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<front>
<journal-meta>
<journal-id journal-id-type="publisher">WESD</journal-id>
<journal-title-group>
<journal-title>Wind Energy Science Discussions</journal-title>
<abbrev-journal-title abbrev-type="publisher">WESD</abbrev-journal-title>
<abbrev-journal-title abbrev-type="nlm-ta">Wind Energ. Sci. Discuss.</abbrev-journal-title>
</journal-title-group>
<issn pub-type="epub">2366-7621</issn>
<publisher><publisher-name></publisher-name>
<publisher-loc>Göttingen, Germany</publisher-loc>
</publisher>
</journal-meta>
<article-meta>
<article-id pub-id-type="doi">10.5194/wes-2026-138</article-id>
<title-group>
<article-title>Brief communication: On the local disagreement of two engineering models for the radial induced velocity of non-planar wind turbine rotors</article-title>
</title-group>
<contrib-group><contrib contrib-type="author" xlink:type="simple"><name name-style="western"><surname>Schaffarczyk</surname>
<given-names>Alois Peter</given-names>
<ext-link>https://orcid.org/0000-0002-9357-3232</ext-link>
</name>
<xref ref-type="aff" rid="aff1">
<sup>1</sup>
</xref>
</contrib>
<contrib contrib-type="author" xlink:type="simple"><name name-style="western"><surname>Masare</surname>
<given-names>Bhima</given-names>
</name>
<xref ref-type="aff" rid="aff1">
<sup>1</sup>
</xref>
</contrib>
</contrib-group><aff id="aff1">
<label>1</label>
<addr-line>Kiel University of Applied Sciences, Mechanical Engineering, 24149 Kiel, Germany</addr-line>
</aff>
<pub-date pub-type="epub">
<day>03</day>
<month>09</month>
<year>2026</year>
</pub-date>
<volume>2026</volume>
<fpage>1</fpage>
<lpage>4</lpage>
<permissions>
<copyright-statement>Copyright: &#x000a9; 2026 Alois Peter Schaffarczyk</copyright-statement>
<copyright-year>2026</copyright-year>
<license license-type="open-access">
<license-p>This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit <ext-link ext-link-type="uri"  xlink:href="https://creativecommons.org/licenses/by/4.0/">https://creativecommons.org/licenses/by/4.0/</ext-link></license-p>
</license>
</permissions>
<self-uri xlink:href="https://wes.copernicus.org/preprints/wes-2026-138/">This article is available from https://wes.copernicus.org/preprints/wes-2026-138/</self-uri>
<self-uri xlink:href="https://wes.copernicus.org/preprints/wes-2026-138/wes-2026-138.pdf">The full text article is available as a PDF file from https://wes.copernicus.org/preprints/wes-2026-138/wes-2026-138.pdf</self-uri>
<abstract>
<p>Blade element momentum (BEM) codes traditionally neglect the radial induced velocity &lt;em&gt;u&lt;sub&gt;r&lt;/sub&gt;&lt;/em&gt;, although for the strongly deflected or prebent blades of current multi-MW turbines it produces an additional in-plane driving force. Two engineering models proposed by Li et al. (2022) &amp;mdash; the smooth correction attributed to Madsen, driven by the area-averaged thrust coefficient, and the superposition of discrete semiinfinite vortex cylinders formed from the converged annulus inductions &amp;mdash; were implemented independently in an open-source blade design and analysis code. For the NREL 5 MW reference rotor with an operational tip deflection of 6 m both models change thrust and power by less than 1 %, and their integral effects on the power coefficient agree to within 0.05 %. This agreement conceals a pronounced local disagreement: at mid-span the vortex-cylinder &lt;em&gt;u&lt;sub&gt;r&lt;/sub&gt;&lt;/em&gt; changes sign, following the non-monotonic radial derivative of the axial induction, and at the outermost section it grows logarithmically with the number of blade sections, &lt;em&gt;u&lt;sub&gt;r&lt;/sub&gt;&lt;/em&gt; ≃ 0.78 ln &lt;em&gt;N&lt;sub&gt;sec&lt;/sub&gt;&lt;/em&gt; + 3.05 (m s&lt;sup&gt;&amp;minus;1&lt;/sup&gt;), in close agreement with the analytic slope |&lt;em&gt;&amp;gamma;&lt;/em&gt;&lt;em style=&quot;position: relative;&quot;&gt;&lt;sub&gt;t&lt;/sub&gt;&amp;nbsp; &amp;nbsp;&lt;sup class=&quot;up&quot; style=&quot;position: absolute; top: 0px; left: 2px;&quot;&gt;tip&lt;/sup&gt;&lt;/em&gt;|/2&lt;em&gt;&amp;pi; &lt;/em&gt;= 0.82, reaching 6.6 m s&lt;sup&gt;&amp;minus;1&lt;/sup&gt; at &lt;em&gt;N&lt;sub&gt;sec&lt;/sub&gt;&lt;/em&gt; = 70 against 2.3 m s&lt;sup&gt;&amp;minus;1&lt;/sup&gt; for the Madsen correction. We show that the integral agreement is systematic rather than coincidental, and that local load predictions near the tip of non-planar rotors are model- and grid-dependent and require an explicit, resolution-independent regularisation.&lt;em style=&quot;position: relative;&quot;&gt;&lt;br /&gt;&lt;/em&gt;</p>
</abstract>
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