We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Published online by Cambridge University Press: 22 February 2022
Introduction
Except for Chapter 14 that discussed spectral methods, all earlier chapters have discussed the method of finite difference that still remains a popular method for solving partial differential equations. While employing the finite difference method, the differential form of the conservation law of fluid flow was utilized and the partial derivatives present in the differential form of the conservation equations were approximated using the appropriate finite difference expressions. In this chapter, the finite volume method will be introduced and applied to solve fluid flow problems. The basis of the finite volume method is the utilization of the conservation law of fluid flow in the integral form.
Integral Form of Conservation Law
The physical (“fluid” in this case) system is governed by a set of conservation laws that include conservation of mass, conservation of momentum, and conservation of energy. These conservation laws are mostly written in differential form. The method of finite difference requires the differential form of the conservation laws, whereas the finite volume method entails the integral form of the conservation laws for a fixed physical domain. Assume the existence of a physical domain, Ω, with the boundary of the domain, indicated by Ω. Then, the canonical conservation equation assuming that the physical domain is fixed is of the following form:
where U is the conserved state, is the flux of the conserved state, is the outward directed unit normal on the boundary of the domain, and S denotes a source term. By applying the Gauss divergence theorem, the aforementioned conservation law can be written as a partial differential equation. The Gauss divergence theorem states that
Equation (15.1) then becomes
Applying the Leibnitz rule for differentiation under the integral sign and combining the volume integrals, one obtains
As Equation (15.4) must be valid for any arbitrary domain, Ω, the integrand that appears in Equation (15.4) must be zero everywhere, or equivalently,
Equation (15.5) specifies the conservation law in the differential form as a partial differential equation.
To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.
