“Stokes approximation and artificial time”的意思、由来-开放百科全书

词条

Stokes approximation and artificial time

释义

Navier-Stokes equation
Stokes approximation
References
External links

{{Multiple issues|{{technical|date=January 2015}}{{Orphan|date=December 2014}}
}}

This article provides an error analysis of time discretization applied to spatially discrete approximation of the stationary and nonstationary Navier-Stokes equations. The nonlinearity of the convection term is the main problem in solving a stationary or nonstationary Navier-Stokes equation or Euler equation problems. Stoke incorporated ‘the method of artificial compressibility’ to solve these problems.

Navier-Stokes equation

Stokes approximation

The Stokes approximation is developed from the Navier-Stokes equations by omission of the convective term. For small Reynolds numbers in the incompressible flow, this approximation is more useful. Then incompressible Navier Stokes equation can be written as-

Here linear diffusion term dominates the convection term.

In the stationary problem neglecting the convection term, we get-

Many theorems can be proved by using this process.

The main problem with the solution of the incompressible flow equation is the decoupling of the continuity and momentum equation due to the absence of pressure or density term. Chorin proposed the solution for this problem of the pressure decoupling; this approach is called artificial compressibility.

In the above equation stoke assume that at, non-stationary Navier Stokes problem converge towards the solution of the correspondent stationary problem. This solution will not depend upon the function .

If this is used for the above equation consisting of Navier stokes equation and continuity equations with time derivative of pressure, then the solution will be same as the stationary solution of the original Navier Stoke problem.

This process also introduce the new term artificial time as t→∞. Artificial compressibility method is combined with a dual time stepping procedure which involves iteration in pseudo-time within each physical time step. This guarantees a convergence towards the solution for the incompressible flow problem.

References

{{citation | first=R. | last=Ansorge | authorlink=Rainer Ansorge | title=Mathematical Models of Fluid dynamics }}

External links

{{cite journal | url = http://www.sciencedirect.com/science/article/pii/S0378383905001250 | doi=10.1016/j.coastaleng.2005.09.020 | volume=53 | title=A discussion of artificial compressibility | journal=Coastal Engineering | pages=93–98| year=2006 | last1=Madsen | first1=P.A | last2=Schäffer | first2=H.A }}
http://www.books.google.com/books?isbn=3527627979

1 : Fluid dynamics

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。