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class="firstHeading mw-first-heading"><span class="mw-page-title-main">Stagnation point</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/SPvortex.JPG/600px-SPvortex.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/SPvortex.JPG/800px-SPvortex.JPG 2x" data-file-width="2003" data-file-height="1202" /></a><figcaption>Photo showing stagnation point and attached <a href="/wiki/Vortex" title="Vortex">vortex</a> at an un-<a href="/wiki/Aircraft_fairing" title="Aircraft fairing">faired</a> wing-root to fuselage junction on a <a href="/wiki/Schempp-Hirth_Janus" title="Schempp-Hirth Janus">Schempp-Hirth Janus</a> C <a href="/wiki/Glider_(sailplane)" title="Glider (sailplane)">glider</a></figcaption></figure> <p>In <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a>, a <b>stagnation point</b> is a point in a flow field where the local <a href="/wiki/Velocity" title="Velocity">velocity</a> of the fluid is zero.<sup id="cite_ref-Clancy_1-0" class="reference"><a href="#cite_note-Clancy-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: § 3.2">: § 3.2 </span></sup> The <a href="/wiki/Bernoulli%27s_principle" title="Bernoulli's principle">Bernoulli equation</a> shows that the <a href="/wiki/Static_pressure" title="Static pressure">static pressure</a> is highest when the velocity is zero and hence static pressure is at its maximum value at stagnation points: in this case static pressure equals <a href="/wiki/Stagnation_pressure" title="Stagnation pressure">stagnation pressure</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Clancy_1-1" class="reference"><a href="#cite_note-Clancy-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: § 3.5">: § 3.5 </span></sup> </p><p>The Bernoulli equation applicable to <a href="/wiki/Incompressible_flow" title="Incompressible flow">incompressible flow</a> shows that the stagnation pressure is equal to the <a href="/wiki/Dynamic_pressure" title="Dynamic pressure">dynamic pressure</a> and static pressure combined.<sup id="cite_ref-Clancy_1-2" class="reference"><a href="#cite_note-Clancy-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: § 3.5">: § 3.5 </span></sup> In <a href="/wiki/Compressible_flow" title="Compressible flow">compressible flows</a>, stagnation pressure is also equal to total pressure as well, provided that the fluid entering the stagnation point is brought to rest <a href="/wiki/Isentropic_flow" class="mw-redirect" title="Isentropic flow">isentropically</a>.<sup id="cite_ref-Clancy_1-3" class="reference"><a href="#cite_note-Clancy-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: § 3.12">: § 3.12 </span></sup> </p><p>A plentiful, albeit surprising, example of such points seem to appear in all but the most extreme cases of fluid dynamics in the form of the "<a href="/wiki/No-slip_condition" title="No-slip condition">no-slip condition</a>" - the assumption that any portion of a flow field lying along some boundary consists of nothing but stagnation points (the question as to whether this assumption reflects reality or is simply a mathematical convenience has been a continuous subject of debate since the principle was first established). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Pressure_coefficient">Pressure coefficient</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stagnation_point&action=edit&section=1" title="Edit section: Pressure coefficient"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This information can be used to show that the <a href="/wiki/Pressure_coefficient" title="Pressure coefficient">pressure coefficient</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc37470431fbf62081b69ba870ad3f855178361" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.721ex; height:2.843ex;" alt="{\displaystyle C_{p}}"></span> at a stagnation point is <a href="/wiki/Unity_(mathematics)" class="mw-redirect" title="Unity (mathematics)">unity</a> (positive one):<sup id="cite_ref-Clancy_1-4" class="reference"><a href="#cite_note-Clancy-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: § 3.6">: § 3.6 </span></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{p}={p-p_{\infty } \over q_{\infty }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{p}={p-p_{\infty } \over q_{\infty }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/088b3639374d872db212885571d4c89d8f0ed1a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.71ex; height:5.676ex;" alt="{\displaystyle C_{p}={p-p_{\infty } \over q_{\infty }}}"></span></dd></dl> <p>where: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc37470431fbf62081b69ba870ad3f855178361" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.721ex; height:2.843ex;" alt="{\displaystyle C_{p}}"></span> is pressure coefficient</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> is <a href="/wiki/Static_pressure" title="Static pressure">static pressure</a> at the point at which pressure coefficient is being evaluated</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4b1d64d7d5bc5f0a8490d9844565f5cb2b83802" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:3.134ex; height:2.009ex;" alt="{\displaystyle p_{\infty }}"></span> is static pressure at points remote from the body (<a href="/wiki/Freestream" title="Freestream">freestream</a> static pressure)</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e27d15fde1bb9e5e8819f7c119b5c019adf615f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.912ex; height:2.009ex;" alt="{\displaystyle q_{\infty }}"></span> is <a href="/wiki/Dynamic_pressure" title="Dynamic pressure">dynamic pressure</a> at points remote from the body (freestream dynamic pressure)</dd></dl> <p>Stagnation pressure minus freestream static pressure is equal to freestream dynamic pressure; therefore the pressure coefficient <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc37470431fbf62081b69ba870ad3f855178361" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.721ex; height:2.843ex;" alt="{\displaystyle C_{p}}"></span> at stagnation points is +1.<sup id="cite_ref-Clancy_1-5" class="reference"><a href="#cite_note-Clancy-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: § 3.6">: § 3.6 </span></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Kutta_condition">Kutta condition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stagnation_point&action=edit&section=2" title="Edit section: Kutta condition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>On a streamlined body fully immersed in a <a href="/wiki/Potential_flow" title="Potential flow">potential flow</a>, there are two stagnation points—one near the leading edge and one near the trailing edge. On a body with a sharp point such as the <a href="/wiki/Trailing_edge" title="Trailing edge">trailing edge</a> of a <a href="/wiki/Wing" title="Wing">wing</a>, the <a href="/wiki/Kutta_condition" title="Kutta condition">Kutta condition</a> specifies that a stagnation point is located at that point.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Streamlines,_streaklines_and_pathlines" class="mw-redirect" title="Streamlines, streaklines and pathlines">streamline</a> at a stagnation point is perpendicular to the surface of the body. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stagnation_point&action=edit&section=3" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Stagnation_point_flow" title="Stagnation point flow">Stagnation point flow</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stagnation_point&action=edit&section=4" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Clancy-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Clancy_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Clancy_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Clancy_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Clancy_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Clancy_1-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Clancy_1-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text">Clancy, L.J. (1975), <i>Aerodynamics</i>, Pitman Publishing Limited, London. <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-273-01120-0" title="Special:BookSources/0-273-01120-0">0-273-01120-0</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFoxMcDonald,_A._T.2003" class="citation book cs1">Fox, R. W.; McDonald, A. T. (2003). <i>Introduction to Fluid Mechanics</i> (4th ed.). Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-20231-2" title="Special:BookSources/0-471-20231-2"><bdi>0-471-20231-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Fluid+Mechanics&rft.edition=4th&rft.pub=Wiley&rft.date=2003&rft.isbn=0-471-20231-2&rft.aulast=Fox&rft.aufirst=R.+W.&rft.au=McDonald%2C+A.+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStagnation+point" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Anderson, John D. (1984) <i>Fundamentals of Aerodynamics</i>, section 4.5 McGraw-Hill Inc. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-001656-9" title="Special:BookSources/0-07-001656-9">0-07-001656-9</a></span> </li> </ol></div></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol 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