The circulation around the vortex sheet is the sum of the strengths of the elemental 4 vortices. b IS . There is a discontinuity change in the tangential component of velocity across the sheet Let r to be the circulation along the dashed line
The circulation around the vortex sheet is the sum of the strengths of the elemental vortices. ds b a There is a discontinuity change in the tangential component of velocity across the sheet. Let to be the circulation along the dashed line
U 2 dn 2 T=-(v,dn-u,ds-v,dn +u,ds) O厂 F=(1-l2)s+(v1-v2)dm
( ) v2dn u1ds v1dn u2ds or (u1 u2 )ds (v1 v2 )dn
as T= yds so rs=(u -u2)ds+(v1-v2)dn As the top and bottom of the dashed line approach the vortex sheet, dn>0, u, u2 become the velocity components tangential to the vortex sheet immediately above and below the sheet. ys=(u-u2)ds or y=(l1-l2) The local jump in the tangential velocity across the vortex sheet is equal to the local sheet strength
as ds so ds (u1 u2 )ds (v1 v2 )dn As the top and bottom of the dashed line approach the vortex sheet, , become the velocity components tangential to the vortex sheet immediately above and below the sheet. dn 0 1 2 u ,u ds (u u )ds 1 2 or ( ) u1 u2 The local jump in the tangential velocity across the vortex sheet is equal to the local sheet strength
o Philosophy of airfoil theory for inviscid, incompressible flows Step 1. Replace the airfoil surface with a vortex sheet of strength r(s) ?(s) Airfoil of s arbitrary shape and thickness Step 2. Find a suitable distribution of r(s) such that the wall boundary condition can be satisfied. That is the combination of the free stream flow and the vortex sheet will make the vortex sheet(the surface of the airfoil) a streamline of the flow
Philosophy of airfoil theory for inviscid, incompressible flows. Step 1. Replace the airfoil surface with a vortex sheet of strength (s) Step 2. Find a suitable distribution of such that the wall boundary condition can be satisfied. That is, the combination of the free stream flow and the vortex sheet will make the vortex sheet(the surface of the airfoil) a streamline of the flow. (s)
Step 3. Calculate the circulation around the airfoil and then get the lift by Kutta-Joukowski theorem T=lrds Note 1. There are no general analytical solution r(s) for an airfoil with arbitrary shape and thickness. This should be solved numerically with suitable digital computers. Vortex pane/ method (Sec. 4.9) Note 2. Physical significance of the vortex sheet which has been used to replace the surface of the airfoil surface. Boundary layer is a highly viscous region, the vorticity inside the boundary layer is finite
Step 3. Calculate the circulation around the airfoil, and then get the lift by Kutta-Joukowski theorem ds L V Note 1. There are no general analytical solution for an airfoil with arbitrary shape and thickness. This should be solved numerically with suitable digital computers. Vortex panel method (Sec. 4.9) (s) Note 2. Physical significance of the vortex sheet which has been used to replace the surface of the airfoil surface. Boundary layer is a highly viscous region, the vorticity inside the boundary layer is finite