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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 $\approx 1$). 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 $x^{\ast }=5$ (3% TI) transect, the root mean square error at the lowest grid resolution was 0.1232 (m/s). For the axial velocity at $x^{\ast }=5$ (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 ($U_{\mathrm{\infty }}$). 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 $\approx 1$). 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.

Free stream velocity ($U_{\mathrm{\infty }}$) per grid resolution with computational cells and error against value at zero grid resolution

resolution (m)	# cells	$U_{\mathrm{\infty }}$ (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

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 ($I_{\mathrm{\infty }}$). 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 $\approx 1$). 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.

Free stream turbulence intensity ($I_{\mathrm{\infty }}$) per grid resolution with computational cells and error against value at zero grid resolution

resolution (m)	# cells	$I_{\mathrm{\infty }}$ (%)	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

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 ($U_{1.2D}$). 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 $\approx 1$). 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.

Wake centerline velocity 1.2 diameters downstream ($U_{1.2D}$) per grid resolution with computational cells and error against value at zero grid resolution

resolution (m)	# cells	$U_{1.2D}$ (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

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 ($I_{1.2D}$). 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 $\approx 1$). TI at zero grid resolution is 10.01%. The grid resolution required for a fine- grid GCI of 1.0% is 0.05952m.

Wake centerline TI 1.2 diameters downstream ($I_{1.2D}$) per grid resolution with computational cells and error against value at zero grid resolution

resolution (m)	# cells	$I_{1.2D}(\mathrm{\%})$	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

Wake TI error against value at zero grid resolution per grid resolution

Validation

At zero grid resolution, the normalised deficit of $U_{1.2D}$, (${\gamma }_{0(1.2D)}$) 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 $y$-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).

Root mean square error (RMSE) for the normalised velocity, $u_0^{\ast }$, 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

Normalised velocity, $u_0^{\ast }$, (m/s) per grid resolution comparison. Experimental data reverse engineered from (Mycek et al. 2014, fig. 11a).

Normalised velocity deficit, ${\gamma }_0$, (%) 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).

Root mean square error (RMSE) for the normalised velocity, $u_0^{\ast }$, 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

Normalised velocity, $u_0^{\ast }$, (m/s) per grid resolution comparison. Experimental data reverse engineered from (Mycek et al. 2014, fig. 11b).

Normalised velocity deficit, ${\gamma }_0$, (%) per grid resolution comparison. Experimental data reverse engineered from (Mycek et al. 2014, fig. 11b).

Axial velocity at $x^{\ast }=5$ (3% TI)

The root mean square error (RMSE) for this transect at the finest grid resolution of 0.0625m was 0.1232 (m/s).

Root mean square error (RMSE) for the normalised velocity, $u_0^{\ast }$, 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

Normalised velocity, $u_0^{\ast }$, (m/s) per grid resolution comparison. Experimental data reverse engineered from (Mycek et al. 2014, fig. A12a).

Normalised velocity deficit, ${\gamma }_0$, (%) per grid resolution comparison. Experimental data reverse engineered from (Mycek et al. 2014, fig. A12a).

Axial velocity at $x^{\ast }=5$ (15% TI)

The root mean square error (RMSE) for this transect at the finest grid resolution of 0.0625m was 0.05688 (m/s).

Root mean square error (RMSE) for the normalised velocity, $u_0^{\ast }$, 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

Normalised velocity, $u_0^{\ast }$, (m/s) per grid resolution comparison. Experimental data reverse engineered from (Mycek et al. 2014, fig. A12a).

Normalised velocity deficit, ${\gamma }_0$, (%) 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 (%).

Root mean square error (RMSE) for the turbulence intensity, $I_0$, 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

Turbulence intensity, $I_0$, (%) 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 (%).

Root mean square error (RMSE) for the turbulence intensity, $I_0$, 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

Turbulence intensity, $I_0$, (%) per grid resolution comparison. Experimental data reverse engineered from (Mycek et al. 2014, fig. 11d).

References

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.