The evaluation of the Biot-Savart integral

A computational method is described for evaluating the Biot-Savart integral. The approach emphasizes the transformation of the involved integrand into suitable forms, from which integral theorems can be used to reduce the volume integral into line integrals. This method is applied to the case where the density of vorticity (current) distributed over a volumetric element bounded by planar surfaces (straight lines in two-dimensional) is constant and/or linear. The resulting expressions for the volume integral involve closed-form expressions for line integrals along the edges of the element. The evaluation of the line integrals is treated independently for each of the edges as opposed to direct numerical integration. The closed-form formulas are expressed in terms of geometric parameters of the element edges. The versatility of the proposed scheme is demonstrated by applying it to two examples: (i) two-dimensional lid-driven cavity flows; (ii) a magnetic field induced by a toroidal tokamak coil. A systematic extension to the general cases where the vorticity distribution is of higher-order polynomial form is also presented.