non-Boussinesq Fluid Research Articles

Numerical analysis of Poiseuille-Rayleigh-Bénard convection in supercritical carbon dioxide

The supercritical carbon dioxide (S–CO2) Brayton cycle is an important energy conversion technology for the fourth generation of nuclear energy. Since the printed circuit heat exchanger (PCHE) used in the S–CO2 Brayton cycle has narrow channels, Rayleigh-Bénard (RB) convection is likely to exist in the tiny channels. However, there are very few studies on RB convection in supercritical fluids. Current research on RB convection mainly focuses on conventional fluids such as water and air that meet the Boussinesq assumption. It is necessary to study non-Boussinesq fluids. PRB convection refers to RB convection that is affected by horizontal incoming flow. In this paper, the computational fluid dynamics simulation method is used to study the PRB convection phenomenon of non-Boussinesq fluid-supercritical carbon dioxide. The result shows that the inlet Reynolds number (Re) of the horizontal incoming flow significantly affects the PRB convection. When the inlet Re remains unchanged, with the increase of Rayleigh number (Ra), the steady-state convective pattern of the fluid layer is shown in order: horizontal flow, local traveling wave, traveling wave convection. If Ra remains unchanged, as the inlet Re increases, three convection patterns of traveling wave convection, local traveling wave, and horizontal flow will appear in sequence. To characterize the relationship between traveling wave convection and horizontal incoming flow, this paper proposes the relationship between critical Reynolds number and relative Rayleigh number (r).

Oscillatory instability in a weakly non-Boussinesq fluid layer with a deformable surface

We investigate the oscillatory convective instability in a liquid layer with a deformable free surface. We provide linear stability analysis within a non-Boussinesq approach as our aim is to examine the influence of deformability of the free surface on the Rayleigh–Bénard–Marangoni instability. Within this approach, fluid is assumed to be isothermally incompressible; the density variations are accounted for in the continuity equation and in the buoyancy and inertial terms of the momentum equations. The numerical results show significant differences in stability behavior as compared to results obtained using the Boussinesq approach. Moreover, a novel oscillatory mode of instability is revealed for the zero Marangoni number and zero gravity. This result could not be obtained within the framework of the conventional Boussinesq approximation. Thorough investigation of the novel oscillatory mode let us propose the mechanism of this mode. It is connected with the capillary wave that enforced with thermal expansion of fluid. Weakly nonlinear analysis shows that supercritical branching of standing rolls is possible.

Rayleigh–Bénard–Marangoni convection in a weakly non-Boussinesq fluid layer with a deformable surface

The influence of surface deformations on the Rayleigh–Bénard–Marangoni instability of a uniform layer of a non-Boussinesq fluid heated from below is investigated. In particular, the stability of the conductive state of a horizontal fluid layer with a deformable surface, a flat isothermal rigid lower boundary, and a convective heat transfer condition at the upper free surface is considered. The fluid is assumed to be isothermally incompressible. In contrast to the Boussinesq approximation, density variations are accounted for in the continuity equation and in the buoyancy and inertial terms of the momentum equations. Two different types of temperature dependence of the density are considered: linear and exponential. The longwave instability is studied analytically, and instability to perturbations with finite wavenumber is examined numerically. It is found that there is a decrease in stability of the system with respect to the onset of longwave Marangoni convection. This result could not be obtained within the framework of the conventional Boussinesq approximation. It is also shown that at Ma = 0 the critical Rayleigh number increases with Ga (the ratio of gravity to viscous forces or Galileo number). At some value of Ga, the Rayleigh–Bénard instability vanishes. This stabilization occurs for each of the density equations of state. At small values of Ga and when deformation of the free surface is important, it is shown that there are significant differences in stability behavior as compared to results obtained using the Boussinesq approximation.

Исследование конвективных структур в небуссинесковской жидкости вблизи порога устойчивости. Часть первая — анализ упрощенных моделей

Исследование конвективных структур в небуссинесковской жидкости вблизи порога устойчивости. Часть первая — анализ упрощенных моделей

Non-Boussinesq laminar mixed convection in a non-isothermal fin array

A numerical simulation of laminar mixed convection in a shrouded vertical non-isothermal fin array has been investigated. Convective medium is treated as non-Boussinesq fluid with temperature dependent properties. Computations are performed for the parametric range: Grashof number = 5.3 × 104 to 2.3 × 106, Reynolds number = 100 to 2200, dimensionless fin spacing = 0.1 to 0.5, and dimensionless clearance = 0.0 to 0.4. Computations demonstrate substantial deviation in heat transfer, as high as 18%, resulting from variable property non-Boussinesq fluid flow over non-isothermal fin compared to constant property Boussinesq fluid flow over isothermal fin. Optimum heat transfer coefficient is found to exist in a range of fin height from 0.04 m to 0.05 m, while dimensionless optimum fin spacing lies in a range from 0.2 to 0.3. Finally, induced natural convection velocity component and overall Nusselt number are well correlated with the governing parameters of the problem.

Three dimensions simulation for the problem of a layer of non-Boussinesq fluid heated internally with prescribed heat flux on the lower boundary and constant temperature upper surface

The purpose of this paper is to study the effect of a heat source on the solution to the equations for an incompressible heat conducting viscous fluid. The validity of both the linear instability and global nonlinear energy stability thresholds are tested using three dimensional simulation. Our results show that the linear threshold accurately predicts on the onset of instability in the basic steady state. However, the required time to arrive at the steady state increases significantly as the Rayleigh number tends to the linear threshold.

On the Stability of Three-Dimensional Disturbances in Stratified Flow with Lateral and Vertical Shear

Howard's semi-circle theorem is generalized for three-dimensional disturbances in a non-Boussinesq stratified fluid with a mean flow varying horizontally and vertically. In case of Boussinesq approximation Howard's and Rayleigh's theorem are extended for three-dimensional disturbances in a fluid with a mean flow varying only with respect to the latitude.

Schlieren measurements of internal waves in non-Boussinesq fluids

Previous experiments that have examined the generation of internal gravity waves by a monochromatic source have been restricted to small amplitude forcing in Boussinesq stratified fluids. Here we present measurements of internal waves generated by a circular cylinder oscillating at large amplitude in a non-Boussinesq fluid. The ‘synthetic schlieren’ optical measurement technique (Sutherland et al. in J Fluid Mech 390:93–126, 1999) is extended to stratifications in which the index of refraction of the fluid may vary nonlinearly with density. The method is applied to examine disturbances in approximately uniformly stratified ambient fluids consisting either of sodium chloride (NaCl) or sodium iodide (NaI) solutions whose concentrations increase to near-saturation at the bottom of the tank. In particular, we report upon the first extensive measurements of the optical properties of NaI solutions as they depend upon concentration and density. Applying the results to experiments, we find that large amplitude forcing generates a patch of oscillatory turbulence surrounding the cylinder, thereby increasing the effective cylinder size and decreasing the amplitude of the waves in comparison with the predictions of linear theory. We parameterize the influence of the turbulent boundary layer in terms of an effective cylinder radius and forcing amplitude.

Influence of Nonlinear Concentration Profiles on Flow Patterns of Solutal Rayleigh–Bénard–Marangoni Convection

The effect of nonlinear concentration profiles on the flow patterns of solutal Rayleigh–Bénard–Marangoni convection has been investigated. It has been demonstrated theoretically that similar to the effect of non-Boussinesq fluids in Rayleigh–Bénard convection, the effect of nonlinear concentration profiles in Rayleigh–Bénard–Marangoni convection gives rise to the formation of hexagonal convection.

RAYLEIGH-BÉNARD CONVECTION OF WATER IN A PERFECTLY CONDUCTING CUBICAL CAVITY: EFFECTS OF TEMPERATURE-DEPENDENT PHYSICAL PROPERTIES IN LAMINAR AND TURBULENT REGIMES

ABSTRACT The effects of a non-Boussinesq fluid are discussed for Rayleigh-Bénard convection in a cubical cavity with perfectly conducting side walls at low and high Rayleigh numbers using water as a convecting fluid. Non-Boussinesq simulations were calculated with a variation of 62% and 40% of the thermal expansion coefficient and viscosity, respectively, between cold and hot plates. These variations are well above the 10% usually accepted to consider the Boussinesq approximation. The overall laminar and turbulent flow topologies obtained are not significantly affected by the effects of the physical properties variation with temperature.

Hexagonal pattern instabilities in rotating Rayleigh-Bénard convection of a non-Boussinesq fluid: experimental results.

Motivated by the Küppers-Lortz instability of roll patterns in the presence of rotation, we have investigated the effects of rotation on a hexagonal pattern in Rayleigh-Bénard convection. While several theoretical models have been developed, experimental data cannot be found in the literature. In order to check the validity of the predictions and to study the effects of rotation on the behavior of the system, we present experimental results for a non-Boussinesq Rayleigh-Bénard convection with rotation about the vertical axis. Rotation introduces an additional control parameter, namely the dimensionless rotation rate Omega= 2 pi f d(2)/nu, where f is the rotation rate (in Hz), d is the thickness of the cell, and nu is the kinematic viscosity. We observe that the cell rotation induces a slow rotation of the pattern in the opposite direction (approximately Omega x 10(-4) ) in the rotating frame. Moreover, it tends to destroy the convective pattern. No oscillation of the hexagonal pattern over the range of its existence (Omega< or =6) has been observed.

Multi-mode spatio-temporal instability in non-Boussinesq convection

It is shown that nonlinear (non-Boussinesq) fluid property variations caused by large temperature differences between the walls of a vertical channel are responsible for the appearance of physically distinct types of instability in mixed convection flows: the previously known shear and new buoyancy-induced instabilities. Shear instability dominates the forced convection regimes, while the buoyancy instability prevails in nearly natural convection states. The most challenging situation requiring elaborate theoretical analysis and numerical verification arises in a mixed convection regime where both instabilities compete, forming a wide variety of possible flow patterns. Each of the instabilities is found to undergo the transition from a convective state (when disturbances grow and propagate away from their source) to absolute (when disturbances grow and occupy the complete flow domain). A number of regions in a complete parameter space corresponding to qualitatively different flow scenarios are identified, and accurate boundaries separating them are computed. Various patterns are illustrated by the direct evaluation of the Fourier integrals representing disturbances.

PRESSURE EFFECTS ON NATURAL CONVECTION FOR NON-BOUSSINESQ FLUID IN A RECTANGULAR ENCLOSURE

Numerical predictions of pressure effects on natural convection for a non-Boussinesq fluid in the rectangular enclosures are presented. A solution method based on a compressible flow model is employed to simultaneously determine the absolute pressure, density, temperature, and velocity distributions in the enclosures. Discretization equations are derived from the integral mass, momentum, and energy equations on a staggered grid. The fluid pressure in the enclosure is varied from 20 to 300 kPa such that the flow behavior in a vacuum or pressurized system can be observed. Physical situations investigated also include cases in a wide range of wall temperature difference associated with respective length scales, corresponding to an equivalent modified Rayleigh number ranging from 10 4 to 10 6 . The validity of the incompressible flow model coupled with the Boussinesq approximation for the fluid density, which is commonly used in the existing studies of the buoyant flows, is discussed.

Spin-up of a liquid with a density maximum in a cylinder

A study is made of the spin-up of a viscous non-Boussinesq fluid in a vertically mounted cylinder. The density ( ρ) of the fluid becomes maximum at temperature T m, and a quadratic density–temperature relation is used. The fluid is stratified by imposing a vertical temperature contrast, with the temperature at the bottom endwall disk being T m. Comprehensive numerical solutions to the time-dependent Navier–Stokes equations are acquired. Due to the vertically non-uniform stratification, spin-up proceeds faster near the bottom endwall than near the top endwall. The meridional circulation is more intense near the bottom endwall. Detailed descriptions of evolutions of both azimuthal and meridional flows are given. Major differences in dynamic characteristics are illustrated between a homogeneous fluid, a Boussinesq fluid, and the present non-Boussinesq fluid which has a density maximum.

Coexistence of patterns on a ramp of overcriticality

We consider nonlinear pattern-forming systems in the case when the local overcriticality parameter ϵ is ramped, smoothly matching the subcritical region to the overcritical one. We assume that the nonlinear interactions between spatial modes favor the roll pattern, the resonant hexagon-generating interaction being weak as usual. If the ramp is sufficiently small, we find that a stripe of hexagons is sandwiched between semi-infinite regions of the uniform state (corresponding to the deeply subcritical range) and rolls (corresponding to the strongly overcritical one). At a certain threshold value of the ramp, the hexagons disappear via a super- or subcritical bifurcation, depending upon their orientation. In the latter case, coexistence of two different patterns, with and without the hexagonal component, is possible. These transitions between different patterns can easily be realized experimentally in terms of the classical Rayleigh-Bénard convection in a slightly non-Boussinesq fluid.

Nature of roll to spiral-defect-chaos transition

We investigate the nature of the parallel-roll to spiral-defect-chaos (SDC) transition in Rayleigh-Benard convection, based on the generalized Swift-Hohenberg model. We carry out extensive, systematic numerical studies by, on one branch, increasing the control parameter gradually from the parallel roll regime to the SDC regime and, on the other branch, decreasing it in the opposite manner. We find that the data of several time-averaged global quantities all form hysteretic loops from the two branches. We also discuss several possible scenarios for the transition and analyze our data for SDC accordingly. We conclude that the roll-to-SDC transition is first-order in character and that the correlation length diverges at the conduction to convection onset. We further postulate that this transition can be understood somewhat similar to the hexagon-to-roll transition in non-Boussinesq fluids. Finally we comment on the differences between our conclusion and those in two experiments.

Quasiperiodic patterns in Rayleigh-Bénard convection under gravity modulation

We investigate nonlinear three-dimensional convection under gravity modulation. Depending upon modulation frequency and amplitude, unstable modes in harmonic or subharmonic temporal resonance with the mechanical drive can be observed. We focus on a situation where both modes compete. In non-Boussinesq fluids the dominating interaction mechanism is a nonlinear triad wave-vector resonance. We construct a minimum Galerkin model which describes the competition between several regular convection patterns. Our results indicate that squares or quasiperiodic patterns with a 12-fold rotational axis are the preferred convection structures.

On Large-Scale Vortical Structures in (Incompressible) Fluids with Thermal Expansion

This paper extends the analysis of Ref. 1 to non-Boussinesq fluids whose thermal equation of state depends negligibly on pressure but non-negligibly on temperature. It is demonstrated that these fluids are more stable than the Boussinesq fluids under the same conditions. We study the effect of the helicity on the stability of the Bénard problem and demonstrate that there is no separatrix that divides the instability regimes into two classes. Here, the phenomenon of inverse energy cascade is present at all values of helicities.

Observations of Spanwise Symmetry Breaking for Unsteady Mixed Convection in Horizontal Ducts

Three-dimensional mixed convection in a horizontal rectangular duct heated from below was studied numerically for a non-Boussinesq fluid. The duct was of aspect ratio four, with cold side-wall boundary conditions. Results were obtained at a reduced temperature of 2.33, Rayleigh number of 130,700, and Reynolds numbers of 5 and 20. Using the Re = 20 solution as initial conditions, the Re = 5 solution developed asymmetries about the duct centerline plane, with a streamwise periodic generation and dissolution of the inner pair of vortices. The consequence of these coherent vortical structures was a complex time-dependent distribution of the Nusselt number on the lower wall, with implications in the area of chemical vapor deposition. Specifically, when compared with the steady flow case occurring at Re = 20, an increased spatial uniformity of the time-averaged Nusselt number along the lower duct surface resulted.

Convective instabilities in rarefied gases by direct simulation Monte Carlo method

Benard and thermal stress instabilities in a rarefied gas are investigated by the direct simulation Monte Carlo (DSMC) method. Particular emphasis is given to the numerical aspects of DSMC (accuracy, convergence, etc.), that were not investigated in the previous studies and to the refining of the simulation results by using a special data processing (filtering) procedure. In the stratified rarefied gases with hightemperature gradients the Boussinesq approximation is not valid and it is shown that the onset of instabilities in the non-Boussinesq fluid (rarefied gas) is not determined by a single nondimensional parameter (Rayleigh number). Benard convection in the stratified rarefied gas is analyzed in different geometries, including the case with curvilinear boundaries. The occurrence of thermal stress convection in rarefied gases and in a continuum flow regime under zero gravity conditions was demonstrated using the DSMC method.