whucsu / nsfnets

PINN in solving Navier–Stokes equation

NSFnets

PINN in solving Navier–Stokes equation

tiny differences between those four NSFnets

VP_NSFnets1.py is the code to simulate Kovasznay flow(2 dimension without time)
VP_NSFnets2.py is the code to simulate cylinder wake(2 dimension with time)
VP_NSFnets3.py is the code to simulate Beltrami flow(3 dimension with time)
VP_NSFnets4.py is the code to simulate Turbulent channel flow(3 dimension with time)

loss function defined in the NSFnets

$\begin{aligned} L &=L_{e}+\alpha L_{b}+\beta L_{i} \ L_{e} &=\frac{1}{N_{e}} \sum_{i=1}^{4} \sum_{n=1}^{N_{e}}\left|e_{V P i}^{n}\right|^{2} \ L_{b} &=\frac{1}{N_{b}} \sum_{n=1}^{N_{b}}\left|\mathbf{u}^{n}-\mathbf{u}{b}^{n}\right|^{2} \ L{i} &=\frac{1}{N_{i}} \sum_{n=1}^{N_{i}}\left|\mathbf{u}^{n}-\mathbf{u}_{i}^{n}\right|^{2} \end{aligned}$

\begin{aligned} u(x, y, z, t)=&-a\left[e^{a x} \sin (a y+d z)+e^{a z} \cos (a x+d y)\right] e^{-d^{2} t} \ v(x, y, z, t)=&-a\left[e^{a y} \sin (a z+d x)+e^{a x} \cos (a y+d z)\right] e^{-d^{2} t} \ w(x, y, z, t)=&-a\left[e^{a z} \sin (a x+d y)+e^{a y} \cos (a z+d x)\right] e^{-d^{2} t} \ p(x, y, z, t)=&-\frac{1}{2} a^{2}\left[e^{2 a x}+e^{2 a y}+e^{2 a z}+2 \sin (a x+d y) \cos (a z+d x) e^{a(y+z)}\right.\ &+2 \sin (a y+d z) \cos (a x+d y) e^{a(z+x)} \ &\left.+2 \sin (a z+d x) \cos (a y+d z) e^{a(x+y)}\right] e^{-2 d^{2} t} \end{aligned}

$e_{V P 1}=\partial_{t} u+u \partial_{x} u+v \partial_{y} u+w \partial_{z} u+\partial_{x} p-1 / \operatorname{Re}\left(\partial_{x x}^{2} u+\partial_{y y}^{2} u+\partial_{z z}^{2} u\right)$

analytic solution for case 3 Beltrami flow

$$\begin{aligned} u(x, y, z, t)=&-a\left[e^{a x} \sin (a y+d z)+e^{a z} \cos (a x+d y)\right] e^{-d^{2} t} \\ v(x, y, z, t)=&-a\left[e^{a y} \sin (a z+d x)+e^{a x} \cos (a y+d z)\right] e^{-d^{2} t} \\ w(x, y, z, t)=&-a\left[e^{a z} \sin (a x+d y)+e^{a y} \cos (a z+d x)\right] e^{-d^{2} t} \\ p(x, y, z, t)=&-\frac{1}{2} a^{2}\left[e^{2 a x}+e^{2 a y}+e^{2 a z}+2 \sin (a x+d y) \cos (a z+d x) e^{a(y+z)}\right.\\ &+2 \sin (a y+d z) \cos (a x+d y) e^{a(z+x)} \\ &\left.+2 \sin (a z+d x) \cos (a y+d z) e^{a(x+y)}\right] e^{-2 d^{2} t} \end{aligned}$$