[mod] misc

README.rst

@@ -82,7 +82,8 @@ Quick start
 
 #. Set initial condition
 
-   A solenoidal velocity field should be given.
+   Although the scalar field can be arbitrary, the velocity field should be solenoidal.
+   ``main.py`` offers several examples.
 
    .. code-block:: console
 
@@ -106,7 +107,7 @@ Quick start
 
       bash exec.sh
 
-This may take a few minutes, depending on your machine spec.
+   This may take a few minutes, depending on your machine spec.
 
 The flow fields are saved under ``output/save/`` in `NPY <https://numpy.org/devdocs/reference/generated/numpy.lib.format.html>`_ format.
 Note that these velocities are in the spectral domain; you need to perform the inverse Fourier transform (and the normalisation) to recover the velocities in the physical domain.
@@ -117,8 +118,6 @@ If proper Python libraries are installed, you can visualise the flow fields by
 
    python3 visualise/2d.py
 
-This should give you a similar movie to the one you can see above.
-
 *************
 3D simulation
 *************

docs/source/equation/main.rst

@@ -15,10 +15,10 @@ and the momentum balance:
 .. math::
 
    \der{u_i}{t}
-   +
-   u_j \der{u_i}{x_j}
    =
    -
+   u_j \der{u_i}{x_j}
+   -
    \der{p}{x_i}
    +
    \frac{1}{Re}
@@ -32,9 +32,10 @@ Also a passive scalar field :math:`T` is transported, which is governed by the a
 .. math::
 
    \der{T}{t}
-   +
-   u_j \der{T}{x_j}
    =
+   -
+   u_j \der{T}{x_j}
+   +
    \frac{1}{Re Sc}
    \der{}{x_j} \der{T}{x_j},
 

docs/source/equation/spectral.rst

@@ -1,3 +1,6 @@
+
+.. _spectral:
+
 ################################
 Equations in the spectral domain
 ################################
@@ -44,46 +47,49 @@ The external forcing terms :math:`\wav{a_x}{\kx \ky}` and :math:`\wav{a_y}{\kx \
 
    \lx \ly
    \left[
+      -
       I \kx \frac{2 \pi}{\lx}
       \sum_{\kx_0^{\prime} + \kx_1^{\prime} = \kx}
       \sum_{\ky_0^{\prime} + \ky_1^{\prime} = \ky}
       \wav{\ux}{\kx_0^{\prime} \ky_0^{\prime}}
       \wav{\ux}{\kx_1^{\prime} \ky_1^{\prime}}
-      +
+      -
       I \ky \frac{2 \pi}{\ly}
       \sum_{\kx_0^{\prime} + \kx_1^{\prime} = \kx}
       \sum_{\ky_0^{\prime} + \ky_1^{\prime} = \ky}
       \wav{\uy}{\kx_0^{\prime} \ky_0^{\prime}}
       \wav{\ux}{\kx_1^{\prime} \ky_1^{\prime}}
-      -
+      +
       \wav{a_x}{\kx \ky}
    \right],
 
    \lx \ly
    \left[
+      -
       I \kx \frac{2 \pi}{\lx}
       \sum_{\kx_0^{\prime} + \kx_1^{\prime} = \kx}
       \sum_{\ky_0^{\prime} + \ky_1^{\prime} = \ky}
       \wav{\ux}{\kx_0^{\prime} \ky_0^{\prime}}
       \wav{\uy}{\kx_1^{\prime} \ky_1^{\prime}}
-      +
+      -
       I \ky \frac{2 \pi}{\ly}
       \sum_{\kx_0^{\prime} + \kx_1^{\prime} = \kx}
       \sum_{\ky_0^{\prime} + \ky_1^{\prime} = \ky}
       \wav{\uy}{\kx_0^{\prime} \ky_0^{\prime}}
       \wav{\uy}{\kx_1^{\prime} \ky_1^{\prime}}
-      -
+      +
       \wav{a_y}{\kx \ky}
    \right],
 
    \lx \ly
    \left[
+      -
       I \kx \frac{2 \pi}{\lx}
       \sum_{\kx_0^{\prime} + \kx_1^{\prime} = \kx}
       \sum_{\ky_0^{\prime} + \ky_1^{\prime} = \ky}
       \wav{\ux}{\kx_0^{\prime} \ky_0^{\prime}}
       \wav{  T}{\kx_1^{\prime} \ky_1^{\prime}}
-      +
+      -
       I \ky \frac{2 \pi}{\ly}
       \sum_{\kx_0^{\prime} + \kx_1^{\prime} = \kx}
       \sum_{\ky_0^{\prime} + \ky_1^{\prime} = \ky}
@@ -134,27 +140,24 @@ Summary
 .. math::
 
    \der{\wav{\ux}{\kx \ky}}{t}
-   +
-   \wav{h_x}{\kx \ky}
    =
+   \wav{h_x}{\kx \ky}
    -
    I \kx \frac{2 \pi}{\lx} \wav{p}{\kx \ky}
    -
    \frac{1}{Re} \left[ \left( \kx \frac{2 \pi}{\lx} \right)^2 + \left( \ky \frac{2 \pi}{\ly} \right)^2 \right] \wav{\ux}{\kx \ky},
 
    \der{\wav{\uy}{\kx \ky}}{t}
-   +
-   \wav{h_y}{\kx \ky}
    =
+   \wav{h_y}{\kx \ky}
    -
    I \ky \frac{2 \pi}{\ly} \wav{p}{\kx \ky}
    -
    \frac{1}{Re} \left[ \left( \kx \frac{2 \pi}{\lx} \right)^2 + \left( \ky \frac{2 \pi}{\ly} \right)^2 \right] \wav{\uy}{\kx \ky},
 
    \der{\wav{T}{\kx \ky}}{t}
-   +
-   \wav{g}{\kx \ky}
    =
+   \wav{g}{\kx \ky}
    -
    \frac{1}{Re Sc} \left[ \left( \kx \frac{2 \pi}{\lx} \right)^2 + \left( \ky \frac{2 \pi}{\ly} \right)^2 \right] \wav{T}{\kx \ky},
 
@@ -164,60 +167,63 @@ where
 
    \wav{h_x}{\kx \ky}
    \equiv
+   -
    I \kx \frac{2 \pi}{\lx}
    \sum_{\kx_0^{\prime} + \kx_1^{\prime} = \kx}
    \sum_{\ky_0^{\prime} + \ky_1^{\prime} = \ky}
    \wav{\ux}{\kx_0^{\prime} \ky_0^{\prime}}
    \wav{\ux}{\kx_1^{\prime} \ky_1^{\prime}}
-   +
+   -
    I \ky \frac{2 \pi}{\ly}
    \sum_{\kx_0^{\prime} + \kx_1^{\prime} = \kx}
    \sum_{\ky_0^{\prime} + \ky_1^{\prime} = \ky}
    \wav{\uy}{\kx_0^{\prime} \ky_0^{\prime}}
    \wav{\ux}{\kx_1^{\prime} \ky_1^{\prime}}
-   -
+   +
    \wav{a_x}{\kx \ky},
 
    \wav{h_y}{\kx \ky}
    \equiv
+   -
    I \kx \frac{2 \pi}{\lx}
    \sum_{\kx_0^{\prime} + \kx_1^{\prime} = \kx}
    \sum_{\ky_0^{\prime} + \ky_1^{\prime} = \ky}
    \wav{\ux}{\kx_0^{\prime} \ky_0^{\prime}}
    \wav{\uy}{\kx_1^{\prime} \ky_1^{\prime}}
-   +
+   -
    I \ky \frac{2 \pi}{\ly}
    \sum_{\kx_0^{\prime} + \kx_1^{\prime} = \kx}
    \sum_{\ky_0^{\prime} + \ky_1^{\prime} = \ky}
    \wav{\uy}{\kx_0^{\prime} \ky_0^{\prime}}
    \wav{\uy}{\kx_1^{\prime} \ky_1^{\prime}}
-   -
+   +
    \wav{a_y}{\kx \ky},
 
    \wav{g}{\kx \ky}
    \equiv
+   -
    I \kx \frac{2 \pi}{\lx}
    \sum_{\kx_0^{\prime} + \kx_1^{\prime} = \kx}
    \sum_{\ky_0^{\prime} + \ky_1^{\prime} = \ky}
    \wav{\ux}{\kx_0^{\prime} \ky_0^{\prime}}
    \wav{  T}{\kx_1^{\prime} \ky_1^{\prime}}
-   +
+   -
    I \ky \frac{2 \pi}{\ly}
    \sum_{\kx_0^{\prime} + \kx_1^{\prime} = \kx}
    \sum_{\ky_0^{\prime} + \ky_1^{\prime} = \ky}
    \wav{\uy}{\kx_0^{\prime} \ky_0^{\prime}}
    \wav{  T}{\kx_1^{\prime} \ky_1^{\prime}}.
 
-To eliminate the pressure from the momentum equation, I consider to compute the inner product of the wave vector and the momentum balance, namely the sum of
+To eliminate the pressure from the momentum equation, I consider the inner product of the wave vector and the momentum balance, namely the sum of
 
 .. math::
 
    I \kx \frac{2 \pi}{\lx}
    \der{\wav{\ux}{\kx \ky}}{t}
-   +
+   =
    I \kx \frac{2 \pi}{\lx}
    \wav{h_x}{\kx \ky}
-   =
+   +
    \left( \kx \frac{2 \pi}{\lx} \right)^2 \wav{p}{\kx \ky}
    -
    \frac{1}{Re} \left[
@@ -233,10 +239,10 @@ and
 
    I \ky \frac{2 \pi}{\ly}
    \der{\wav{\uy}{\kx \ky}}{t}
-   +
+   =
    I \ky \frac{2 \pi}{\ly}
    \wav{h_y}{\kx \ky}
-   =
+   +
    \left( \ky \frac{2 \pi}{\ly} \right)^2 \wav{p}{\kx \ky}
    -
    \frac{1}{Re} \left[
@@ -250,9 +256,10 @@ Since the temporal derivative terms and the diffusive terms are zero because of
 
 .. math::
 
+   -
    I \kx \frac{2 \pi}{\lx}
    \wav{h_x}{\kx \ky}
-   +
+   -
    I \ky \frac{2 \pi}{\ly}
    \wav{h_y}{\kx \ky}
    =
@@ -269,9 +276,8 @@ which is used to eliminate the pressure, giving
 
    \der{\wav{\ux}{\mkx \mky}}{t}
    =
-   -
    \wav{h_x}{\mkx \mky}
-   +
+   -
    \frac{\mkx}{\mkx^2 + \mky^2}
    \left(
       \mkx \wav{h_x}{\mkx \mky}
@@ -283,19 +289,18 @@ which is used to eliminate the pressure, giving
 
    \der{\wav{\uy}{\mkx \mky}}{t}
    =
-   -
    \wav{h_y}{\mkx \mky}
-   +
+   -
    \frac{\mky}{\mkx^2 + \mky^2}
    \left(
       \mkx \wav{h_x}{\mkx \mky}
       +
       \mky \wav{h_y}{\mkx \mky}
    \right)
    -
-   \frac{1}{Re} \left( \mkx^2 + \mky^2 \right) \wav{\uy}{\mkx \mky},
+   \frac{1}{Re} \left( \mkx^2 + \mky^2 \right) \wav{\uy}{\mkx \mky}.
 
-where
+Recall that
 
 .. math::
 

docs/source/numerical_method/adv.rst

@@ -2,5 +2,38 @@
 Advective terms
 ###############
 
-TBA
+********
+Overview
+********
+
+To evaluate the convolution sums efficiently, instead of computing
+
+.. math::
+
+   \sum_{\ix_0^{\prime} + \ix_1^{\prime} = \ix}
+   \sum_{\iy_0^{\prime} + \iy_1^{\prime} = \iy}
+   \wav{p}{\ix_0^{\prime} \iy_0^{\prime}}
+   \wav{q}{\ix_1^{\prime} \iy_1^{\prime}},
+
+where :math:`p` and :math:`q` are the arguments and whose computational cost is :math:`\mathcal{O} \left( N_x^2 \times N_y^2 \right)`, I consider their representations in the physical domain, i.e. computing the product:
+
+.. math::
+
+   p q
+
+directly, whose cost is :math:`\mathcal{O} \left( N_x \log N_x \times N_y \log N_y \right)`, which is known as the transform method.
+
+**************
+Implementation
+**************
+
+#. ``src/fluid/physical.c``
+
+   This file contains a function to compute the description of each flow field in the physical place by performing the multi-dimensional inverse Fourier transform.
+
+#. ``src/fluid/slope.c``
+
+   This file contains several functions which evaluate the right-hand-side terms of the Runge-Kutta scheme.
+   In particular, ``convolute`` computes the convolution sum (product of the given two arrays), while ``compute_adv`` computes the advective terms by multiplying pre-factors corresponding to the spatial derivative in each direction and the pre-factors.
+   Finally ``project_velocity`` is necessary for the velocity field to satisfy the divergence-free condition (see the :ref:`equations <spectral>`, where two terms are involved in each direction to describe the advection).