Chapter 10 - Coanda effect

1. Problems

(i) Riley(1987) gives one way of computing a Prandtl-Batchelor flow which might be applied to a channel flow if one assumes that there is potential flow through the channel with vorticity being confined to finite regions on the walls. If the channel has a symmetric expansion is it possible to compute an asymmetric Prandtl-Batchelor flow or are all solutions necessarily symmetric? Is it possible to compute asymmetric exterior Prandtl-Batchelor flows where the geometry and upstream outer flow is symmetric?

(ii) Can the $\epsilon\sim R^{-1/7}$ theory be applied in a symmetric channel in the same way as the $\epsilon << R^{-1/7}$ theory was applied by Borgas & Pedley; that is, as the limit of an asymmetric channel becoming symmetric? My experience with computed solutions of the two coupled non-linear boundary layer equations in a symmetric geometry is that when they do converge, they converge to a symmetric solution. Is it possible to compute asymmetric solutions by starting from an asymmetric initial condition or by introducing some form of psuedo time marching, for instance by solving

$$u_t+uu_X+vu_\eta = -P_L'(X)+u_{\eta\eta},\ {\tilde u}_t+{\tilde u}{\tilde u}_X+{\tilde v}{\tilde u}_\xi = -P_U'(X)+{\tilde u}_{\xi\xi},$$ (1)

with $P_L$ and $P_U$ appropriately related? This could be done by either an explicit scheme or an implicit scheme using a suitable modification for the Newton iteration matrix.

2. Software

(i) bpgen.f generate solutions to Borgas & Pedley's variant of the Falkner-Skan equation (10.17)
(ii) channel.f time marching solution of Navier-Stokes equations for flow thorough a sudden channel expansion using a rectangular grid. Time stepping of vorticity can be by central, upwind, Lax-Wendroff or Quickest algorithms. Stream function-vorticity equation solved using multigrid. You will need header file step.h

3. Corrections

(1) Equation (10.19) is missing a $z$ multiplying $\gamma (\tau )$

$$N(z)\sim {1\over 2}\zeta (\tau)z^2+\gamma (\tau )z +o(z)\ \ {\rm as}\ \ z\to\infty$$

Return to Contents page