{"id":228,"date":"2015-08-14T14:23:22","date_gmt":"2015-08-14T19:23:22","guid":{"rendered":"http:\/\/charles-oneill.com\/blog\/?p=228"},"modified":"2015-08-14T14:23:22","modified_gmt":"2015-08-14T19:23:22","slug":"circulation-question","status":"publish","type":"post","link":"https:\/\/charles-oneill.com\/blog\/circulation-question\/","title":{"rendered":"Circulation Question"},"content":{"rendered":"<p>\\(\\)In the spring, a colleague asked a question regarding the computation of circulation about an airfoil. In response, I created a comparison of methods to calculate the circulation about an NACA 0012 airfoil at 10 degrees angle of attack, Mach 0.16 and a Reynolds number of 6 million. Data was generated with the FUN3D CFD solver with an SA turbulence model. The visualization is through TecPlot360. The pressure coefficient, \\(C_p \\), field is given below:<\/p>\n<p><a href=\"http:\/\/charles-oneill.com\/blog\/wp-content\/uploads\/2015\/08\/0012-CP-a10.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-258\" src=\"http:\/\/charles-oneill.com\/blog\/wp-content\/uploads\/2015\/08\/0012-CP-a10.png\" alt=\"0012-CP-a10\" width=\"462\" height=\"229\" srcset=\"https:\/\/charles-oneill.com\/blog\/wp-content\/uploads\/2015\/08\/0012-CP-a10.png 941w, https:\/\/charles-oneill.com\/blog\/wp-content\/uploads\/2015\/08\/0012-CP-a10-300x149.png 300w, https:\/\/charles-oneill.com\/blog\/wp-content\/uploads\/2015\/08\/0012-CP-a10-800x396.png 800w\" sizes=\"(max-width: 462px) 100vw, 462px\" \/><\/a><\/p>\n<p>The definition of vorticity is: $$ \\omega = \\nabla\\times V$$ In 2D, expanding gives $$\\omega = \\frac{dV_y}{dx} &#8211; \\frac{dV_x}{dy} $$Computing the vorticity in the domain gives the following field.<a href=\"http:\/\/charles-oneill.com\/blog\/wp-content\/uploads\/2015\/08\/0012-w-a10.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-261 \" src=\"http:\/\/charles-oneill.com\/blog\/wp-content\/uploads\/2015\/08\/0012-w-a10-300x145.png\" alt=\"0012-w-a10\" width=\"476\" height=\"240\" \/><\/a><\/p>\n<p>Three methods will be used to determine the circulation \\(\\Gamma \\) about the airfoil.<\/p>\n<ul>\n<li>Method 1: Compute from lift coefficient. The CFD solver estimates a lift coefficient of 1.09. From the Kutta-Joukowski theorem, lift per unit span is proportional to circulation. Solving for circulation gives $$ \\Gamma = \\frac{1}{2} V_o \\rho C_l $$ Computing gives \\(\\Gamma = 1.04 \\)<\/li>\n<li>Method 2: Integrate vorticity in domain. $$\\Gamma = \\int\\int_A \\omega dA $$ Computing gives \\(\\Gamma = 1.03 \\)<\/li>\n<li>Method 3: Contour integral of velocity. A coarse numerical integral with 16 points manually sampled from the CFD computed velocity components \\[\\Gamma\u00a0 =\u00a0 &#8211; \\oint\\limits_S {V \\cdot ds} \\]. <a href=\"http:\/\/charles-oneill.com\/blog\/wp-content\/uploads\/2015\/08\/0012-pts.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-262 size-medium\" src=\"http:\/\/charles-oneill.com\/blog\/wp-content\/uploads\/2015\/08\/0012-pts-300x180.png\" alt=\"0012-pts\" width=\"300\" height=\"180\" srcset=\"https:\/\/charles-oneill.com\/blog\/wp-content\/uploads\/2015\/08\/0012-pts-300x180.png 300w, https:\/\/charles-oneill.com\/blog\/wp-content\/uploads\/2015\/08\/0012-pts.png 479w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/a>This computation gives a circulation of \\(\\Gamma = 1.01\\).<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In the spring, a colleague asked a question regarding the computation of circulation about an airfoil. In response, I created a comparison of methods to calculate the circulation about an NACA 0012 airfoil at 10 degrees angle of attack, Mach 0.16 and a Reynolds number of 6 million. Data was generated with the FUN3D CFD [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/charles-oneill.com\/blog\/wp-json\/wp\/v2\/posts\/228"}],"collection":[{"href":"https:\/\/charles-oneill.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/charles-oneill.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/charles-oneill.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/charles-oneill.com\/blog\/wp-json\/wp\/v2\/comments?post=228"}],"version-history":[{"count":5,"href":"https:\/\/charles-oneill.com\/blog\/wp-json\/wp\/v2\/posts\/228\/revisions"}],"predecessor-version":[{"id":264,"href":"https:\/\/charles-oneill.com\/blog\/wp-json\/wp\/v2\/posts\/228\/revisions\/264"}],"wp:attachment":[{"href":"https:\/\/charles-oneill.com\/blog\/wp-json\/wp\/v2\/media?parent=228"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/charles-oneill.com\/blog\/wp-json\/wp\/v2\/categories?post=228"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/charles-oneill.com\/blog\/wp-json\/wp\/v2\/tags?post=228"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}