Note on the Theorems of Bjerknes and Crocco Page: 4 of 5
This report is part of the collection entitled: National Advisory Committee for Aeronautics Collection and was provided to UNT Digital Library by the UNT Libraries Government Documents Department.
Extracted Text
The following text was automatically extracted from the image on this page using optical character recognition software:
NACA TN No. 1073
3
THEOREM OF CROCCO
The theorem of Crocco may be obtained from the L
theorem of Bjerknes by a vector transformation. Since
T = , the left-hand side of equation (3) may be written
pq x curl R
p
Take the curl of this expression, omitting the constant R;
cur curl Cur curl
curl P xuQ p div div pq
P P / p
our v - (p ) curl q
For two-dimensional flow the three first terms on the
right of this equation are identically zero since the
vector curl 4 is perpendicular to the vector q and
since div p = 0. With reference to equation (35) it is
therefore seen that
curl 0
(P* V) = O
or that curl is constant along the streamlines. The
useful result obtained.then is that the rotation remains
proportional to the absolute pressure p along each and
all streamlines. This is the theorem of Crocco.
Similarly, for flow with rotation symmetry, the
following expression may be written:
pqr x curl q R
pr
where r is the radius from the center of symmetry. By
the same reasoning cul is shown to be constant along
each sreamlne This expr session, when translate
each streamline. This expression, when translated into
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.
Theodorsen, Theodore. Note on the Theorems of Bjerknes and Crocco, report, May 1946; (https://digital.library.unt.edu/ark:/67531/metadc54965/m1/4/: accessed April 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.