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Brian Spalding: Some Contributions to Computational Fluid Dynamics During the Period 1993 to 2004 Read chapter Read first chapter

2020 | OriginalPaper | Chapter

Some Observations on Thermodynamic Basis of Pressure Continuum Condition and Consequences of Its Violation in Discretised CFD

Author : A. W. Date

Published in: 50 Years of CFD in Engineering Sciences

Publisher: Springer Singapore

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Abstract

CFD is concerned with solution of Navier–Stokes (NS) equations in discretised space. It is important, therefore, to ensure that the discretised equations and their solutions obey the continuum condition embedded in Stokes’s stress–strain laws for an isotropic continuum fluid. In this paper, it is shown that adherence to this condition leads to three important conceptual/algorithmic outcomes: 1. Prevention of zig-zag pressure distribution when NS equations are solved for incompressible flow of a single fluid on colocated grids. 2. Prevention of loss of volume/mass at large times when NS equations are solved for interfacial incompressible flows of multi-fluids within single-fluid formalism. 3. Evaluation of surface tension force in interfacial flows without using phenomenology embedded in the definition of the surface tension coefficient. All the above benefits are justified on the basis of a thermodynamic principle rarely invoked in discretised CFD. A few problems are solved by way of case studies.

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previous chapter Brian Spalding: Some Contributions to Computational Fluid Dynamics During the Period 1993 to 2004
next chapter The SUPER Numerical Scheme for the Discretization of the Convection Terms in Computational Fluid Dynamics Computations
Footnotes
1
This can be appreciated from the Avogadro’s number which specifies that at normal temperature and pressure, a gas will contain \(6.022 \times 10^{26}\) molecules per kmol. Thus in air, for example, there will be \(10^{16}\) molecules per \(\text {mm}^{3}\).
 
2
In [27], symbol \(\overline{\sigma } = (\sum _{i=1}^{3}\,\sigma _{xi})/3\) is used. Here, \(\overline{p} = - \overline{\sigma }\) is preferred. Both \(\overline{p}\) and q are newly introduced to serve a pedagogic purpose.
 
3
It is important to recognise that in discretised CFD, the incompressible condition (\(\bigtriangledown \,.\,V_{f}\) = 0) is defined in terms of CV face velocities \(u_{fi}\) as shown in Fig. 1. In fact, when this definition is explicitly implemented, there results the SIMPLE staggered grid procedure of Patankar and Spalding [19]. Further, \(u_{fi}\) must satisfy momentum equations. In a continuum, \(u_{fi}\) and \(u_{i}\) fields coincide but in a discretised space, it is important to distinguish them. This will become apparent in the next section.
 
4
In passing we note that in all three cases, it can be verified that the quantity q is invariant under rotation of the coordinate system or interchange of axes. This property ensures isotropy [27].
 
5
Analysis of the discretised equations presented in the next section shows that \(\lambda = 0.5\).
 
6
Equation 16 is validated in Eqs. 39–43 for a two-dimensional flow.
 
7
In deriving Eq. 20, it is assumed that \(\sum \,A_{k}\,u^{'}_{fi,k} = 0\). This is consistent with the staggered grid practice [19].
 
8
Incidentally, in the literature, several different types of interpolations have been proposed. Some of these are given below by way of example.
  • Rhie and Chow [24] (1D Pressure gradient interpolation)
    $$\begin{aligned} u_{f1,e}= & {} \overline{u}_{1,e} - \frac{\Delta V}{AP^{u}}\,\left[ \frac{\partial p}{\partial x_{1}}\,|_{e} - \overline{\frac{\partial p}{\partial x_{1}}}\,|_{e} \, \right] \nonumber \\ \text{ where } \overline{\frac{\partial p}{\partial x_{1}}}\,|_{e}= & {} \frac{1}{2}\,\left[ \frac{\partial p}{\partial x_{1}}\,|_{P} + \frac{\partial p}{\partial x_{1}}\,|_{E} \,\right] \end{aligned}$$
    (33)
  • Peric [8] (1D Mom-Outflow interpolation)
    $$\begin{aligned} u_{f1,e} = \frac{1}{2}\,\left[ \frac{\sum A_{k}\,u_{1,k}}{AP^{u_{1}}}\,|_{P} + \frac{\sum A_{k}\,u_{1,k}}{AP^{u_{1}} }\,|_{E} \,\right] - \frac{\Delta V}{AP^{u}}\,\frac{\partial p}{\partial x_{1}}\,|_{e} \end{aligned}$$
    (34)
  • Thiart [34] (Power Law Scheme [20])
    $$\begin{aligned} u_{f1,e}= & {} \theta \,u_{1,P} + ( 1 - \theta )\,u_{1,E} \text{ where } \nonumber \\ \theta ( Pc_{e} )= & {} \left[ Pc_{e} - 1 + \text{ max }(0,-Pc_{e})\right] /Pc_{e} \nonumber \\+ & {} \text{ max }\left\{ 0, ( 1 - 0.1|Pc_{e}|)^{5}\right\} /Pc_{e} \end{aligned}$$
    (35)
    where cell-face Reynolds/Peclet number \(Pc_{e} = ({\rho _{m}\,u_{f1}\Delta x_{1}}/{\mu })_{e}\).
 
9
Equations 39–43 justify the assertion made in Eqs. 15 and 16 for a two-dimensional flow.
 
10
This is unlike the staggered grid practice in which the mass error is estimated from discretised version of Eq. 1.
 
11
Incidentally, the superficial viscosity is now evaluated as \(\mu _{m} = F\,\mu _{a} + ( 1 - F )\,\mu _{b}\).
 
12
In all problems, the convective terms are discretised using a Total Variation Diminishing (TVD) scheme [14] to minimise interface smearing around \(F = 0.5\). Implementation details are given in [17].
 
13
Volume error is defined as
$$\begin{aligned} \text{ Error }\,(t) = \left( \sum \,F_{i,j}\,\Delta V_{i,j}\right) / \left( \sum \,F^{0}_{i,j}\,\Delta V_{i,j}\right) \end{aligned}$$
(72)
where \(F^{0}\) is the initial F-distribution at \(t = 0\) and \(\Delta V_{i,j}\) is the volume of the cell surrounding node (i, j).
 
14
This ignores the fact that \(\sigma \) is essentially a property of a specified fluid pair (a, b).
 
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Metadata
Title
Some Observations on Thermodynamic Basis of Pressure Continuum Condition and Consequences of Its Violation in Discretised CFD
Author
A. W. Date
Copyright Year
2020
Publisher
Springer Singapore
Book
50 Years of CFD in Engineering Sciences

Print ISBN: 978-981-15-2669-5

Electronic ISBN: 978-981-15-2670-1

Copyright Year: 2020

DOI
https://doi.org/10.1007/978-981-15-2670-1_2

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