Investigation of the Sidewall Boundary Layers in the Bidirectional Vortex Liquid Rocket Engine

To complement previous studies on the bidirectional vortex, we attempt to characterize viscous effects in both the axial and radial directions along the sidewall with standard asymptotic techniques. The actual boundary layer present in the chamber will be a composite of both axial and tangential shearing layers. Since the tangential velocity is completely dominant, we expect the defining characteristics of the composite boundary layer to be congruent with the tangential layer in both thickness and axial invariance. The analytic analysis is outlined first by, the formulation of the boundary layer equations via Prandtl’s method. Next, asymptotic techniques are applied to linearize and rigorously truncate the governing equations from PDEs to more manageable ODEs. A scaling transformation is applied to resolve the rapid changes near the wall. Due to the nature of the outer solution, a dependent variable transformation is applied to recover constant boundary conditions. The viscous corrections are matched to the outer solution via Prandtl’s matching principle. We see a similar form in all three wall corrections; the axial and radial presented here and the swirl previously formulated. This can be expected to some extent because of the similarity of the asymptotic assumptions and the linearization techniques used in all three cases. Although the assumption that curvature terms can be neglected is never made, they are found to be asymtotically small and the problem then parallels the case of a one-dimensional Cartesian boundary layer. It can be seen that all viscous corrections along the wall are strongly dependent on the value of the vortex Reynolds number, V. This parameter shows up naturally in all vector directions. With the new corrected solutions, other key features of the flowfield can be revisited.