Compressible Natural Convection Research Articles

A playground for compressible natural convection with a nearly uniform density

In the quest to understand the basic universal features of compressible convection, one would like to disentangle genuine consequences of compression from spatial variations of transport properties. For instance, one may choose to consider a fluid with uniform dynamic viscosity, but, then, compressible effects will generate a density gradient and consequently the kinematic viscosity will not be uniform. In the present work, we consider a very peculiar equation of state, whereby entropy is solely dependent on density, so that a nearly isentropic fluid domain is nearly isochoric. Within this class of equations of state, there is a thermal adiabatic gradient and a key property of compressible convection is still present, namely its capacity to viscously dissipate a large fraction of the thermal energy involved, of the order of the well-named dissipation number. In the anelastic approximation, under the assumption of an infinite Prandtl number, the number of governing parameters can be brought down to two, the Rayleigh number and the dissipation number. This framework is proposed as a playground for compressible convection, an opportunity to extend the vast corpus of theoretical analyses on the Oberbeck–Boussinesq equations regarding stability, bifurcations or the determination of upper bounds for the turbulent heat transfer. Here, in a two-dimensional geometry, we concentrate on the structure of numerical solutions. For all Rayleigh numbers, a change in the vertical temperature profile is observed in the range of dissipation number between $0$ and less than $0.4$ , associated with the weakening of ascending plumes. For larger dissipation numbers, the heat flux dependence on this number is found to be well predicted by Malkus's model of critical layers. For dissipation numbers of order unity, and large Rayleigh numbers, dissipation becomes related to the entropy heat flux at each depth, so that the vertical dissipation profile can be predicted, and so does the total ratio of dissipation to convective heat flux.

An Interpolated Bounce Back Thermable Method for Simulating Solid Particles Dynamics in a Viscous Medium

In this paper we discuss the mathematical and computer modeling of non-isothermal two-phase flows with suspended particles. Natural convection between an outer cubical cavity and an inner hot sphere is investigated. To simulate heat fluxes loaded with particles, a thermal model of the lattice Boltzmann equation in combination with the interpolated bounce back method (TLBM-IBB) has been developed. In TLBM-IBB, IBB is used to process liquid-solid interfaces, and TLBM is used to simulate the heat flow of a fluid. The momentum exchange method is used to calculate the hydrodynamic force on the particle surface. Simulation performed for a range of Rayleigh numbers . The accuracy and efficiency of the existing method is demonstrated by the example of solving the test problem of natural convection around a stationary particle and three-dimensional compressible natural convection in a square cavity filled with air, which has a hot wall on the left and a cold wall on the right, and two horizontal walls are adiabatic. The results obtained are in good agreement with the experimental and numerical results of other authors.

Development of unsteady natural convection in a square cavity under large temperature difference

To investigate how the nonuniform fluid density distribution caused by large temperature variations affects the development of unsteady natural convection, we perform a series of direct numerical simulations of two-dimensional compressible natural convection in an air-filled square cavity. The cavity has a hot wall on the left and a cold wall on the right, and two horizontal walls are adiabatic. The simulations are done using a kinetic approach based on a modeled Boltzmann equation, from which the fully compressible Navier–Stokes–Fourier equations are recovered. No Boussinesq approximation or low Mach number approximation is made. An extra source term is introduced to adjust the fluid Prandtl number. Simulations are performed for a range of Rayleigh numbers (107−109) with a fixed dimensionless temperature difference of ε=0.6 to determine the critical Rayleigh number and study the development of unsteady flow. To illustrate the instability mechanism, instantaneous fluctuation field, time trace of temperature, and velocity at selected monitoring points, the spectrum and other statistics are presented and discussed. As expected, significant differences are observed between the instability of compressible natural convection and the Boussinesq-type natural convection. With a large temperature difference, the transition to unsteady flow is asymmetric for the flows near the hot wall and cold wall. For the Rayleigh number range we studied, the cold wall region is dominated by low-frequency impact instability of the boundary thermal jet at the bottom corner. For the hot wall region, besides the upper corner impact instability, a boundary layer instability featuring high-frequency oscillations is observed.

Designing a consistent implementation of the discrete unified gas-kinetic scheme for the simulation of three-dimensional compressible natural convection

Discrete unified gas-kinetic scheme (DUGKS) has been developed as a robust and accurate approach for thermal compressible flow simulations; however, designing an efficient and accurate lattice velocity model to take full advantage of DUGKS remains a challenge. In this study, we apply DUGKS to simulate three-dimensional compressible natural convection in an enclosure with a large temperature difference, without making the Boussinesq approximation. The Chapman–Enskog analysis indicates that the fourth-order moments of equilibrium is needed for the heat flux evaluation in the energy equation, implying that the fourth-order Hermite expansion of equilibrium and thus at least an eighth-order Gauss–Hermite quadrature are needed for accurate simulation of the Navier–Stokes–Fourier system. For this purpose, a highly efficient lattice velocity model, D3Q77A9, is derived, which provides a Gauss–Hermite quadrature of ninth-order accuracy in three dimensions. The accuracy of this D3Q77A9 model is demonstrated by simulating compressible natural convection flows in both two-dimensional and three-dimensional cavities. An error analysis is performed to emphasize the importance of combining a quadrature with an adequate degree of precision and a proper order of Hermite expansion of the equilibrium distribution.

An improved discrete unified gas kinetic scheme for simulating compressible natural convection flows

Discrete unified gas-kinetic scheme (DUGKS) has been developed recently as a general method for simulating flows at all Knudsen numbers. In this study, we extend DUGKS to simulate fully compressible thermal flows. We introduce a source term to the Boltzmann equation with the Bhatnagar-Gross-Krook (BGK) collision model [1] to adjust heat flux and thus the Prandtl number. The fully compressible Navier-Stokes equations can be recovered by the current model. As a mesoscopic CFD approach, it requires an accurate mesoscopic implementation of the boundary conditions. Using the Chapman-Enskog approximation, we derive the “bounce-back” expressions for both temperature and velocity distribution functions, which reveal the need to consider coupling terms between the velocity and thermal fields. To validate our scheme, we first reproduce the Boussinesq flow results by simulating natural convection in a square cavity with a small temperature difference (ϵ=0.01) and a low Mach number. Then we perform simulations of steady natural convection (Ra=1.0×106) in a square cavity with differentially heated side walls and a large temperature difference (ϵ=0.6), where the Boussinesq approximation becomes invalid. Temperature, velocity profiles, and Nusselt number distribution are obtained and compared with the benchmark results from the literature. Finally, the unsteady compressible natural convection with Ra=5.0×109,ϵ=0.6 is studied and the turbulent fluctuation statistics are computed and analyzed.

Numerical solutions of 2-D steady compressible natural convection using high-order flux reconstruction

Low Mach number flows are common and typical in industrial applications. When simulating these flows, performance of traditional compressible flow solvers can deteriorate in terms of both efficiency and accuracy. In this paper, a new high-order numerical method for two-dimensional (2-D) state low Mach number flows is proposed by combining flux reconstruction (FR) and preconditioning. Firstly, a Couette flow problem is used to assess the efficiency and accuracy of preconditioned FR. It is found that the FR scheme with preconditioning is much more efficient than the original FR scheme. Meanwhile, this improvement still preserves the numerical accuracy. Using this new method and without the Boussinesq assumption, classic natural convection is directly simulated for cases of small and large temperature differences. For the small temperature difference, a p and h refinement study is conducted to verify the grid convergence and accuracy. Then, the influence of the Rayleigh number (Ra) is analyzed. By comparing with the reference results, the numerical results of preconditioned FR is very close to that calculated by incompressible solvers. Furthermore, a large temperature difference test case is calculated and analyzed, indicating this method is not limited by the Boussinesq assumption and is also applicable to heat convection with large temperature differences.

Validation of a two-dimensional gas-kinetic scheme for compressible natural convection on structured and unstructured meshes

Gas-kinetic schemes (GKS) have been developed as a kinetic Finite-Volume approach to computational fluid dynamics. The GKS a priori allows to obtain approximate solutions of the fully compressible Navier-Stokes equations. In our contribution we show simulation results of compressible natural convection at large temperature differences and low Mach numbers beyond the applicable range of the Boussinesq approximation. The simulations were performed on non-uniform quadrilateral and unstructured triangular grids. The dependence of the critical Rayleigh number on the temperature difference for compressible Rayleigh-Bénard convection, predicted by theory, is accurately reproduced. Moreover, heat transfer in a buoyancy driven square cavity with differentially heated sides at large Rayleigh numbers and large temperature differences is investigated. Temperature and velocity profiles as well as Nusselt numbers show good agreement with benchmark results in literature. After validating the scheme for thermal compressible convection, we investigate unsteady natural convection at a Rayleigh number of Ra=5⋅109 and at a large temperature difference of Th/Tc=4. We find that compressibility has a leading influence on the stability of the boundary layers, such that the flow at the heated wall becomes unstable, whereas the flow at the cooled wall remains stable. This phenomenon has not yet received much attention.

Numerical investigation of laminar compressible natural convection flow in asymmetrically and isothermally heated open-ended inclined channel

This study presents a numerical investigation of laminar compressible natural convection flow induced under high-temperature difference in an open-ended inclined parallel-walled channel heated isothermally and asymmetrically at the top wall (hot plate facing downwards) while the lower plate is considered adiabatic. The investigation is carried out with air (Pr=0.72) as the working fluid for a temperature difference of 110 °C from the ambient, over of range of modified Rayleigh number 6.2×101 to 1.6×104 and inclination of channel varying between 10 and 90° with respect to horizontal position. For the simulation, compressible governing equations without Boussinesq approximation is solved by employing new modified all-speed Roe scheme with matching preconditioning method, dual time-stepping technique and utilizing the modified Local One-dimensional Inviscid (LODI) relations suitable for compressible natural convection flow as non-reflecting boundary conditions at the inlet and outlet of the channel. Average Nusselt number based on inter-plate spacing increases with the increase of modified Rayleigh number and also with the increase of inclination from the horizontal position. This increase in average Nusselt number can be described by the variation of the level of thermal saturation inside the channel. Significant reduction in average Nusselt number is observed in the combined zone of lower angle of inclination and lower modified Rayleigh number. Visualization of fluid flow (temperature contour and velocity magnitude contour) indicates the existence of both fully developed flow and developing flow regime within the span of investigated modified Rayleigh number. Based on the investigation, a composite correlation between average Nusselt number and product of modified Rayleigh number and the sine of the angle of inclination considering as a single parameter is presented which can be conveniently used for designing of engineering application dealing with natural convection heat transfer under high-temperature difference.

A generalized Characteristic-Based Split projection method for Navier-Stokes with real fluids

A new computational approach for the general numerical simulation of fluid flows is presented and demonstrated for compressible natural convection flows with real fluid properties. The Characteristic-Based Split finite element method is generalized, and abbreviated as GCBS, in the present work to introduce both pressure-expansion effects in addition to thermal-expansion effects on density within the projection step, in contrast to recent works that preserve only one of those two expansion effects. Moreover, we keep the fluid in thermodynamic equilibrium to support a real equation of state and fluid model. The method retains the full Navier-Stokes equations, transitioning smoothly between compressible and incompressible flows, without special artificial-compressibility methods, special low-Mach-number methods, or other modeling treatments. Neither isothermal or isobaric assumptions are made. The non-linear solution obtained is validated by the linearized solution during the early time period of the simulation. Natural convection in a cavity is explored for its convenience in testing across known benchmark flow regimes, from classical Boussinesq flow to the full Navier-Stokes equations with real fluid models. Results for natural convection in a cavity across Rayleigh numbers, non-dimensional adiabatic temperature gradients, and temperature differences, are obtained across flow regimes with the GCBS method. Classical Boussinesq, Thermodynamic Boussinesq, low-Mach number, and real-gas flows are presented, demonstrating the generalization of the method across flow regimes. Thus, our contribution is to generalize the Characteristic-Based Split method to capture the full Navier-Stokes equations while demonstrating the capability with compressible natural convection.

Numerical solutions for a compressible dusty fluid flow along a vertical wavy cone

The purpose of this paper is to establish the detailed parametric study for compressible natural convection flow of two-phase dusty fluid along a vertical wavy cone. Typical sinusoidal surface is used to elucidate the heat transport phenomena for the compressible dusty fluid. The governing equations are transformed into parabolic coordinates and the singularity of the leading edge is removed. The resulting equations are then solved straightforwardly by implicit finite difference method. The influence of compressibility, dusty fluid and the transverse curvature of the surface are discussed in detail in terms of local heat transfer rate, skin friction coefficient, velocity and temperature distributions. In order to ensure the accuracy, comparison with some special cases are also presented and the agreement is found to be very good. The key observation from the present study is that the half-cone angle, ϕ, and mass concentration parameter, Dρ, extensively promote the rate of heat transfer into the compressible boundary layer region.

Natural Convection Flow of a Compressible Gas Along a Vertical Wavy Surface

The intent of this paper is to establish the detailed parametric study for compressible natural convection flow along a vertical wavy plate. A typical sinusoidal surface is used to elucidate the heat transport phenomena for the compressible gas. From the present analysis, we will investigate whether the presence of roughness element disturbs the compressible gas flow and alters the physical characteristics associated with the wavy surface or not. The numerical solutions are obtained after converting the governing equations into a suitable coordinate system. The results are interpreted for the parameters that emerge from the compressibility of the participating gas and transverse curvature of the surface. To ensure the accuracy, the present numerical results are also compared graphically with some special cases and are found in good agreement. The key observation from the present analysis is that the compressible gas promotes the heat transfer rate near the wavy surface.

Compressible dusty gas along a vertical wavy surface

This analysis deals with the numerical solutions for the compressible natural convection flow of two-phase dusty gas. In particular, it gives the solutions for the flow having spherical particles suspended in the gas (air) over the surface with sharp lateral curvatures. The governing equations are converted into dimensionless equations by using a set of suitable continuous transformations and solved through the implicit finite difference method. The effect of compressibility, dusty gas and sinusoidal waveform are discussed in detail in terms of local heat transfer rate, skin friction coefficient, velocity and temperature distributions. This investigation reveals the fact that the air–particle mixture reduces the rate of heat transfer, significantly.

Entropy generation for compressible natural convection with high gradient temperature in a square cavity

Entropy generation due to heat transfer and fluid friction irreversibility has been investigated in a square cavity subjected to different side wall temperatures for compressible and incompressible natural convection flows. Based on the obtained velocity and temperature values, the distributions of local entropy generation, average entropy generation and average Bejan number are determined and compared for compressible and incompressible regimes. It is found that the entropy generated for compressible flow always is more than incompressible flow. The study is performed for Ra = 10 4–10 8, ε = 0.01(incompressible regime) and 0.6 (compressible regime), Ge = 10 −5 and Pr = 0.7.

An algorithm for low Mach number unsteady flows

An algorithm is proposed for the simulation of unsteady viscous stratified compressible flows. The advantage of the method is its capability to deal with a broad range of subsonic Mach numbers, including nearly incompressible flows, with a single modelling, based on the fully compressible Navier–Stokes equations. The method is second-order accurate both in space and time. To remove the stiffness of the numerical problem due to the large disparity between the flow and the acoustic wave speeds at low Mach number, an approximate Newton method, based on artificial compressibility, is used. After a detailed description of the method, the accuracy of the algorithm is checked by computing compressible and incompressible benchmark test cases. Results of flows over a backward facing step with or without stratification effects for a Reynolds number of 800 are compared with the steady solution of incompressible methods. The computation of a compressible natural convection allows to evaluate the accuracy of the method for flow including significant compressibility effects, whereas numerical results of a Poiseuille–Bénard channel flow for a Reynolds number of 10 demonstrate the efficiency of the present algorithm to compute unsteady flows.

Compressibility effects in laminar free convection from a vertical cone

In this paper an analysis of the laminar compressible natural convection about a vertical cone is presented. The governing boundary layer equations are suitably transformed and studied in terms of a compressibility variable expansion. The numerical solutions for the first three terms have been obtained for the ratio of specific heatsγ=1.4, and a Prandtl number Pr=0.72, as characteristics of a diatomic gas, and for a range of values of the wall-ambient temperature difference ofA=0.2 to 1000. The numerical results obtained allow an evaluation of the skin friction and heat transfer at the surface of the cone. It is found that the effect of the compressibility is to increase the heat transfer rate at the wall for small values of the wall-ambient temperature difference parameter and to decrease it for large values of this parameter. The skin friction, however, is always reduced.

Heat transfer in mixed laminar convection in a reacting fluid

The influences of natural convection on a reaction occurring in a fluid flowing laminarly in a vertical tube and on heat transfer parameters have been examined. A numerical solution of the dimensionless continuity, momentum, heat and reactant mass balance equations was carried out using a finite difference technique. The principal assumptions under which solution was obtained were: “in compressible natural convection”, a first order reaction and a constant wall temperature equal to the inlet temperature of the fluid. The results bring out the fact that natural convection may significantly affect conversion, the final effect depending essentially on the heat of reaction. In the case of the local and average Nusselt numbers, natural convection always causes appreciable variations. The values of bulk properties and Nu number, both local and average, the axial distribution of the velocity and radial profiles of temperature and reactant concentration are given for some typical situations.