Equations for Determination of the Strength of a Bound Vortex Sheet

Abstract

By using the formulas given in the preceding chapter for an infinite vortex chain, we can obtain expressions for the disturbed velocities in the fluid about a cascade of thin profiles. We replace each profile by a continuous distribution of vortex sheets of strength Γ (s). We use the coordinate system ξ, ŋ, where the Oŋ axis is the front of the cascade (Fig. 1.8 and 1.11). Let ξ̄ ₀ and ŋ ₀ be the dimensionless coordinates defined in (7.51) for any point of the plane; then for the disturbed velocity at this point we have

$$\left. {{}_{{W_\eta }\left( {{{\bar \xi }_0},{{\bar \eta }_0}} \right) = \frac{1}{{2t}}\int {\gamma \left( s \right){w_\eta }\left( {{{\bar \zeta }_0} - \bar \zeta ,{{\bar \eta }_0} - \bar \eta } \right)ds,} }^{{W_{\bar \varsigma }}\left( {{{\bar \xi }_0},{{\bar \eta }_0}} \right) = \frac{1}{{2t}}\int {\gamma \left( s \right){w_\zeta }\left( {{{\bar \zeta }_0} - \bar \zeta ,{{\bar \eta }_0} - \bar \eta } \right)ds} }} \right\}$$

((8.1))

since the differential γ(s)ds will correspond in dimension to the circulation Γ₊ of Eqs. (7.51). In (8.1), the integration is taken along the length of each profile and the expressions for w _ξ and W _η are given by Eqs. (7.50).