scholarly journals A Robust Numerical Solution of the Stochastic Collection–Breakup Equation for Warm Rain

2007 ◽  
Vol 46 (9) ◽  
pp. 1480-1497 ◽  
Author(s):  
Olivier P. Prat ◽  
Ana P. Barros
Keyword(s):  
Numerical Solution ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Strong Dependence ◽  
Discrete Equations ◽  
Warm Rain ◽  
Wide Range ◽  
Short Time ◽  
Rain Rates ◽  
The Impact

Abstract The focus of this paper is on the numerical solution of the stochastic collection equation–stochastic breakup equation (SCE–SBE) describing the evolution of raindrop spectra in warm rain. The drop size distribution (DSD) is discretized using the fixed-pivot scheme proposed by Kumar and Ramkrishna, and new discrete equations for solving collision breakup are presented. The model is evaluated using established coalescence and breakup parameterizations (kernels) available in the literature, and in that regard this paper provides a substantial review of the relevant science. The challenges posed by the need to achieve stable and accurate numerical solutions of the SCE–SBE are examined in detail. In particular, this paper focuses on the impact of varying the shape of the initial DSD on the equilibrium solution of the SCE–SBE for a wide range of rain rates and breakup kernels. The results show that, although there is no dependence of the equilibrium DSD on initial conditions for the same rain rate and breakup kernel, there is large variation in the time that it takes to reach steady state. This result suggests that, in coupled simulations of in-cloud motions and microphysics and for short time scales (<30 min) for which transient conditions prevail, the equilibrium DSD may not be attainable except for very heavy rainfall. Furthermore, simulations for the same initial conditions show a strong dependence of the dynamic evolution of the DSD on the breakup parameterization. The implication of this result is that, before the debate on the uniqueness of the shape of the equilibrium DSD can be settled, there is critical need for fundamental research including laboratory experiments to improve understanding of collisional mechanisms in DSD evolution.

Progressive Mesh Densification Method for Numerical Solution of Mixed Elastohydrodynamic Lubrication

2015 ◽  
Vol 138 (2) ◽  
Author(s):  
Wei Pu ◽  
Jiaxu Wang ◽  
Dong Zhu
Keyword(s):  
Surface Roughness ◽  
Numerical Solution ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Solution Process ◽  
Elastohydrodynamic Lubrication ◽  
Operating Conditions ◽  
Machined Surface ◽  
Progressive Mesh ◽  
Wide Range

Numerical solution of mixed elastohydrodynamic lubrication (EHL) is of great importance for the study of lubrication formation and breakdown, as well as surface failures of mechanical components. However, converged and accurate numerical solutions become more difficult, and solution process with a fixed single discretization mesh for the solution domain appears to be quite slow, especially when the lubricant films and surface contacts coexist with real-machined roughness involved. Also, the effect of computational mesh density is found to be more significant if the average film thickness is small. In the present study, a set of sample cases with and without machined surface roughness are analyzed through the progressive mesh densification (PMD) method, and the obtained results are compared with those from the direct iteration method with a single fixed mesh. Besides, more numerical analyses with and without surface roughness in a wide range of operating conditions are conducted to investigate the influence of different compound modes in order to optimize the PMD procedure. In addition, different initial conditions are used to study the effect of initial value on the behaviors of this transient solution. It is observed that, no matter with or without surface roughness considered, the PMD method is stable for transient mixed EHL problems and capable of significantly accelerating the EHL solution process while ensuring numerical accuracy.

scholarly journals The Impact of Initial Conditions in N-Body Simulations of Debris Discs

2015 ◽  
Vol 32 ◽  
Author(s):  
E. Thilliez ◽  
S. T. Maddison
Keyword(s):  
Numerical Simulations ◽  
Initial Conditions ◽  
Parent Body ◽  
Dust Trapping ◽  
Physical Context ◽  
Disc Width ◽  
Wide Range ◽  
Belt Width ◽  
New Generation ◽  
The Impact

AbstractNumerical simulations are a crucial tool to understand the relationship between debris discs and planetary companions. As debris disc observations are now reaching unprecedented levels of precision over a wide range of wavelengths, an appropriate level of accuracy and consistency is required in numerical simulations to confidently interpret this new generation of observations. However, simulations throughout the literature have been conducted with various initial conditions often with little or no justification. In this paper, we aim to study the dependence on the initial conditions of N-body simulations modelling the interaction between a massive and eccentric planet on an exterior debris disc. To achieve this, we first classify three broad approaches used in the literature and provide some physical context for when each category should be used. We then run a series of N-body simulations, that include radiation forces acting on small grains, with varying initial conditions across the three categories. We test the influence of the initial parent body belt width, eccentricity, and alignment with the planet on the resulting debris disc structure and compare the final peak emission location, disc width and offset of synthetic disc images produced with a radiative transfer code. We also track the evolution of the forced eccentricity of the dust grains induced by the planet, as well as resonance dust trapping. We find that an initially broad parent body belt always results in a broader debris disc than an initially narrow parent body belt. While simulations with a parent body belt with low initial eccentricity (e ~ 0) and high initial eccentricity (0 < e < 0.3) resulted in similar broad discs, we find that purely secular forced initial conditions, where the initial disc eccentricity is set to the forced value and the disc is aligned with the planet, always result in a narrower disc. We conclude that broad debris discs can be modelled by using either a dynamically cold or dynamically warm parent belt, while in contrast eccentric narrow debris rings are reproduced using a secularly forced parent body belt.

New Travelling Wave Solutions of Burgers Equation with Finite Transport Memory

2010 ◽  
Vol 65 (8-9) ◽  
pp. 633-640 ◽  
Author(s):  
Rathinasamy Sakthivel ◽  
Changbum Chun ◽  
Jonu Lee
Keyword(s):  
Evolution Equations ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Diffusion Processes ◽  
Initial Conditions ◽  
Travelling Wave ◽  
Nonlinear Evolution Equations ◽  
Wave Solutions ◽  
Wide Range ◽  
Finite Memory

The nonlinear evolution equations with finite memory have a wide range of applications in science and engineering. The Burgers equation with finite memory transport (time-delayed) describes convection-diffusion processes. In this paper, we establish the new solitary wave solutions for the time-delayed Burgers equation. The extended tanh method and the exp-function method have been employed to reveal these new solutions. Further, we have calculated the numerical solutions of the time-delayed Burgers equation with initial conditions by using the homotopy perturbation method (HPM). Our results show that the extended tanh and exp-function methods are very effective in finding exact solutions of the considered problem and HPM is very powerful in finding numerical solutions with good accuracy for nonlinear partial differential equations without any need of transformation or perturbation

scholarly journals The Status of Multi-Dimensional Core-Collapse Supernova Models

2016 ◽  
Vol 33 ◽  
Author(s):  
B. Müller
Keyword(s):  
Initial Conditions ◽  
Massive Stars ◽  
Core Collapse ◽  
Solar Mass ◽  
Convective Boundary ◽  
3D Simulations ◽  
Core Collapse Supernova ◽  
Wide Range ◽  
Supernova Explosions ◽  
The Impact

AbstractModels of neutrino-driven core-collapse supernova explosions have matured considerably in recent years. Explosions of low-mass progenitors can routinely be simulated in 1D, 2D, and 3D. Nucleosynthesis calculations indicate that these supernovae could be contributors of some lighter neutron-rich elements beyond iron. The explosion mechanism of more massive stars remains under investigation, although first 3D models of neutrino-driven explosions employing multi-group neutrino transport have become available. Together with earlier 2D models and more simplified 3D simulations, these have elucidated the interplay between neutrino heating and hydrodynamic instabilities in the post-shock region that is essential for shock revival. However, some physical ingredients may still need to be added/improved before simulations can robustly explain supernova explosions over a wide range of progenitors. Solutions recently suggested in the literature include uncertainties in the neutrino rates, rotation, and seed perturbations from convective shell burning. We review the implications of 3D simulations of shell burning in supernova progenitors for the ‘perturbations-aided neutrino-driven mechanism,’ whose efficacy is illustrated by the first successful multi-group neutrino hydrodynamics simulation of an 18 solar mass progenitor with 3D initial conditions. We conclude with speculations about the impact of 3D effects on the structure of massive stars through convective boundary mixing.

scholarly journals Mathematical modeling of temperature changes impact on artificial ice islands

2021 ◽  
Vol 13 (1) ◽  
pp. 79-86
Author(s):  
Maxim V. Muratov ◽  
◽  
Vladimir A. Biryukov ◽  
Denis S. Konov ◽  
Igor B. Petrov ◽  
...  
Keyword(s):  
Numerical Solution ◽  
Stefan Problem ◽  
Initial Conditions ◽  
Economic Feasibility ◽  
The Arctic ◽  
Seasonal Temperature ◽  
Time Step ◽  
Temperature Changes ◽  
The Stability ◽  
The Impact

The article is devoted to the numerical solution of the Stefan problem for thermal effects on an artificial ice island. For modern tasks of the development of the Arctic, associated with the exploration and production of minerals, it is important to create artificial ice islands in the Arctic shelf, due to the speed of their construction, economic feasibility and other factors. The most important task for the exploitation of such islands is their stability, including against melting. This paper discusses the issue of the stability of ice islands to melting. For this, the Stefan problem on the change in the phase state of matter is formulated. An enthalpy solution method is constructed, and the applicability of this method is considered. For the numerical solution, the Peasman-Reckford scheme is used, which is unconditionally spectrally stable in the two-dimensional case, which allows to freely choose the time step. In addition, the developed approach takes into account the flow of water and the flow of melted water, which is important in the task at hand. The developed computational algorithms are parallelized for use on modern multiprocessor computing systems An approach is implemented for modeling thermal processes in the thickness of an arbitrary mass of substances, taking into account arbitrary initial conditions, environmental conditions, tidal currents of water, and solar radiation. This approach was used to calculate the temperature distribution in the thickness of the ice island, as well as to study the impact of seasonal temperature changes on the stability of the island.

Laminar spread of a circular liquid jet impinging axially on a rotating disc

2019 ◽  
Vol 864 ◽  
pp. 449-489 ◽  
Author(s):  
B. Scheichl ◽  
A. Kluwick
Keyword(s):  
Numerical Solutions ◽  
Film Flow ◽  
Thin Liquid Film ◽  
Jet Impingement ◽  
Flow Regimes ◽  
Careful Analysis ◽  
Rotating Disc ◽  
Viscous Shear ◽  
Wide Range ◽  
The Impact

The steady laminar annular spread of a thin liquid film generated by a circular jet which impinges perpendicularly in direction of gravity on the centre of a rotating disc is examined both analytically and numerically. Matched asymptotic expansions of the flow quantities provide the proper means for studying the individual flow regimes arising due to the largeness of the Reynolds number formed with the radius of the jet, its slenderness and the relative magnitude of the centrifugal body force. This is measured by a suitably defined Rossby number, $Ro$. The careful analysis of jet impingement predicts a marked influence of gravity and surface tension on the film flow, considered in the spirit of a shallow-water approach, only through the vorticity imposed by the jet flow. Accordingly, associated downstream conditions are disregarded as the local Froude and Weber numbers are taken to be sufficiently large. Hence, the parabolic problem shaped from the governing equations in a rigorous manner describes the strongly supercritical spread of a developed viscous film past an infinite disc, essentially controlled by $Ro$. Its numerical solutions are discussed for a wide range of values of $Ro$. The different flow regimes reflecting varying effects of viscous shear and centrifugal force are elucidated systematically to clarify the surprising richness of flow phenomena. Special attention is paid to the cases $Ro\gg 1$ and $Ro\ll 1$. The latter, referring to relatively high disc spin, implies a delicate breakdown of the asymptotic flow structure, thus requiring a specific analytical and numerical treatment. Finally, the impact of gravity and capillarity and thus of the disc edge on the film flow is envisaged in brief.

scholarly journals PIECEWISE OPTIMAL FRACTIONAL REPRODUCING KERNEL SOLUTION AND CONVERGENCE ANALYSIS FOR THE ATANGANA–BALEANU–CAPUTO MODEL OF THE LIENARD’S EQUATION

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040007
Author(s):  
SHAHER MOMANI ◽  
OMAR ABU ARQUB ◽  
BANAN MAAYAH
Keyword(s):  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Reproducing Kernel Hilbert Space ◽  
Weakly Nonlinear ◽  
Analytical Technique ◽  
Wide Range ◽  
Fractional Models ◽  
Weakly Nonlinear Waves ◽  
Fractional Solution

In this paper, an attractive reliable analytical technique is implemented for constructing numerical solutions for the fractional Lienard’s model enclosed with suitable nonhomogeneous initial conditions, which are often designed to demonstrate the behavior of weakly nonlinear waves arising in the oscillating circuits. The fractional derivative is considered in the Atangana–Baleanu–Caputo sense. The proposed technique, namely, reproducing kernel Hilbert space method, optimizes numerical solutions bending on the Fourier approximation theorem to generate a required fractional solution with a rapidly convergent form. The influence, capacity, and feasibility of the presented approach are verified by testing some applications. The acquired results are numerically compared with the exact solutions in the case of nonfractional derivative, which show the superiority, compatibility, and applicability of the presented method to solve a wide range of nonlinear fractional models.

Similarity solutions and viscous gravity current adjustment times

2019 ◽  
Vol 874 ◽  
pp. 285-298 ◽  
Author(s):  
Thomasina V. Ball ◽  
Herbert E. Huppert
Keyword(s):  
Differential Equation ◽  
Similarity Solution ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Nonlinear Partial Differential Equations ◽  
Gravity Current ◽  
Similarity Solutions ◽  
Partial Differential ◽  
Initial Shape ◽  
Wide Range

A wide range of initial-value problems in fluid mechanics in particular, and in the physical sciences in general, are described by nonlinear partial differential equations. Recourse must often be made to numerical solutions, but a powerful, well-established technique is to solve the problem in terms of similarity variables. A disadvantage of the similarity solution is that it is almost always independent of any specific initial conditions, with the solution to the full differential equation approaching the similarity solution for times $t\gg t_{\ast }$, for some $t_{\ast }$. But what is $t_{\ast }$? In this paper we consider the situation of viscous gravity currents and obtain useful formulae for the time of approach, $\unicode[STIX]{x1D70F}(p)$, for a number of different initial shapes, where $p$ is the percentage disagreement between the radius of the current as determined by the full numerical solution of the governing partial differential equation and the similarity solution normalised by the similarity solution. We show that for any initial shape of volume $V,\unicode[STIX]{x1D70F}\propto 1/(\unicode[STIX]{x1D6FD}V^{1/3}\unicode[STIX]{x1D6FE}_{0}^{8/3}p)$ (as $p\downarrow 0$), where $\unicode[STIX]{x1D6FD}=g\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}/(3\unicode[STIX]{x1D707})$, with $g$ representing the acceleration due to gravity, $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$ the density difference between the gravity current and the ambient, $\unicode[STIX]{x1D707}$ the dynamic viscosity of the fluid that makes up the gravity current and $\unicode[STIX]{x1D6FE}_{0}$ the initial aspect ratio. This framework can used in many other situations, including where it is not an initial condition (in time) that is studied but one valid for specified values at a special spatial coordinate.

scholarly journals On the solution of asymptotic impact problems with significant localised indentation

2016 ◽  
Vol 231 (5) ◽  
pp. 807-822 ◽  
Author(s):  
Akuro Big-Alabo ◽  
Matthew P Cartmell ◽  
Philip Harrison
Keyword(s):  
Numerical Solutions ◽  
Nonlinear Models ◽  
Infinite Plate ◽  
Nonlinear Odes ◽  
Wide Range ◽  
Contact Models ◽  
Scope Of Application ◽  
Impact Problems ◽  
The Impact ◽  
Impact Events

A major challenge in studying impact problems analytically is solving the governing equations of impact events, which are mostly in the form of nonlinear ODEs. This paper focuses on the solution of nonlinear models for impact problems in asymptotic cases, where local indentation is significant. The asymptotic cases consist of both half-space and infinite plate impacts, which cover a wide range of practical impact events. A so-called force–indentation linearisation method (FILM), first described in a previous study, is reformulated here in a more general form in order to broaden its scope of application. The generalisation of the FILM facilitates stable and convergent solutions even when complex nonlinear contact models are used to estimate the impact force. Simulations based on the FILM approach are validated using numerical solutions.

scholarly journals Social and Economic Security in the System of State Protectedness

2019 ◽  
Vol 5 (4) ◽  
pp. 280-287
Author(s):  
D. Shvaiba
Keyword(s):  
Social Policy ◽  
Financial Economic ◽  
Economic Security ◽  
Political Preferences ◽  
Wide Range ◽  
Deformation Processes ◽  
The Many ◽  
Short Time ◽  
The Impact ◽  
State Protection

Taking as a basis the scientific study of different concepts in the theory of security, it is necessary to assume that the inaccessibility of the threat in the absolute sense is impossible. In fact, there may not be a certain type of threat to a particular object in a specific period of time (if there is not yet or there is no longer a corresponding danger factor). It is necessary to take into account that interests are only a small part of a wide range of objects of state protection. This share differs subjectively and interacts with the implemented financial, economic and social policy, the productivity of which is largely dependent on the impact of individual groups of people and parties (based on socio–political preferences). In addition — it is quite a mobile category, which has the ability to change qualitatively. It is obvious that the danger is one of the many destructive moments of security, along with those of which have already been discussed, for example threat, challenge, risk, decline, crisis, cataclysm, destruction, deformation processes, etc. It is necessary to clarify that the danger in the context of the ‘security triad’ is always modified: in a short time, they have all chances to transform from the present into the probable and vice versa.

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