scholarly journals Impacts of Updraft Size and Dimensionality on the Perturbation Pressure and Vertical Velocity in Cumulus Convection. Part II: Comparison of Theoretical and Numerical Solutions and Fully Dynamical Simulations

2016 ◽  
Vol 73 (4) ◽  
pp. 1455-1480 ◽  
Author(s):  
Hugh Morrison
Keyword(s):  
Vertical Velocity ◽  
Numerical Solutions ◽  
Pressure Effects ◽  
Cumulus Convection ◽  
Moist Convection ◽  
Gray Zone ◽  
Thermal Buoyancy ◽  
Wide Range ◽  
Perturbation Pressure ◽  
2D And 3D

Abstract This paper compares simple theoretical expressions relating vertical velocity, perturbation pressure, updraft size, and dimensionality for cumulus convection, derived in Part I, with numerical solutions of the anelastic buoyant perturbation pressure Poisson equation and vertical velocity w. A range of thermal buoyancy profiles representing shallow to deep moist convection are tested for both two-dimensional (2D) and three-dimensional (3D) updrafts. The theoretical expressions give similar results for w and perturbation pressure difference from updraft top to base Δp compared to the numerical solutions over a wide range of updraft radius R. The theoretical expressions are also consistent with 2D and 3D fully dynamical updraft simulations initiated by warm bubbles of varying width. Implications for nonhydrostatic modeling in the “gray zone,” with a horizontal grid spacing Δx of O(1–10) km where convection is generally underresolved, are discussed. The theoretical and numerical solutions give a scaling of updraft velocity with R (~Δx) consistent with fully dynamical 2D and 3D simulations in the gray zone, with a rapid decrease of maximum w at relatively small R and a slower decrease at large R. These results suggest that an incorrect representation of perturbation pressure may be an important contributor to biases in convective strength at these resolutions. The theoretical solutions also provide a concise physical interpretation of the “virtual mass” coefficient in convection parameterizations and can be easily incorporated into these schemes to provide a consistent scaling of perturbation pressure effects with R, updraft height, and the buoyancy profile.

scholarly journals Impacts of Updraft Size and Dimensionality on the Perturbation Pressure and Vertical Velocity in Cumulus Convection. Part I: Simple, Generalized Analytic Solutions

2016 ◽  
Vol 73 (4) ◽  
pp. 1441-1454 ◽  
Author(s):  
Hugh Morrison
Keyword(s):  
Poisson Equation ◽  
Vertical Velocity ◽  
Analytic Solutions ◽  
Direct Result ◽  
Cumulus Convection ◽  
Pressure Poisson Equation ◽  
Thermal Buoyancy ◽  
Wide Range ◽  
Perturbation Pressure ◽  
2D And 3D

Abstract This study investigates relationships between vertical velocity, perturbation pressure, updraft size, and dimensionality for cumulus convection. Generalized theoretical expressions are derived from approximate analytic solutions of the governing momentum and mass continuity equations for both two-dimensional (2D) and axisymmetric quasi-three-dimensional (3D) steady-state updrafts. These expressions relate perturbation pressure and vertical velocity to updraft radius R, height H, and thermal buoyancy. They suggest that the vertical velocity at the level of neutral buoyancy is reduced from perturbation pressure effects by factors of and in 2D and 3D, respectively, where is a nondimensional length, with somewhat different scalings lower in the updraft (α is a parameter equal to the ratio of vertical velocity horizontally averaged across the updraft to that at the updraft center). They also indicate that updrafts are weaker in 2D than 3D, all else being equal, with a difference of up to a factor of 2 in vertical velocity for as a direct result of differences in mass continuity between 2D and axisymmetric 3D flow. Differences between these expressions and other analytic solutions, including those derived from single normal mode Fourier/Fourier–Bessel expansion of the buoyant perturbation pressure Poisson equation, are discussed. Part II of this study compares the theoretical expressions with numerical solutions of the buoyant perturbation pressure Poisson equation for a wide range of thermal buoyancy profiles representing shallow-to-deep moist convection and also with fully dynamical 2D and 3D updraft simulations.

Internal-inertia waves in a fluid of variable depth

1973 ◽  
Vol 73 (1) ◽  
pp. 205-213 ◽  
Author(s):  
W. D. McKee
Keyword(s):  
Exact Solutions ◽  
Surface Waves ◽  
Internal Waves ◽  
Vertical Velocity ◽  
General Problem ◽  
Three Dimensional ◽  
Variable Depth ◽  
Internal Inertia ◽  
Perturbation Pressure ◽  
Rotating Stratified Fluid

AbstractWaves in a rotating, stratified fluid of variable depth are considered. The perturbation pressure is used throughout as the dependent variable. This proves to have some advantages over the use of the vertical velocity. Some previous three-dimensional solutions for internal waves in a wedge are shown to be incorrect and the correct solutions presented. A WKB analysis is then performed for the general problem and the results compared with the exact solutions for a wedge. The WKB solution is also applied to long surface waves on a rotating ocean.

Convective and turbulent motions in non-precipitating Cu. Part 1: Method of separation of convective and turbulent motions

2021 ◽  
Author(s):  
Mark Pinsky ◽  
Eshkol Eytan ◽  
Ilan Koren ◽  
Orit Altaratz ◽  
Alexander Khain
Keyword(s):  
Velocity Field ◽  
Vertical Velocity ◽  
Isotropic Turbulence ◽  
Convective Velocity ◽  
Cloud Dynamics ◽  
Wide Range ◽  
Homogeneous And Isotropic Turbulence ◽  
Microphysical Processes ◽  
Bin Microphysics ◽  
Parameter Values

AbstractAtmospheric motions in clouds and cloud surrounding have a wide range of scales, from several kilometers to centimeters. These motions have different impacts on cloud dynamics and microphysics. Larger-scale motions (hereafter referred to as convective motions) are responsible for mass transport over distances comparable with cloud scale, while motions of smaller scales (hereafter referred to as turbulent motions) are stochastic and responsible for mixing and cloud dilution. This distinction substantially simplifies the analysis of dynamic and microphysical processes in clouds. The present research is Part 1 of the study aimed at describing the method for separating the motion scale into a convective component and a turbulent component. An idealized flow is constructed, which is a sum of an initially prescribed field of the convective velocity with updrafts in the cloud core and downdrafts outside the core, and a stochastic turbulent velocity field obeying the turbulent properties, including the -5/3 law and the 2/3 structure function law. A wavelet method is developed allowing separation of the velocity field into the convective and turbulent components, with parameter values being in a good agreement with those prescribed initially. The efficiency of the method is demonstrated by an example of a vertical velocity field of a cumulus cloud simulated using SAM with bin-microphysics and resolution of 10 m. It is shown that vertical velocity in clouds indeed can be represented as a sum of convective velocity (forming zone of cloud updrafts and subsiding shell) and a stochastic velocity obeying laws of homogeneous and isotropic turbulence.

NUMERICAL SOLUTIONS OF 2D AND 3D SLAMMING PROBLEMS

2021 ◽  
Vol 153 (A2) ◽  
Author(s):  
Q Yang ◽  
W Qiu
Keyword(s):  
Free Surface ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Type Equation ◽  
The Body ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Highly Nonlinear ◽  
2D And 3D

Slamming forces on 2D and 3D bodies have been computed based on a CIP method. The highly nonlinear water entry problem governed by the Navier-Stokes equations was solved by a CIP based finite difference method on a fixed Cartesian grid. In the computation, a compact upwind scheme was employed for the advection calculations and a pressure-based algorithm was applied to treat the multiple phases. The free surface and the body boundaries were captured using density functions. For the pressure calculation, a Poisson-type equation was solved at each time step by the conjugate gradient iterative method. Validation studies were carried out for 2D wedges with various deadrise angles ranging from 0 to 60 degrees at constant vertical velocity. In the cases of wedges with small deadrise angles, the compressibility of air between the bottom of the wedge and the free surface was modelled. Studies were also extended to 3D bodies, such as a sphere, a cylinder and a catamaran, entering calm water. Computed pressures, free surface elevations and hydrodynamic forces were compared with experimental data and the numerical solutions by other methods.

Buoyancy effects on large-scale motions in convective atmospheric boundary layers: implications for modulation of near-wall processes

2018 ◽  
Vol 856 ◽  
pp. 135-168 ◽  
Author(s):  
S. T. Salesky ◽  
W. Anderson
Keyword(s):  
Boundary Layer ◽  
Velocity Field ◽  
Boundary Layers ◽  
Vertical Velocity ◽  
Amplitude Modulation ◽  
Large Scale ◽  
Streamwise Velocity ◽  
Small Scale ◽  
Convective Conditions ◽  
Wide Range

A number of recent studies have demonstrated the existence of so-called large- and very-large-scale motions (LSM, VLSM) that occur in the logarithmic region of inertia-dominated wall-bounded turbulent flows. These regions exhibit significant streamwise coherence, and have been shown to modulate the amplitude and frequency of small-scale inner-layer fluctuations in smooth-wall turbulent boundary layers. In contrast, the extent to which analogous modulation occurs in inertia-dominated flows subjected to convective thermal stratification (low Richardson number) and Coriolis forcing (low Rossby number), has not been considered. And yet, these parameter values encompass a wide range of important environmental flows. In this article, we present evidence of amplitude modulation (AM) phenomena in the unstably stratified (i.e. convective) atmospheric boundary layer, and link changes in AM to changes in the topology of coherent structures with increasing instability. We perform a suite of large eddy simulations spanning weakly ($-z_{i}/L=3.1$) to highly convective ($-z_{i}/L=1082$) conditions (where$-z_{i}/L$is the bulk stability parameter formed from the boundary-layer depth$z_{i}$and the Obukhov length $L$) to investigate how AM is affected by buoyancy. Results demonstrate that as unstable stratification increases, the inclination angle of surface layer structures (as determined from the two-point correlation of streamwise velocity) increases from$\unicode[STIX]{x1D6FE}\approx 15^{\circ }$for weakly convective conditions to nearly vertical for highly convective conditions. As$-z_{i}/L$increases, LSMs in the streamwise velocity field transition from long, linear updrafts (or horizontal convective rolls) to open cellular patterns, analogous to turbulent Rayleigh–Bénard convection. These changes in the instantaneous velocity field are accompanied by a shift in the outer peak in the streamwise and vertical velocity spectra to smaller dimensionless wavelengths until the energy is concentrated at a single peak. The decoupling procedure proposed by Mathiset al.(J. Fluid Mech., vol. 628, 2009a, pp. 311–337) is used to investigate the extent to which amplitude modulation of small-scale turbulence occurs due to large-scale streamwise and vertical velocity fluctuations. As the spatial attributes of flow structures change from streamwise to vertically dominated, modulation by the large-scale streamwise velocity decreases monotonically. However, the modulating influence of the large-scale vertical velocity remains significant across the stability range considered. We report, finally, that amplitude modulation correlations are insensitive to the computational mesh resolution for flows forced by shear, buoyancy and Coriolis accelerations.

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.

scholarly journals RELIABLE ITERATIVE METHODS FOR SOLVING 1D, 2D AND 3D FISHER’S EQUATION

2021 ◽  
Vol 22 (1) ◽  
pp. 138-166
Author(s):  
Othman Mahdi Salih ◽  
Majeed AL-Jawary
Keyword(s):  
Iterative Methods ◽  
Numerical Solutions ◽  
Adomian Decomposition Method ◽  
Nonlinear Problems ◽  
Variational Iteration Method ◽  
Absolute Error ◽  
Fisher’S Equation ◽  
Adomian Decomposition ◽  
Fisher's Equation ◽  
2D And 3D

In the present paper, three reliable iterative methods are given and implemented to solve the 1D, 2D and 3D Fisher’s equation. Daftardar-Jafari method (DJM), Temimi-Ansari method (TAM) and Banach contraction method (BCM) are applied to get the exact and numerical solutions for Fisher's equations. The reliable iterative methods are characterized by many advantages, such as being free of derivatives, overcoming the difficulty arising when calculating the Adomian polynomial boundaries to deal with nonlinear terms in the Adomian decomposition method (ADM), does not request to calculate Lagrange multiplier as in the Variational iteration method (VIM) and there is no need to create a homotopy like in the Homotopy perturbation method (HPM), or any assumptions to deal with the nonlinear term. The obtained solutions are in recursive sequence forms which can be used to achieve the closed or approximate form of the solutions. Also, the fixed point theorem was presented to assess the convergence of the proposed methods. Several examples of 1D, 2D and 3D problems are solved either analytically or numerically, where the efficiency of the numerical solution has been verified by evaluating the absolute error and the maximum error remainder to show the accuracy and efficiency of the proposed methods. The results reveal that the proposed iterative methods are effective, reliable, time saver and applicable for solving the problems and can be proposed to solve other nonlinear problems. All the iterative process in this work implemented in MATHEMATICA®12. ABSTRAK: Kajian ini berkenaan tiga kaedah berulang boleh percaya diberikan dan dilaksanakan bagi menyelesaikan 1D, 2D dan 3D persamaan Fisher. Kaedah Daftardar-Jafari (DJM), kaedah Temimi-Ansari (TAM) dan kaedah pengecutan Banach (BCM) digunakan bagi mendapatkan penyelesaian numerik dan tepat bagi persamaan Fisher. Kaedah berulang boleh percaya di kategorikan dengan pelbagai faedah, seperti bebas daripada terbitan, mengatasi masalah-masalah yang timbul apabila sempadan polinomial bagi mengurus kata tak linear dalam kaedah penguraian Adomian (ADM), tidak memerlukan kiraan pekali Lagrange sebagai kaedah berulang Variasi (VIM) dan tidak perlu bagi membuat homotopi sebagaimana dalam kaedah gangguan Homotopi (HPM), atau mana-mana anggapan bagi mengurus kata tak linear. Penyelesaian yang didapati dalam bentuk urutan berulang di mana ianya boleh digunakan bagi mencapai penyelesaian tepat atau hampiran. Juga, teorem titik tetap dibentangkan bagi menaksir kaedah bentuk hampiran. Pelbagai contoh seperti masalah 1D, 2D dan 3D diselesaikan samada secara analitik atau numerik, di mana kecekapan penyelesaian numerik telah ditentu sahkan dengan menilai ralat mutlak dan baki ralat maksimum (MER) bagi menentukan ketepatan dan kecekapan kaedah yang dicadangkan. Dapatan kajian menunjukkan kaedah berulang yang dicadangkan adalah berkesan, boleh percaya, jimat masa dan boleh guna bagi menyelesaikan masalah dan boleh dicadangkan menyelesaikan masalah tak linear lain. Semua proses berulang dalam kerja ini menggunakan MATHEMATICA®12.

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^

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