Plane subsonic forms with negative pressure gradients at the surface (Q2759536)

The condition for the pressure coefficients \(dCp(x)/dx<0\) makes impossible separation of boundary layer, prevents its turbulization and consequently reduces hydrodynamic resistance. In the paper, two-dimensional forms (hydrodynamic airfoils) with very short zone of positive pressure gradient, are obtained. The profiles are calculated from the formula relating the potential \(\phi(x,y)\) and the stream function \(\psi(x,y)\) with the intensity of sources and drains, \(W(z)=\phi(x,y)+ i\psi(x,y)= z+\int_0^1 q(\xi)\ln(z-\xi) d\xi\), \(z=x+iy\). First approximation for \(q(x)\) is obtained on the basis of similarity of plane and axisymmetrical problems. Coordinate \(Y(x)\) of the upper airfoil surface was calculated using the equation \(\psi(x, Y(x))={\text Im\,}W(s)=0\), which was solved numerically. After determination of function \(Y(x)\), components of velocity and \(Cp\) at the surface are calculated using equations \(v_x(x,y)-iv_y(x,y)=dW(z)/dz\), \(Cp(x)= 1-v_x^2(x,Y(x)) -v_y^2(x,Y(x))\). Examples of calculated airfoils are presented.