8.1
general documentation
Variables
Field names

Variables

char * est_error_pre_2
 
char * est_error_der_2
 
char * est_error_cor_2
 
char * est_error_tot_2
 

Detailed Description

Variable Documentation

◆ est_error_cor_2

est_error_cor_2

Error estimator for Navier-Stokes: correction.

The estimator $ \eta^{\,corr}_{\,i,k}(\vect{u}^{\,n+1})$ comes directly from the mass flow calculated with the updated velocity field:

\begin{eqnarray*} \eta^{\,corr}_{\,i,k}(\vect{u}^{\,n+1})= |\Omega_i|^{\,\delta_{\,2,k}}\ |div (\rho^n \vect{u}^{n+1}) - \Gamma| \end{eqnarray*}

  • The velocities $\vect{u}^{n+1}$ are taken at the cell centers, the divergence is calculated after projection on the faces. $ \,\delta_{\,2,k}$ represents the Kronecker symbol.
  • The first family, k=1, is the absolute raw value of the divergence of the mass flow minus the mass source term. The second family, $k=2$, represents a physical property and allows to evaluate the difference in $kg.s^{\,-1}$.
  • Ideally, it is equal to zero when the Poisson equation is solved exactly and the projection from the mass flux at the faces to the velocity at the cell centers is made in a set of functions with null divergence.

◆ est_error_der_2

est_error_der_2

Error estimator for Navier-Stokes: drift.

The estimator $\eta^{\,der}_{\,i,k}(\vect{u}^{\,n+1})$ is based on the following quantity (intrinsic to the code):

\begin{eqnarray*} \eta^{\,der}_{\,i,k}(\vect{u}^{\,n+1}) &=& {|\Omega_i|}^{(k-2)/2} || \divs (\text{corrected mass flow after the pressure step}) - \Gamma||_{{L}^{2}(\Omega_i)} \\ &=& {|\Omega_i|}^{(1-k)/2} | \divs (\text{corrected mass flow after the pressure step})- \Gamma| \end{eqnarray*}

  • Ideally, it is equal to zero when the Poisson equation related to the pressure is solved exactly.

◆ est_error_pre_2

est_error_pre_2

Error estimator for Navier-Stokes: prediction.

After the velocity prediction step (yielding $\vect{u}^*$), the estimator $\eta^{\,pred}_{\,i,k}(\vect{u}^*)$, local variable calculated at every cell $ \Omega_i $, is created from $\vect{\mathcal R}^{\,pred}(\vect{u}^*)$, which represents the residual of the equation solved during this step: $\vect{u}$ and $ P $:

\begin{eqnarray*} \vect{\mathcal R}^{\,pred}(\vect{u}^*) & = & \rho^n \dfrac{\vect{u}^*-\vect{u}^n}{\Delta t} + \rho^n \vect{u}^n \cdot \gradt (\vect{u}^*) - \divv \left((\mu+\mu_t)^n \gradt (\vect{u}^*) \right) + \grad(P^n) \\ & - & \text{rest of the right-hand member } (\vect{u}^n, P^n, \text{other variables}^n) \end{eqnarray*}

  • By definition: $ \eta^{\,pred}_{\,i,k}(\vect{u}^*)= {|\Omega_i|}^{\,(k-2)/2}\ ||\vect{\mathcal R}^{\,pred}(\vect{u}^*)|| _{{IL}^{2}(\Omega_i)} $
  • The first family, k=1, suppresses the volume $ |\Omega_i| $ which intrinsicly appears with the norm $ {IL}^{2}(\Omega_i) $.
  • The second family, k=2, exactly represents the norm $ {IL}^{2}(\Omega_i) $. The size of the cell therefore appears in its calculation and induces a weighting effect.
  • $ \eta^{\,pred}_{\,i,k}(\vect{u}^*)$ is ideally equal to zero when the reconstruction methods are perfect and the associated system is solved exactly.

◆ est_error_tot_2

est_error_tot_2

Error estimator for Navier-Stokes: total.

The estimator $ \eta^{\,tot}_{\,i,k}(\vect{u}^{\,n+1})$, local variable calculated at every cell $\Omega_i$, is based on the quantity $\vect{\mathcal R}^{\,tot}(\vect{u}^{\,n+1})$, which represents the residual of the equation using the updated values of $\vect{u}$ and $P$:

\begin{eqnarray*} \vect{\mathcal R}^{\,pred}(\vect{u}^*) & = & \rho^n \dfrac{\vect{u}^*-\vect{u}^n}{\Delta t} + \rho^n \vect{u}^n \cdot \gradt (\vect{u}^*) - \divv \left((\mu+\mu_t)^n \gradt (\vect{u}^*) \right) + \grad(P^n) \\ & - & \text{rest of the right-hand member } (\vect{u}^n, P^n, \text{other variables}^n) \end{eqnarray*}

  • By definition: $ \eta^{\,tot}_{\,i,k}(\vect{u}^{\,n+1})= {|\Omega_i|}^{\,(k-2)/2}\ ||\vect{\mathcal R}^{\,tot}(\vect{u}^{\,n+1})|| _{{I\hspace{-.25em}L}^{2}(\Omega_i)} $
  • The mass flux in the convective term is recalculated from $\vect{u}^{n+1}$ expressed at the cell centers (and not taken from the updated mass flow at the faces).
  • As for the prediction estimator:
    • The first family, k=1, suppresses the volume $ |\Omega_i| $ which intrinsicly appears with the norm $ {IL}^{2}(\Omega_i) $.
    • The second family, k=2, exactly represents the norm $ {IL}^{2}(\Omega_i) $. The size of the cell therefore appears in its calculation and induces a weighting effect.