> For the complete documentation index, see [llms.txt](https://docs.aviumtechnologies.com/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.aviumtechnologies.com/ellipsoid.md). # Ellipsoid Example simulation of an ellipsoid An ellipsoid with dimensions (semiaxes) $$L\_{x}=2a=8$$m, $$L\_{y}=2b=4$$m, $$L\_{z}=2c=2$$m at an angle of attack of $$\alpha=15$$deg and sideslip angle of $$\beta=10$$deg is considered. The magnitude of the freestream velocity is $$|\mathbf{V}\_{ref}|=1$$m/s. An unstructured mesh of 14924 elements is used. Similarly to the sphere, an analytical solution for the perturbation and total potential around the ellipsoid exists \[1]. The perturbation (doublet) potential is given by: $$ \phi = \phi\_{x} + \phi\_{y} +\phi\_{z} = xuabc\frac{\alpha\_{\lambda}}{2-\alpha\_{\lambda}} + yvabc\frac{\beta\_{\lambda}}{2-\beta\_{\lambda}} + zwabc\frac{\gamma\_{\lambda}}{2-\gamma\_{\lambda}}\text{,} $$ where $$\alpha{\lambda}$$, $$\beta{\lambda}$$ and $$\gamma\_{\lambda}$$ are integrals given by: $$ \alpha\_{\lambda}=\int\_{\lambda}^{\infty} \frac{d\lambda}{(a^2+\lambda)\sqrt{(a^2+\lambda)(b^2+\lambda)(c^2+\lambda)}}\\ \beta\_{\lambda}=\int\_{\lambda}^{\infty} \frac{d\lambda}{(b^2+\lambda)\sqrt{(a^2+\lambda)(b^2+\lambda)(c^2+\lambda)}}\ \gamma\_{\lambda}=\int\_{\lambda}^{\infty} \frac{d\lambda}{(c^2+\lambda)\sqrt{(a^2+\lambda)(b^2+\lambda)(c^2+\lambda)}}\text{.} $$ The solution of the $$\alpha{\lambda}$$, $$\beta{\lambda}$$ and $$\gamma{\lambda}$$ integrals is obtained with the aid of the incomplete elliptic integrals as explained in \[2,4]. When the potential is evaluated at the surface of the ellipsoid $$\lambda$$ is set to 0. Numerical values of the incomplete elliptic integrals, along with values of the $$\alpha{\lambda}$$, $$\beta{\lambda}$$, $$\gamma{\lambda}$$ integrals for any combinations of $$a$$, $$b$$, and $$c$$ are given in \[3]. The total potential is given by: $$ \Phi = \Phi\_{x} + \Phi\_{y} + \Phi\_{z} = (\phi\_{x} + xu) + (\phi\_{y} + yv) + (\phi\_{z} + zw)\text{,} $$ where $$\Phi$$ is the total potential, $$\phi$$is the perturbation (doublet) potential, $$a$$,$$b$$, and $$c$$ are the semiaxes of the ellipsoid,$$x$$, $$y$$, and $$z$$ are the centroid coordinates of an element on the ellipsoid surface, $$u$$,$$v$$,and $$w$$ are velocity components relative to the ellipsoid. The image blow shows the ellipsoid pressure coefficient. ![](https://81269019-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-MV8htLPXN-_fWc_anY_%2F-Mh5CLnhy5-cutrZvbQQ%2F-Mh5DZSyv-25wAafKJ5x%2Fellipsoid_cp.png?alt=media\&token=e7137d99-ed62-4b69-b785-7bdff7c48a28) The images below compare the perturbation potential and the total potential with the analytical solution. As there is no analytical solution for the pressure coefficient at $$\alpha=15$$deg and $$\beta=10$$deg, the analytical solution for the pressure coefficient at $$\alpha=\beta=0$$deg is compared to the one from a simulation at the same conditions. Good agreement between the simulation and the analytical solution is observed. ## Files {% file src="/files/-MgbFgyzWrOrV5s0RXOH" %} ## References \[1] Craig, T., "On the Motion of an Ellipsoid in a Fluid", American Journal of Mathematics, Vol.2, No.3, 1879, pp. 260-279\ \[2] Byrd, P.F., "Handbook of Elliptic Integrals for Engineers and Scientists", Springer-Verlag, 1971, pp. 1-7, pp. 8-41\ \[3] Jones, R.T., "Classical Aerodynamic Theory", University Press of the Pacific, 2005\ \[4] Viana, F. and Fuanda, T. and Joseph, D.D. and Tashiro, N. and Sonoda, Y. "Potential flow of a second-order fluid over a tri-axial ellipsoid", J. Appl. Math., Vol.4, 2005, pp. 341-364