scholarly journals Optimal convection cooling flows in general 2D geometries

2017 ◽  
Vol 814 ◽  
pp. 484-509 ◽  
Author(s):  
S. Alben
Keyword(s):  
Numerical Solutions ◽  
Horizontal Layer ◽  
Rectangular Domain ◽  
Fixed Rate ◽  
Cold Surface ◽  
Cooling Flows ◽  
Wide Range ◽  
Convection Cooling ◽  
Hot Surface ◽  
Conformal Coordinates

We generalize a recent method for computing optimal 2D convection cooling flows in a horizontal layer to a wide range of geometries, including those relevant for technological applications. We write the problem in a conformal pair of coordinates which are the pure conduction temperature and its harmonic conjugate. We find optimal flows for cooling a cylinder in an annular domain, a hot plate embedded in a cold surface, and a channel with a hot interior and cold exterior. With a constraint of fixed kinetic energy, the optimal flows are all essentially the same in the conformal coordinates. In the physical coordinates, they consist of vortices ranging in size from the length of the hot surface to a small cutoff length at the interface of the hot and cold surfaces. With the constraint of fixed enstrophy (or fixed rate of viscous dissipation), a geometry-dependent metric factor appears in the equations. The conformal coordinates are useful here because they map the problems to a rectangular domain, facilitating numerical solutions. With a small enstrophy budget, the optimal flows are dominated by vortices that have the same size as the flow domain.

scholarly journals Conduction of heat through rarefied gases.—II

1911 ◽  
Vol 84 (574) ◽  
pp. 576-585 ◽  
Keyword(s):  
Thermal Conductivity ◽  
Kinetic Theory ◽  
The Other ◽  
Cold Surface ◽  
Surrounding Water ◽  
Original Apparatus ◽  
Wide Range ◽  
Palladium Surface ◽  
Hot Surface ◽  
Different Gases

In a previous paper* measurements were described of the thermal conductivity of twelve different gases in what may he termed a “free-path vacuum,” that is, at pressures so low that the molecules conducting the heat from the hot to the cold surface do not, as a rule, experience mutual encounters. It was shown ,that at sufficiently low pressure the conductivity of all the gases was proportional to the pressure, and the conductivity K (defined as the calories dissipated per second, per 0·01 mm. pressure of gas per square centimetre of hot surface, per 1° difference of temperature between the latter and its surroundings) was compared with the theoretical conductivity Q, as calculated approximately from the molecular heat and mean molecular velocity of the gas, by means of the kinetic theory, on the assumption that the heat interchange between the molecule and the surface it impinges upon was perfect. For argon and neon the ratio K/Q (1·09 and 1·04 respectively) was in gratifying agreement with this assumption, but for all the other gases the ratio was less than unity. For these the ratio was greater than 0·66, except for helium and hydrogen, for which the very low values 0·5 and 0·25 respectively were found. The suggestion was hazarded that possibly for these light gases the interchange of heat is imperfect, owing to the greater velocities at which the molecules move. In the present paper some of the earlier measurements have been repeated with the original apparatus, with better provision for keeping the tempera­ture of the surrounding water jacket uniform. Then the apparatus was rebuilt to allow of the measurements of the conductivity to be taken over a wide range of temperature, both of the hot and cold surface, and the effect of variation of the temperature on the ratio K/Q was examined. Measure­ments were confined to the three gases—hydrogen, helium, and argon Incidentally the effect on the conductivity of hydrogen of using a hot palladium surface instead of one of platinum was examined. The result of these new experiments has been to negative the suggestion already referred to, that the discrepancies between the found and calculated conductivities might be due to imperfect interchange of energy on impact. The ratio K/Q appears to diminish as the temperature at which the experiments are performed decreases and to increase as the temperature is increased, and no explanation of this can at present be suggested.

Multigrid Numerical Solutions of Non-Isothermal Laminar Recirculating Flows

1999 ◽  
Author(s):  
Marcelo J. S. de Lemos ◽  
Maximilian S. Mesquita
Keyword(s):  
Numerical Solutions ◽  
Rectangular Domain ◽  
Optimal Number ◽  
Computational Effort ◽  
Temperature Fields ◽  
Two Dimensional ◽  
Discretization Scheme ◽  
Multigrid Algorithm ◽  
Finite Volume Discretization ◽  
Velocity And Temperature Fields

Abstract The present work investigates the efficiency of the multigrid numerical method applied to solve two-dimensional laminar velocity and temperature fields inside a rectangular domain. Numerical analysis is based on the finite volume discretization scheme applied to structured orthogonal regular meshes. Performance of the correction storage (CS) multigrid algorithm is compared for different inlet Reynolds number (Rein) and number of grids. Up to four grids were used for both V- and W-cycles. Simultaneous and uncoupled temperature-velocity solution schemes were also applied. Advantages in using more than one grid is discussed. Results further indicate an increase in the computational effort for higher Rein and an optimal number of relaxation sweeps for both V- and W-cycles.

scholarly journals Derivation of Global Parametric Performance of Mixed Flow Hydraulic Turbine Using CFD

1970 ◽  
Vol 7 ◽  
pp. 60-64 ◽  
Author(s):  
Ruchi Khare ◽  
Vishnu Prasad Prasad ◽  
Sushil Kumar
Keyword(s):  
Fluid Dynamics ◽  
Computational Fluid Dynamics ◽  
Numerical Solutions ◽  
Guide Vane ◽  
Hydraulic Turbine ◽  
Francis Turbine ◽  
Mixed Flow ◽  
Flow Equations ◽  
Laboratory Setup ◽  
Wide Range

The testing of physical turbine models is costly, time consuming and subject to limitations of laboratory setup to meet International Electro technical Commission (IEC) standards. Computational fluid dynamics (CFD) has emerged as a powerful tool for funding numerical solutions of wide range of flow equations whose analytical solutions are not feasible. CFD also minimizes the requirement of model testing. The present work deals with simulation of 3D flow in mixed flow (Francis) turbine passage; i.e., stay vane, guide vane, runner and draft tube using ANSYS CFX 10 software for study of flow pattern within turbine space and computation of various losses and efficiency at different operating regimes. The computed values and variation of performance parameters are found to bear close comparison with experimental results.Key words: Hydraulic turbine; Performance; Computational fluid dynamics; Efficiency; LossesDOI: 10.3126/hn.v7i0.4239Hydro Nepal Journal of Water, Energy and EnvironmentVol. 7, July, 2010Page: 60-64Uploaded date: 31 January, 2011

scholarly journals Homotopy Perturbation Method

2017 ◽  
pp. 24-61
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Perturbation Method ◽  
Exact Solutions ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Homotopy Perturbation ◽  
Wide Range ◽  
Transfer Problems

The homotopy perturbation method (HPM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing on the problem. It has been attempted to show the capabilities and wide-range applications of the homotopy perturbation method in comparison with the previous ones in solving heat transfer problems. The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations.

An Approximate Method for Transient Radial Flow

1962 ◽  
Vol 2 (03) ◽  
pp. 225-256 ◽  
Author(s):  
G. Rowan ◽  
M.W. Clegg
Keyword(s):  
Infinite Series ◽  
Analytical Solutions ◽  
Radial Flow ◽  
Numerical Solutions ◽  
Basic Theory ◽  
Variable Pressure ◽  
Flow Problems ◽  
Disturbed Zone ◽  
Approximate Analytical Solutions ◽  
Wide Range

Abstract The basic equations for the flow of gases, compressible liquids and incompressible liquids are derived and the full implications of linearising then discussed. Approximate solutions of these equations are obtained by introducing the concept of a disturbed zone around the well, which expands outwards into the reservoir as fluid is produced. Many important and well-established results are deduced in terms of simple functions rather than the infinite series, or numerical solutions normally associated with these problems. The wide range of application of this approach to transient radial flow problems is illustrated with many examples including; gravity drainage of depletion-type reservoirs; multiple well systems; well interference. Introduction A large number of problems concerning the flow of fluids in oil reservoirs have been solved by both analytical and numerical methods but in almost all cases these solutions have some disadvantages - the analytical ones usually involve rather complex functions (infinite series or infinite integrals) which are difficult to handle, and the numerical ones tend to mask the physical principles underlying the problem. It would seem appropriate, therefore, to try to find approximate analytical solutions to these problems without introducing any further appreciable errors, so that the physical nature of the problem is retained and solutions of comparable accuracy are obtained. One class of problems will be considered in this paper, namely, transient radial flow problems, and it will be shown that approximate analytical solutions of the equations governing radial flow can be obtained, and that these solutions yield comparable results to those calculated numerically and those obtained from "exact" solutions. It will also be shown that the restrictions imposed upon the dependent variable (pressure) are just those which have to be assumed in deriving the usual diffusion-type equations. The method was originally suggested by Guseinov, whopostulated a disturbed zone in the reservoir, the radius of which increases with time, andreplaced the time derivatives in the basic differential equation by its mean value in the disturbed zone. In this paper it is proposed to review the basic theory leading to the equations governing the flow of homogeneous fluids in porous media and to consider the full implications of the approximation introduced in linearising them. The Guseinov-type approximation will then be applied to these equations and the solutions for the flow of compressible and incompressible fluids, and gases in bounded and infinite reservoirs obtained. As an example of the application of this type of approximation, solutions to such problems as production from stratified reservoirs, radial permeability discontinuities; multiple-well systems, and well interference will be given. These solutions agree with many other published results, and in some cases they may be extended to more complex problems without the computational difficulties experienced by other authors. THEORY In order to review the basic theory from a fairly general standpoint it is proposed to limit the idealising assumptions to the minimum necessary for analytical convenience. The assumptions to be made are the following:That the flow is irrotational.That the formation is of constant thickness.Darcy's Law is valid.The formation is saturated with a single homogeneous fluid. SPEJ P. 225^

NUMERICAL SOLUTIONS OF SMOLUCHOWSKI'S EQUATIONS FOR THE COAGULATION OF UNCHARGED AEROSOLS

1965 ◽  
Vol 43 (8) ◽  
pp. 2312-2318 ◽  
Author(s):  
J. M. Beeckmans
Keyword(s):  
Particle Size ◽  
Numerical Solutions ◽  
Coagulation Factor ◽  
Dispersion Parameter ◽  
Particle Size Range ◽  
Mean Particle Size ◽  
A Value ◽  
Wide Range ◽  
Differential Sedimentation ◽  
Smoluchowski’S Equation

Smoluchowski's equations for the coagulation of uncharged aerosol particles were programmed for solution by electronic computer. Terms representing differential sedimentation, turbulence, and mean aggregate density in solid aerosols were included. The effect of heterogeneity in the particle-size distribution of the aerosols on their rate of coagulation was illustrated by means of a slip-corrected coagulation factor Fc, which assumes a value of unity in all non-turbulent homogeneous aerosols. Curves of Fc vs. σg, the geometrical standard deviation, were calculated for aerosols of various mean particle-size. The effects due to turbulence, and to differential sedimentation, were illustrated in a similar manner. It was also found that the process of coagulation gives rise to a degree of dispersion which is independent of the original dispersion parameter, and depends only slightly on the mean particle-size of the aerosol over a wide range of particle-sizes. In the particle-size range in which differential sedimentation is inappreciable, the relatively constant value of the dispersion parameter implies that heterogeneous aerosols must obey the simplified integrated form of Smoluchowski's equation, which is applicable to homogeneous aerosols. The coagulation constant exceeds that predicted by the simple theory by about 10% for liquid aerosols of 0.1 μ or less.

Application of minimal mathematical models for the dynamics of the signaling pathway of the p53 - miRNA to the analysis of laboratory data

2021 ◽  
pp. 4-49
Author(s):  
Софья Дмитриевна Сенотрусова ◽  
Ольга Фалалеевна Воропаева ◽  
Юрий Иванович Шокин
Keyword(s):  
Mathematical Models ◽  
Signaling Pathway ◽  
Numerical Solutions ◽  
Laboratory Data ◽  
Minimal Models ◽  
Basic Model ◽  
P53 Signaling ◽  
Wide Range ◽  
P53 Signaling Pathway

Работа посвящена практическому использованию минимальных математических моделей динамики сигнального пути p53 для описания достаточно широкого круга лабораторных экспериментов, в которых взаимодействие p53 и белковингибиторов p53 опосредуется микроРНК, образующими с p53 петлю положительной обратной связи. Представлены базовая модель, разработанные на ее основе новые минимальные модели, алгоритм численного решения прямых и обратных коэффициентных задач и результаты сопоставления полученных численных решений с экспериментальными данными о динамике уровней белков p53, p21, Bax, белков-ингибиторов Mdm2, Wip1, Sirt1 и различных микроРНК (miR-16, miR-34a, miR-192, miR-194, miR-215) в условиях стрессовых воздействий. С привлечением полученных математических моделей исследованы базовые механизмы функционирования сигнального пути p53 в условиях, приближенных к условиям конкретных лабораторных экспериментов in vitro и in vivo. Продемонстрированы синергический эффект гиперактивации сигнального пути p53, в котором задействованы микроРНК, и механизмы бимодального переключения. Показана ключевая роль p53-зависимых микроРНК в реализации некоторых гипотетических терапевтических стратегий, связанных с управлением механизмом активации апоптоза клеток. В рамках принятой базовой модели даны оценки вероятности рассогласования в диагностике дегенеративных заболеваний, основанной на анализе уровня p53зависимых микроРНК и p53, при слабой и умеренной дерегуляции микроРНК. This study addresses the practical use of minimal mathematical models of the dynamics of a hypothetical system of the p53 signaling pathway to describe a fairly wide range of laboratory experiments. In such system, the interaction of p53 and p53 inhibitor proteins is mediated by microRNAs that form a positive feedback loop with p53. A basic model, new minimal models developed on its basis, an algorithm for the numerical solution of direct and inverse coefficient problems, and the results of comparing the obtained numerical solutions with experimental data on the dynamics of the levels of p53, p21, Bax proteins, inhibitor proteins Mdm2, Wip1, Sirt1, and various microRNAs (miR-16, miR-34a, miR-192, miR-194, miR-215) under stress conditions are presented. In numerical experiments, the main mechanisms of the p53 signaling pathway were investigated. A synergistic effect of hyperactivation of the p53 signaling pathway and bimodal switching mechanisms has been demonstrated. We show the key role of p53-dependent microRNAs in the implementation of some hypothetical therapeutic strategies associated with the control mechanism for activation of cells apoptosis. Within the framework of the accepted basic model, we estimated the probability of mismatch in the diagnosis of the patient’s status. The status is based on the analysis of the level of p53-dependent microRNAs and p53, with weak and moderate deregulation of microRNAs.

scholarly journals Constitutive relations for compressible granular flow in the inertial regime

2019 ◽  
Vol 874 ◽  
pp. 926-951 ◽  
Author(s):  
D. G. Schaeffer ◽  
T. Barker ◽  
D. Tsuji ◽  
P. Gremaud ◽  
M. Shearer ◽  
...  
Keyword(s):  
Steady State ◽  
Granular Flow ◽  
Numerical Solutions ◽  
Volume Fraction ◽  
Constitutive Relations ◽  
Practical Interest ◽  
Time Dependent ◽  
Direct Result ◽  
Wide Range ◽  
Inertial Regime

Granular flows occur in a wide range of situations of practical interest to industry, in our natural environment and in our everyday lives. This paper focuses on granular flow in the so-called inertial regime, when the rheology is independent of the very large particle stiffness. Such flows have been modelled with the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$-rheology, which postulates that the bulk friction coefficient $\unicode[STIX]{x1D707}$ (i.e. the ratio of the shear stress to the pressure) and the solids volume fraction $\unicode[STIX]{x1D719}$ are functions of the inertial number $I$ only. Although the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$-rheology has been validated in steady state against both experiments and discrete particle simulations in several different geometries, it has recently been shown that this theory is mathematically ill-posed in time-dependent problems. As a direct result, computations using this rheology may blow up exponentially, with a growth rate that tends to infinity as the discretization length tends to zero, as explicitly demonstrated in this paper for the first time. Such catastrophic instability due to ill-posedness is a common issue when developing new mathematical models and implies that either some important physics is missing or the model has not been properly formulated. In this paper an alternative to the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$-rheology that does not suffer from such defects is proposed. In the framework of compressible $I$-dependent rheology (CIDR), new constitutive laws for the inertial regime are introduced; these match the well-established $\unicode[STIX]{x1D707}(I)$ and $\unicode[STIX]{x1D6F7}(I)$ relations in the steady-state limit and at the same time are well-posed for all deformations and all packing densities. Time-dependent numerical solutions of the resultant equations are performed to demonstrate that the new inertial CIDR model leads to numerical convergence towards physically realistic solutions that are supported by discrete element method simulations.

Numerical solutions and analysis of diffusion for new generalized fractional Burgers equation

2013 ◽  
Vol 16 (3) ◽  
Author(s):  
Yufeng Xu ◽  
Om Agrawal
Keyword(s):  
Finite Difference ◽  
Diffusion Process ◽  
Weight Function ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Weight Functions ◽  
Linear Recurrence ◽  
Scale Function ◽  
Special Cases ◽  
Wide Range

AbstractIn this paper, numerical solutions of Burgers equation defined by using a new Generalized Time-Fractional Derivative (GTFD) are discussed. The numerical scheme uses a finite difference method. The new GTFD is defined using a scale function and a weight function. Many existing fractional derivatives are the special cases of it. A linear recurrence relationship for the numerical solutions of the resulting system of linear equations is found via finite difference approach. Burgers equations with different fractional orders and coefficients are computed which show that this numerical method is simple and effective, and is capable of solving the Burgers equation accurately for a wide range of viscosity values. Furthermore, we study the influence of the scale and the weight functions on the diffusion process of Burgers equation. Numerical simulations illustrate that a scale function can stretch or contract the diffusion on the time domain, while a weight function can change the decay velocity of the diffusion process.

scholarly journals The Flow of a Glacier in a Channel of Rectangular, Elliptic or Parabolic Cross-Section

1965 ◽  
Vol 5 (41) ◽  
pp. 661-690 ◽  
Author(s):  
J. F. Nye
Keyword(s):  
Cross Section ◽  
Numerical Solutions ◽  
Maximum Shear Stress ◽  
Surface Velocity ◽  
Channel Section ◽  
Ice Surface ◽  
Centre Line ◽  
Symmetry Principles ◽  
Wide Range ◽  
Flow Law

AbstractNumerical solutions are found for the steady rectilinear flow of ice, obeying Glen’s non-linear flow law, down uniform cylindrical channels of rectangular, semi-elliptic and parabolic cross-section. The results are also directly applicable to the pumping of a non-Newtonian fluid down a pipe. There is assumed to be no slip of the ice on the channel surface. Certain results on the centre-line velocity in symmetrical channels may be derived purely from dimensional and symmetry principles. An analytical solution due to Dr. W. Chester is given for a semi-elliptic channel section which departs only slightly from a semi-circle. Contrary to a view sometimes held, the maximum shear stress at the ice surface in a parabolic channel and in some elliptical channels does not always occur at the edge. With the flow law, strain-rate proportional to (stress)3, the velocity averaged across the ice surface, which is easily measured with a line of stakes, is close to the average velocity over the whole section for a wide range of parabolic sections; the hydrological importance of this result is that the discharge may be inferred without the need to measure the velocity at depth. Arguments are given to show that the result still holds when there is slipping on the bed and when the power in the flow law differs somewhat from 3, Depending on the amount of bed slip and the shape of the channel section, the kinematic wave velocity for a range of parabolic channels is between 2.0 and 2.3 times the centre-line velocity of the ice, and between 2.0 and 3.5 times the mean surface velocity of the ice.

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