add basic stats to cheat sheet

documentation/_static/overview.rst

@@ -32,7 +32,43 @@ following:
 
 .. math::
 
-    E = \frac{1}{2} \sum_{\mathbf{k}} \hat{\mathbf{u}} \cdot \hat{\mathbf{u}}^*
+    E = \frac{1}{2} \sum_{\mathbf{k}} \hat{\mathbf{u}} \cdot \hat{\mathbf{u}}^*, \hskip .5cm
+    \varepsilon = \nu \sum_{\mathbf{k}} k^2 \hat{\mathbf{u}} \cdot \hat{\mathbf{u}}^*, \hskip .5cm
+    \varepsilon_{\textrm{inj}} = \sum_{\mathbf{k}} \hat{\mathbf{u}} \cdot \hat{\mathbf{f}}^*
+
+In fact, code generated by
+:class:`NavierStokes <NavierStokes.NavierStokes>` computes the velocity
+and vorticity cospectra (9 components each):
+
+.. math::
+
+    \sum_{k \leq \|\mathbf{k}\| \leq k+dk}\hat{u_i} \cdot \hat{u_j}^*, \hskip .5cm
+    \sum_{k \leq \|\mathbf{k}\| \leq k+dk}\hat{\omega_i} \cdot \hat{\omega_j}^*
+
+In all honesty, this is overkill for homogenous and isotropic flows, but
+in principle we will look at more complicated flows.
+The energy spectrum :math:`E(t, k)` and the enstrophy spectrum
+:math:`\frac{1}{2}\omega^2(t, k)` are then computed in
+postprocessing from these tensors, and the enstrophy spectrum is used to
+compute the dissipation :math:`\varepsilon(t)`.
+The other standard quantities computed in postprocessing are as follows
+(consistent with [Ishihara]_):
+
+.. math::
+
+    U_{\textrm{int}}(t) = \sqrt{\frac{2E(t)}{3}}, \hskip .5cm
+    L_{\textrm{int}}(t) = \int \frac{dk}{k} \frac{\pi}{2U_{int}^2} E(t, k), \hskip .5cm
+    T_{\textrm{int}}(t) =
+    \frac{L_{\textrm{int}}(t)}{U_{\textrm{int}}(t)}
+
+    u_K = (\nu \varepsilon)^{1/4}, \hskip .5cm
+    \eta_K = \left(\frac{\nu^3}{\varepsilon}\right)^{1/4}, \hskip .5cm
+    \tau_K = \left(\frac{\nu}{\varepsilon}\right)^{1/2}
+
+    Re = \frac{U_{\textrm{int}} L_{\textrm{int}}}{\nu}, \hskip .5cm
+    \lambda = \sqrt{\frac{15 U_{\textrm{int}}^2}{\varepsilon}}, \hskip
+    .5cm
+    R_{\lambda} = \frac{U_{\textrm{int}} \lambda}{\nu}
 
 -----------
 Conventions