The Prandtl–Glauert transformation is a mathematical technique which allows solving certain
compressible
In thermodynamics and fluid mechanics, the compressibility (also known as the coefficient of compressibility or, if the temperature is held constant, the isothermal compressibility) is a measure of the instantaneous relative volume change of a f ...
flow problems by
incompressible
In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An eq ...
-flow calculation methods. It also allows applying incompressible-flow data to compressible-flow cases.
Mathematical formulation
Inviscid compressible flow over slender bodies is governed by linearized compressible small-disturbance potential equation:
:
together with the small-disturbance flow-tangency boundary condition.
:
is the freestream Mach number, and
are the surface-normal vector components. The unknown variable is the perturbation potential
, and the total velocity is given by its gradient plus the freestream velocity
which is assumed here to be along
.
:
The above formulation is valid only if the small-disturbance approximation applies,
:
and in addition that there is no transonic flow, approximately stated by the requirement that the local Mach number not exceed unity.
:
The Prandtl–Glauert (PG) transformation uses the Prandtl–Glauert factor
. It consists of scaling down all ''y'' and ''z'' dimensions and angle of attack by the factor of
the potential by
and the ''x'' component of the normal vectors by
:
:
This
geometry will then have normal vectors whose x components are reduced by
from the original ones:
:
The small-disturbance potential equation then transforms to the Laplace equation,
:
and the flow-tangency boundary condition retains the same form.
:
This is the incompressible potential-flow problem about the transformed
geometry. It can be solved by incompressible methods, such as thin airfoil theory, vortex lattice methods, panel methods, etc. The result is the transformed perturbation potential
or its gradient components
in the transformed space. The physical linearized pressure coefficient is then obtained by the inverse transformation
:
which is known as Göthert's rule
Results
For two-dimensional flow, the net result is that
and also the lift and moment coefficients
are increased by the factor
:
:
where
are the incompressible-flow values for the original (unscaled)
geometry. This 2D-only result is known as the Prandtl Rule.
For three-dimensional flows, these simple
scalings do NOT apply. Instead, it is necessary to work with the scaled
geometry as given above, and use the Göthert's Rule to compute the
and subsequently the forces and moments. No simple results are possible, except in special cases. For example, using
Lifting-Line Theory
The Prandtl lifting-line theory is a mathematical model in aerodynamics that predicts lift distribution over a three-dimensional wing based on its geometry. It is also known as the Lanchester–Prandtl wing theory.
The theory was expressed indepen ...
for a flat elliptical wing, the lift coefficient is
:
where ''AR'' is the wing's aspect ratio. Note that in the 2D case where ''AR'' → ∞ this reduces to the 2D case, since in incompressible 2D flow for a flat airfoil we have
as given by
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. ...
.
Limitations
The PG transformation works well for all freestream Mach numbers up to 0.7 or so, or once transonic flow starts to appear.
History
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 ...
had taught the transformation in his lectures earlier, however the first publication was in 1928 by
Hermann Glauert
Hermann Glauert, FRS (4 October 1892 – 6 August 1934) was a British aerodynamicist and Principal Scientific Officer of the Royal Aircraft Establishment, Farnborough until his death in 1934.
Early life and education
Glauert was born in Shef ...
. The introduction of this relation allowed the design of aircraft which were able to operate in higher subsonic speed areas. Originally all these results were developed for 2D flow. Göthert eventually realized in 1946 that the geometric distortion induced by the PG transformation renders the simple 2D Prandtl Rule invalid for 3D, and properly stated the full 3D problem as described above.
The PG transformation was extended by
Jakob Ackeret
Jakob Ackeret, FRAeS (17 March 1898 – 27 March 1981) was a Swiss aeronautical engineer. He is widely viewed as one of the foremost aeronautics experts of the 20th century.
Birth and education
Jakob Ackeret was born in 1898 in Switzerland. He ...
to supersonic-freestream flows. Like for the subsonic case, the supersonic case is valid only if there are no transonic effect, which requires that the body be slender and the freestream Mach is sufficiently far above unity.
Singularity
Near the sonic speed
the PG transformation features a
singularity. The singularity is also called the
Prandtl–Glauert singularity
The Prandtl–Glauert singularity is a theoretical construct in flow physics, often incorrectly used to explain vapor cones in transonic flows.
It is the prediction by the Prandtl–Glauert transformation that infinite pressures would be experie ...
, and the flow resistance is calculated to approach infinity. In reality, aerodynamic and thermodynamic perturbations get amplified strongly near the sonic speed, but a singularity does not occur. An explanation for this is that the linearized small-disturbance potential equation above is not valid, since it assumes that there are only small variations in Mach number within the flow and absence of compression shocks and thus is missing certain nonlinear terms. However, these become relevant as soon as any part of the flow field accelerates above the speed of sound, and become essential near
The more correct nonlinear equation does not exhibit the singularity.
See also
*
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 ...
*
Hermann Glauert
Hermann Glauert, FRS (4 October 1892 – 6 August 1934) was a British aerodynamicist and Principal Scientific Officer of the Royal Aircraft Establishment, Farnborough until his death in 1934.
Early life and education
Glauert was born in Shef ...
*
Jakob Ackeret
Jakob Ackeret, FRAeS (17 March 1898 – 27 March 1981) was a Swiss aeronautical engineer. He is widely viewed as one of the foremost aeronautics experts of the 20th century.
Birth and education
Jakob Ackeret was born in 1898 in Switzerland. He ...
References
Citations
Sources
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{{DEFAULTSORT:Prandtl-Glauert transformation
Aerodynamics