Sensitivity Analysis to Control the Far-Wake Unsteadiness Behind Turbines
Abstract
:1. Introduction
2. Methodology
2.1. Turbine Modelling
2.2. Discrete Linear Instability Analysis
2.3. Discrete Adjoints
2.4. Discrete Sensitivities
2.5. Numerical Flow Solutions Using a High Order Method
2.6. Summary of the Methodology for Stability and Sensitivity Analysis
- Compute a steady base flow using a numerical solver. High order h/p methods are preferred to obtain find solutions with low numerical errors. We time march the compressible Navier–Stokes Equations (1), until the residual falls below . If we do not obtain convergence, the case is considered unstable and we do not perform the stability analysis.
- Having computed all direct eigenvalues/eigenvectors, we select the least stable and compute its adjoint, see Section 2.3.
- With the direct and adjoint modes, it is possible to compute the Hessian matrix associated to the sensitivities detailed in Section 2.4. Subsequently, the sensitivities to base flow modifications and to localised forcing can be calculated using Equations (5).
3. Results
3.1. Preliminaries
3.2. Mesh Convergence
3.3. The Onset of Wake Unsteadiness
3.4. Stability Analysis
3.5. Wake Control Using Sensitivity Analysis
3.5.1. Sensitivity Analysis
3.5.2. Wake Stabilisation
4. Wake Control at Higher Reynolds Numbers
5. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
Appendix A
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P-Order | % Error () | % Error () | ||
---|---|---|---|---|
3 | −0.002287 | 13.611 | 3.0998769 | 0.002 |
4 | −0.002019 | 0.298 | 3.099801 | 9.678 × |
5 | −0.002013 | - | 3.099798 | - |
Turbine Thrust | Reynolds Number Re | |||||
---|---|---|---|---|---|---|
1000 | 1500 | 2000 | 2500 | 3000 | 3500 | |
0.50 | S | S | S | S | S | S |
0.75 | S | S | S | S | S | S |
1.00 | S | S | S | S | S | U |
1.20 | S | S | U | U | U | U |
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Ferrer, E.; Browne, O.M.F.; Valero, E. Sensitivity Analysis to Control the Far-Wake Unsteadiness Behind Turbines. Energies 2017, 10, 1599. https://doi.org/10.3390/en10101599
Ferrer E, Browne OMF, Valero E. Sensitivity Analysis to Control the Far-Wake Unsteadiness Behind Turbines. Energies. 2017; 10(10):1599. https://doi.org/10.3390/en10101599
Chicago/Turabian StyleFerrer, Esteban, Oliver M.F. Browne, and Eusebio Valero. 2017. "Sensitivity Analysis to Control the Far-Wake Unsteadiness Behind Turbines" Energies 10, no. 10: 1599. https://doi.org/10.3390/en10101599