Sidewall Boundary Layers of the Bidirectional Vortex

Abstract

This study seeks to resolve the sidewall boundary layers forming in the axial and radial directions of a bidirectional vortex chamber. Our analysis is initiated by the formulation of the laminar boundary-layer equations via an order of magnitude reduction of the incompressible Navier-Stokes equations at the wall. Asymptotic theory is then applied to linearize and systematically truncate the governing equations, thus converting them from partial differential equations to more manageable ordinary differential equations. Scaling transformations are additionally applied to resolve the rapid changes arising near the sidewall. Because of the spatial character of the outer solutions, further transformations of the dependent variables are undertaken to secure the axially changing outer conditions. Through the use of matched-asymptotic expansions, we recover similar boundary-layer structures in all three orthogonal directions: the axial and radial components presented here, and the wall-tangential boundary layer obtained previously. This behavior is consistent with the resultant velocity being dominated by its tangential component and with the tangential boundary layer being axially invariant. These factors cause the axial layer to remain uniform in the streamwise direction. Based on the ensuing asymptotic results, viscous corrections at the wall are seen to be mainly dependent on the vortex Reynolds number, V. The latter combines the swirl number, the Reynolds number, and the chamber aspect ratio. Having obtained the three components of the velocity, essential flow characteristics, such as pressure, vorticity, swirling intensity, and wall shear stresses, are evaluated and discussed. Nomenclature A i = inlet area a = chamber radius b = chamber outlet radius, b < a l = chamber aspect ratio, L=a p = normalized pressure, p=U 2 Q i = normalized volumetric flow rate, Q i =Ua 2 1 Q i = inlet volumetric flow rate Re = injection Reynolds number, Ua= 1=" r, z = normalized radial or axial coordinates, r=a, z=a S = swirl number, ab=A i s = scaled transformation variable, = U = average inflow velocity in the tangential direction, u a; L u = normalized velocity u r ; u z ; u =U u = normalized swirl/spin/tangential velocity, u =U V = vortex Reynolds number, Q i Rea=L "l 1 = constant, 1 6 2 1 ' 0:644934 = normalized outlet radius, b=a = minus rescaled layer w = wall boundary-layer thickness, w =a " = perturbation parameter, 1=Re =Ua = transformed variable, r 2 = inflow parameter, Q i =2l 2l 1 = kinematic viscosity, = = density = modified swirl number, Q 1 i S= Subscripts i = inlet property r = radial component z = axial component = azimuthal component Superscripts o = outer (inviscid) solution = dimensional variables

Unified Framework for Modeling Swirl Dominated Helical Motions

“…Their approximation was found to be uniformly valid from the core to the sidewall, despite their use of a strictly ad hoc assumption near the sidewall. Along similar lines, Batterson and Majdalani [64] have regularized the axial and radial momentum equations and provided sidewall corrections in the off-swirl directions. These approaches will be revisited here and reconstructed in the neighborhood of the sidewall.…”

“…Because the radial velocity in its inviscid form already vanishes at the sidewall, we find that the role of the viscous correction is limited to providing a natural smoothing to the corresponding curves, thus causing not only u r but also its derivative to vanish as r → 1. This adjustment is what we refer to as 'smoothing,' which is well described by Batterson and Majdalani [64]. In reference to the radial motion, it may be instructive to discuss two particular points of interest: the crossflow velocity (u r ) cross along the mantle interface and the most negative radial velocity (i.e.…”

“…Despite its inherent simplicity and cold-flow basis, this so-called complex lamellar model helped to dispel some of the skepticism that lingered at that point against the physicality of cyclonic motions within a right-cylindrical VCCWC. The ensuing profile 4 American Institute of Aeronautics and Astronautics quickly became the launchpad for a number of complementary sequels in which the treatment of boundary layers could be partially addressed [60][61][62][63][64]. In separate work, Majdalani [65] subdivided the hybrid rocket flowfield within VIHRE into a leading-order approximation, for which the sidewall could be kept inert, and a first-order correction, which proved to be commensurate with the wall blowing speed relative to the tangential injection velocity.…”

Inversion of the Fundamental Thermodynamic Equations for Isentropic Nozzle Flow Analysis

“…Taking a cue from the largest, we then collect the leading-order quantities and discount the trailing elements as per Eq. (26). We promptly identify S 0 ¼ en ð Þ 1=a and write…”

Biglobal Instability of the Bidirectional Vortex. Part 2: Complex Lamellar and Beltramian Motions

“…Later, uniformly valid asymptotic solutions were constructed to resolve these deficiencies in the tangential 4 and axial velocities. 5 2 American Institute of Aeronautics and Astronautics Further advancements have been made to these solutions including the effects of multidirectional motion 6,7 and compressibility. 8,9 Most recently, the Bragg-Hawthorne equation (BHE) 10 has been considered as the opening point for the discussion of new solutions for the bidirectional vortex.…”