The Calculation of Pressure on Slender Airplanes in Subsonic and Supersonic Flow Page: 2 of 12
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REPORT 1185-NATIONAL ADVISORY COMMITTEE FOR AERONATUICS
enough so that nearly everywhere in the fluid the magnitude
of the perturbation velocity vector divided by the speed of
the free stream is much less than one; that is,(la)
UO .
Moreover, large supersonic Mach numbers are to be avoided
and as a measure of this condition the inequalityA 2 2+2+Uo
Uo(lb)
is imposed.
Consider, next, the linearized partial differential equation
governing weakly disturbed isentropic fluid flow. In terms
of the perturbation velocity potential '(x, y, z), the lowest
order approximation consistent with inequalities (la) and
(lb) is(1--Mo2)9= c+ Pt+ 0=O
(2)
where the subscripts denote partial differentiation with
respect to the indicated variable.
Consider, finally, the expression for the pressure coefficient
that is again consistent to the lowest order with inequalities
(la) and (ib). By expanding the pressure-velocity relation
for steady isentropic flow and neglecting higher-order terms,
one finds
Sp--o 2 (1-Mo)u2++w2
2 Uo Uo
whore p and p are pressure and density, respectively, and
the subscript 0 refers to conditions in the free stream. It
follows from inequalities (la) and (ib) that pressure co-
efficient can be expressed in the form2 Uo Uo+w2
p U0T U02(3)
Equation (3) is the simplest general expression for pressure
coefficient that is still entirely consistent with the assump-
tions basic to the development of equation (2).
Special solutions applying to problems of the class indi-
cated can be obtained by appropriate simplification of general
solutions to equation (2). Such a procedure will be discussed
in the next section. The pressure coefficient is then deter-
mined by substituting these results into equation (3). The
simplifications that can be made in evaluating the pressure
on the surface of the airplane will also be discussed.
THE REDUCED SOLUTIONS
Subsonic.-As it applies to subsonic flow, equation (2)
can be written in its normalized form aspOxz= + 9? '4+'=O
(4)
The analysis of equation (4) can be interpreted as applying
to the condition M0=0 but one can extend the solutions
throughout the subsonic Mach number range by applying
the Prandtl-Glauert rule.A well-known solution to equation (4), resulting from an
application of Green's 'theorem, is given by the expression(5)
pxyz)= - 6 ( dSz
4? ) \nwhere dS is the element of surface area on the airplane or its
vortex wake, r equals (y-y)+ (z-z), and b/bn' is the
derivative normal to the surface S1. When this solution is
applied to boundary-value problems for slender configurations
it can be simplified considerably.
For example, when the airplane shape is slender it is
justifiable to introduce simplifications in the form of the
derivative b/bn' and the differential area dS1. The operator
b/bn' can be expressed as
' n b
where n1, n2, and n3 are the direction cosines between a normal
to the surface SL and the x, y, and z axes, respectively. The
differential area dS can be expressed as
dEldx
where ds is a differential length along the surface in a yz
plane. If the shape is slender, n, is small and can be neglected
relative to either unity or 1n22+na.
By means of these simplifications, equation (5) can be
approximated by the expression (from now on, the configura-
tion will be considered to lie along.the positive x, axis with its
foremost part in the x=O0 plane)p(x,yz)= 1 dxf ( .
47r o , an (x-zz)2+P2(6)
where b/bn represents n7b/by+Tb/bz, the normal derivative
to a section in the yz plane, and s is the curve bounding this
section.
If the wing-body configuration is slender, the ratio
[r/(x-x-)]2 is small over almost all of its surface and vortex
wake provided the point x, y, z is on or in the vicinity of these
surfaces. This implies the approximation(x-zZ1)2+r s [-x1
(7)
can be used.-to simplify further equation (6). However,
since, in the limiting case of r=O, equation (6) is a divergent
integral, it is necessary to introduce this approximation
with some care.
First let us consider in equation (6) only the portion of the
integral multiplying qb/bn. Designating this by p((x,y,z)
one can readily show
1 b 'd f l(x,sI)(x-x) (br/) (8a)
(4x,y,z)s dxo'J r1(x r de(8)
which, with the approximation given by equation (7), re-
duces to ..
l(Xyz- O dx X- Inr,i d (8b)
4r OSfo , 0 -En646
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Heaslet, Max A. & Lomax, Harvard. The Calculation of Pressure on Slender Airplanes in Subsonic and Supersonic Flow, report, November 28, 1953; (https://digital.library.unt.edu/ark:/67531/metadc65547/m1/2/: accessed April 23, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.