Abstract
The additional complexity of the Navier–Stokes equations and special features for incompressible flows are considered in this chapter and the next; here we cover basic issues, the features of the equations, and the methods of solution. The staggered and collocated variable arrangements, the pressure equation, and the pressure-velocity coupling for incompressible flows using the fractional-step and SIMPLE algorithms are described in detail. Other approaches (the PISO algorithm, streamfunction-vorticity, and artificial compressibility) are also briefly described. The initial and boundary conditions for the Navier–Stokes equations and their implementation on Cartesian grids are also covered.
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Notes
- 1.
In flows where the density of the fluid varies with height in a gravitational field, the flow is said to be stratified, and the flow may carry heavy fluid up or light fluid down so that it now has a density different from its surroundings; the fluid then has not only kinetic energy, but also energy as a result of its position, called potential energy. The result is buoyant forces that are touched on later in this chapter and are very important in both meteorology and oceanography.
- 2.
In this case the second-order extrapolation \(p^{n+1}=(3/2) p^{n+1/2}-(1/2) p^{n-1/2}\) is used.
- 3.
Coupled (or monolithic) solvers are also available and most commercial packages now offer them; discussion of solver alternatives can be found in the code documentation or in the literature, e.g., Heil et al. (2008) or Malinen (2012). See also Chap. 11 for a brief description of one such method.
- 4.
This relation was first derived by Raithby and Schneider (1979) and later re-discovered by Perić (1985) using different arguments.
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Ferziger, J.H., Perić, M., Street, R.L. (2020). Solution of the Navier–Stokes Equations: Part 1. In: Computational Methods for Fluid Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-319-99693-6_7
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DOI: https://doi.org/10.1007/978-3-319-99693-6_7
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