Computational Fluid
Dynamics 2D
Discretization
Forward - approximates the derivative using the next spatial grid point
Backward - estimate the spatial derivative using the point behindxi
Central - use the two points on each side ofxi
Taylor expansion
" Forward
" Backward
" https://www.khanacademy.org/math/integral-
calculus/sequences_series_approx_calc/taylor-series/v/generalized-
taylor-series-approximation
" http://pl.wikipedia.org/wiki/Wz%C3%B3r_Taylora
Extension from 1D to 2D
" In 2D space, a rectangular (uniform) grid is defined by the points with
coordinates:
" Defineuij
" All derivatives are based on the 2D Taylor expansion of a mesh point
value arounduij
Extension from 1D to 2D
" First-order partial derivative in the x-direction, a finite-difference
formula is:
" The same for y direction
2-D Linear Convection
" timestep will be discretized as a forward difference and both spatial
steps will be discretized as backward differences (Euler's method)
" discretization of the PDE
" iindex movement in xdirection, jindex movement in ydirection
2-D Linear Convection
" Solve equation for unknown
" Initial conditions
Python code
" Boundary conditions
2-D Convection (non linear)
" 2D Convection, represented by the pair of coupled partial differential
equations
2D convection
" Rearranging both equations
" Initial Conditions
Python code
" Boundary Conditions
2D convection Application
2D Diffusion
Python code
Application
" Yazdipour et al. 2D modelling of the effect of grain size on hydrogen diffusion in X70 steel.
Computational Materials Science 56 (2012) 49 57
" Model set up
" Results
Python code
Burgers' Equation in 2D
" Burgers' equation can generate discontinuous solutions from an initial condition that is smooth,
i.e., can develop "shocks.
" Coupled set of PDEs and discritization
Application
" Shock capturing  testing numerical schemes
" http://nbviewer.ipython.org/github/barbagroup/CFDPython/blob/ma
ster/lessons/19_Odd_Even_Decoupling.ipynb
2D Laplace Equation
" Laplace's equation has the features typical of diffusion phenomena.
For this reason, it has to be discretized withcentral differences, so
that the discretization is consistent with the physics we want to
simulate
2D Laplace Equation
" Laplace Equation does not have a time dependence
" Laplace equation calculates the equilibrium state of a system under
the supplied boundary conditions
" Laplace Equation can be the steady-state heat equation
" We will iteratively solve for untill it meets a specified condition
" The system will reach equilibrium only as the number of iterations
tends INF but we can approximate the equilibrium state by iterating
until the change between one iteration and the next isverysmall.
Python code
2D Laplace Equation
" Using second-order central-difference schemes in both directions is
the most widely applied method for the Laplace operator. It is also
known as the five-point difference operator, alluding to its stencil
" initial state, p = 0 every where
Analitical Solution
" Boundary conditions
Application
" Laplace s Equation describes steady state situations such as:
" steady state temperature distributions
" steady state stress distributions
" steady state potential distributions (it is also called the potential equation)
" steady state flows, for example in a cylinder, around a corner
" They can be used to accurately describe the behavior of electric,
gravitational, and fluid potentials
" http://users.aber.ac.uk/ruw/teach/260/laplace.php
2D Poisson Equation
" Poisson's equation is obtained from adding a source term to the right-
hand-side of Laplace's equation
" there is some finite value inside the field that affects the solution.
Poisson's equation acts to "relax" the initial sources in the field.
" discretized form
2D Poisson Equation
Python code
" An equation for patianj
" initial state of of p = 0 everywhere
" boundary conditions
" source term consists of two initial spikes inside the domain
Application
" Laplace eqaution with a source term  reaction (exotermic reaction)
Cavity Flow with Navier-Stokes
" The momentum equation in vector form for a velocity field
" differential equations: two equations for the velocity componentsu,v
and one equation for pressure
Cavity Flow with Navier-Stokes
" Discretized equations
" discretized pressure-Poisson equation
Cavity Flow with Navier-Stokes
" Rearrange the equations
Cavity Flow with Navier-Stokes
" The initial conditio u,v,p = 0 everywhere
" Boundary conditions
Python code
Application
" Supersonic combustor
Channel Flow with Navier-Stokes
" Add a source term to the u-momentum equation, to mimic the effect
of a pressure-driven channel flow
Channel Flow with Navier-Stokes
" Discretized equations
Channel Flow with Navier-Stokes
" Re-arrange these equations
Channel Flow with Navier-Stokes
" Initial conditions u,v,p = 0
" Boundary conditions
Python code
CFL condition
" Ensure stability of numerical simulations
Kolejne wykłady
" OpenFoam
" http://openfoam.org/ ; https://github.com/OpenFOAM
" VirtualBox
" https://www.virtualbox.org/wiki/Downloads
" CAELinux
" http://www.caelinux.com/CMS/
" Instalacja Linuxa na VB
" http://www.wikihow.com/Install-Ubuntu-on-VirtualBox
" Tutorials
" http://www.tfd.chalmers.se/~hani/kurser/OS_CFD/ + od 2007  2014
" C++
" https://www.youtube.com/user/ReelLearning/playlists
https://www.youtube.com/watch?v=Rub-JsjMhWY , in one video
Projekt OpenFoam
" Podział na grupy 2-3 osobowe
" Tematy
" Tutorial OpenFoam (temat związany z silnikami lub spalaniem)
" Cavity flow (C++ lub Java)
" Deflagration and Detonation in channel (Hydrogen-air, Methane-air)