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 1.032 (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 35.85%, a 28.18% 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 15.2%, an error of 69.42% against the measured value of 21.9% for the 3% ambient TI experiment. The simulated ambient TI, at zero grid resolution, is 5.394%. For the centreline velocity (3% TI) transect, the root mean square error at the lowest grid resolution was 0.1954 (m/s). For the centreline velocity (15% TI) transect, the root mean square error at the lowest grid resolution was 0.1172 (m/s). For the axial velocity at :math:`x^*=5` (3% TI) transect, the root mean square error at the lowest grid resolution was 0.1167 (m/s). For the axial velocity at :math:`x^*=5` (15% TI) transect, the root mean square error at the lowest grid resolution was 0.06263 (m/s). For the centreline turbulence intensity (3% TI) transect, the root mean square error at the lowest grid resolution was 5.785 (%). For the centreline turbulence intensity (15% TI) transect, the root mean square error at the lowest grid resolution was 8.987 (%). 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 0.9994 was achieved (asymptotic range is indicated by a value :math:`\approx 1`). The free stream velocity at zero grid resolution is 0.8053m/s. The grid resolution required for a fine-grid GCI of 1.0% is 0.2808m. .. 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 0.757747 0.0591016 0.5 1152 0.783403 0.0272444 0.25 9216 0.80086 0.00556725 0.125 73728 0.804829 0.000638789 0.0625 589824 0.805285 7.32951e-05 ============== ======= ====================== =========== .. 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 14.95 was achieved (asymptotic range is indicated by a value :math:`\approx 1`). The free stream velocity at zero grid resolution is 5.394%. The grid resolution required for a fine-grid GCI of 1.0% is 0.03819m. .. 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 8.18034 0.516566 0.5 1152 6.4764 0.20067 0.25 9216 4.85727 0.0995036 0.125 73728 4.94206 0.0837847 0.0625 589824 5.28091 0.0209649 ============== ======= ==================== ========= .. 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 1.032 was achieved (asymptotic range is indicated by a value :math:`\approx 1`). The free stream velocity at zero grid resolution is 0.5166m/s. The grid resolution required for a fine-grid GCI of 1.0% is 0.03873m. .. 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 0.74141 0.435113 0.5 1152 0.678241 0.312841 0.25 9216 0.591398 0.144743 0.125 73728 0.542273 0.0496528 0.0625 589824 0.525421 0.017033 ============== ======= ====================== ========= .. 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 0.8836 was achieved (asymptotic range is indicated by a value :math:`\approx 1`). TI at zero grid resolution is 15.2%. The grid resolution required for a fine-grid GCI of 1.0% is 0.006075m. .. 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 8.29067 0.454552 0.5 1152 7.66767 0.49554 0.25 9216 8.75471 0.424023 0.125 73728 12.1812 0.19859 0.0625 589824 13.786 0.093009 ============== ======= ===================== ======== .. 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 35.85%, a 28.18% 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 0.1954 (m/s). .. table:: Root mean square error (RMSE) for the normalised velocity, :math:`u^*_0`, per grid resolution. ============== ========== resolution (m) RMSE (m/s) ============== ========== 1 0.419804 0.5 0.335561 0.25 0.26347 0.125 0.22455 0.0625 0.195446 ============== ========== .. 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 0.1172 (m/s). .. table:: Root mean square error (RMSE) for the normalised velocity, :math:`u^*_0`, per grid resolution. ============== ========== resolution (m) RMSE (m/s) ============== ========== 1 0.212807 0.5 0.149742 0.25 0.106792 0.125 0.10218 0.0625 0.117195 ============== ========== .. 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 0.1167 (m/s). .. table:: Root mean square error (RMSE) for the normalised velocity, :math:`u^*_0`, per grid resolution. ============== ========== resolution (m) RMSE (m/s) ============== ========== 1 0.213179 0.5 0.174207 0.25 0.144064 0.125 0.128447 0.0625 0.116704 ============== ========== .. 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 0.06263 (m/s). .. table:: Root mean square error (RMSE) for the normalised velocity, :math:`u^*_0`, per grid resolution. ============== ========== resolution (m) RMSE (m/s) ============== ========== 1 0.0609747 0.5 0.0330307 0.25 0.0432415 0.125 0.053325 0.0625 0.0626312 ============== ========== .. 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 5.785 (%). .. table:: Root mean square error (RMSE) for the turbulence intensity, :math:`I_0`, per grid resolution. ============== ======== resolution (m) RMSE (%) ============== ======== 1 8.68781 0.5 9.4221 0.25 9.19553 0.125 7.04457 0.0625 5.78451 ============== ======== .. 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 8.987 (%). .. table:: Root mean square error (RMSE) for the turbulence intensity, :math:`I_0`, per grid resolution. ============== ======== resolution (m) RMSE (%) ============== ======== 1 12.3491 0.5 13.0006 0.25 12.6459 0.125 10.322 0.0625 8.98737 ============== ======== .. 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.