Datasets:
Update README.md
Browse files
README.md
CHANGED
@@ -62,22 +62,25 @@ $$
|
|
62 |
where \\(\Delta = \nabla \cdot \nabla\\) is the spatial Laplacian, \\(b\\) is the buoyancy, \\(u = (u_x,u_y)\\) the (horizontal and vertical) velocity, and \\(p\\) is the pressure, \\(\vec{e}_z\\) is the unit vector in the vertical direction, with the additional constraints \\(\int p = 0\\) (pressure gauge).
|
63 |
|
64 |
The boundary conditions vertically are as follows:
|
65 |
-
|
|
|
66 |
\begin{align*}
|
67 |
b(z=0) = Lz ~~~,~~~ b(z=Lz) = 0
|
68 |
\\
|
69 |
u(z=0) = u(z=Lz) = 0
|
70 |
\end{align*}
|
71 |
-
|
72 |
|
73 |
These PDE are parameterized by the Rayleigh and Prandtl numbers through \\(\kappa\\) and \\(\nu\\).
|
74 |
-
|
|
|
75 |
\begin{align*}
|
76 |
\text{(thermal diffusivity)} ~~~~~~~ \kappa & = \big(\text{Rayleigh} * \text{Prandtl}\big)^{-\frac12}
|
77 |
\\
|
78 |
\text{(viscosity)} ~~~~~~~ \nu & = \bigg(\frac{\text{Rayleigh}}{\text{Prandtl}}\bigg)^{-\frac12}.
|
79 |
\end{align*}
|
80 |
-
|
|
|
81 |
![Gif](https://users.flatironinstitute.org/~polymathic/data/the_well/datasets/rayleigh_benard/gif/buoyancy_good_normalized.gif)
|
82 |
<!-- <div style="transform: rotate(90deg);">
|
83 |
<img src="https://users.flatironinstitute.org/~polymathic/data/the_well/datasets/rayleigh_benard/gif/buoyancy_normalized.gif" alt="Rotated GIF">
|
|
|
62 |
where \\(\Delta = \nabla \cdot \nabla\\) is the spatial Laplacian, \\(b\\) is the buoyancy, \\(u = (u_x,u_y)\\) the (horizontal and vertical) velocity, and \\(p\\) is the pressure, \\(\vec{e}_z\\) is the unit vector in the vertical direction, with the additional constraints \\(\int p = 0\\) (pressure gauge).
|
63 |
|
64 |
The boundary conditions vertically are as follows:
|
65 |
+
|
66 |
+
$$
|
67 |
\begin{align*}
|
68 |
b(z=0) = Lz ~~~,~~~ b(z=Lz) = 0
|
69 |
\\
|
70 |
u(z=0) = u(z=Lz) = 0
|
71 |
\end{align*}
|
72 |
+
$$
|
73 |
|
74 |
These PDE are parameterized by the Rayleigh and Prandtl numbers through \\(\kappa\\) and \\(\nu\\).
|
75 |
+
|
76 |
+
$$
|
77 |
\begin{align*}
|
78 |
\text{(thermal diffusivity)} ~~~~~~~ \kappa & = \big(\text{Rayleigh} * \text{Prandtl}\big)^{-\frac12}
|
79 |
\\
|
80 |
\text{(viscosity)} ~~~~~~~ \nu & = \bigg(\frac{\text{Rayleigh}}{\text{Prandtl}}\bigg)^{-\frac12}.
|
81 |
\end{align*}
|
82 |
+
$$
|
83 |
+
|
84 |
![Gif](https://users.flatironinstitute.org/~polymathic/data/the_well/datasets/rayleigh_benard/gif/buoyancy_good_normalized.gif)
|
85 |
<!-- <div style="transform: rotate(90deg);">
|
86 |
<img src="https://users.flatironinstitute.org/~polymathic/data/the_well/datasets/rayleigh_benard/gif/buoyancy_normalized.gif" alt="Rotated GIF">
|