Research output: Contribution to journal › Article › peer-review
Calculation of the induced velocities in lifting line analyses of propellers and turbines. / Wood, D. H.; Okulov, V. L.; Vaz, J. R.P.
In: Ocean Engineering, Vol. 235, 109337, 01.09.2021.Research output: Contribution to journal › Article › peer-review
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TY - JOUR
T1 - Calculation of the induced velocities in lifting line analyses of propellers and turbines
AU - Wood, D. H.
AU - Okulov, V. L.
AU - Vaz, J. R.P.
N1 - Funding Information: The work of DHW and VLO is part of a project on hydrokinetic turbines supported by the Ministry of Education and Science of the Russian Federation under contract no. 075-15-2019-1923 . JRPV thanks CNPq, Brazil , CAPES, Brazil , PROCAD project (no. 88881.200549/2018-01 ), and PROPESP/UFPA, Brazil for financial support. Publisher Copyright: © 2021 Elsevier Ltd Copyright: Copyright 2021 Elsevier B.V., All rights reserved.
PY - 2021/9/1
Y1 - 2021/9/1
N2 - Lifting line analyses of propellers and horizontal-axis turbines require the axial and circumferential velocities induced by the helicoidal vorticity shed from the blades. These velocities can be found from the analytic solution for a helical vortex of constant radius and pitch due to Kawada and Hardin. This solution, however, involves infinite series of products of Bessel functions and their derivatives, whose evaluation is computationally intensive partly because the number of terms required for a specified accuracy increases without bound as the vortex is approached. We compare three closed-form approximations to the Kawada–Hardin equations. The first, due to Kawada and rediscovered by Lerbs, involves asymptotic expansions for large pitch whereas the second is a more general approximation derived by Wrench and subsequently by Okulov. The last uses additional terms found by Okulov. The three have comparable evaluation times but the third is more accurate. The accuracy of the approximations is assessed for N equispaced and identical helical vortices where N is the number of blades. We provide, for the first time, approximate “remainders” for the Kawada–Hardin equations which allow an assessment of the number of terms in the series required to achieve a specified accuracy. As a test case for assessing the calculations of induced velocity, we consider the design of a hydrokinetic turbine blade to avoid cavitation at two different operating conditions with different vortex pitch. The use of the approximated induced velocities is compared to Prandtl's well-known tip loss factor. Near the hub and tip the blade shape is altered considerably by the induced velocities and the method used to calculate them.
AB - Lifting line analyses of propellers and horizontal-axis turbines require the axial and circumferential velocities induced by the helicoidal vorticity shed from the blades. These velocities can be found from the analytic solution for a helical vortex of constant radius and pitch due to Kawada and Hardin. This solution, however, involves infinite series of products of Bessel functions and their derivatives, whose evaluation is computationally intensive partly because the number of terms required for a specified accuracy increases without bound as the vortex is approached. We compare three closed-form approximations to the Kawada–Hardin equations. The first, due to Kawada and rediscovered by Lerbs, involves asymptotic expansions for large pitch whereas the second is a more general approximation derived by Wrench and subsequently by Okulov. The last uses additional terms found by Okulov. The three have comparable evaluation times but the third is more accurate. The accuracy of the approximations is assessed for N equispaced and identical helical vortices where N is the number of blades. We provide, for the first time, approximate “remainders” for the Kawada–Hardin equations which allow an assessment of the number of terms in the series required to achieve a specified accuracy. As a test case for assessing the calculations of induced velocity, we consider the design of a hydrokinetic turbine blade to avoid cavitation at two different operating conditions with different vortex pitch. The use of the approximated induced velocities is compared to Prandtl's well-known tip loss factor. Near the hub and tip the blade shape is altered considerably by the induced velocities and the method used to calculate them.
KW - Blade element analysis
KW - Horizontal axis turbine
KW - Induced velocities
KW - Lifting line
KW - Propeller
UR - http://www.scopus.com/inward/record.url?scp=85108669244&partnerID=8YFLogxK
U2 - 10.1016/j.oceaneng.2021.109337
DO - 10.1016/j.oceaneng.2021.109337
M3 - Article
AN - SCOPUS:85108669244
VL - 235
JO - Ocean Engineering
JF - Ocean Engineering
SN - 0029-8018
M1 - 109337
ER -
ID: 29136954