Download On the Coefficients in Meteor Physics Equations

On the Coefficients in Meteor Physics Equations
Daria Yu. Khanukaeva
Department of Applied Mathematics, Moscow Institute of Physics and Technology (State University),
Institutsky 9, 141700 Dolgoprudny, Moscow region, Russia
Abstract. Drag coefficients of meteoroids have been investigated in various flow regimes. Some new simple
approximations for the drag coefficient and convective heat transfer coefficient of the meteoroid are proposed. Analytic
solutions of the meteoroid drag equation were obtained using these approximations. Some numerical solutions with all
variable coefficients have also been obtained. The comparison of solutions with constant and variable coefficients shows
significant differences, depending on the size of meteoroid.
INTRODUCTION
Basic equations of physics of meteoritic phenomena are the drag equation [1]
1
dV
= − C D ρV 2 A
m
(1)
2
dt
and the ablation equation [1]
1
dm
= − C H ρV 3 A ,
Q
(2)
2
dt
where m – meteoroid mass, V – its velocity, А – middle square area, ρ – density of the atmosphere, Q – effective
enthalpy of ablation, CD and CH – drag and heat transfer coefficients respectively, which are under consideration in
the present work.
Mechanical characteristics of the body: position, velocity, deceleration, mass, kinetic energy, strength etc. are the
unknown quantities in meteoritic physics problems. The dependences of atmosphere density on height and crosssectional area on body mass and density are given, as well as coefficients in equations should be set. The problem of
meteoroid motion with constant coefficients has been accurately discussed in literature. The solution of the
problem, including ablation and mechanical fragmentation, was obtained even in the case of a non-isothermal
atmosphere [2].
In fact, these coefficients are not constant over the meteoroid trajectory. The values of CD/2 and CH are
respectively the part of momentum and energy of the flow transferred to the body. They depend essentially on the
flow regime realized. The range of flow regimes over meteoroids changes from free-molecule in upper atmosphere
to continuum with thin shock wave at low altitudes.
Drag and heat transfer coefficients can be found as the solution of aerodynamic problem in the frame of some
particular flow model. But this solution may be incorrect on some parts of the trajectory. An alternative variant is
the search of some approximations of the coefficients, applicable for any regime. At first, we will concentrate on the
drag coefficient, and then use the results in the analysis of the convective heat transfer coefficient. The value of the
drag coefficient for simple geometric forms can be found analytically in free-molecule and continuum limits [3, 4].
Empirical dependences, based on experimental data, are used for the interpolation of these values in transitional
regime. The review of the investigations on this method is given in Ref. [5].
The existing approximations are rather bulky expressions, not convenient for the application in meteoritic
problems. The present work is devoted to the search of some simple analytical dependence of the drag coefficient
on the dimensionless parameters of the problem, in particular on the Reynolds number, and to the solution of the
problem accounting this dependence. Special attention is paid to the free-molecule regime. Some analytic solutions
are obtained. Also some extension of the results has been done in order to consider the ablation of meteoroids more
adequately.
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
726
THE DRAG COEFFICIENT IN VARIOUS REGIMES
It is sufficient to use the Newton theory in order to determine the pressure on the surface of meteoroid, flowing
with the stable hypersonic continuum stream [1, 4]. In this case the drag coefficient will depend only on the body
geometry.
In free-molecule regime the drag coefficient is determined by the nature of the particle-surface interactions,
rather then body geometry [3].
It was mentioned in Ref. [6], that incident particles of free-molecule stream can drive out sufficient amount of
atoms or molecules from the crystal lattice of the meteoroid material to produce a sort of microexplosion on its
surface. Though the speed of the crashed mass outbreak is less then the thermal velocity, corresponding to the
evaporation temperature, the resulting “jet” efficiency is larger in comparison with the case of gas outbreak.
Therefore the process leads to the growth of the deceleration and drag coefficient. Considering this effect, the
expression for the drag coefficient in free-molecule flow regime C Dfm was shown in [6] for hypersonic velocities to
reduce to
2Q∗  M mat  V 2
9 Qvap 
+
+
C Dfm = 2 +
(3)

,

2V  2М a  2Qvap 4 Q∗ 


where Ma=29 g/mol – air molecular weight, Qvap – effective enthalpy of evaporation, Q* – effective enthalpy of
“crashing” of meteoroid lattice material, which molecular weight – Mmat.
Equation (1) with the drag coefficient defined by (3) appears to have an analytic solution. It can be written in
general form, but here for the sake of shortness and simplicity concrete values of parameters are used. For iron and
stony meteoroids Mmat≈2Ма , Q*≈0.6 km2/s2, Qvap≈8 km2/s2. Using these values, the solution of the problem of
meteoroid ballistics in the isothermal atmosphere, which trajectory is a straight line, inclined to horizon with θ, may
be presented as
40
V =
[V ] = [Ve ] = km / s ,
− 10,
(4)

40 
0.675 hA 
1 +
 exp
ρ − 1
 m sin θ 
 Ve + 10 
where Ve – entrance velocity of the meteoroid, h – scale height of the atmosphere (for the Earth h=7 km).
In transitional flow regime the drag coefficient is often approximated as a function of the Reynolds number,
recommended as the main parameter in [5]
ρVr
,
(5)
Re 0 =
µ(T0 )
γ −1 2
where r – the body size, µ – the dynamical viscosity coefficient. T0 = T∞ (1 +
Μ ) – the stagnation temperature,
2
where Т∞ is the free stream temperature, γ – ratio of specific heats, М – Mach number. The model of gas-particles
interactions involved defines the dependence of viscosity on the temperature. Here we use the hard spheres model,
which gives the viscosity coefficient as the square root of the temperature.
NEW APPROXIMATION
Simple dependence СD(Re) was offered by G.A. Tirskiy (oral report). It looks like С D ≅ 1 + e − Re . This formula
qualitatively models the change of the drag coefficient for the spherical shape: in continuum flow Re→∞ and СD→1,
in free-molecule flow Re→0 and СD→2, in agreement with known results [3, 4]. Function СD(Re), as it is known
from works [1, 3, 5] has a derivative close to zero in the vicinity of Re=0. But the function е-Re has a negative and
very large derivative, that is the curve drops sharply in the vicinity of Re=0. Besides, the convexity of the graph of
function е-Re is downcast (the second derivative is positive), but real curves have negative convexity. The function
2
2
е − Re has the demanded first and second derivatives. Therefore it seems reasonable to write it as е − а Re , where а<1
is a free coefficient which can be determined to fit the experimental data.
In order that the formula discussed can be applied to the calculation of the drag coefficients of not only spherical
bodies, we will use a general notation of aerodynamical variables in transitional flow regime. Then we obtain
727
(
)
2
С D = C Dc + C Dfm − C Dc e − a Re .
(6)
c
D
fm
D
The values of drag coefficient in continuum C and free-molecule C
limits are calculated separately,
according to known formulas [3, 4]. Reynolds number is defined by formula (5).
CD
2.5
2
1.5
1
0.5
Re 0
0
1
10
100
1000
Circles – М=24.6, ТW/T∞=4.5;
Triangulars – М=19, ТW/T∞=2.5;
Crosses – М=15, ТW/T∞=1.6.
FIGURE 1. Theoretical with а=0.001 (solid line) and experimental data for the drag coefficient of a sphere.
The calculations were fulfilled using the spherical shape of the meteoroid. According to experimental data of
work [7] the value of free coefficient was found to be equal to а=10-3. The experiments in [7] were fulfilled under
different Mach numbers and ratios of body surface ТW and free stream temperatures. Fig. 1 contains these series of
experimental points and theoretical curve provided that а=0.001. Free-molecule and continuum limits were taken
traditionally equal to C Dfm = 2 and C Dc = 0.92 .
SOLUTIONS AND ANALYSIS
It was found that the analytic solution of equation (1) with drag coefficient defined by (6), may be written in the
following form

hAρ  c
π fm
erf ( Dρ)  
(7)
V = Ve exp−
C D − C Dc
C D +
 ,
2
Dρ  
 2m sin θ 
(
where D is the product of known constants and body size r =
D=
r
µ∞
)
A/ π :
2γRT∞
a , γ=1.4, Т∞=284 К, µ∞=1.7⋅10-5 Pа⋅s.
( γ − 1) M a
If the value of C Dfm , defined by (3), is substituted in expression (6) for CD , one can obtain a numerical solution of
equation (1) over the whole range of heights including the specialties of free-molecule flow regime.
Fig. 2 represents the meteoroid velocity change with height under various drag coefficient definitions. Cases a)
and b) correspond to different values of entrance mass of the meteoroid me=3⋅103 kg and me=3⋅10-6 kg respectively,
other initial parameters being the same, namely entrance velocity Ve=21 km/s and inclination angle counted from the
horizon, θ=30°.
The classical solution of equation (1) with constant drag coefficient looks like
728
 C ρ hA 
(8)
Vсl = Ve exp − D
.
 2m sin θ 
Plotting curve 2 on Fig. 2a), a numerical calculation of equation (1) including gravity was used in order to avoid
full coincidence of curves 1 and 2. This means that solution (7) barely differs from classical one for a large body. It
is rather clear that the change of the drag coefficient gives the change of the deceleration in two times. For massive
bodies the absolute value of the drag force acceleration is very small almost over the whole trajectory. Therefore
classical solution (8) can be used, when the velocity change of massive meteoroids is discussed. In this case CD=1
should be taken, as large bodies move predominantly in continuum flow regime.
1 – solution (8) provided CD=1.5
2 – solution (7)
fm
3 – numerical solution with variable CD and CD ,
4 – solution (4)
FIGURE 2. Change of velocity with height of large (a) and small (b) meteoroid.
The classical solution gives the most essential divergence for small bodies at high altitudes. One can determine
the critical size of meteoroid rc, for which taking into account dependence (6) gives the correction to the classical
value Vcl no less then 10% at the altitude z=100 km. Taking the ratio of velocities, defined by expressions (7) and
(8), equal to 0.9, we obtain the equation for rc . With the meteoroid density being δ=1 g/sm3, it takes form
erf ( Dρ)
= 0.032 .
rc Dρ
Here coefficient D also depends on rc . A numerical solution of this equation for the isothermal atmosphere gives
the value of r=27 mkm at the altitude z=100 km. As the drag of micrometeoroids takes place at higher altitudes in
free-molecule flow regime, the dependence (3) should be used. This will result in even larger difference. The drag
will become more intensive as the entrance velocity becomes larger.
The curves on Fig. 2b) shows the velocity change of a small meteoroid. According to its size, it belongs to the
class of meteoroids, representing more than one half of 70 million cosmic bodies entering the atmosphere daily. As
it was predicted, solution (4) has given much stronger deceleration, than the others.
However, it is worth keeping in mind, that the meteoritic body’s motion is accompanied by the intensive
ablation. Also, what is essential is the mass loss due to the driving out of groups of meteoroid particles ought to be
considered in free-molecule regime along with probable melting and evaporation. As Q*<<Qvap , the rate of the
729
ablation will be an order of magnitude higher than it is usually estimated in the classical theory of meteoritic
phenomena. Consequently, the meteor track may be much shorter. The value of Q* was introduced into the general
scheme of the effective enthalpy calculation, taken from [8]. According to equation (2), the solution of the problem
including meteoroid mass change gives a complete mass loss at the altitude of ~70 km for the small body, under
consideration. It means, that the curves of velocity change terminate at this altitude and do not reach zero, as it
follows from the solution without ablation.
The mass change and velocity change (curve 5) of the considered small meteoroid are given on Fig. 3. Curve 1
for the classical solution (8) and curve 3 for the solution with the variable drag coefficient and constant mass are
given on Fig. 3a), for the sake of comparison. Variable heat transfer coefficient, used in the calculations of
equations (1-2), is discussed below.
FIGURE 3. Velocity and mass change of small body (r~1 mm).
THE HEAT TRANSFER COEFFICIENT
The heat transfer to a meteor body is realized by the convective and radiative mechanisms [1]. Following the
traditional way, we calculate the heat transfer coefficient of the meteoroid as a simple sum of convective CHcon and
radiative CHrad components. As possible nonequilibrium states of the gas in the shock layer give no more than 30%
change in the heat transfer to the body [9], they are not considered here. In meteor physics the equilibrium heat
flows are usually approximated as functions of the air density, velocity and size of the body [1]. We use the
approximation of the radiative heat transfer coefficient, given in [8].
It is proposed to use analogous procedures for approximation of the drag and convective heat transfer
coefficients. The mechanism of convective heat transfer is similar to the mechanism of momentum transfer. The
value of CHcon decreases along the trajectory [8], that means it changes with flow regime, which may be
characterized by the Reynolds number. So, the coefficient may be written in the form analogous to (6). In freemolecule limit total heat transfer coefficient is equal to unit [1]. Therefore for the convective component we have to
fm
c
= 1 − C Hrad and in continuum regime C Hcon
~ 1 / Re , according to [1,8]. Then for any regime, including
write C Hcon
transitional one, we write
(
)
2
0
С Hc = b / Re 0 + 1 − C Hrad − b / Re 0 e − c Re ,
where unknown coefficients b and c should be determined to fit the numerical data.
730
(9)
The convective component, calculated accordingly expression (9) was added to with the radiative component to
find total heat transfer coefficient, which was substituted into equation (2). The change of components and full
coefficient, found in the process of numerical solution of system (1,2), is presented on Fig. 4(a-c). Unlike the drag
coefficient the heat transfer coefficient substantially influences the solution not only for small, but also for massive
bodies. Curve 2 on Fig. 4d) gives the change of mass with height, obtained in the numerical calculations for massive
body (R~1 m). Curves 1 and 3 are given for the sake of comparison. They correspond to the solutions with CH=1
and CH=0.01 respectively, which are rather common estimates, used in meteoritic problems. Thus, it can be seen
from Fig. 4, that the heat transfer coefficient changes dramatically on the trajectory of the meteoroid. It is not
recommended to use constant values of this coefficient in considerations of real meteoritic falls.
FIGURE 4. Change of convective, radiative and full heat transfer coefficient with height of massive body (R~1 m) and its mass
change, calculated with various values of CH.
One more aspect of the problem should be mentioned. It is a shielding effect, produced by ablated vapors of the
body and diminishing the heat flux to the surface. The convective heat transfer is predominantly subjected to this
phenomenon. We used special method of the effective enthalpy calculation, which is partly suitable for this case.
But its accurate discussion is beyond the scope of the present work.
CONCLUSIONS
The behavior of the meteoroid drag coefficient has been considered in various flow regimes. An effect of
microexplosions on the surface of the body on meteoroids drag was taken into account in free-molecule flow. The
analytic solution of the drag equation was found with this condition. Simple and convenient approximation of the
drag coefficient for any flow regime has been offered. Analytic solutions of the drag equation were found provided
various values of the drag coefficient and compared with the numerical solution, obtained using variable drag
fm
coefficient, with free-molecule limit value CD being variable. It was found the critical size of a micrometeoroid,
for which the variability of the drag coefficient is substantial. The present investigation may serve as the
justification of constant drag coefficient usage for massive meteoroids and as the proof of impossibility of this
simplification in case of small particles.
In modeling the meteoroid entry it often comes to deal with “middle” sizes. In this case the dependences offered
in this work are especially convenient. The altitudes, where the boundaries of flow regimes are situated, are not
known beforehand, and monitoring the transitions between the regimes is a rather laborious problem. The formulae,
given in the present work, fully exclude this problem, as they are universal for all regimes.
731
The idea of the drag coefficient calculation was extended to the calculation of the heat transfer coefficient. The
offered approximation was used in the solution of the problem of the meteoroid motion in the atmosphere with
variable mass. The importance of accurate calculation of the heat transfer coefficient was demonstrated in the
calculation of the meteoroid mass change with height.
The results of the present investigation may be used in modeling of meteoroids entry without restrictions on their
sizes and also of artificial objects, moving in the atmosphere with cosmic velocities.
ACKNOWLEDGMENTS
The work was supported by RFBR grant №00-01-06-522.
REFERENCES
1. Bronshten, V. A., Physics of Meteoric Phenomena, Reidel, Dordrecht, 1983, 356p.
2. Tirskiy, G. A., Khanukaeva, D. Yu., “The Model of Interaction of Cosmic Bodies With Nonisothermal Atmosphere,” in
Near-Earth Astronomy of the XXI Century, edited by M. A. Smirnov at al., INASAS Conference Proceedings, Geos, Moscow,
2001, pp. 367-378. (In Russian)
3. Bird, G. А., Molecular Gas Dynamics, Clarendon Press, Oxford, 1976, 320p.
4. Lunev, V. V., Hypersonic Aerodynamics, Mashinostroenie, Мoscow, 1975, 327p. (In Russian)
5. Khlopkov, Yu. I., Statistical Modeling in Physical Gasdynamics, MIPT Press, Moscow, 1998, 140p. (In Russian)
6. Stanukovich, K. P., News of AS USSR, Ser. Mech. and Mash 5, 3-8 (1960). (In Russian)
7. Kussoy, M. I., Hortsman, C. C., AIAA Journal 8, N2, 315-320 (1970).
8. ReVelle, D. O., Planetary Sciences SR-76-1, 1-90 (1976).
9. Tirskiy, G. A., Shcherbak V. G., Chemical and Thermodynamical Nonequilibrium Air Flows under Small and Moderate
Reynolds Numbers, MSU Press, Moscow, 1988, 240p. (In Russian)
732