Longitudinal rarefied gas flow over a plate

Abstract

Results are given in the present article from a numerical solution of the two-dimensional problem of longiCazdinal supersonic flow of a rarefied gas over a plate of infinite span and finite length, The problem is solved for a kinetic equation [1] approximating the Boltzmann equation [2, 3]. A precise numerical method (method of characteristics) Chat the author developed and applied earlier to the problem of transverse flow over a plate [4] is used in the present study. The results of the calculations are compared with the existing experimental and theoretical work of other authors. The agreement withthe datais satisfactory, The first attempt at a theoretical solution of the problem in the kinetic context was undertaken in [5], There the authors solved the problem of a semiinfinite plate for the Krook equation by an approximate numerical method, using the formal analogy with one-dimensional nonsteady-state problems. The problem was stated and solved as a parabolic problem with initial data in a certain cross section in front of the leading edge. The results of calculations were given for a Mach number M = 1.5. The same ntm~.erical method was later applied to an ellipsoidal model [6] for M = 1.5 and 5.5. For M = 5.5, however, t~e results of the calculations are given only with respect to the pressure coefficient. Some preliminary (in the author's own words) results of a numerical solution of the problem of longitudinal flow over a plate of finite length for the Krook equation have been published in [7]. By contrast with [5, 6], the problem was stated at the outset as a boundary-value problem in both variables. A new difference scheme was used. However, the details of the numerical algorithm are not entirely clear from the description given in I7]. An example is given of the flow field calculations for M = 1.5 and a Knudsen number K = 0.5. A complete description of the results has never, to the author's knowledge, been published. For large Mach numbers the problem of longitudinal flow past a plate of finite length has been solved by the Monte Carlo method [8, 9]. The authors of [8] cite the approximateness of the method used in [5, 6] and the departure of their results from the calculations carried out in [10] by the method of [5] for hypersonic flow. In [8] the calculated and experimental density profiles were compared and found to be .in good agreement. The results of a determination of the dependence of the friction and heat-transfer co~fficients on the Mach number, Knudsen number, and temperature ratio are not given. It is important to point out the large statistical scatter of the results obtained by the Monte Carlo method. For example, the scatter of the data about the average for the pressure on the surface of a plate at M = 24 is about 15%. The main series of calculations in the present study is carried out for a velocity ratio U = 4.6, which corresponds to M = 5.5 in a diatomic gas or to M -~ 5 in a monatomic gas. At these values of M all the effects of hypersonic flow are fully established. The viscosity- coefficient is assumed to have a power-law dependence on the temperature, with power exponent 0.816, and the Prsxldtl number a = ~. The results 'of the calculations are qualitatively consistent with the data of [8] and with the results of density profile measurements, but discrepancies also exist in the results, particularly for cold plates. The analytical dependences of the friction coefficient concur with the experimental data. For comparison, calculations are carried out for M = 1.5 in a monatomic gas. The hat,are of the flow for M = 1.5 differs significantly from the case M = 5. The flow field calculations give results different from [5, 6] as well as from [7], but the behavior of the friction coefficient is close to that obtained in [5]. The heat-transfer coefficient does not match the results of [5].