# Background An ideal CFD mesh has an infinite number of cells, which is not practical - a finite number of cells must be used --> discretization error - linear interpolation of profiles - mesh refinement study can estimate discretization error - run CFD model for meshes of varying size - tool for choosing the optimal mesh that balances - accuracy - computational expense # Discretization Error Consider a general flow quantity $\phi$ (temperature, velocity, pressure, etc.). Over a small distance, the profile of $\phi$ can be represented by a Taylor series expansion: $ \phi = c_0 + c_1x+c_2x^2+c_3x^3 + \dots +c_nx^n $ CFD codes use linear variation between cells: $ \phi_\text{CFD} = c_0+c_1x $ Discretization error is given by the difference between the Taylor series representation and the CFD approximation: $ e = \phi - \phi_\text{CFD}=\boxed{c_2+c_3x^3+\dots+c_nx^n} $ The error is proportional to distance squared ($x^2$) - reducing distance by 1/2 reduces error by 1/4 - second-order accurate, $p=2$ In reality, upwind schemes and gradient limiters make CFD codes a bit less than second-order accurate Often think of mesh refinement as number of cells Should be though of in terms of cell size --> representative cell length $h$ $ h = \frac{1}{N}\sum_\text{cells}V_p^{1/3} $ # Mesh Density Representative cell length should differ by at least 30% between each mesh $ r = \frac{h_\text{coarse}}{h_\text{medium}} > 1.3\qquad r= \frac{h_\text{medium}}{h_\text{fine}} > 1.3 $ # Richardson Extrapolation Having found the solution variable $\phi$ for each representative cell length, a curve can be fitted to the data and $\phi$ can be extrapolated at $h=0$ (an infinitely fine mesh). A power law function is used since CFD codes are not quite second-order accurate: $ \phi = \phi_0+ch^p $ - $p=2$ implies a perfect mesh, $p<2$ implies real meshes Write this for two meshes: $ \phi_1 = \phi_0+c{h_1}^p\qquad \phi_2 = \phi_0+c{h_2}^p $ Combine the equations by eliminating $c$ and rearranging for $\phi_0$ $ c=\frac{\phi_1-\phi_0}{{h_1}^p} $ $ \phi_0 = \frac{(h_2/h_1)^p\phi_1-\phi_2}{(h_2/h_1)^p-1} $ Introduce refinement ratio $r_{21} = h_2/h_1$ $ \phi_0 = \frac{r_{21}^p\phi_1-\phi_2}{r_{21}^p-1} $ # Quantifiying Error Relative error: $ e_{21}=\bigg|\frac{\phi_2-\phi_1}{\phi_1}\bigg| $ Compares error between fine mesh and medium mesh Better to use extrapolated relative error: $ e^\text{extra}_{21}=\bigg|\frac{\phi_1-\phi_0}{\phi_0}\bigg| $ Compares fine mesh to infinitely fine mesh Grid Convergence Index: $ GCI_{21}=\frac{1.25e_{21}}{r^p-1} $ An improvement upon relative error