Example simulation of an ellipsoid

An ellipsoid with dimensions (semiaxes) Lx=2a=8L_{x}=2a=8m, Ly=2b=4L_{y}=2b=4m, Lz=2c=2L_{z}=2c=2m at an angle of attack of α=15\alpha=15deg and sideslip angle of β=10\beta=10deg is considered. The magnitude of the freestream velocity is Vref=1|\mathbf{V}_{ref}|=1m/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:

ϕ=ϕx+ϕy+ϕz=xuabcαλ2αλ+yvabcβλ2βλ+zwabcγλ2γλ,\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 αλ\alpha{\lambda}, βλ\beta{\lambda} and γλ\gamma_{\lambda} are integrals given by:

αλ=λdλ(a2+λ)(a2+λ)(b2+λ)(c2+λ)βλ=λdλ(b2+λ)(a2+λ)(b2+λ)(c2+λ)γλ=λdλ(c2+λ)(a2+λ)(b2+λ)(c2+λ).\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 αλ\alpha{\lambda}, βλ\beta{\lambda} and γλ\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 λ\lambda is set to 0. Numerical values of the incomplete elliptic integrals, along with values of the αλ\alpha{\lambda}, βλ\beta{\lambda}, γλ\gamma{\lambda} integrals for any combinations of aa, bb, and cc are given in [3]. The total potential is given by:

Φ=Φx+Φy+Φz=(ϕx+xu)+(ϕy+yv)+(ϕz+zw),\Phi = \Phi_{x} + \Phi_{y} + \Phi_{z} = (\phi_{x} + xu) + (\phi_{y} + yv) + (\phi_{z} + zw)\text{,}

where Φ\Phi is the total potential, ϕ\phiis the perturbation (doublet) potential, aa,bb, and cc are the semiaxes of the ellipsoid,xx, yy, and zz are the centroid coordinates of an element on the ellipsoid surface, uu,vv,and ww are velocity components relative to the ellipsoid. The image blow shows the ellipsoid pressure coefficient.

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 α=15\alpha=15deg and β=10\beta=10deg, the analytical solution for the pressure coefficient at α=β=0\alpha=\beta=0deg is compared to the one from a simulation at the same conditions.

Good agreement between the simulation and the analytical solution is observed.



[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

Last updated