
1.1
Sept 4, 2015 18:01:56 GMT
Post by matthorr on Sept 4, 2015 18:01:56 GMT
GIVEN: This semester we will primarily concentrate on examining how to numerically solve just a few simple scalar PDE model problems and solving them on equalspaced Cartesian meshes.
REQUIRED: Please note the four main scalar PDE model problems that we will examine this semester and describe some of their main features. Please explain why we will confine ourselves to equalspaced Cartesian meshes when in reality most CFD uses meshes that have variable spacing and nonCartesian meshes.
(Thankfully, as discussed in class these limitations can be overcome without too much difficulty and indeed will be removed in ENAE685.)



1.1
Sept 8, 2015 18:40:55 GMT
Post by matthorr on Sept 8, 2015 18:40:55 GMT
SOLUTION
Convection. Convection/advection is a hyperbolic PDE. The general form is \begin{equation} \frac{\partial{u}}{\partial{t}} + a\frac{\partial{u}}{\partial{x}} = 0 \end{equation} where \(a\) is the propagation speed. The solution is periodic with a phase shift and no damping or dissipation. The initial condition (IC) is required at \(t = 0\), and there are two types of boundary conditions (BCs). One is a physical BC at one end and undisturbed at the other (off to infinity), or with periodic flow (biconvection) where the flow that exits one end reenters at the other.
Diffusion. Diffusion/dissipation is a parabolic PDE. The general form is \begin{equation} \frac{\partial{u}}{\partial{t}} + \nu\frac{\partial^2{u}}{\partial{x^2}} = 0 \end{equation} where \(\nu\) is a positive damping coefficient. The solution is sinusoidal with exponential decay given constant boundary conditions. The damping increases with higher frequencies. The IC is required at \(t = 0\) and BCs are needed at \(x = 0, L\). Dirichlet BCs give \(u\), Neuman BCs give \(\frac{\partial{u}}{\partial{x}}\) and Robin BCs give \(au + b\frac{\partial{u}}{\partial{x}}\).
ConvectionDiffusion. This is a combination of the two above types and is a parabolic PDE. The general form is \begin{equation} \frac{\partial{u}}{\partial{t}} + a\frac{\partial{u}}{\partial{x}} = \nu\frac{\partial^2{u}}{\partial{x^2}} \end{equation} The equation can be either convection (\(R_\Delta \gg 2\)) or diffusion dominated (\(R_\Delta \ll 2\)), with Mesh Reynold's Number defined as \(R_\Delta \doteq \frac{a\Delta x}{\nu} \). There will be both diffusion and a phase shift, with the higher frequencies again damping out the fastest. The form of the solution is dependent on \(R_\Delta\).
Poisson's Equation. Poisson's Equation is an elliptic PDE with imaginary eigenvalues. The general form is \begin{equation} \frac{\partial^2{\phi}}{\partial{x^2}} + \frac{\partial^2{\phi}}{\partial{y^2}} = f(x,y) \end{equation} BCs are required on all sides but the IC is not necessary as there is no time derivative. It can be solved using iterative relaxation methods until the solution satisfies both the governing equation and all BCs.
We will use equalspaced Cartesian meshes to simplify the development of the solution methods we will use (so the methods will be independent of the actual physical reality). We can do this because a nonequal spaced, nonCartesian mesh in the physical world can be easily transformed or morphed (transmogrified!) into an equalspaced Cartesian mesh when setting up the problem, and the reverse can be done to translate the solution back into physical reality.

