Terms Of Richardson Number Research Articles

Numerical analysis of coupled Kelvin–Helmholtz and Rayleigh–Taylor instability on inclined walls

The front tracking method was used to study the 2D Kelvin–Helmholtz (K–H) instability on an inclined wall for three-component immiscible fluids. Coupled effects between K–H instability and Rayleigh–Taylor (R–T) instability were studied by analyzing the effect of inclination angle, Atwood number (At), and Richardson number (Ri) on interface evolution. The results show that the coupling of R–T instability has an important influence on the development of K–H instability. The R–T instability first affects the lower interface and then the upper interface at different inclination angles, and it is also observed that the critical time of the coupled effect is earlier with an inclined wall. The R–T instability promotes the development of upper and lower interfaces at different At numbers. In addition, the billow height increases with the increase in At number and the influence of R–T instability on the upper interface can be neglected when the dimensionless time is less than critical time t = 0.6. The R–T instability has little effect on the different surface tension in terms of Richardson number (Riσ).

Two-phase mixed convection heat transfer and entropy generation analysis of a non-Newtonian nanofluid inside a cavity with internal rotating heater and cooler

In this study, mixed convection inside a square cavity filled with Cu-water nanofluid is simulated using the Eulerian two-phase mixture model. The base fluid is considered to be non-Newtonian and obeys from the power law. Two rotating cylinders are placed inside the heat exchanger and heat is transferred through the nanofluid from the hot cylinder to the cold one. In addition to the natural convection heat transfer inside the heat exchanger, rotation of the internal cylinders can provide forced convection heat transfer. The direction of rotation of cylinders can strengthen or weaken the natural convection effects in different regions, hence, four cases with various rotation directions for the cylinders are considered. The effects of changes in the value of imposed angular velocity (in terms of Richardson number), non-Newtonian power-law index, Rayleigh number and nanofluid volume fraction on both the heat transfer and entropy generation are discussed. The results have been illustrated that the shear-thinning or shear-thickening behavior of the non-Newtonian nanofluid can noticeably change the effect of forced convection on the heat transfer efficiency. Besides, regarding the values of Rayleigh number, Richardson number and non-Newtonian power-law index, the addition of nanoparticles into the non-Newtonian base fluid might adversely affect the heat transfer efficiency. In addition, to investigate irreversibilities, total entropy generation is discussed. It is indicated that shear-thinning nanofluids result in higher values of total entropy generation.

Influence of air change rates on indoor CO2 stratification in terms of Richardson number and vorticity

Above rules-of-thumb ventilation standard based on air change rate per hour (ACH), it is necessary to assess the indoor airflow and indoor air quality (IAQ) with respect to simple dimensionless numbers from the practical point of view. Hence, this work aims at investigating the influence of air change rates on indoor CO2 stratification and removal with dimensionless time scale ratio (proportional to Ri/α, Richardson number Ri, mass flux ratio α) and vorticity by applying experiment and simulation methods. We firstly carried out a series of experiments with different ACHs along with constant CO2 pollutant sources, further for simulation validation. Next, numerical simulation was employed to investigate five different ACHs (ranged from 4 to 28) on indoor CO2 concentration for two different vent-inlet-size cases. It was found that as the increase of ACH, the averaging outlet CO2 concentration was decreased until ACH equal to 16, both cases starting showing asymptotic behavior. When looking at indoor CO2 distribution, the case with larger vent-inlet size showed clear stratification even with higher ACH value (i.e., 28), due to the relatively smaller vorticity value (i.e., 0.47, with 3.14 for small-vent-size case) and larger Ri magnitude (37 times) compared to the small-vent-size case. Specifically, it is beneficial for indoor CO2 removal to keep time scale ratio Ri/α below 10−3 and vorticity above 1 rather than merely following with high ACH values. These findings will be of great importance for practical applications to design and control ventilation systems in the perspective of health and energy efficiency.

Stability Impact on Wake Development in Moderately Complex Terrain

This paper uses a year of SCADA data from Whitelee Wind Farm near Glasgow to investigate wind turbine wake development in moderately complex terrain. Atmospheric stability measurements in terms of Richardson number from a met mast at an adjoining site have been obtained and used to assess the impact of stability on wake development. Considerable filtering of these data has been undertaken to ensure that all turbines are working normally and are well aligned with the wind direction. A group of six wind turbines, more or less in a line, have been selected for analysis, and winds within a 2 degree direction sector about this line are used to ensure, as far as possible, that all the turbines investigated are fully immersed in the wake/s of the upstream turbine/s. Results show how the terrain effects combine with the wake effects, with both being of comparable importance for the site in question. Comparison has been made with results from two commercial CFD codes for neutral stability, and reasonable agreement is demonstrated. Richardson number has been plotted against wind shear and turbulence intensity at a met mast on the wind farm that for the selected wind direction is not in the wake of any turbines. Good correlations are found indicating that the Richardson numbers obtained are reliable. The filtered data used for wake analysis were split according to Richardson number into two groups representing slightly stable to neutral, and unstable conditions. Very little difference in wake development is apparent. A greater difference can be observed when the data are separated simply by turbulence intensity, suggesting that, although turbulence intensity is correlated with stability, of the two it is the parameter that most directly impacts on wake development through mixing of ambient and wake flows.

Effects of wall confinement and rheology of non‐Newtonian nanofluids on mixed convection phenomenon of a square cylinder in a vertical channel

AbstractThe mixed convective momentum and heat transfer phenomena of confined square cylinders in non‐Newtonian nanofluids are numerically investigated. The experimental thermophysical properties of alumina‐water‐based nanofluids are adopted from literature and these nanofluids obey shear‐thinning power‐law type non‐Newtonian behavior. The square cylinder is confined in a vertical channel with a confinement ratio of 0.1333. The flow is assumed to be two‐dimensional and the fluid is allowed to flow in upward direction across the confined square cylinder in the vertical channel. The aiding/opposing buoyancy in the flow is incorporated in terms of Richardson number (Ri) in the range of –2 to 2. The ranges of other dimensionless parameters considered are: Reynolds number, Re: 1 to 40; and volume fraction of nanoparticles, ϕ: 0.005 to 0.045. This range of volume fraction of nanoparticles (i.e., ϕ = 0.005 to 0.045) corresponds to the power‐law index (n) of a non‐Newtonian nanofluid in the range of n = 0.88 to 0.5, respectively. Prior to obtaining new results, the solution methodology is validated with existing literature counterparts. Finally, effects of the Reynolds number, Richardson number, and the rheology of non‐Newtonian nanofluids on streamline patterns, surface pressure, surface vorticity, drag coefficients, isotherm contours, local and average Nusselt numbers are delineated.

Numerical study on mixed convection cooling of solar cells with nanofluid

In the present study, the mixed convection heat transfer for a nanofluid flow inside solar cell cooling system was numerically investigated. To this end, an inclined two-dimensional channel with insulated walls was taken under study. The solar cells were assumed to locate on one side of channel where the nanofluid enters at constant temperature. The study has been carried out for the Richardson numbers in the range 0.01⩽Ri⩽20, with solid volume fraction 0⩽φ⩽0.04, Reynolds number 100 and for Prandtl number 7.02. The effects of solid volume fraction and solar cells location in terms of Richardson number on the flow and temperature fields and also the heat transfer rate were studied. The results showed that the presence of nanoparticles led to an increase in temperature gradient and heat transfer near the solar cells. It was also observed that the maximum heat transfer in solar cells occurs in minimum inclination angle of channel at a certain relative distance between the cells.

Numerical analysis of mixed convection at various walls speed ratios in two-sided lid-driven cavity partially heated and filled with nanofluid

In this study, the problem of laminar mixed convection in a two-sided lid-driven square cavity partially heated and filled with Cu/water nanofluid, is analyzed using a numerical procedure based on the finite volume method with the help of a full multigrid solver. Two heat sources of dimensionless length B maintained at high temperature (TH) are embedded in the center of the two side walls of the cavity. The top and bottom walls of the enclosure are considered with low temperature (TC), while the remaining boundary parts of the enclosure are thermally insulated. The top and bottom walls of the cavity can slide in the same or opposite direction with various speed ratios named λ. A parametric study is conducted and a set of graphical results is presented and discussed to illustrate the effects of λ (−2≤λ≤2), the Richardson number Ri (0.01≤Ri≤100) and the solid volume fraction ϕ (0≤ϕ≤0.1) on the flow and heat transfer characteristics. The results show that varying the speed ratio has significant effects on both the flow structure and the heat transfer. It is found that heat transfer enhancement can be provided by the presence of nanoparticles and that this is accentuated by decreasing the Richardson number. Special attention has been given to predict a characteristic Richardson number of equilibrium, namely Rie, leading to an equal heat transfer rate balance between the two opposite sources. Finally, multiple correlations in terms of Richardson number and volume fraction nanoparticles have been established at various speed ratios.

An empirical study on the mixed convection transfer and pressure drop of HTO/CuO nanofluid in inclined tubes

In this paper, the mixed convective flow of heat transfer oil (HTO)–CuO nanofluid in inclined tubes is studied experimentally. The flow regime is laminar and the surface temperature is constant. The effect of nanoparticles mass concentration on the heat transfer rate and the pressure drop is studied as Richardson number is between 0.1 and 0.7, and the nanoparticles mass concentration is 0%, 0.5%, 1.0% and 1.5%. The experiments were performed in four inclination angles, e.g. 0°, 30°, 60°, and 90°. The results show that the combined free and forced convection heat transfer rate increases significantly by the nanoparticle concentration. According to the experimental results, two new correlations are proposed to predict nanofluid flow Nusselt number in terms of Richardson number and nanoparticles mass concentration with the maximum error of 16%. The pressure drop of heat transfer oil (HTO)–CuO nanofluid flow is also investigated and a new correlation is extracted to estimate the flow Darcy friction factor in inclined tubes. As the heat transfer enhancement methods usually accompany with an increase in the pressure drop, the system performance index is also presented. Accordingly, the maximum performance index is 1.58 which is achieved with 1.5% nanoparticles mass concentration and an inclination angle of 30° at Richardson number of 0.7. This result shows that using nanoparticles is more in favor of the heat transfer enhancement rather than in the increase of the pressure drop.

Numerical simulation and analysis of confined turbulent buoyant jet with variable source

AbstractIn this work, experimental and numerical investigations are undertaken for confined buoyant turbulent jet with varying inlet temperatures. Results of the experimental work and numerical simulations for the problem under consideration are presented. Four cases of different variable inlet temperatures and different flow rates are considered. The realizable k-ɛ turbulence model is used to model the turbulent flow. Comparisons show good agreements between simulated and measured results. The average deviation of the simulated temperature by realizable k-ɛ turbulent model and the measured temperature is within 2%. The results indicate that temperatures along the vertical axis vary, generally, in nonlinear fashion as opposed to the approximately linear variation that was observed for the constant inlet temperature that was done in a previous work. Furthermore, thermal stratification exits, particularly closer to the entrance region. Further away from the entrance region the variation in temperatures becomes relatively smaller. The stratification is observed since the start of the experiment and continues during the whole course. Numerical experiments for constant, monotone increasing and monotone decreasing of inlet temperature are done to show its effect on the buoyancy force in terms of Richardson number.

Analysis of stability parameters in relation to precipitation associated with pre-monsoon thunderstorms over Kolkata, India

The upper air RS/RW (Radio Sonde/Radio Wind) observations at Kolkata (22.65N, 88.45E) during pre-monsoon season March–May, 2005–2012 is used to compute some important dynamic/thermodynamic parameters and are analysed in relation to the precipitation associated with the thunderstorms over Kolkata, India. For this purpose, the pre-monsoon thunderstorms are classified as light precipitation (LP), moderate precipitation (MP) and heavy precipitation (HP) thunderstorms based on the magnitude of associated precipitation. Richardson number in non-uniformly saturated (R i *) and saturated atmosphere (R i ); vertical shear of horizontal wind in 0–3, 0–6 and 3–7 km atmospheric layers; energy-helicity index (EHI) and vorticity generation parameter (VGP) are considered for the analysis. The instability measured in terms of Richardson number in non-uniformly saturated atmosphere (\(R_{i}^{\mathrm {\ast } })\) well indicate the occurrence of thunderstorms about 2 hours in advance. Moderate vertical wind shear in lower troposphere (0–3 km) and weak shear in middle troposphere (3–7 km) leads to heavy precipitation thunderstorms. The wind shear in 3–7 km atmospheric layers, EHI and VGP are good predictors of precipitation associated with thunderstorm. Lower tropospheric wind shear and Richardson number is a poor discriminator of the three classified thunderstorms.

Theoretical and experimental analysis of ceiling-jet flow in corridor fires

In tunnels or long corridors, the combustion products of the fire are confined to spread in one or two directions, forming a ceiling-jet flow. For safety assessment and emergency treatment, it is important to investigate and understand the behavior of the ceiling-jet flow. In this paper, a simple model has been presented, in terms of Richardson number and non-dimensional ceiling-jet thickness, to predict the temperature and the velocity of fire-induced ceiling-jet in a rectangular corridor. Besides, the location of hydraulic jump, occurring in ceiling-jet flow, has been estimated theoretically. In order to validate the theoretical predictions, a series of reduced-scale fire experiments were conducted in a 5 m long corridor. The predicted results, concerning non-dimensional excess temperature, agree favorably with experimental data in different fuels and heat release rates of the fire tests. Finally, the scaling issue has also been discussed and validated.

Discrepancy and applicability of various similarity functions in flux calculations under stable conditions

Based on data obtained during the Huaihe River Basin Experiment (HUBEX) in 1999, this study intends to detect the quantitative discrepancies in the momentum (τo), sensible heat (Ho) and latent heat (Eo) fluxes among six sets of similarity functions with the aerodynamic method. It also aims to clarify the applicability of the functions under stable conditions. The relative discrepancy was studied with the normalized transfer coefficients for τo, Ho and Eo, namely CD,CH and CQ, respectively. Except for one set of functions that adopted a rather small von Karman’s constant (0.365), the relative discrepancy in τo among the other functions was less than 10%, while that in Ho(Eo) sometimes reached 25% when the bulk Richardson number (RiB) was less than 0.07. The absolute discrepancy in the fluxes was studied with statistical computations. Among the six sets of functions, the discrepancy in τoHo and Eo sometimes reached 0.03 kg m−1 s−2, 4W m−2 and 10 W m−2, respectively, and the discrepancy in the energy balance ratio sometimes exceeded 0.1. Furthermore, when RiB exceeded the critical value (RiC) for a specific set of functions, no fluxes could be derived with the functions. It is therefore suggested that RiB be compared with RiC before computing the fluxes if RiB is less than RiC. Finally, two sets of nonlinear similarity functions are recommended, due to their unlimited applicability in terms of RiB.

On the bulk Richardson number and flux–profile relations in an atmospheric surface layer under weak wind stable conditions

For describing the surface fluxes in the atmospheric dispersion models, the commonly used linear universal similarity functions φm and φh for nondimensional wind and temperature profiles (J. Atmos. Sci. 28 (1971) 181; Boundary-Layer Meteorol. 7 (1974) 363) are analyzed under weak wind stable conditions. The stability parameter z/L (z is the height above the ground and L is the Obukhov length) is not known a priori. On the other hand, the bulk Richardson number (RiB) is readily available from profile measurements. Thus, a systematic mathematical analysis is carried out to analyze the applicability of the linear functions in terms of RiB. These linear functions are found to be valid as long as RiB is smaller than Prtγ/β2 where Prt is the turbulent Prandtl number and β and γ are constants appearing in the linear functions of φm and φh. The linear functional forms for φm and φh underestimate the Obukhov length resulting in the large value of the stability parameter when RiB⩾Prtγ/β2 occurring in weak wind stable conditions. Using the data from plume validation field experiment conducted by Electric Power Research Institute at Kincaid, it is found that in 70% of the weak wind cases, the RiB is larger than Prtγ/β2. The similarity functions proposed by Beljaars and Holtslag (J. Appl. Meteorol. 30 (1991) 327) are shown to perform well in weak wind stable conditions for all the values of RiB. The computed surface fluxes are also compared with those based on turbulence measurements taken in Cooperative Atmospheric Surface Exchange Study.

Statistics of the Richardson Number: Mixing Models and Finestructure

Abstract Parameterization of the dissipation rate of turbulent kinetic energy (ϵ) in terms of Richardson number (Ri = N2/S2) is examined for a variety in internal wave environments. Previous work with these data suggests a scaling of ϵ ∼ E2N2g(w) to within a factor of 2, where E represents a low-wavenumber shear spectral density, N the background buoyancy frequency, and g(w) a dependence upon average wave frequency content. The alternative Richardson-number-based parameterization of Kunze et al. is also shown to collapse the dissipation data to within a factor of 2. On the basis of a number of theoretical Richardson number probability distributions, however, the nominal (E, N) scaling of the Kunze et al. model is determined to be E2N3. The difference between the nominal (N3) and observed (N2) scaling is hypothesized to be an effect of turbulent momentum and buoyancy fluxes on the internal wave shear and strain profiles associated with the shear instability. For a statistically homogeneous subset of ...

Vertical mixing in the bottom mixed layer near the continental shelf break in the East China Sea

The strength of the vertical mixing in the bottom mixed layer near the continental shelf break in the East China Sea was directly measured with the Micro-Scale Profiler (MSP). It has been shown that there is no significant statistical relation between the turbulent energy dissipation e and the degree of the stratificationN2. It seems that the vigorous turbulence occurs not constantly but intermittently in the bottom mixed layer so that a large variation of e is found depending on the time. In contrast to e, the coefficient of the vertical eddy diffusivityKz is mostly determined byN such thatKz is large in the bottom mixed layer and small in the thermocline. Large value ofKz in the bottom mixed layer is also found in the time series ofKz estimated in terms of Richardson number calculated from the data obtained with electromagnetic current meters. The value ofKz more than 10 cm2s−1 frequently occur in the layer of 20–25 m thick just above the bottom.

Instabilities in a Stratified Fluid Having One Critical Level. Part II: Explanation of Gravity Wave Instabilities Using the Concept of Overreflection

We show that there is a one-to-one correspondence between the values of the parameters Ri (Richardson number at the critical level, where mean flow equals phase speed), k (horizontal wavenumber) and cr (phase speed) for which gravity wave instabilities in the presence of a rigid lower boundary were found in Part I and the parameters Ri, k and cr for which overreflection of a neutral wave incident on the shear zone from below would occur. A simple formula involving the reflection coefficient, vertical group velocity of the mode, and the ground to shear layer distance can give quantitative estimates of the growth rate that are especially accurate for small growth rates. The maximum growth rates and corresponding reflection coefficients can be parameterized in a more physically meaningful manner in terms of the Richardson number at the critical level than in terms of the minimum Richardson number in the background wind profile.

A Numerical Study of Mass and Energy Transfer of Surface, Surface Boundary Layer and Planetary Boundary Layer

The horizontal inhomogeneity is a common occurrence in the lower layer of the atmosphere. For example, in land-lake interface problems, and in flows over finite areas of irrigated land, abrupt changes of surface conditions, i. e., roughness, temperature and humidity occur simultaneously, and the profiles are no longer in equilibrium. The principal purpose of this paper is to present these phenomena quantitatively from a theory.In this study, we used the same numerical method as Estoque and Bhumralkar (1970) describing a finite-difference procedure for numerical integrating the planetary boundary-layer equations. According to the similarity theory expressed as those equations, the thermal stability was assessed in terms of Richardson number (Ri). The evaluation of the Ri is open to question because it is based on the doubtful assumption that the stability condition of the planetary boundary layer is equal with height. If the surface of the earth is heterogeneous as mentioned above, the flow pattern, temperature and humidity field must change with its surface characters. In this case, one should have a more the appropriate parameter in describing the internal boundary layer.In order to take the varying stability conditions with height into consideration, Hayakawa et al. (1979) suggested to use a gradient Richardson number (Rig) for the stability parameter. The flow pattern with the Rig shows that the prevailing flow accelerates on encountering the windward edge of a localized heat source (LHS) and decelerates on leaving the leeward edge. As a consequence of the horizontal velocity variations there is a downwind above LHS and an upward motion in most part of the leeward area. The characteristics of this flow pattern with the Rig is completely contrary to Hayakawa's results (1978) based on Rib. It can be that the reason for these serious differences are due to the different diffusion coefficient caused by a different definition of Ri.The profiles of the diffusion coefficient (K) obtained by use of Rig show its peaks at reasonable levels, and the values of K and the peak height increase with the scale of LHS. These characteristics agree qualitatively well with the observation results of the planetary boundary layer.The perturbation of potential temperature based on the Rig can not develop in an altitude so high and horizontally so extensive as the case with Rib.By an analysis of the condition that the perturbation of potential temperature vanishes in the atmospheric boundary layer, we have reached to the conclusion that the inertial term is more important in the “cross-over” phenomenon than the bouyant force.We define ZH, which is decided from the potential temperature profiles (Fig. 6) and Ri profiles (Fig. 2), as a thickness of the atmospheric boundary layer, and have examined the growth of the internal boundary layer thickness with fetch (x). We have compared these values of thickness obtained from the simulation experiment with those obtained from analytic expressions by other researchers. As a result, the best linear fit was obtained from Kimura's expression (ZH∝R-1/6·x), and Sutton's expression (ZH∝K1/2·x1/2·U-1/2) followed.When air moves from a cool area to a hot one, the horizontal temperature gradient ∂T/∂x becomes positive above the hot area, then the gradient of vertical sensible heat flux ∂H/∂z becomes positive above that area. On the other hand, when air moves from a hot area to a cool one, ∂T/∂x and ∂H/∂z become negative above the cool area.

Thermal Instability in Plane Poiseuille Flow

The conditions marking the onset of longitudinal columnar vortices due to buoyant forces are studied for fully developed laminar flow between two infinite horizontal parallel plates with nonlinear basic temperature profile. The wall temperatures at the bottom and top plates, T1 and T2, respectively, are assumed to vary linearly in the main flow direction. The nonlinear basic temperature distribution and connective motion due to longitudinal disturbance component give rise to the influence on stability criteria: This influence may be expressed by a characteristic parameter representing the effect of longitudinal temperature gradient. Numerical values for critical Rayleigh numbers based on temperature difference, T1 − T2, are found for various Prandtl numbers and the parameter μ characterizing the effect of longitudinal wall temperature gradient. An increase in value for μ reduces the critical Ra further to a value less than 1708 when T1 > T2, and this tendency becomes pronounced as Pr increases. Results for the cases T1 ≦ T2 also show that the vortex rolls can be caused by the effect of longitudinal temperature gradient. Tentative discussion in terms of Richardson number is made to define the region where columnar vortices have priority of appearance over two-dimensional Tollmien-Schlichting waves. The computed secondary flow streamlines and perturbation temperatures show that the mode of convection motion is also affected by the parameter μ.