boundary layer is edge depending on the chordwise location of updated interactively with the potential flow the control surface being modelled. Several solution. As the time marching pr-ogresses, such sections are stacked spanwise to model

the three dimensional wing. The span stations are chosen such that the inboard and outboard boundaries of the control surface are located in between successive computational stations.

Analysis Procedure: Typically, the un- steady analysis is done in two phases. In the first phase, the converged steady flow sol- ution is obtained for the given free stream Mach number and incidence at the mean po- sition of the control surface. The time step used for this computation is chosen to opti- mize convergence to the steady flow solution. In the second phase, the unsteady solution is obtained starting from the converged steady flow solution. For this analysis, the time step is based on physical considerations such as number of time steps per cycle of oscillation. The total number of time steps needed for the analysis is determined by monitoring the re- sponse of the unsteady aerodynamic loads in time; the normal procedure is to continue the solution until the aerodynamic loads become periodic in time. It has been found from ex- perience that two complete cycles of sinusoidal control surface motion are ade- quate for the solution to reach a steady state periodic condition. The computations for the unsteady flow were made typically for two complete cycles of oscillation, with 200 time steps per cycle.

The unsteady results presented in this paper are limited to sinusoidal motion of the control surfaces. Unsteady pressure data is presented in the form of real and imaginary parts (F5 wing), or magnitude and phase dis- tributions for the (HARW), to match with the available experimental data. Numerical val- ues for these representations are estimated by post-processing the pressure data con)- puted as a functioli of time 'during the un- steady analysis.

Results and Discussion

Numerical results have been obtained for two different wings with available exper- imental data for static and oscil!atory control surface deflection. The first wing tested is the F5 wing with a flap for which experimental data are reported in Ref. 11. This wing has an aspect ratio of 3.16, leading edge sweep of nearly 32" and uses thin wing sections (thickness ratio of 4.8%). The inboard flap extends from the root to 11 = 0.586 and extends

Figure 1. A sketch of the High Aspect Ratio Wing (HARW) shows the plan form and location of the control surfaces. Results are presented in this pa- per for control surfaces 4 and 9.

over 18% of the chord near the trailing edge. A sketch of the wing can be seen in figure 2.

The second wing fo i which iesiilts have been obtained is a swept high aspect ratio wing (HARW). This wing has a greater variety of available experirnerital data and is compu- tationally more challenging because of the thick supercritical wing sections and associ- ated effects of viscosity and separation. It has an Aspect Ratio of 10.76, and a leading edge sweep of 28.8" with the wing thickness ratio varying between 16% (at the root) and 12% (at the tip). Experimental results for this trans- port aircraft configuration are reported in Ref- erences 12 and 13, and include static and oscillatory control surface data for many dif- ferent combinations. Figure 1 shows a sketch of the wing plan form along with the location of the various control surfaces used in the experimental studies, Computations were

Figure 2. For the F5 wing, the lifting pressures ob- tained from inviscid full potential analysis at the mid span location of the flap are in good agreement with experimental data for steady flap deflections of 0' and 0.5' for M, = 0.90 and a = 0.0".

carried out to study the outboard aileron (control surface no. 9) and the leading edge control surface (no. 4). Both these control surfaces are located between span stations 71 = 0.589 and q- = 0.794. The aileron covers 20% of the cliorcl near the trailing edge, while the leading edge device extends over 15% of the wing seciion.

First, some results are preserlted for steady control surface deflections. Figure 2 shows the effect of a flap deflection of 0.5" on the lifting pressure (AC, = Ckw" - C;pper) for the F5 wing at a free stream Mach number of 0.9 and zero angle of attack. It is seen that the inviscid computations using the present method agree quite well with experimental data at a span location of 34.1% which corre- sponds approximately to the mid-span lo- cation for the flap.

Next, the lifting pressures are correlated with experimental data for the high aspect ra- tio wing. Figure 3 shows the effect of deflect- ing the outboard aileron when the free stream Mach nurnber is 0.601 and cr = 0.012". For all the cases analyzed, inviscid theory predicts higher values of overall loading on the wing as compared to the experiments. When the effect of viscosity is included, using the inter- active boundary layer, the agreement be- tween theory and experiment improves considerably. The cornputed values match the experiment quite well for negative and low positive aileron deflections but deviate from the experimental data as the aileron deflection is increased. This discrepancy is believed to be due to thickening of the boundary layer and development of larger separation regions.

Figure 4 shows the effect of aileron de- flection on the lifting pressure distribution for a higher free stream Mach number of 0.78 when the flow becomes transoriic. In this case also, the inviscid arlalysis over-predicts the lifting pressures. The results improve significantly when viscosity is included in the analysis, however, the agreement between the analysis and experiment is not as good as in the previous case when the flow was fully subsonic. Also, the solution with interactive boundary layer requires additional iterations for this czse compared to the previous case.

The effect of deflecting the leading edge control surface is shown in Figure 5 for the HARW at a freestream Mach number of 0.602. The inviscid analysisLpredicts an overall load-

Figure 3. The computed lifting pressures at the mid span location of the aileron are cornpared with ex- perimental data for steady aileron deflections ranging between -6" and +6" for M, = 0.601 and n = 0.014". While inviscid computations (dashed lines) overpredict the lifting pressures, including viscous corrections (solid lines) improves the cor- relation with experiment considerably.

ing higher than that from experiment. How- ever, when the viscous corrections - are included, the agreement with experiment is very good. No computatioris were made at the higher Mach nurnber for the leading edge control surface.

Figures 2 'through 5 clearly demonstrate that the technique of representing control surface deflections by the equivalent body motion is satisfactory for modelling various

control surfaces with small deflections. In addition, they also indicate that while iriviscid analysis might be adequate for the evaluation of loads due to thin wings, including viscosity in the computational model is very imp'ortant for accurate prediction of steady aerodynamic loads due to thick supercritical wings. The control surface hinge-moment is a very im- portant parameter of interest in practical ap- plications. As shown in Figure 6 for the aileron on the HARW at Ad, = 0.601, viscosity significantly alters the hinge-moment pre- dicted by the analysis.

Figure 4. Lifting pressures over the aileron from the present analysis with viscous corrections (solid lines) correlate better with experimental data than the inviscid analysis (dashed lines), for steady positive and negative aileron deflections at M, = 0.78 and a = 0.014".

Kigure 5. The loads predicted by the present nethod for steady deflections of the leading edge :ontrol surface, at the mid span location, are com- 3ared with experimental data for (M, = 0.602, r =0.013"). The inviscid analysis (dashed line) ~verpredicts the load while the analysis with inter- active boundary layer (solid line) correlates well uith experiment.

Another point of interest is the effect of control surface deflection on the chordwise wing load distribution. When the control sur- face is deflected, it induces a change in the pressure at other chordwise locations of the

Figure 6. The aileron hinge moment computed by the inviscid full potential analysis is significantly lower (more nose down) than that obtained from analysis with interactive boundary layer. The latter also shows non-linear behavior (M,=0.601, a = 0.0").

wing section; however, the effect on the local lift and pitching moment are very different for the various control surfaces. For the F5 wing, when the flap is deflected down, the increase in loading is highest near the hinge line and tapers off at locations away frorn the hinge; the effect near the leading edge is almost negligible (see figure 2). This causes in- creased lift accompanied by a nose down pitching moment. On the other hand, a posi- tive deflection of the aileron on the HARW not only increases the lifting pressure over the control surface itself, but also increases the load, more or less uniformly, over the re- mainder of the wing section forward of the hinge (see figure 7). This results in a nose down pitching moment accompanied by a significant increase in the sectional lift. It is interesting to note that the effect of deflecting the trailing edge control surface is different for the F5 and the HARW. Based on experimental data for the HARW for the inboard flap (control surface no. 6) it appears that this difference is due to the spanwise locations of the device, i.e., inboard for the flap versus outboard for the aileron, as well as due to differences in the free stream Mach number. However, other influences such as those due to thick- ness and camber cannot be ruled out.

When the leading edge control surface on the HARW is given a positive deflection, the changes in the pressure distribution are largely confined to the region of the control surface itself and the immediate vicinity of the hinge on the fixed part of the wing section (see figure 8). This primarily produces a

Figure 7. A positive deflection of the aileron causes a nose down pitching moment accompanied by an increase in lift. Computational results using full potential analysis with viscous corrections for the HARW at M, = 0.601, a = 0.014".

nose-up pitching moment with a very sn~a l l effect on the local lift coefficient.

Next, results are presented for oscillating control surfaces. As mentioned before, the full potential analysis was done in the inviscid mode as well as with various levels of viscous correction, namely, (i) with a frozen boundary layer, (ii) with a boundary layer interpolated from those at the extreme positions, and (iii) with a- continuously updated boundary layer.

Figure 9 presents the real and imaginary parts of the oscillatory surface pressure dis- tribution at the flap mid-span location for the

Figure 8. When the leading edge control surface is given a positive deflection, i t causes a nose up pitching moment with a rather small effect on the local lift coefficient. Computational results using full potential analysis with viscous corrections for the HARW at M, = 0.602, u = 0.013".

-2.0 , IMAGINARY

Figure 9. Computational results using inviscid full potential analysis are in very good agreement with experimental data for the real part and fair agree- ment for the imaginary part for the F5 wing at M, = 0.90, u = O.On, for the case of the flap oscil- lating at 20 Hz. (k=0.139).

F5 wing when the flap is oscillating at 20 Hz. (reduced frequency of 0.139 based on the root chord) for M, = 0.90 and a = 0.0". It is seen that the real part is larger than the imaginary. The results of the present inviscid analysis for the real part are in very good agreement with the experimental data. However, the pre- dicted imaginary part of the pressure does not agree quite as well with experiment.

The rest of the data for oscillating control surfaces presented here is for the HARW. For this wing, the case studied extensively corre- sponds to the aileron oscillating at 5 Hz. (re- duced frequency of 0.136 based on the root chord) with an amplitude of 4.02" (zero mean deflection) at a free stream Mach number of 0.601, at a = 0". Figure 10 compares the mag- nitude and phase of the lifting pressures (AC P - - CFwer - C y r ) obtained from the present analysis with that from experiment at the

3 case mid-span location of the aileron. For thi- with a sub-critical free stream flow, it was found that there is virtually no difference be- tween the inviscid analysis and that with the frozen houndary layer; hence only the iriviscid results are shown. The addition of a frozen

boundary layer effectively acts as a change to the camber distribution but does not influence the unsteady response. However, both these analyses overpredict the magnitude of the lifting pressure by a significant margin. The phase angles predicted by these analyses are lower than those from experiment by about 15" to 20". There is a substantial improvement in the correlation of the magnitude of the un- steady pressure with experiment when the steady flow boundary layer is computed for aileron deflections corresponding to the mean and two extreme positions, and the viscous effects for the oscillating aileron obtained by interpolation of this data. However, using this approach does not improve the phase re- lationship.

Figure 10. Comparison of the magnitude (upper) and phase (lower) of the lifting pressures obtained from computational analysis with that from exper- iment for the HARW at M,, = 0.601, a = O.On, for the case of the aileron oscillating at 4.99 Hz. (k=0.136). The computational results were obtained using (i) inviscid full potential analysis (chain dashed line), (ii) full potential analysis with an interpolated boundary layer (dashed line) and (iii) full potential analysis with a boundary layer updated every time step (solid line). Results from the doublet lattice method (dotted line) are also shown.

When the boundary layer is updated at every time step, the best correlation with ex- perimental data is obtained. The magnitude of the lifting pressure matches that from ex- periment along most of the wing chord with a very respectable match over the aileron itself. The correlation of the phase angles also im- proves, however, the discrepancy between computation and experiment remains at about 10" and the trend of the phase over the aileron does not match with experiment. A further irnprovernent in the agreement with exper- imental phase angles is obtained when the time step is decreased such that there are 400 time steps per cycle of oscillation (see figure 11). In this case, the phase angle matches with that from experiment forward of the hinge, and has the correct trend over the

aileron itself. Even though the boundary layer is based on a two dimensional steady flow analysis, the interaction with the unsteady inviscid part of the analysis is sufficient to yield an overall result that is very much im- proved and agrees favorably with experiment.

Results from the doublet lattice method are also shown in figure 10. It is interesting to note that results from this lifting surface method are in good agreement with exper- imental data for chord locations forward of the aileron; however, this method yields the poorest correlation with experiment.

When the aileron is oscillated at higher frequencies (10 Hz. and 15 Hz.) the advantage of using the fu ll-potential analysis becomes more noticeable. At these higher frequencies,

/

Figure 11. The correlation of the phase angles be- tween analysis and experiment itnproves signif- icantly when 400 time steps per cycle of oscillation are used (solid line). The magnitude is comparable to that obtained with 200 steps per cycle (dashed line). Data data presented is for the HARW at M, = 0.601, a = 0.0", when the aileron is oscillating at 4.99 Hz. (k= 0.136).

Figure 12. When the free stream Mach number is 0.785 (a = 0.0") and the aileron is oscillating at 15.04 Hz. (k = 0.31 3), the magnitude (upper) and phase (lower) of the lifting pressures computed from the inviscid full potential analysis (dashed line) or full potential analysis with frozen boundary layer (solid line) are in better agreement with ex- periment than results from the doublet lattice method (dotted line).

Figure 13. For the HARW at M, = 0.602, a = O.On, when the leading edge control surface is oscillated at 5.01 Hz. (k=0.136), the magnitude (upper) and phase (middle) of the lifting pressures obtained from full potential analysis are in good agreement with experiment except near the trailing edge. The lower plot shows the phase angles modified to in- crease the clarity of the comparisons.

the prediction of phase angles from the doublet lattice method becomes increasingly inaccurate, whereas the full-potential methed with viscous corrections maintains its accu- racy.

The case of a supercritical free stream Mach number is shown in Figure 12. The

magnitude and phase of the lifting pressure for the aileron oscillating at 15.04 Hz. at M, = 0.785 are presented for the full-potential analysis with frozen boundary layer, inviscid full-potential analysis and the doublet lattice method, along with experimental data. For this case, when the flow field includes regions of supersonic flow, the results from the doublet lattice method are not as good as be- fore. When the full-potential method is used, the inviscid analysis and that with a frozen boundary layer yield slightly different results, with the latter correlating quite well with ex- perimental data. For this case, the procedure with continuous update of the boundary layer ran into convergence difficulties.

From the above results, it is clear that viscous effects play a dominant part in the unsteady aerodynamics of trailing edge con- trol surfaces for wings with thick supercriticai sections. Their influence is stronger in transonic flow than for subsonic flow. How- ever, even a comparatively simple model of the viscous effects can result in a significant improvement in the prediction of the unsteady response of the control surface.

Figure 13 presents data for the HARW with a leading edge control surface oscillating at 5.01 Hz. when the freestream Mach number is 0.602 and a = 0.0". As before, the magnitude and phase distributions of the lifting pressure are shown. In this case, the lifting pressures are either nearly in phase (over the control surface) or almost exactly out of phase. Be- cause of this, presenting the phase data di- rectly yields a plot with a rather poor resolution. In order to improve the resolution of the phase comparisons, an additional plot which represents the phase angles in a modi- fied form is also shown in figure 13. In this plot, the angles are selectively shifted by 180" to yield modified phase angles that vary smoothly over the chord of the wing. As seen before for the case of the trailing edge control surface, the results of inviscid full-potential analysis and those using a frozen boundary layer are almost identical. However, in this case, the magnitudes of the lifting pressure predicted by the full-potential analysis are in quite good agreement with experimental data over most of the chord length, except near the trailing edge. The phase angles are also pre- dicted quite accurately by the full-potential analysis everywhere except near the trailing edge.

The generally much better agreement of the unsteady data for the case of the oscillat- ing leading edge control surface, in contrast to the trailing edge surface, is believed to be due to the smaller influence of viscous effects because of the thin boundary layer. It is rather interesting to note that for the case of the leading edge control device, the unsteady pressure magnitudes display a double peak in the vicinity of the leading edge. Moreover, these pressures are either nearly in phase (as over the leading edge control surface itself) or almost exactly out of phase (over the rest of the wing section). When the data is ex- pressed are real and imaginary parts, it is found that the distribution of the real part is very similar to that of the incremental lifting pressure due to control surface deflection in steady flow. The imaginary part is very small in magnitude, however, its sign determines the phase angle r es~~ l t i ng in the distribution shown in figure 13.

Computational Time

All the analysis was done on the Cray X-MP. The typical computational time for the steady flow analysis is of the order of 60 sec- onds for inviscid analysis and of the order of 90 seconds for analysis including a boundary layer. The unsteady part of the analysis re- quires about 220 seconds for inviscid analysis or when using a frozen boundary layer. When the boundary layer is updated at every time step, the computational time is about 600 seconds for 400 time steps. All the above times are for subcritical f l o \~s . For higher Mach numbers, when the flow is supercritical, additional grid points are used and the com- putational times increase by a factor of ap- proximately 30% to 50%.

Conclusion

A method based on the transonic full po- tential equation for ttie prediction of steady and unsteady loads due to control surfaces on three dimensional wings has been developed. The validity of the simple approach to model- ling the control surfaces, using an equivalent body velocity without changing the grid, has been established by correlation with exper- imental data for control devices located at the trailing edge as well as at the leading edge. It has been shown that modelling of viscous effects is a key requirement for estimating the influence of control surfaces accurately in

steady flow for wings with thick supercritical sections. The effect of viscosity is more pro- nounced in transonic flows than in purely subsonic flows. The effect of a steady de- flection on the sectional lift and moment has been shown to be very different for the trailing edge and leading edge control devices. The spanwise location of the control surface also seems to affect the overall influence of the deflection.

When the wing is thin, inviscid analysis yields reasonably accurate results for oscil- lating flaps. On the other hand, when the wing has thick supercritical sections, inviscid anal- ysis is not adequate to predict the unsteady loads due to oscillating trailing edge control surfaces. However, these results can be im- proved substantially by the judicious use of viscous corrections obtained from an interac- tive boundary layer. It is generally found that ttie accuracy of the unsteady results improves as more sophisticated viscous corrections are used - the phase angles are usually more sensitive than are the magnitudes. For the leading edge control surface, the inviscid analysis is already in quite good agreement with experiment indicating a much smaller role played by viscosity.

Acknowledgements

The author would like to thank Professor L. N. Sankar (Georgia Tech) for his help with development of the full potential code, Dr. T. Cebeci, Mr. K. C. Chang and Mr. G. Wang (Douglas Aircraft Co.) for their assistance in using the inverse boundary layer code, and Mr. J. P. Giesing (Douglas Aircraft Co.) for many interesting and useful discussions. This research was conducted under the independ- ent research and development program of McDonnell Douglas Corporation.

References

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