Navier-Stokes equations

Nonlinear strong form

\[\begin{split} \begin{align*} &\text{Find}~\textbf{u}(\textbf{x}, t): \Omega\times[0, \infty) \to \mathbb{R}^d~\text{and}~p(\textbf{x}, t): \Omega\times[0, \infty) \to \mathbb{R}~\text{such that} \\ &\mathbb{IBVP}\begin{cases} \nabla\cdot\textbf{u} = 0 & \\ \rho \left(\frac{\partial\textbf{u}}{\partial t}+\textbf{u}\cdot\nabla\textbf{u}\right)=-\nabla p + \nabla\cdot\tau + \textbf{f} & \forall(\textbf{x}, t)\in\Omega\times[0,\infty) \\ \textbf{u}=\textbf{u}_0 & \forall(\textbf{x},t)\in\Omega\times\{0\} \\ p=p_0 & \forall(\textbf{x},t)\in\Omega\times\{0\} \\ \textbf{u} = \textbf{u}_{\text{E}} & \forall \textbf{x}\in\partial\Omega_{\text{E}} \times[0, \infty) \\ (-p\mathsf{I}+\tau)\cdot\textbf{n} = \boldsymbol{\tau}_{\text{N}} & \forall(\textbf{x},t)\in\partial\Omega_{\text{N}}\times[0, \infty)~,~\partial\Omega_{\text{N}}=\partial\Omega/\partial\Omega_{\text{E}} \end{cases} \\ &\text{given} \\ &\mathbb{S}\begin{cases} \Omega & \text{domain}\\ \textbf{u}_0(\textbf{x}) & \text{velocity initial condition}\\ p_0(\textbf{x}) & \text{pressure initial condition}\\ \textbf{u}_{\text{E}}(\textbf{x}, t)~,~\partial\Omega_{\text{E}} & \text{velocity essential boundary condition} \\ \boldsymbol{\tau}_{\text{N}}(\textbf{x}, t)~,~\partial\Omega_{\text{N}} & \text{traction natural boundary condition} \\ \tau(\textbf{u}) & \text{deviatoric stress constitutive relation} \\ \textbf{f}(\textbf{x}, t) & \text{body force} \\ \rho(\textbf{x}, t) & \text{density} \end{cases} \end{align*} \end{split}\]

Linearized weak forms

Incremental pressure correction scheme

\[\begin{split} \begin{aligned} &\text{Find} \\ &\widetilde{\textbf{u}}^{n+1}\in V_{\textbf{u}}, \\ &p^{n+1}\in V_p, \\ &\textbf{u}^{n+1}\in V_{\textbf{u}} \\ &\text{such that} \\ &\mathbb{F} \begin{cases} \begin{align*} F_1(\widetilde{\textbf{u}}^{n+1}, \textbf{v}) &= \int_\Omega\text{d}\Omega~\mathcal{D}_\rho(\rho)\textbf{v}\cdot\frac{\widetilde{\textbf{u}}^{n+1}-\textbf{u}^n}{\Delta t^n} + \mathcal{D}_\rho(\rho)\textbf{v}\cdot\mathcal{D}_{\textbf{u}}(\textbf{u}\cdot\nabla\textbf{u}) \\ &\qquad\quad +\varepsilon(\textbf{v}):(-p^n\mathsf{I} + \mathcal{D}_\tau(\boldsymbol{\tau})) -\textbf{v}\cdot\mathcal{D}_{\textbf{f}}(\textbf{f}) \\ &\quad -\int_{\partial\Omega}\text{d}\Gamma~\textbf{v}\cdot(-p^n\mathsf{I} + \mathcal{D}_\tau(\boldsymbol{\tau}))\cdot\textbf{n} \\ &=0 \quad\forall \textbf{v} \in V_{\textbf{u}} \end{align*} \\ F_2(p^{n+1}, q) = \int_\Omega\text{d}\Omega~\nabla q\cdot\nabla p^{n+1} - \nabla q\cdot\nabla p^n + q\rho\frac{\nabla\cdot\widetilde{\textbf{u}}^{n+1}}{\Delta t^n} =0 \quad\forall q\in V_p \\ F_3(\textbf{u}^{n+1}, \textbf{v}) = \int_\Omega\text{d}\Omega~\textbf{v}\cdot\rho\frac{\textbf{u}^{n+1}-\widetilde{\textbf{u}}^{n+1}}{\Delta t^n} + \textbf{v}\cdot\nabla p^{n+1} - \textbf{v}\cdot\nabla p^n = 0 \quad\forall \textbf{v} \in V_{\textbf{u}} \end{cases} \end{aligned} \end{split}\]

Chorin’s scheme

\[\begin{split} \begin{aligned} &\text{Find} \\ &\widetilde{\textbf{u}}^{n+1}\in V_{\textbf{u}}, \\ &p^{n+1}\in V_p, \\ &\textbf{u}^{n+1}\in V_{\textbf{u}} \\ &\text{such that} \\ &\mathbb{F} \begin{cases} \begin{align*} F_1(\widetilde{\textbf{u}}^{n+1}, \textbf{v}) &= \int_\Omega\text{d}\Omega~\mathcal{D}_\rho(\rho)\textbf{v}\cdot\frac{\widetilde{\textbf{u}}^{n+1}-\textbf{u}^n}{\Delta t^n} + \mathcal{D}_\rho(\rho)\textbf{v}\cdot\mathcal{D}_{\textbf{u}}(\textbf{u}\cdot\nabla\textbf{u}) \\ &\qquad\quad + \nabla\textbf{v}\cdot\nabla\mathcal{D}_{\tau}(\boldsymbol{\tau}) -\textbf{v}\cdot\mathcal{D}_{\textbf{f}}(\textbf{f}) \\ &=0 \quad\forall \textbf{v} \in V_{\textbf{u}} \end{align*} \\ F_2(p^{n+1}, q) = \int_\Omega\text{d}\Omega~\nabla q\cdot\nabla p^{n+1} + q\rho\frac{\nabla\cdot\widetilde{\textbf{u}}^{n+1}}{\Delta t^n}=0 \quad\forall q\in V_p \\ F_3(\textbf{u}^{n+1}, \textbf{v}) = \int_\Omega\text{d}\Omega~\textbf{v}\cdot\rho\frac{\textbf{u}^{n+1}-\widetilde{\textbf{u}}^{n+1}}{\Delta t^n} + \textbf{v}\cdot\nabla p^{n+1} =0 \quad\forall \textbf{v} \in V_{\textbf{u}} \end{cases} \end{aligned} \end{split}\]

Streamfunction-vorticity formulation

General definitions

\[\begin{split} \begin{align*} &\textbf{u}=\nabla\times\boldsymbol{\psi} \iff \nabla\cdot\textbf{u}=0~, \\ &\tau(\textbf{u}) = \frac{\mu}{2}\left(\nabla\textbf{u} + \nabla\textbf{u}^{\mathsf{T}}\right)~, \\ &\boldsymbol{\omega}=\nabla\times\textbf{u}=\nabla\times(\nabla\times\boldsymbol{\psi}) \\ &\text{and}~\nabla\rho=\nabla\mu=\textbf{0} \\ &\implies\rho\left(\frac{\partial\boldsymbol{\omega}}{\partial t}+(\nabla\times\boldsymbol{\psi})\cdot\nabla\boldsymbol{\omega} - \boldsymbol{\omega}\cdot\nabla(\nabla\times\boldsymbol{\psi})\right)=\mu\nabla^2\boldsymbol{\omega} + \nabla\times\textbf{f} \end{align*} \end{split}\]

Two-dimensional Cartesian definitions

\[\begin{split} \begin{align*} &\boldsymbol{\psi}=\psi\textbf{e}_z\implies\textbf{u}=\frac{\partial\psi}{\partial y}\textbf{e}_x - \frac{\partial\psi}{\partial x}\textbf{e}_y \\ &\boldsymbol{\omega}=\omega\textbf{e}_z=\left(\frac{\partial u_y}{\partial x} - \frac{\partial u_x}{\partial y}\right)\textbf{e}_z \\ &\textbf{f}=f_x\textbf{e}_x + f_y\textbf{e}_y \end{align*} \end{split}\]

Strong form

\[\begin{split} \begin{align*} &\text{Find}~\psi(\textbf{x}, t): \Omega\times[0, \infty) \to \mathbb{R}~\text{and}~\omega(\textbf{x}, t): \Omega\times[0, \infty) \to \mathbb{R}~\text{such that} \\ &\mathbb{IBVP}\begin{cases} \nabla^2\psi =\omega & \\ \rho\left(\frac{\partial\omega}{\partial t}+\left(-\frac{\partial\psi}{\partial y}, \frac{\partial\psi}{\partial x}\right)\cdot\nabla\omega\right) =\mu\nabla^2\omega + \frac{\partial f_y}{\partial x} - \frac{\partial f_x}{\partial y} & \forall(\textbf{x}, t)\in\Omega\times[0,\infty) \\ \omega(\textbf{x}, t=0)=\omega_0 & \forall \textbf{x}\in\Omega \\ \psi=\psi_{\text{D}} & \forall \textbf{x}\in\partial\Omega_{\text{D}, \psi} \times [0,\infty] \\ \textbf{n}\cdot\nabla\psi = \psi_{\text{N}} & \forall\textbf{x}\in\partial\Omega_{\text{N}, \psi} \times [0,\infty]~,~\partial\Omega_{\text{N}, \psi}=\partial\Omega/\partial\Omega_{\text{D}, \psi} \\ \omega=\omega_{\text{D}} & \forall \textbf{x}\in\partial\Omega_{\text{D},\omega} \times [0,\infty] \\ \textbf{n}\cdot\nabla\omega = \omega_{\text{N}} & \forall\textbf{x}\in\partial\Omega_{\text{N},\omega}\times[0,\infty]~,~\partial\Omega_{\text{N},\omega}=\partial\Omega/\partial\Omega_{\text{D},\omega} \end{cases}~. \end{align*} \end{split}\]