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2017 | OriginalPaper | Buchkapitel

3. Numerical Simulation of Incompressible Flows

verfasst von : Takeo Kajishima, Kunihiko Taira

Erschienen in: Computational Fluid Dynamics

Verlag: Springer International Publishing

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Abstract

For compressible and incompressible flows, there is a difference in how the numerical solution techniques are formulated, based on whether or not the mass conservation equation includes a time-derivative term. Fluid motion is described by the conser4 vation equations for mass, momentum, and energy. For incompressible flow, the 5 conservation equation for kinetic energy can be derived from the momentum conser6 vation equation. Hence, we only need to be concerned with the mass and momentum 7 conservation equations. Furthermore, if the temperature field is not a variable of 8 interest, we do not need to consider the internal energy in the formulation. We do 9 note that the treatment of momentum conservation should be consistent with the 10 conservation of kinetic energy in a discrete manner, as it influences the achievement 11 of reliable solution and numerical stability.

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Vorheriges Kapitel Finite-Difference Discretization of the Advection-Diffusion Equation

Nächstes Kapitel Incompressible Flow Solvers for Generalized Coordinate System

The variable P can be regarded as a Lagrange multiplier that is needed to enforce the incompressibility constraint [5, 24].

Numerically solving stiff differential equations requires excessively small time steps due to the disparity in the magnitude of the characteristic eigenvalues [17, 19]. In the case of low-Mach-number flow, the time step necessary to satisfy the CFL condition with the acoustic speed c is much smaller than that with the advective speed u, leading to the requirement of very small time steps to be used for time integration.

We make a distinction between the regular and collocated grids. The regular grid places all variables at the same location. On the other hand, the collocated grid places the fluxes between where u, v, and p are positioned. For further details, see Sect. 4.4.

Let us consider an interval of $-1 \le x \le 1$, discretized nonuniformly into N cells with velocity defined on

$$\begin{aligned} x_j^u = - \cos \frac{\pi (j-1/2)}{N}, \quad j = \frac{1}{2}, \frac{3}{2}, \dots , N+\frac{1}{2}. \end{aligned}$$

In this case, there are two ways to place the pressure variables [22]. One way is to use the midpoints of where the velocity variables are positioned

$$\begin{aligned} x_j^p = \frac{1}{2} (x^u_{j-\frac{1}{2}} + x^u_{j+\frac{1}{2}}), \quad j = 1,2,\dots ,N. \end{aligned}$$

In this case, the distance between discrete pressure $\tilde{\Delta }_{j+\frac{1}{2}}$ is described in the text above. On the other hand, one can use

$$\begin{aligned} x_j^p = - \cos \frac{\pi (j-1/2)}{N}, \quad j = 1,2,\dots , N \end{aligned}$$

as a mapping function. This approach allows for a smooth distribution of

$$\begin{aligned} \widetilde{\Delta }_{j+\frac{1}{2}} = - x_j^p + x_{j+1}^p \end{aligned}$$

but such grid distribution based on a functional description is limited to spacial cases and is not common.

Grid stretching can be incorporated for certain cases. See Cain et al. [2] for details.

The term upwind is commonly used. However, the use of the term upstream may be more appropriate so that applications are not limited to air.

As in cases where fine grids are required near wall boundaries (e.g., near-wall turbulence).

Here, we have not specified a pressure boundary condition for p but utilized the zero-gradient boundary condition for $\phi $. To determine the divergence-free (solenoidal) velocity field, the pressure boundary condition for p is not needed.

for three-dimensional flows, s is the discrete streamfunction vector (or vector potential); also see [3].

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Titel: Numerical Simulation of Incompressible Flows
verfasst von: Takeo Kajishima
Kunihiko Taira
Verlag: Springer International Publishing
Buch: Computational Fluid Dynamics
Print ISBN: 978-3-319-45302-6

Electronic ISBN: 978-3-319-45304-0

Copyright-Jahr: 2017
DOI: https://doi.org/10.1007/978-3-319-45304-0_3

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