Jae Min Hyun

A numerical study of three-dimensional natural convection in a differentially heated cubical enclosure

A high-resolution, finite difference numerical study is reported on three-dimensional steady-state natural convection of air, for the Rayleigh number range 103 ⩽ Ra ⩽ 106, in a cubical enclosure, which is heated differentially at two vertical side walls. The details of the three-dimensional flow and thermal characteristics are described. Extensive use is made of state-of-the-art numerical flow visualizations. The existence of the transverse z-component velocity, although small in magnitude, is clearly shown. Comparison of the present three-dimensional results with the two-dimensional solutions is conducted. The three-dimensional data demonstrate reasonable agreement with the experimental measurements.

Numerical simulation of three-dimensional flow structure in a driven cavity

Investigations have been made of three-dimensional flows of an incompressible viscous fluid in a square cubic cavity. The flows are driven by the sliding upper surface of the cavity. Numerical solutions are obtained by directly integrating the full, three-dimensional, time-dependent Navier-Stokes equations. The three-dimensional flow structure is examined in detail over a wide range of the Reynolds number Re. One primary finding of these three-dimensional numerical simulations indicates that steady solutions are attained at lower values of Re, but the flow becomes unsteady at higher values, say, when Re exceeds approximately 2000. Due to the profound influence of the endwall effects, three-dimensional flows show substantial differences from two-dimensional solutions: for two-dimensional flow situations, steady solutions are known to exist for up to Re = 10000. The three-dimensional flow structure displays qualitatively distinct features in the low-Re and high-Re regimes. The demarcation separating these two regimes appears to lie in the neighborhood of Re = 2000–3000. One principal characteristic is that the Taylor-Gortler-like vortices are discernible for the high-Re regimes, although these have not been clearly captured in the numerical results for the low-Re regimes. Critical assessments of the present numerical results have been made by cross-checking the data with the available experimental measurements for three-dimensional cavity flows. The comparisons demonstrate broad qualitative agreement between the present numerical computational results and the laboratory measurement data.

Analyses of three-dimensional flow calculations in a driven cavity

Abstract Comprehensive numerical solutions of three-dimensional flows of an incompressible viscous fluid confined in a square cubic cavity are presented. The flow is maintained by the upper surface of the cavity, which slides in its own plane at a constant speed. Extensive numerical solutions to the governing Navier-Stokes equations were acquired over a wide range of the Reynolds numbers.Re ⩽ 2000. The previous report depicted the global structures of these complex three-dimensional flows at varying Re. The preceding account demonstrated broad agreement between the numerical results and the available experimental observations in gross flow characteristics. The present paper continues to give expanded description of the flows, in particular, the detailed velocity profiles along the vertical and horizontal symmetry lines. The changes in the main character of flow with increasing Re are elaborated. The conspicuous three-dimensionalities, at high Re exemplified by substantial transverse gradients of the velocities, as a result of the Taylor-Gortler-like vortices and the comer vortices, are delineated. Taking advantage of the wealth of the numerical results, diagnostic studies of the major terms in the momentum equations are conducted. These studies reveal the dominant dynamic balances in various parts of the flow field. The characterizations of the three-dimensional nature of the flow, based on the present numerical results, are in accord with the prior experimental visualizations.

A NUMERICAL STUDY OF 3D NATURAL-CONVECTION IN A CUBE - EFFECTS OF THE HORIZONTAL THERMAL-BOUNDARY CONDITIONS

A high-resolution, three-dimensional finite-difference numerical study of natural convection flows of a viscous fluid in a differentially heated cubical box is reported. The vertical sidewalls of the enclosure are maintained at constant temperatures of different values. The other vertical walls (the end walls) are thermally insulated. For the horizontal walls, two kinds of thermal boundary conditions are specified: adiabatic and perfectly conducting. Computations have been performed for an air-filled cavity for Rayleigh numbers of 105 and 106. The specific effects of the horizontal thermal boundary conditions on the flow structure are examined in detail. In the case of conducting walls, heat transfer through the horizontal walls enhances the convective flow activities. The numerically predicted velocity and temperature profiles in the symmetry planes are consistent with previous experimental measurements and computations.

Numerical Study of Natural Convection in a Differentially Heated Cavity With Internal Heat Generation: Effects of the Aspect Ratio

Flows in an enclosure that are generated by both internal and external heating are found in several important thermal engineering applications. Examples include heat removal from postaccident nuclear reactors and thermal control of nuclear waste stored underground. There exists a large body of previous studies on natural convection in cavities in which either internal heat generation or external heating is present. However, analysis of flows that involve both modes of heating has attracted far less attention in the past. In the present paper, systematically organized high-resolution numerical calculations of natural convection in a differentially heated rectangular cavity with internal heat generation are reported.

Transient Stewartson layers of a rotating compressible fluid

An analysis of the azimuthal velocity (ν) structure in transient Stewartson layers of a rapidly rotating compressible fluid is made. The specific problem formulation is for the case when the rotating sidewall of a cylindrical container is given a small impulsive increase in rotation rate. A two-variable matched asymptotic technique is employed to describe the azimuthal velocity in the Stewartson layer. The Laplace transforms are applied to obtain an approximate analytical solution for ν. This solution at large times is consistent with the previously established steady-state behavior. Also, when the fluid compressibility vanishes, this solution recovers the features of incompressible-fluid flows. When the compressibility effect is appreciable, the influence of the local viscosity effect becomes noticeable, and the thicknesses of boundary layers increase. At a given radial location, with the increase of the compressibility effect, the magnitude of ν becomes larger. The impact of the compressibility effect is to shorten the transitory time needed to reach the steady state. These results are in qualitative agreement with the assertions of the preceding studies.

Numerical solutions for thermally driven compressible flows in a rapidly rotating cylinder

Numerical studies are made of flows of a gas in a rapidly rotating cylindrical container. The reference Ekman number is small, and the peripheral Mach number is O(1). The internal flows are generated by applying a small temperature gradient on the boundaries of the cylinder. Analyses are made of comprehensive and systematically organized numerical results, which have been acquired by solving the complete, compressible Navier-Stokes equations. The results are examined to recall the effects of the Ekman number and of the cylinder aspect ratio. The existence of the short-bowl flow regime is verified by cross-checking the numerical data with the previous analytical predictions. The characteristic details of this flow regime are described. The diagnostic studies by using the numerical results exhibit the predominant dynamic balance in the flow field. The changeover in the character of the flow is scrutinized by varying the aspect ratio λ. For large values of λ, the long-bowl approach solution is shown to be consistent with the numerical results. The demarcation between these two regimes is assessed by reviewing the numerical data.

Transient sidewall shear layers of compressible fluid in a rapidly rotating circular cylinder

An analysis is made of the structure of transient sidewall shear layer of a rapidly rotating compressible gas in a cylinder. The fluid motion is caused by a mechanical perturbation at the sidewall. Considerations are given to the fluid flow over the spin-up time scale based on the local Ekman number. It is assumed that the ratio of the thickness of the Stewartson layer to the scale height of the basic density field is small but finite. A leading-order linearized formulation for transient Stewartson layers is obtained for general-type time-dependent mechanical perturbations at the cylindrical sidewall. As specific examples, two canonical cases of the sidewall perturbations are dealt with: (1) the case of a step change, and (2) the case of a sinusoidal oscillation. The theoretical solutions acquired are shown to be in agreement with the available results. Physical explanations are offered to describe the main features of transient sidewall layers.

Effects of the sidewall thermal conditions on the gas flows in a rapidly rotating cylinder

Numerical studies are presented of the flows of a gas in a rapidly rotating cylindrical container. The reference Ekman number is small, and the peripheral Mach number is O(1). Fluid motions are induced by small differences in the boundary temperatures. In order to assess the effects of thermal boundary conditions at the sidewall on the flow structure and temperature field, three types of conditions at the sidewall are adopted, i.e. a linearly varying, an insulated, and an isothermal temperature condition. Analyses are made of comprehensive and systematically organized numerical results, which have been acquired by solving the complete, compressible Navier-Stokes equations. Contour maps of the temperature and stream function are constructed. In the short-bowl regime, the closed circulation near the sidewall is strongly affected by the thermal conditions; the importance of the work done by compression produced in the radial motions is emphasized. However, the axial flow in the inner inviscid region is found to be mainly controlled by the Ekman suction. In the long-bowl regime, the global flow structure is considerably influenced by the sidewall thermal condition. This is due principally to the significant diffusion of momentum in the radial direction.

FLOW IN A CYLINDER DRIVEN BY ROTATING SPLIT-DISK ENDWALLS

Flow of an incompressible viscous fluid contained in a cylindrical vessel (radius R, height H) is considered. Each of the cylinder endwalls is split into two parts which rotate steadily about the central axis with different rotation rates: the inner disk (r < r1) rotating at Ω1, and the outer annulus (r1 < r < R) rotating at Ω2. Numerical solutions to the axisymmetric Navier–Stokes equations are secured for small system Ekman numbers E (≡ v/(ΩH2)). In the linear regime, when the Rossby number Ro (≡ 2(Ω2 − Ω1)/(Ω1 + Ω2)) ≤ 1, the numerical results are shown to be compatible with the theoretical prediction as well as the available experimental measurements. Emphasis is placed on the results in the nonlinear regime in which Ro is finite. Details of the structures of azimuthai and meridional flows are presented by the numerical results. For a fixed Ekman number, the gross features of the flow remain qualitatively unchanged as Ro increases. The meridional flows are characterized by two circulation cells. The shear layer is a region of intense axial flow toward the endwall and of vanishing radial velocity. The thicknesses of the shear layer near r = r1 and the Ekman layer on the endwall scale with E1/4 and E1/2, respectively. The numerical results are consistent with these scalings.