Constant Richardson Number Research Articles

Impact of Reynolds and Prandtl Numbers Coupled with Viscous Dissipation on Mixed Convection in A Square Cavity

Efficient thermal management in industrial manufacturing and electronic cooling systems can be achieved by comprehending characteristics of Reynolds and Prandtl numbers in mixed convection scenarios, which aids in the optimization of heat transfer systems through the utilization of forced and natural convection effects. The impact of Reynolds and Prandtl numbers on mixed convection in a square cavity connected to a moving heated horizontal plate is investigated in this research paper. The convective behavior under different conditions was evaluated by discretizing the flow governing equations, which included the momentum and energy equations, using the finite difference method. The research assessed various fluids, including air (Pr = 0.7), liquid metal (Pr = 0.01), and oil (Pr = 10). The Reynolds number ranged from 0.001 to 100, the Eckert number ranged from 0.01 to 40, and a constant Richardson number (Ri = 1) was maintained throughout. The results revealed that the Reynolds number substantially impacts the velocity and temperature characteristics, especially when coupled with a Prandtl number over one and when viscous dissipation remains constant. The utmost velocity that can be attained within the cavity is significantly diminished as the Reynolds number rises, underscoring the critical importance of dynamic fluid properties in determining heat transfer efficiency and fluid flow characteristics. The research unveiled the critical importance of Reynolds and Prandtl numbers in the field of fluid dynamics concerning enhancing heat transfer attributes for engineering purposes, thereby guaranteeing the effectiveness of thermal systems.

Mixed convection of Water-Aluminum oxide nanofluid in an inclined lid-driven cavity containing a hot elliptical centric cylinder

The mixed convection in a lid-driven cavity containing a hot elliptical centric cylinder was studied by using of Water-Aluminum oxide (Al2O3) nanofluid. The problem was simulated at different Richardson numbers (0.1, 1, 10, 100), different volume fractions (0, 0.1%, and 0.2%), different angles (0°, 15°, and 45°), and temperature difference of 50°C. The obtained results show that by Increase in Richardson number in a constant volume fraction and cavity angle decreases the Nusselt number, because increased Richardson number reduces the nanofluid velocity and increases the shear force. Also, increased in Richardson number reduces the decreasing trend of Nusselt number, because at the Richardson numbers of 0.1–10, the main heat transfer mechanism is forced convection; Also, increase in the temperature difference between the cold walls and the cylinder at a constant Richardson number, volume fraction and cavity angle increase the Nusselt number and heat transfer in such a way that the increasing trend of the Nusselt number depends on volume fraction, fluid velocity, and cavity angle.

Numerical study of entropy generation due to coupled laminar and turbulent mixed convection and thermal radiation in an enclosure filled with a semitransparent medium.

The effect of radiation on laminar and turbulent mixed convection heat transfer of a semitransparent medium in a square enclosure was studied numerically using the Finite Volume Method. A structured mesh and the SIMPLE algorithm were utilized to model the governing equations. Turbulence and radiation were modeled with the RNG k-ε model and Discrete Ordinates (DO) model, respectively. For Richardson numbers ranging from 0.1 to 10, simulations were performed for Rayleigh numbers in laminar flow (104) and turbulent flow (108). The model predictions were validated against previous numerical studies and good agreement was observed. The simulated results indicate that for laminar and turbulent motion states, computing the radiation heat transfer significantly enhanced the Nusselt number (Nu) as well as the heat transfer coefficient. Higher Richardson numbers did not noticeably affect the average Nusselt number and corresponding heat transfer rate. Besides, as expected, the heat transfer rate for the turbulent flow regime surpassed that in the laminar regime. The simulations additionally demonstrated that for a constant Richardson number, computing the radiation heat transfer majorly affected the heat transfer structure in the enclosure; however, its impact on the fluid flow structure was negligible.

The Impact of Gradient Wind Imbalance on Potential Intensity of Tropical Cyclones in an Unbalanced Slab Boundary Layer Model

Abstract The assumption of gradient wind balance is customarily made so as to derive the theoretical upper-bound intensity of a mature tropical cyclone. Emanuel's theory of hurricane potential intensity (E-PI) makes use of this assumption, whereas more recent studies by Bryan and Rotunno demonstrate that the effect of unbalanced flow can result in maximum winds that are well in excess of E-PI (superintensity). The existence of supergradient winds has been verified in a slab boundary layer model developed by Smith. Here, the authors apply the slab boundary layer model within the framework of classical E-PI theory to investigate the sensitivity of supergradient winds to the radius of maximum gradient wind (RMGW) and four nondimensional model parameters. The authors find that the Rossby number, the drag coefficient, and the modified Rankine decay parameter all have a considerable influence on the strength of the unbalanced flow. In contrast, the ratio of surface exchange coefficients has little noticeable effect on superintensity. The inclusion of horizontal momentum diffusion leads to a weaker superintensity, but the qualitative features of the model remain similar. To further elucidate these findings, the authors use the boundary layer model to examine the modified E-PI theory proposed by Emanuel and Rotunno. They assume a constant Richardson number for the outflow. The boundary layer model driven by the modified E-PI solution depends on just three model parameters rather than the four parameters used in the classical E-PI framework. Despite this apparent advantage, the results obtained in the framework of the modified E-PI are less realistic than those computed with the classical E-PI approach.

Mixed convection heat transfer performance of water-based nanofluids in lid-driven cavity with wavy surfaces

A numerical investigation is performed into the mixed convection heat transfer characteristics of water-based nanofluids confined within a lid-driven cavity. In modeling the cavity, it is assumed that the left and right walls have a wavy surface, while the upper and lower walls are both flat. In addition, it is assumed that the left wavy-wall has a constant heat flux, the right wavy-wall is maintained at a low temperature, and the upper and lower walls are both insulated and move horizontally. The analysis considers three different nanofluids, namely Cu–water, Al2O3–water and TiO2–water. In performing the analysis, the governing equations are modeled using the Boussinesq approximation and are solved numerically using the finite-volume method. The simulations focus on the respective effects of the nanoparticle volume fraction, the type of nanofluid, the Richardson number, the Grashof number and the wavy surface geometry parameters on the mean Nusselt number. The results show that for all considered values of the Richardson number, the mean Nusselt number increases with an increasing volume fraction of nanoparticles. In addition, it is shown that the Cu–water nanofluid yields the best heat transfer performance of the three nanofluids. Furthermore, it is shown that the mean Nusselt number increases with an increasing Grashof number given a constant Richardson number. Finally, the results show that for a given nanofluid, the mean Nusselt number can be optimized via an appropriate tuning of the wavy surface geometry parameters.

Self-Stratification of Tropical Cyclone Outflow. Part I: Implications for Storm Structure

Abstract Extant theoretical work on the steady-state structure and intensity of idealized axisymmetric tropical cyclones relies on the assumption that isentropic surfaces in the storm outflow match those of the unperturbed environment at large distances from the storm’s core. These isentropic surfaces generally lie just above the tropopause, where the vertical temperature structure is approximately isothermal, so it has been assumed that the absolute temperature of the outflow is nearly constant. Here it is shown that this assumption is not justified, at least when applied to storms simulated by a convection-resolving axisymmetric numerical model in which much of the outflow occurs below the ambient tropopause and develops its own stratification, unrelated to that of the unperturbed environment. The authors propose that this stratification is set in the storm’s core by the requirement that the Richardson number remain near its critical value for the onset of small-scale turbulence. This ansatz is tested by calculating the Richardson number in numerically simulated storms, and then showing that the assumption of constant Richardson number determines the variation of the outflow temperature with angular momentum or entropy and thereby sets the low-level radial structure of the storm outside its radius of maximum surface winds. Part II will show that allowing the outflow temperature to vary also allows one to discard an empirical factor that was introduced in previous work on the intensification of tropical cyclones.

FORMING PLANETESIMALS BY GRAVITATIONAL INSTABILITY. I. THE ROLE OF THE RICHARDSON NUMBER IN TRIGGERING THE KELVIN-HELMHOLTZ INSTABILITY

Gravitational instability (GI) of a dust-rich layer at the midplane of a gaseous circumstellar disk is one proposed mechanism to form planetesimals, the building blocks of rocky planets and gas giant cores. Self-gravity competes against the Kelvin-Helmholtz instability (KHI): gradients in dust content drive a vertical shear which risks overturning the dusty subdisk and forestalling GI. To understand the conditions under which the disk can resist the KHI, we perform 3D simulations of stratified subdisks in the limit that dust particles are small and aerodynamically well coupled to gas. This limit screens out the streaming instability and isolates the KHI. Each subdisk is assumed to have a vertical density profile given by a spatially constant Richardson number Ri. We vary Ri and the midplane dust-to-gas ratio mu and find that the critical Richardson number dividing KH-unstable from KH-stable flows is not unique; rather Ri_crit grows nearly linearly with mu for mu=0.3-10. Only for disks of bulk solar metallicity is Ri_crit ~ 0.2, close to the classical value. Our results suggest that a dusty sublayer can gravitationally fragment and presumably spawn planetesimals if embedded within a solar metallicity gas disk ~4x more massive than the minimum-mass solar nebula; or a minimum-mass disk having ~3x solar metallicity; or some intermediate combination of these two possibilities. Gravitational instability seems possible without resorting to the streaming instability or to turbulent concentration of particles.

Dust Sedimentation and Self‐sustained Kelvin‐Helmholtz Turbulence in Protoplanetary Disk Midplanes

We perform numerical simulations of the Kelvin-Helmholtz instability in the midplane of a protoplanetary disk. A two-dimensional corotating slice in the azimuthal-vertical plane of the disk is considered, where we include the Coriolis force and the radial advection of the Keplerian rotation flow. Dust grains, treated as individual particles, move under the influence of friction with the gas, while the gas is treated as a compressible fluid. The friction force from the dust grains on the gas leads to a vertical shear in the gas rotation velocity. As the particles settle around the midplane due to gravity, the shear increases, and eventually the flow becomes unstable to the Kelvin-Helmholtz instability. The Kelvin-Helmholtz turbulence saturates when the vertical settling of the dust is balanced by the turbulent diffusion away from the midplane. The azimuthally averaged state of the self-sustained Kelvin-Helmholtz turbulence is found to have a constant Richardson number in the region around the midplane where the dust-to-gas ratio is significant. Nevertheless, the dust density has a strong nonaxisymmetric component. We identify a powerful clumping mechanism, caused by the dependence of the rotation velocity of the dust grains on the dust-to-gas ratio, as the source of the nonaxisymmetry. Our simulations confirm recent findings that the critical Richardson number for Kelvin-Helmholtz instability is around unity or larger, rather than the classical value of 1/4.

Modeling of gravity-wave tail spectra in the middle atmosphere via numerical and Doppler-spread methods

Wind fluctuations in the middle atmosphere exhibit an Eulerian spectral tail approximating to the form m −3 at large vertical wavenumbers m. The tail is taken here to result from a characteristically Eulerian nonlinearity in the governing fluid dynamic equations, a nonlinearity that is absent from a Lagrangian description. It is modeled here via a previously described analytic transformation from assumed model Lagrangian input spectra. The present work gives numerical expression to the earlier analytic results for the intensity of the tail. The resultant tail spectra are found to represent well the relevant data at reasonable values of Richardson number. Under an assumption of constant Richardson number, they exhibit saturation as height increases, as do the observations. Previously obtained approximate results, including those of the Doppler-spread theory of saturation, are found to represent well the new numerical results. Implications for that theory and for the associated Doppler-spread parameterization are discussed in a companion paper.

The gravitational instability in the dust layer of a protoplanetary disk:: axisymmetric solutions for nonuniform dust density distributions in the direction vertical to the midplane

The gravitational instability in the dust layer of a protoplanetary disk with nonuniform dust density distributions in the direction vertical to the midplane is investigated. The linear analysis of the gravitational instability is performed. The following assumptions are used: (1) One fluid model is adopted, that is, difference of velocities between dust and gas are neglected. (2) The gas is incompressible. (3) Models are axisymmetric with respect to the rotation axis of the disk. Numerical results show that the critical density at the midplane is higher than the one for the uniform dust density distribution by Sekiya (1983, Prog. Theor. Phys. 69, 1116–1130). For the Gaussian dust density distribution, the critical density is 1.3 times higher, although we do not consider this dust density distribution to be realistic because of the shear instability in the dust layer. For the dust density distribution with a constant Richardson number, which is considered to be realized due to the shear instability, the critical density is 2.85 times higher and is independent of the value of the Richardson number. Further, if a constant Richardson number could decrease to the order of 0.001, the gravitational instability would be realized even for the dust to gas surface density ratio with the solar abundance. Our results give a new restriction on planetesimal formation by the gravitational instability.

Shear instabilities in the dust layer of the solar nebula II. Different unperturbed states

In the previous paper (Sekiya and Ishitsu, 2000), the shear instability in the dust layer of the solar nebula was investigated by using the constant Richardson number solution (Sekiya, 1998) as the unperturbed state, and the growth rate of the most unstable mode was calculated to be much less than the Keplerian angular velocity as long as the Richardson number was larger than 0.1. In this paper, we calculate the growth rate using different unperturbed states: (1) a sinusoidal density distribution, and (2) a constant density around the midplane with sinusoidal transition regions. An unperturbed state of this paper is considered to correspond to the first stage of shear instability that occurs as a result of dust settling in a laminar phase of the solar nebula. On the other hand, the unperturbed state of the previous paper corresponds to the quasi-equilibrium state (Sekiya, 1998) which is attained by the turbulent mixing. The results show that the growth rate is much larger than the Keplerian angular velocity, in contrast to the previous result.

Universality of the Modeled Small-Scale Response of the Upper Tropical Ocean to Squall Wind Forcing

Abstract The upper ocean response to idealized surface wind forcing that is representative of conditions observed during the TOGA-COARE Intensive Observation Period is studied by numerical simulations using a second-moment closure model. A set of experiments is described with a variety of squall-like wind stress distributions and linear initial stratification in the ocean. Several physical regimes of turbulent mixing and decay during and after wind forcing are described. Differences in the structure of the upper and lower parts of the mixing layer are analyzed. The results indicate an exponential decay of turbulent kinetic energy (TKE) with time after surface forcing is removed, and TKE source terms continue to play an important role. The velocity and density structure after the squall are found to be universal, with a nearly constant Richardson number throughout the mixing layer. It is demonstrated that this implies that the mixed layer depth is determined by the initial buoyancy frequency and total mome...