[CFD | ENG] Introduction to the finite volume method in CFD

In this video, we will introduce the most popular and efficient numerical method used for the solution of conservation equations in Computational Fluid Dynamics (CFD): the finite volume method (FVM).

We will start by describing the main features of the FVM and how we discretise some of the terms of every conservation equation: source, diffusive and convective terms. Then, we will apply the fundamentals of the FVM to solve the simplified Navier-Stokes equations for simple shear flow and shear flow with pressure gradient, where we will illustrate how we generate our computational mesh and how we treat mathematically both boundary and interior cells.

Lastly, we will derive our matrix linear system to find a numerical solution for the velocity profile of a Newtonian fluid, and we will compare our FVM with 1) analytical equations obtained from solving the Navier-Stokes equations and 2) with the numerical solution obtained using the finite difference method (FDM).

Timestamps
00:00 Introduction and overview of the presentation
01:29 Introduction to the finite volume method
05:09 Features of the finite volume method
11:37 The finite volume method and the Navier-Stokes equations
15:45 Discretisation of the source term
18:16 Discretisation of the diffusive term
23:38 Discretisation of the convective term
24:27 Description of the geometries for simple shear flow and shear flow with pressure gradient
29:05 Finite volume method for shear flows
33:57 Computation of the surface integrals for interior cells
39:59 Computation of the surface integrals for boundary cells (top plate)
43:30 Computation of the surface integrals for boundary cells (bottom plate)
45:04 Summary of coefficients for all cells
46:08 Sketch of the computational mesh used for shear flows
47:55 Linear system of equations and assembling of the matrix Au=b
52:05 Approximate solution for the velocity profile for the simple shear flow case
54:47 Comparison between analytical equations and numerical solutions (FVM and FDM) 1
56:02 Approximate solution for the velocity profile for the shear flow with pressure gradient case
57:24 Comparison between analytical equations and numerical solutions (FVM and FDM) 2
58:49 References and final comments

This is the last video of the course called "Introduction to CFD", which I am offering for free in this channel.

I encourage you to watch all the videos of the course. Enjoy the content.

Links of interest
1st peer-reviewed paper that describes viscoelastic flow simulations:
2nd peer-reviewed paper that describes viscoelastic flow simulations:
3rd peer-reviewed paper that describes complex-fluid flow simulations: