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radians, Equation (4.13) reduces to dz Voo,n = Voo (a - dx (4.14) %3D Equation (4.14) gives the expression for V to be used in Equation (4.12). Keep in mind that, in Equation (4.14)a is in radians) Returning to Equation (4.12), let us develop an expression for w'(s) in terms of the strength of the vortex sheet. Refer again to Figure 4.22b. Here, the vortex sheet is along the chord line, and w'(s) is the component of velocity normal to the camber line induced by the vortex sheet. Let w(x) denote the component of velocity normal to the chord line induced by the vortex sheet, as also shown in Figure 4.22b. If the airfoil is thin, the camber line is close to the chord line, and it is consistent with thin airfoil theory to make the approximation that w'(s) ~ w(x) (4.15) An expression for w(x) in terms of the strength of the vortex sheet is easily obtained from Equation (4.1), as follows. Consider Figure 4.24, which shows the vortex sheet along the chord line. We wish to calculate the value of w(x) at the location x. Consider an elemental vortex of strength y dg located at a distance g from the origin along the chord line, as shown in Figure 4.24. The strength of the vortex sheet y varies with the distance along the chord; that is, y = The velocity dw at point x induced by the elemental vortex at point & is given by Equation (4.1) as y(E). y(5) dž dw = - (4.16) 27 (x – 5) In turn, the velocity w(x) induced at point x by all the elemental vortices along the chord line is obtained by integrating Equation (4.16) from the leading edge (5 = 0) to the trailing edge (5 = c): lo y(5) dg w(x) = (4.17) 27 (x - 5) Combined with the approximation stated by Equation (4.15), Equation (4.17) gives the expression for w'(s) to be used in Equation (4.12). (安越sheet 6的速な) oe ele o E W. Figure 4.24 Calculation of the induced velocity at the chord

radians, Equation (4.13) reduces to dz Voo,n = Voo (a - dx (4.14) %3D Equation (4.14) gives the expression for V to be used in Equation (4.12). Keep in mind that, in Equation (4.14)a is in radians) Returning to Equation (4.12), let us develop an expression for w'(s) in terms of the strength of the vortex sheet. Refer again to Figure 4.22b. Here, the vortex sheet is along the chord line, and w'(s) is the component of velocity normal to the camber line induced by the vortex sheet. Let w(x) denote the component of velocity normal to the chord line induced by the vortex sheet, as also shown in Figure 4.22b. If the airfoil is thin, the camber line is close to the chord line, and it is consistent with thin airfoil theory to make the approximation that w'(s) ~ w(x) (4.15) An expression for w(x) in terms of the strength of the vortex sheet is easily obtained from Equation (4.1), as follows. Consider Figure 4.24, which shows the vortex sheet along the chord line. We wish to calculate the value of w(x) at the location x. Consider an elemental vortex of strength y dg located at a distance g from the origin along the chord line, as shown in Figure 4.24. The strength of the vortex sheet y varies with the distance along the chord; that is, y = The velocity dw at point x induced by the elemental vortex at point & is given by Equation (4.1) as y(E). y(5) dž dw = - (4.16) 27 (x – 5) In turn, the velocity w(x) induced at point x by all the elemental vortices along the chord line is obtained by integrating Equation (4.16) from the leading edge (5 = 0) to the trailing edge (5 = c): lo y(5) dg w(x) = (4.17) 27 (x - 5) Combined with the approximation stated by Equation (4.15), Equation (4.17) gives the expression for w'(s) to be used in Equation (4.12). (安越sheet 6的速な) oe ele o E W. Figure 4.24 Calculation of the induced velocity at the chord

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
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Above is part of contents of aerodynamics, I want to know the reason why velocity w(x) is negative in equation(4.17)

CHAPTER 4 Incompressible Flow over Airfoils radians, Equation (4.13) reduces to dz Voon = Voo (a dx (4.14) Equation (4.14) gives the expression for V to be used in Equation (4.12). Keep in mind that, in Equation (4.14)a is in radians Returning to Equation (4.12), let us develop an expression for w'(s) in terms of the strength of the vortex sheet. Refer again to Figure 4.22b. Here, the vortex sheet is along the chord line, and w'(s) is the component of velocity normal to the camber line induced by the vortex sheet. Let w(x) denote the component of velocity normal to the chord line induced by the vortex sheet, as also shown in Figure 4.22b. If the airfoil is thin, the camber line is close to the chord line, and it is consistent with thin airfoil theory to make the approximation that w'(s) ~ w(x) (4.15) An expression for w(x) in terms of the strength of the vortex sheet is easily obtained from Equation (4.1), as follows. Consider Figure 4.24, which shows the vortex sheet along the chord line. We wish to calculate the value of w(x) at the location x. Consider an elemental vortex of strength y dg located at a distance E from the origin along the chord line, as shown in Figure 4.24. The strength of the vortex sheet y varies with the distance along the chord; that is, y = y(§). The velocity dw at point x induced by the elemental vortex at point is given by Equation (4.1) as y(5) dE dw = -; (4.16) 2n (x - 5) In turn, the velocity w(x) induced at point x by all the elemental vortices along the chord line is obtained by integrating Equation (4.16) from the leading edge (5 = 0) to the trailing edge ( = c): w(x) = - | y(5) dE :- 27(x-E) (4.17) Combined with the approximation stated by Equation (4.15), Equation (4.17) gives the expression for w'(s) to be used in Equation (4.12). (强 sheet 的速な) Figure 4.24 Calculation of the induced velocity at the chord line.
Transcribed Image Text:CHAPTER 4 Incompressible Flow over Airfoils radians, Equation (4.13) reduces to dz Voon = Voo (a dx (4.14) Equation (4.14) gives the expression for V to be used in Equation (4.12). Keep in mind that, in Equation (4.14)a is in radians Returning to Equation (4.12), let us develop an expression for w'(s) in terms of the strength of the vortex sheet. Refer again to Figure 4.22b. Here, the vortex sheet is along the chord line, and w'(s) is the component of velocity normal to the camber line induced by the vortex sheet. Let w(x) denote the component of velocity normal to the chord line induced by the vortex sheet, as also shown in Figure 4.22b. If the airfoil is thin, the camber line is close to the chord line, and it is consistent with thin airfoil theory to make the approximation that w'(s) ~ w(x) (4.15) An expression for w(x) in terms of the strength of the vortex sheet is easily obtained from Equation (4.1), as follows. Consider Figure 4.24, which shows the vortex sheet along the chord line. We wish to calculate the value of w(x) at the location x. Consider an elemental vortex of strength y dg located at a distance E from the origin along the chord line, as shown in Figure 4.24. The strength of the vortex sheet y varies with the distance along the chord; that is, y = y(§). The velocity dw at point x induced by the elemental vortex at point is given by Equation (4.1) as y(5) dE dw = -; (4.16) 2n (x - 5) In turn, the velocity w(x) induced at point x by all the elemental vortices along the chord line is obtained by integrating Equation (4.16) from the leading edge (5 = 0) to the trailing edge ( = c): w(x) = - | y(5) dE :- 27(x-E) (4.17) Combined with the approximation stated by Equation (4.15), Equation (4.17) gives the expression for w'(s) to be used in Equation (4.12). (强 sheet 的速な) Figure 4.24 Calculation of the induced velocity at the chord line.
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