Applications of Modern Hydrodynamics to Aeronautics Part 1: Fundamental Concepts and the Most Important Theorems. Part 2: Applications Page: 10 of 56
56 p. : ill.View a full description of this report.
Extracted Text
The following text was automatically extracted from the image on this page using optical character recognition software:
REPORT NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS.
For r =o this expression becomes zero or infinite, according as n is greater or less than 11, i. e.,
according as the angle a is less or greater than r(- 1800). Figures 6 and 7 give the streamlines
4 a 3
for a= 450 anld a- 270, corresponding to n=4 and . In the case of figure 7 the velocity,
as just explained, becomes infinite at the corner. It would bie expected that in the case of
the actual flow some effect due to friction would enter. In fact there are observed at such
corners, at the beginning of the motion, great velocities, and immediately thereafter the for-
mation of vortices, by which the motion is so changed that the velocity
at the corner becomes finite.
It must also be noted that with an equatiorf
Sp +iq= p (c +iy) (11)
I the ?x-y plane can be mapped upon the p-q plane, since to every pair
of values x,y a pair of values p,q corresponds, to every point of the x-y
plane corresponds a point of the p-q plane, and therefore also to every
,. .-n .. .. element of a line or to every curve in the former plane a linear element
S" an" e ,,ilih hj hand a curve in the latter plane. The transformation keeps all angles
othir unchanged, i. e., corresponding lines intersect in both figures at the same
angle.
By inverting the function of equation (11) we can write
x +iy = x P +iq)
and therefore deduce from equation (8) that
4,+ +i~ f [x(p +i)] )1= (p + iq) (12)
c and 'P are connected therefore with p and q by an equation of the type of equation (8), and
hence, in the p-q plane, are potential and stream functions of a flow, and further of that flow
which arises fromt he transformation of the 4, I network in the -y plarW into the p-q plane.
This is a powerful method used to obtain by transformation from a known simple flow
new types of flow for other given boundaries. Applications of this will be given in section 14
11. The discussion of the principles of the hydrodynanjics of nonviscous fluids to be
applied by us Imay ie stopped here. I ad1i but one conlsidera-
tion, which has reference to a very usef:'l theore:n for oitaiing ,
the forces in fluid motion, namely the so-called "momentum theo-
rein for stationary motions."'
We have to apply to fluid motion the tlleo',em of general
mechanics, which states that the rate of change with the time /
of the linear mnonmer-u;n is equal to tihe resultant of all the ex- /
ternal forces. To do this, consider a definite portion of the
fluid separated from time rest of the fluid by a closed surface. /
This surface may, in accordance witih the spirit of the theorem,
be considered as a "fluid surface," i. e., made up always of . .-Unipran ar o-ow around plane
the same fluid particles. Ie must now state in a formula the walls making an angle 270 wnith eac
change of the momentum of the fluid within the surface. If, as
we shall assume, the flow is stationary, then after a time dt every fluid particle in the interior
will be replaced by another, which has the same velocity as had tile former. On the boundary,
however, owing to its displacement, mass will pass out at the side where the fluid is approaching,
and a corresponding mass will enter on the side away from which the flow takes place. If dS
is the area of an element of surface, and v, the component of the velocity in the direction of
the outward drawn normal at this element, then at this point dm= pdS . vn dt. If we wish
to derive the component of the " impulse "-defined as the time rate of thile change of momen-
tum-for any direction s, the contribution to it of the element of surface is
dm
dJ,= v-dt =pdS . v,, (13)10
Upcoming Pages
Here’s what’s next.
Search Inside
This report can be searched. Note: Results may vary based on the legibility of text within the document.
Tools / Downloads
Get a copy of this page or view the extracted text.
Citing and Sharing
Basic information for referencing this web page. We also provide extended guidance on usage rights, references, copying or embedding.
Reference the current page of this Report.
Prandtl, L. Applications of Modern Hydrodynamics to Aeronautics Part 1: Fundamental Concepts and the Most Important Theorems. Part 2: Applications, report, 1979%; (https://digital.library.unt.edu/ark:/67531/metadc53396/m1/10/: accessed March 28, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.