On the drag and heat transfer coecients in. free-molecular ow. Carlo Cercignani, Maria Lampis, Andrea Lentati

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1 On the drag and heat transfer coecients in free-molecular ow Carlo Cercignani, Maria Lampis, Andrea Lentati Dipartimento di Matematica, Politecnico di Milano, Milano, Italy Abstract. We have calculated and evaluated numerically drag, lift and heat transfer coecients in free molecule ow according to a new scattering kernel; some results have been compared with available experimental data. 1 Introduction The scattering kernel approach to the problem of writing boundary conditions for the Boltzmann equation has been introduced many years ago [1] and particular kernels have been constructed [2]. In this paper a new kernel is introduced, exploiting a purely mathematical method already known [1], but never used until recently [3]. This approach is useful for reformulating rigorously, in the frame of the scheme of the scattering kernel theory, reemission models which approximate satisfactorily the experimental results while do not satisfy the fundamental properties of the scattering kernels. An example is the boundary condition introduced many years ago by Nocilla [4], which assumes as distribution function for re-emitted molecules a "shifted Maxwellian". 2 The new scattering kernel We recall that a scattering kernel gives the probability density that a molecule impinging on a point x of a wall with velocity c 0, is reemitted pratically at the same point with velocity c. We assume as x-axis the normal outward axis to the wall and introduce the shifted Mawellian [5] R 0 (c 0! c) = (2=) 2 w b(1 + h)(1 + a2 )?2 exp[? w h(1 + h + a 2 )(1 + h)?1 (1? a 2 )?1 c 02 x ] c x exp[? w (1? a 2 )?1 [(1 + h)(c x + a(1 + h)?1 c 0 x )2 + j c t? ac' t j 2 ]] (1:1) containing the three parameters?1 < a < 1, h > 0, b > 0; while w = (2RT w )?1. Then we introduce the following scattering kernel [5]: R(c 0! c) = R 0 (c 0! c) + c x f 0 (c)(1? H(?c 0 ))(1? H(c))=I (1:2) 1

2 where On the drag and heat transfer coecients in free-molecular ow and H(c 0 ) and I are obtained from f 0 (c) = (2=) 2 w exp(? wc 2 ) (1:3) K(c;?c 0 ) = [c x f 0 (c)]?1 R 0 (c 0! c) (1:4) H(c 0 ) = I = With some calculations we obtain K(c; c 0 )f 0 (c)c x dc (1:5) c x f 0 (c)(1? H(c))dc (1:6) H(c 0 ) = H 0 (c 0 x ) + a1=2 w c xh 1 (c 0 x ) (1:7) H 0 (c 0 x) = b exp (? w (h + a 2 )(1? a 2 )?1 c 02 x ) (1:8) H 1 (c 0 x) = 1=2 b(1? a 2 )?1=2 (1 + h)?1=2 exp[? w h(1 + a 2 + +h)(1 + h)?1 (1? a 2 )?1 c 02 x ][1 + erf[a 1=2 w (1? a 2 )?1=2 (1 + h)?1=2 c 0 x]] (1:9) I = 1? (K 01 + ak 11 ) (1:10) where K ij are functions dened in the Appendix. We have?1 < a < 1, h > 0; moreover the parameters a; b and h must be linked in such a way that H(c x ) 1: An approximate estimate provides b b max = [1 + a[2(eh)?1 (1 + h + a 2 )?1 ] 1=2 ]?1 (1:11) 3 Drag, lift and heat transfer coecients. We calculate the coecients of drag, lift and heat transfer in free molecule ow. As usual [7,8] we assume as incident distribution function f 1 = (2RT 1 )?3=2 n 1 exp[? 1 (c? v 1 ) 2 ] (2:1) where n 1 is the number density, 1 = m=(2kt 1 ). Let us denote with f? and f + the restrictions of f(c) respectively to negative and positive values of c x, while f 0 denotes f(c 0 ). The incident and reemitted uxes i and r of a quantity are given by r = i = c0x<0 j c 0x j 0 f 1?0 dc0 = c x f 1 + dc = c x P n P n f 1? dc (2:2) dc c0x<0 j c 0x j R(c 0! c)f?0 1 dc0 =

3 Carlo Cercignani, Maria Lampis, Andrea Lentati c0x<0 0x = j c j f?0 1 dc0 R(c 0! c)dc (2:3) and P n (h) = h(?c x ). Then we dene the coecients C N = (c x )=(v 2 1 =2) = ( i(c x ) + r (c x ))=(v 2 1 =2) = C Ni + C Nr (2:4a) C = (c y )=(v 2 1 =2) = ( i(c y )? r (c y ))=(v 2 1 =2) = C i? C r (2:4b) C H = (c 2 =2)=(v 2 1 =2) = ( i(c 2 =2)? r (c 2 =2))=(v 2 1 =2) = C Hi? C Hr (2:4c) where the expression of the incident quantities C Ni ; C i ; C Hi for a given attack angle are well known [7], while our new kernel provides C Nr =?1=2 S?2 [2a 2 (1 + h)?1 (T 1 =T w ) 1= (1? a 2 )(T w =T 1 ) 1= a(1 + h)? ( 1=2 =2)(T w =T 1 ) 1=2 (1? K 2 )(1? K 1 )?1 [exp(?s 2 )+ 1=2 S (1 + erf(s ))? ( a(T 1 =T w ) 1=2 12 )]] C r = 2?1=2 S?1 cos [a 2 (T 1 =T w ) 1= ] (2:5a) (2:5b) C Hr = (2 1=2 S 3 )?1 [2a 3 (1+h)?3 (T 1 =T w ) 1=2 14 +[2a 3 (1+C 2 )(T 1 =T w ) 1=2 + +a(1?a 2 )(5+2h)(1+h)?1 (T w =T 1 ) 1=2 ] 12 +2a 2 (1+h)?2 03 +[2a 2 (1+C 2 ) +2(1?a 2 )(2+h)(1+h)?1 (T w =T 1 )] 01 +(2?K 1?K 3 )(1?K 1 )?1 (T w =T 1 ) [exp(?s 2 ) + 1=2 S (1 + erf(s ))? 2 01? 2a(T w =T 1 ) 12 ]] (2:5c) where S = S sin ; C = S cos and the drag and lift coecients are referred to the frontal area as in [6]: C D = (C N sin + C cos )= sin (2:6a) C L = (C N cos? C sin )= sin (2:6b) In Fig. 1, 2 we make a comparison of our results in the case = =2 with those given by the DR model and with some experimental data [6] regarding the He?Au interaction. The coecients are reported as functions of the ratio r = T w =T 0, where T 0 = T 1 (1 + 2S 2 =5). The values of the parameters are chosen giving a; h and the corresponding b max. For lower values of b we obtain values comprised between those given by b max and b = 0 (the latter corresponds to diused reemission (DR)). An important characteristic of the experimental data is that the value r c of r for which C H = 0 is 1.27; other data [6], regarding dierent values of the attack angle and not reported here, show that r is independent of. We remind that the CL model [2] in order to give r independent of, and consequently

4 On the drag and heat transfer coecients in free-molecular ow the accommodation coecient of the normal energy n equal to that of the tangential energy t (2? t ), implies a relationship between the two coecients that reduces the model to be an one-parameter model. With the present model, a good agreement with the data is attained for a close to one (in Fig. 1 a = 0:9) and b = b max. Other numerical calculations show that the value of r c increases with : we have obtained r c = 1:27; 1:32; 1:35; 1:36 for = 45 0, 60 0, 75 0, 90 0, respectively. Concerning C D it appears that, in general, the values predicted are higher than the experimental ones, although it is dicult to estimate the dierence, since the experimental error is not indicated in [6]. In any case the dierences between the values obtained according to the various theories are not so high in percentage to allow a denitive conclusion. Moreover it seems us worthwhile to do some remarks on the classical theory of freemolecular ows [7]. The latter introduces the accommodation coecient n of normal momentum p according to the well known denition p r = n p w + (1? n )p i ; (2:7) where the index w refers to diuse reemission. It happens that p w may be equal to p i for some value of the ratio r depending on the attack angle [8]. In the hypersonic limit, and for normal incidence, we get 4? C D (90 0 ) = n (2? (2r=5) 1=2 ) (2:8) which implies C D = 4 for r = 10= for any model of reemission, if the possibility that n tends to 1 is disregarded. The experimental data for r close to 3 are sensibly lower than 4, while our theoretical curves approach this value. Moreover, from the experimental data, using Eq.(2.8) we can calculate n. We nd, for instance [9], that n for r = 1:5 ranges from 0.72 to 1.2; for higher r; n assumes values greater than 1 increasing with r, for instance the value n = 3:16 is attained for r = 2:7: We remark that for decreasing the values of n it is necessary to have higher values of C D, as do the theoretical curves. For instance C D = 3:9 for r = 2:7 gives n = 0:632: On the other hand it is clear, from Eq.(2.8), that, approaching the critical value r = 10=; a small error in the measurement of C D produces a large error on the value deduced for n. At this point, if we give credit to the fact that the experimental data are so low, we are forced to conclude that the denition of the accommodation coecient for normal momentum is not good, since forces p r to be a linear combination of two quantities that may be equal and then produces strange results incompatible with experiments. We can examine also the experimental data [6] and the classical theory for other angles of attack. The formula shows that for any it exists a value of r for which the coecient of n in the expression of p r vanishes so

5 Carlo Cercignani, Maria Lampis, Andrea Lentati that C D depends only on t, whose values can then be deduced from the measurements of C D. This value in the hypersonic limit is (10=) sin 2, in correspondence of which C D has the simplied expression C D = 4 sin t cos 2 (2:9) which gives C D sin 2, if positive values of t are assumed. While sensible results for t are deduced from the data for the lower values of the attack angle, in the case of = 0 a negative value of t is obtained, and it is clear that the data for C D should be increased in order to give t 0. All these inconvenients can be ascribed or to an underestimation of C D or to a bad denition of n. At this purpose we recall [1] that n is connected to the ux of c x, which is a quantity whose sign changes if the normal to the wall is inverted, at dierence with c y and c 2. We remark that in order to collect our results in few gures we have drawn curves corresponding to constant values of the parameters of the model, although these could be functions of the ratio T w =T 0 [9]. This choice makes it easier to compare our results with those of the paper quoted in [6] and does not prevent a correct comment about some of the main features of the results, such as the recovery temperature and the fact that the values of C D supplied by the theory are higher than the experimental ones. Finally we remark that it is easy to construct kernels that provide values of C D lesser than 4 for r close to 10=. A well known example is the elasto-diuse re-emission [10]. This kernel provides t = 1, E = 0, and, in the hypersonic limit, C D = 2 sin + 4=3 for any value of T w =T 0 : this formula, with easy calculations, gives for n a result dicult to be understood. Clearly this kernel cannot be considered a realistic model, but only a mathematical example. It is easy to construct kernels of this kind, exploiting the technique used in the present paper, but we have not enough space here for dealing this argument, about which we have some preliminary results; moreover we think that it is better to get a clarication on experiments before introducing new kernels of this kind. 4 Appendix We give here the denitions of some functions used in the main paper. K 0j = b[(1? a 2 )=(1 + h)] (j+1)=2 (3:1a) K 1j = 1=2 b (1 + h)(j+1)=2 (1? a 2 ) (j+1)=2 E j+1 (0) + F j+1 (a(1 + h)?1 ) [(1 + h) 2? a 2 ] (j+2)=2 E j (0) (3:1b) E n (x) = 1 x y n exp(?y 2 )dy (3:2)

6 On the drag and heat transfer coecients in free-molecular ow where # = arcsin x. 0i = 1 F n (x) = 2?1=2 n+1 exp(? 2 )d y>0 1i = 1=2 (1 + h)?1=2 (1? a 2 )?1=2 0 # 0 cos n #d# (3:3) y i exp[(t 1 =T w )(h+a 2 )(1?a 2 )?1 y 2 ] exp(?(y?s ) 2 )dy (3:4) y>0 y i exp[(t 1 =T w )h(1 + h+ a 2 )(1 + h)?1 (1? a 2 )?1 y 2 ] exp(?(y? S ) 2 )[1+ erf[(t 1 =T w ) 1=2 ay(1 + h)?1=2 (1? a 2 )?1=2 ]]dy (3:5) Acknowledgment This work has been performed in the frame of the activity of G.N.F.M. of C.N.R., and supported by M.U.R.S.T. (40% and 60%) and by Marcel Dassault Aviation. References 1. Cercignani, C. (1988):The Boltzmann Equation and Its Applications, Springer. 2. Cercignani, C., Lampis, M. (1971): Kinetic Models for Gas-Surface Interaction, Transport Theory Stat. Phys., 1, Cercignani, C. (1990): Scattering Kernels for Gas-Surface Interaction, Proceedings of the Workshop on Hypersonic Flows for Reentry Problems, 1, 9, INRIA, Antibes. 4. Nocilla, S. (1963): The Surface Re-emission Law in Free Molecule Flow, Rareed Gas Dynamics, Laurmann Ed., , Academic Press. 5. Cercignani, C., Lampis, M., Lentati, A.: A new Scattering Kernel in Kinetic Theory of Gases, to be published. 6. Bellomo, N., Dankert, C., Legge, H., Monaco, R. (1985): Drag, Heat Flux, and Recovery Factor Measurements in Free Molecular Hypersonic ow and Gas-Surface Interaction Analysis, Rareed Gas Dynamics, Belotserkovskii et al., Eds., 1, , Plenum Press. 7. Schaaf, S. A. (1963): Mechanics of Rareed Gases, Handbuch der Physik 8/2, S. Flugge Ed., Springer, Berlin, Legge, H. (1992): Heat transfer and Forces on a LiF Single-Crystal Disc in Hypersonic Flow, DLR-IB A 14 Report 9. Cercignani, C., Frezzotti, A. (1989): Numerical Simulation of Supersonic Rareed Gas Flows Past a Flat Plate: Eects of the Gas-Surface Interaction Model on the Floweld, Rareed Gas Dynamics, Muntz, E.P. et al. Eds., , AIAA, Washington, DC. 10. Klinc, T., Kuscer, I. (1972): Slip Coecients for General Gas-Surface Interactions, Phys. Fluids, 15,

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