Convection-diffusion of a Gaussian hill problem

Kratos version: 8.0

Source files: source

Case Specification

This example is taken from [Donea, J., & Huerta, A. (2003). Finite Element Methods for Flow Problems. Section 5.6.1] and adapted to run on a two-dimensional mesh. We solve the transient convection diffusion equation $\frac{\partial \phi}{\partial t} + v \cdot \nabla \phi + \phi \nabla \cdot v - \nabla \cdot k \nabla \phi = f$ , where specific initial conditions are set. We refer to the above reference for further details.

The problem is solved exploiting the Runge-Kutta 4 time integration explicit method, and it can be run with four different stabilizations:

quasi-static algebraic subgrid scale (QSASGS)
quasi-static orthogonal subgrid scale (QSOSS)
dynamic algebraic subgrid scale (DASGS)
dynamic orthogonal subgrid scale (DOSS)

Results

We present the temporal evolution of $\phi$ for the DOSS case.

temperature

We can observe the results we obtain are consistent with both the reference [Donea, J., & Huerta, A. (2003). Finite Element Methods for Flow Problems. Section 5.6.1].

Moreover, we also compared our numerical solution against the analytical solution $\frac{5}{7\sigma} \exp(-(\frac{x-x_0-vt}{l\sigma})^2)$ , where $\sigma=\sqrt{1+4kl^2}$ , $x_0=\frac{2}{15}$ , $k=\exp(-3)$ the diffusivity, $l=\frac{7\sqrt{2}}{300}$ , $v=1$ the convective velocity and $t$ the simulation time. All physical quantities unit measures are expressed according to the Convection Diffusion application and the materials.json file. The $l^2$ norm we obtain is 0.001575 for the DOSS element.