Grid Convergence Study (Linux) ============================== Summary ------- This is a grid convergence study of 5 cases. The case with the finest grid resolution, of 0.0625m, achieved an asymptotic ratio of 1.003 (asymptotic range is indicated by a value :math:`\approx 1`). At zero grid resolution, the normalised velocity deficit measured 1.2 diameters downstream from the turbine was 43.04%, a 13.78% error against the measured value of 49.92%. For the centreline velocity transect, the root mean square error at the lowest grid resolution was 0.2004. For the axial velocity at :math:`x^*=5` transect, the root mean square error at the lowest grid resolution was 0.1232. Grid Convergence Studies ------------------------ Free Stream Velocity ~~~~~~~~~~~~~~~~~~~~ This section presents the convergence study for the free stream velocity (:math:`U_\infty`). For the final case, with grid resolution of 0.0625m, an asymptotic ratio of 1.63 was achieved (asymptotic range is indicated by a value :math:`\approx 1`). The free stream velocity at zero grid resolution is 0.8046m/s. The grid resolution required for a fine-grid GCI of 1.0% is 0.03862m. .. table:: Free stream velocity (:math:`U_\infty`) per grid resolution with computational cells and error against value at zero grid resolution ============== ======= ================ ========== resolution (m) # cells :math:`U_\infty` error ============== ======= ================ ========== 1 144 0.756681 0.0595775 0.5 1152 0.790921 0.0170232 0.25 9216 0.793301 0.0140648 0.125 73728 0.794949 0.0120173 0.0625 589824 0.797055 0.00939936 ============== ======= ================ ========== .. figure:: u_infty_convergence.png :alt: Free stream velocity error against value at zero grid resolution per grid resolution :width: 3.64in Free stream velocity error against value at zero grid resolution per grid resolution Wake Velocity ~~~~~~~~~~~~~ This section presents the convergence study for the wake centerline velocity measured 1.2 diameters downstream from the turbine (:math:`U_{1.2D}`). For the final case, with grid resolution of 0.0625m, an asymptotic ratio of 1.003 was achieved (asymptotic range is indicated by a value :math:`\approx 1`). The free stream velocity at zero grid resolution is 0.4583m/s. The grid resolution required for a fine-grid GCI of 1.0% is 0.1524m. .. table:: Wake centerline velocity 1.2 diameters downstream (:math:`U_{1.2D}`) per grid resolution with computational cells and error against value at zero grid resolution ============== ======= ================ =========== resolution (m) # cells :math:`U_{1.2D}` error ============== ======= ================ =========== 1 144 0.736009 0.605977 0.5 1152 0.644324 0.405919 0.25 9216 0.515944 0.125793 0.125 73728 0.45951 0.00265517 0.0625 589824 0.458319 5.60439e-05 ============== ======= ================ =========== .. figure:: u_wake_convergence.png :alt: Wake velocity error against value at zero grid resolution per grid resolution :width: 3.64in Wake velocity error against value at zero grid resolution per grid resolution Validation ~~~~~~~~~~ At zero grid resolution, the normalised deficit of :math:`U_{1.2D}`, (:math:`\gamma_{0(1.2D)}`) is 43.04%, a 13.78% error against the measured value of 49.92%. Wake Transects -------------- This section presents axial velocity transects along the turbine centreline and at cross-sections along the :math:`y`-axis. Errors are reported relative to the experimental data given in (Mycek et al. 2014). Centreline velocity ~~~~~~~~~~~~~~~~~~~ The root mean square error (RMSE) for this transect at the finest grid resolution of 0.0625m was 0.2004. .. table:: Root mean square error (RMSE) for the normalised velocity, :math:`u^*_0`, per grid resolution. ============== ======== resolution (m) RMSE ============== ======== 1 0.401335 0.5 0.271788 0.25 0.239064 0.125 0.190003 0.0625 0.200371 ============== ======== .. figure:: transect_u0_0.png :alt: Normalised velocity, :math:`u^*_0`, (m/s) per grid resolution comparison. Experimental data reverse engineered from (Mycek et al. 2014, fig. 11a). :width: 5.68in Normalised velocity, :math:`u^*_0`, (m/s) per grid resolution comparison. Experimental data reverse engineered from (Mycek et al. 2014, fig. 11a). .. figure:: transect_gamma0_0.png :alt: Normalised velocity deficit, :math:`\gamma_0`, (%) per grid resolution comparison. Experimental data reverse engineered from (Mycek et al. 2014, fig. 11a). :width: 5.68in Normalised velocity deficit, :math:`\gamma_0`, (%) per grid resolution comparison. Experimental data reverse engineered from (Mycek et al. 2014, fig. 11a). Axial velocity at :math:`x^*=5` ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The root mean square error (RMSE) for this transect at the finest grid resolution of 0.0625m was 0.1232. .. table:: Root mean square error (RMSE) for the normalised velocity, :math:`u^*_0`, per grid resolution. ============== ======== resolution (m) RMSE ============== ======== 1 0.201806 0.5 0.134912 0.25 0.135605 0.125 0.118328 0.0625 0.123211 ============== ======== .. figure:: transect_u0_1.png :alt: Normalised velocity, :math:`u^*_0`, (m/s) per grid resolution comparison. Experimental data reverse engineered from (Mycek et al. 2014, fig. A12a). :width: 5.68in Normalised velocity, :math:`u^*_0`, (m/s) per grid resolution comparison. Experimental data reverse engineered from (Mycek et al. 2014, fig. A12a). .. figure:: transect_gamma0_1.png :alt: Normalised velocity deficit, :math:`\gamma_0`, (%) per grid resolution comparison. Experimental data reverse engineered from (Mycek et al. 2014, fig. A12a). :width: 5.68in Normalised velocity deficit, :math:`\gamma_0`, (%) per grid resolution comparison. Experimental data reverse engineered from (Mycek et al. 2014, fig. A12a). References ---------- .. container:: references csl-bib-body hanging-indent :name: refs .. container:: csl-entry :name: ref-mycek2014 Mycek, Paul, Benoît Gaurier, Grégory Germain, Grégory Pinon, and Elie Rivoalen. 2014. “Experimental Study of the Turbulence Intensity Effects on Marine Current Turbines Behaviour. Part I: One Single Turbine.” *Renewable Energy* 66: 729–46. https://doi.org/10.1016/j.renene.2013.12.036.