A numerical study of separation on a spheroid at incidence

The three-dimensional incompressible, steady and laminar flow field around a prolate spheroid at incidence is considered. The parabolized Navier-Stokes equations are solved numerically. The method can handle vortex types as well as bubble type flow separation because the pressure is one of the dependent variables. Here, the distribution of the skin friction is reported for two test cases. The first test case is a prolate spheroid of aspect ratio of 4:1 at 6 degrees incidence and Reynolds number of 1 million (based on half the major axis). The second case is a spheroid with a 6:1 aspect ratio at 10 degrees incidence and Reynolds number of 0.8 x 1 million. The properties of the flow field near the body are discussed on the basis of the pattern of the skin friction lines, and the shape of the separation lines. Favorable agreement with experimental results is obtained

Natural Convection Heat Transfer in Enclosures With Multiple Vertical Partitions

of air or vacuum, N 2 = n + IK is the complex refractive index of the lamina material, and 9 2 is the (complex) angle of refraction, which is related to 9 t by Snell's law: N, sin #, = N 2 sin 9 2 . Since r 2l = -r i2 , the reflectance at both interfaces is equal to p = r n rf 2 , where * denotes the complex conjugate. The internal transmittance T is related to the (complex) phase change 6 by r = exp . After carefully examining the transmittance formulae of a lamina, this work shows that the geometric-optics formula may result in a significant error for a highly absorbing medium even in the incoherent limit (when interference effects are negligible). Introduction Consider the transmission of electromagnetic radiation through a lamina with smooth and parallel surfaces. In the incoherent limit when radiation coherence length is much smaller than the thickness of the lamina, the transmittance (or reflectance) may be obtained either by tracing the multiply reflected radiant power fluxes (ray-tracing method) or by separating the power flux at each interface into an outgoing component and an incoming component (net-radiation method), viz. ( where p is the reflectance at the interface and r is the internal transmittance. This formula is also called the geometric-optics formula since it is obtained without considering interference effects. For a plane wave, p equals the square of the absolute value of the complex Fresnel reflection coefficient (i.e., the ratio of the reflected electric field to the incident electric field at the interface). The Fresnel reflection coefficient is (Heavens, 1965) r\ 2 = { cos 9 2 -N 2 cos f?i JVi cos (2) N, cos 0, -N 2 cos 6*2 , . , for s -polarization ,7V, cos 9 t + N 2 cos 9 2 where 9 l is the angle of incidence, /V, = 1 is the refractive index where d is the lamina thickness and X is the wavelength in vacuum. In the coherent limit, the transmittance of a lamina may be obtained from thin-film optics (i.e., wave optics) either by tracing the reflected and transmitted waves (Airy's method) or by separating the electric fields into a forward-propagating component (forward wave) and a backward-propagating component (backward wave), viz. (Heavens, 1965; Analysis and Discussion The power transmittance at the interface between the air (or vacuum) and the medium (lamina) is where (1 + r !2 ) is the Fresnel transmission coefficient. The power transmittance at the second interface between the medium and the air can be obtained by exchanging the subscripts 1 and 2 in Eq. (6). At normal incidence, r 12 = (1 -n -('K)/(1 If both K and Im(r 21 ) are nonzero, T 2 \ =t= 1 -p. As discussed by Journal of Heat Transfer AUGUST 1997, Vol. 119/645 Copyright © 1997 by ASME Zhang The above equation is identical to Eq. (5). However, it is not a simple replacement of (1 -p) 2 in Eq. As an example, suppose the lamina is a LaA10 3 wafer of thickness d = 100 p,m. The optical constants are calculated from the Lorentian dielectric function determined by (1) and the transmittance for a LaA10 3 lamina at wavelengths from 9 to 14 p,m at normal incidence are shown in The difference between the wave-optics formula and the incoherent formula is shown in For a highly absorbing lamina (i.e., r < § 1), multiple reflections may be neglected. The transmittance obtained from Eq. (1), when multiple reflections are negligible, is (1 -pfr. The transmittance calculated from Eq. (8) for T < 1 is where the last expression is for normal incidence only. Eq. Concluding Remarks By inspecting the energy balance at the second interface, this work reveals an implicit assumption associated with Eq. Certain important applications require the determination of transmittance below 10~4. Examples are in the characterization of attenuation filters, bandpass filters, and materials with strong absorption bands Acknowledgments This work has been supported by the University of Florida through a start-up fund and an Interdisciplinary Research Initiative award. / Vol. 119, AUGUST 1997 Transactions of the ASME A. A., 1994, "Modelling of the Reflectance of Silicon," Infrared Physics and Technology, Vol. 35, pp. 701 -708. Frenkel, A" and Zhang, Z. M" 1994, "Broadband High Optical Density Filters in the Infrared," Optics Letters, Vol. 19, pp. 1495-1497 Gentile, T. R., Frenkel, A" Migdall, A. L., and Zhang, Z. M" 1995, "Neutral Density Filter Measurements at the National Institute of Standards and Technology," Spectrophotometry, Luminescence and Colour; Science and Compliance, C. Burgess and D. G. Jones, eds., Elsevier, Amsterdam, The Netherlands, pp. 129-139. Grossman, E. N" and McDonald, D. G" 1995, "Partially Coherent Transmittance of Dielectric Lamellae," Optical Engineering, Vol. 34, pp. 1289-1295. Heavens, O. S., 1965, Optical Properties of Thin Solid Films, Dover Publications, Inc., New York, chap. 4, pp. 46-95. Knittl, Z" 1976, Optical of Thin Films, John Wiley & Sons, Inc., NY, pp. 203-204. Salzberg, B., 1948, "A Note on the Significance of Power Reflection," American Journal of Physics, Vol. 16, pp. 444-446. Siegel, R" and Howell, J. R., 1992, Thermal Radiation Heat Transfer, 3rd ed" Hemisphere Publishing Corp., Washington D.C., chap. 4, p. 120, and chap. 18, pp. 928-930. Yeh, P., 1988, Optical Waves in Layered Media, John Wiley & Sons, Inc., New York, chap. 4, pp. 83-101. Zhang, Z. M., 199

Large Eddy Simulation Of Turbulent Channel Flows

In order to be of use in engineering situations, LES models must be able to reproduce near-wall flows. We approach this by refining the mesh close to the wall in a manner that reflects the anisotropic nature of nearwall turbulence. This requires the SGS modelling to be able to deal with anisotropic meshes as well as the anisotropic turbulence near the walls. The technique is then applied to fully-developed turbulent flow in a channel. We show that the first and second order statistical moments of the resolvable velocity are well reproduced, with the differential stress model of Deardorff performing particularly well. INTRODUCTION Large Eddy Simulation (LES) is based on the decomposition of the dependent variables into large- or Grid Scale (GS) components and small- or Sub Grid Scale (SGS) scale components, which represent the unresolved fraction of the turbulence, with the effect of the latter on the former being accounted for by SGS models. Away from walls LES performs well even on ..