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 3% ambient turbulence intensity (TI) experiment. At zero grid resolution the turbulence intensity measured 1.2 diameters downstream from the turbine was 10.01%, an error of 45.71% against the measured value of 21.9% for the 3% ambient TI experiment. The simulated ambient TI, at zero grid resolution, is 5.47%. For the centreline velocity (3% TI) transect, the root mean square error at the lowest grid resolution was 0.2004 (m/s). For the centreline velocity (15% TI) transect, the root mean square error at the lowest grid resolution was 0.09075 (m/s). For the axial velocity at :math:`x^*=5` (3% TI) transect, the root mean square error at the lowest grid resolution was 0.1232 (m/s). For the axial velocity at :math:`x^*=5` (15% TI) transect, the root mean square error at the lowest grid resolution was 0.05688 (m/s). For the centreline turbulence intensity (3% TI) transect, the root mean square error at the lowest grid resolution was 6.543 (%). For the centreline turbulence intensity (15% TI) transect, the root mean square error at the lowest grid resolution was 10.31 (%). 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` (m/s) error ============== ======= ====================== ========== 1 144 0.756681 0.0595774 0.5 1152 0.790921 0.0170231 0.25 9216 0.793301 0.0140647 0.125 73728 0.794949 0.0120171 0.0625 589824 0.797055 0.00939923 ============== ======= ====================== ========== .. 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 Free Stream Turbulence Intensity ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This section presents the convergence study for the free stream turbulence intensity (:math:`I_\infty`). For the final case, with grid resolution of 0.0625m, an asymptotic ratio of 2.029 was achieved (asymptotic range is indicated by a value :math:`\approx 1`). The free stream velocity at zero grid resolution is 5.47%. The grid resolution required for a fine-grid GCI of 1.0% is 0.003269m. .. table:: Free stream turbulence intensity (:math:`I_\infty`) per grid resolution with computational cells and error against value at zero grid resolution ============== ======= ==================== ========= resolution (m) # cells :math:`I_\infty` (%) error ============== ======= ==================== ========= 1 144 4.60414 0.158243 0.5 1152 3.47658 0.364391 0.25 9216 5.24653 0.0407978 0.125 73728 5.18668 0.0517407 0.0625 589824 5.27265 0.0360232 ============== ======= ==================== ========= .. figure:: ti_infty_convergence.png :alt: Free stream turbulence intensity error against value at zero grid resolution per grid resolution :width: 3.64in Free stream turbulence intensity 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}` (m/s) 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 0.00265518 0.0625 589824 0.458319 5.6044e-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 Wake Turbulence Intensity ~~~~~~~~~~~~~~~~~~~~~~~~~ This section presents the convergence study for the wake centerline turbulence intensity (TI) measured 1.2 diameters downstream from the turbine (:math:`I_{1.2D}`). For the final case, with grid resolution of 0.0625m, an asymptotic ratio of 1.036 was achieved (asymptotic range is indicated by a value :math:`\approx 1`). TI at zero grid resolution is 10.01%. The grid resolution required for a fine- grid GCI of 1.0% is 0.05952m. .. table:: Wake centerline TI 1.2 diameters downstream (:math:`I_{1.2D}`) per grid resolution with computational cells and error against value at zero grid resolution ============== ======= ===================== ========== resolution (m) # cells :math:`I_{1.2D} (\%)` error ============== ======= ===================== ========== 1 144 4.63007 0.537339 0.5 1152 4.96932 0.50344 0.25 9216 12.3321 0.23229 0.125 73728 10.4663 0.0458488 0.0625 589824 10.0981 0.00904952 ============== ======= ===================== ========== .. figure:: ti_wake_convergence.png :alt: Wake TI error against value at zero grid resolution per grid resolution :width: 3.64in Wake TI 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 (3% TI) ~~~~~~~~~~~~~~~~~~~~~~~~~~~ The root mean square error (RMSE) for this transect at the finest grid resolution of 0.0625m was 0.2004 (m/s). .. table:: Root mean square error (RMSE) for the normalised velocity, :math:`u^*_0`, per grid resolution. ============== ========== resolution (m) RMSE (m/s) ============== ========== 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). Centreline velocity (15% TI) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The root mean square error (RMSE) for this transect at the finest grid resolution of 0.0625m was 0.09075 (m/s). .. table:: Root mean square error (RMSE) for the normalised velocity, :math:`u^*_0`, per grid resolution. ============== ========== resolution (m) RMSE (m/s) ============== ========== 1 0.203884 0.5 0.136543 0.25 0.0818053 0.125 0.101083 0.0625 0.0907545 ============== ========== .. 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. 11b). :width: 5.68in Normalised velocity, :math:`u^*_0`, (m/s) per grid resolution comparison. Experimental data reverse engineered from (Mycek et al. 2014, fig. 11b). .. 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. 11b). :width: 5.68in Normalised velocity deficit, :math:`\gamma_0`, (%) per grid resolution comparison. Experimental data reverse engineered from (Mycek et al. 2014, fig. 11b). Axial velocity at :math:`x^*=5` (3% TI) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The root mean square error (RMSE) for this transect at the finest grid resolution of 0.0625m was 0.1232 (m/s). .. table:: Root mean square error (RMSE) for the normalised velocity, :math:`u^*_0`, per grid resolution. ============== ========== resolution (m) RMSE (m/s) ============== ========== 1 0.201806 0.5 0.134912 0.25 0.135605 0.125 0.118328 0.0625 0.123211 ============== ========== .. figure:: transect_u0_2.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_2.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). Axial velocity at :math:`x^*=5` (15% TI) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The root mean square error (RMSE) for this transect at the finest grid resolution of 0.0625m was 0.05688 (m/s). .. table:: Root mean square error (RMSE) for the normalised velocity, :math:`u^*_0`, per grid resolution. ============== ========== resolution (m) RMSE (m/s) ============== ========== 1 0.0474432 0.5 0.0520709 0.25 0.0470543 0.125 0.0609816 0.0625 0.056879 ============== ========== .. figure:: transect_u0_3.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_3.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). Centreline turbulence intensity (3% TI) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The root mean square error (RMSE) for this transect at the finest grid resolution of 0.0625m was 6.543 (%). .. table:: Root mean square error (RMSE) for the turbulence intensity, :math:`I_0`, per grid resolution. ============== ======== resolution (m) RMSE (%) ============== ======== 1 12.0893 0.5 11.121 0.25 6.18045 0.125 6.62332 0.0625 6.54326 ============== ======== .. figure:: transect_I0_0.png :alt: Turbulence intensity, :math:`I_0`, (%) per grid resolution comparison. Experimental data reverse engineered from (Mycek et al. 2014, fig. 11c). :width: 5.68in Turbulence intensity, :math:`I_0`, (%) per grid resolution comparison. Experimental data reverse engineered from (Mycek et al. 2014, fig. 11c). Centreline turbulence intensity (15% TI) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The root mean square error (RMSE) for this transect at the finest grid resolution of 0.0625m was 10.31 (%). .. table:: Root mean square error (RMSE) for the turbulence intensity, :math:`I_0`, per grid resolution. ============== ======== resolution (m) RMSE (%) ============== ======== 1 15.5431 0.5 14.7203 0.25 9.71568 0.125 10.3019 0.0625 10.3063 ============== ======== .. figure:: transect_I0_1.png :alt: Turbulence intensity, :math:`I_0`, (%) per grid resolution comparison. Experimental data reverse engineered from (Mycek et al. 2014, fig. 11d). :width: 5.68in Turbulence intensity, :math:`I_0`, (%) per grid resolution comparison. Experimental data reverse engineered from (Mycek et al. 2014, fig. 11d). 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.