DEPENDENCE OF THE PROFILE AERODYNAMIC CHARACTERISTICS ON THE SURFACE HEATING AT SUBSONIC VELOCITY OF FLOWING AROUND

Actuality of task on perfection of modern air transport is shown. The brief review of some works sanctified to theoretical, experimental and numeral researches of heat exchange on aerodynamic descriptions of wing profile at his flowing around is conducted. The mathematical model of the conjugated task about hydrodynamics and heat exchange transfer taking into account k-e models of turbulence is brought. The two-dimensional simulation of the subsonic flowing around of NACA-23012 profile by viscous incompressible liquid by the finite elements method with the use of standard k-e turbulence models in the isothermal raising, and also at heating (cooling) bottom (top) profile surfaces is executed. As a result of numeral simulation graphic dependences of lift coefficient on the Reynolds numbers in the range of Re = 104 ÷ 107 for the cases of the isothermal flowing around, and also at the successive heating top and bottom surfaces corresponding to the difference of temperatures 100 К are got. For Re = 104 ÷ 105 curves of aerodynamic coefficients at the simultaneous heating of top and cooling bottom surfaces of profile in relation to the temperature of the stream «at infinity» for the difference of surfaces temperatures lying in the range ΔТ = 0 ÷ 100 К are brought. Influence of increase of profile lift coefficient on the brought dependence ΔСу = ΔСу(ΔТ) at Re = 104; 105 is rated. The optimal thermal mode on a criterion "minimum drag - maximal lift" – simultaneous heating of bottom profile surface and cooling of his topside is recommended

Ключові слова: нестислива течія в'язкої рідини, метод скінченних елементів, модель турбулентності, обтікання профілю, теплообмін, температура, аеродинамічний опір, підйомна сила. Introduction. In the modern world an air transport occupies a leading role in civil, trade and economic and soldiery aims. There is a growing need to improve the energy efficiency of air transport, which, in turn, is achieved by improving the aerodynamic characteristics of aircraft and their elements.
There are two basic directions of perfection the aerodynamic quality of aircrafts [3]. The first method consists in the improvement of geometry: in particular -profiles of wings; in general -all aircraft by exceptionally geometrical methods. Presently this direction did not yet exhaust itself; however, every winning of aerodynamic quality is given with constantly increasing complication of calculations.
Other method is based on the use of active power facilities the management of flowing around. This direction is most perspective, because allows to overcome the theoretical barrier of increase the aerodynamic descriptions of profile.
One of methods by the management of flowing around, as authors of work mark [3][4][5][6], there is influence of heat transfer on flowing around -one of possible power methods of increase of aerodynamic quality of subsonic aircrafts. Unlike other power methods, for example, such as admission of energy in a supersonic zone with the purpose of reduction of impedance, this method being based on heating/cooling of the streamlined surfaces has a greater practical and theoretical value.
In accordance with the theory of influence of weak heat transfer on the lift of flat surfaces [9], heating of bottom surface increases lift, and heating of top surface -decreases it. A most effect will be realized at the simultaneous heating of bottom and cooling top surfaces of body. The author notes that areas of energy supply and removal are the sources of new turbulences formation. Circulations on the top side of profile at removal of energy form additional velocity 0 V   , that, in turn, results in the additional decompressing on this surface and, as a result, to the increase of lift. Circulation at supply of energy to the bottom surface of profile initiates appearance of additional velocity 0 V   , which also results in the increase of pressure and lift.
However, if to heat up not surface of a profile, and jet, flowing around a profile, then the improvement of aerodynamic characteristics of profile is not observed. According to the authors of [2], the heating of the jet, directed perpendicularly to the stream flowing around the wing, does not affect its lift. The paper presents the results of an experimental research of influence the stream temperature simulating jet stream of an aircraft engine on the magnitude of the lift of an isolated triangular wing at undisturbed flow near the screen.
The tests were carried out in a wind tunnel with an open working part using the installation that provides heating of the jet. It is shown that the lift of wing does not change at heating of stream to the temperatures in a range 25-600 0 С.
Formulation of the problem. Flowing around of aerodynamic profile of NACA-23012 by viscous incompressible liquid for the numbers of Reynolds, changing from 10 4 to 10 7 , in the isothermal formulation, and also at heating (cooling) bottom (top) profile surfaces is consider. Influence on a profile lift and drag from difference of temperatures top and bottom surfaces, equal 30, 60 and 100 0 C was investigated. The got results of problem with heating were compared to the results of analogical isothermal problem.
In the case of a laminar isothermal incompressible flow the system of equations included the equation of motion in the form Navier-Stokes and the continuity equation [8]: The system of motion equations in the form of Reynolds and the continuity equation were used to describe the turbulent isothermal flow [8]: where i x -Cartesian coordinates; ttime (problem parameter); i u -Cartesian velocity vector components; i uaveraged velocity vector components; i upulsating velocity vector components; ppressure; density;  and t molecular and turbulent kinematic viscosity coefficients. The system (3-4) was supplemented by a «k-»-model of turbulence, according to which the turbulent viscosity determined from the system solution [8] , . ; .
For the coupled problems of hydrodynamics and heat transfer, the energy equation is added to the equations listed [1,7]: where Т -fluid temperature; аcoefficient of conductivity; Pr   turbulent analogue of the Prandtl number. The boundary conditions were set as follows.
In the entrance section -plane-parallel flow with velocity  V ; on the top and bottom boundaries of the computational domain -simulation of flow past "at infinity": In the output section -zero pressure: .
On the surface of profile is a condition of adhesion: The temperature Tп was set on the profile surface, which differed from the ambient flow temperature: Aerodynamic drag and lift coefficients were determined by formulas (14,15), respectively [10]: where X -drag force; Y -lift force; X C -drag coefficient; Y C -lift coefficient;  Vflow velocity at infinity; L h S PL *  -wing area in plan; 1 L  -wing span (equal to one for two-dimensional setting); h -profile chord.
Results. The Figure 1 shows the curves for the lift coefficient of the profile Cy on the Reynolds number Re for three variants of heating the profile surface: curves 1 and 3 correspond to the heating of the bottom and top surfaces, respectively; curve 2 is constructed for isothermal flow at a temperature difference of the surface ΔТ = 0.
From the curves in Fig. 1 it follows that heating the bottom profile surface increases the lift coefficient, and heating the top surface reduces it, which fully coincides with the theoretical conclusions of the work [9].
The values of the lift coefficient for an isothermal flow (curve 2, Fig. 1) were compared with the corresponding results of [10] at Re = Re = 4.4·10 5 . The difference did not exceed 3.6%, which suggests a satisfactory accuracy of calculations. Analysis of the curves in Fig. 1 also shows that the effect of increasing/decreasing the lift coefficient during heating of the lower/upper surface of the profile practically ceases to appear at high flow velocities corresponding to Reynolds numbers Re > 10 5 . In this flow region, all three curves coincide with each other, which corresponds to the isothermal regime of flow around of the profile. Fig. 2 shows the groups of curves of the aerodynamic coefficients Cx and Cy of the profile for the Reynolds numbers Re = 10 4 ÷ 10 5 with the simultaneous action of heating the bottom and cooling the top surfaces of the profile -the most optimal heating mode, as indicated in [9].
Curves in Fig. 2 obtained at the temperature difference between the top and bottom surfaces equal 0, 30, 60 and 100 0 С.
Analysis of fig. 2 shows that at flowing around of profile with Reynolds numbers Re = 10 4 ÷ 10 5 , corresponding to the area of the thermal effect action, with the simultaneous action of heating the bottom and cooling the top surfaces relative to the freestream temperature, a decrease in the drag coefficient and an increase in the lift coefficient of the profile is obtained, right proportional to the temperature difference between the heated and cooled surfaces. For the most complete analysis of the results in Fig. 3 shows a quantitative estimate of the increase in the lift coefficient from the temperature difference of the profile surface with the simultaneous effect of heating the top and cooling its bottom surfaces.
Curves in fig. 3 show an increase in the lift coefficient Cy (values as a percentage) for Re = 10 4 (left scale of the y-axis) and Re = 10 5 (right scale of the y-axis).
The behavior of the curves in Fig. 3 confirms the effect of heating/cooling of the profile surfaces when it flows around for Re = 10 4 and 10 5 to increase the lift coefficient, which shows up itself to a greater degree at the value Re = 10 4 . Thus, the effect of heating/cooling of the bottom/top profile surfaces, respectively, increases its lift, the greater the magnitude of the temperature difference between the bottom and top profile surfaces.
Conclusions. Undertaken investigations and analysis of the got results in solution of the problem about the influence of heating/cooling the profile surface on its aerodynamic drag and lift coefficients in a flow around profile at subsonic velocity allowed formulating the following conclusions: 1) heating/cooling the profile surface influences on a change of its aerodynamic characteristics; 2) the highest aerodynamic characteristics of the profile according to the criterion "minimum coefficient of drag -maximum coefficient of lift" were achieved while simultaneously heating the bottom surface and cooling the top surface of the profile compared to the isothermal problem; 3) the effect indicated in point 2 of the conclusions increases with increasing temperature difference on the top and bottom surfaces of the profile; 4) influence of heating/cooling of the profile surface on its aerodynamic characteristics at flowed around subsonic flow decreases with increasing Reynolds number. When Re > 10 5 heat transfer already has almost no effect on the change in lift and drag of the profile.