A method for simulating the atmospheric entry of long-range ballistic missiles Page: 3 of 7
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A METHOD FOR SIMULATING ATMOSPHERIC ENTRY
(b) Average rate of heat transfer per unit area
dHa. O/PoVe _ , 02*3
d.C_ 4/pVs e nOe m ine.-
dt 4(a) Rate of heat transfer to stagnation point
dH, o c ,,
-= const . V. e e 2m sin *
dt r(4)
(5)
According to equation (1) the density of the air in the test
chamber must vary exponentially with the distance corre-
sponding to altitude in the atmosphere. The more general
implication is, of course, that the test chamber must duplicate
variations of p/po in the atmosphere, whatever they may be,
although the absolute magnitudes of p and po may be quite
different from those in the atmosphere. The static tempera-
ture of the air in the test chamber is, as in the case of the
atmosphere, presumed to be small- by comparison to missile
recovery temperature (see ref. 2 in connection with this point
as it relates to the derivation of eqs. (2) through (5)).
It will be stipulated now that model and missile be geome-
trically similar in structure and configuration, and made of
the same material. Furthermore, the condition is imposed
that the model enter the test chamber at the same speed and
temperature as the missile enters the atmosphere. Finally,
it is required that model and missile have the same Reynolds
numbers (based on local conditions outside the boundary
layer) at corresponding points fly' in their trajectories. By
corresponding points it is meant where the product fly' is the
same for model and missile. It should be recognized, of
course, that, in general, 9 and y' will individually be grossly
different for model and missile.
It follows from these requirements and equation (2) that
the heat transfer per unit mass Q/m to model and missile
will be the same at corresponding points fly' provided V is
the same, since C/ and G/'S/jCDA are the same. But from
equation (3) the velocity V will be the same at corresponding
fly' only if OCDp0A/flm sin 0, is the same. If the subscript mo
refers to model and mi to missile, then the last provision may
be written.(pD)mo (m '/sin 0e,)m
(6)
But model and missile Reynolds numbers, velocities, and
disturbed air temperatures 5 are the same, hence(poD).= (poD)mo
and equation (6) may be written
sm * m si Oe as(7)
(8)SThis observation with regard to disturbed air temperatures is easily verified by considering
flow near the surface of a missile with a stagnation point. Assuming for simplicity that air
in the disturbed flow behaves Ideally, we have at the stagnation point T.s VP2C, since
MC>I. Then the temperature of the air Just outside the boundary layer is given by the
expression Tr- 1 Mr Hence if Vand Mi are the same for model and missile.
then Ta and Vi are the same, independent of ambient air temperature. We are assured of
equal M f's by the freeze principle (see footnote 4).This expression fixes the length L of the test chamber in
terms of the portion of the atmosphere to be simulated
therein and the ratio of model to missile size." If equation
(8) and the previously set forth requirements are satisfied,
then model and missile should experience equal heat transfer
per unit mass, and hence equal average temperature rise at
corresponding points in their trajectories. These quantities
are significant, of course, because they tend to determine
whether a missile will melt or perhaps burn during flight.
The next question is how do the heat-transfer rates com-
pare in the case of model and missile? It is easily deduced
from equations (4) and (5) and the conditions for equal
heat transfer per unit mass that
._D, dH,
dt D dt (9)
and
dHs)D, )dH' (10)
dt noD,,odt )=
h o..
at corresponding points fly'. That is to say, the average
and stagnation point heat-tran,sfer rates are higher for the
model in proportion to the ratio of missile to model size.
But perhaps the foremost importance of heat-transfer rates
is, as discussed earlier, in how they influence thermal stresses
in the missile structure and, for example, lead to ablation
of surface material. Evidently, then, it would be most
desirable if equations (9) and (10) implied equal thermal
stresses in model and missile. This possibility is easily
checked using modified equilibrium thermal-stress equations
for an unrestrained isotropic elastic body (see ref. (4) and
sketch).* Since model and missile velocities are the same at corresponding points in their trajectorIes,
It follows from equation (8) that the time of flight in the test chamber is reduced below that of
atmospheric entry by the ratio (D4./DO.
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Eggers, A. J., Jr. A method for simulating the atmospheric entry of long-range ballistic missiles, report, September 15, 1955; (https://digital.library.unt.edu/ark:/67531/metadc60825/m1/3/: accessed March 28, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.