Equations in the spectral domain¶
I apply the Fourier series expansion to the governing equations to describe them in the spectral domain.
Incompressibility¶
\[\lx \ly
\left(
I \kx \frac{2 \pi}{\lx} \wav{\ux}{\kx \ky}
+
I \ky \frac{2 \pi}{\ly} \wav{\uy}{\kx \ky}
\right)
=
0.\]
Temporal evolution¶
\[ \begin{align}\begin{aligned}\lx \ly
\der{\wav{\ux}{\kx \ky}}{t},\\\lx \ly
\der{\wav{\uy}{\kx \ky}}{t},\\\lx \ly
\der{\wav{T}{\kx \ky}}{t}.\end{aligned}\end{align} \]
Advective terms¶
The external forcing terms \(\wav{a_x}{\kx \ky}\) and \(\wav{a_y}{\kx \ky}\) are included.
\[ \begin{align}\begin{aligned}\lx \ly
\left[
-
I \kx \frac{2 \pi}{\lx}
\sum_{\kx_0^{\prime} + \kx_1^{\prime} = \kx}
\sum_{\ky_0^{\prime} + \ky_1^{\prime} = \ky}
\wav{\ux}{\kx_0^{\prime} \ky_0^{\prime}}
\wav{\ux}{\kx_1^{\prime} \ky_1^{\prime}}
-
I \ky \frac{2 \pi}{\ly}
\sum_{\kx_0^{\prime} + \kx_1^{\prime} = \kx}
\sum_{\ky_0^{\prime} + \ky_1^{\prime} = \ky}
\wav{\uy}{\kx_0^{\prime} \ky_0^{\prime}}
\wav{\ux}{\kx_1^{\prime} \ky_1^{\prime}}
+
\wav{a_x}{\kx \ky}
\right],\\\lx \ly
\left[
-
I \kx \frac{2 \pi}{\lx}
\sum_{\kx_0^{\prime} + \kx_1^{\prime} = \kx}
\sum_{\ky_0^{\prime} + \ky_1^{\prime} = \ky}
\wav{\ux}{\kx_0^{\prime} \ky_0^{\prime}}
\wav{\uy}{\kx_1^{\prime} \ky_1^{\prime}}
-
I \ky \frac{2 \pi}{\ly}
\sum_{\kx_0^{\prime} + \kx_1^{\prime} = \kx}
\sum_{\ky_0^{\prime} + \ky_1^{\prime} = \ky}
\wav{\uy}{\kx_0^{\prime} \ky_0^{\prime}}
\wav{\uy}{\kx_1^{\prime} \ky_1^{\prime}}
+
\wav{a_y}{\kx \ky}
\right],\\\lx \ly
\left[
-
I \kx \frac{2 \pi}{\lx}
\sum_{\kx_0^{\prime} + \kx_1^{\prime} = \kx}
\sum_{\ky_0^{\prime} + \ky_1^{\prime} = \ky}
\wav{\ux}{\kx_0^{\prime} \ky_0^{\prime}}
\wav{ T}{\kx_1^{\prime} \ky_1^{\prime}}
-
I \ky \frac{2 \pi}{\ly}
\sum_{\kx_0^{\prime} + \kx_1^{\prime} = \kx}
\sum_{\ky_0^{\prime} + \ky_1^{\prime} = \ky}
\wav{\uy}{\kx_0^{\prime} \ky_0^{\prime}}
\wav{ T}{\kx_1^{\prime} \ky_1^{\prime}}
\right].\end{aligned}\end{align} \]
Pressure-gradient terms¶
\[ \begin{align}\begin{aligned}- \lx \ly I \kx \frac{2 \pi}{\lx} \wav{p}{\kx \ky},\\- \lx \ly I \ky \frac{2 \pi}{\ly} \wav{p}{\kx \ky}.\end{aligned}\end{align} \]
Diffusive terms¶
\[ \begin{align}\begin{aligned}- \frac{1}{Re} \lx \ly \left[ \left( \kx \frac{2 \pi}{\lx} \right)^2 + \left( \ky \frac{2 \pi}{\ly} \right)^2 \right] \wav{\ux}{\kx \ky},\\- \frac{1}{Re} \lx \ly \left[ \left( \kx \frac{2 \pi}{\lx} \right)^2 + \left( \ky \frac{2 \pi}{\ly} \right)^2 \right] \wav{\uy}{\kx \ky},\\- \frac{1}{Re Sc} \lx \ly \left[ \left( \kx \frac{2 \pi}{\lx} \right)^2 + \left( \ky \frac{2 \pi}{\ly} \right)^2 \right] \wav{ T}{\kx \ky}.\end{aligned}\end{align} \]
Forcing terms¶
They are embedded in the advective terms.
Summary¶
\[I \kx \frac{2 \pi}{\lx} \wav{\ux}{\kx \ky}
+
I \ky \frac{2 \pi}{\ly} \wav{\uy}{\kx \ky}
=
0.\]
\[ \begin{align}\begin{aligned}\der{\wav{\ux}{\kx \ky}}{t}
=
\wav{h_x}{\kx \ky}
-
I \kx \frac{2 \pi}{\lx} \wav{p}{\kx \ky}
-
\frac{1}{Re} \left[ \left( \kx \frac{2 \pi}{\lx} \right)^2 + \left( \ky \frac{2 \pi}{\ly} \right)^2 \right] \wav{\ux}{\kx \ky},\\\der{\wav{\uy}{\kx \ky}}{t}
=
\wav{h_y}{\kx \ky}
-
I \ky \frac{2 \pi}{\ly} \wav{p}{\kx \ky}
-
\frac{1}{Re} \left[ \left( \kx \frac{2 \pi}{\lx} \right)^2 + \left( \ky \frac{2 \pi}{\ly} \right)^2 \right] \wav{\uy}{\kx \ky},\\\der{\wav{T}{\kx \ky}}{t}
=
\wav{g}{\kx \ky}
-
\frac{1}{Re Sc} \left[ \left( \kx \frac{2 \pi}{\lx} \right)^2 + \left( \ky \frac{2 \pi}{\ly} \right)^2 \right] \wav{T}{\kx \ky},\end{aligned}\end{align} \]
where
\[ \begin{align}\begin{aligned}\wav{h_x}{\kx \ky}
\equiv
-
I \kx \frac{2 \pi}{\lx}
\sum_{\kx_0^{\prime} + \kx_1^{\prime} = \kx}
\sum_{\ky_0^{\prime} + \ky_1^{\prime} = \ky}
\wav{\ux}{\kx_0^{\prime} \ky_0^{\prime}}
\wav{\ux}{\kx_1^{\prime} \ky_1^{\prime}}
-
I \ky \frac{2 \pi}{\ly}
\sum_{\kx_0^{\prime} + \kx_1^{\prime} = \kx}
\sum_{\ky_0^{\prime} + \ky_1^{\prime} = \ky}
\wav{\uy}{\kx_0^{\prime} \ky_0^{\prime}}
\wav{\ux}{\kx_1^{\prime} \ky_1^{\prime}}
+
\wav{a_x}{\kx \ky},\\\wav{h_y}{\kx \ky}
\equiv
-
I \kx \frac{2 \pi}{\lx}
\sum_{\kx_0^{\prime} + \kx_1^{\prime} = \kx}
\sum_{\ky_0^{\prime} + \ky_1^{\prime} = \ky}
\wav{\ux}{\kx_0^{\prime} \ky_0^{\prime}}
\wav{\uy}{\kx_1^{\prime} \ky_1^{\prime}}
-
I \ky \frac{2 \pi}{\ly}
\sum_{\kx_0^{\prime} + \kx_1^{\prime} = \kx}
\sum_{\ky_0^{\prime} + \ky_1^{\prime} = \ky}
\wav{\uy}{\kx_0^{\prime} \ky_0^{\prime}}
\wav{\uy}{\kx_1^{\prime} \ky_1^{\prime}}
+
\wav{a_y}{\kx \ky},\\\wav{g}{\kx \ky}
\equiv
-
I \kx \frac{2 \pi}{\lx}
\sum_{\kx_0^{\prime} + \kx_1^{\prime} = \kx}
\sum_{\ky_0^{\prime} + \ky_1^{\prime} = \ky}
\wav{\ux}{\kx_0^{\prime} \ky_0^{\prime}}
\wav{ T}{\kx_1^{\prime} \ky_1^{\prime}}
-
I \ky \frac{2 \pi}{\ly}
\sum_{\kx_0^{\prime} + \kx_1^{\prime} = \kx}
\sum_{\ky_0^{\prime} + \ky_1^{\prime} = \ky}
\wav{\uy}{\kx_0^{\prime} \ky_0^{\prime}}
\wav{ T}{\kx_1^{\prime} \ky_1^{\prime}}.\end{aligned}\end{align} \]
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
\[I \kx \frac{2 \pi}{\lx}
\der{\wav{\ux}{\kx \ky}}{t}
=
I \kx \frac{2 \pi}{\lx}
\wav{h_x}{\kx \ky}
+
\left( \kx \frac{2 \pi}{\lx} \right)^2 \wav{p}{\kx \ky}
-
\frac{1}{Re} \left[
\left( \kx \frac{2 \pi}{\lx} \right)^2
+
\left( \ky \frac{2 \pi}{\ly} \right)^2
\right]
I \kx \frac{2 \pi}{\lx} \wav{\ux}{\kx \ky}\]
and
\[I \ky \frac{2 \pi}{\ly}
\der{\wav{\uy}{\kx \ky}}{t}
=
I \ky \frac{2 \pi}{\ly}
\wav{h_y}{\kx \ky}
+
\left( \ky \frac{2 \pi}{\ly} \right)^2 \wav{p}{\kx \ky}
-
\frac{1}{Re} \left[
\left( \kx \frac{2 \pi}{\lx} \right)^2
+
\left( \ky \frac{2 \pi}{\ly} \right)^2
\right]
I \ky \frac{2 \pi}{\ly} \wav{\uy}{\kx \ky}.\]
Since the temporal derivative terms and the diffusive terms are zero because of the incompressibility, I obtain
\[-
I \kx \frac{2 \pi}{\lx}
\wav{h_x}{\kx \ky}
-
I \ky \frac{2 \pi}{\ly}
\wav{h_y}{\kx \ky}
=
\left[
\left( \kx \frac{2 \pi}{\lx} \right)^2
+
\left( \ky \frac{2 \pi}{\ly} \right)^2
\right]
\wav{p}{\kx \ky},\]
which is used to eliminate the pressure, giving
\[ \begin{align}\begin{aligned}\der{\wav{\ux}{\mkx \mky}}{t}
=
\wav{h_x}{\mkx \mky}
-
\frac{\mkx}{\mkx^2 + \mky^2}
\left(
\mkx \wav{h_x}{\mkx \mky}
+
\mky \wav{h_y}{\mkx \mky}
\right)
-
\frac{1}{Re} \left( \mkx^2 + \mky^2 \right) \wav{\ux}{\mkx \mky},\\\der{\wav{\uy}{\mkx \mky}}{t}
=
\wav{h_y}{\mkx \mky}
-
\frac{\mky}{\mkx^2 + \mky^2}
\left(
\mkx \wav{h_x}{\mkx \mky}
+
\mky \wav{h_y}{\mkx \mky}
\right)
-
\frac{1}{Re} \left( \mkx^2 + \mky^2 \right) \wav{\uy}{\mkx \mky}.\end{aligned}\end{align} \]
Recall that
\[ \begin{align}\begin{aligned}\mkx
\equiv
\kx \frac{2 \pi}{\lx},\\\mky
\equiv
\ky \frac{2 \pi}{\ly}.\end{aligned}\end{align} \]