The small time behaviour for the impulsively started flow past a semi- infinite flat plate
Let \((x,y)\) be the system of Cartesian coordinates in which the half plate is represented by \(y=0\), \(x>0\). The flow is started impulsively with a uniform velocity \(V\) in the direction of the positive \(x\)-axis. Neglecting the inertia terms the equations to be solved are \(\omega_ t=\nu\nabla^ 2\omega\), \(\omega=\nabla^ 2\Psi\) (\(\omega\)= vorticity, \(\Psi\)= stream function), with the conditions \(\Psi=\Psi_ y=0\) for \(y=0\), \(x>0\), \(t>0\), \(\Psi=\Psi_{yy}=0\) for \(y=0\), \(x<0\), \(t>0\), \(\Psi=U_ y\) for \(t=0\), \(y\neq 0\). The initial condition reduces to be homogeneous by taking the transformation \(\Psi=U_ y+\Psi\), \(\Omega=\nabla^ 2\Psi\). The condition for \(\Psi_ y\) on \(y=0\), \(x>0\) is modified to \(\Psi_ y=- Ue^{-\delta x}\) in order to eliminate singularities in the subsequent treatment, letting \(\delta\) tend to zero at the end of the calculation. The equations for \(\Omega\), \(\Psi\) are solved by taking Laplace and Fourier transforms (in this order) and finally the inverse transforms in the reverse order, thus obtaining approximated expressions of \(\Psi\), \(\Omega\). An accurate discussion of the results follows, estimating the time for which each order of approximation provides a satisfactory description of the phenomenon. Finally a similar analysis is developed for the case in which the initial flow has a non-zero inclination with respect to the plate.
scientific article; zbMATH DE number 91583