2.5 Lifting-line theory

Lifting-line theory is a powerful tool for analyzing finite wings in aerodynamics. It bridges the gap between 2D airfoil theory and complex 3D wing flows, providing insights into lift distribution, induced drag, and wing planform effects.

This theory, developed by Ludwig Prandtl, replaces the wing with a vortex filament along the quarter-chord line. It considers circulation distribution, downwash, and induced angle of attack to predict wing performance and optimize designs for various applications.

Lifting-line theory fundamentals

• Lifting-line theory provides a mathematical model to analyze the aerodynamic characteristics of finite wings
• Developed by Ludwig Prandtl in 1918, it forms the foundation for understanding the behavior of three-dimensional wings
• Lifting-line theory bridges the gap between two-dimensional airfoil theory and the complex flow patterns around finite wings

Prandtl's lifting-line concept

Top images from around the web for Prandtl's lifting-line concept

• 

  Frontiers | Structure and Topology Analysis of Separated Vortex in Forward-Swept Blade View original

  Is this image relevant?

• 

  Lift (force) - Wikipedia View original

  Is this image relevant?

• 

  Vortices enable the complex aerobatics of peregrine falcons | Communications Biology View original

  Is this image relevant?

• 

  Frontiers | Structure and Topology Analysis of Separated Vortex in Forward-Swept Blade View original

  Is this image relevant?

• 

  Lift (force) - Wikipedia View original

  Is this image relevant?

1 of 3

Top images from around the web for Prandtl's lifting-line concept

• 

  Frontiers | Structure and Topology Analysis of Separated Vortex in Forward-Swept Blade View original

  Is this image relevant?

• 

  Lift (force) - Wikipedia View original

  Is this image relevant?

• 

  Vortices enable the complex aerobatics of peregrine falcons | Communications Biology View original

  Is this image relevant?

• 

  Frontiers | Structure and Topology Analysis of Separated Vortex in Forward-Swept Blade View original

  Is this image relevant?

• 

  Lift (force) - Wikipedia View original

  Is this image relevant?

1 of 3

• Prandtl's lifting-line concept replaces the wing with a single bound vortex filament along the quarter-chord line
• Assumes that the vortex filament is straight and perpendicular to the freestream velocity
• Vortex strength varies along the wingspan to account for the spanwise lift distribution
• Trailing vortices are shed from the bound vortex, forming a horseshoe vortex system

Vortex filament representation

• The vortex filament is an idealized representation of the circulation around the wing
• Circulation ($\Gamma$) is a measure of the vortex strength and is related to the lift generated by the wing
• The vortex filament induces a velocity field around the wing, affecting the local flow conditions
• The induced velocity is perpendicular to the vortex filament and its magnitude depends on the circulation strength

Bound vortex vs trailing vortices

• The bound vortex represents the circulation around the wing and is responsible for generating lift
• Trailing vortices are shed from the wing tips and extend downstream, forming a vortex wake
• Trailing vortices are a consequence of the finite wingspan and the pressure difference between the upper and lower surfaces
• The strength of the trailing vortices is equal to the change in circulation along the wingspan

Helmholtz's vortex theorems

• Helmholtz's vortex theorems describe the behavior of vortices in an inviscid, incompressible flow
• The first theorem states that the strength of a vortex filament remains constant along its length
• The second theorem states that a vortex filament cannot end in a fluid; it must form a closed loop or extend to infinity
• These theorems are fundamental to the concept of circulation and the formation of the horseshoe vortex system

Kutta-Joukowski theorem

• The Kutta-Joukowski theorem relates the lift generated by a wing to the circulation around it
• Lift per unit span ($L'$) is given by: $L' = \rho_{\infty} V_{\infty} \Gamma$, where $\rho_{\infty}$ is the freestream density and $V_{\infty}$ is the freestream velocity
• The theorem assumes that the flow leaves the trailing edge smoothly, satisfying the Kutta condition
• It provides a link between the circulation distribution and the lift distribution along the wingspan

Circulation distribution

• The circulation distribution along the wingspan determines the lift distribution and the induced drag of the wing
• Different circulation distributions result in different aerodynamic characteristics and performance

Elliptical vs non-elliptical distributions

• An elliptical circulation distribution is considered optimal as it minimizes induced drag for a given lift
• Elliptical distribution results in a constant downwash velocity along the wingspan
• Non-elliptical distributions, such as triangular or rectangular, result in higher induced drag
• Non-elliptical distributions can be advantageous in certain situations, such as for stall behavior or structural considerations

Relation to lift distribution

• The lift distribution along the wingspan is directly related to the circulation distribution
• Elliptical circulation distribution results in an elliptical lift distribution
• Non-elliptical circulation distributions lead to non-elliptical lift distributions
• The lift distribution affects the spanwise loading and the overall aerodynamic performance of the wing

Effect on induced drag

• Induced drag is a consequence of the trailing vortices and the downwash they create
• Elliptical circulation distribution minimizes induced drag for a given lift and wingspan
• Non-elliptical distributions result in higher induced drag compared to the elliptical case
• The induced drag coefficient ($C_{D,i}$) depends on the lift coefficient ($C_L$) and the wing aspect ratio ($AR$): $C_{D,i} = \frac{C_L^2}{\pi AR}$

Induced angle of attack

• The presence of trailing vortices and the resulting downwash modifies the local flow conditions around the wing
• The downwash induces an additional angle of attack, known as the induced angle of attack ($\alpha_i$)

Downwash velocity

• The downwash velocity ($w$) is the vertical component of the induced velocity caused by the trailing vortices
• Downwash velocity varies along the wingspan, with the maximum value occurring at the wing tips
• The downwash velocity is related to the circulation distribution and the wingspan
• It reduces the effective angle of attack seen by the wing sections

Effective angle of attack

• The effective angle of attack ($\alpha_{eff}$) is the actual angle of attack experienced by the wing sections
• It is the difference between the geometric angle of attack ($\alpha$) and the induced angle of attack ($\alpha_i$): $\alpha_{eff} = \alpha - \alpha_i$
• The induced angle of attack is given by: $\alpha_i = \frac{w}{V_{\infty}}$, where $V_{\infty}$ is the freestream velocity
• The effective angle of attack determines the local lift coefficient and the overall lift distribution

Induced drag coefficient

• The induced drag coefficient ($C_{D,i}$) quantifies the additional drag caused by the trailing vortices
• It is a function of the lift coefficient ($C_L$) and the wing aspect ratio ($AR$): $C_{D,i} = \frac{C_L^2}{\pi AR}$
• The induced drag coefficient increases with the square of the lift coefficient
• Higher aspect ratio wings have lower induced drag coefficients for a given lift coefficient

Finite wing theory

• Finite wing theory extends the concepts of lifting-line theory to predict the aerodynamic characteristics of real wings
• It takes into account the spanwise variation of circulation and the presence of trailing vortices

Lifting-line equation derivation

• The lifting-line equation is derived by applying the Biot-Savart law to the vortex filament system
• It relates the circulation distribution ($\Gamma(y)$) to the downwash velocity distribution ($w(y)$) along the wingspan
• The lifting-line equation is an integral equation that can be solved numerically or analytically using simplifying assumptions
• The solution of the lifting-line equation provides the circulation distribution and the aerodynamic characteristics of the wing

Fourier series solution

• The circulation distribution can be represented as a Fourier series expansion
• The Fourier series coefficients ($A_n$) are determined by solving a system of linear equations
• The Fourier series solution provides an analytical expression for the circulation distribution
• It allows for the calculation of the lift distribution, induced drag, and other aerodynamic parameters

Glauert's trigonometric series

• Glauert proposed a simplified approach to solve the lifting-line equation using a trigonometric series expansion
• The circulation distribution is represented as a series of sine functions: $\Gamma(y) = 4bV_{\infty}\sum_{n=1}^{\infty}A_n\sin(n\theta)$, where $b$ is the wingspan and $\theta$ is the spanwise coordinate
• The coefficients ($A_n$) are determined by solving a system of linear equations
• Glauert's method provides a more efficient solution compared to the general Fourier series approach

Multhopp's quadrature method

• Multhopp's quadrature method is a numerical approach to solve the lifting-line equation
• It discretizes the wingspan into a series of control points and applies a quadrature formula to evaluate the integrals
• The method reduces the lifting-line equation to a system of linear algebraic equations
• Multhopp's method is more versatile than analytical methods and can handle arbitrary wing planforms and twist distributions

Wing planform effects

• The wing planform, which includes aspect ratio, taper ratio, and sweep angle, has a significant impact on the aerodynamic characteristics
• Lifting-line theory can be used to analyze the effects of different wing planforms on lift distribution and induced drag

Aspect ratio influence

• Aspect ratio ($AR$) is defined as the square of the wingspan divided by the wing area: $AR = \frac{b^2}{S}$
• Higher aspect ratio wings have lower induced drag coefficients for a given lift coefficient
• Increasing the aspect ratio reduces the intensity of the trailing vortices and the associated downwash
• High aspect ratio wings are more efficient but may face structural and weight limitations

Taper ratio considerations

• Taper ratio ($\lambda$) is the ratio of the tip chord to the root chord: $\lambda = \frac{c_{tip}}{c_{root}}$
• Tapered wings have a reduced chord length towards the wing tips compared to the wing root
• Taper ratio affects the spanwise lift distribution and the stall behavior of the wing
• Moderate taper ratios (0.3 to 0.5) are commonly used to balance aerodynamic and structural considerations

Sweep angle impact

• Sweep angle ($\Lambda$) is the angle between the wing leading edge and a perpendicular to the aircraft centerline
• Swept wings are used to delay the onset of compressibility effects at high Mach numbers
• Sweep angle modifies the effective velocity component perpendicular to the leading edge, reducing the effective Mach number
• Swept wings have a spanwise component of flow, which affects the lift distribution and the stall behavior

Winglets and end plates

• Winglets and end plates are devices added to the wing tips to reduce the intensity of the trailing vortices
• They work by modifying the pressure distribution at the wing tips and reducing the spanwise flow
• Winglets and end plates effectively increase the aspect ratio of the wing without increasing the wingspan
• They can provide a reduction in induced drag, leading to improved fuel efficiency and performance

Limitations and extensions

• Lifting-line theory, while powerful, relies on several simplifying assumptions that limit its applicability in certain situations
• Various extensions and modifications have been proposed to address these limitations and improve the accuracy of the theory

Assumptions and simplifications

• Lifting-line theory assumes a straight vortex filament along the wing quarter-chord line
• It neglects the effects of wing thickness, viscosity, and compressibility
• The theory assumes a high aspect ratio wing with small angles of attack
• These assumptions limit the accuracy of lifting-line theory for low aspect ratio wings, high angles of attack, and transonic conditions

Prandtl's classical lifting-line theory

• Prandtl's classical lifting-line theory is the original formulation based on the assumptions mentioned above
• It provides a good approximation for high aspect ratio wings at low angles of attack
• Classical lifting-line theory does not account for the effects of wing sweep or dihedral
• It assumes a planar wake and does not consider the deformation of the vortex sheet downstream

Weissinger's extended lifting-line model

• Weissinger's extended lifting-line model is an improvement over Prandtl's classical theory
• It accounts for the non-planar shape of the vortex sheet and the curvature of the trailing vortices
• Weissinger's model allows for the analysis of swept and non-planar wings
• It provides more accurate results for wings with moderate to high sweep angles and non-planar configurations

Vortex lattice method comparison

• The vortex lattice method (VLM) is a numerical approach that discretizes the wing into a lattice of vortex panels
• VLM can handle complex wing geometries, including sweep, taper, and twist
• It accounts for the non-linear effects of wing thickness and compressibility to some extent
• VLM is more computationally intensive than lifting-line theory but provides higher accuracy for complex wing configurations
• Lifting-line theory remains a valuable tool for preliminary design and analysis, while VLM is used for more detailed aerodynamic simulations