Richardson number

This study presents a semi-analytic non-iterative solution for the Monin-Obukhov similarity equations under unstable surface conditions. The solution is represented in terms of the non-dimensional Monin-Obukhov stability parameter z/L. This parameter is given as a function of the bulk Richardson number and other surface parameters including the heat and momentum roughness lengths which are generally assumed to be different in this formulation. The proposed formulations give results that are both quantitatively and qualitatively consistent with the fully iterated numerical solution for a wide range of surface parameters.

In this study, data from 794 GPS dropsondes deployed by research aircraft in 13 hurricanes are analyzed to study the characteristic height scales of the hurricane boundary layer. The height scales are defined in a variety of ways: the height of the maximum total wind speed, the inflow layer depth, and the mixed layer depth. The height of the maximum wind speed and the inflow layer depth are referred to as the dynamical boundary layer heights, while the mixed layer depth is referred to as the thermodynamical boundary layer height. The data analyses show that there is a clear separation of the thermodynamical and dynamical boundary layer heights. Consistent with previous studies on the boundary layer structure in individual storms, the dynamical boundary layer height is found to decrease with decreasing radius to the storm center. The thermodynamic boundary layer height, which is much shallower than the dynamical boundary layer height, is also found to decrease with decreasing radius to the storm center. The results also suggest that using the traditional critical Richardson number method to determine the boundary layer height may not accurately reproduce the height scale of the hurricane boundary layer. These different height scales reveal the complexity of the hurricane boundary layer structure that should be captured in hurricane model simulations.

Assisting and opposing flows in a mixed convection boundary layer flow over an isothermal vertical plate are studied for the case of variable physical properties and uniform free stream. Fluid viscosity and thermal conductivity are assumed to be linear functions of temperature. Using local similarity the flow and heat transfer quantities are found to be functions of four parameters, i.e. Richardson number, Prandtl number, a viscosity variation parameter and a thermal conductivity variation parameter. Numerical solutions are obtained by two methods, a shooting technique and Nachtsheim-Swigert technique, for selected values of parameters appropriate for the fluids considered and specific temperatures of the plate and ambient fluid. For assisting flows, there exist solutions for all values of Richardson number while for opposing flows solutions exist only for a finite set of its values and, in addition, there also exist dual solutions. Important flow and heat transfer quantities of practical interest are determined and the influence of different parameters is discussed.

This work is concerned with the numerical study of laminar, steady MHD mixed convection flow, and entropy generation analysis of 𝐴𝑙 2 𝑂 3 -water nanofluid flowing in a lid-driven trapezoidal enclosure. The aspect ratio of the cavity is taken very small. The cavity is differentially heated to study the fluid flow, heat, and mass transfer rate. The adiabatic upper wall of the enclosure is allowed to move with a constant velocity along the positive 𝑥-direction. The second-order finite difference approximation is employed to discretize the governing partial differential equations, and a stream-function velocity formulation is used to solve the coupled non-linear partial differential equations numerically. The simulated results are plotted graphically through streamlines, isotherms, entropy generation, Nusselt number, and Sherwood number. The computations indicate that the average Nusselt number and average Sherwood number are decreasing functions of Hartmann number, aspect ratio, and nanoparticle volume fraction. Significant changes in streamlines, temperature and concentration contours for high Richardson number are observed.

This work presents observations of diurnal-inertial motions forced by a sea-land breeze regime near critical latitudes (30 3 7 10 3 N=S) for resonance on the Portuguese shelf. Wind conditions were determined with an Aanderaa meteorological buoy placed in the Bay of Setu ´bal during the Maritime Rapid Environmental Assessment 2004 (MREA04) sea trial. Currents data for this period were recorded from two Barny ADCP (acoustic Doppler current profiler) deployments located on the north and south sides of the bay, respectively. Both ADCPs covered almost the full depth of water ( $ 112 m) with 24 bins of 4 m size at both sites. Dominant motions exhibited a two-layer configuration in response to energetic out-of-phase oscillations driven by diurnal-inertial currents. Hence, simultaneous CTD (conductivity, temperature, depth) data were considered to assess the strength of observed motions in promoting diapycnal mixing. With this aim, ADCP and CTD data were used to analyse the vertical shear (S 2 ), squared buoyancy frequency (N 2 ) and gradient Richardson number (Ri) of the water column in terms of two main layers with highly stratified and vertically sheared flows during diurnal-inertial events. Observations show a strong variation in the vertical density field near both ADCP deployments that involved high vertical shears distributed into two layers associated with diurnal-inertial currents. The results of our analysis highlight the role of wind-forced diurnal-inertial motions in shelf areas near the critical latitudes, where they can greatly contribute to triggering diapycnal mixing at the scale of a few days, owing to sea-land breeze forcing.

The American Physical Society Jets, plumes and particle-laden jets in two-dimensional environments J.R. LANDEL, C.P. CAULFIELD, BPI & DAMTP, University of Cambridge, A.W. WOODS, BPI, University of Cambridge -Results on the experimental investigation of liquid jets and plumes in a quasi Hele-Shaw cell are presented. Few experimental studies have been conducted when jets are constrained in a narrow gap whose length is two orders of magnitude smaller than the length scales of the other two dimensions. In this configuration, the dynamics shown by the jets is very rich when parameters such as the initial flow-rate and the buoyancy are changed. Furthermore, different behaviors have been observed for the front of the jet and the flow in steady state. In particular, the models for the rise of the jet and the expansion must be slightly modified between the two cases. PIV techniques have been used to measure accurately the flow field of the jets and to allow accurate comparison with the theoretical models. Finally, the results of this investigation on jets and plumes serve as a basis for more complex experiments involving particleladen jets. A better understanding of liquid jets with varying buoyancy proves to be useful to the study of this two-phase flow.

Results on the experimental investigation of liquid jets and plumes in a quasi Hele- Shaw cell are presented. Few experimental studies have been conducted when jets are constrained in a narrow gap whose length is two orders of magnitude smaller than the length scales of the other two dimensions. In this configuration, the dynamics shown by the jets is very

1 Département de Génie Mécanique, Université Mentouri Constantine, Route de Ain El Bey, 25000, Constantine, Algérie 2 Département Génie Mécanique, Université Oum El Bouaghi, Algérie 3 Département des Sciences et Techniques, Faculté des Sciences et de la Technologie, Centre Universitaire Abdelhafid Boussouf – Mila, Algérie 4 Département de Génie des Transport, Université Mentouri Constantine, Route de Ain El Bey, 25000, Constantine, Algérie 5 Département Génie Mécanique, Ecole National Polytechnique de Constantine ENPC, Constantine, Algérie

The order of accuracy of any convergent time integration method for systems of differential equations can be increased by using the sequence acceleration method known as Richardson extrapolation, as well as its variants (classical Richardson extrapolation and multiple Richardson extrapolation). The original (classical) version of Richardson extrapolation consists in taking a linear combination of numerical solutions obtained by two different time-steps with time-step sizes h and h/2 by the same numerical method. Multiple Richardson extrapolation is a generalization of this procedure, where the extrapolation is applied to the combination of some underlying numerical method and the classical Richardson extrapolation. This procedure increases the accuracy order of the underlying method from p to p + 2, and with each repetition, the order is further increased by one. In this paper we investigate the convergence of multiple Richardson extrapolation in the case where the underlying numerical method is an explicit Runge-Kutta method, and the computational efficiency is also checked.

A vertical gradient of suspended sediment concentration often exists in estuaries, particularly within the bottom boundary layer where sediment erosion and deposition take place. This results in a vertical density gradient and hence modification of the flow. However, this important effect has often been ignored in past sediment studies. Because of this and because of other empirical assumptions, exisiting erosion models cannot be used as predictive tools. This paper employs a simplified second‐order closure model to simulate the effect of sediment‐induced stratification on bottom boundary layer dynamics, and particularly the erosion process. Numerical models, which employ the concept of Richardson number dependent eddy viscosity, have been developed in the past for stratified flows. These models require a large number of data for tuning parameters of the eddy viscosity formula. The simplified second‐order closure model used here consists of the dynamic equations of motion for mean v...

In this paper we present two new results. The first concerns the proper identification of the critical Richardson number Ri(cr) above which there is no longer turbulent mixing. Thus far, all studies have assumed that: Ri(cr) = Ri (cr) = 1/4.

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Three autonomous profiling Electromagnetic Autonomous Profiling Explorer (EM-APEX) floats were air deployed one day in advance of the passage of Hurricane Frances (2004) as part of the Coupled Boundary Layer Air–Sea Transfer (CBLAST)-High field experiment. The floats were deliberately deployed at locations on the hurricane track, 55 km to the right of the track, and 110 km to the right of the track. These floats provided profile measurements between 30 and 200 m of in situ temperature, salinity, and horizontal velocity every half hour during the hurricane passage and for several weeks afterward. Some aspects of the observed response were similar at the three locations—the dominance of near-inertial horizontal currents and the phase of these currents—whereas other aspects were different. The largest-amplitude inertial currents were observed at the 55-km site, where SST cooled the most, by about 2.2°C, as the surface mixed layer deepened by about 80 m. Based on the time–depth evolutio...

In the present work, laminar mixed convection of a Newtonian fluid around a hot obstacle in a square cavity with moving vertical walls is studied numerically. The objective of this study is to analyze the effect of the Richardson number (0 ≼ Ri ≼ 10) and Reynolds number (50 ≼ Re ≼ 500) on both hydrodynamic and thermal characteristics around a hot obstacle in the enclosure. The analysis of the obtained results shows that the heat transfer is enhanced for high values of Richardson and Reynolds numbers.

In the present work, laminar mixed convection of TiO 2 -Water nanofluid around a hot obstacle in a square cavity with moving vertical walls is studied numerically. The objective of this study is to analyze the effect of the Richardson number (0 ≼Ri≼ 10), Reynolds number (50 ≼ Re ≼ 500) and the nanoparticles volume fraction (0% ≼ ≼ 5%) on both hydrodynamic and thermal characteristics around a hot obstacle in the enclosure. The analysis of the obtained results shows that the heat transfer is enhanced for high values of Richardson and Reynolds numbers. In addition, the volume fraction of nanoparticles has a significant effect on the heat transfer within the cavity.

The present numerical study consists of a mixed convection heat transfer, taking place within a ventilated cubic cavity crossed by an alumina-water nanofluid. The ventilation is assured by three openings, practiced on the walls of the cavity. So, the cold nanofluid enters by an opening placed at the top of the left vertical wall and exits by two opening: one is placed at the bottom of the right vertical wall, and the second opening occupied several possibility of location along the four walls of the cavity. All the enclosure's walls are maintained at a same temperature, higher than that of the entering fluid, except of the side walls which are adiabatic. The governing equations are discretized using the finite volume method and the SIMPLER algorithm to treat the coupling velocity-pressure. The method line by line is used to solve iteratively the algebraic equations. The simulations were carried for different values of the Richardson number (0 ^Ri ^10), the Hartmann number (0 ^Ha ^100) and a solid volume fractions u = 4% using Koo-Kleinstreuer-Lee model for the evaluation of effective thermal conductivity and dynamic viscosity of nanofluid. The results are presented in the form of streamlines, isotherms, velocity vectors and average Nusselt number.

The present work relates a numerical study of mixed convection around a horizontal cylinder heated and equipped with fins and placed in a cylindrical cavity. In addition, the length of the cylinder is assumed to be infinite. Moreover, the heat transfer by radiation from the hot surface is assumed to be negligible, buoyancy effects are also considered, with the Boussinesq approximation. The forced circulation is induced by the inner cylinder in rotation with an angular velocity (ω), with its axis at the center of the annular space. Searches are made for various combinations of dimensionless numbers; Number of Reynolds (Re), number of Grashof (Gr). A finite volume scheme is used to solve the equations governing our system, which are continuity, bidimensional and energy, by the SIMPLE algorithm. The results found indicates that the heat exchange on the outer and inner cylinder characterized by the number of Nusselt decreases by increasing the Reynolds number and for low Reynolds number...

In this work, mixed convection in a Bingham plastic fluid from a heated hemi-sphere has been investigated numerically in the so-called aiding buoyancy configuration. The results reported herein, spanning the following ranges of conditions: Reynolds number, 1 6 Re 6 100, Prandtl number, 1 6 Pr 6 100, Richardson number, 0 6 Ri 6 10 and Bingham number, 0 6 Bn 6 100, are used to describe the momentum and heat transfer characteristics of a hemi-sphere oriented with its curved surface towards the oncoming fluid stream. In particular, flow characteristics are analyzed in terms of the streamlines and the morphology of fluid-like/solid-like regions and the corresponding heat transfer characteristics are examined via isotherm contours, local Nusselt number and average Nusselt number values to delineate the role of each of these parameters. The flow separates itself from the hemi-sphere at a much lower Reynolds numbers as compared to that for a sphere due to the sudden loss of the rear surface. The average value of the Nusselt number is influenced by four non-dimensional parameters, namely, Reynolds number, Prandtl number, Bingham number and Richardson number. Finally, the present results have been consolidated using the Colburn j-factor for predicting the value of the average Nusselt number in a new application.

A Finite-Volume based POD-Galerkin reduced order modeling strategy for steady-state Reynolds averaged Navier-Stokes (RANS) simulation is extended for low-Prandtl number flow. The reduced order model is based on a full order model for which the effects of buoyancy on the flow and heat transfer are characterized by varying the Richardson number. The Reynolds stresses are computed with a linear eddy viscosity model. A single gradient diffusion hypothesis, together with a local correlation for the evaluation of the turbulent Prandtl number, is used to model the turbulent heat fluxes. The contribution of the eddy viscosity and turbulent thermal diffusivity fields are considered in the reduced order model with an interpolation based data-driven method. The reduced order model is tested for buoyancy-aided turbulent liquid sodium flow over a vertical backward-facing step with a uniform heat flux applied on the wall downstream of the step. The wall heat flux is incorporated with a Neumann boundary condition in both the full order model and the reduced order model. The velocity and temperature profiles predicted with the reduced order model for the same and new Richardson numbers inside the range of parameter values are in good agreement with the RANS simulations. Also, the local Stanton number and skin friction distribution at the heated wall are qualitatively well captured. Finally, the reduced order simulations, performed on a single core, are about 10 5 times faster than the RANS simulations that are performed on eight cores.

A numerical study of double-diffusive mixed convection within a horizontal rotating annulus has been investigated. The outer cylinder is fixed but the inner cylinder is considered to rotate in clockwise and anti-clockwise directions to introduce the forced convection effect. In addition, the solutal and thermal buoyancy forces are sustained by maintaining the inner and outer cylinder at uniform temperatures and concentrations but their values for the inner are higher than the outer. The flow is considered laminar regime under steady state conditions. The transport equations for the continuity, momentum, energy and mass transfer are solved using the finite volume technique. The considered domains in this investigation are: -15 ≤ N ≤ 15, 0.01 ≤ Ri ≤ 100 and 0.01 ≤ Pr ≤ 100. While the thermal Grashof number, Lewis number and the radius ratio are kept constant at values equal to10 4 , 1 and 2 respectively. The effect of the selected parameters on the local and average Nusselt and Sherwood numbers are presented and studied. Finally, this investigation concerned with selection the best direction of the inner cylinder rotation to enhance both heat and mass transfer. A comparison was made with the published results and a good agreement was found.

In this work mixed convection in a horizontal channel with the lower wall heated at a uniform heat flux is studied experimentally. The experiments are performed in air. The analysis is accomplished for several heat fluxes and forced air velocities. The Reynolds numbers investigated are between 10 and 500, these being in the laminar regime. The Richardson number Ri=Gr/Re 2 holds values in the range 0.1-100. Flow visualization is performed to detect the flow patterns into the channel. The wall temperature distribution is given as a function of the Gr/Re 2 values.

Convective heat transfer can be enhanced passively by changing flow geometry and boundary conditions or by improving the thermal conductivity of the working fluid, for example, introducing suspended small solid nanoparticles. In this paper, a numerical investigation on laminar mixed convection in a water-Al 2 O 3 -based nanofluid, flowing in a triangular cross-sectioned duct, is presented. The duct walls are assumed at uniform temperature, and the single-phase model has been employed in order to analyze the nanofluid behaviour. The hydraulic diameter is equal to 0.01 m. A fluid flow with different values of Richardson number and nanoparticle volume fractions has been considered. Results show the increase of average convective heat transfer coefficient and Nusselt number for increasing values of Richardson number and particle concentration. However, also wall shear stress and required pumping power profiles grow significantly.

Laminar mixed convection in a two-dimensional symmetrically and partially heated vertical channel is investigated. The heaters are located on both walls and uniform temperature is applied on the heated sections. The number of heaters is considered as 1, 4, 8, and 10. Aluminum oxide/water nanofluid is considered as working fluid and the inlet velocity is uniform. The continuity, momentum and energy equations with appropriate boundary conditions are solved in dimensionless form, numerically. The study is performed for Richardson number of 0.01 and 10, Reynolds number of 100 and 500, and nanofluid volume fraction of 0% and 5%. Based on the obtained velocity and temperature distributions, the local and mean Nusselt number is calculated and plotted for different cases. The variation of the mean Nusselt number with the number of the heated portions is also discussed. It is found that the addition of nanoparticles into the base fluid increases mean Nusselt number but the rate of increase depends on Reynolds, Richardson numbers and number of heated portions. It is possible to increase mean Nusselt number 138% by increasing Reynolds number from 100 to 500, Richardson number from 0.01 to 10 and number of heated portions from 1 to 10 when volume fraction value is 5%.

In this work numerical investigation of mixed convection by non-concentric different positions of rotating circular cylinder in square enclosure is investigated. The assumption of using six rotating cylinders at constant temperature Th and square closed enclosure with temperature Tc for all walls are maintained. Continuity, Momentum and energy governing equations are solved by using finite volume technique using Ansys-Fluent 16 commercial code with two dimension and steady state case. Richardson number (Ri=Gr/Re 2 ) varying over the wide range of 0.1, 1, 10,100, 1000 and ∞. This research investigate the effect of changing the rotating circular cylinder positions along the vertical centerline on the fluid flow and heat transfer inside the enclosure. Six different positions are tested. The phenomenon inside the square enclosure is mathematically analyzed. A streamlines, isothermal patterns, local Nusselt numbers, and average Nusselt numbers are calculated. The analytical results give that, the Richardson numbers is playing the dominant role on temperature distributions and flow patterns inside the enclosure. The average Nusselt number variation demonstrates the best location of rotating cylinder which gives best heat transfer value. The bottom left corner (P4) is the best location for hot rotating cylinders. While the rotating in middle right (P6) is the inferior position.

Understanding mixed convection in engineering applications such as heat exchangers, electronics cooling devices, and solar energy collectors have urged researchers to investigate this phenomenon deeper. This study investigates the fluid flow and heat transfer pattern in a two-dimensional (2D) rectangular cavity with sinusoidal heating on the moving top lid numerically. The bottom wall is kept cool while the vertical walls are insulated. The effect of Hartmann number, 𝐻𝑎 on the thermal characteristics and fluid flow are analyzed for Richardson number, 𝑅𝑖 = 1 which indicate mixed convection dominated regime. The governing equations are solved numerically using a SIMPLE algorithm with the finite volume method. The numerical results are displayed in streamlines and isotherms plots. The value of the Nusselt number indicating the heat transfer rate is also discussed. It is found that 𝐻𝑎 has a significant effect on the heat transfer process and fluid flow. It can be seen clearly when the value of 𝐻𝑎 = 30, the rate of heat transfer dropped significantly on the cold wall. Generally, the heat transfer rate decreases with the increase of 𝐻𝑎.

In November-December 1984 we carried out an intensive 12-day upper ocean sampling program on the equator at 140øW as part of the Tropic Heat Experiment. From our observations we constructed hourly averaged profiles of temperature, salinity, at, turbulent kinetic energy dissipation rate, and horizontal velocity. These data were used to examine the correspondence between hydrographic and velocity fields and to compare the measured turbulent dissipations with the calculated Richardson numbers. We found that the core of the Equatorial Undercurrent tracked a density surface (a t = 25.25) on times as short as 1 hour. The variability in both hydrographic and velocity fields was greatest at the semidiurnal frequency. The supertidal energy was not significantly different from the Garrett-Munk mid-latitude level once latitudinal scaling was removed from the Garrett-Munk model parameters. Horizontal velocity spectra were found to be contaminated by displacement of the background shear. Turbulent dissipation was dominated by a dirunal cycle, with high values of dissipation occurring at night above the undercurrent core. Shear and buoyancy frequency, calculated over 12-m vertical scales, were observed to track each other above the core and were dominated by a diurnal period above 40 m and by a semidiurnal period below 40 m. When shear and buoyancy frequency were combined to form a Richardson number, neither diurnal nor semidiurnal cycles were present. Above the undercurrent core, the Richardson numbers were uniformly small (0.3 to 0.6). 1. INTRODUCTION Studies have suggested that space and time variability in the equatorial ocean may be quite different from variability at mid-latitudes. Internal wave spectra from the equatorial Indian Ocean indicate more energy at frequencies above the tidal than is predicted by the Garrett-Munk (GM) universal internal wave spectrum [Eriksen, 1980]. Other deviations from GM [Garrett and Munk, 1972, 1975] include less spatial coherence [Wunsch and Webb, 1979; Eriksen, 1980] and an excess of horizontal kinetic energy over potential energy at subtidal frequencies [Eriksen, 1980]. One obvious difference is that at the equator there is no Coriolis parameter to limit the frequency bandwidth of internal waves. From their fine-structure measurements, Toole and Hayes [1984] found enhanced shear and strain variance and a greater proportion of low values of the Richardson number on the equator. McPhaden [1985] noted anomalously high fine-scale temperature and density variance confined within 1 ø of the equator. Microstructure measurements indicated a peak in turbulent kinetic energy dissipation rate within 1 ø of the equator [Crawford, 1982], but more recent results [Moumet al., 1986b-I have shown this peak to be an artifact of the sampling limitations of the previous study. In November-December 1984 we carried out an intensive 12-day sampling program near the equator at 140øW as part of the Tropic Heat experiment. Vertical profiles of temperature, conductivity, and small-scale shears were obtained every 10 min (on average) with the rapid sampling vertical Copyright 1986 by the American Geophysical Union. Paper number 6C0326. 0148-0227/86/006C-0326505.00 profiler (RSVP). Vertical profiles of horizontal currents were obtained every 30 s with a shipboard acoustic Doppler current profiler (ADCP). From these data we constructed hourly averaged profiles of temperature T, salinity S, at, kinetic energy dissipation rate e, and horizontal velocity. Using these series of profiles we made the following observations and comparisons: 1. We examined the correspondence between hydrographic and velocity fields. We found that the core of the Equatorial Undercurrent (EUC) tracked a density surface (a t = 25.25) on times as short as 1 hour. The core tracked the salinity maximum almost as well. 2. We compared the high-frequency variability with that observed at other latitudes in terms of the Garrett-Munk model. We found that (1) the time variability in both hydrographic and velocity fields was greatest at the semidiurnal frequency, (2) the supertidal energy was not significantly different from the Garrett-Munk mid-latitude level once the latitudinal scaling was removed from the Garrett-Munk model parameters, and (3) the horizontal velocity spectra were contaminated by displacement of the background shear. 3. We compared calculated Richardson numbers with the observed values of turbulent kinetic energy dissipation rate and found that (1) the dissipation was dominated by a diurnal cycle, with high values occurring at night above the undercurrent core, (2) shear and buoyancy frequency calculated over 12-m scales tracked each other extremely well, with a diurnal cycle above 40 m and a semidiurnal cycle below 40 m and (3) the Richardson number was uniformly small (between 0.3 and 0.6) above the undercurrent core. For the 12-day period, the Richardson number displayed neither a diurnal nor a semidiurnal periodicity. (In longer times series, periods were observed when the Richardson number cycled diurnally near the surface.) 12,887 12,888

The measurements taken during the Vertical Transport and Mixing Experiment (VTMX, October, 2000) on a northeastern slope of Salt Lake Valley, Utah, were used to calculate the statistics of velocity fluctuations in a katabatic gravity current in the absence of synoptic forcing. The data from ultrasonic anemometer-thermometers placed at elevations 4.5 and 13.9 m were used. The contributions of small-scale turbulence and waves were isolated by applying a high-pass digital (Elliptical) filter, whereupon the filtered quantities were identified as small-scale turbulence and the rest as internal gravity waves. Internal waves were found to play a role not only at canonical large gradient Richardson numbers (Ri g > 1), but sometimes at smaller values (0.1 < Ri g < 1), in contrast to typical observations in flatterrain stable boundary layers. This may be attributed, at least partly, to (critical) internal waves on the slope, identified by Princevac et al. , which degenerate into turbulence and help maintain an active internal wave field. The applicability of both Monin-Obukhov (MO) similarity theory and local scaling to filtered and unfiltered data was tested by analyzing rms velocity fluctuations as a function of the stability parameter z/L, where L is the Obukhov length and z the height above the ground. For weaker stabilities, z/L < 1, the MO similarity and local scaling were valid for both filtered and unfiltered data. Conversely, when z/L > 1, the use of both scaling types is questionable, although filtered data showed a tendency to follow local scaling. A relationship between z/L and Ri g was identified. Eddy diffusivities of momentum K M and heat K H were dependent on wave activities, notably when Ri g > 1. The

A conceptual model of the physical behavior of a shallow (6 m deep) micro-tidal estuary (Alfacs Bay) is proposed, based on the interpretation of a field data set, and subsequently tested against the results of three-dimensional hydrodynamic simulations. At seasonal timescales, the buoyancy associated with freshwater inflows dominates the tidal forcing, yielding a strongly stratified two-layered system, with the surface and the bottom layers flowing in opposite directions (classical estuarine circulation). Wind controls the physical behavior of the bay at shorter (days to weeks) timescales. Three scenarios or states have been defined, depending on the strength of stratification relative to the predominant direction and magnitude of the wind forcing, parameterized through the Wedderburn number, W. For weak winds (scenario 1), with W ≫ 1, mixing occurs as a consequence of stirring and convective cooling, and the mixed layer deepens slowly. For strong winds, with W ≪ 1/2, mixing is fast and is largely driven by shear at the pycnocline. Two scenarios are further identified for W ≪ 1/2 depending on the directionality of the winds: one for persistent NW winds (scenario 2) and another for diurnal SW winds (scenario 3). In scenario 2, the water is pushed laterally, overturning the stratification and generating transverse density gradients. In scenario 3, the estuarine circulation is weakened and even reversed, yielding strong longitudinal density gradients. The bay waters relax quickly (within 10-18 h) back to the original state, after the wind ceases, as a result of the horizontal density gradients developed under wind forcing. Bay-sea exchange rates are shown to decrease significantly in the low W scenarios, the magnitudes of these changes being largely dependent on wind direction. These scenarios have important implications for the ecology of the bay, including the occurrence of phenomena such as harmful algal blooms.

The critical Richardson number, Ric, is used in studies of stably stratified turbulence as a measure of flow laminarization. The accepted range of Ric is between 0.2 and 1. A growing body of experimental and observational data indicates, however, that turbulence survives for Ri ≫ 1. This result is supported by a new spectral theory of turbulence that accounts for strong anisotropy and waves. The anisotropization results in the enhanced horizontal mixing of both momentum and scalar. Internal wave contribution preserves vertical momentum mixing above its molecular level. In the absence of laminarization, Ric becomes devoid of its conventional meaning. Copyright © 2007 Royal Meteorological Society

The mixed convection of heat transfer and fluid flow in a -lid drivencubical cavity filled with air is investigated numerically in this study. The computational procedure is based on the finite volume method and a full multigrid acceleration solver. The top wall of the cavity is maintained at a constant high -temperature Th, and it can move with a constant velocity U0. The bottom wall is immobile and maintained at a cold temperature Tc. While, the remaining boundary parts of the cavity are motionless and kept thermally insulated. Several numerical simulations were conducted to investigate mixed convection heat transfer in a sliding cubical cavity for a range of Reynolds numbers from 1000 to 5000 and Richardson numbers from 0.001 to 10. The influence of mixed convection parameters, Reynolds number, Richardson number, and heat transfer rate on the flow behavior was analyzed through parametric studies. The results include flow and heat transfer characteristics, iso-surfaces, and streamlines for the entire range of Richardson numbers and Reynolds numbers investigated. The study shows that as Reynolds number is increased beyond a critical value, the flow becomes unstable and bifurcates.

The mixed convection of heat transfer and fluid flow in a -lid drivencubical cavity filled with air is investigated numerically in this study. The computational procedure is based on the finite volume method and a full multigrid acceleration solver. The top wall of the cavity is maintained at a constant high -temperature Th, and it can move with a constant velocity U0. The bottom wall is immobile and maintained at a cold temperature Tc. While, the remaining boundary parts of the cavity are motionless and kept thermally insulated. Several numerical simulations were conducted to investigate mixed convection heat transfer in a sliding cubical cavity for a range of Reynolds numbers from 1000 to 5000 and Richardson numbers from 0.001 to 10. The influence of mixed convection parameters, Reynolds number, Richardson number, and heat transfer rate on the flow behavior was analyzed through parametric studies. The results include flow and heat transfer characteristics, iso-surfaces, and streamlines for the entire range of Richardson numbers and Reynolds numbers investigated. The study shows that as Reynolds number is increased beyond a critical value, the flow becomes unstable and bifurcates.

Currents and vertical mixing characteristics were investigated on the basis of time series of current meter and temperature data from a summer-stratified period in Lake Ontario. The experimental set up consisted of seven current meters distributed in one vertical line from 12 m below the surface to 1 m above the lake bottom at a total depth of 143 m. The period considered for the analysis was from June to September, 1991. The currents showed pronounced oscillations with two significant kinetic energy peaks, one at about 17 hours due to inertial motions, and one at 10 days, probably due to meteorological forcing. The current shear in the hypolimnion was strong enough to overcome stability and generate turbulence (Richardson numbers below 0.25) and there was probably turbulence enough available to keep the matter (almost neutral buoyant particles) in the whole Nepheloid bottom layer in suspension. In the thermocline region the turbulence was mainly damped (Richardson numbers above I ) , but some events with lower Richardson numbers were also calculated indicating increased mixing during these events. By analysing filtered and unfiltered current meter data it was found that the shear-generated turbulence in the hypolimnion was mainly due to the meteorologically forced currents. In the thermocline region, however, the vertical shcar associated with the inertial oscillation had a greater impact on the mixing.

In the present study, numerical simulation of magnetohydrodynamic (MHD) mixed convection heat transfer and fluid flow has been analyzed in a lid-driven enclosure provided with a constant flux heater. Governing equations were solved via differential quadrature (DQ) method. Moving wall of the enclosure has constant temperature and speed. The calculations were performed for different Richardson number ranging from 0.1 to 10, constant heat flux heater length from 0.2 to 0.8, location of heater center from 0.1 to 0.9, Hartmann number from 0 to 100 and aspect ratio from 0.5 to 2. Two different magnetic field directions were tested as vertical and horizontal. It was found that results of DQ method show good agreement with the results of literature. The magnetic field was more effective when it applied horizontally than that of vertical way. In both direction of magnetic field, it reduced the flow strength and heat transfer. Thus, it can be used as an important control parameter for heat an...

It is shown that the observationally determined roughness relation z,, = au$g in which g is the acceleration of gravity, u* is the friction velocity in air, and a = 0.0185 (Wu, 1982) for the wind profile over the sea surface relative to the surface current, is consistent with the existence of a Richardson Number criterion at the air-sea interface in which the critical Richardson Number, Ri, = 1, such that all the shear energy is converted into potential energy.

The preliminary results of a statistical analysis of the atmospheric stability effects on the growth of sea waves are given in this paper. In unstable conditions (air colder than water) an increase of vertical mixing is experienced due to turbulence which makes the wind profile more uniform with height (deviating it from the logarithmic law). This circumstance increases the wind stress on the sea surface and thus gives rise to higher wave energy. In order to quantify the wave energy increase, the analysis of wave, wind and temperature data was performed, on the basis of the SWAN (Sea WAve monitoring Network) data set regarding both the Tyrrhenian and Adriatic Sea. The statistical analysis was first performed in terms of relative deviation of non-dimensional wave energy as a function of the Bulk Richardson Number in terms of the wind velocity U 10. Then another stability parameter, the Monin-Obukhov length, was related to the friction velocity u *. This approach needed the implementation of a numerical procedure to estimate the set of Monin-Obukhov parameters, which account for atmospheric stability.

This study investigates the cooling of a central processing unit (CPU) using a nano-encapsulated phase change material (NEPCM)-water mixture in a trapezoidal cavity with rotating cylinders and baffles. A numerical model based on the finite element method (FEM) is employed to solve the governing equations. The system is subjected to a sinusoidal temperature profile from the CPU and a constant magnetic field. Key parameters examined include Reynolds number (Re: 10-100), Richardson number (Ri: 0.1-10), Hartmann number (Ha: 5-80), NEPCM volume fraction (ϕ: 0.015-0.035), Lewis number (Le: 0.1-10), buoyancy ratio (Nz: 1-5), NEPCM fusion temperature (θ f : 0.1-0.9), and Stefan number (Ste: 0.1-0.9). Results show that increasing Re and Ri significantly enhances heat and mass transfer, with the average Nusselt number (Nu av) increasing by up to 80.5 % and average Sherwood number (Sh av) by up to 147.9 %. The magnetic field suppresses convection, reducing Nu av by 12.7 % and Sh av by 39.5 % as Ha increases. Increasing ϕ improves heat transfer (Nu av up by 32.5 %) with minimal effect on mass transfer. Le strongly influences mass transfer, with Sh av increasing by 284.6 % as Le increases. The NEPCM fusion temperature exhibits a non-monotonic effect on Nu av , with an optimal value at θ f = 0.5. In conclusion, the study reveals complex interactions between parameters, with Re, Ri, and Le having the most significant impacts on system performance. These findings provide valuable insights for optimizing CPU cooling systems using NEPCM-water mixtures and magnetohydrodynamic (MHD) effects.

Fluid flow in the lid driven cavity is a well-known phenomenon in the realm of fluid flow and heat transfer. Different cavity shapes such as square, circular, trapezoidal, and hexagonal have already been studied. However, most of the parametric analyses have previously been done using one specific cavity shape design. The present study intends to generalize the cavity design for a larger range of applications starting from the selection of cavity form due to the enhancement of heat transfer inside that selected cavity for different boundary conditions. Numerical studies using a finite element solver have been carried out to investigate the physics of fluid flow and heat transmission in a cavity where a wall of the enclosure is moving. In this research, the average Nusselt number for six distinct shaped lid-driven cavities is studied (e.g., square, rhombus, circular, rounded rectangular, trapezoidal and hexagonal). The hexagonal-shaped cavity has the largest Nusselt number of all of ...

Local similarity theory is suggested based on the Brunt-Väisälä frequency and the dissipation rate of turbulent kinetic energy instead the turbulent fluxes used in the traditional Monin-Obukhov similarity theory. Based on dimensional analysis (Pi theorem), it is shown that any properly scaled statistics of the small-scale turbulence are universal functions of a stability parameter defined as the ratio of a reference height z and the Dougherty-Ozmidov length scale which in the limit of z-less stratification is linearly proportional to the Obukhov length scale. Measurements of atmospheric turbulence made at five levels on a 20-m tower over the Arctic pack ice during the Surface Heat Budget of the Arctic Ocean experiment (SHEBA) are used to examine the behaviour of different similarity functions in the stable boundary layer. It is found that in the framework of this approach the non-dimensional turbulent viscosity is equal to the gradient Richardson number whereas the non-dimensional turbulent thermal diffusivity is equal to the flux Richardson number. These results are a consequence of the approximate local balance between production of turbulence by the mean flow shear and viscous dissipation. The turbulence framework based on the Brunt-Väisälä frequency and the dissipation rate of turbulent kinetic energy may have practical advantages for estimating turbulence when the fluxes are not directly available.

The chemical reaction of heat and mass transfer in an unsteady magnetohydrodynamics (MHD) mixed convection stagnation point flow towards a vertical stretching plate embedded in porous medium was studied. Richardson number and concentration were taken into consideration in this study. By selecting suitable similarity variables, the non-linear partial differential equations (PDEs) are transformed into a set of ordinary differential equations (ODEs), and then solved numerically using the boundary value problem solver (bvp4c) in MATLAB software. The solution is dependent on the governing parameters including Prandtl number, reaction rate parameter, Schmidt number, thermal Richardson number and buoyancy ratio parameter. The numerical results are depicted graphically in velocity, temperature and concentration profiles. The findings have shown that as the chemical reaction parameter increases, the temperature profile and Sherwood number also increases.

In the present study, fluid flow, heat transfer, and entropy generation for mixed convection inside a water-filled square cavity were investigated numerically. The sidewalls of the cavity, which move upwards, are kept at low-temperature Tc while only a part in the center of the bottom wall is kept at high-temperature Th and the remaining parts are kept adiabatic. The governing equations, in stream function-vorticity form, are discretized and solved using the finite difference method. Particular attention was paid to the influence of the Prandtl numbers of 5.534, 3.045 and 2, corresponding respectively to the water temperatures of 303.15 K, 333.15 K and 363.15 K. The numerical results are presented in the form of streamlines, isotherms, and entropy generation contours for different values of the Richardson numbers at an arbitrary Reynolds number Re=10 2. Besides this, the evolution of the average Nusselt number and the average entropy generation is also reported. The obtained results show interesting behaviors of the flow and thermal fields, which mainly involve stable symmetric and non-symmetric steady-state solutions, as well as unsteady regimes, depending on specific values of the Richardson and Prandtl numbers. It is additionally observed that the average Nusselt number increases and the average entropy generation decreases when both the Richardson and Prandtl numbers increase.

We present airplane measurements of the stably strattied nocturnal boundary layer obtained during the Severe Environmental Storms and Mesoscale Experiment (SESAME) in 1979. The cases presented here were obtained over rolling terrain in central Oklahama, with a mean slope of about 0.003. The results are in general agreement with previous modeling and observational studies for the mean and turbulence structure of the nocturnal boundary layer, with the exception that the eddy diffusivity of heat, and consequently the flux Richardson number are less than expected. DONALD H. LENSCHOW ET AL.

Purpose The purpose of this paper is to investigate the mixed convection of a two-phase water–aluminum oxide nanofluid in a cavity under a uniform magnetic field. Design/methodology/approach The upper wall of the cavity is cold and the lower wall is warm. The effects of different values of Richardson number, Hartmann number, cavitation length and solid nanoparticles concentration on the flow and temperature field and heat transfer rate were evaluated. In this paper, the heat flux was assumed to be constant of 10 (W/m2) and the Reynolds number was assumed to be constant of 300 and the Hartmann number and the volume fraction of solid nanoparticles varied from 0 to 60 and 0 to 0.06, respectively. The Richardson number was considered to be 0.1, 1 and 5. Aspect ratios were 1, 1.5 and 2. Findings Comparison of the results of this paper with the results of the numerical and experimental studies of other researchers showed a good correlation. The results were presented in the form of veloci...