Abstract:
Unique aeroplane wing profiles substantially increasing the aerodynamic qualities of the wing are proposed. The advantage of the proposed profiles and novel method for forming lifting force for a wing on the basis of said profiles is the complete shifting of the interaction of the windstream onto the lower contour, the complete liberation of the upper contour from interaction with the windstream, leading to the elimination of wave drag—an insurmountable defect in wings with a classic profile, and a substantial increase in lifting force for the wing. Novel solutions are given which were the basis for a basically novel interpretation of the process of flow around a wing by the windstream and of the formation of excess pressure along the lower surface.

Description:
RELATED APPLICATIONS 
       [0001]    This application is a continuation of International Patent Application No. PCT/RU2011/000744, filed Sep. 29, 2011, which claims priority to Russian Patent Application No. 2010144348, filed Nov. 1, 2010, both of which are incorporated herein by reference in their entirety. 
     
    
     FIELD OF THE INVENTION 
       [0002]    The invention refers to aerodynamics and can be used to create an aircraft, as well as rotors for helicopters, propellers for piston airplanes and propeller screws for water transport. 
       BACKGROUND OF THE INVENTION 
       [0003]    There is a large number of wing profiles known [S. T. Kashafutdinov, V. N. Lushin, Atlas of the aerodynamic characteristics of wing profiles, Novosibirsk, 1994]. They are united by one common disadvantage—forming lifting force for a wing by means of the creation of a vacuum on the upper contour of the wing with the part of the windstream. 
         [0004]    Known is a method for forming lifting power, where a wing with NACA-0012 profile [Helicopters of countries around the world. Edited by V. G. Lebed, 1994] by the angle of incidence σ=0° does not form lifting power at all as the front edge divides the windstream into two equal parts: onto the upper and lower contour. Only by the angle of incidence σ≧1° symmetry breaking occurs in the distribution of the windstream, which leads to a difference in pressure between the upper and lower surfaces of the wing. 
         [0005]    There is also a method for forming lifting power known, where a wing with NACA-23012 profile [Helicopters of countries around the world. Edited by V. G. Lebed, 1994] is asymmetric, and most of the windstream is directed onto the upper contour which is subjected to uniform compression in the AB area, gains a large amount of kinetic energy and represents in the BC area a thin (0.5-2 mm) high-speed stream with two main functions: a dynamic barrier between the upper surface of the wing and unperturbed atmosphere above the BC and a gas jet pump, rapidly outflowing the air molecules out of the BCD area and creating a vacuum with a critical limit in it, by reaching this limit the stream BC falls to the wing surface BD with an impact. As a result, the BCD area is filled with air until it reaches the unperturbed air pressure at the flight level of an aircraft, and speed stream BC is restored again. This is one cycle of wave drag of the upper surface of the wing in the area of negative angles of incidence BCD. The process is self-oscillating and while an aircraft nearing the speed of sound it becomes a major obstacle to develop high speeds. 
         [0006]    There is a profile known which differs from a classic one with geometric features similar to the element of our profile. There is a wing (FIG. 1) known from the U.S. Pat. No. 6,378,802 (IPC: B64C 30/00, published on Apr. 30, 2002) taken as a prototype for claims 1, 3 and 4 of the invention. The main difference of the prototype from a classic profile is that the acute angle of its front edge does not divide the windstream into 2 parts onto the upper and lower contour, like it does the rounded front edge of the classic profile. According to FIG. 1 from U.S. Pat. No. 6,378,802 and its description, forming lifting force for such profile involves only front and back sections, which constitutes 32% onto upper and lower contour of the wing, whereas aviation age-long experience proved that lifting force is always proportionate to the complete area S of a wing. 
         [0007]    The disadvantage of the prototype is low efficiency of forming lifting force caused by occurrence of wave drag onto the upper contour of the wing which reduces its lifting force by 1 unit of the wing area. 
         [0008]    There is also a symmetrical plane-wedge profile of a wing known from Pat. RU No. 2207967 (IPC: B64C 23/06, released on Jul. 10, 2003). It was taken as a prototype for a wing profile according to claim  2 . 
         [0009]    The disadvantage of such wing is existence of 2 terminating at right angle tailing edges, which create the basis of powerful turbulent resistance that decreases aircraft efficiency. 
       SUMMARY OF THE INVENTION 
       [0010]    The aim of the proposed invention is rising efficiency of forming lifting force through elimination of wave drag onto the upper contour of the wing and lift benefit by 1 unit of the wing area. Another aim is liberation of the wing from aerodynamic flutter. 
         [0011]    These aims can be obtained by the method for forming lifting force for an aircraft with a longitudinal axis and a wing, which has a part of its upper contour of the profile as a straight line, includes creation of an acute angle of the front edge, straight line of the upper contour is parallel to the longitudinal axis of an aircraft, meanwhile the sharp front edge directs the windstream onto the lower contour of the wing. 
         [0012]    To realize method for an aircraft with a longitudinal axis and a wing a wing profile was created. It has sharp front and tailing edges, as well as the upper and lower contours, meanwhile said lower contour is rectilinear from the front to the tailing edge, and said upper contour has a rectilinear section parallel to the longitudinal axis of an aircraft and connected with tailing edge by a flat curve. 
         [0013]    Other alternative of an aircraft wing profile which can realize the claimed method is a wing profile of an aircraft with a longitudinal axis and a wing which has sharp front and tailing edges, as well as the upper and lower contours, partially represented by parallel lines, the above mentioned rectilinear sections of the upper and lower contours are connected with the front and tailing edges by flat curves, whereas the upper contour is parallel to the longitudinal axis of an aircraft. 
         [0014]    A third alternative of an aircraft wing profile which can realize the claimed method is a wing profile of an aircraft with a longitudinal axis and a wing which has sharp front and tailing edges, as well as the upper and lower contours, whereas the upper contour has a rectilinear section, and the above mentioned rectilinear section of the upper contour is parallel to the longitudinal axis of an aircraft, and the lower contour is represented by a flat curve connecting the front and tailing edge of a wing profile. 
         [0015]    It&#39;s rather difficult to define lifting force for a wing with the proposed profiles on the basis of known equations. Therefore a new equation is proposed which considers height of the master cross-section of the wing, chord length, air pressure at the flight level and linear velocity of air molecules as follows: 
         [0000]    
       
         
           
             
               
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                   i 
                 
               
             
             , 
             N 
             , 
           
         
       
     
         [0000]    where 
         [0016]    Y i —lifting force for a wing, N. 
         [0017]    S i =L i ·b i —area of a wing, m 2 . 
         [0018]    L i —wingspan, m. 
         [0019]    b i —chord length, m. 
         [0020]    ρ i —air density at the flight level, kg/m 3 . 
         [0021]    υ μi —linear velocity of air molecules, m/s. 
         [0022]    υ i —speed of an aircraft, ms. 
         [0023]    h i —height of the master cross-section of a wing, average, m. 
         [0000]    
       
         
           
             
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               = 
               
                 
                   
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             , 
           
         
       
     
         [0024]    P 0i —air pressure at the flight level, N/m 2 , 
         [0025]    lifting force coefficient (C y ) is calculated by the following equation: 
         [0000]    
       
         
           
             
               
                 C 
                 yi 
               
               = 
               
                 
                   
                     
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                 1 
               
             
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         [0000]    where 
         [0026]    m i —all-up weight of an aircraft, kg, 
         [0027]    g i —Gravitational acceleration, m/s 2 . 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0028]    The invention is explained in figures where: 
           [0029]      FIG. 1  illustrates the proposed aircraft with a wing profile according to claim  2 , where 
           [0030]    AD=b is a chord and lifting surface of the wing; 
           [0031]    AD 1 =b 1 —outer chord, 
           [0032]    AC 1 —horizontal section of the upper contour. 
           [0033]    C 1 D—section of the flat curve forming the tailing edge of a wing, 
           [0034]    DD 1 =h—height of the master cross-section, 
           [0035]    CC 1 —maximal thickness of a wing, 
           [0036]    angle DAC 1 =β—angle of divergence of the upper and lower contours at the front edge. 
           [0037]      FIG. 2  illustrates an aircraft with a wing profile according to claim  3  of the claim, where 
           [0038]    AD=b—a chord without any function load by this profile: 
           [0039]    AD 1 =b 1 —outer chord, 
           [0040]    AB—a flat curve connecting upper and lower horizontal sections AC 1  and BD and forming a nose of the profile; 
           [0041]    BB 1 =CC 1 =DD 1 =h—height of the master cross-section, 
           [0042]    α—angle of incidence on the master cross-section at the curve AB, 
           [0043]    angle BAB 1 =β—angle of divergence of the upper and lower contours at the front edge; 
           [0044]    C 1 D—a curve forming the tailing edge of a wing, 
           [0045]    MN—a tangent line to the middle point of the curve AB. 
           [0046]    Setting angle of the wing with this profile is 0, so is angle of incidence on the lower lifting surface BD. 
           [0047]      FIG. 3  illustrates an aircraft with a wing profile according to claim  4 , where 
           [0048]    AC 1 —straight line of the upper contour, 
           [0049]    AD—a flat curve connecting the front and tailing edges, 
           [0050]    C 1 D—a flat curve connecting the straight line of the upper contour with the tailing edge. 
       
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
       [0051]    The proposed wing profiles provide interaction of the windstream with the lower contour only, which is represented by segment (AD) connecting the front edge (A) with the tailing edge (D) and simultaneously being a chord (b). In this case on the upper contour (AC 1 D) there is no speed stream as the sharp front edge directs all windstream onto the lower contour (AD). The main part of the upper contour is represented by a straight line (AC 1 ), and its tail section (C 1 D) smoothly descends to the tailing edge. Pressure at the upper contour (AC 1 ) is almost equal to the pressure of unperturbed air at the flight level, while the upper surface is parallel to speed vector of an aircraft, which is a qualitatively new and essential feature of the proposed method. The function of forming lifting force for a wing completely shifts onto the lower contour (AD). The following results are achieved: 
         [0052]    1) Complete liberation of the upper contour of the wing from interaction with the windstream. 
         [0053]    2) Shifting of the interaction of the wing with the environment completely onto the lower contour. 
         [0054]    3) Efficient use of the wall boundary layer for lifting force increase. 
         [0055]    4) Introduction of thickness (h), angle of incidence (α), wall boundary layer thickness (Δh), linear velocity of air molecules (υ M ) in the analysis and calculation of lifting force for a wing. 
         [0056]    5) Liberation of the wing from wave drag—an insurmountable defect in wings with a classic profile. 
         [0057]    6) Minimal frontal drag of the wing and its high aerodynamic quality. 
         [0058]    A dynamic parameter used for calculation of lifting force for a wing with classic aerodynamics is dynamic pressure which is applied to the empirically selected lifting force coefficient (Cy), and lifting force (Y) is calculated by the formula [Encyclopedia of physics. Vol. 3, page 670, 1992]: 
         [0000]        Y=C   y ·ρυ 2   ·s/ 2 ,N , where (1)
 
         [0059]    ρ—air density, kg/m 3 , 
         [0060]    υ—speed of an aircraft, m/s, 
         [0061]    s—area of a wing, m 2 . 
         [0062]    The following equation is true for an aircraft on cruise flight: 
         [0000]        Y=m·g,N , where (2) 
         [0063]    m—weight of an aircraft, kg, 
         [0064]    g—Gravitational acceleration at the flight level, m/s 2 ; 
         [0065]    after equating the right parts (1) and (2) and solving the equation for C y  one will get the following: 
         [0000]    
       
         
           
             
               
                 
                   
                     C 
                     y 
                   
                   = 
                   
                     
                       m 
                       · 
                       g 
                     
                     
                       
                         ρυ 
                         2 
                       
                       · 
                       
                         s 
                         / 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
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         [0066]    Some important parameters are not considered in formulas (1), (2) and (3), such as thickness of a wing (h), angle of incidence (a), pressure on the upper surface of a wing (P B ), pressure on the lower surface of a wing (P H  ), velocity of air molecules (υ M ), thickness of the wall boundary layer (Δh). The biggest paradox, however, is the contradiction between (1) and (3). According to (1), the greater lifting force coefficient (C y &gt;1)—the greater lifting force for a wing and the easier it is for an aircraft to take off, the shorter the take-off path etc. But according to (3), if C y &gt;1, the weight of an aircraft is greater than lifting force for a wing and it cannot take off. 
         [0067]    Therefore the calculation above shows that classic aerodynamics lacks a theory of flow around a wing which moves through unperturbed air. 
         [0068]    There is a corresponding mathematic model for a wing with the patented profile proposed. It is based on the assumption that lifting force for a wing is a result of difference in pressure between upper (P B ) and lower (P H ) surfaces and it can be expressed in the following equation (4): 
         [0000]        Y =( P   B   −P   H )· s,N   (4)
 
         [0069]    Since pressure on the upper surface of a wing with the proposed profile B-1 is always equal to pressure of unperturbed air (P 0i ) at the flight level (P Oi =P 0i ), after expanding (4) one will get: 
         [0000]    
       
         
           
             
               
                 
                   
                     
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                       i 
                     
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                   where 
                 
               
               
                 
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         [0070]    P 0i —unperturbed air pressure at the flight level, N/m 2 , 
         [0071]    ρ i —unperturbed air density at the flight level, kg/m 3 , 
         [0072]    υ i —speed of an aircraft, m/s. 
         [0073]    υ μi —linear velocity of air molecules at the flight level, m/s. 
         [0074]    Under normal conditions (t=0° C., P 0 =101 325 Pa) velocity of air molecules is υ μi =47131.725 m/s. [D. H. Baziev Fundamentals of a unified theory of physics. Moscow, Pedagogics, 1994, p. 619] 
         [0075]    tgβ=h/b 1 —relation between average height of the master cross-section and outer chord, 
         [0076]    h—height of the master cross-section ( FIG. 2 ), m, 
         [0000]    
       
         
           
             
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               = 
               
                 
                   
                     4 
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         [0077]    β—angle of divergence of the upper and lower contours at the front edge of a wing, 
         [0078]    s=L·b—area of a wing, m 2 , 
         [0079]    L—wingspan, m, 
         [0080]    b—chord of a wing, AD ( FIGS. 2 and 3 ), m. 
         [0081]    b 1 —outer chord AD 1  ( FIGS. 2 and 3 ), m. 
         [0082]    Introducing values Y i =c y ·m i g i  and tgβ in (5), one gets a completed equation for lifting force for a wing with the proposed profile B-1. It does not have any coefficients, since all physical and geometric parameters have been taken into account, which take part in forming lifting force for a wing (Y) for subsonic speeds of an aircraft (υ≦1M): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
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         [0083]    where c y ≧1.01—lifting force coefficient of a wing. 
         [0084]    From (6) it follows that in take-off mode the right part of an aircraft must be higher than the left one, i.e. lifting force is greater than take-off weight of an aircraft. And on cruise flight weight and lifting force of an aircraft become equal. Meanwhile the value of lifting force in (6) always takes a negative sign which shows that this force is directed against the gravitational force vector, i.e. upwards. 
         [0000]    
       
         
           
             
               
                 
                   
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         [0085]    equation for lifting force for a wing for aircraft speeds (υ&gt;1M), where M is Mach number, y=1.36805912 is adiabatic coefficient of air in the wall boundary layer by υ&gt;1M 
         [0086]    The following are examples of practical use of the invention. 
       Example 1 
       [0087]      FIG. 1  illustrates a wing profile, where AD is a chord and the lower contour; AC 1 D is the upper contour; CC 1  is the largest thickness of a profile; DD 1 =h—height of the master cross-section of a wing; angle CAC 1 =β—angle of divergence of the upper and lower contours. As one can see in  FIG. 2 , the proposed variant has an acute-angled front edge with the following features: 
         [0088]    1) Exceptionally acute nose angle, CAC 1 =B, which is the angle of divergence of the upper and lower contours, while the front edge of a wing (A) for supersonic aircrafts is extremely sharp like blade. 
         [0089]    2) The lower contour (AD)—chord (b)—is a straight line forming a high-speed wall boundary layer, which has a large amount of kinetic energy and causes excess pressure along the lower surface of a wing (AD). A wing with this profile has minimal frontal drag and maximal lifting force and as a result extremely high aerodynamic quality against the prototype. 
         [0090]    The main part of the upper contour (AC 1 ) is represented by a horizontal straight line parallel to the motion vector of the aircraft wing or to the aircraft main longitudinal axis. The tail section of the upper contour from the point of the largest thickness (C 1 ) of a profile up to the tailing edge (D) is performed as a flat curve (C 1 D). Because of the sharp front edge (A), which is the beginning of the upper contour, the interaction of the windstream with the upper contour is completely avoided, which leads to the elimination of wave drag and liberation from aerodynamic flutter in all flight modes of the aircraft. 
       Example 2 
       [0091]      FIG. 2  illustrates a wing profile, where A is a moderately sharp front edge, B is the beginning of the lifting surface of a wing (BD), AB is a flat curve connecting the lower and upper contours forming the front edge, C 1 D is a flat curve connecting the upper contour with the tailing edge. 
         [0092]    Distinctive features of this profile are as follows: 
         [0093]    1) The main parts of the upper contour AC 1  and the lower contour BD can be parallel or not, it depends on the radius of curvature AB ( FIG. 2 ) and the height of the master cross-section. 
         [0094]    2) The sharp front edge directs all windstream under the wing onto the lower contour because there is no angle of incidence in the upper contour which is caused by parallel alignment of the upper contour to the longitudinal axis of the aircraft. 
         [0095]    3) The windstream interacts only with the lower contour (ABD) which has no segment with negative angle of incidence. Also, as studies showed, a high-speed wall boundary layer is formed along the lower contour at speed υ≦0.6 M at speed υ&gt;0.6 M the wall layer ends at point (B), but because of the windstream a densified underlayer is formed under the wing, this underlayer supports the lifting surface of the wing (BD), as a result specific lifting force for a wing with this profile is two times greater than of the prototype. This feature becomes apparent when a wing moves through unperturbed air. 
         [0096]    This is the basic profile, which can be used to design a series of profiles by changing angle of divergence of the upper and lower contours between 0° and 90°, and also by changing the height of the master cross-section widely. Supersonic aircrafts are equipped with wings with sharp front edges and acceptably low value of the master cross-section height, which depends on several technical conditions. Heavy-duty aircrafts are equipped with this profile or its variations, in this case height of the master cross-section depends on take-off weight and speed on the flight strip at the moment of take-off. The upper contour of the wing profile (AC 1 ) is parallel to the motion vector of the aircraft or to the aircraft main longitudinal axis. Thus, setting angle of the upper surface of a wing with the proposed profile is 0°, while setting angle of a wing with the classic profile is always greater than zero and changes between 2° and 6°. 
       Example 3 
       [0097]      FIG. 3  illustrates a wing profile, where A is a sharp front edge, AC 1  is a rectilinear section of the upper contour, C 1 D is a flat curve connecting the upper contour with the tailing edge, and AD is a flat curve connecting the front and tailing edges forming the lower contour. 
         [0098]    Concept of the invention has been confirmed by the practical realization of the method. 
       Example of Realization of the Proposed Method for Forming Lifting Force for a Wing and Devices for Realizing Said Method 
       [0099]    In order to confirm the realizability of the method and efficiency of the devices, four wing models with profiles according to  FIG. 1  and  FIG. 2  and NACA-23015 profile with the same geometric parameters (wingspan, chord and wing thickness) were constructed. 
         [0100]    The test model was mounted on an AC commutator motor shaft with capacity of W=400 W, and speed n=14 000 rpm. The motor with the wing was installed on a massive platform which was fixed on an electronic balance pan “Nikoteks NPV-15 kg” with tolerance Δ=±0.005 kg. The balance pan was shielded by a large impenetrable duralumin disk. 
         [0101]    The wing models were made of magnesium-aluminum alloy, their surface was thoroughly polished. 
         [0102]    Experimental studies confirmed higher efficiency of wings with proposed profiles compared to the prototype representing a wing with the classic profile forming lifting force mainly through creation of exhaustion along the upper contour. The results are shown in tables 1-4 (see APPENDIX). Specific lifting force for a wing (Y s , N/m 2 ) as a function of speed x is accepted as the control dynamic parameter. Let us compare the wing with the profile according to  FIG. 1  with other wings: with the profile according to  FIG. 2  and NACA-23015 profile assuming that wing motion speeds through unperturbed air are equal: 
         [0103]    1) υ 3 =25.068 m/s (B-1, table 1), Y s3 =247.944 N/m 2 , 
         [0104]    υ 1 =25.917 m/s (NACA, table 2), Y s1 =64.378 N/m 2 , 
         [0105]    k 1 =Y s3 /Y s1 =3.85. 
         [0106]    2) υ 11 =62.777 m/s (B-1, table 1), Y s11 =1724.982 N/m 2 , 
         [0107]    υ 5 =62.207 m/s (NACA, table 2), Y s5 =287.807 N/m 2 , 
         [0108]    k 2 =Y s11 /Y s5 =5.993. 
         [0109]    3) υ 9 =69.309 m/s (B-2, table 3), Y s9 =1105.787 N/m 2 , 
         [0110]    υ 6 =69.309 m/s (NACA, table 2), Y s6 =355.972 N/m 2 , 
         [0111]    k 3 =Y s9 /Y s6 =3.106. 
         [0112]    4) υ 10 =56.516 m/s (B-1, table 1), Y s10 =1388.486 N/m 2 , 
         [0113]    υ 6 =56.413 m/s (B-2, table 3), Y s6 =708.158 N/m 2 , 
         [0114]    k 4 =Y s10 /Y 6 =1.9607. 
         [0115]    As ensues from this comparison of experimental results, the wing with the profile according to  FIG. 1  indicates a substantial advantage in all four examples over the prototype and the wing with profile according to  FIG. 2 , it is reflected by coefficient k. 
         [0116]    Analysis of the results confirms that the proposed method for forming lifting force for a wing and series of profiles based on  FIG. 2  for realizing said method are considerably better than the classic method and the classic profile. 
         [0117]    Based on the above, one can make a conclusion that the proposed method for forming lifting force for a wing and devices for realizing said method can be implemented in practice with reaching the indicated technical result. 
       BIBLIOGRAPHY 
       [0000]    
       
         A. M. Volodko, M. P. Verkhozin, V. A. Gorshkov Helicopters. Guidebook. Moscow, Military edition, 1992. 
         E. I. Ruzhitsky Helicopters. Moscow, Victoria, AST, 1997. 
         Helicopters of countries around the world. Edited by V. G. Lebed, Moscow, 1994. 
         D. H. Baziev Fundamentals of a unified theory of physics. Moscow, Pedagogics, 1994, 640 pages. 
         V. N. Dalin Specifications and construction of helicopters. Moscow, 1983. 
         T. I. Ligum, S. Y. Skripchenko, L. A. Chulsky, A. V. Shishmarev, S. I. Yurovsky Aerodynamics of the Tu-154 airliner. Moscow, Transport, 1977. 
         S. T. Kashafutdinov, V. N. Lushin Atlas of the aerodynamic characteristics of wing profiles, Novosibirsk, 1994. 
         Encyclopedia of physics. Moscow, 1992, Vol. 3. 
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
               
                 Testing results of a wing with a profile according to Fig. 2 
               
               
                 Wing geometry: L = 0.322 m; b = 0.04 m; h = 6 mm, S = 0.01288 m 2 ; S m  = 0.001 932 m 2 ; m 1  = 0.275 kg; G 1  = 
               
               
                 m 1  · g M  = 2.699331N; α = 30°. Laboratory conditions: P 0  = 98791.875 Pa; t 0  = 15° C., p 0  = 1.19496 kg/m 3 . 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                   
                   
                   
                 Excess 
                   
                   
                   
                   
               
               
                   
                   
                 Average 
                   
                 pressure 
                   
                   
                   
                 Wall 
               
               
                   
                   
                 circum- 
                   
                 along the 
                 Pressure 
                   
                   
                 boundary 
               
               
                   
                   
                 ferential 
                 Lifting 
                 lower 
                 along the 
                 Wall 
                   
                 layer 
               
               
                   
                 Rotational 
                 speed 
                 force for a 
                 surface, 
                 lower 
                 boundary 
                   
                 thickness, 
               
               
                   
                 frequency 
                 u = 2πR · n, 
                 wing 
                 Pa 
                 surface 
                 layer 
                   
                 mm 
               
               
                 No. 
                 n, rps 
                 m/s 
                 Y, N 
                 ΔP = Y/S 
                 P N  = ΔP + P 0.   
                 speed, m/s 
                 β = υ u /u 
                 Δh = h/β 
               
               
                   
               
               
                  1 
                 26.667 
                 18.179 
                 0.932 496 
                 72.399 
                 98 864.774 
                 287.636 
                 15.822 424 
                 0.379 208 
               
               
                  2 
                 44.258 
                 30.172 
                 2.453 938 
                 190.989 
                 98 989.864 
                 287.808 
                 9.538 920 
                 0.629 002 
               
               
                  3 
                 60.133 
                 40.994 
                 4.711 560 
                 365.804 
                 99 157.678 
                 288.062 
                 7.026 939 
                 0.853 856 
               
               
                  4 
                 68.167 
                 46.471 
                 6.233 002 
                 483.928 
                 99 275.803 
                 288.234 
                 6.202 440 
                 0.967 359 
               
               
                  5 
                 75.592 
                 51.533 
                 7.656 286 
                 594.432 
                 99 386.307 
                 288.394 
                 5.596 302 
                 1.072 136 
               
               
                  6 
                 82.750 
                 56.413 
                 9.177 727 
                 712.557 
                 99 504.432 
                 288.566 
                 5.115 232 
                 1.112 967 
               
               
                  7 
                 89.500 
                 61.014 
                 10.601011 
                 823.059 
                 99 614.935 
                 288.726 
                 4.732 123 
                 1.267 929 
               
               
                  8 
                 95.917 
                 65.389 
                 12.367 846 
                 960.237 
                 99 752.112 
                 288.924 
                 4.418 549 
                 1.357 912 
               
               
                  9 
                 101.667 
                 69.309 
                 14.330 996 
                 1112.655 
                 99 904.530 
                 289.145 
                 4.177 827 
                 1.438 219 
               
               
                 10 
                 107.583 
                 73.342 
                 15.312 572 
                 1188.864 
                 99 980.738 
                 289.255 
                 3.943 926 
                 1.521 327 
               
               
                 11 
                 114.417 
                 78.000 
                 17.030 328  
                 1322.230 
                 100 114.105 
                 289.448 
                 3.710 875 
                 1.616 869 
               
               
                 12 
                 119.333 
                 81.352 
                 19.091 636 
                 1482.269 
                 100 274.144 
                 289.679 
                 3.560 816 
                 1.685 007 
               
               
                 13 
                 123.333 
                 84.079 
                 20.465 841 
                 1588.963 
                 100 380.838 
                 289.834 
                 3.447 158 
                 1.740 564 
               
               
                 14 
                 127.500 
                 86.920 
                 21.594 652 
                 1676.603 
                 100 468.478 
                 289.960 
                 3.335 942 
                 1.798 592 
               
               
                   
               
               
                         No. 
                   Rotational force N F = m 1  · and · 2π · n 
                     Frontal drag X, N 
                   Frontal drag coefficient c x  = X/F 
                   Lifting force coefficient c y    =  Y/G 1   
                     Aerodynamic quality K = c y /c x   
                 
                   
                     
                       
                           
                         
                           
                             
                               Wing 
                             
                           
                           
                             
                               efficiency 
                             
                           
                           
                             
                               
                                 η 
                                 = 
                                 
                                   
                                     F 
                                     - 
                                     X 
                                   
                                   F 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                     Rotational inertia F i  = F − X 
                     Inertial coefficient k i  = F i /X 
               
               
                   
               
               
                  1 
                 837.639 
                 98.723 
                 0.117 859 
                 0.345 454 
                 2.931 082 
                 0.882 141 
                 738.915 
                 7.484 712 
               
               
                  2 
                 2307.323 
                 104.488 
                 0.045 285 
                 0.909 091 
                 20.074 881 
                 0.954 715 
                 2202.835 
                 21.082 164 
               
               
                  3 
                 4259.374 
                 112.162 
                 0.026 333 
                 1.745 484 
                 66.284 013 
                 0.973 667 
                 4147.212 
                 36.975 227 
               
               
                  4 
                 5473.546 
                 116.944 
                 0.021 365 
                 2.309 091  
                 108.076 290 
                 0.978 635 
                 5356.601 
                 45.805 000 
               
               
                  5 
                 6730.911 
                 121.885 
                 0.018 108 
                 2.836 364 
                 156.632 647 
                 0.981 892 
                 6609.025 
                 54.223 037 
               
               
                  6 
                 8066.029 
                 127.132 
                 0.015 761 
                 3.400 000 
                 215.715 639 
                 0.984 238 
                 7938.896 
                 62.451 216 
               
               
                  7 
                 9435.504 
                 132.504 
                 0.014 043 
                 3.927 272 
                 279.660 471 
                 0.985 957 
                 9303.001 
                 70.209 855 
               
               
                  8 
                 10837.094 
                 138.025 
                 0.012 736 
                 4.581 817 
                 359.743 242 
                 0.987 264 
                 10699.069 
                 77.515 407 
               
               
                  9 
                 12175.371 
                 143.318 
                 0.011 771 
                 5.309 085 
                 451.026 351 
                 0.988 229 
                 12032.053 
                 83.953 689 
               
               
                 10 
                 13633.560 
                 148.997 
                 0.010 929 
                 5.672 726 
                 519.056 366 
                 0.989 071 
                 13484.553 
                 90.501 948 
               
               
                 11 
                 15420.474 
                 155.996 
                 0.010 116 
                 6.309 089 
                 623.662 829 
                 0.989 884 
                 15264.477 
                 97.851 471 
               
               
                 12 
                 16774.182 
                 161.355 
                 0.009 619 
                 7.072 726 
                 735.266 497 
                 0.990 381 
                 16612.827 
                 102.958 006 
               
               
                 13 
                 17917.582 
                 165.835 
                 0.009 356 
                 7.581 817 
                 819.075 704 
                 0.990743 
                 17 751.726 
                 107.031 587 
               
               
                 14 
                 19148.839 
                 170.673 
                 0.008 912 
                 8.000 000 
                 897.567 952 
                 0.991087 
                 18978.167 
                 111.195 994 
               
               
                   
               
               
                 V g0  = m 0 /p 0  = 4.025 801 031 · 10 −26  m 3 ; d g0  = {square root over (6 V g0 /π)} = 4.252 241 23686 · 10 −9  m; f 0  = φ · T = 6.002 135 1087 · 10 12  s −1.  υ μ0  = 2d g0  · f 0  = 51045.052837 m/s. 
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 2 
               
               
                   
               
               
                 Testing results of a wing with a profile according to Fig. 2 
               
               
                 Wing geometry: L = 0.364 m; b = 0.045 m; S = 0.01638 m 2 ; S m  = L · h = 0.00364 m 2 ; h 2  = 10 mm; 
               
               
                 m 2  = 0.55 kg; G 2  = 5.398663N; α = 30°. 
               
               
                 Laboratory conditions: P 0  = 10258.0 Pa; t 0  = 16° C., p 0  = 1.2085 kg/m 3 . 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                   
                   
                   
                 Excess 
                   
                   
                 Wall 
                   
               
               
                   
                   
                 Average 
                   
                 pressure 
                   
                   
                 boundary 
                 Wall 
               
               
                   
                   
                 circum- 
                   
                 along the 
                 Pressure 
                 Wall 
                 layer 
                 boundary 
               
               
                   
                   
                 ferential 
                 Lifting 
                 lower 
                 along the 
                 boundary 
                 acceleration 
                 layer 
               
               
                   
                 Rotational 
                 speed 
                 force for a 
                 surface 
                 lower surface 
                 layer 
                 factor 
                 thickness, 
               
               
                   
                 frequency 
                 u = 2πR · 
                 wing 
                 Pa 
                 P M  = ΔP + P 0.   
                 speed, m/s 
                 m/s 
                 mm 
               
               
                 No. 
                 n, rps 
                 n, m/s 
                 Y, N 
                 ΔP = Y/S 
                 Pa 
                 υ π  = {square root over (P i /p 0 )} 
                 β = υ u /u 
                 Δh = h/β 
               
               
                   
               
               
                 1 
                 17.675 
                 13.160 
                 1.128 811 
                 68.914 
                 100 326.914 
                 288.128 
                 21.894 222 
                 0.456 741 
               
               
                 2 
                 24.667 
                 18.366 
                 2.355 780 
                 143.821 
                 100 401.821 
                 288.235 
                 15.693 972 
                 0.637 187 
               
               
                 3 
                 33.333 
                 24.818 
                 4.318 930 
                 263.671 
                 100 521.671 
                 288.407 
                 11.620 899 
                 0.860 518 
               
               
                 4 
                 41.667 
                 31.023 
                 6.723 789 
                 410.487 
                 100669.488 
                 288.618 
                 9.303 356 
                 1.074 881 
               
               
                 5 
                 50.000 
                 37.228 
                 9.815 751 
                 599.252 
                 100857.252 
                 288.888 
                 7.759 978 
                 1.288 663 
               
               
                 6 
                 58.333 
                 43.432 
                 13.545 736 
                 826.968 
                 101084.968 
                 289.214 
                 6.659 017 
                 1.501 723 
               
               
                 7 
                 66.500 
                 49.513 
                 17.570 194 
                 1072.661 
                 101330.661 
                 289.566 
                 6.848 276 
                 1.709 905 
               
               
                 8 
                 74.833 
                 55.717 
                 22.183 597 
                 1354.310 
                 101612.310 
                 289.968 
                 5.204 297 
                 1.921 489 
               
               
                   
               
               
                         No. 
                   Rotational force, N F = m 1  · and · 2π · n  
                     Frontal drag X, N 
                   Frontal drag coefficient c x  = X/F 
                   Lifting force coefficient c y  = Y/G 1   
                     Aerodynamic quality K = c y /c x   
                 
                   
                     
                       
                           
                         
                           
                             
                               Wing 
                             
                           
                           
                             
                               efficiency 
                             
                           
                           
                             
                               
                                 η 
                                 = 
                                 
                                   
                                     F 
                                     - 
                                     X 
                                   
                                   F 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                     Rotational  inertia F i  = F − X 
                     Inertial coefficient k i  = F i /X 
               
               
                   
               
               
                 1 
                 803.818 
                 185.685 
                 0.231 004 
                 0.209 091 
                 0.905 139 
                 0.768 996 
                 618.133 
                 3.328 926 
               
               
                 2 
                 1565.974 
                 188.750 
                 0.120 563 
                 0.436 364 
                 3.619 379 
                 0.879 437 
                 1376.823 
                 7.294 412 
               
               
                 3 
                 2858.799 
                 193.940 
                 0.067 839 
                 0.800 000 
                 11.792 480 
                 0.932 160 
                 2664.859 
                 13.740 600 
               
               
                 4 
                 4467.087 
                 200.391 
                 0.044 860 
                 1.245 454 
                 27.763 140 
                 0.955 140 
                 4266.636 
                 21.291 576 
               
               
                 5 
                 6432.537 
                 208.291 
                 0.032 381 
                 1.818 182 
                 56.149 894 
                 0.967 619 
                 6224.245 
                 29.882 442 
               
               
                 6 
                 8755.213 
                 217.635 
                 0.024 858 
                 2.509 091 
                 100.936 962 
                 0.975 142 
                 8537.577 
                 39.228 810 
               
               
                 7 
                 11378.459 
                 228.168 
                 0.020 052 
                 3.254 545 
                 162.300 182 
                 0.979 947 
                 11150.291 
                 48.868 772 
               
               
                 8 
                 14408.655 
                 240.331 
                 0.016 679 
                 4.109 091 
                 246.354 169 
                 0.983 320 
                 14168.324 
                 58.953 459 
               
               
                   
               
               
                 V g0  = m 0 /p 0  = 3.980 596 0695 · 10 −26  m 3 ; d g0  = {square root over (6V g0 /π)} = 4.236 265 41834 · 10 −9  m; f 0  = φ · T = 6.022 795 902 · 10 12  s −1.  υ μ0  = 2d g0  ·  
               
               
                 f 0  = 51028.323 m/s. 
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 3 
               
               
                   
               
               
                 Testing results of a wing with a profile according to Fig. 1 
               
               
                 Wing geometry: L = 0.346 m; b = 0.04 m; S = 0.01384 m 2 ; S m  = 0.002 076 m 2 ; m 3  = 0.204 kg; G 3  = 2.002432N; 
               
               
                 a = 9°56′. 
               
               
                 Laboratory conditions: P 0  = 99591.809 Pa; t 0  = 18° C., p 0  = 1.19222 kg/m 3 . 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                   
                 u = 2πR · n, 
                   
                   
                 P m  = ΔP + P 0.   
                 υ u  = {square root over (P i /p 0 )}, 
                   
                   
               
               
                 No. 
                 n, rps 
                 m/s 
                 Y, N 
                 ΔP = Y/S 
                 Pa 
                 m/s 
                 β = υ π /u 
                 Δh = h/β 
               
               
                   
               
               
                  1 
                 21.475 
                 15.395 
                 1.177 890 
                 85.108 
                 99676.917 
                 289.147 
                 18.781 895  
                 0.319 456 
               
               
                  2 
                 28.458 
                 20.402 
                 2.208 544 
                 159.577 
                 99751.386 
                 289.255 
                 14.177 789 
                 0.423 197 
               
               
                  3 
                 34.967 
                 25.068 
                 3.435 513 
                 248.231 
                 99840.039 
                 289.384 
                 11.543 951 
                 0.519 753 
               
               
                  4 
                 41.700 
                 29.895 
                 5.055 112 
                 365.254 
                 99957.063 
                 289.553 
                 9.685 677 
                 0.619 471 
               
               
                  5 
                 48.750 
                 34.949 
                 6.969 183 
                 503.554 
                 100095.363 
                 289.753 
                 8.290 754 
                 0.723 698 
               
               
                  6 
                 54.917 
                 39.370 
                 9.030 491 
                 652.492 
                 100244.301 
                 289.969 
                 7.365 228 
                 0.814 639 
               
               
                  7 
                 61.417 
                 44.030 
                 11.435 350 
                 826.254 
                 100418.063 
                 290.220 
                 6.591 421 
                 0.910 274 
               
               
                  8 
                 67.917 
                 48.690 
                 13.938 366 
                 1007.107 
                 100598.916 
                 290.481 
                 5.965 937 
                 1.005 709 
               
               
                  9 
                 73.500 
                 52.693 
                 16.686 776 
                 1205.692 
                 100797.501 
                 290.768 
                 5.518 153 
                 1.087 320 
               
               
                 10 
                 78.833 
                 56.516 
                 19.238 872 
                 1390.092 
                 100981.901 
                 291.034 
                 5.149 584 
                 1.165 143 
               
               
                 11 
                 87.567 
                 62.777 
                 23.901 354 
                 1726.976 
                 101318.785 
                 291.518 
                 4.643 722 
                 1.292 067 
               
               
                 12 
                 93.750 
                 67.210 
                 27.287 787 
                 1971.661 
                 101563.470 
                 291.871 
                 4.342 668 
                 1.381 639 
               
               
                 13 
                 100.000 
                 71.691 
                 31.655 797 
                 2287.268 
                 101879.077 
                 292.324 
                 4.077 553 
                 1.471 470 
               
               
                 14 
                 105.000 
                 75.276 
                 34.944 073 
                 2524.861 
                 102116.669 
                 292664 
                 3.887 886 
                 1.543 255 
               
               
                   
               
               
                         No. 
                   Rotational force, N F = m 1  ·  and · 2π · n 
                     Frontal drag X, N 
                   Frontal drag coefficient c x  = X/F 
                   Lifting  force coefficient c y  = Y/G 1   
                     Aerodynamic quality K = c y /c x   
                 
                   
                     
                       
                           
                         
                           
                             
                               Wing 
                             
                           
                           
                             
                               efficiency 
                             
                           
                           
                             
                               
                                 η 
                                 = 
                                 
                                   
                                     F 
                                     - 
                                     X 
                                   
                                   F 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                     Rotational inertia F i  = F − X 
                     Inertial coefficient k i  = F i /X 
               
               
                   
               
               
                  1 
                 423.763 
                 37.324 
                 0.088 079 
                 0.588 235 
                 6.678 512 
                 0.911 921 
                 386.438 
                 10.353 471 
               
               
                  2 
                 744.196 
                 38.583 
                 0.051 845 
                 1.102 942 
                 21.273 645 
                 0.948 155 
                 705.612 
                 18.288 105 
               
               
                  3 
                 1123.539 
                 40.073 
                 0.035 667 
                 1.715 686 
                 48.102819 
                 0.964 333 
                 1083.465 
                 27.037 070 
               
               
                  4 
                 1597.882 
                 41.938 
                 0.026 246 
                 2.524 510 
                 96.184 359 
                 0.973 753 
                 1555.943 
                 37.100 210 
               
               
                  5 
                 2183.833 
                 44.241 
                 0.020 258 
                 3.480 392 
                 171.798 910 
                 0.979 741 
                 2139.592 
                 48.361 942 
               
               
                  6 
                 2771.292 
                 46.553 
                 0.016 798 
                 4.509 804 
                 268.467 018 
                 0.983 202 
                 2724.739 
                 58.529 643 
               
               
                  7 
                 3466.149 
                 49.287 
                 0.014 219 
                 5.710 784 
                 401.616 928 
                 0.985 781 
                 3416.863 
                 69.326 054 
               
               
                  8 
                 4238.657 
                 52.322 
                 0.012 344 
                 6.960 784 
                 563.903 928 
                 0.987 656 
                 4186.336 
                 80.011 553 
               
               
                  9 
                 4964.212 
                 55.182 
                 0.011 116 
                 8.333 333 
                 749.669 001 
                 0.988 884 
                 4909.029 
                 88.960 284 
               
               
                 10 
                 5710.702 
                 58.118 
                 0.010 177 
                 9.607 843 
                 944.066 637 
                 0.989 823 
                 5652.584 
                 97.260 000 
               
               
                 11 
                 7046.137 
                 63.373 
                 0.008 994 
                 11.936 275 
                 1327.132 044 
                 0.991 006 
                 6982.763 
                 110.184 778 
               
               
                 12 
                 8076.351 
                 67.422 
                 0.008 348 
                 13.627 451 
                 1632.414 391 
                 0.991 652 
                 8008.929 
                 118.788 685 
               
               
                 13 
                 9189.136 
                 71.813 
                 0.007 815 
                 15.808 823 
                 2022.886 344 
                 0.992 185 
                 9117.323 
                 126.959 322 
               
               
                 14 
                 10131.083 
                 75.519 
                 0.007 454 
                 17.450 980 
                 2341.082 126 
                 0.992 546 
                 10055.563 
                 133.151 899 
               
               
                   
               
               
                 V g0  = m 0 /p 0  = 4.035 053 26198 · 10 −26  m 3 ; d g0  = {square root over (6V g0 /π)} = 4.255 496 29232 · 10 −9  m; f 0  = P 0 V g0 /h = 6.064 624 12486 · 10 12  s −1.  υ μ0  = 2d g0  · f 0  = 51615.971 m/s. 
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 4 
               
               
                   
               
               
                 Testing results of a wing with NACA-23015 profile 
               
               
                 Wing geometry: L = 0.322 m; b = 0.04 m; S = 0.01288 m 2 ; S m  = 0.001 932 m 2 ; h = 6 mm; 
               
               
                 m 4  = 0.2405 kg; G 4  = 2.360 688N; α = 1°. 
               
               
                 Laboratory conditions: P 0  = 98781.875 Pa; t 0  = 15° C., p 0  = 1.19496 kg/m 3 . 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                   
                   
                 Lifting 
                   
                   
                   
                   
                 Lifting force, 
               
               
                   
                   
                   
                 force, 
                   
                   
                   
                   
                 theoretical 
               
               
                   
                   
                   
                 experimental 
                   
                 β 
                   
                   
                 value 
               
               
                   
                   
                 u = 2πR · n, 
                 value 
                 ΔP = Y/S, 
                 Theoretical  
                 υ π  = β · u, 
                 Δh = h/β, 
                 Y = (P B  − P H ) · S, 
               
               
                 No. 
                 n, rps 
                 m/s 
                 Y, N 
                 Pa 
                 value 
                 m/s 
                 mm 
                 N 
               
               
                   
               
               
                  1 
                 38.017 
                 25.917 
                 0.834 339 
                 64.772 
                 11.101 149 
                 287.708 
                 0.540 485 
                 −0.834 339 
               
               
                  2 
                 53.333 
                 36.358 
                 1.472 363 
                 114.313 
                 7.924 002 
                 288.104 
                 0.757 184 
                 −1.472 363 
               
               
                  3 
                 65.875 
                 44.908 
                 2.110 386 
                 163.849 
                 6.416 140 
                 288.136 
                 0.935 142 
                 −2.110 386 
               
               
                  4 
                 79.167 
                 53.970 
                 2.895 646 
                 224.817 
                 5.343 972 
                 288.414 
                 1.122 760 
                 −2.895 646 
               
               
                  5 
                 91.250 
                 62.207 
                 3.729 985 
                 289.596  
                 4.640 678 
                 288.683 
                 1.292 914 
                 −3.729 985 
               
               
                  6 
                 101.667 
                 69.309 
                 4.613 403 
                 358.183 
                 4.170 215 
                 289.033 
                 1.438 774 
                 −4.613 403 
               
               
                  7 
                 112.000 
                 76.352 
                 5.693 136 
                 442.013 
                 3.788 395 
                 289.251 
                 1.583 784 
                 −5.693 136 
               
               
                  8 
                 121.667 
                 82.943 
                 6.772 868 
                 525.844 
                 3.491 587 
                 289.603 
                 1.718 416 
                 −6.772 868 
               
               
                  9 
                 132.500 
                 90.329 
                 7.852 601 
                 609.6730 
                 3.209 490 
                 289.910 
                 1.869 456 
                 −7.852 601 
               
               
                 10 
                 139.833 
                 95.328 
                 8.785 097 
                 682.072 
                 3.042 419 
                 290.028 
                 1.972 114 
                 −8.785 097 
               
               
                   
               
               
                             No. 
                         F = m 1  · and · 2π · n, N 
                             X, N 
                             c x  = X/F 
                             c y  = Y/G 4   
                             K = c y /c x   
                           
       η   =       F   -   X     F         
 
                     Pressure along the upper surface of a wing P B  = p 0 υ π   2  − ΔP 
                 Pressure along the  lower surface of a wing P H  = p 0 υ π   2.  Pa 
               
               
                   
               
               
                  1 
                 1488.872 
                 183.296 
                 0.123 111 
                 0.353 430 
                 2.870 824 
                 0.876 889 
                 98 849.437 
                 98 914.209 
               
               
                  2 
                 2930.157 
                 189.585 
                 0.064701 
                 0.623 701 
                 9.639 699 
                 0.935 298 
                 99 072.396 
                 99 186.709 
               
               
                  3 
                 4470.327 
                 195.545 
                 0.043 743 
                 0.893 971 
                 20.436 908 
                 0.956 257 
                 99 044.556 
                 99 208.408 
               
               
                  4 
                 6456.417 
                 203.525 
                 0.031 523 
                 1.226 611 
                 38.911 616 
                 0.968 477 
                 99 175.216 
                 99 400.033 
               
               
                  5 
                 8577.625 
                 212.014 
                 0.024 717 
                 1.580041 
                 63.925 089  
                 0.975 283 
                 99 295.598 
                 99 585.193 
               
               
                  6 
                 10 647.915 
                 220.409 
                 0.020699 
                 1.954 262 
                 94.410 123 
                 0.979 300 
                 99 469.168 
                 99 827.351 
               
               
                  7 
                 12 922.109 
                 229.423 
                 0.017 754 
                 2.411 643 
                 135.834 215 
                 0.982 246 
                 99 536.041 
                 99 978.054 
               
               
                  8 
                 15 249.215 
                 238.807 
                 0.015 660 
                 2.869 023 
                 183.203 781 
                 0.984 339 
                 99 695.124 
                 100 220.968 
               
               
                  9 
                 18085.812 
                 250.095 
                 0.013 828 
                 3.326 403 
                 207.475 506 
                 0.986 172 
                 99 824.104 
                 100 433.777 
               
               
                 10 
                 20143.044 
                 258.154 
                 0.012 816 
                 3.721 414 
                 290.374 787 
                 0.987 184 
                 99 833.332 
                 100 515.405 
               
               
                   
               
               
                 V g0  = m 0 /p 0  = 4.025 801 031 · 10 −26  m 3 ; d g0  = {square root over (6 V g0 /π)} = 4.252 241 23686 · 10 −9  m; f 0  = P 0 V g0 h = 6.001 510 47643 · 10 12  s −1.  υ μ0  = 2d g0 ; f 0  = 51039.741 m/s 
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
             
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 5 
               
             
             
               
                   
               
               
                 Excess pressure along the upper and lower surfaces 
               
               
                 of a wing with NACA-23015 profile 
               
             
          
           
               
                   
                 u, m/s 
               
             
          
           
               
                 ΔP, Pa 
                 25.917 
                 44.908 
                 62.207 
                 69.309 
                 76.352 
                 82.943 
                 95.328 
               
               
                   
               
             
          
           
               
                 ΔP B   
                 67.562 
                 262.680 
                 513.723 
                 687.293 
                 754.166 
                 913.249 
                 1051.458 
               
               
                 ΔP H   
                 132.334 
                 426.530 
                 803.318 
                 1045.476 
                 1196.179 
                 1439.093 
                 1733.530 
               
               
                 ΔP B  −− 
                 −64.772 
                 −163.85 
                 −289.595 
                 −358.183 
                 −442.013 
                 −525.844 
                 −682.072 
               
               
                 ΔP H   
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
             
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 6 
               
             
             
               
                   
               
               
                 Excess pressure along the lower surface of a wing with a profile according to FIG. 2 
               
             
          
           
               
                   
                 u, m/s 
               
             
          
           
               
                 ΔP, Pa 
                 30.172 
                 46.471 
                 61.014 
                 69.309 
                 78.000 
                 84.079 
                 86.92 
               
               
                   
               
             
          
           
               
                 ΔP B   
                 98791.8 
                 98791.87 
                 98791.87 
                 98791.87 
                 98791.87 
                 98791.87 
                 98791.87 
               
               
                 ΔP H   
                 98982.86 
                 99275.80 
                 99614.93 
                 99904.530 
                 100114.10 
                 100380.83 
                 100468.48 
               
               
                 ΔP B  − 
                 −190.989 
                 −483.928 
                 −823.060 
                 −1112.655 
                 −1322.230 
                 −1588.963 
                 −1676.603 
               
               
                 ΔP H