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Numerical Methods In Fluid Mechanics
Fluid motion is governed by the Navier–Stokes equations, a set of coupled and nonlinear partial differential equations derived from the basic laws of conservation of mass, momentum and energy. The unknowns are usually the flow velocity, the pressure and density and temperature. The analytical solution of this equation is impossible hence scientists resort to laboratory experiments in such situations. The answers delivered are, however, usually qualitatively different since dynamical and geometric similitude are difficult to enforce simultaneously between the lab experiment and the prototype. Furthermore, the design and construction of these experiments can be difficult (and costly), particularly for stratified rotating flows. Computational fluid dynamics (CFD) is an additional tool in the arsenal of scientists. In its early days CFD was often controversial, as it involved additional approximation to the governing equations and raised additional (legitimate) issues. Nowadays CFD is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fluid Motion
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation. Fluid dynamics offers a systematic structure—which underlies these practical disciplines—that embraces empirical and semi-empirical laws derived from flow measurement and used to solve practical problems. The solution to a fluid dynamics problem typically involves the calculation of various properties of the fluid, such as flow velocity, pressure, density, and temperature, as functions of space and time. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability. Among the many open questions are the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Differential Equations
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast with the term partial differential equation which may be with respect to ''more than'' one independent variable. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where , ..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of the unknown function of the variable . Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathematics are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wave Function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi, respectively). The wave function is a function of the degrees of freedom corresponding to some maximal set of commuting observables. Once such a representation is chosen, the wave function can be derived from the quantum state. For a given system, the choice of which commuting degrees of freedom to use is not unique, and correspondingly the domain of the wave function is also not unique. For instance, it may be taken to be a function of all the position coordinates of the particles over position space, or the momenta of all the particles over momentum space; the two are related by a Fourier tran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dissipation
In thermodynamics, dissipation is the result of an irreversible process that takes place in homogeneous thermodynamic systems. In a dissipative process, energy (internal, bulk flow kinetic, or system potential) transforms from an initial form to a final form, where the capacity of the final form to do thermodynamic work is less than that of the initial form. For example, heat transfer is dissipative because it is a transfer of internal energy from a hotter body to a colder one. Following the second law of thermodynamics, the entropy varies with temperature (reduces the capacity of the combination of the two bodies to do work), but never decreases in an isolated system. These processes produce entropy at a certain rate. The entropy production rate times ambient temperature gives the dissipated power. Important examples of irreversible processes are: heat flow through a thermal resistance, fluid flow through a flow resistance, diffusion (mixing), chemical reactions, and electric cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convection–diffusion Equation
The convection–diffusion equation is a combination of the diffusion equation, diffusion and convection (advection equation, advection) equations, and describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection. Depending on context, the same equation can be called the advection–diffusion equation, drift velocity, drift–diffusion equation, or (generic) scalar transport equation. Equation General The general equation is \frac = \mathbf \cdot (D \mathbf c) - \mathbf \cdot (\mathbf c) + R where * is the variable of interest (species concentration for mass transfer, temperature for heat transfer), * is the diffusivity (also called diffusion coefficient), such as mass diffusivity for particle motion or thermal diffusivity for heat transport, * is the velocity field that the quantity is moving with. It is a function of time and space. For example, in advection, might be t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Difference Method
In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are discretized, or broken into a finite number of steps, and the value of the solution at these discrete points is approximated by solving algebraic equations containing finite differences and values from nearby points. Finite difference methods convert ordinary differential equations (ODE) or partial differential equations (PDE), which may be nonlinear, into a system of linear equations that can be solved by matrix algebra techniques. Modern computers can perform these linear algebra computations efficiently which, along with their relative ease of implementation, has led to the widespread use of FDM in modern numerical analysis. Today, FDM are one of the most common approaches to the numerical solution of PDE, along with finite element metho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Element Method
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The FEM is a general numerical method for solving partial differential equations in two or three space variables (i.e., some boundary value problems). To solve a problem, the FEM subdivides a large system into smaller, simpler parts that are called finite elements. This is achieved by a particular space discretization in the space dimensions, which is implemented by the construction of a mesh of the object: the numerical domain for the solution, which has a finite number of points. The finite element method formulation of a boundary value problem finally results in a system of algebraic equations. The method approximates the unknown function over the domain. The sim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aerodynamics
Aerodynamics, from grc, ἀήρ ''aero'' (air) + grc, δυναμική (dynamics), is the study of the motion of air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dynamics and its subfield of gas dynamics. The term ''aerodynamics'' is often used synonymously with gas dynamics, the difference being that "gas dynamics" applies to the study of the motion of all gases, and is not limited to air. The formal study of aerodynamics began in the modern sense in the eighteenth century, although observations of fundamental concepts such as aerodynamic drag were recorded much earlier. Most of the early efforts in aerodynamics were directed toward achieving Aircraft#Heavier than air – aerodynes, heavier-than-air flight, which was first demonstrated by Otto Lilienthal in 1891. Since then, the use of aerodynamics through mathematical analysis, empirical approximations, wind tunnel experimentation, and computer simu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Methods
Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations. The idea is to write the solution of the differential equation as a sum of certain " basis functions" (for example, as a Fourier series which is a sum of sinusoids) and then to choose the coefficients in the sum in order to satisfy the differential equation as well as possible. Spectral methods and finite element methods are closely related and built on the same ideas; the main difference between them is that spectral methods use basis functions that are generally nonzero over the whole domain, while finite element methods use basis functions that are nonzero only on small subdomains ( compact support). Consequently, spectral methods connect variables ''globally'' while finite elements do so ''locally''. Partially for this reason, spectral methods have excellent error properties, with the so-called "exponential convergence" being the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Element Methods
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The FEM is a general numerical method for solving partial differential equations in two or three space variables (i.e., some boundary value problems). To solve a problem, the FEM subdivides a large system into smaller, simpler parts that are called finite elements. This is achieved by a particular space discretization in the space dimensions, which is implemented by the construction of a mesh of the object: the numerical domain for the solution, which has a finite number of points. The finite element method formulation of a boundary value problem finally results in a system of algebraic equations. The method approximates the unknown function over the domain. The simple e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Difference Methods
In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating Derivative, derivatives with Finite difference approximation, finite differences. Both the spatial domain and time interval (if applicable) are Discretization, discretized, or broken into a finite number of steps, and the value of the solution at these discrete points is approximated by solving algebraic equations containing finite differences and values from nearby points. Finite difference methods convert ordinary differential equations (ODE) or partial differential equations (PDE), which may be Nonlinear partial differential equation, nonlinear, into a system of linear equations that can be solved by matrix algebra techniques. Modern computers can perform these linear algebra computations efficiently which, along with their relative ease of implementation, has led to the widespread use of FDM in modern numerical analysis. Today, FDM are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
