The present investigation has performed a number of direct numerical simulations (DNS) in three dimensional fully turbulent incompressible flows, e.g. the periodic hill and the stepped cylinder, with adaptive mesh refinement(AMR). The implementation is carried out in Nek5000, an open-source, highly scalable and portable code based on the spectral element method (SEM). Initially, we have defined a coarse mesh with the aim to refine only the regions pointed out by the chosen error indicator. In the literature, we have three main strategies for mesh adaptation in spectral element methods. They consist in moving the existing gridpoints (r-refinement), locally increasing the number of grid points (h-refinement) or locally increasing the polynomial order (p-refinement). The h-refinement has been used in our study for its flexibility and relative ease of implementation. Currently, it is the only one implemented in Nek5000.

Once the refinement approach has been chosen, we have to draw up how the committed error has been defined and measured. Indeed errors resulting from the discretization and resolution of partial differential equations with SEM arise from different sources. Modelling error, which occurs when the mathematical model does not match nature, and roundoff error, which is due to the finite accuracy of computers, are supposed to be prevented. In practice, we assumed the mathematical model consistency and the adequate finite accuracy of computers, guaranteed for example by double-precision floating-point arithmetic. So we are left with the truncation error, which arises because the solution is approximated by a finite spectral expansion, and the quadrature error, related to the discrete integration. In order to estimate these errors, we present the methods that have been used. The spectral error indicator developed by C.Mavriplis relies on the local properties of the solution. For simplicity we introduce here the formulation for a 1D solution $u(x)$ and its spectral transformation $u(x) = \sum_{k=0}^{\infty} \hat{u}_k L_k(x)$, where $L_k$ is the Legendre polynomial at order $k$, which is the polynomial basis implemented in Nek5000, and $\hat{u}_k$ is the $k$-th spectral coefficient. Looking at the $N$-order of the polynomial expansion, we obtain the spectral element solution $ (\boldsymbol{u_N},p_N) $ and the error indicator, which is $ \epsilon_{ind} \approx \| u-u_N \|_{L^2} $, reads:

$ \epsilon_{ind} = \biggl( \int_N^{\infty} \frac{\hat{u}(k)^2}{\frac{2k+1}{2}}dk + \frac{\hat{u}_N^2}{\frac{2N+1}{2}} \biggr)^{\frac{1}{2}} $

where the decay of the spectral element coefficients is assumed to be $\hat{u}_k \sim \hat{u} = c exp(-\sigma k)$. Previous works pointed out that this indicator is the best to track the error in flow with high gradients and aims to uniformly reduce the error on the solution. On the other hand, we have implemented the adjoint error estimator. These kinds of error estimators are goal-oriented and generate an estimation of the committed error based on a functional of interest. In this case, we used the method of the adjoint-weighted residuals, which combines local info in the form of strong residuals and the global sensitivity given by the computation of the dual problem. Firstly we need to define the functional of interest $J$. In an unsteady turbulent flow, the functional of physical interest is commonly a time-averaged quantity, e.g. stresses, mean fluxes, drag or lift coefficients. The averaging time $T$ is chosen long enough for the statistics convergence. An estimation of the error on the functional is computed as:

\[\begin{eqnarray} \delta J &=& |J({u},p) - J({u}_N,p_N) | \\ &<& \sum_{e=1}^E \frac{1}{T} \int_O^T \biggl[ | R_1(\boldsymbol{u}_N,p_N) |_{L^2(\Omega_e)} \cdot | \boldsymbol{u}^{+} - \boldsymbol{u}^{+}_N |_{L^2(\Omega_e)} \\ &&\quad \mbox{}+ | R_2(\boldsymbol{u}_N,p_N) |_{\Gamma_e} \cdot | \boldsymbol{u}^{+} - \boldsymbol{u}^{+}_N |_{\Gamma_e} \\ &&\quad+ \| R_3(\boldsymbol{u}_N,p_N) \|_{L^2(\Omega_e)} \cdot \| p^{+} -p^{+}_N \|_{L^2(\Omega_e)} \biggr] dt \end{eqnarray}\]

where $R_1$ and $R_3$ are the strong residuals of the momentum and continuity equations, $R_2$ correspond to the jump in the stresses at the interface between elements (usually neglected). The terms $| \boldsymbol{u}^{+} - {u}^{+}_N |$ and $ \|p^{+} -p^{+}_N \|$ represent the interpolation error on the adjoint field and are estimated with the spectral error indicator.

The comparison between different error-driven approaches is carried out for 2D/3D steady and unsteady turbulent flow too. Eventually, the latest application in moderately complex geometry is presented, i.e. the 3D stepped cylinder which consists of two attached cylinders with different diameters