Numerical simulation of 1D compressible flows

Date
2014-12
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Abstract

From clouds twirling overhead to the endlessly lapping waves of the oceans, fluids and gases have important impacts on everyday living. However, as ordinary and commonplace fluid and gaseous phenomena are, simulating them can be a difficult proposition due to the interconnectedness of forces such as convection, diffusion, turbulence, and compression. In fact, many problems in fluid dynamics remains unanswered - such as a universal expression for turbulence.

Despite its difficulties, computational fluid mechanics offers a wide variety of applications. Through use of simulation engineers can ensure their bridges will withstand tempestuous conditions without the need of costly physical models, surgeons can practice their procedures in real-time environments without worry of killing a patient, and audiences in cinemas everywhere can be immersed into a fantasy world aided by renderings of realistic fluid and gaseous flow.

In this paper the derivation of the Euler Equations used to model such flows will be demonstrated. This will illustrate the forces that are at work in a fluid flow, possibly providing implications that have numerical benefits. Necessary conditions for numerical stability will then be established, giving rise to implication that are essential to computational fluid dynamics. Additionally the Rankine-Hugonoit conditions, a set of three equations that must hold true for the characteristics in a shock wave, will be defined and derived.

The primary sample problem analyzed in this paper is the Sod Shock Tube, a classic problem used in the testing of numerical methods for accuracy due to the existence of an analytic solution. In this paper we will take a look at the analytic solution to this problem, as well as analyzing the characteristics that composes its solution. Then, to better understand the nature of the characteristics, we will demonstrate how such characteristics arise in traffic flow.

Finally, an analysis of two numerical methods that may be used to approximate compressible flow in one dimension will be performed; The Lax Method and the Lax-Wendroff Method. Using the analytic solution as a reference, we will see how well these function perform as a means of approximating the nature of the Sod shock tube. Furthermore, necessary conditions for stability and convergence for both methods will be established, which will allow us to anticipate the numerical results.

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Keywords
Numerical simulation, Fluid dynamics, Finite element methods, Compressible flow
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