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3 - Numerical solutions of the Navier–Stokes equations

Published online by Cambridge University Press:  07 October 2011

Grétar Tryggvason
Affiliation:
University of Notre Dame, Indiana
Ruben Scardovelli
Affiliation:
Università degli Studi, Bologna, Italy
Stéphane Zaleski
Affiliation:
Université de Paris VI (Pierre et Marie Curie)
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Summary

The one-field formulation of the Navier–Stokes equations described in Chapter 2, where a single set of equations is used to describe the motion of all the fluids present, allows us to use numerical methods developed for single-phase flows. There are, however, two complications: the material properties (usually density and viscosity) generally vary from one fluid to the other and to set these properties we must construct an indicator function that identifies each fluid. We must usually also find the surface tension at the interface. The advection of the indicator function is the topic of Chapters 4 to 6 and finding the surface tension will be dealt with in Chapter 7. In this chapter we discuss numerical methods to solve the Navier–Stokes equations, allowing for variable density and viscosity. We will use the finite-volume method and limit the presentation to regular Cartesian grids. Since the multiphase flows considered in this book all involve relatively low velocities, we will assume incompressible flows.

For any numerical solution of the time-dependent Navier–Stokes equations it is necessary to decide:

  1. (i) how the grid points, where the various discrete approximations are stored, are arranged;

  2. (ii) how the velocity field is integrated in time;

  3. (iii) how the advection and the viscous terms are discretized;

  4. (iv) how the pressure equation, resulting from the incompressibility condition, is solved; and

  5. (v) how boundary conditions are implemented.

These tasks can be accomplished in a variety of ways, but the approach outlined here has been widely used for multiphase flow simulations and results in a reasonably accurate and robust numerical method.

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Publisher: Cambridge University Press
Print publication year: 2011

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