Airfoil smoothing using unconditional optimization taking into account aerodynamic criteria

. The smoothing of airfoils in sections of the surface of an aircraft wing is considered. The issue of using the distribution of pressure and Mach number along the chord of an airfoil as criteria for smoothing efficiency is being studied. Smoothing is considered as a problem of unconditional minimization of a quadratic objective function. The arguments of the objective function are the second derivatives of the tabular function interpolating the contour of the airfoil. Minimization is performed using the gradient descent method.


Introduction
When designing the wing surface, airfoils are used as the curves of its cross sections.The optimal shape of the used airfoils ensures low aerodynamic drag and high lift and, therefore, maximum aerodynamic quality of the aircraft.
Methods for optimizing the airfoil and wing surface to improve aerodynamic characteristics are discussed in [1,2,3,4].Another approach to optimization, in which airfoils and wings are designed that correspond to the required aerodynamic characteristics, is described in [5,6,7].
Airfoils, as a rule, are tabulated curves.During the design of the wing surface, various modifications are made, as a result of which the smoothness of the airfoil curves may be disrupted.To eliminate emerging defects, smoothing is required.
The works [8,9,10,11] indicate that the presence of surface roughness in transonic modes leads to a significant change in the pressure distribution along the chord of the airfoil compared to a smooth surface with the formation of local supersonic flow regions.In this case, significant differences in the pressure coefficient ∆C p are observed, which leads to a significant increase in drag.Modern passenger long-haul aircraft perform cruising flights at transonic speeds.
Thus, in addition to obtaining the optimal shape of the airfoil, an urgent task is to smooth it during the design of the wing surface.
In this case, to assess the efficiency of smoothing, it is advisable to use aerodynamic criteria.As such criteria, one can use the smoothness of the distribution graph of the pressure coefficient   along the airfoil chord and the distribution of the free-stream Mach number along it.To obtain these characteristics, it is possible to simulate the flow around a smoothed airfoil in a transonic flight mode in the CAE system.[12,13] In Figure 1 shows the upper half of a symmetrical airfoil with irregularities in smoothness and its curvature diagram, plotted in a CAD system.As can be seen from Fig. 1-2 there are significant differences in the pressure coefficient ∆  , and their location has some correlation with the location of irregularities on the profile.From Fig. 3 it can be seen that profile irregularities lead to the formation of two shock waves with local sections of supersonic flow.
Thus, the results of flow simulation confirm the influence of irregularities on aerodynamic characteristics described in the above-mentioned works.
Previous works [14,15,16] describe developed smoothing procedures that made it possible to eliminate unspecified changes in the sign of curvature and ensure convexity throughout the entire contour.After ensuring the convexity of the contour of the smoothed airfoil, it was necessary to solve the problem of ensuring a smooth change in its curvature.

Smoothing via unconditional optimization
The contour smoothing problem was presented as a problem of minimizing a quadratic function of several variables.
The values of the second derivatives of the function  = (), interpolating the tabulated curve of airfoil at its nodes are used as arguments of the quadratic function.
The quadratic objective function was presented as . (1) Here the vector of ordinates of the tabulated function   = (  ) interpolating the contour is  = ( 1 ,  2 , … ,   )  , the transposition sign is , weighting coefficients are   , second derivatives of the  function at the -th point are   (  ).
The quadratic objective function has continuous first derivatives, which makes it possible to apply first-order minimization methods, for example, gradient ones.To apply such minimization methods, the objective function must be presented in the form of an explicit dependence on the ordinates of the nodes of the smoothed contour.For this purpose, the values of the second derivative   (  ) of the function   = (  ) at each -th point were expressed using difference-free formulas of numerical differentiation [17,18] through the values of the function at five points.
After discarding the infinitesimal remainder terms, the formula for expressing the second derivative of the function   = (  ) through the values of the function at five points has the form: where ℎ =   −  −1 .
By substituting the second derivative formula ( 2) into (1), we obtain a function of , specified explicitly: For the resulting objective function, a minimization algorithm was developed using the gradient descent method with a constant step [19,20].Based on this algorithm, a computer program was developed, with the help of which the airfoil under consideration was smoothed.

Results
In Fig. 4 shows a smoothed profile curve constructed in the CAD system and its curvature diagram.

Fig. 4. Airfoil curvature diagram after smoothing
As can be seen from Figure 4, smoothing made it possible to obtain a fairly smooth graph of the curvature of the airfoil, however, an unintended concavity appeared in the tail part of the airfoil.
For a smoothed profile, flow simulation was also carried out in cruising flight mode.In Fig. 5-6 shows how the pressure coefficient   and Mach number  are distributed along the chord of the airfoil.As can be seen from the figures, the elimination of irregularities led to the disappearance of the second shock wave and a uniform distribution of the supersonic flow region along the airfoil chord.The change in flow rates and pressure coefficient   along the profile chord occurs evenly, without sudden changes.

Discussion
The results of the smoothing performed confirm the correctness of the choice of the objective function and the effectiveness of the developed smoothing technique for the tasks of ensuring a smooth change in the curvature of the airfoil.However, the technique needs to be refined to prevent the appearance of unforeseen inflections in the curve [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37].
The results of the flow simulation confirm the correctness of the assumption about the possibility of using graphs of the distribution of the pressure coefficient   and the Mach number M along the airfoil chord in cruising flight mode as criteria for smoothing efficiency.However, their use is not sufficient to meet the requirements for the quality of the wing surface [38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54].In the case considered, despite the disappearance of the second shock wave and the achieved more uniform pressure distribution along the airfoil chord, as a result of smoothing, an unintended concavity of the airfoil arose.[18]

Conclusions
The smoothing of tabulated curves describing airfoils is considered.It is proposed to use the distribution of the pressure coefficient and Mach number along the chord of the airfoil as criteria for smoothing efficiency.
Smoothing is presented as a problem of minimizing a quadratic function of many variables.The values of the second derivatives of the function interpolating the tabulated airfoil curve are used as arguments to the quadratic function.Minimization is performed using the gradient descent method with a constant step.
A computer smoothing program has been developed using the developed methodology.The profile under consideration was smoothed using the developed program.
The effectiveness of the developed smoothing technique has been confirmed.As a disadvantage of the technique, the possibility of unforeseen concavities occurring when the smoothness of the curvature graph of the smoothed airfoil is achieved.
The possibility of using the proposed aerodynamic criteria was confirmed, but their insufficiency for a comprehensive assessment of smoothing efficiency was revealed.
Directions for further research have been identified to prevent the appearance of unintended inflections in the airfoil curve and to expand the set of used smoothing efficiency criteria.

Fig. 2 .
Fig. 2. Graph of pressure coefficient distribution along the chord of the airfoil under consideration

Fig. 3 .
Fig. 3. Mach number distribution along the chord of the airfoil under consideration

Fig. 5 .
Fig. 5. Graph of pressure distribution along the profile chord after smoothing

Fig. 6 .
Fig. 6.Mach number distribution along the airfoil chord after smoothing