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

Hydrodynamic instability research has begun to shift from the one-dimensional analysis of the Orr-Sommerfeld, local nonparallel, and parabolic type to the two-dimensional, biglobal analysis. Classic methods, such as the local nonparallel approach, specify a one-dimensional amplitude function a priori so that streamwise and azimuthal variations become characterized by a complex wave and integer mode number, respectively. This formulation is heavily dependent on the so-called parallel flow assumption. In brief, this property requires the streamlines of the baseflow to be parallel or nearly parallel, a condition that may be quite restrictive at times. It can therefore be seen that the use of biglobal analysis becomes necessary to encompass a broader range of baseflows. The relatively novel, two-dimensional characterization of the hydrodynamic waveform gives rise to a more complete analysis that intrinsically captures spatial instability behavior while remaining unrestricted to special geometries, such as elongated rocket chambers. The biglobal approach only requires periodicity in the tangential direction and seems to be tailor-made for cylindrical axisymmetric flows, such as those arising in the bidirectional vortex engine. Moreover, for short rocket chambers such as those associated with bidirectional vortex engines, the parallel flow assumption cannot be justified, and one finds the biglobal analysis as a more suitable alternative to the local nonparallel (LNP) approach used extensively in solid rocket motor stability investigations. However, being multi-dimensional, the biglobal technique requires more effort than its predecessor, specifically in the analytical derivation, numerical discretization, and spectral collocation tools necessitated by the treatment of the resulting partial differential equations. The present work serves to offer a detailed description of the theoretical framework and numerical tools needed to tackle this problem in the context of a confined vortex chamber such as that investigated by NASA/ORBITEC.

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In this article, a viscous approximation is obtained for the linear, inviscid, Beltramian motion that may be engendered in a confined, bidirectional vortex chamber. Using the theory of matched-asymptotic expansions, viscous corrections are developed near the core region, where a forced vortex prevails, and near the cylindrical wall, where the no-slip requirement holds. Through proper scaling and variable transformations, a uniformly valid composite solution is subsequently constructed from which Majdalani's helical profile may be recovered in the inviscid limit. The latter was derived directly from first principles (see Majdalani, J., "Exact Eulerian Solutions of the Cylindrical Bidirectional Vortex," AIAA Paper 2009-5307, Denver, Colorado, Aug. 2009). However, because of its exact Eulerian nature, the strictly inviscid profile could not account for the effects of viscous stresses near the axis of rotation. The present approximation overcomes this deficiency by ensuring both the velocity adherence condition at the wall and the solid-body rotation of the forced vortex region. Being driven in large part by the inviscid character, the viscous-rectified swirl velocity component is also seen to exhibit small variations in the axial direction while continuing to vanish, as it should, at the sidewall. The advent of a viscous approximation enables us to quantify the size of the core and wall boundary layers, improve our prediction of the vorticity and pressure distributions, and relate most relevant flow features to the vortex Reynolds number.

Having established the framework for biglobal hydrodynamic instability of an incompressible mean flowfield in Part 1 of this two-paper series, the present focus is turned towards applications. To this end, the instability of the bidirectional vortex motion is analyzed using the biglobal approach. Three distinct mean flow profiles are considered, specifically, the complex-lamellar, linear, and nonlinear Beltramian motions. Their spectral characteristics and eigensolutions are computed and compared to one another, as well as to the one-dimensional local nonparallel (LNP) approach. Our findings suggest that hydrodynamic waves produce visible oscillations around the mean flow streamlines. Their amplitudes are fairly insignificant in the case of axisymmetric oscillations but can become quite pronounced in the asymmetric cases. Overall, we find this class of helical flows to be quite stable, especially with successive increases in swirl intensity (or reductions in the inflow parameter κ). Nonetheless, regions that are most susceptible to instabilities seem to occur where shearing is most appreciable, for instance, where streamline curvatures are abrupt. The region near the headwall is thus identified as a site that may potentially exhibit flow breakdown. Several parametric cases are examined, and these show that increasing the swirl intensity of the injected stream reduces the number of unstable modes. We also find that the aspect ratio can influence the stability spectra and that an aspect ratio of L/R = 1.5 may be near-optimal. Most simulations are carried out for N = 50, given CPU runtime limitations. From a practical design perspective, suppressing unstable modes in the vortex engine may be realized by tightly securing an axisymmetric flow configuration both geometrically and dynamically.