Analytical solutions for turbulent Boussinesq fountains in a linearly stratified environment

2011 ◽  
Vol 691 ◽  
pp. 487-497 ◽  
Author(s):  
Rabah Mehaddi ◽  
Olivier Vauquelin ◽  
Fabien Candelier
Keyword(s):  
Theoretical Model ◽  
Asymptotic Analysis ◽  
Flow Rate ◽  
Vertical Velocity ◽  
Analytical Solutions ◽  
Boussinesq Approximation ◽  
Singular Case ◽  
Maximal Height ◽  
Analytical Integration ◽  
Analytical Expressions

AbstractThis paper theoretically investigates the initial up-flow of a vertical turbulent fountain (round or plane) in a linearly stratified environment. Conservation equations (volume, momentum and buoyancy) are written under the Boussinesq approximation assuming an entrainment proportional to the vertical velocity of the fountain. Analytical integration leads to exact values of both density and flow rate at the maximal height reached by the fountain. This maximal height is expressed as a function of the release conditions and the stratification strength and plotted from a numerical integration in order to exhibit overall behaviour. Then, analytical expressions for the maximal height are derived from asymptotic analysis and compared to experimental correlations available for forced fountains. For weak fountains, these analytical expressions constitute a new theoretical model. Finally, modified expressions are also proposed in the singular case of an initially non-buoyant vertical release.

scholarly journals Quantitative Monitoring of Dynamic Blood Flows Using Coflowing Laminar Streams in a Sensorless Approach

2021 ◽  
Vol 11 (16) ◽  
pp. 7260
Author(s):  
Yang Jun Kang
Keyword(s):  
Blood Flow ◽  
Flow Rate ◽  
Blood Viscosity ◽  
Measurement Errors ◽  
Present Method ◽  
Test Fluid ◽  
Constant Flow Rate ◽  
Rbc Aggregation ◽  
Sinusoidal Flow ◽  
Analytical Expressions

Determination of blood viscosity requires consistent measurement of blood flow rates, which leads to measurement errors and presents several issues when there are continuous changes in hematocrit changes. Instead of blood viscosity, a coflowing channel as a pressure sensor is adopted to quantify the dynamic flow of blood. Information on blood (i.e., hematocrit, flow rate, and viscosity) is not provided in advance. Using a discrete circuit model for the coflowing streams, the analytical expressions for four properties (i.e., pressure, shear stress, and two types of work) are then derived to quantify the flow of the test fluid. The analytical expressions are validated through numerical simulations. To demonstrate the method, the four properties are obtained using the present method by varying the flow patterns (i.e., constant flow rate or sinusoidal flow rate) as well as test fluids (i.e., glycerin solutions and blood). Thereafter, the present method is applied to quantify the dynamic flows of RBC aggregation-enhanced blood with a peristaltic pump, where any information regarding the blood is not specific. The experimental results indicate that the present method can quantify dynamic blood flow consistently, where hematocrit changes continuously over time.

scholarly journals Simple and Accurate Analytical Solutions of the Electrostatically Actuated Curled Beam Problem

2014 ◽  
Author(s):  
Mohammad I. Younis
Keyword(s):  
Analytical Solutions ◽  
Galerkin Approximation ◽  
Single Mode ◽  
Beam Model ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Analytical Formulas ◽  
Euler Bernoulli Beam ◽  
Analytical Expressions ◽  
Multi Mode

We present analytical solutions of the electrostatically actuated initially deformed cantilever beam problem. We use a continuous Euler-Bernoulli beam model combined with a single-mode Galerkin approximation. We derive simple analytical expressions for two commonly observed deformed beams configurations: the curled and tilted configurations. The derived analytical formulas are validated by comparing their results to experimental data in the literature and numerical results of a multi-mode reduced order model. The derived expressions do not involve any complicated integrals or complex terms and can be conveniently used by designers for quick, yet accurate, estimations. The formulas are found to yield accurate results for most commonly encountered microbeams of initial tip deflections of few microns. For largely deformed beams, we found that these formulas yield less accurate results due to the limitations of the single-mode approximations they are based on. In such cases, multi-mode reduced order models need to be utilized.

An Approximate Method for Non -Darcy Radial Gas Flow

1964 ◽  
Vol 4 (02) ◽  
pp. 96-114 ◽  
Author(s):  
G. Rowan ◽  
M.W. Clegg
Keyword(s):  
Linear Equation ◽  
Flow Rate ◽  
Analytical Solutions ◽  
Gas Flow ◽  
Darcy’S Law ◽  
Darcy's Law ◽  
Non Linear Equation ◽  
Approximate Analytical Solutions ◽  
Non Linear ◽  
Flow Law

Abstract Approximate analytical solutions for non-Darcy radial gas flow are derived for bounded and infinite reservoirs producing at either constant rate or constant pressure. These analytical solutions are compared with published results for non-Darcy flow obtained on digital and analogue computers, and the agreement is shown to be very good. Some observations on the interpretation of gas well tests are made. Introduction The flow of gases in porous media is a problem that has been the subject of much study in recent years, and many methods have been proposed for solving the non-linear equations associated with it. The assumption that the flow satisfies Darcy's Law (1) leads to a non-linear equation of the form (2) in a homogeneous medium, assuming an equation of state(3) It has been observed, however, that the linear relationship between the flow rate and pressure gradient is only approximately valid even at low flow rates, and that as the flow rate increases the deviations from linearity also increase. It has been suggested by a number of authors that Darcy's Law should be replaced by a quadratic flow law of the form (4) This form of equation was first suggested by Forchheimer and, later, Katz and Cornell, and Irmay, developed a similar equation. Houpeurt derived this form of equation using the concept of an idealized pore system in which each channel consists of sequences of truncated cones giving rise to successive restrictive orifices along the channel. This type of representation leads to a quadratic flow law of type, for all fluids, but it is found that the quadratic term is only significant in the case of gas flow. The methods of Houpeurt for solving gas flow problems will be discussed further in another section of this paper. Solutions of the non-linear equation for Darcy gas flow may be classified as either computer (digital and analogue), or approximate analytical ones. The former include the well-known solutions of Bruce et al., and Aronofsky and Jenkins, but the latter solutions, apart from the simple linearization of equation [2] to yield a diffusion equation in p2, are not so well-known. SPEJ P. 96ˆ

A Theoretical Model for Ship–Wave Impact Generated Sea Spray

2020 ◽  
Vol 143 (4) ◽  
Author(s):  
Shafiul Mintu ◽  
David Molyneux ◽  
Bruce Colbourne
Keyword(s):  
Theoretical Model ◽  
Field Measurements ◽  
Operating Conditions ◽  
Ship Motions ◽  
Wave Impact ◽  
Fishing Vessels ◽  
Semi Empirical ◽  
Wave Induced ◽  
Analytical Expressions ◽  
Ship Speed

Abstract Spray generated by ships traveling in cold oceans often leads to topside icing, which can be dangerous to vessels. Estimation of the spray flux is a first step in predicting icing accumulation. The amount of spray water, the duration of exposure to the spray, and the frequency at which the spray is generated are all important parameters in estimating the spray flux. Most existing spray flux formulae are based on field observations from small fishing vessels. They consider meteorological and oceanographic parameters but neglect the vessel behavior. Ship heave and pitch motions, together with ship speed, determine the frequency of spray events. Thus, the existing formulae are not generally applicable to different sizes and types of vessels. This paper develops simple methods to quantify spray properties in terms that can be applied to vessels of any size or type. Formulae to estimate water content and spray duration are derived based on principles of energy conservation and dimensional analysis. To estimate spray frequency considering ship motions, a theoretical model is proposed. The model inputs are restricted to ship’s principal particulars, operating conditions, and environmental conditions. Wave-induced motions are estimated using semi-empirical analytical expressions. A novel spray threshold is developed to separate deck wetness frequency from spray frequency. Spray flux estimates are validated against full-scale field measurements available in the open literature with reasonable agreement.

Analytical solutions and expressions of the propagator for Bloch equations

2020 ◽  
Vol 34 (18) ◽  
pp. 2050158
Author(s):  
Heung-Ryoul Noh
Keyword(s):  
Analytical Solutions ◽  
Initial Conditions ◽  
Secular Equation ◽  
Magnetization Vector ◽  
Bloch Equations ◽  
Optical Bloch Equations ◽  
Parameter Spaces ◽  
Pulse Spectroscopy ◽  
Analytical Expressions ◽  
Composite Laser

In this paper, we present analytical solutions to the Bloch equations. After solving the secular equation for the eigenvalues, derived from the Bloch equations, analytical solutions for the temporal evolution of the magnetization vector are obtained at arbitrary initial conditions. Subsequently, explicit analytical expressions of the propagator for the Bloch equations and optical Bloch equations are obtained. Compared to the results of existing analytical studies, the present results are more succinct and rigorous, and they can predict the behavior of the propagator in different regions of parameter spaces. The analytical solutions to the propagator can be directly used in composite laser-pulse spectroscopy.

scholarly journals Air exchange through doorways. The effect of temperature difference, turbulence and ventilation flow

1977 ◽  
Vol 79 (1) ◽  
pp. 141-154 ◽  
Author(s):  
O. M. Lidwell
Keyword(s):  
Theoretical Model ◽  
Temperature Difference ◽  
Effect Of Temperature ◽  
Air Mass ◽  
Excess Air ◽  
Outflow Velocity ◽  
Simple Theoretical Model ◽  
Air Supply ◽  
Air Mixing ◽  
Analytical Expressions

SUMMARYAnalytical expressions have been derived for the exchange of air across doorways or similar apertures, in terms of the temperature difference between the spaces on both sides of the opening and the net volume of air flowing through this as a result of unbalanced air supply or extract. A simple allowance for turbulence which gives reasonable correspondence with observation is included. The formulae, which assume complete air mixing on both sides of the doorway up to the plane of the aperture, predict outflows from the warmer side, when there is an excess air supply to this side, which are progressively smaller than those observed as the temperature difference rises above 2–3 °C and the volume of excess air supply increases to produce an averaged outflow velocity greater than 0·1–0·15m/s. This seems to be due to lack of mixing of the warm outflowing air with the cooler air mass. A correction factor for this can be deduced as a function of the pressure difference due to the excess air supply. The limiting magnitude and general form of this function are compatible with a simple theoretical model of the air flow patterns involved.

Action of Flow Rate of Mobile Phase in Chromatography

2013 ◽  
Vol 641-642 ◽  
pp. 35-38
Author(s):  
Da Li Zhang ◽  
Jia Jun Ke ◽  
Li Zhu Lu
Keyword(s):  
Theoretical Model ◽  
Flow Rate ◽  
Mobile Phase ◽  
Plate Theory ◽  
Theoretical Plate ◽  
Chromatographic Peak ◽  
Slip Mechanism ◽  
Plate Number ◽  
Theoretical Plate Number ◽  
Two Phases

Chromatography has found an increasingly wide utilization in scientific and technological fields, even in some cases, has become indispensable methods for analysis and separation. Compared with its application, some common queries in chromatography still lack for good theoretical explanations, for instance, the action of flow rate of mobile phase on symmetry of peak. As is familiar to people, an increase in flow rate of mobile phase always causes theoretical plate number to decrease under normal conditions. At the same time, maybe the symmetry of chromatographic peak obviously increases, on the contrary. This result is self-contradictory. Why does the theoretical plate number not increase under the condition? Utilizing our prior paper on five plate numbers in chromatography, this paper points out why theoretical plate number decreases with an increase in symmetry of peak when flow rate of mobile phase increases. Based upon gas-solid chromatography results, the relation between flow rate of mobile phase and frequency of partition of solute, and the relation between flow rate and plate number are deduced. From the angle of theoretical model, this paper discusses the relation between solute partition in static mobile phase such as partition between two phases in plate theory and solute partition in flowing mobile phase such as Martin-Synge partition in slip mechanism, and their characteristics.

scholarly journals Matched Asymptotic Expansions to the Circular Sitnikov Problem with Long Period

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Marco Rosales-Vera
Keyword(s):  
Analytical Solutions ◽  
Asymptotic Expansions ◽  
Matched Asymptotic Expansions ◽  
Sitnikov Problem ◽  
Third Body ◽  
The Third ◽  
Long Period ◽  
Large Oscillation ◽  
Analytical Expressions

The circular Sitnikov problem is revisited, using matched asymptotic expansions. In the case of large oscillation periods, approximate analytical expressions for the period and the orbit of the third body are found. The results are compared with those described in the literature and show that the movement of the third body can be well described by two analytical solutions, the inner and outer solutions.

Fluid Flow and Solute Transport though a Fracture Intersecting a Canister - Analytical Solutions for the Parallel Plate Model

2003 ◽  
Vol 807 ◽  
Author(s):  
L. Liu ◽  
I. Neretnieks
Keyword(s):  
Fluid Flow ◽  
Solute Transport ◽  
Flow Rate ◽  
Analytical Solutions ◽  
Parallel Plate ◽  
Spent Nuclear Fuel ◽  
Volumetric Flow Rate ◽  
Single Fracture ◽  
Plate Model ◽  
Specific Scenario

ABSTRACTIn this paper, we are concerned with a specific scenario where a large fracture intersects, at its center, a canister that contains spent nuclear fuel. Assuming that a nuclide is free to release from the canister into groundwater flowing through the fracture, a detailed formulation of the volumetric flow rate and the equivalent flow rate are made for the parallel plate model. The formulas proposed have been validated by numerical examinations; they are not only simple in forms but also universal in applications where the flow may be taken normal, inclined or parallel to the axis of the canister. Of great importance, they provide a convenient way to predict the average properties of fluid flow and solute transport through a single fracture with spatially variable apertures.

scholarly journals Mixed thermal convection: fundamental issues and analysis of the planar case

2015 ◽  
Vol 87 (3) ◽  
pp. 1865-1885 ◽  
Author(s):  
JACQUES PADET ◽  
RENATO M. COTTA ◽  
EMILIA C. MLADIN ◽  
COLETTE PADET
Keyword(s):  
Free Convection ◽  
Mixed Convection ◽  
Forced Convection ◽  
Analytical Solutions ◽  
Thermal Convection ◽  
Boussinesq Approximation ◽  
Reference Temperature ◽  
Clear Definition ◽  
Planar Case ◽  
Definition Of

This paper aims to renew interest on mixed thermal convection research and to emphasize three issues that arise from the present analysis: (i) a clear definition of the reference temperature in the Boussinesq approximation; (ii) a practical delimitation of the three convective modes, which are the forced convection (FC), mixed convection (MC) and natural (or free) convection (NC); (iii) and, finally, a uniform description of the set FC/MC/NC in the similarity framework. The planar case, for which analytical solutions are available, allows a detailed illustration of the answers here advanced to the above issues.

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