Applications of Modern Hydrodynamics to Aeronautics. Part 1: Fundamental Concepts and the Most Important Theorems. Part 2: Applications Page: 1 of 56
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REPORT No. 116.
APPLICATIONS OF MODERN HYDRODYNAMICS TO AERONAUTICS.
By L. PRANDTL.
FUNDAMENTAL CONCEPTS AND THE MOST IMPORTANT THEOREMS.
1. All actual fluids show internal friction (viscosity), yet the forces due to viscosity, with
the dimensions and velocities ordinarily occurring in practice, are so very small in comparison
with the forces due to inertia, for water as well as for air, that we seem justified, as a first ap-
proximation, in entirely neglecting viscosity. Since the consideration of viscosity in the
mathematical treatment of the problem introduces difficulties which have so far been overcome
only in a few specially simple cases, we are forced to neglect entirely internal friction unless we
wish to do without the mathematical treatment.
We must now ask how far this is allowable for actual fluids, and how far not. A closer
examination shows us that for the interior of the fluid we can immediately apply our knowl-
edge of the motion of a nonviscous fluid, but that care must be taken in considering the layers
of the fluid in the immediate neighborhood of solid bodies. Friction between fluid and solid
body never comes into consideration in the fields of application to be treated here, because it
is established by reliable experiments that fluids like water and air never slide on the surface
of the body; what happens is, the final fluid layer immediately in contact with the body is
attached to it (is at rest relative to it), and all the friction of fluids with solid bodies is therefore
an internal friction of the fluid. Theory and experiment agree in indicating that the transition
from the velocity of the body to that of the stream in such a case takes place in a thin layer of
the fluid, which is so much the thinner, the less the viscosity. In this layer, which we call the
boundary layer, the forces due to viscosity are of the same order of magnitude as the forces due
to inertia, as may be seen without difficulty.' It is therefore important to prove that, however
small the viscosity is, there are always in a boundary layer on the surface of the body forces
due to viscosity (reckoned per unit volume) which are of the same order of magnitude as those
due to inertia. Closer investigation concerning this shows that under certain conditions there
may occur a reversal of flow in the boundary layer, and as a consequence a stopping of the fluid
in the layer which is set in rotation by the viscous forces, so that, further on, the whole flow is
changed owing to the formation of vortices. The analysis of the phenomena which lead to the
formation of vortices shows that it takes place where the fluid experiences a retardation of flow
along the body. The retardation in some cases must reach a certain finite amount so that a
reverse flow arises. Such retardation of flow occurs regularly in the rear of blunt bodies; there-
fore vortices are formed there very soon after the flow begins, and consequently the results
which are furnished by the theory of nonviscous flow can not be applied. On the other hand,
in the rear of very tapering bodies the retardations are often so small that there is no noticeable
formation of vortices. The principal successful results of hydrodynamics apply to this case.
Since it is these tapering bodies which offer specially small resistance and which, therefore,
have found special consideration in aeronautics under similar applications, the theory can be
made useful exactly for those bodies which are of most technical interest.
1 From this consideration one can calculate the approximate thickness of the boundary layer for each special case.
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Prandtl, L. Applications of Modern Hydrodynamics to Aeronautics. Part 1: Fundamental Concepts and the Most Important Theorems. Part 2: Applications, report, 1979?~; (digital.library.unt.edu/ark:/67531/metadc53396/m1/1/: accessed October 22, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.