User:Lothar.brendel

Lothar Brendel

"Admin" of this Wiki

Scratch Pad

2D flow in polar coordinates $(\rho ,\varphi )$

• vorticity: $\omega =(\overrightarrow{\mathrm{\nabla }}\times \overrightarrow{v})\cdot {\overrightarrow{e}}_z={\partial }_{\rho }{v}_{\varphi }+\frac{v_{\varphi }-{\partial }_{\varphi }{v}_{\rho }}{\rho }$
• its gradient: $\overrightarrow{\mathrm{\nabla }}\omega ={\overrightarrow{e}}_{\rho }{\partial }_{\rho }\omega +\frac{{\overrightarrow{e}}_{\varphi }}{\rho }{\partial }_{\varphi }\omega$

Energy equation

$$\begin{array}{rl}{E}^{\prime }& =\rho e+\frac{\rho }{2}{\overrightarrow{u}}^2\\ E& =E^{\prime }+\rho \varphi \\ {\partial }_{t}E& ={\partial }_t{E}^{\prime }+\varphi {\partial }_t\rho \\ & =-\overrightarrow{\mathrm{\nabla }}\cdot ((E+P)\overrightarrow{u})\\ & =-\overrightarrow{\mathrm{\nabla }}\cdot ((E^{\prime }+P)\overrightarrow{u}+\varphi \rho \overrightarrow{u})\\ & =-\overrightarrow{\mathrm{\nabla }}\cdot ((E^{\prime }+P)\overrightarrow{u})-\varphi \overrightarrow{\mathrm{\nabla }}\cdot (\rho \overrightarrow{u})-\overrightarrow{u}\cdot \rho \overrightarrow{\mathrm{\nabla }}\varphi \\ \iff {\partial }_t{E}^{\prime }& =-\overrightarrow{\mathrm{\nabla }}\cdot ((E^{\prime }+P)\overrightarrow{u})+\overrightarrow{u}\cdot \overrightarrow{f}\end{array}$$

Florian's eCDF

Let $p_n$ be the empirical probabilities and let $p_n=0$ for $n_{i-1}<n<n_i$. Then, being a step-function, the eCDF fulfills $$c(n_i)-c(n_{i-1})\stackrel{\ast }{=}c(n_i)-c(n_i-1)=c(n_i)-c(n_i-ϵ)=p_{n_i}.$$ Florian's modified eCDF replaces the step-function $c$ by a piecewise linear function $\tilde{c}$ with $\tilde{c}(n_i)=c(n_i)$, from which one obtains fictious empirical probabilities ${\tilde{p}}_n=\tilde{c}(n)-\tilde{c}(n-1)$ for $n_{i-1}<n\le n_i$. Due to the "natural" $\mathrm{\Delta }n=1$ of the Steigungsdreieck, those ${\tilde{p}}_n$ (being all the same because of the linearity) coincide with the slope $$s_{n_i}=\frac{\tilde{c}(n_i)-\tilde{c}(n_{i-1})}{n_i-n_{i-1}}=\frac{c(n_i)-c(n_{i-1})}{n_i-n_{i-1}},$$ which implies $$\sum _{n=n_{i-1}+1}^{n_i}{\tilde{p}}_n=(n_i-n_{i-1}){\tilde{p}}_n=(n_i-n_{i-1})s_{n_i}=c(n_i)-c(n_{i-1})=p_{n_i}.$$ Hence, the effect of using $\tilde{c}$ is replacing the "bar" $p_{n_i}$ and the zeros to its left by smaller "bars" of equal height ${\tilde{p}}_n$ while preserving probability, which of course implies $\sum _k{\tilde{p}}_k=1$. In other words, the scheme is a left-biased coarse graining.

*: even though in general $n_{i-1}\ne n_i-1$

Hamiltons principle

• Phys.SE: Modified Hamilton's Principle overconstraining a system by imposing too many boundary conditions
• Phys.SE: Boundary conditions for calculus of variations in phase space and under canonical transformations