Syozo Kubo

Asymptotic Theory of an Optically Thick Radiating Gas Flow Past a Smooth Boundary at Moderate Radiation Strength

Asymptotic behaviours of a grey, inviscid, non-oonducting gas flow past a black body with arbitrary smooth shape are studied analytically. It is shown that the flow field is composed of three parts; an isentropic outer region, a diffusion type (Rosseland type) middle layer and a purely radiative inner layer. These two layers are thin, in which boundary layer treatments are made. Formal solutions in the inner layer and reduced equations with reduced boundary conditions ( e.g. slip coefficients) in the other regions are obtained to the first orders of expansion with respect to the inverse square root of a large global optical thickness. The results are compared with those by the moment method (P. Cheng; AIAA J. 4 (1966) 238) and its modification. The moment method does not give the inner layer. It gives smaller temperature slip coefficients. Its curvature effects on the middle layer are different from those by the present exact treatment.

Steady One-Dimensional Radiative Heat Transfer with Emission, Reflection and External Radiation

This paper deals theoretically with a radiative heat transfer through an absorbing-emitting medium which occupies a semi-infinite space bounded by a plane wall. A steady one-dimensional problem is considered. The medium is assumed to be grey and to be in a local thermodynamic equilibrium. Effects due to the emission from the wall, specular and diffuse reflection at the wall and external beam radiation through the wall are considered in contrast with the Milen problem, in which no radiation comes into the medium through a purely transparent boundary. The general solution is obtained. The solution is studied in detail in five special cases to show each effect of these boundary conditions. They are the cases of pure emission, pure beam radiation input, pure reflection, a uniform asymptotic field and emission accompanied by diffuse reflection. In the second case, a localized temperature change is found in the neighbourhood of the wall, even when the asymptotic field has a uniform temperature.

Secondary Flow Induced by a Circular Cylinder Oscillating in Two Directions

This paper deals theoretically with the two-dimensional flow induced by a circular cylinder translationally oscillating in an unbounded visous fluid otherwise at rest. The oscillation is composed of a harmonic oscillation with a frequency normal to the axis of the cylinder and another harmonic one with double frequency perpendicular to the former and the axis. Steady secondary flow is studied in detail by the method of successive approximations. It is shown that a uniform flow is induced at infinity. The direction of the uniform flow can change according as a Reynolds number. This is qualitatively similar to the result of the available experimental result on the flow around a circular cylinder which performs an eccentric relational oscillation. The direction depends ends also on the type of the motion of the cylinder, which traces, in general, a Lissajouss figure.

Effects of Anisotropic Scattering on Steady One-Dimensional Radiative Heat Transfer through an Absorbing-Emitting Gray Medium

Effects of anisotropic scattering on temperature distribution of a steady one-dimensional radiative heat transfer through an absorbing, emitting gray medium in a semi-infinite space are analyzed by method of matched asymptotic expansions. An external radiation beam is imposed through a transparent boundary. Rayleigh scattering is considered as a typical model of an anisotropic scattering. The medium is assumed to be non-heat conducting, not to move and to have constant physical properties, The results show that no qualitative difference is shown between the temperature distribution in a scattering medium and that in a non-scattering medium. Only a little quantitative difference is shown for various values of the beam angle. The difference is in 10 percent of the non-scattering value at most.

Steady Radiating Gas Flow Past a Semi-Infinite Flat Plate at a Constant Temperature for an Optically Thick Case

A general asymptotic theory of a radiating gas which has been developed by the same author is applied to the flow past a semi-infinite flat plate at a constant temperature. It is assumed that the global optical thickness of the gas is large and that the convective heat in the flow is of the same order of magnitude as the radiative one. The gas is assumed to have constant properties. The general theory shows that the flow field is composed of an isentropic outer region, a diffusive (Rosseland type) middle layer and a radiative inner layer. It is also shown that the solutions in the inner layer are described by linear combinations as coefficients. Similar solutions of universal functions by using specific values of the middle layer solutions are numerically obtained in the middle layer by making a boundary layer treatment. Solutions in the outer region are obtained for a subsonic flow. Actual analyses are carried out up to the first two orders of expansion with respect to the inverse square root of the glob...

Stagnation Point Flow of a Radiating Gas of a Large Optical Thickness

This paper deals theoretically with the stagnation point flow of a radiating gas when the global optical thickness is large. The radiative heat transfer is assumed to be of the same order of magnitude as the convective one and the Mach number is assumed to be small A method of solution is used in which the isentropic outer flow, the diffusive (Rosseland type) layer flow and the radiation layer flow are considered successively. Actual analysis has been carried out to the second approximation correct to the order of the inverse square root of the Bouguer number. Effects due to the radiation slip are studied in detail. Numerical results are also obtained for the distributions of the flow velocity and the temperature in the diffusive layer and the radiation layer. They are compared with those of a radiative boundary layer flow on a flat plate.

Theory of Unsteady Radiative Heat Transfer in a Semi-Infinite Space

Theoretical investigation is made on the unsteady radiative heat transfer through an absorbing and emitting gray medium which occupies a semi-infinite space bounded by a semitransparent gray plate. The system is initially in a uniform state. At an instance the temperature of the plate is changed to a constant value in addition to the impose of a beam radiation through the plate. They are main-tained hereafter. It is found that the problem has two stages of time evolution: the initial stage and the large time stage, whose time scales as well as spatial scales are different each other. At first, the temperature rises rapidly in the radiation layer. The solution is solved in a form of time series expansion. In the large time stage, the temperature rises gradually in the asymptotic region. The solution is solved by the matched asymptotic expansions.

One Dimensional Steady Flow of a Radiating Gas Past a Heat Source

One dimensional steady flow of a grey gas past a grey non-reflecting heat source is studied. This linearized problem is not only a model of radiative furnace but also one of the simplest problems which concern the interaction between the convection and the thermal radiation. The conductivity and the viscosity are neglected. The kernel substitution solution is obtained and its correction is tried. The exact form for weak radiation and near the heat source for arbitrary radiation strength, the solution by diffusion approximation without slip and numerical results are obtained and compared with each other. The temperature profile shows almost symmetric feature for weak radiation. But when radiation is strong, it gradually shifts far up-stream against the convective cooling and rapidly attains its asymptotic value downstream. Radiation slips are also calculated.

Effect of Scattering on Unsteady Radiative Heat Transfer

Theoretical investigation is made on the effect of the isotropic scattering on an unsteady radiative heat transfer through an absorbing, emitting and scattering medium in a semi-infinite space bounded by a semitransparent plate. The system is initially in a uniform state. At an instance the temperature of the plate is changed to a constant value in addition to the impose of a beam radiation through the plate. They are maintained hereafter. It is found out that the effect of the scattering is practically regarded as a similarity effect, in which the standard length scale is chosen the inverse of the extinction coefficient, when the albedo of scattering is 0 ∼0.8.

Flow of a Radiating Gas Induced by Buoyant Force in Radiation Layer

This paper deals theoretically with a flow of a grey radiating gas induced by buoyant force in a semi-infinite space bounded by a flat plate, which is parallel to the direction of the gravity, An external beam radiation is imposed normally through the transparent plate. The gas is assumed to be viscous, non-heat conducting and to have constant physical properties. The radiation field assumed to be in local thermodynamic equilibrium. The case is considered that the temperature and the flow velocity are uniform in a region away from the plate. The solution is obtained by method of matched asymptotic expansions. A steady flow is obtained in which the buoyant force balances against the viscous force. The flow is induced not only in the neighbourhood of the plate, but also in a region away from the plate, though the temperature change is localized in the former.