But this value does vary dramatically on meteoroids trajectories for the most part of them. The drag and heat-transfer coefficients in the basic equations of PTM ...
Earth, M o o n , and Planets (2004) 95: 433-439 D O I 10.1007/s 11038-005-9032-z
© Springer 2005
CALCULATION OF VARIABLE DRAG AND HEAT-TRANSFER COEFFICIENTS IN METEORIC PHYSICS EQUATIONS D. YU. KHANUKAEVA Institute of Mechanics, Moscow State M.V. Lomonosov University, 119192 Moscow, Russia (E-mail: dukh@,imec.msu.ru)
Michurinskiy-1,
(Received 13 October 2004; Accepted 30 May 2005)
Abstract. T h e conservation of the ablation parameter is one of the main assumptions of the classical physical theory of meteors (PTM). But this value does vary dramatically on meteoroids trajectories for the most part of them. The drag and heat-transfer coefficients in the basic equations of P T M are considered f r o m the gasdynamical point of view and mathematically modeled in the present paper. Analytical approximations of the drag and heat transfer coefficients valid for any flow regimes and separate approximation of the drag coefficient for free-molecule regime are offered and discussed. They use the Reynolds n u m b e r as the main parameter characterizing flow regimes. T h e variations of coefficients result in the variation of the ablation parameter by 100% for meteoroids, passing all the flow regimes. The importance of correct calculation of the coefficients for meteoroids motion modeling was demonstrated by the numerical examples. Keywords: Meteoroids, drag coefficient approximation, heat transfer coefficient approximation, freemolecule flow regime, continuum flow regime
1. Introduction The classical physical theory of meteors (PTM) (Bronshten, 1983) and many modern works assume conservation of the ablation parameter a — CH/QCD that, in fact, implies conservation of specific heat of ablation Q, drag C D and heat transfer C H coefficients of meteoroids. By definition the values of C D /2 and C H are, respectively, the fraction of the momentum and energy of the oncoming flow transferred to the body. Therefore they depend essentially on the realizing flow regime, which, in turn, is highly dependent on body size. So, there are small particles, which move only in free-molecule regime and large bodies, which move in continuum regime. In general case the range of gasdynamic flow regimes over meteoroids changes from free-molecule in upper atmosphere to continuum with boundary layer at low altitudes. Frozen, non-equilibrium and equilibrium chemical regimes are realized on meteoroids trajectories. The heat transfer to the body consists of convective and radiative parts, which may be of different order of magnitude depending
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on the flow conditions. Thus, the drag and heat transfer coefficients are multivariable functions varying considerably. Strictly speaking, the problem of these dependences search must be formulated in the framework of the multicomponent flow over some body; it is described by the Navier-Stokes equations and the equation of radiative transport. The solution of mentioned set of equations is time consuming even for modern computers. Besides, it gives the sought quantities at a particular free stream condition, that is, in a particular point of the trajectory of the body moving with given velocity. But ready values of the drag and heat transfer coefficients are required in meteoric problems. So, it is rather topical to find some dependencies of the coefficients on the defining parameters and to express them in analytical form. The present work is devoted to the discussion of existing and new approximations of the drag and heat transfer coefficients and their application to the calculations. The first contribution to this problem is the work of D.O. ReVelle (ReVelle, 1976) where these coefficients were presented as functions of the Knudsen number. But determining the Knudsen number represents a separate and not so simple problem, and the functions mentioned were given as a set of formulas, fitting the numerical and observational data. In the present work the Reynolds number is used as the main parameter, which characterizes the flow regime; gas dynamical considerations are involved. The following definition of the Reynolds number is used here: Re = p Vrj //(Jo), where p is the free-stream density, V - meteoroid velocity, r - its radius, p(Tq) is the air viscosity, calculated with the stagnation temperature To = T(l +XY~Ma2), T - free stream temperature, у is the ratio of specific 5 heats, Ma - the Mach number, ц(Т0) = ц • {Та/Т)03, p= 1.7 x 10~ Pa s, со depends on the model of air molecules interaction potential.
2. The Drag Coefficient The free-molecule С $ and continuum C£> limits of the drag coefficient for meteoroids are well known in classical PTM (Bronshten, 1983). The former is determined by the nature of the particle-surface interactions, the latter depends on the body geometry. The approximation for the transitional regime, which includes these limit cases, was offered and analyzed in (Khanukaeva, 2003). It has the following form:
(1) where a = 0.001 is the free constant, found on the basis of experimental data (Kussoy and Hortsman, 1970).
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Approximation (1) allows an analytical solution of the drag equation for the case of со = 0.5 in the expression for the viscosity. The analysis of that solution was made in (Khanukaeva, 2003). For large bodies generating bolides it has demonstrated negligible (less than 1 %) difference between the velocities calculated with variable and constant drag coefficient. And this coefficient variation gives more than 10% correction to the dynamics of particles of less than 20-30 microns in size. In addition to the above, the effect of sputtering, being much discussed today, also affects micrometeoroids deceleration in the upper layers of the atmosphere. Though the velocity of the sputtered mass is less then the thermal velocity corresponding to the evaporation temperature (Stanukovich, 1960), the resulting recoil momentum is larger in comparison with the case of molecules outbreak. Therefore the process leads to the growth of the deceleration and may be expressed in terms of the drag coefficient increase (Stanukovich, 1960):
where Ma = 29 g mol - 1 - air molecular weight, g v a p - effective enthalpy of evaporation, Q* - effective enthalpy of "crashing" of meteoroid lattice material with molecular weight of Mmat. Using approximation (2) one can obtain an analytical solution of the drag equation in free-molecule regime (Khanukaeva, 2003). Approximation (2) can be used as a limit expression in approximation (1) instead of the traditional constant value of 2. Such a dependence of the drag coefficient on the Reynolds fiumber is presented in Figure 1. It was obtained in the process of numerical computation of a meteoroid deceleration with the following parameters: M m a t « 2M a , Q* & 0.6 km 2 s~2, g v a p « 8 km 2 s - 2 ,
Co
4 3
2 1 0
0.1
1
10
100
1000
Figure 1. The result of numerical calculation of C D according to (1) with
defined by (2).
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T = 284 К, у — 1.4. The initial velocity of 35 km s _ 1 was used. For faster bodies the increase of the value of will be higher. Approximation (1) is valid for any Reynolds numbers from the very small, corresponding to the free-molecule regime, to the infinite limit, indicating the continuum flow. This universality makes it very convenient for the application to the meteoroids' deceleration description.
3. The Heat Transfer Coefficient Heat transfer to a meteoric body is calculated here as a simple sum of convective C Hco n and radiative CHrad components. The search of universal approximations for each of them is hardly a solvable problem, because they vary in several orders of magnitude on meteoroids trajectories and depend on many factors. Still the efforts are being done. The general scheme is the following: (1) approximation of the coefficient in the stagnation point of the body; (2) approximation of heat flux distribution over the body surface; (3) taking into account the vapor shielding effect. The review of the problem of radiative heat fluxes calculations can be found in (Piluygin and Tirskiy, 1989; Stulov et al., 1995) and it cannot be seen as promising for meteoric problems. The approximations existing for the coefficient in the stagnation point are valid only for the restricted intervals of velocities, bodies' sizes and air densities. It was found out that the distribution over the body surface to depend weakly on the body size and to depend very strongly on the body velocity. The dependence was formulated in I cos"^ (pds
(Apshtein et al., 1986) as follows C Hra d = Cm-ad ( 0 ) ^ — 5
, where cp is
the meridian angle, S is the cross-section area, C Hra d(0) is the value of the coefficient in the stagnation point, n(V) = 1.811 + 1/(0.051 V-0.43) is the approximation parameter. The existing methods of the vapor shielding consideration are also applicable only to several specific cases. The approximations of the integral radiative heat transfer coefficient, developed in (Baldwin and Sheaffer, 1971; Suttles et al., 1974; Biberman et al., 1979), demonstrate acceptable correlation only for moderate velocities (15-40 km s _ 1 ) at altitudes from 40 to 70 km. The discrepancy at lower altitudes is more significant the higher the velocities. Figure 2 represents the comparison of these results for two different velocities. The review of numerical solutions of radiation gas-dynamics problems, including non-equilibrium radiation can be found in (Park, 1999). Though the accuracy and complexity of numerical solutions are rather high they are not formulated as analytical expressions for the coefficient or for the heat transfer and, thus, cannot be introduced in P T M equations. And even the
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Figure 2. The change of C H r a d with height, according to approximations of Baldwin and Sheaffer, 1971 (x), Suttles et al., 1974 ( • ) and numerical computations of Biberman et al. (1979) ( - ) .
existing approximations are rarely applied. So, the problem of adequate modeling of radiative heat transfer coefficient for meteoric bodies is opened. Better success is achieved in convective heat transfer modeling. Shielding effect was considered and formalized for the first time in (Adams, 1959). It may be expressed in terms of the effective enthalpy of ablation or as a correction to the very coefficient value. The distribution over the surface depends neither on body size, nor on its velocity and for a sphere looks like I cos^ (pels
(Murzinov, 1966) Снсоп = C'Hcon(0) it£ -^—, Снсоп(О) is the value of the coefficient in the stagnation point. The last quantity has been approximated separately in the continuum and free-molecule regimes in many works. Here it is offered the universal dependence of C Hc on(0) on flow conditions as a function of Reynolds number defined above: Снсоп(О) - b/VWe
+ ( l - CHrad - b/VR^je-cRe\
(3)
where C^ c o n = 1 — C^rad, as the total heat transfer coefficient is equal to unity (Bronshten, 1983) in the free-molecule limit; and for continuum limit Cfjcon ~ 1 /\/Яе, the approximation known in boundary layer theory. Free coefficients b and с have been determined to fit the numerical data, collected in (Alexandrov, 2003).
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The effect of chemical regime change on the convective heat transfer was investigated in many papers devoted to spacecrafts entries. The correction of heat flux due to chemical reactions comes to 30%. This question pertaining to meteoric problems has been considered in (Khanukaeva, 2004). The study demonstrated the accuracy of formula (3) within 30%, and it confirms its sufficient efficiency. The importance of the adequate heat transfer coefficient modeling is demonstrated by Figure 3. Figure 3a represents the curves of mass change with height, obtained with constant and variable values of C H for the parameters of Vitim bolide (entrance velocity Ve = 15 km s - 1 , entrance angle 9 — 34.3°, density 5 = 3500 kg m~ 3 , entrance mass M e = 5 x 104 kg). The convective heat transfer coefficient (Figure 3b) was calculated according to formula (3) (it is worth, mentioning here, that C H c 0 n = Снсоп(О) for a spherical shape), approximation of (Suttles et al., 1974) was used for the radiative heat transfer coefficient (Figure 3c). Total coefficient (Figure 3d) is obviously far from being constant.
4. Conclusion The problem of ablation parameter variation on meteoroids trajectories was discussed. The influence of gasdynamic flow regimes on the coefficients was modeled in the form of the dependences on the Reynolds number. There are other factors affecting the drag and heat transfer to meteoroids, such as fragmentation effects, body surface temperature variation, the specifics of gas molecules interactions model, in particular, more realistic values for со instead of 0.5 used in the present work. They were assumed to be of secondary
Figure 3. Changes of mass and heat transfer coefficients with height for Vitim bolide parameters.
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importance and were not considered in the present work. They may be taken into account in future investigations. The main conclusion is that the assumption of a = const is too strong. The approximations of acceptable accuracy were offered for the drag and convective heat transfer coefficients. They are valid for any gasdynamic flow regime and presented in forms convenient for the exploitation. Analytical and numerical solutions of drag and ablation equations of PTM were obtained. The analysis has demonstrated that the variation of the drag coefficient is essential only for small particles, while the variation of the heat transfer coefficient is very significant for any meteoroids. The existing approximations for the radiative heat transfer coefficient are seemed to be inadequate for meteoric conditions. So the detailed studies in the field of radiative gas dynamics for meteoric velocities may be useful.
Acknowledgement The work was supported by grants: RFBR N03-01-00-542, N04-01-00-874; LSS N1899.2003.1.
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