Momentum equation in Nek5000

Nek routines to compute the different terms in the momentum equation in strong and weak formulation.

NS

$$ \begin{align} \frac{\partial u_i}{\partial t} &= -\frac{\partial p}{\partial x_i} + \frac{1}{Re}\Delta u_i + f_i\ - u_j \frac{\partial u_i}{\partial x_j}\\ \frac{\partial u_i}{\partial x_i}&=0 \end{align} $$

Semi-discrete NS in Nek5000

$$ \begin{align} \mathbf{M}\frac{\partial u_i}{\partial t} &= \mathbf{D}_i^T p - \frac{1}{Re}\mathbf{K}u_i + \mathbf{M}f_i\ - \mathbf{C}u_i\\ \mathbf{D}_i u_i&=0 \end{align} $$

• $\mathbf{C}$: Convection operator
• $\mathbf{M}$: Mass matrix
• $\mathbf{K}$: Stiffness matrix
• $\frac{\partial u_i}{\partial t}$ : Computed using BDF2, considering current solution vx and two previous time steps vxlag. Then scaled by the mass matrix
• $\mathbf{D}_i^T p$ : opgradt(px,py,pz,pr) (in navier1.f) where pr is defined in the pressure mesh and the outpouts px, py and pz are defined in the velocity mesh.
• $\mathbf{K} u_i$ : wlaplacian(lapu,u,diff,1) (in navier1.f)
• $M f_i$ : makeuf will call user defined forces and put them in BFX, BFY and BFZ
• The convective term is computed in nek as: convop(convu,u_i) and then scaled by the mass matrix (check advab routine in navier1.f)

Time integration: Fractional step method

$$ \begin{align} \mathbf{H}u_{i}^&=\mathbf{D}_{i}^Tp^{n} + h_i^{n+1}\ \frac{b_0}{\Delta t}\mathbf{D}i\mathbf{M}^{-1}\mathbf{D}{i}^T(p^{n+1}-p^{n})&= \mathbf{D}_i u_i^\ u_{i}^{n+1} &= u_i^* + \frac{\Delta t}{b_0}\mathbf{M}^{-1}\mathbf{D}_{i}^T(p^{n+1}-p^{n}) \end{align} $$

• Helmoltz operator: $\mathbf{H}= \frac{b_0}{\Delta t}\mathbf{M} + \frac{1}{Re}\mathbf{K}$
• $h_i^{n+1}=-\sum_{j=1}^k\frac{b_j}{\Delta t} \mathbf{M}u_i^{n+1-j} - \sum_{j=1}^k a_j\mathbf{C}u_i^{n+1-j} + \mathbf{M}f_i^n$