The Kriging-based genetic algorithm is applied to aerodynamic design problems. The Kriging model, one of the response surface models, represents a relationship between the objective function (output) and design variables (input) using stochastic process. The kriging model drastically reduces the computational time required for objective function evaluation in the optimization (optimum searching) process. 'Expected improvement (EI)' is used as a criterion to select additional sample points. This makes it possible not only to improve the accuracy of the response surface but also to explore the global optimum efficiently. The functional analysis of variance (ANOVA) is conducted to evaluate the influence of each design variable and their interactions to the objective function. Based on the result of the functional ANOVA, designers can reduce the number of design variables by eliminating those that have small effect on the objective function. In this paper, the present method is applied to a two-dimensional airfoil design and the prediction of flap's position in a multi-element airfoil, where the lift-to-drag ratio (L/D) is maximized.

The effect of curvature and torsion on the flow in a helical pipe of circular cross-section is studied numerically by the spectral method. The calculations are carried out for 0 < s <0.6, 0 < f30 < 1.4 and 500 < o; <2000, where {) is the non-dimensional curvature, f30 the ratio of torsion to square root of curvature, and D n the Dean number. The results obtained indicate large effects of torsion on the flow: The conventional two-vortex secondary flow is distorted to become almost one single recirculating cell when f30 ;::: 0.8. The flux through the pipe at the given Dean number and curvature first decreases from that of the toroidally curved pipe as f30 increases from zero, reaches a minimum at f30 ~0.8, and then increases to values larger than that of the toroidally curved pipe. The minimum value decreases as {) increases.

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The effect of torsion on the flow in a helical tube of circular cross-section is experimentally investigated over a range of Reynolds numbers from about 500 to 20000. Three cases of dimensionless curvature fi were studied, i.e. fi = 0.01, 0.05 and 0.1. The torsional parameter/30, which is defined as /30 = z/v/~ with dimensionless torsion r, was considered for seven cases between 0.45 and 1.72. The results reveal rather large effect of torsion on the flow: the friction factor of the flow at a fixed ~ deviates from that of the toroidally curved tube without torsion as/30 increases, and it decreases toward that of the straight tube as/30 further increases. It is also found that the torsion has a destabilizing effect on the flow. The critical Reynolds number at the onset of turbulence varies with/30 and has a minimum at/30 -~ 1.3. The minimum critical Reynolds number at 6 = 0.1 is about 800.

Dual solutions, i.e. two-vortex and four-vortex solutions, and their stability of flow through a slightly curved circular tube are numerically investigated by the spectral method in the range 96 '" Is.;«;10000. where D" is the Dean number. It is found that the two-vortex solution is stable in response to any small disturbances, while the four-vortex solution is unstable to asymmetric disturbances. Time evolution of the unsteady four-vortex flow is also studied by a numerical simulation of the Navier-Stokes equation when D" = 1000. The four-vortex flow eventually turns into a two-vortex flow.

The Couette flow and a thermal problem of a rarefied nitrogen gas between two platinum walls are considered to investigate the characteristics of the reflected gas molecule at a solid surface. The analysis is based on the molecular dynamics (MD) method for the gas-wall interaction together with the direct simulation Monte-Carlo (DSMC) method for the motion of gas molecules. The accommodation coefficients of momentum, translational, and rotational energies of the molecule are obtained. The velocity and rotational energy distributions of the molecule at the wall surface are also obtained. It is found that the Maxwell-type distribution function consisting of specular and diffuse reflections well describes the distribution function of the reflected molecules if the accommodation coefficient involved is chosen properly. It is also found that the flow and temperature fields subject to the Maxwell-type reflection conditions decomposed into each direction of the space coordinates result in good agreements with those of the DSMC combined with the MD calculation.

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A theoretical study is made of thermophoresis of a solid sphere in a rarefied gas in which a uniform temperature gradient and a uniform velocity at infinity exist. The analysis is carried out on the basis of the linearized Bhatnager–Gross–Krook (BGK) equation, from which simultaneous integral equations for the density, flow velocity, and temperature are derived. These equations are solved numerically over a wide range of Knudsen numbers covering the area from the slip flow to the nearly free molecular flow. A formula for the variation of the thermophoretic force acting on the sphere versus the Knudsen number is obtained for any value of thermal conductivity of the sphere when there is no imposed flow at infinity. The thermophoretic velocity of a suspended sphere in a gas is also calculated. The flow patterns as well as the distributions of temperature are shown.