chapter 12 wings of finite span
TRANSCRIPT
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CHAPTER 12 WINGS OF FINITE SPAN ___________________________________________________________________________________________________________
12.1 FLOW OVER A THREE-DIMENSIONAL WING
The images below from Van Dyke’s Album of Fluid Motion depict low speed flow over a lifting wing of finite span. The images include a view from the side, a plan view from above and a series of images showing the vortex wake in a series of planes normal to the wing wake at increasing downstream distances from the wing. The span of the wing is b and the chord is called C .
(a)
(b)
(c)
Figure 12.1 Images of the flow past a finite span wing at low speed. From An Album of Fluid Motion by M. Van Dyke.
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The wings are not all the same. Figure 12.1a shows a wing in water visualized by colored dye. Figure 12.1b shows smoke lines in the flow over a wing in air. And figure 12.1c shows the wake of a wing towed in water visualized by very small hydrogen bubbles emitted by a fine wire just downstream of the wing trailing edge.
These images beautifully illustrate the three-dimensional flow over the wing. In each case the wing has lift leading to reduced pressure above the wing and increased pressure below compared to the free stream pressure. The consequences of this pressure difference are illustrated in the figures.
The dyelines in Figure 12.1a emanate from the boundary layer on the lower surface of the wing. At the wing tip, vorticity from the lower and upper surface of the wing separates and rolls up to form a trailing vortex.
Figure 12.1b illustrates the same process as viewed from above looking down on the suction surface of the wing. Smoke lines introduced ahead of the wing are positioned so that the lines pass just below the wing. The upstream influence of the elevated pressure below the wing leads to a divergence of the smoke lines ahead of the wing. If the smoke lines had been positioned to pass above the wing one would instead see a convergence driven by the low pressure on the suction surface. This very important effect can also be clearly seen in the smoke lines that leave the trailing edge of the wing and diverge outward to join the vortex rollup from the wing tips.
Figure 12.2 schematically shows the flow on the centerline of a three-dimensional wing.
Figure 12.2 – Velocity field normal to a wing comprising a transverse bound vortex of circulation Γ plus downwash generated by a semi-infinite system of free vortices in the wake.
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Near the wing the bound circulation due to lift leads to an up-wash ahead of the wing and downwash behind the wing similar to the flow produced by a two-dimensional lifting wing of infinite span.
A very important effect is generated by the flow due to the vortex pair that comprises the wake. The semi-infinite sheet of vorticity distributed in the wake produces a downward velocity component in the free-stream ahead of the wing, at the wing and far downstream as illustrated in Figure 12.3.
Figure 12.3 Upstream and downstream effect of the wake of a finite span lifting wing.
The downwash by the wake leads to a reduction in the angle-of-attack of the wing relative to the free stream, reducing the lift. In addition the downwash rotates the oncoming flow vector at the wing leading to a component of drag as shown in Figure in 12.3. The change in angle of attack due to the downwash generated by the wake is
α i = ArcTanUz 0,0,0( )
U∞
⎛⎝⎜
⎞⎠⎟< 0 (12.1)
where Uz 0,0,0( ) is the z component of velocity induced by the vortex system of the wake at the mid span of the wing. The downwash produced by the circulation of the bound vortex of the wing is zero at the wing and therefore does not contribute to the induced drag.
The span-wise divergence and convergence of the flow at the wing trailing edge seen in Figure 12.1b is illustrated in Figure 12.4.
Figure 12.4 Span-wise flow in a plane perpendicular to the wing trailing edge
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The general direction of the flow about the wing is from high pressure to low pressure. At the wing trailing edge where the pressures above and below the wing are equal the stream-wise vorticity produced by the spanwise flow over the wing convects into the wake where it feeds the rolling up vortices as can be seen in figure 12.1b.
The connection between the bound vorticity on the wing and the free vorticity in the wake can be determined using the contour shown in Figure 12.5.
(a)
(b)
Figure 12.5 Contour used to connect the circulation bound to a lifting wing, the span-wise flow at the wing trailing edge and free vorticity in the wake.
The idea behind Figure 12.5 is that the circulation about a contour that lies entirely in a potential flow is zero since the potential is continuous over such a contour.
Uic dC!∫ = ∇Φic dC = dΦ =!∫ Φ final − Φinitial!∫ = 0 (12.2)
Follow the arrows closely in Figure 12.5a. The contour CA passes over the wing in a downstream direction and, before reaching the vortex sheet, connects to contour CB . Contour CB then turns 90 degrees and upward almost reversing direction. From just above the vortex sheet CB passes around and below the wake vortex sheet approaching the sheet from below. Contour CB connects to CA then turns back 90 degrees, nearly
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reversing direction again and passes under the wing where CA closes. At no point does the contour penetrate the vortex sheet that is discontinuous in the potential because of the discontinuity in span-wise velocity across the sheet. Using (12.2) we can write
Uic dC!∫ = Uic dC
CA∫ + Uic dC
CB∫ = 0 (12.3)
To clarify this idea, in Figure 12.5b the contours are imagined to be shrunk-wrapped about the wing and the trailing edge vortex sheet while remaining imbedded entirely in potential flow. Equation (12.3) implies
ΓTrailingEdge y( ) = −ΓWing y( ) (12.4)
By the way it is not necessary to shrink-wrap the contour to get to this result. Figure 12.4a is enough, either contour gets to the result (12.4).
12.2 CIRCULATION AND PRESSURE
Figure 12.6 shows a cross-section of a finite-span wing moving at a constant velocity U∞ in an incompressible flow.
Figure 12.6 Airfoil cross-section
The dashed line with length C connects the leading edge of the wing to the trailing edge and is called the chord line. The section lift (lift per unit span) is
dLdy
= PLower1
1+ dzLowerdx
⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟
1/2
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
− PUpper1
1+dzUpperdx
⎛⎝⎜
⎞⎠⎟
2⎛
⎝⎜
⎞
⎠⎟
1/2
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
⎛
⎝
⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟
0
C y( )∫ dx (12.5)
where zLower x, y( ) and zUpper x, y( ) are the z coordinates of the lower and upper surfaces of the wing at the spanwise position y . The Bernoulli equation is
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P∞ +12ρU∞
2 = PLower +12ρ ULower( )2 = PUpper + 12 ρ UUpper( )2 (12.6)
If the airfoil is relatively thin we can approximate the pressure by
P∞ +12ρU∞
2 ≅ PLower + ρU∞uLower = PUpper + ρU∞uUpper (12.7)
The lift per unit span is then
dLdy
= ρU∞ uUpper − uLower( )0
C
∫ dx = ρU∞Γ y( ) (12.8)
where Γ y( ) is the circulation about the airfoil at the section y .
Γ = Uic dC
C!∫ (12.9)
12.3 FORCES AND MOMENTS ON A WING
Figure 12.7 shows the relationship between the various forces that act on a differential slice dy at some spanwise location on a three-dimensional wing with span b .
Fig 12.7 Differential forces on a section of a 3-D wing
The differential normal force at a given spanwise location is
dF⊥ y( ) = ρUR y( )Γ y( )dy (12.10)
The differential lift and induced drag at the position y are
dL y( ) = dF⊥ y( )Cos α i( ) = dF⊥ y( ) U∞
UR 0, y,0( ) = ρU∞Γ y( )dy (12.11)
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and
dDi y( ) = −dF⊥ y( )Sin α i( ) = −dF⊥ y( )Uz 0, y,0( )UR 0, y,0( ) = −ρUz 0, y,0( )Γ y( )dy (12.12)
The induced flow angle is generally very small so we can apply the approximation α i ≅Uz 0, y,0( ) /U∞ . The differential induced drag is related to the differential lift by
dDi y( ) = −α i y( )dL y( ) (12.13)
The lift and drag are found by integrating the above differential relations
L = ρU∞ Γ y( )dy−b /2
b /2
∫ (12.14)
Di = −ρ Uz 0, y,0( )Γ y( )dy−b /2
b /2
∫ (12.15)
The rolling moment vector is aligned with the positive x-axis and also involves integrating the circulation.
Mx = MRoll = ydL y( )−b /2
b /2
∫ = ρU∞ yΓ y( )−b /2
b /2
∫ dy (12.16)
The yawing moment is aligned with the positive z-axis and, like drag, its integral involves the circulation and downwash velocity.
Mz = MYaw = − ydDi y( )−b /2
b /2
∫ = ρ yUz 0, y,0( )−b /2
b /2
∫ Γ y( )dy (12.17)
12.4 LIFTING LINE THEORY
Assume that the flow over a finite wing can be treated as locally two-dimensional with the flow geometry depicted in Figure 12.8 applied at each position y along the span of the wing. This is a good assumption for thin wings of large aspect ratio. The aspect ratio is defined as
AR =b2
S (12.18)
where S is the geometric planform area of the wing, ie not the area projected on the x, y{ } plane but the area in the plane of the chord line of the wing – the largest area of the
wing. The relative free stream velocity vector at each span-wise position is
UR 0, y,0( ) = U∞ ,0,Uz 0, y,0( ){ } (12.19)
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Figure 12.8 Wing cross section at spanwise position y .
The fundamental problem of finite wing theory is that even when the basic parameters of the wing are known (Lift curve slope, etc.) the circulation at a given angle of attack with respect to the free stream is not known since it is determined by the downwash that in turn depends on the circulation. In this section and the next we will derive the fundamental equation that expresses this mutual dependence.
In Figure 12.8 the force on the wing is perpendicular to the local direction of the flow over the wing leading to a downstream tilting of the normal force relative to the free stream. As a consequence a three-dimensional lifting body will always have drag, even when the flow is inviscid. The downwash produces induced drag and also reduces the effective angle-of-attack of the wing leading to a reduction in lift. Figure 12.9 depicts the model flow we will use to determine the downwash velocity produced by a finite wing.
Figure 12.9 Wing and trailing vortex sheet model for inviscid lifting line theory.
The flow is inviscid and the vorticity shed into the wake at the trailing edge of the wing is assumed to form a smooth vortex sheet across which the transverse velocity (in the y-direction) is discontinuous. The sheet lies in the x, y( ) plane and extends to infinity in the x-direction. It is formed by summing an infinite system of streamwise vortex lines each characterized by a differential amount of circulation. The bound vorticity attached to the wing comprises the boundary layer on the wing. The boundary layer is viewed as an infinite system of differential vortices with their axes aligned with the y-direction. The induced motion of the bound vortices produces the circulation about the wing and the
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wing lift, but is assumed to contribute nothing to the downwash at the wing to the extent that the wing can be viewed purely as a “lifting line” of span b aligned with the y-axis.
Before we develop a model of the system shown in Figure 12.9 it is useful to recall the development of the Poisson equation solution described in Chapter 10. The flow field generated by each vortex element that makes up the wake vortex sheet will be derived below. This will allow us to build toward the relatively complex system shown in Figure 12.9. The Poisson equation for the vector potential is
∇2A x, y, z,t( ) = −Ω x, y, z,t( ) (12.20)
In Cartesian coordinates, (12.20) is equivalent to three scalar equations relating the Cartesian components of the vector potential to the Cartesian components of the vorticity.
∇2Ax = −Ωx ∇2Ay = −Ωy ∇2Az = −Ωz (12.21)
The fundamental solution of (12.20) at vector position x due to a point source of vorticity of strength Ω xs ,t( )dxsdysdzs at source point xs is
dA = −Ω xs ,t( )dxsdysdzs
4π x − xs (12.22)
where G x , xs( ) = −1 / 4π x − xs( ) is the Green’s function for Laplace’s equation.
Figure 12.10 Vorticity source distribution surrounded by irrotational flow
The general solution of (12.20) is obtained by integrating over the source distribution illustrated in Figure 12.10.
A x ,t( ) = 14π
Ω xs ,t( )x − xs
dxs dys dzs−∞
∞
∫−∞
∞
∫−∞
∞
∫ (12.23)
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In the examples below the vorticity field will be specified and (12.23) will be used to generate the vector potential. The velocity field, which is the primary quantity of interest, is generated from the vector potential using U = ∇ × A . The examples that apply to the wing wake have vorticity distributions that are extended to infinity by a limiting process. Although logarithmic singularities appear in the solutions they do not affect the velocity field generated from U = ∇ × A .
Example 1 – Vector potential of two semi-infinite lines of vortex monopoles.
Figure 12.11 shows two lines of vortex monopoles of uniform strength aligned in the x-direction and displaced a distance y0 from either side of the x, z{ } plane. The length of the distribution in the x-direction is a . We are interested in the vector potential for a→∞ .
Figure 12.11 Two parallel semi-infinite vortex lines of opposite sign
The vorticity source distribution for the vortex line terminating at x, y, z{ } = 0, y0 ,0{ } is
Ω+ x ,t( ) = Γu x( )δ y − y0( )δ z( ),0,0{ } (12.24)
and the vector potential is A+ = Ax+ ,0,0{ } where
Ax+ =
14π
Γu xs( )δ ys − y0( )δ zs( )x − xs( )2 + y − ys( )2 + z − zs( )2( )1/2
dxs dys dzs−∞
∞
∫−∞
∞
∫−∞
∞
∫ =
14π
Γ
x − xs( )2 + y − y0( )2 + z2( )1/20
∞
∫ dxs = lima→∞
−Γ4π
Lnx − a + x − a( )2 + y − y0( )2 + z2
x + x2 + y − y0( )2 + z2⎛
⎝⎜⎜
⎞
⎠⎟⎟
(12.25)
In the limit a→∞ the vector potential is
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Ax+ =
Γ4π
Ln x + x2 + y − y0( )2 + z2( ) − Ln y − y0( )2 + z2( ) − lima→∞Ln
12a
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
(12.26)
The vorticity source distribution for the vortex line terminating at x, y, z{ } = 0,−y0 ,0{ } is
Ω− x( ) = -Γu x( )δ y + y0( )δ z( ),0,0{ } (12.27)
and the vector potential is A− = A−x ,0,0{ } where
Ax− =
14π
−Γu xs( )δ ys + y0( )δ zs( )x − xs( )2 + y − ys( )2 + z − zs( )2( )1/2
dxs dys dzs−∞
∞
∫−∞
∞
∫−∞
∞
∫ =
−14π
Γ
x − xs( )2 + y + y0( )2 + z2( )1/20
∞
∫ dzs = lima→∞
Γ4π
Lnx − a + x − a( )2 + y + y0( )2 + z2
x + x2 + y + y0( )2 + z2⎛
⎝⎜⎜
⎞
⎠⎟⎟
(12.28)
In the limit a→∞ the vector potential of the semi-infinite vortex line at x, y, z{ } = 0,−y0 ,0{ } is
Ax− =
Γ4π
−Ln x + x2 + y + y0( )2 + z2( ) + Ln y + y0( )2 + z2( ) + lima→∞Ln
12a
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
(12.29)
The vector potential of the system is obtained by linear superposition of the vector potentials contributed by each line, A = A+ + A− . Notice that when we add (12.26) and (12.29) the logarithmic singularities at a→∞ cancel and the overall vector potential is A = Ax ,0,0{ } where
Ax =−Γ4π
Lnx + x2 + y + y0( )2 + z2( )1/2 y − y0( )2 + z2( )x + x2 + y − y0( )2 + z2( )1/2⎛
⎝⎞⎠ y + y0( )2 + z2( )
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
(12.30)
The velocity field generated by (12.30) is U = 0,Uy ,Uz{ } where
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Uy =−Γ
4π
−z y + y0( )2 + z2( ) − 2z x + x 2 + y − y0( )2 + z2( )1/ 2( ) x 2 + y − y0( )2 + z2( )1/ 2
x 2 + y − y0( )2 + z2( )1/ 2 x + x 2 + y − y0( )2 + z2( )1/ 2( ) y + y0( )2 + z2( )+
z y − y0( )2 + z2( ) + 2z x + x 2 + y + y0( )2 + z2( )1/ 2( ) x 2 + y + y0( )2 + z2( )1/ 2
x 2 + y + y0( )2 + z2( )1/ 2 x + x 2 + y + y0( )2 + z2( )1/ 2( ) y − y0( )2 + z2( )
⎛
⎝
⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟
Uz =Γ
4π
− y − y0( ) y + y0( )2 + z2( ) − 2 y + y0( ) x + x 2 + y − y0( )2 + z2( )1/ 2( ) x 2 + y − y0( )2 + z2( )1/ 2
x 2 + y − y0( )2 + z2( )1/ 2 x + x 2 + y − y0( )2 + z2( )1/ 2( ) y + y0( )2 + z2( )+
y + y0( ) y − y0( )2 + z2( ) + 2 y − y0( ) x + x 2 + y + y0( )2 + z2( )1/ 2( ) x 2 + y + y0( )2 + z2( )1/ 2
x 2 + y + y0( )2 + z2( )1/ 2 x + x 2 + y + y0( )2 + z2( )1/ 2( ) y − y0( )2 + z2( )
⎛
⎝
⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟
(12.31)
The downwash velocity generated from (12.31) on the line x, z{ } = 0,0{ } is
Uz 0, y,0( ) = Γ2π
y0y2 − y0
2( )⎛
⎝⎜
⎞
⎠⎟ (12.32)
plotted in Figure 12.12.
Figure 12.12 Downwash induced by two parallel semi-infinite vortex lines of opposite sign on the line x, z{ } = 0,0{ } viewed from the vortex wake (positive x).
At the center of the lifting line the downwash velocity is
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Uz 0,0,0( ) = −Γ2π y0
(12.33)
The downwash velocity generated on the line y, z{ } = 0,0{ } is
Uz x,0,0( ) = Γ2π y0
1− 2 x / y0 + 1+ x / y0( )2( )1/2⎛⎝
⎞⎠ 1+ x / y0( )2( )1/2
1+ x / y0( )2( )1/2 x / y0 + 1+ x / y0( )2( )1/2⎛⎝
⎞⎠
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
(12.34)
plotted below.
Figure 12.13 Downwash induced by two parallel semi-infinite vortex lines of opposite sign on the line y, z{ } = 0,0{ } .
Notice that half of the downwash occurs ahead of x = 0 and half occurs behind. In the far wake the downwash is
limz→−∞
Uy 0,0, z( ) = −Γπ y0
(12.35)
The tangential induced velocity of a single line vortex at a distance r = y0 is Uθr / Γ = 1 / 2π( ) . The far wake value (12.35) is just twice this velocity with contributions from the two line vortices of opposite sign equidistant from the x-axis..
Example 2 – Vector potential of a continuous distribution of vortex lines superimposed on a uniform flow.
Figure 12.14 shows a smooth distribution of vortex lines attached to the x-axis. This is the system we will use to model the wake vortex sheet.
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Figure 12.14 Continuous distribution of vortex lines attached to the y-axis.
The incremental circulation of the free vorticity generating the wake can be related to the incremental circulation of the bound vorticity on the wing using the contour shown in figure 12.15
Figure 12.15 Contour used to relate the incremental circulation on the wing to the incremental circulation shed into the wake
The contour in Figure 12.15 never penetrates the wake vortex sheet. Since the contour is imbedded entirely in potential flow we can write
Uic dC = Uic dC
CI∫!∫ + Uic dC
CII∫ + Uic dC
CIII∫ + Uic dC
CIV∫ = 0 (12.36)
Assume the contours are brought close together, separated only by a very small distance dy along the span of the wing. Then
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Uic dCCI∫ = Γ > 0
Uic dCCIII∫ = −Γ −
dΓdy
dy
Uic dCCII∫ + Uic dC
CIV∫ = −
dΓT .E
dydy
(12.37)
Insert (12.37) into (12.36). The result is
dΓT .E . = −dΓ (12.38)
The differential circulation at the trailing edge determines the local strength of the wake vortex sheet and is the appropriate source term for the Poisson equation governing the vector potential. The differential vector potential arising from the vortex sheet is therefore dA = dAx ,0,0{ } where
dAx =14π
dΓT .E . y0( )Lnx + x2 + y − y0( )2 + z2( )1/2⎛
⎝⎞⎠
y − y0( )2 + z2( )⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
(12.39)
Using (12.38)
dAx = −14π
dΓ y0( )Lnx + x2 + y − y0( )2 + z2( )1/2⎛
⎝⎞⎠
y − y0( )2 + z2( )⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
(12.40)
where Γ y0( ) is the circulation on the wing at station y0 . The full vector potential including the uniform velocity from the left is A = Ax ,0,U∞y{ } where Ax is
Ax = −14π
dΓ y0( )dy0
Lnx + x2 + y − y0( )2 + z2( )1/2⎛
⎝⎞⎠
y − y0( )2 + z2( )⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
−b /2
b /2
∫ dy0 (12.41)
and the integration is over the span of the wing. The lifting effect of the wing is assumed to be all concentrated on the lifting line on the y-axis between y = −b / 2 and y = b / 2 . The corresponding velocity field is U = U∞ ,Uy ,Uz{ } where
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Uy x, y, z( ) = −14π
dΓdy0−b /2
b /2
∫ ×
z y − y0( )2 + z2( )2 − 2z x + x2 + y − y0( )2 + z2( )1/2⎛⎝
⎞⎠ x2 + y − y0( )2 + z2( )1/2 y − y0( )2 + z2( )
x2 + y − y0( )2 + z2( )1/2 x + x2 + y − y0( )2 + z2( )1/2⎛⎝
⎞⎠ y − y0( )2 + z2( )2
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟dy0
Uz x, y, z( ) = 14π
dΓdy0−b /2
b /2
∫ ×
y − y0( ) y − y0( )2 + z2( )2 − 2 y − y0( ) x + x2 + y − y0( )2 + z2( )1/2⎛⎝
⎞⎠ x2 + y − y0( )2 + z2( )1/2 y − y0( )2 + z2( )
x2 + y − y0( )2 + z2( )1/2 x + x2 + y − y0( )2 + z2( )1/2⎛⎝
⎞⎠ y − y0( )2 + z2( )2
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟dy0
(12.42)
12.5 PRANDTL’S EQUATION OF FINITE WING THEORY
From the expression for the vertical velocity component in (12.42) the velocity vector on the lifting line, x, z{ } = 0,0{ } , −b / 2 < y < b / 2 is U = U∞ ,0,Uz{ } . From (12.42) at any spanwise station on the lifting line the downwash velocity is
Uz 0, y,0( ) = −14π
dΓ y0( )dy0
⎛⎝⎜
⎞⎠⎟−b /2
b /2
∫1
y − y0
⎛⎝⎜
⎞⎠⎟dy0 (12.43)
where the circulation distribution on the wing Γ y( ) may or may not be symmetric but must go to zero at the wing tips. The velocity relative to the wing is UR y( ) = U∞ ,0,Uz 0, y,0( ){ } with magnitude
UR y( ) = U∞2 +Uz 0, y,0( )2( )1/2 (12.44)
The angle of attack of the flow relative to the wing is
αR y( ) = α y( ) +α i y( ) (12.45)
where α y( ) is the angle of attack of the wing measured from the zero lift line as shown in Figure 12.8 and α i y( ) is the angle of attack reduction due to the downwash induced by the wake. Recall
α i y( ) = ArcTan Uz 0, y,0( )U∞
⎛⎝⎜
⎞⎠⎟< 0 (12.46)
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At small angles of attack the circulation about an infinite wing is proportional to the relative free stream speed and the relative angle of attack.
Γ y( ) = K y( )UR y( )αR y( ) (12.47)
where K y( ) is a shape factor that depends on the form and size of the airfoil section at the station y . Two-dimensional airfoil theory tells us that
K y( ) = 12a0 y( )C y( ) (12.48)
where C y( ) is the wing chord at a given spanwise location and
a0 y( ) = dCL
dαy( )⎛
⎝⎜⎞⎠⎟ (12.49)
is the slope of the lift coefficient versus angle of attack curve of a two-dimensional (infinite) wing with the same airfoil section. Combine (12.43), (12.45), (12.46) and (12.47) to generate
Γ y( ) = 12a0 y( )C y( )UR y( ) ×
α y( ) − ArcTan 14πU∞
dΓ y0( )dy0
⎛⎝⎜
⎞⎠⎟−b /2
b /2
∫1
y − y0( )⎛
⎝⎜⎞
⎠⎟dy0
⎛
⎝⎜
⎞
⎠⎟
⎛
⎝⎜
⎞
⎠⎟
(12.50)
Equation (12.50) is an integro-differential equation for the circulation. Since the downwash speed is small compared to the free stream speed we can approximate (12.50) using UR ≅U∞ and ArcTan Uz 0, y,0( ) /U∞( ) ≅Uz 0, y,0( ) /U∞ . The equation simplifies to
Γ y( ) = 12a0 y( )C y( ) U∞α y( ) − 1
4πdΓ y0( )dy0
⎛⎝⎜
⎞⎠⎟−b /2
b /2
∫1
y − y0( )⎛
⎝⎜⎞
⎠⎟dy0
⎛
⎝⎜
⎞
⎠⎟ (12.51)
Equation (12.51) is known as Prandtl’s equation of finite wing theory and is the fundamental equation that needs to be solved for Γ y( ) subject to the condition that the circulation falls to zero at the wing tips.
Γb2
⎛⎝⎜
⎞⎠⎟= Γ −
b2
⎛⎝⎜
⎞⎠⎟= 0 (12.52)
To study solutions of (12.51) lets first consider a case where the circulation distribution is known.
18
12.6 ELLIPTIC LIFT DISTRIBUTION
Let’s look at the case of a wing with an elliptic lift distribution. Let the circulation along the span be described by
Γ y( )Γ0
= 1− 2yb
⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟
1/2
(12.53)
with derivative
dΓ y( )dy
= −4Γ0
b
2yb
⎛⎝⎜
⎞⎠⎟
1− 2yb
⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟
1/2 (12.54)
This situation is illustrated in Figure 12.16
Figure 12.16 Elliptical load distribution
The total lift in this case is
L =12ρU∞Γ0b 1− 2y
b⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟
1/2
d 2yb
⎛⎝⎜
⎞⎠⎟−1
1
∫ = ρU∞π4Γ0b
⎛⎝⎜
⎞⎠⎟
(12.55)
and the downwash velocity is
Uz 0, y,0( ) = Γ0
2πb
2y0b
⎛⎝⎜
⎞⎠⎟
1− 2y0b
⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟
1/2−1
1
∫1
2yb
− 2y0b
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟d 2y0
b⎛⎝⎜
⎞⎠⎟
(12.56)
19
Let 2y / b = Sin θ( ) and 2y0 / b = Sin θ0( )
Uz 0,θ,0( ) = Γ0
2πbSin θ0( )Cos θ0( )
1− Sin2 θ0( )( )1/2 Sin θ( ) − Sin θ0( )( )−π /2
π /2
∫ dθ0 (12.57)
Equation (12.57) simplifies to
Uz 0,θ,0( ) = Γ0
2πbSin θ0( )
Sin θ( ) − Sin θ0( )( )−π /2
π /2
∫ dθ0 (12.58)
The downwash is
Uz 0,θ,0( ) = Γ0
2πb−θ0 + Tan θ( )Ln Cos θ0 +θ( )
Sin θ0 −θ( )⎛
⎝⎜⎞
⎠⎟⎛
⎝⎜
⎞
⎠⎟
π /2
−π /2
(12.59)
When (12.59) is evaluated at the limits, the result is.
Uz 0, y,0( ) = −Γ0
2b (12.60)
In this special case of a circulation distribution given by (12.53) the downwash is constant along the lifting line. The integral treated above is a special case of the following integral.
Cos nφ( )
Cos φ( ) − Cos θ( )0
π
∫ dφ = πSin nθ( )Sin θ( ) (12.61)
Actually this useful integral is modified by letting φ = !φ + π / 2 and θ = !θ + π / 2 .
Cos n !φ + π / 2( )( )Cos !φ + π / 2( ) − Cos !θ + π / 2( )0
π
∫ d !φ + π / 2( ) =
Sin n !φ( )Sin !φ( ) − Sin θ( )−π /2
π /2
∫ d !φ = πCos n !θ( )Cos !θ( )
(12.62)
The downwash velocity everywhere is
20
Uz x, y, z( ) = 14π
dΓdy0−b /2
b /2
∫ ×
y − y0( ) y − y0( )2 + z2( )2 − 2 y − y0( ) x + x2 + y − y0( )2 + z2( )1/2⎛⎝
⎞⎠ x2 + y − y0( )2 + z2( )1/2 y − y0( )2 + z2( )
x2 + y − y0( )2 + z2( )1/2 x + x2 + y − y0( )2 + z2( )1/2⎛⎝
⎞⎠ y − y0( )2 + z2( )2
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟dy0
(12.63)
On the x-axis, y, z{ } = 0,0{ } , the downwash velocity is
Uz x,0,0( ) = −14π
dΓdy0−b /2
b /2
∫y02 − 2 x + x2 + y0
2( )1/2( ) x2 + y02( )1/2
y0 x2 + y02( )1/2 x + x2 + y0
2( )1/2( )⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ dy0
(12.64)
Let ς = y0 / b and !x = x / b . The downwash along the x-axis for an elliptic wing is
Uz !x,0,0( ) = Γ0
πb1
1− 4ς 2( )1/2−1/2
1/2
∫ς 2 − 2 !x + !x2 + ς 2( )1/2( ) !x2 + ς 2( )1/2
!x2 + ς 2( )1/2 !x + !x22 + ς 2( )1/2( )⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ dς
(12.65)
Equation (12.65) integrates to
Uz !x,0,0( ) = Γ0
2πb−ArcSin
2y0b
⎛⎝⎜
⎞⎠⎟− EllipticF ArcSin
2y0b
⎛⎝⎜
⎞⎠⎟,− 1
2xb
⎛⎝⎜
⎞⎠⎟2
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟−1/2
1/2
(12.66)
where EllipticF is the elliptic integral of the first kind. Equation (12.66) evaluates to
Uz !x,0,0( ) = −Γ0
πbπ2+ Sign x( ) × EllipticK −
12xb
⎛⎝⎜
⎞⎠⎟2
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
(12.67)
where EllipticK is the complete elliptic integral of the first kind. In the far wake
limx→∞
Uz x,0,0( ) = −Γ0
b (12.68)
21
One half of the downwash occurs ahead of the lifting line and one half occurs downstream. The downwash velocity along the x-axis, (12.67) , is plotted below.
Figure 12.17 Downwash induced along the x-axis by a continuous distribution of semi-infinite vortex lines of attached to the y-axis for the case of elliptic loading. Note the somewhat larger magnitude compared to two single vortex lines.
Substitute the circulation distribution (12.53) and the downwash (12.60) into (12.51).
Γ0 1−2yb
⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟
1/2
=12a0 y( )C y( ) U∞α y( ) − Γ0
2b⎛⎝⎜
⎞⎠⎟
(12.69)
There is an infinite variety of airfoils with different lift slopes a0 y( ) , chord distributions C y( ) and angle of attack distributions α y( ) that can generate an elliptic lift distribution. However if we assume the wing has the same cross-section geometry all along the span and that the angle of attack is constant as well, then a0 and α are constant and (12.69) can be solved for the chord distribution.
C y( ) = C0 1−2yb
⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟
1/2
(12.70)
where
C0 =4bΓ0
2bU∞a0α − a0Γ0( ) (12.71)
is the chord of the wing at midspan. The result (12.70) tells us that to achieve an elliptic lift distribution on an untwisted wing the chord distribution should also be elliptic. Perhaps the most famous example of an aircraft with an elliptic wing is the British Spitfire used in World War II. The image below shows the aircraft seen from the side and below during a turn.
22
Figure 12.18 British Spitfire showing elliptic planform wing. Note the wing is formed from two ellipses of different minor axis. This shifts the major axis and center of lift forward.
Solve for the circulation at midspan.
Γ0 =2bU∞a0C0αa0C0 + 4b
(12.72)
Introduce the lift coefficient and use (12.55). Equation (12.72) becomes
CL =L
12ρU∞
2S=π2ρU∞bρU∞
2SΓ0 = π b
2
S1
1+ 4ba0C0
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟α (12.73)
Recall the definition of the wing aspect ratio.
AR =b2
S (12.74)
For an ellipse with major axis b and minor axis C0 the area is S = πC0b / 4 . For this shape the aspect ratio is AR = 4b /πC0 . Equation (12.73) for the lift coefficient becomes
CL =a0α
1+ a0πAR
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
(12.75)
23
an elegantly simple result. Recall that a0 = limα→0dCL / dα( ) for a wing of infinite aspect
ratio. The result (12.75) tells us how to adjust the lift curve slope for a wing of similar shape but finite aspect ratio.
If the wing is thin, theory tells us that a0 = 2π . In this classical case (12.75) becomes
CL
2πα=
AR
2 + AR
⎛⎝⎜
⎞⎠⎟
(12.76)
plotted in Figure 12.19
Figure 12.19 The effect of aspect ratio on the lift slope of a thin elliptical wing.
The result (12.76) tells us that the lift curve slope of a finite wing is always lower than that of its two-dimensional counterpart. This is the price exacted by the work needed to create the wake.
12.7 DRAG DUE TO LIFT OF AN ELLIPTIC WING
So far we have focused on the lift characteristics and plan-form shape of an elliptical wing. To complete the picture we need to relate the downwash velocity at the lifting line to the both the lift and induced drag of the wing. From Figure 12.3 the lift is
L = F⊥Cos α i( ) = F⊥ U∞
U∞2 +Uz 0,0,0( )2( )1/2
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
(12.77)
where F⊥ is the force on the wing perpendicular to the flow velocity vector at the centerline position of the wing, U 0,0,0( ) = U∞ ,0,Uz 0,0,0( ){ } . The induced drag is
24
Di = F⊥Sin −α i( ) = F⊥−Uz 0,0,0( )
U∞2 +Uz 0,0,0( )2( )1/2
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
(12.78)
For an elliptically loaded wing
F⊥ =π4
⎛⎝⎜
⎞⎠⎟ρ U∞
2 +Uz 0,0,0( )2( )1/2 Γ0b (12.79)
The lift is
L =π4
⎛⎝⎜
⎞⎠⎟ρ U∞
2 +Uz 0,0,0( )2( )1/2 Γ0bU∞
U∞2 +Uz 0,0,0( )2( )1/2
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟=
π4
⎛⎝⎜
⎞⎠⎟ρU∞Γ0b (12.80)
and the induced drag is
Di =π4
⎛⎝⎜
⎞⎠⎟ρ U∞
2 +Uz 0,0,0( )2( )1/2 Γ0b−Uz 0,0,0( )
U∞2 +Uz 0,0,0( )2( )1/2
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟= −
π4
⎛⎝⎜
⎞⎠⎟ρUz 0,0,0( )Γ0b
(12.81)
Insert the downwash velocity from (12.60) into (12.81).
Di =π8
⎛⎝⎜
⎞⎠⎟ρΓ0
2 (12.82)
The lift coefficient of the wing is
CL =L
12ρU∞
2S=π2Γ0bU∞S
(12.83)
where S is the planform area of the wing. The drag coefficient is
CDi=
Di
12ρU∞
2S=π4
Γ02
U∞2S
(12.84)
Use (12.83) to replace Γ0 in (12.84). The result is the classical relation for the induced drag or drag due to lift.
25
CDi=1πCL2 Sb2
⎛⎝⎜
⎞⎠⎟=1πCL2
AR
(12.85)
where AR = b2 / S is the aspect ratio of the wing. Equation (12.85) is plotted below for
AR = 5
Figure 12.20 Lift to drag parabola for an elliptical wing with aspect ratio AR = 5
Using (12.76)
CDi
4πα 2 = AR1
2 + AR
⎛⎝⎜
⎞⎠⎟
2
(12.86)
plotted below
Figure 12.21 The effect of aspect ratio on the induced drag slope of a thin elliptical wing
26
The lift to drag ratio of the wing as a function of aspect ratio can be determined by combining (12.76) and (12.86).
CLαCDi
= 1+ AR
2 (12.87)
It is clear from the results of lifting line theory for elliptic wings that an aircraft designed to stay aloft with very little power and/or for long periods of time needs to have a large aspect ratio. This is typically the case for gliders and high altitude reconnaissance aircraft such as the U2 shown below.
Figure 12.22 Solar powered aircraft, left and U2 reconnaissance aircraft, right.
The unique feature of the elliptically loaded wing is that the downwash is constant along the span. This leads to simple analytical results for the lift, drag and shape of the wing. The really fortunate thing about these results is that they are not only elegant but important as well. The main reason is that all slender wings, whether they are rectangular, diamond or trapezoidal shaped can be viewed as modest variations away from the elliptic case. As for non-slender wings the elliptic case tells us how far from two-dimensional the wing behavior is as the aspect ratio becomes small.
12.8 GENERAL WING LOADINGS
Now return to (12.51) rewritten here for convenience.
Γ y( ) = 12a0 y( )C y( ) U∞α y( ) − 1
4πdΓ y0( )dy0
⎛⎝⎜
⎞⎠⎟−b /2
b /2
∫1
y − y0( )⎛
⎝⎜⎞
⎠⎟dy0
⎛
⎝⎜
⎞
⎠⎟ (12.88)
We want to now consider more general distributions of Γ y( ) subject to the condition that Γ −b / 2( ) = 0 and Γ b / 2( ) = 0 . To approach this we will use a generalization of the result (12.68) for an elliptical wing. That is that the downwash from the trailing vortex pair at x→∞ is twice the downwash felt at the lifting line.
Uz 0,0,0( ) = Uz ∞,0,0( )2
(12.89)
27
Assume that (12.89) holds at every point along the span of the wing. Let
Uz 0, y,0( ) = limx→∞
Uz x, y,0( )2
(12.90)
Now (12.88) becomes
Γ y( ) = 12a0 y( )C y( ) U∞α y( ) + lim
x→∞
Uz x, y,0( )2
⎛⎝⎜
⎞⎠⎟
(12.91)
If we could just figure out the distribution of downwash in the far wake as a function of y we should be able to infer the distribution of circulation on the wing from (12.91).
Figure 12.23 depicts a large control volume containing the wake produced by an aircraft. In the far wake the mutual induction of the vortex sheet as well as viscous effects leads to a roll up and diffusion of the sheet leaving the wing trailing edge. Far downstream of the aircraft the wake is intersected by a plane perpendicular to the free stream direction. This plane is called the Trefftz plane after Erich Trefftz (1888-1937) a German applied mathematician who worked with Prandtl as well as his uncle Carl Runge and others at the famed Gottingen school of mathematics and physics in the early 1900s.
Figure 12.23 Trefftz plane intersecting the rolled up wake far behind an aircraft.
Figure 12.24 depicts the much more idealized case we will actually treat.
28
Figure 12.24 Trefftz plane intersecting the flat, straight vortex sheet from a wing
In Figure 12.24 the flow is assumed to be irrotational everywhere except within the vortex sheet. The velocity field in the y, z{ } plane is given by
limx→∞
Uy x, y, z( ) = 12π
dΓdy0−b /2
b /2
∫z
y − y0( )2 + z2( )⎛
⎝⎜⎜
⎞
⎠⎟⎟dy0
limx→∞
Uz x, y, z( ) = −12π
dΓdy0−b /2
b /2
∫y − y0( )
y − y0( )2 + z2( )⎛
⎝⎜⎜
⎞
⎠⎟⎟dy0
(12.92)
If the wing loading is elliptic then (12.92) with (12.53) becomes
Uy y, z( ) = −2Γ0
πb
2y0b
⎛⎝⎜
⎞⎠⎟
1− 2y0b
⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟
1/2−1
1
∫2zb
⎛⎝⎜
⎞⎠⎟
2yb
− 2y0b
⎛⎝⎜
⎞⎠⎟2
+ 2zb
⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
d 2y0b
⎛⎝⎜
⎞⎠⎟
Uz y, z( ) = 2Γ0
πb
2y0b
⎛⎝⎜
⎞⎠⎟
1− 2y0b
⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟
1/2−1
1
∫2yb
− 2y0b
⎛⎝⎜
⎞⎠⎟
2yb
− 2y0b
⎛⎝⎜
⎞⎠⎟2
+ 2zb
⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
d 2y0b
⎛⎝⎜
⎞⎠⎟
(12.93)
Let 2y / b = Cos θ( ) and 2y0 / b = Cos θ0( )
29
Uy y, z( ) = 2Γ0
πb
Cos θ0( ) 2zb
⎛⎝⎜
⎞⎠⎟
Cos θ( ) − Cos θ0( )( )2 + 2zb
⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
dθ0−π
0
∫
Uz y, z( ) = −2Γ0
πbCos θ0( ) Cos θ( ) − Cos θ0( )( )Cos θ( ) − Cos θ0( )( )2 + 2z
b⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
dθ0−π
0
∫
(12.94)
The velocity field (12.94) is shown in the Trefftz plane in Figure 12.25.
Figure 12.25 Flow in the Trefftz shown for an elliptically loaded wing.
Note the anti-symmetry of the flow direction across the vortex sheet and the discontinuity in the span-wise velocity component implied by the flow pattern in Figure 12.25.
uy y,0+( ) = −uy y,0−( ) (12.95)
The difference in spanwise velocity across the sheet is directly related to the circulation gradient on the wing
dΓ y( )dy
= uy y,0+( ) − uy y,0−( ) = 2uy y,0+( ) (12.96)
30
Since the flow in the Trefftz plane is irrotational everywhere except within the vortex sheet the velocity field can be generated from a potential u = ∇φ y, z( ) with velocity components
limx→∞
Uy x, y, z( ) = uy y, z( ) = ∂φ∂y
limx→∞
Uz x, y, z( ) = uz y, z( ) = ∂φ∂z
(12.97)
The circulation on the wing and velocity potential in the Trefftz plane are connected by
dΓ y( )dy
= uy y,0+( ) − uy y,0−( ) = 2 ∂φ∂y
y,0+( ) (12.98)
Therefore
Γ y( ) = 2φ y,0+( ) or Γ y( ) = −2φ y,0−( ) (12.99)
Note that the downwash velocity
limx→∞
Uz x, y,0( ) = uz y,0( ) = ∂φ∂z
y,0( ) (12.100)
is continuous across the vortex sheet as can be seen in Figure 12.25. The Prandtl equation (12.91) can be expressed in terms of the velocity potential as
Γ y( ) = 12a0 y( )C y( ) U∞α y( ) + 1
2∂φ∂z
y,0( )⎛⎝⎜
⎞⎠⎟ (12.101)
The problem boils down to determining the Trefftz plane potential φ y, z( ) . The problem formulation is as follows.
1) The potential satisfies Laplace’s equation
∇2φ =∂2φ∂y2
+∂2φ∂z2
= 0 (12.102)
2) The velocity goes to zero at large distances from the vortex sheet.
limy2 + z2 →∞
∇φ → 0 (12.103)
3) The velocity potential is an odd function of z .
φ y, z( ) = −φ y,−z( ) (12.104)
31
The condition (12.104) comes from
φ y,0+( ) = −φ y,0−( ) for −b / 2 < y < b / 2 (12.105)
and
φ y,0( ) = 0 for y < −b / 2 and y > b / 2 (12.106)
In addition, at the end points of the vortex sheet
φ −b2,0⎛
⎝⎜⎞⎠⎟= φ b
2,0⎛
⎝⎜⎞⎠⎟= 0 (12.107)
4) The Prandtl equation provides the boundary condition
2φ y,0+( ) = 12 a0 y( )C y( ) U∞α y( ) + 12∂φ∂z
y,0( )⎛⎝⎜
⎞⎠⎟ for −b / 2 < y < b / 2 (12.108)
where (12.99) has been used.
Since we are solving a potential flow problem in all the solutions are harmonic functions and we can use methods of complex analysis. Define the complex variable
ξ = y + iz = ρeiϑ = ρ Cos ϑ( ) + iSin ϑ( )( ) (12.109)
where i = −1 and the complex potential
F ξ( ) = φ y, z( ) + iψ y, z( ) (12.110)
The velocity components in the y, z{ } plane are related to the potentials by
uy y, z( ) = ∂φ∂y
uz y, z( ) = ∂φ∂z
(12.111)
and
uy y, z( ) = ∂ψ∂z
uz y, z( ) = −∂ψ∂y
(12.112)
The potentials satisfy the Cauchy-Riemann conditions
∂φ∂y
=∂ψ∂z
∂φ∂z
= −∂ψ∂y
(12.113)
32
The derivative of an analytic function is independent of the path in the complex plane along which Δξ → 0 . Therefore the complex velocity can be either
W ξ( ) = dFdξ
=dFdy
1dξ / dy
⎛⎝⎜
⎞⎠⎟=∂φ y, z( )
∂y+ i
∂ψ y, z( )∂y
= uy y, z( ) − iuz y, z( ) (12.114)
or
W ξ( ) = dFdξ
=dFdz
1dξ / dz
⎛⎝⎜
⎞⎠⎟= −i
∂φ y, z( )∂z
+∂ψ y, z( )
∂z= uy y, z( ) − iuz y, z( ) (12.115)
Either derivative generates the same velocity field.
In this section we will be primarily concerned with the velocity potential φ z, y( ) . In the search for F ξ( ) it is convenient to use conformal (angle preserving) mapping to map the ξ plane to a new complex plane η defined by the conformal transformation
ξ η( ) = η +b / 4( )2η
(12.116)
where
η = p + iq = reiθ = r Cos θ( ) + iSin θ( )( ) (12.117)
and
r = p2 + q2 θ = ArcTan qp
⎛⎝⎜
⎞⎠⎟
(12.118)
The inverse of (12.116) is
η =ξ2±12
ξ2 − b2
⎛⎝⎜
⎞⎠⎟2
(12.119)
The real and imaginary parts of ξ and η are related by
y = p 1+ b4
⎛⎝⎜
⎞⎠⎟2 1p2 + q2
⎛
⎝⎜⎞
⎠⎟ z = q 1− b
4⎛⎝⎜
⎞⎠⎟2 1p2 + q2
⎛
⎝⎜⎞
⎠⎟ (12.120)
or
33
y = Cos θ( ) r +b4
⎛⎝⎜
⎞⎠⎟2 1r
⎛
⎝⎜⎞
⎠⎟ z = Sin θ( ) r − b
4⎛⎝⎜
⎞⎠⎟2 1r
⎛
⎝⎜⎞
⎠⎟ (12.121)
The mapping in cylindrical coordinates is
ρ =b4
16r2+ r2 +
b2
2Cos 2θ( ) ϑ = ArcTan
r − b2
4r
r + b2
4r
Tan θ( )⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
(12.122)
The mapping (12.116) takes the line z = 0 , −b / 2 < y < b / 2 in the ξ plane to the circle
ηSheet =b4
⎛⎝⎜
⎞⎠⎟Cos θ( ) + iSin θ( )( ) = b
4⎛⎝⎜
⎞⎠⎟eiθ (12.123)
in the η plane as shown in Figure 12.26.
Figure 12.26 Mapping the vortex sheet to a circle
The flow we are seeking is outside the vortex sheet in the ξ plane and outside the circle in the η plane. Let the complex potential in the η plane be
G η( ) = ϕ r,θ( ) + iζ r,θ( ) (12.124)
Under the conformal mapping (12.116) the complex potential in the ξ plane is
G η( ) = F ξ η( )( ) (12.125)
and the complex potential in the ξ plane is generated from
F ξ( ) = G η ξ( )( ) (12.126)
34
Following (12.126) the scalar potentials in the two planes are related by
φ y, z( ) = ϕ r y, z( ),θ y, z( )( ) (12.127)
where y, z( ) and r,θ( ) are related by (12.121)
Now work out G η( ) . Assume that G η( ) is of the form
G η( ) = bn + ianηn
⎛⎝⎜
⎞⎠⎟n=1
∞
∑ (12.128)
In polar coordinates (12.117) the complex potential is
G η( ) = bnCos nθ( ) + anSin nθ( )( ) + i anCos nθ( ) − bnSin nθ( )( )rn
⎛
⎝⎜⎞
⎠⎟n=1
∞
∑ (12.129)
and the scalar potential in the η plane is
ϕ r,θ( ) = bnCos nθ( ) + anSin nθ( )( )rn
⎛
⎝⎜⎞
⎠⎟n=1
∞
∑ (12.130)
The conditions (12.104), (12.105), (12.106) and (12.107) in the ξ plane imply the requirement
ϕ r,θ( ) = −ϕ r,−θ( ) for r ≥ b / 4 (12.131)
ϕ r,0( ) = ϕ r,π( ) for r > b / 4 (12.132)
Conditions (12.131) and (12.132) imply that the coefficients of the symmetric terms in (12.130) must all be zero. Now
ϕ r,θ( ) = anSin nθ( )rnn=1
∞
∑ (12.133)
The complex potential in the η plane is
G η( ) = i anηn
⎛⎝⎜
⎞⎠⎟n=1
∞
∑ (12.134)
From (12.127) the scalar potential in the ξ plane is
35
φ y, z( ) = ϕ r y, z( ),θ y, z( )( ) = anSin nθ y, z( )( )r y, z( )nn=1
∞
∑ (12.135)
From (12.126) the complex velocity is
W ξ( ) = uy y, z( ) − iuz y, z( ) = dFdξ
=dGdη
1dξ / dη
⎛⎝⎜
⎞⎠⎟
(12.136)
Substitute (12.134) into (12.136) and differentiate (12.116). The complex velocity in the ξ plane is
uy y, z( ) − iuz y, z( ) = −i nanηn+1
⎛⎝⎜
⎞⎠⎟n=1
∞
∑ η2
η2 − b4
⎛⎝⎜
⎞⎠⎟2
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
(12.137)
The result (12.137) can be expressed in terms of r,θ( )
uy y, z( ) − iuz y, z( ) = −i nane−nθ
rn+1⎛⎝⎜
⎞⎠⎟n=1
∞
∑ r2
r2eiθ − b4
⎛⎝⎜
⎞⎠⎟2
e− iθ
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
(12.138)
Separate the right hand side of (12.138) into real and imaginary parts.
uy y, z( ) − iuz y, z( ) = −inane
− inθ
rn+1⎛⎝⎜
⎞⎠⎟n=1
∞
∑ r2
r2eiθ − b4
⎛⎝⎜
⎞⎠⎟2
e− iθ
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
r2e− iθ − b4
⎛⎝⎜
⎞⎠⎟2
eiθ
r2e− iθ − b4
⎛⎝⎜
⎞⎠⎟2
eiθ
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
=
−
nan
r2 + b4
⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟Cos nθ( )Sin θ( ) + r2 − b
4⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟Sin nθ( )Cos θ( )
⎛
⎝⎜
⎞
⎠⎟ +
i r2 − b4
⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟Cos nθ( )Cos θ( ) − r2 + b
4⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟Sin nθ( )Sin θ( )
⎛
⎝⎜
⎞
⎠⎟
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
rn−1 r4 + b4
⎛⎝⎜
⎞⎠⎟4
− 2 b4
⎛⎝⎜
⎞⎠⎟2
r2Cos 2θ( )⎛
⎝⎜⎞
⎠⎟
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
n=1
∞
∑
(12.139)
36
Rearrange the complex velocity (12.139).
uy y, z( ) − iuz y, z( ) = ∂φ y, z( )∂y
− i∂φ y, z( )
∂z=
−
nan r2 + b4
⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟Cos nθ( )Sin θ( ) + r2 − b
4⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟Sin nθ( )Cos θ( )
⎛
⎝⎜
⎞
⎠⎟
⎛
⎝⎜
⎞
⎠⎟
rn−1 r4 + b4
⎛⎝⎜
⎞⎠⎟4
− 2 b4
⎛⎝⎜
⎞⎠⎟2
r2Cos 2θ( )⎛
⎝⎜⎞
⎠⎟
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
n=1
∞
∑
−i
nan r2 − b4
⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟Cos nθ( )Cos θ( ) − r2 + b
4⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟Sin nθ( )Sin θ( )
⎛
⎝⎜
⎞
⎠⎟
⎛
⎝⎜
⎞
⎠⎟
rn−1 r4 + b4
⎛⎝⎜
⎞⎠⎟4
− 2 b4
⎛⎝⎜
⎞⎠⎟2
r2Cos 2θ( )⎛
⎝⎜⎞
⎠⎟
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
n=1
∞
∑
(12.140)
The downwash velocity in the ξ plane is
∂φ y, z( )∂z
=
nan r2 − b4
⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟Cos nθ( )Cos θ( ) − r2 + b
4⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟Sin nθ( )Sin θ( )
⎛
⎝⎜
⎞
⎠⎟
⎛
⎝⎜
⎞
⎠⎟
rn−1 r4 + b4
⎛⎝⎜
⎞⎠⎟4
− 2 b4
⎛⎝⎜
⎞⎠⎟2
r2Cos 2θ( )⎛
⎝⎜⎞
⎠⎟
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
n=1
∞
∑ (12.141)
Refer back to the Prandtl equation (12.108). We need to determine ∂φ y,0( ) / ∂z .
∂φ y, z( )∂z z=0
=
nan r2 − b4
⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟Cos nθ( )Cos θ( ) − r2 + b
4⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟Sin nθ( )Sin θ( )
⎛
⎝⎜
⎞
⎠⎟
⎛
⎝⎜
⎞
⎠⎟
rn−1 r4 + b4
⎛⎝⎜
⎞⎠⎟4
− 2 b4
⎛⎝⎜
⎞⎠⎟2
r2Cos 2θ( )⎛
⎝⎜⎞
⎠⎟
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
n=1
∞
∑
r=b /4
=
−nanSin nθ( )Sin θ( )b4
⎛⎝⎜
⎞⎠⎟n+1
1− Cos 2θ( )( )
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
n=1
∞
∑ = −nan
2 b4
⎛⎝⎜
⎞⎠⎟n+1
Sin nθ( )Sin θ( )
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
n=1
∞
∑
(12.142)
37
Now
∂φ y, z( )
∂z z=0
= −nan
2 b4
⎛⎝⎜
⎞⎠⎟n+1
Sin nθ( )Sin θ( )
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
n=1
∞
∑ (12.143)
The Prandtl equation (12.108) also requires φ y,0+( ) which we can get from (12.135).
φ y, z( ) z=0+ =anSin nθ y, z( )( )
r y, z( )nn=1
∞
∑r=b /4
=anSin nθ( )
b4
⎛⎝⎜
⎞⎠⎟n
n=1
∞
∑ (12.144)
Now substitute (12.143) and (12.144) into (12.108) repeated here in a slightly different form for convenience.
φ y,0+( )U∞b
−18a0 y( ) C y( )
b⎛⎝⎜
⎞⎠⎟1U∞
∂φ∂z
y,0( ) = 14a0 y( ) C y( )
b⎛⎝⎜
⎞⎠⎟α y( ) for −
b2< y < b
2
(12.145)
The steps are
φ y,0+( )U∞b
−18a0 y( ) C y( )
b⎛⎝⎜
⎞⎠⎟1U∞
∂φ∂z
y,0( ) = 14a0 y( ) C y( )
b⎛⎝⎜
⎞⎠⎟α y( )
1U∞b
anSin nθ( )b4
⎛⎝⎜
⎞⎠⎟n
n=1
∞
∑ +18a0 y( ) C y( )
b⎛⎝⎜
⎞⎠⎟1U∞
nan
2 b4
⎛⎝⎜
⎞⎠⎟n+1
Sin nθ( )Sin θ( )
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
n=1
∞
∑ =14a0 y( ) C y( )
b⎛⎝⎜
⎞⎠⎟α y( )
anSin nθ( )
U∞bb4
⎛⎝⎜
⎞⎠⎟n
n=1
∞
∑ Sin θ( ) + na0 y( )C y( )4b
⎛⎝⎜
⎞⎠⎟= a0 y( ) C y( )
4b⎛⎝⎜
⎞⎠⎟α y( )Sin θ( )
anSin nθ( )
U∞bb4
⎛⎝⎜
⎞⎠⎟n Sin θ( ) + na0 y( )C y( )
4b⎛⎝⎜
⎞⎠⎟
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
n=1
∞
∑ = a0 y( ) C y( )4b
⎛⎝⎜
⎞⎠⎟α y( )Sin θ( )
(12.146)
Finally we have the solution (12.135)
38
φ y, z( ) = anSin nθ y, z( )( )r y, z( )nn=1
∞
∑ (12.147)
The coefficients in the series in (12.147) are determined from the last relation in (12.146).
anSin nθ( )
U∞bb4
⎛⎝⎜
⎞⎠⎟n Sin θ( ) + n a0 y( )C y( )
4b⎛⎝⎜
⎞⎠⎟
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
n=1
∞
∑ =a0 y( )C y( )
4b⎛⎝⎜
⎞⎠⎟α y( )Sin θ( ) (12.148)
Equation (12.148) is evaluated on the vortex sheet 2y / b = Cos θ( ) . The range of θ is −π < θ < 0 . Equation (12.148) for the coefficients becomes
anSin nθ( )
U∞bb4
⎛⎝⎜
⎞⎠⎟n Sin θ( ) +
na0b2Cos θ( )⎛
⎝⎜⎞⎠⎟C
b2Cos θ( )⎛
⎝⎜⎞⎠⎟
4b
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
n=1
∞
∑ =
a0 y( )C b2Cos θ( )⎛
⎝⎜⎞⎠⎟
4b
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟α b2Cos θ( )⎛
⎝⎜⎞⎠⎟Sin θ( )
(12.149)
Check (12.147) and (12.149) against the case of an elliptical wing
C y( ) = C0 1−2yb
⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟
1/2
= C0Sin θ( ) (12.150)
39
anSin nθ( )
U∞bb4
⎛⎝⎜
⎞⎠⎟n Sin θ( ) + na0 y( )C y( )
4b⎛⎝⎜
⎞⎠⎟
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
n=1
∞
∑ = a0 y( ) C y( )4b
⎛⎝⎜
⎞⎠⎟α y( )Sin θ( )
C y( ) = C0Sin θ( )
anSin nθ( )
U∞bb4
⎛⎝⎜
⎞⎠⎟n Sin θ( ) + na0 y( )C0Sin θ( )
4b⎛⎝⎜
⎞⎠⎟
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
n=1
∞
∑ = a0 y( ) C0Sin θ( )4b
⎛⎝⎜
⎞⎠⎟α y( )Sin θ( )
anSin nθ( )
U∞bb4
⎛⎝⎜
⎞⎠⎟n 1+ na0C0
4b⎛⎝⎜
⎞⎠⎟
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
n=1
∞
∑ = a0C0Sin θ( )4b
⎛⎝⎜
⎞⎠⎟α
an = 0,n = 2,3,...
1 = a0a1
C0
4⎛⎝⎜
⎞⎠⎟U∞
b4
⎛⎝⎜
⎞⎠⎟α −
a0C0
4b= C0
a04a1
U∞b4
⎛⎝⎜
⎞⎠⎟α −
a04b
⎛⎝⎜
⎞⎠⎟
C0 =4b
a0U∞
a1b2
⎛⎝⎜
⎞⎠⎟2
α −1⎛
⎝⎜⎞
⎠⎟
(12.151)
Compare the last equation in (12.151) to
C0 =4bΓ0
2bU∞a0α − a0Γ0( ) =4b
a02bU∞
Γ0
α −1⎛⎝⎜
⎞⎠⎟
=
C0 =4b
a0U∞
Γ0b / 8( )b2
⎛⎝⎜
⎞⎠⎟2
α −1⎛
⎝⎜⎞
⎠⎟
(12.152)
The results are identical if we choose the first coefficient in the series (12.147) to be
a1 =Γ0b8
(12.153)
For the elliptic wing the downwash at infinity is
40
∂φ y, z( )∂z z=0
= −8a1b2
⎛⎝⎜
⎞⎠⎟
a1 =Γ0b8
∂φ y, z( )∂z z=0
= −Γ0
b⎛⎝⎜
⎞⎠⎟
(12.154)
which is twice the downwash at the wing.
That is some reassurance that (12.148) is correct. Since y = bCos θ( ) / 2 we can let C y( ) = C θ( ) , a0 y( ) = a0 θ( ) and α y( ) = α θ( ) . The Prandtl equation becomes
anSin nθ( )
U∞bb4
⎛⎝⎜
⎞⎠⎟n Sin θ( ) + n a0 θ( )C θ( )
4b⎛⎝⎜
⎞⎠⎟
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
n=1
∞
∑ =a0 θ( )C θ( )
4b⎛⎝⎜
⎞⎠⎟α θ( )Sin θ( ) (12.155)
Finally the solutions for the circulation and downwash of a general wing shape are from (12.99), (12.100), (12.144) and (12.143)
Γ y( )2U∞b
= AnSin nθ( )n=1
∞
∑ (12.156)
Uz 0, y,0( )
U∞
= − nAnSin nθ( )Sin θ( )
⎛⎝⎜
⎞⎠⎟n=1
∞
∑ (12.157)
where
An =an
U∞bb4
⎛⎝⎜
⎞⎠⎟n (12.158)
12.9 FORCES ON A GENERAL WING
In section 12.4 we summarized the relations for the forces and moments on a wing. The differential lift and induced drag at a given span-wise location are
dL y( ) = ρU∞Γ y( )dy (12.159)
and
dDi y( ) = ρUz 0, y,0( )Γ y( )dy (12.160)
41
Let
y =b2Cos θ( ) and dy = − b
2Sin θ( )dθ
(12.161)
The range on y is−b / 2 < y < b / 2 and −π < θ < 0 . Now the differential lift of a general wing is
dL y( ) = ρU∞2b2 AnSin nθ( )
n=1
∞
∑ Sin θ( )dθ (12.162)
and the induced drag is
dDi y( ) = ρU∞2b2 nAnSin nθ( )( )
n=1
∞
∑ AmSin mθ( )m=1
∞
∑⎛⎝⎜
⎞⎠⎟dθ (12.163)
The lift and drag are found by integrating the above differential relations
L = ρU∞2b2 An Sin nθ( )
−π
0
∫n=1
∞
∑ Sin θ( )dθ (12.164)
Di = ρU∞2b2 nAnAm
m=1
∞
∑ Sin nθ( )−π
0
∫n=1
∞
∑ Sin mθ( )dθ (12.165)
The integrals that appear in (12.164) and (12.165) are easily evaluated and lead to a considerable simplification.
Sin nθ( )Sin mθ( )−π
0
∫ dθ = 0 if n ≠ m , Sin nθ( )Sin mθ( )−π
0
∫ dθ =π2
if n = m (12.166)
Now
L =π2ρU∞
2b2A1 (12.167)
and
Di =π2ρU∞
2b2 nAn2
n=1
∞
∑ (12.168)
Interestingly the lift only depends on the first coefficient in the series (12.156) for the circulation. Whereas, the drag depends on every term in the series; a much more difficult quantity to evaluate. The rolling moment vector is aligned with the positive x-axis and also involves integrating the circulation.
42
Mx = MRoll = ρU∞ yΓ y( )−b /2
b /2
∫ dy = −12ρU∞
2b3 An Sin nθ( )−π
0
∫n=1
∞
∑ Cos θ( )Sin θ( )dθ (12.169)
The integral in (12.169) is
Sin nθ( )Cos θ( )Sin θ( )−π
0
∫ dθ = 0 all n ≠ 2 , Sin 2θ( )Cos θ( )Sin θ( )−π
0
∫ dθ =π4
(12.170)
The rolling moment only depends on the second term in the series.
Mx = MRoll = ρU∞ yΓ y( )−b /2
b /2
∫ dy = −π8ρU∞
2b3A2 (12.171)
The yawing moment is aligned with the positive z-axis and, like drag, its integral involves the circulation and downwash velocity.
Mz = MYaw = − ydDi y( )−b /2
b /2
∫ = ρ yUz 0, y,0( )−b /2
b /2
∫ Γ y( )dy (12.172)
which becomes
Mz = MYaw = ρ yUz 0, y,0( )
−b /2
b /2
∫ Γ y( )dy =12ρU∞
2b3 nAnAmm=1
∞
∑ Sin nθ( )−π
0
∫n=1
∞
∑ Sin mθ( )Sin θ( )Cos θ( )dθ (12.173)
The integral in (12.173) can be evaluated.
Sin nθ( )Sin mθ( )Cos θ( )−π
0
∫ dθ =π4
if m = n +1
Sin nθ( )Sin mθ( )Cos θ( )−π
0
∫ dθ = 0 otherwise (12.174)
Finally
Mz = MYaw =π8ρU∞
2b3 2n +1( )AnAn+1n=1
∞
∑ (12.175)
Not surprisingly, the yawing moment depends on all the coefficients in the series since it is directly related to differences in drag along the wing. The corresponding force coefficients are
43
CL =L
12ρU∞
2S= π b2
S⎛⎝⎜
⎞⎠⎟A1 = πAR( )A1
CDi=
Di
12ρU∞
2S= π b2
S⎛⎝⎜
⎞⎠⎟
nAn2
n=1
∞
∑ = πAR( ) nAn2
n=1
∞
∑
CMRoll =Mx
12ρU∞
2Sb= −
π4
b2
S⎛⎝⎜
⎞⎠⎟A2 = −
π4AR
⎛⎝⎜
⎞⎠⎟A2
CMYaw =Mz
12ρU∞
2Sb=π4
b2
S⎛⎝⎜
⎞⎠⎟
2n +1( )AnAn+1n=1
∞
∑ =π4AR
⎛⎝⎜
⎞⎠⎟
2n +1( )AnAn+1n=1
∞
∑
(12.176)
12.10 MINIMUM INDUCED DRAG WING
The coefficients in (12.176) depend on the distributions of a0 y( ),C y( ),α y( ) along the span of the wing. The lift only depends on the the first coefficient. The drag depends on a sum of squares of all the coefficients. Clearly the lowest drag occurs when An = 0 for all n > 1. In this case the circulation (12.156) becomes
Γ y( ) = 2U∞bA1Sin θ( ) = 2U∞bA1 1−2yb
⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟
1/2
= 8 a1b1− 2y
b⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟
1/2
(12.177)
Recall (12.153), a1 = Γ0b / 8 . Now
Γ y( ) = Γ0 1−2yb
⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟
1/2
(12.178)
Minimum induced drag occurs when the lift distribution on a wing is elliptic. For an untwisted wing this corresponds to an elliptic chord distribution.
12.11 INDUCED DRAG OF A RECTANGULAR WING
Circulation distributions calculated from the Prandtl equation (12.51) for a rectangular wing are shown in Figure 12.27.
44
Figure 12.27 Circulation distribution along straight wings of various aspect ratios from Prandtl & Tietjens (Applied Hydro and Aeromechanics). The aspect ratio parameter is P = 2 /π( )b /C .
In the limit b / c→ 0 the circulation distribution from (12.51) is elliptic and for small aspect ratios the circulation distribution is almost elliptic. However as the aspect ratio increases the portion of the span over which the circulation decreases to zero becomes a smaller and smaller part of the span and the circulation distribution over most of the span is almost rectangular. In the limit of infinite span the wing becomes two-dimensional and the circulation becomes exactly constant along the span.
Prandtl and Tietjens point out that the induced drag of a rectangular wing does not deviate that much from the drag of an elliptic wing. Figure 12.28 illustrates the lift, downwash and induced drag for a typical rectangular wing.
Figure 12.28 Typical variations of lift, downwash and induced drag for a rectangular wing from Prandtl & Tietjens.
The higher downwash at the wing tips and concomitant higher induced drag is due to the area distribution of a rectangular wing that puts more area at the wing tips than an
45
elliptical wing of the same total area. At the same time the centerline chord of the rectangular wing is smaller producing lower induced drag over the center span so the net effect is a relatively small overall drag increase compared to the elliptical wing. An approximate formula for 1 < 2b /πc < 10 is
Di
Dimin
= 0.99 + 0.015 2bπC
⎛⎝⎜
⎞⎠⎟
(12.179)
plotted in Figure 12.29.
Figure 12.29 Induced drag of a straight wing of varying aspect ratio compared to the induced drag of an elliptic wing Dimin from Prandtl & Tietjens.
Viscosity and the no slip condition create a boundary layer on the wing that is responsible for two additional contributions to the drag. There is a direct contribution from the skin friction integrated over the surface of the wing. In addition, the displacement effect of the boundary layer on the surrounding potential flow modifies the pressure from the pure potential flow solution leading to a drag contribution. Together these two contributions are called profile drag.
When we determined the induced drag of an elliptic wing we developed the relation
CDi =CL2
πSb2
(12.180)
where S = πC0b / 4 is the wing area. We can generalize (12.180) to approximate the induced drag of a rectangular wing with S = Cb . Let
CDi =CL2
πCb
(12.181)
Figure 12.30 shows the Lift-drag parabola of a rectangular wing compared to its calculated lift-induced-drag parabola. It can be seen that when the angle of attack exceeds about 3 degrees most of the drag is induced drag. The relatively small difference between
46
the two curves is the profile drag. Interestingly, the contribution from the profile drag is almost independent of the angle of attack of the wing.
Figure 12.30 Lift-drag parabola for a rectangular wing with aspect ratio parameter P = 2 /π( )b /C = 5 compared to the induced drag of a rectangular wing.
Similar data at a variety of aspect ratios shows that the profile drag coefficient is also relatively insensitive to aspect ratio. This can be used as the basis for an approximate method for converting the lift-drag curve for one aspect ratio wing to another aspect ratio.
The drag coefficient is the sum of induced drag and profile drag.
CD = CDi + CDp (12.182)
Now consider two wings of similar profile but different aspect ratio.
CD1
=CL2
πS1b12 + CDp
CD2=CL2
πS2b22 + CDp
(12.183)
If the profile drag contribution is the same for both wings then the drag coefficient of one wing can be converted to the other using
CD2= CD1
+CL2
πS2b22 −
S1b12
⎛⎝⎜
⎞⎠⎟
(12.184)
47
In Figure 12.31 the transformation formula (12.184) has been used to convert all the data in the left figure for various aspect ratios to the single curve for aspect ratio 5 in the right figure.
Figure 12.31 Left, lift-drag parabola for a rectangular wing of various aspect ratios. Same data scaled to P = 2 /π( )b /C = 5 using (12.184).
A similar approach can be used to to convert the lift data for one aspect ratio wing to another. Here again the idea is to generalize the result obtained for an elliptic wing. Recall the downwash velocity for an elliptic wing (12.60)
Uz 0, y,0( ) = −Γ0
2b (12.185)
The lift for an elliptic wing is
L =π4
⎛⎝⎜
⎞⎠⎟ρU∞Γ0b (12.186)
and the lift coefficient is
CL =L
12ρU∞
2S=
π2
⎛⎝⎜
⎞⎠⎟Γ0
U∞S (12.187)
Combine (12.185) and (12.187).
48
Uz
U∞
= −Γ0
2bU∞
= −CL
πSb2
(12.188)
Recall that one of the effects of the downwash of a finite wing is to decrease the effective angle of attack of the wing and thereby reduce the lift. If a section of a finite wing were to have the same lift as the same section considered to be part of an infinite wing at angle of
attack α0 its angle of attack would have to be increased by −α i =CL
πSb2
. The angle of
attack of the wing would need to be
α = α0 −α i = α0 +CL
πSb2
(12.189)
where Tan Uz /U∞( ) ≅Uz /U∞ has been used. If we compare two wings with the same profile and therefore the same α0 but different aspect ratios we can write
α1 = α0 +
CL
πS1b12
α2 = α0 +CL
πS2b22
(12.190)
Eliminate the common infinite aspect ratio angle of attack α0 between the two wings. The transformation of angle of attack becomes
α2 = α1 +CL
πS2b22 −
S1b12
⎛⎝⎜
⎞⎠⎟
(12.191)
Figure 12.32 shows lift versus-angle-of-attack data for rectangular wings of various aspect ratio transformed to a single aspect ratio using (12.191).
49
Figure 12.32 Left, Lift versus angle-of-attack for a rectangular wing of various aspect ratios. Same data scaled to P = 2 /π( )b /C = 5 using (12.191).
The data mappings in Figures 12.31 and 12.32 have been derived using results for wings with elliptic lift distributions. However the fact that the lift distributions for rectangular wings do not deviate too much from elliptic suggests that the transformation rules (12.184) and (12.191) can be used to correlate aspect ratio data for a wide variety of wing shapes.
12.12 UNSTEADY MOMENTUM INTEGRAL IN A TREFFTZ PLANE FIXED WITH RESPECT TO THE SURROUNDING FLUID
After a period of time has passed, the vast majority of momentum in the minus z-direction generated by the wing is contained in the far wake. Moreover the length of the wake in the positive x-direction that contains virtually all the momentum is wake length =U∞t . Using this idea and the results for the momentum generated by a point force described at the end of Chapter 10, we can approximate the momentum balance by
U∞t Uz−∞
∞
∫−∞
∞
∫ dydz = −23Lρu t( )t (12.192)
or
Uz−∞
∞
∫−∞
∞
∫ dydz = −23
LρU∞
⎛⎝⎜
⎞⎠⎟
(12.193)
50
The integral in (12.193) is carried out in a Trefftz plane that is fixed with respect to the surrounding fluid and increasingly far behind the aircraft as it moves away at speed U∞ in the minus x-direction. This situation is illustrated in Figure 12.33.
Figure 12.33 Aircraft wake with Trefftz plane fixed with respect to the surrounding fluid
If we are far enough behind the aircraft so that any stream-wise velocity components have died off then the wake should be completely described by the two-dimensional flow in the Trefftz plane. Remember the drag and thrust are in perfect balance so the streamwise velocities should die off rapidly.
The integral momentum balance in two dimensions is
DDt
U dAA∫
⎛
⎝⎜⎞
⎠⎟+ UU +
PρI
⎛⎝⎜
⎞⎠⎟in dC + ν n × Ω( )dA
C∫C∫ =
F x ,t( )ρ
⎛⎝⎜
⎞⎠⎟A
∫ dA (12.194)
where C is a fixed circular contour of large radius R surrounding the 2-D momentum source (force) located at the origin. The force divided by density acting on the flow is really a force per unit length with units L3 /T instead of L4 /T as in the 3-D case. In the far field the velocity behaves as U ∼ 1 / R2 , the vorticity is zero and the momentum balance becomes
limR→∞
DDt
U dAA∫
⎛
⎝⎜⎞
⎠⎟+
PρI
⎛⎝⎜
⎞⎠⎟in dC
C∫ =F x ,t( )
ρ⎛⎝⎜
⎞⎠⎟A
∫ dA (12.195)
The force creating the flow is
F x ,t( )
ρ= 0,− I
ρδ t( )δ y( )δ z( )⎧
⎨⎩
⎫⎬⎭
(12.196)
and the total impulse generated by the force is
51
Iρ= 0, I
ρδ t( )dt
0
t
∫⎧⎨⎩
⎫⎬⎭= 0, I
ρ⎧⎨⎩
⎫⎬⎭
(12.197)
with units I / ρ[ ] = Length3 /Time . The 2-D momentum balance in the z-direction is
limR→∞
DDt
Uz dAA∫
⎛
⎝⎜⎞
⎠⎟+ lim
R→∞
PρI
⎛⎝⎜
⎞⎠⎟in dC
C∫z
= −Iρδ t( ) (12.198)
The flow in the far field created by the isolated force (12.196) is a planar dipole with a 3-D vector potential that turns on at t = 0 .
A = Ax ,Ay ,Az{ } = −I2πρ
u t( ) yy2 + z2( ) ,0,0
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪ (12.199)
and a velocity field in the Trefftz plane
U = Uy ,Uz{ } = Iπρ
u t( ) yzy2 + z2( )2
, I2πρ
u t( ) −2y2
y2 + z2( )2+
1y2 + z2( )
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪ (12.200)
The corresponding scalar potential can be determined by integrating (12.200) by quadrature.
Φ =I2πρ
u t( ) zy2 + z2
⎛⎝⎜
⎞⎠⎟
(12.201)
The pressure is determined from the far field balance between pressure and momentum
∂U∂t
+∇Pρ
⎛⎝⎜
⎞⎠⎟= 0⇒ P
ρ= −
∂φ∂t
(12.202)
and
Pρ= −
I2πρ
δ t( ) zy2 + z2
⎛⎝⎜
⎞⎠⎟
(12.203)
The pressure integral in the 2-D momentum balance (12.198) is now
limR→∞
PρI
⎛⎝⎜
⎞⎠⎟in dC
C∫z
= −I2πρ
δ t( ) Sin2 θ( )dθ0
2π
∫ = −12Iρδ t( ) (12.204)
and the momentum balance (12.198) becomes
52
limR→∞
DDt
Uz dAA∫
⎛
⎝⎜⎞
⎠⎟= −
12Iρδ t( ) (12.205)
The integral of the momentum over the Trefftz plane is
Hz = Uz−∞
∞
∫−∞
∞
∫ dydz = −12Iρ
(12.206)
During the brief moment when the line impulsive force creating the vortex pair is turned on, the pressure force at infinity removes one half of the impulse applied to the flow (versus 1/3 in 3-D). Once the force is turned off the pressure at infinity also turns off and the vortex pair drifts downward under its own self-induction.
If the vortex pair produced by the impulsive line force is interpreted as the far wake of a lifting aircraft then the impulse per unit length of wake is interpreted as the downward impulsive force applied by the aircraft to the fluid as it passes over a given position x =U∞t . Using this idea, the lift is related to the impulse by
Iρ=
LρU∞
(12.207)
and the z-momentum per unit length of wake is
Hz = Uz−∞
∞
∫−∞
∞
∫ dydz = −12
LρU∞
(12.208)
Now consider an inviscid model of the wake that does include the downward convection of the vortex pair.
12.13 INVISCID VORTEX PAIR MODEL OF THE WAKE.
Let the wake of the aircraft be modeled as two infinite inviscid line vortices of circulation Γ0 convecting downward by their own mutual induction. Figure 12.34 depicts the situation.
53
Figure 12.34 Inviscid vortex pair.
The vorticity source distribution for a single line vortex is
Ω x ,t( ) = Γ0δ y − b0 / 2( )δ z + a t( )( ),0,0{ } (12.209)
where a t( ) is a positive function of time to be determined that defines the speed with which the vortex pair system drifts downward. The vector potential of (12.209) is A = Ax ,0,0{ } where
Ax =14π
Γ0δ ys − b0 / 2( )δ zs + a t( )( )x − xs( )2 + y − ys( )2 + z − zs( )2( )1/2
dxs dys dzs−∞
∞
∫−∞
∞
∫−∞
∞
∫ =
Ax =Γ0
4π1
x − xs( )2 + y − b0 / 2( )2 + z + a t( )( )2( )1/2−∞
∞
∫ dxs
(12.210)
The integral over xs is
Ax =Γ0
4πlimλ→∞
Lnλ + x + x + λ( )2 + y − b0 / 2( )2 + z + a t( )( )2
−λ + x + x − λ( )2 + y − b0 / 2( )2 + z + a t( )( )2⎛
⎝⎜⎜
⎞
⎠⎟⎟
(12.211)
When the limit λ → ∞ is taken in (12.211) the result is
54
Ax = −Γ0
2πLn y − b0 / 2( )2 + z + a t( )( )2( )1/2 − Ln 2( )⎛
⎝⎞⎠
Ax = −Γ0
2πLn
y − b0 / 2( )2 + z + a t( )( )2( )1/22
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
(12.212)
The vector potential of the vortex pair is generated by superposing the vector potentials of the two vortex lines of opposite sign.
Ax = −Γ0
4πLn
y − b0 / 2( )2 + z + a t( )( )2y + b0 / 2( )2 + z + a t( )( )2
⎛
⎝⎜
⎞
⎠⎟ (12.213)
The corresponding scalar potential of the system is
Φ y, z( ) = Γ0
2πArcTan
z + a t( )y − b0 / 2
⎛⎝⎜
⎞⎠⎟− ArcTan
z + a t( )y + b0 / 2
⎛⎝⎜
⎞⎠⎟
⎛
⎝⎜⎞
⎠⎟ (12.214)
and the velocity field derived from either potential is
U = Uy ,Uz{ } = Γ0b02π y − b0 / 2( )2 + z + a t( )( )2( ) y + b0 / 2( )2 + z + a t( )( )2( )
⎛
⎝⎜⎜
⎞
⎠⎟⎟×
−2y z + a t( )( ), y2 − b02
⎛⎝⎜
⎞⎠⎟2
− z + a t( )( )2⎛
⎝⎜⎞
⎠⎟⎛
⎝⎜
⎞
⎠⎟
Uy =Γ0
2π− z + a( )
z + a( )2 + y − b0 / 2( )2+
z + a( )z + a( )2 + y + b0 / 2( )2
⎛
⎝⎜
⎞
⎠⎟
Uz =Γ0
2πy − b0 / 2( )
z + a( )2 + y − b0 / 2( )2−
y + b0 / 2( )z + a( )2 + y + b0 / 2( )2
⎛
⎝⎜
⎞
⎠⎟
(12.215)
Now, the momentum of the inviscid vortex pair is
H = U dzdy−∞
∞
∫−∞
∞
∫ = Φrn0
2π
∫ dθ
Γ0
2πArcTan
rSin θ( ) + a t( )rCos θ( ) − b0 / 2
⎛⎝⎜
⎞⎠⎟− ArcTan
rSin θ( ) + a t( )rCos θ( ) + b0 / 2
⎛⎝⎜
⎞⎠⎟
⎛
⎝⎜⎞
⎠⎟r Cos θ( ),Sin θ( ){ }
0
2π
∫ dθ
(12.216)
Use the identity
55
ArcTan u( ) − ArcTan v( ) = ArcTan u − v1− uv
⎛⎝⎜
⎞⎠⎟ (12.217)
where
u − v1− uv
=b0 rSin θ( ) + a t( )( )
r2Cos2 θ( ) − b0 / 2( )2 − rSin θ( ) + a t( )( )2 (12.218)
Now the momentum becomes
H = U dzdy−∞
∞
∫−∞
∞
∫ =
Γ0
2πArcTan
b0 rSin θ( ) + a t( )( )r2Cos2 θ( ) − b0 / 2( )2 − rSin θ( ) + a t( )( )2
⎛
⎝⎜
⎞
⎠⎟
⎛
⎝⎜⎜
⎞
⎠⎟⎟r Cos θ( ),Sin θ( ){ }
0
2π
∫ dθ
(12.219)
When the integral is carried out
H = Hy ,Hz{ } = 0,− Γ0b02
⎧⎨⎩
⎫⎬⎭
(12.220)
Equate the z-momentum from (12.208) to the z-momentum from (12.220). The result is
L = ρU∞Γ0b0 (12.221)
We worked out the lift for the case of elliptic loading in (12.55).
L = ρU∞Γ0π4b⎛
⎝⎜⎞⎠⎟ (12.222)
Equating (12.221) and (12.222) leads to the spacing between vortex lines compared to the wing span required to properly model the lift.
b0 =π4b (12.223)
The visualization of the vorticity shed from the trailing edge of the wing in Figure 12.1c confirms that there is a tendency to form relatively thin vortex tubes at least in the near wake of the wing. The plan view of the wing shown in Figure 12.1b also indicates a natural tendency for the trailing vortex pair to move toward one another. For an elliptically loaded wing we might see something like the sketch in Figure 12.35.
56
Figure 12.35 Vortex spacing in the wake of an elliptically loaded wing.
We still need to determine the downward drift a t( ) of the vortex pair. Here we will use the requirement that there is no net pressure on the vortex pair; they are free vortices. The pressure field on the z-axis can be determined from the z-momentum equation.
−∂∂z
Pρ
⎛⎝⎜
⎞⎠⎟=∂Uz
∂t+Uz
∂Uz
∂z=∂Uz
∂adadt
+Uz∂Uz
∂z (12.224)
Equation (12.224) can be integrated to give the pressure on y = 0
−∂∂z
P 0, z( )ρ
⎛⎝⎜
⎞⎠⎟=∂Uz 0, z( )
∂t+Uz 0, z( ) ∂Uz 0, z( )
∂z=
∂Uz 0, z( )∂a
dadt
+Uz 0, z( ) ∂Uz 0, z( )∂z
=∂Uz 0, z( )
∂zdadt
+Uz 0, z( ) ∂Uz 0, z( )∂z
−Pρ−P∞ρ
⎛⎝⎜
⎞⎠⎟= Uz 0, z( ) da
dt+12Uz 0, z( )2⎛
⎝⎜⎞⎠⎟
(12.225)
Now we require that there be no net pressure on the vortex pair.
Pρ−P∞ρ
⎛⎝⎜
⎞⎠⎟−∞
∞
∫ dz = Uz 0, z( ) dadt
+12Uz 0, z( )2⎛
⎝⎜⎞⎠⎟−∞
∞
∫ dz = 0 (12.226)
which gives the drift as
dadt
= −12
Uz 0, z( )2( )−∞
∞
∫ dz
Uz 0, z( )( )−∞
∞
∫ dz (12.227)
57
The z-velocity component on the centerline y = 0 is
Uz 0, z( ) = −Γ0b02π
1
z + a( )2 + b02
⎛⎝⎜
⎞⎠⎟2
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
(12.228)
Substitute (12.228) into (12.227). The drift velocity of the vortex pair is
dadt
=Γ0b04π
1 / z + a( )2 + b02
⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟⎛
⎝⎜
⎞
⎠⎟
2
−∞
∞
∫ dz
1 / z + a( )2 + b02
⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟⎛
⎝⎜
⎞
⎠⎟−∞
∞
∫ dz=
dadt
=Γ0b04π
1b02
⎛⎝⎜
⎞⎠⎟2
1 / z + ab0 / 2
⎛⎝⎜
⎞⎠⎟
2
+1⎛
⎝⎜
⎞
⎠⎟
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
−∞
∞
∫ d z + ab0 / 2
⎛⎝⎜
⎞⎠⎟
1 / z + ab0 / 2
⎛⎝⎜
⎞⎠⎟
2
+1⎛
⎝⎜
⎞
⎠⎟
⎛
⎝⎜⎜
⎞
⎠⎟⎟−∞
∞
∫ d z + ab0 / 2
⎛⎝⎜
⎞⎠⎟
=Γ0
2πb0
(12.229)
in agreement with the well known classical value.
Let’s look at the flow about the vortex pair in a frame of reference moving downward at the speed (12.229). The unsteady velocity potential and vector potential are
Φ y, z( ) = Γ0
2πArcTan
z + Γ0t2πb0
y − b0 / 2
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟− ArcTan
z + Γ0t2πb0
y + b0 / 2
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
(12.230)
Ax = −Γ0
4πLn
y − b0 / 2( )2 + z + Γ0t2πb0
⎛⎝⎜
⎞⎠⎟
2
y + b0 / 2( )2 + z + Γ0t2πb0
⎛⎝⎜
⎞⎠⎟
2
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
(12.231)
The unsteady velocity field in the non-moving frame is from (12.215).
58
Uy =Γ0
2π
− z + Γ0t2πb0
⎛⎝⎜
⎞⎠⎟
z + Γ0t2πb0
⎛⎝⎜
⎞⎠⎟
2
+ y − b02
⎛⎝⎜
⎞⎠⎟2+
z + Γ0t2πb0
⎛⎝⎜
⎞⎠⎟
z + Γ0t2πb0
⎛⎝⎜
⎞⎠⎟
2
+ y + b02
⎛⎝⎜
⎞⎠⎟2
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
Uz =Γ0
2π
y − b02
⎛⎝⎜
⎞⎠⎟
z + Γ0t2πb0
⎛⎝⎜
⎞⎠⎟
2
+ y − b02
⎛⎝⎜
⎞⎠⎟2−
y + b02
⎛⎝⎜
⎞⎠⎟
z + Γ0t2πb0
⎛⎝⎜
⎞⎠⎟
2
+ y + b02
⎛⎝⎜
⎞⎠⎟2
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
(12.232)
Transform (12.230), (12.231) and (12.232) to an observer moving downward at a constant speed with the vortex pair. Let
!y = y
!z = z +Γ0t2πb0
!U !y =Uy
!U !z =Uz +Γ0
2πb0!P = P
!A!x = Ax −Γ0y2πb0
!Φ = Φ +Γ0
2πb0z +
Γ0t2πb0
⎛⎝⎜
⎞⎠⎟
(12.233)
!Φ !y, !z( ) = Γ0
2πArcTan
!z!y − b0 / 2
⎛⎝⎜
⎞⎠⎟− ArcTan
!z!y + b0 / 2
⎛⎝⎜
⎞⎠⎟
⎛
⎝⎜⎞
⎠⎟+
Γ0 !z2πb0
(12.234)
!A!x !y, !z( ) = −Γ0
4πLn
!y − b0 / 2( )2 + !z2!y + b0 / 2( )2 + !z2
⎛
⎝⎜
⎞
⎠⎟ −
Γ0 !y2πb0
(12.235)
The velocity field with respect to the moving (tildaed) observer is steady.
59
!U !y =Γ0
2π− !z
!z2 + !y − b02
⎛⎝⎜
⎞⎠⎟2 +
!z
!z2 + !y + b02
⎛⎝⎜
⎞⎠⎟2
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
!U !z =Γ0
2π
!y − b02
⎛⎝⎜
⎞⎠⎟
!z2 + !y − b02
⎛⎝⎜
⎞⎠⎟2 −
!y + b02
⎛⎝⎜
⎞⎠⎟
!z2 + !y + b02
⎛⎝⎜
⎞⎠⎟2
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
+Γ0
2πb0
(12.236)
The steady streamline pattern seen by the observer drifting downward with the vortex pair is shown in Figure 7.36.
Figure 12.36 Streamline pattern of an inviscid trailing vortex pair as seen by an observer convecting downward with the pair.
In this frame of reference we can see the body of fluid carried downward by the vortex pair in the form of a closed streamline surrounding the two vortex centers.
12.14 EFFECT OF A GROUND PLANE ON THE DOWNWASH VELOCITY
We can solve the problem of a wake vortex sheet placed a height h above a wall by placing a system of image vortices an equal distance −h below the wall as shown in the figure below.
60
Figure 12.37 Continuous distribution of vortex lines with image system beneath the ground plane.
The vector potential of the aggregate system is, by superposition A = Ax ,0,U∞y{ } where Ax is
Ax x, y, z( ) = −14π
dΓ y0( )dy0
⎛⎝⎜
⎞⎠⎟×
0
b /2
∫
Lnx + x2 + y + y0( )2 + z − h( )2( )1/2 y − y0( )2 + z − h( )2( )x + x2 + y − y0( )2 + z − h( )2( )1/2⎛
⎝⎞⎠ y + y0( )2 + z − h( )2( )
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟−
Lnx + x2 + y + y0( )2 + z + h( )2( )1/2 y − y0( )2 + z + h( )2( )x + x2 + y − y0( )2 + z + h( )2( )1/2⎛
⎝⎞⎠ y + y0( )2 + z + h( )2( )
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟
dy0
(12.237)
where the integral is from y0 = 0 to y0 = b / 2 and the vortex sheet is assumed to be anti-symmetric. The downwash velocity of the overall system (object plus image) is
61
Uz x, y, z( ) = Γ 0
2πb
2y0b
⎛⎝
⎞⎠
1 −2y0b
⎛⎝
⎞⎠
2⎛⎝⎜
⎞⎠⎟1/ 20
b / 2
∫ ×
− y − y0( ) y + y0( )2 + z − h( )2( ) − 2 y + y0( ) x + x 2 + y − y0( )2 + z − h( )2( )1/ 2( ) x 2 + y − y0( )2 + z − h( )2( )1/ 2
x 2 + y − y0( )2 + z − h( )2( )1/ 2 x + x 2 + y − y0( )2 + z − h( )2( )1/ 2( ) y + y0( )2 + z − h( )2( )+
y + y0( ) y − y0( )2 + z − h( )2( ) + 2 y − y0( ) x + x 2 + y + y0( )2 + z − h( )2( )1/ 2( ) x 2 + y + y0( )2 + z − h( )2( )1/ 2
x 2 + y + y0( )2 + z − h( )2( )1/ 2 x + x 2 + y + y0( )2 + z − h( )2( )1/ 2( ) y − y0( )2 + z − h( )2( )
⎛
⎝
⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟
dy0 −
Γ 0
2πb
2y0b
⎛⎝
⎞⎠
1 −2y0b
⎛⎝
⎞⎠
2⎛⎝⎜
⎞⎠⎟1/ 20
b / 2
∫ ×
− y − y0( ) y + y0( )2 + z + h( )2( ) − 2 y + y0( ) x + x 2 + y − y0( )2 + z + h( )2( )1/ 2( ) x 2 + y − y0( )2 + z + h( )2( )1/ 2
x 2 + y − y0( )2 + z + h( )2( )1/ 2 x + x 2 + y − y0( )2 + z + h( )2( )1/ 2( ) y + y0( )2 + z + h( )2( )+
y + y0( ) y − y0( )2 + z + h( )2( ) + 2 y − y0( ) x + x 2 + y + y0( )2 + z + h( )2( )1/ 2( ) x 2 + y + y0( )2 + z + h( )2( )1/ 2
x 2 + y + y0( )2 + z + h( )2( )1/ 2 x + x 2 + y + y0( )2 + z + h( )2( )1/ 2( ) y − y0( )2 + z + h( )2( )
⎛
⎝
⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟
dy0
(12.238)
At the origin of the object system x, y, z{ } = 0,0,h{ } (ie., the lifting line) the downwash velocity is
Uz 0,0,h( ) =
−Γ0
πb1
1− 2y0b
⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟
1/20
1
∫ d 2y0b
⎛⎝⎜
⎞⎠⎟−
2y0b
⎛⎝⎜
⎞⎠⎟2
1− 2y0b
⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟
1/22y0b
⎛⎝⎜
⎞⎠⎟2
+ 4 2hb
⎛⎝⎜
⎞⎠⎟2⎛
⎝⎜⎞
⎠⎟0
1
∫ d 2y0b
⎛⎝⎜
⎞⎠⎟
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
(12.239)
which integrates to
Uz 0,0,h( ) = −Γ0
2b
4 hb
⎛⎝⎜
⎞⎠⎟
1+ 4 hb
⎛⎝⎜
⎞⎠⎟2
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
(12.240)
plotted below.
62
Figure 12.38 Effect of the presence of a ground plane on the downwash at the center of the lifting line.
The conclusion from Figure 12.29 is that the ground effect begins to come into play when 4h / b is less than about three, ie, when the height of the lifting line is less than about ¾ of the span of the lifting line.