The Prandtl lifting-line theory is a mathematical model in
aerodynamics
Aerodynamics, from grc, ἀήρ ''aero'' (air) + grc, δυναμική (dynamics), is the study of the motion of air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dy ...
that predicts
lift
Lift or LIFT may refer to:
Physical devices
* Elevator, or lift, a device used for raising and lowering people or goods
** Paternoster lift, a type of lift using a continuous chain of cars which do not stop
** Patient lift, or Hoyer lift, mobil ...
distribution over a three-dimensional
wing
A wing is a type of fin that produces lift while moving through air or some other fluid. Accordingly, wings have streamlined cross-sections that are subject to aerodynamic forces and act as airfoils. A wing's aerodynamic efficiency is e ...
based on its geometry. It is also known as the Lanchester–Prandtl wing theory.
The theory was expressed independently by
Frederick W. Lanchester in 1907, and by
Ludwig Prandtl
Ludwig Prandtl (4 February 1875 – 15 August 1953) was a German fluid dynamicist, physicist and aerospace scientist. He was a pioneer in the development of rigorous systematic mathematical analyses which he used for underlying the science of ...
in 1918–1919 after working with
Albert Betz and
Max Munk
Max Michael Munk (October 22, 1890 – June 3, 1986) was a German aerospace engineer who worked for the National Advisory Committee for Aeronautics (NACA) in the 1920s and made contributions to the design of airfoils.
Education and early career
M ...
.
In this model, the bound vortex loses strength along the whole wingspan because it is shed as a vortex-sheet from the trailing edge, rather than just as a single vortex from the wing-tips.
Introduction
It is difficult to predict analytically the overall amount of lift that a wing of given geometry will generate.
When analyzing a three-dimensional
finite wing
A finite wing is an aerodynamic wing with tips that result in trailing vortices.http://www.aerospaceweb.org/question/aerodynamics/q0167.shtml This is in contrast to an infinite wing. According to John D. Anderson, Jr., finite wings experience ...
, the first approximation to understanding is to consider slicing the wing into cross-sections and analyzing each cross-section independently as a wing in a two-dimensional world. Each of these slices is called an
airfoil, and it is easier to understand an airfoil than a complete three-dimensional wing.
One might expect that understanding the full wing simply involves adding up the independently calculated forces from each airfoil segment. However, it turns out that this approximation is grossly incorrect: on a real wing, the lift over each wing segment (local lift per unit span,
or
) does not correspond simply to what two-dimensional analysis predicts. In reality, the local amount of lift on each cross-section is not independent and is strongly affected by neighboring wing sections.
The lifting-line theory corrects some of the errors in the naive two-dimensional approach by including some of the interactions between the wing slices. It produces the lift distribution along the span-wise direction,
based on the wing geometry (span-wise distribution of chord, airfoil, and twist) and flow conditions (
,
,
).
Principle
The lifting-line theory applies the concept of
circulation and the
Kutta–Joukowski theorem
The Kutta–Joukowski theorem is a fundamental theorem in aerodynamics used for the calculation of lift of an airfoil (and any two-dimensional body including circular cylinders) translating in a uniform fluid at a constant speed large enough so ...
,
:
so that instead of the ''lift'' distribution function, the unknown effectively becomes the distribution of circulation over the span,
.
Modeling the local lift (unknown and sought-after) with the local circulation (also unknown) allows us to account for the influence of one section over its neighbors. In this view, any span-wise change in lift is equivalent to a span-wise change of circulation. According to
Helmholtz's theorems
In fluid mechanics, Helmholtz's theorems, named after Hermann von Helmholtz, describe the three-dimensional motion of fluid in the vicinity of vortex lines. These theorems apply to inviscid flows and flows where the influence of viscous forces are ...
, a vortex filament cannot begin or terminate in the air. Any span-wise ''change in lift'' can be modeled as
the shedding of a vortex filament down the flow, behind the wing.
This shed vortex, whose strength is the derivative of the (unknown) local wing circulation distribution,
, influences the flow left and right of the wing section.
This sideways influence (upwash on the outboard, downwash on the inboard) is the key to the lifting-line theory. Now, if the ''change'' in lift distribution is known at given lift section, it is possible to predict how that section influences the lift over its neighbors: the vertical induced velocity (upwash or downwash,
) can be quantified using the velocity distribution within a
vortex
In fluid dynamics, a vortex ( : vortices or vortexes) is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in th ...
and related to a change in effective angle of attack over neighboring sections.
In mathematical terms, the local induced change of angle of attack
on a given section can be quantified with the integral sum of the downwash induced by every other wing section. In turn, the integral sum of the lift on each downwashed wing section is equal to the (known) total desired amount of lift.
This leads to an
integro-differential equation
In mathematics, an integro-differential equation is an equation that involves both integrals and derivatives of a function.
General first order linear equations
The general first-order, linear (only with respect to the term involving derivati ...
in the form of
, where
is expressed solely in terms of the wing geometry and its own span-wise variation
. The solution to this equation is a function
that accurately describes the circulation (and therefore lift) distribution over a finite wing of known geometry.
Derivation
(Based on.)
Nomenclature:
*
is the
circulation over the entire wing (m²/s)
*
is the 3D
lift coefficient
In fluid dynamics, the lift coefficient () is a dimensionless quantity that relates the lift generated by a lifting body to the fluid density around the body, the fluid velocity and an associated reference area. A lifting body is a foil or a com ...
(for the entire wing)
*
is the
aspect ratio
*
is the freestream
angle of attack (rad)
*
is the freestream velocity (m/s)
*
is the drag coefficient for
induced drag
In aerodynamics, lift-induced drag, induced drag, vortex drag, or sometimes drag due to lift, is an aerodynamic drag force that occurs whenever a moving object redirects the airflow coming at it. This drag force occurs in airplanes due to wings or ...
*
is the
planform efficiency factor
The following are all functions of the wings span-wise station
(i.e. they can all vary along the wing)
*
is the 2D
lift coefficient
In fluid dynamics, the lift coefficient () is a dimensionless quantity that relates the lift generated by a lifting body to the fluid density around the body, the fluid velocity and an associated reference area. A lifting body is a foil or a com ...
(units/m)
*
is the 2D circulation at a section (m/s)
*
is the
chord length of the local section
*
is the local change in angle of attack due to geometric twist of the wing
*
is zero-lift angle of attack of that section (depends on the airfoil geometry)
*
is the 2D lift coefficient slope (units/m⋅rad, and depends on airfoil geometry, see
Thin airfoil theory
An airfoil (American English) or aerofoil (British English) is the cross-sectional shape of an object whose motion through a gas is capable of generating significant lift, such as a wing, a sail, or the blades of propeller, rotor, or turbine. ...
)
*
is change in angle of attack due to
downwash
In aeronautics, downwash is the change in direction of air deflected by the aerodynamic action of an airfoil, wing, or helicopter rotor blade in motion, as part of the process of producing lift.Crane, Dale: ''Dictionary of Aeronautical Terms, thir ...
*
is the local downwash velocity
To derive the model we start with the assumption that the circulation of the wing varies as a function of the spanwise locations. The function assumed is a Fourier function. Firstly, the coordinate for the spanwise location
is transformed by
, where y is spanwise location, and s is the semi-span of the wing.
and so the circulation is assumed to be:
Since the circulation of a section is related to the
by the equation:
but since the coefficient of lift is a function of angle of attack:
hence the vortex strength at any particular spanwise station can be given by the equations:
This one equation has two unknowns: the value for
and the value for
. However, the downwash is purely a function of the circulation only. So we can determine the value
in terms of
, bring this term across to the left hand side of the equation and solve. The downwash at any given station is a function of the entire shed vortex system. This is determined by integrating the influence of each differential shed vortex over the span of the wing.
Differential element of circulation:
Differential downwash due to the differential element of circulation (acts like half an infinite vortex line):
The integral equation over the span of the wing to determine the downwash at a particular location is:
After appropriate substitutions and integrations we get:
And so the change in angle attack is determined by (
assuming small angles):
By substituting equations 8 and 9 into RHS of equation 4 and equation 1 into the LHS of equation 4, we then get:
After rearranging, we get the series of simultaneous equations:
By taking a finite number of terms, equation 11 can be expressed in matrix form and solved for coefficients A. Note the left-hand side of the equation represents each element in the matrix, and the terms on the RHS of equation 11 represent the RHS of the matrix form. Each row in the matrix form represents a different span-wise station, and each column represents a different value for n.
Appropriate choices for
are as a linear variation between
. Note that this range does not include the values for 0 and
, as this leads to a singular matrix, which can't be solved.
Lift and drag from coefficients
The lift can be determined by integrating the circulation terms:
which can be reduced to:
where
is the first term of the solution of the simultaneous equations shown above.
The induced drag can be determined from
which can also be reduced to:
where
are all the terms of the solution of the simultaneous equations shown above.
Moreover, this expression may be arranged as a function of
in the following way :
where
is the
span efficiency factor
Symmetric wing
For a symmetric wing, the even terms of the series coefficients are identically equal to 0, and so can be dropped.
Rolling wings
When the aircraft is rolling, an additional term can be added that adds the wing station distance multiplied by the rate of roll to give additional angle of attack change. Equation 3 then becomes:
where
*
is the rate of roll in rad/sec,
Note that y can be negative, which introduces non-zero even coefficients in the equation that must be accounted for.
When the wing is rolling, the induced drag is altered because the lift vector is rotated at each spanwise station due to rolling rate.
The resulting induced drag for a wing with a rolling rate is
where
*
is the dimensionless rolling rate.
A similar change in induced drag is also present when the wing is flapping, and comprises the main production of thrust for flapping wings.
Control deflection
The effects of control surface deflection can be accounted for by simply changing the
term in Equation 3. For non-symmetric controls such as ailerons the
term changes on each side of the wing.
Elliptical wings
For an elliptical wing with no twist, with:
The chord length is given as a function of span location as:
Also,
This yields the famous equation for the elliptic induced drag coefficient:
where
*
is the value of the wing span,
*
is the position on the wing span, and
*
is the chord.
Decomposed Fourier solution
A decomposed Fourier series solution can be used to individually study the effects of planform, twist, control deflection, and rolling rate.
Useful approximations
A useful approximation is that
:
where
*
is the 3D
lift coefficient
In fluid dynamics, the lift coefficient () is a dimensionless quantity that relates the lift generated by a lifting body to the fluid density around the body, the fluid velocity and an associated reference area. A lifting body is a foil or a com ...
for elliptical circulation distribution,
*
is the 2D lift coefficient slope (see
thin airfoil theory
An airfoil (American English) or aerofoil (British English) is the cross-sectional shape of an object whose motion through a gas is capable of generating significant lift, such as a wing, a sail, or the blades of propeller, rotor, or turbine. ...
),
*
is the
aspect ratio, and
*
is the
angle of attack in radians.
The theoretical value for
is 2
. Note that this equation becomes the
thin airfoil equation if ''AR'' goes to infinity.
As seen above, the lifting-line theory also states an equation for
induced drag
In aerodynamics, lift-induced drag, induced drag, vortex drag, or sometimes drag due to lift, is an aerodynamic drag force that occurs whenever a moving object redirects the airflow coming at it. This drag force occurs in airplanes due to wings or ...
:
[Clancy, L.J., ''Aerodynamics'', Equation 5.7]
:
where
*
is the
induced drag
In aerodynamics, lift-induced drag, induced drag, vortex drag, or sometimes drag due to lift, is an aerodynamic drag force that occurs whenever a moving object redirects the airflow coming at it. This drag force occurs in airplanes due to wings or ...
component of the
drag coefficient
In fluid dynamics, the drag coefficient (commonly denoted as: c_\mathrm, c_x or c_) is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment, such as air or water. It is used in the drag e ...
,
*
is the 3D
lift coefficient
In fluid dynamics, the lift coefficient () is a dimensionless quantity that relates the lift generated by a lifting body to the fluid density around the body, the fluid velocity and an associated reference area. A lifting body is a foil or a com ...
,
*
is the
aspect ratio,
*
is the
Oswald efficiency number
The Oswald efficiency, similar to the span efficiency, is a correction factor that represents the change in drag with lift of a three-dimensional wing or airplane, as compared with an ideal wing having the same aspect ratio and an elliptical lift ...
(or span efficiency factor.) This is equal to 1 for elliptical circulation distribution, and usually tabulated for other distributions.
Interesting solutions
According to lifting-line theory, any wing planform can be twisted to produce an elliptic lift distribution.
Limitations of the theory
The lifting line theory does not take into account the following:
*
Compressible flow
Compressible flow (or gas dynamics) is the branch of fluid mechanics that deals with flows having significant changes in fluid density. While all flows are compressible, flows are usually treated as being incompressible when the Mach number (the r ...
*
Viscous flow
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inter ...
*
Swept wings
*
Low aspect ratio wings
*
Unsteady flow
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...
s
See also
*
Horseshoe vortex
The horseshoe vortex model is a simplified representation of the vortex system present in the flow of air around a wing. This vortex system is modelled by the ''bound vortex'' (bound to the wing) and two '' trailing vortices'', therefore having ...
*
Thin airfoil theory
An airfoil (American English) or aerofoil (British English) is the cross-sectional shape of an object whose motion through a gas is capable of generating significant lift, such as a wing, a sail, or the blades of propeller, rotor, or turbine. ...
*
Vortex lattice method
Notes
References
*
L. J. Clancy
Laurence Joseph Clancy (15 March 1929 - 16 October 2014) was an Education Officer in aerodynamics at Royal Air Force College Cranwell whose textbook ''Aerodynamics'' became standard.
He was born in Egypt to Alfred Joseph Clancy and Agnes Hunter. I ...
(1975), ''Aerodynamics'', Pitman Publishing Limited, London. {{ISBN, 0-273-01120-0
* Abbott, Ira H., and Von Doenhoff, Albert E. (1959), ''Theory of Wing Sections'', Dover Publications Inc., New York. Standard Book Number 486-60586-8
Aerodynamics