Velocity Distribution Functions Near the Leading Edge of a Flat Plate

Abstract

A further example in Sec. VIII uses the modified Newtonian theory technique to investigate the static stability of the Apollo Command Module. Using tunnel results from 0 = 0 and 9 = 30°o nly, the static stability conditions of the module are derived and are found to be close to those obtained using the full range of experimental results. The electron beam fluorescence technique has been used to determine the velocity distribution function within several mean free paths of the leading edge in a rarefied hypersonic helium flow. The Doppler shift and broadening of the 5016 A He line was measured with a computer controlled Fabry-Perot interferometer. For analysis, the velocity distribution function was approximated with a bimodal model consisting of two Gaussian functions, which was convoluted with the appropriate instrument function and natural line profile and then fitted to the experimental data. Macroscopic properties in the vicinity of the leading edge, slip velocities, and temperature jumps have been obtained from the complete velocity distribution functions. Nomenclature A = height of distribution function c -speed of light D = orifice diameter H = enthalpy k = Boltzmann constant Presented as Paper 73-691 at 1 P -2 1 S transition M = Mach number M (A, T) = Maxwellian distribution function m = mass flow m = atomic mass of helium n = weighting of distribution function p = pressure R = gas constant S -speed ratio 5(1) = spectral profile T = temperature u = particle velocity U = mean velocity AC/ =U,-U 2 x = axial distance from leading edge y = height above flat plate z = direction normal to flow in plane of plate surface a = energy accommodation coefficient ft = bevel angle = mean free path = wavelength A 0 = He 5016 A wavelength AA = deviation A -/ 0 r\ = coefficient fj. = viscosity p = density a = tangential momentum accommodation coefficient Downloaded by UNIVERSITY OF MICHIGAN on February 3, 2015 | http://arc.aiaa.org |

Numerical analysis of a supersonic rarefied gas flow past a flat plate

“…͑a͒ Distributions of v 1 /U ϱ along the lines X 1 /l ϱ ϭ const; here, --indicates the present numerical results for M ϱ ϭ8 and T w /T ϱ ϭ1, and ᭺, ᭹, ᭝, and ᭡ the experimental data for M ϱ ϭ8.93 and T w /T ϱ ϭ27.1 in Ref. 9. ͑b͒ Distribution of ( ϱ U ϱ 2 /2) Ϫ1 p 12 on the upper surface of the plate; here, --indicates the present numerical result for M ϱ ϭ3 and 8 and T w /T ϱ ϭ1, and ᭹ the numerical result by the direct simulation Monte Carlo method for M ϱ ϭ4 and T w /T ϱ ϭ1.6 in Ref.…”

“…18 ͑for M ϱ ϭ3 and 8; at least in the range 0рX 1 ϩL/2Շ5l ϱ ) can be considered to give the universal flow field independent of the plate length L, namely, the flow field around the leading edge of a semi-infinite plate (ϪL/2ϽX 1 ,X 2 ϭ0). [8][9][10]13 ͑By the rearrangement of Figs. 12-14, we can also show that, in the case of M ϱ ϭ3 and 8, the distributions of p 12 , p 22 , and e f on the plate for Knϭ0.05 coincide with those for Knϭ0.1 in the range 0рX 1 ϩL/2Շ5l ϱ .)…”

“…[1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] An important aspect of the problem, among others, is the supersonic leading-edge problem. The physical quantities undergo abrupt changes at the leading and trailing edges, and therefore the local Knudsen number there is much larger than the overall Knudsen number.…”

Benchmark numerical simulations of rarefied non-reacting gas flows using an open-source DSMC code

“…Recourse to solutions of the Boltzmann equation must be made, and DSMC has proven to be the most reliable method for this purpose in the transition regime, where non-equilibrium effects dominate the gas behavior but inter-molecular collisions are still important. Different forms of Knudsen number can be required 35 to predict different types of continuum breakdown, e.g., a Knudsen number based on local flow gradient lengths can be used across shock waves [10][11][12].…”

“…Rarefied hypersonic flow over flat plates has been studied theoretically, experimentally, and numerically by many authors, e.g., [33][34][35][36][37][38][39][40]. The extremely simple geometry makes the flat plate one of the most useful test cases for numerical validation purposes.…”