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PINNs-Laminar-Flow-Sudden-Construction-Channel

PINNs-Laminar-Flow-Sudden-Construction-Channel

Paper from what the idea came from: https://arxiv.org/pdf/2002.10558.pdf

Original Implementation (Tensorflow 1): https://github.com/Raocp/PINN-laminar-flow

Paper: https://github.com/sleshworld/PINNs-Laminar-Flow-Sudden-Construction-Channel/blob/main/VKR_TroshinOf.pdf

Channel drawio (4)

Navier-Stokes Equation

$$ \nabla \cdot \textbf{v} = 0 $$ $$ \frac{\partial{\textbf{v}}}{\partial{t}} + (\textbf{v} \cdot \nabla)\textbf{v} = -\frac{1}{\rho}\nabla p + \frac{\mu}{\rho}\nabla^2\textbf{v} + \textbf{g} $$ $$ \textbf{v} = (u, v) $$

This representation of the equations is complicated for PINN because of the presence of second-order derivatives, so we transform:

$$\frac{\partial{\textbf{v}}}{\partial{t}} + (\textbf{v} \cdot \nabla)\textbf{v} = -\frac{1}{\rho}\nabla \sigma + \textbf{g}$$ $$\sigma = p \textbf{I} + \mu(\nabla\textbf{v} + \nabla\textbf{v}^T)$$

, where $\sigma$ - is a stress tensor, and $p = -tr(\sigma)/2$.


$$\sigma = p \textbf{I} + \mu(\nabla\textbf{v} + \nabla\textbf{v}^T), $$

где $\textbf{I}$- unit tensor.

$$ \sigma = \begin{bmatrix} s_{11} & s_{12} \\ s_{21} & s_{22} \end{bmatrix} $$

  • $s_{11} = -p + 2\mu \frac{\partial u}{\partial x}$
  • $s_{22} = -p + 2\mu \frac{\partial v}{\partial y}$
  • $s_{12} = s_{21} = \mu ( \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x})$
  • $p = - \frac{s_{11} + s_{22}}{2} = -tr(\sigma)/2$

In this paper, instead of the continuity equation, the velocity is calculated from the current function. $$[u, v, w]=\nabla \times [0, 0, \psi]$$

uvp_step

White_Arhitecture drawio (1)