In this work, a new exact Euler solution is derived under the same fundamental contingencies of axisymmetric, steady, rotational, incompressible, single phase, non-reactive, and inviscid fluid, which stand behind the ubiquitously used mean flow profile named "Taylor-Culick." In comparison to the latter, which proves to be complex lamellar, the present model is shown to be of the Trkalian type, hence capable of generating a nonzero swirl component that increases linearly in the streamwise direction. This enables us to provide an essential mathematical representation that is appropriate of flow configurations where the bulk gaseous motion is permitted to swirl. Examples abound and one may cite the classic experiments of Dunlap and co-workers, which focus on porous chambers with circular cross-sections, and where the inevitable presence of swirl as a natural flow companion is clearly demonstrated (). From a procedural standpoint, the new Trkalian solution is deduced directly from the Bragg-Hawthorne equation, which has been repeatedly shown to possess sufficient platitude to reproduce several existing profiles such as Taylor-Culick's as special cases. Throughout this study, the fundamental properties and benefits of the present model are highlighted and discussed in the light of existing flow approximations and experiments. Consistent with the original Taylor-Culick mean flow motion, the Trkalian velocity is seen to exhibit both axial and tangential components that increase linearly with the distance from the headwall, and a radial component that remains axially invariant. However, unlike its axial and radial counterparts, which identically satisfy the no slip requirement at the sidewall (even in their purely inviscid form), the tangential velocity must be treated asymptotically so as to capture the evolving sidewall shear layer while resolving the core layer near the centerline. At the outset, a matched-asymptotic expression for the swirl velocity is obtained and discussed. Finally, our Trkalian model is shown to form a subset of the Beltramian class of solutions for which the velocity and vorticity vectors are not only parallel but also directly proportional. This characteristic feature is interesting as it stands in sharp contrast to the complex lamellar nature of the Taylor-Culick flow, where velocity and vorticity vectors remain orthogonal. Nomenclature a = chamber radius B = tangential angular momentum, ru θ B = axially normalized angular momentum, / ( ) B z rU r θ = C = swirl momentum constant 1 c = radial velocity constant multiplier, 1 1 1.9 1/ ( 26 34 ) 2 8 J λ ≈ D = tangential surface parameter H = stagnation pressure head 0 L = chamber length L = chamber aspect ratio, 0 / L a 2 p = normalized pressure, 2 / ( ) w p U ρ Q = normalized volumetric flow rate, 2 / ( ) w Q U a q = auxiliary constant, 1 1 1 1 1 4 ( ) 1.1580647 J λ λ − ≈ Re = crossflow or wall injection Reynolds number, 1 / w U a ν ε − = , , r z θ = radial, tangential, and axial coordinates U = mean tangential (inflow) velocity c U = headwall inj...
In this work, an exact Euler solution is derived under the fundamental contingencies of axisymmetric, steady, rotational, incompressible, single-phase, non-reactive and inviscid fluid, which also stand behind the ubiquitously used mean flow profile named 'Taylor-Culick.' In comparison with the latter, which proves to be complex lamellar, the present model is derived in the context of a Trkalian flow field, and hence is capable of generating a non-zero swirl component that increases linearly in the streamwise direction. This enables us to provide an essential mathematical representation that is appropriate for flow configurations where the bulk gaseous motion is driven to swirl. From a procedural standpoint, the new Trkalian solution is deduced directly from the Bragg-Hawthorne equation, which has been repeatedly shown to possess sufficient latitude to reproduce several existing profiles such as Taylor-Culick's as special cases. Throughout this study, the fundamental properties of the present model are considered and discussed in the light of existing flow approximations. Consistent with the original Taylor-Culick mean flow motion, the Trkalian velocity is seen to exhibit both axial and tangential components that increase linearly with the distance from the headwall, and a radial component that remains axially invariant. Furthermore, the Trkalian model is shown to form a subset of the Beltramian class of solutions for which the velocity and vorticity vectors are not only parallel but also directly proportional. This characteristic feature is interesting, as it stands in sharp contrast to the complex-lamellar nature of the Taylor-Culick motion, where the velocity and vorticity vectors remain orthogonal. By way of verification, a numerical simulation is carried out using a finite-volume solver, thus leading to a favourable agreement between theoretical and numerical predictions.
In this work, an exact, swirl-tolerant, Euler solution describing the mean flow in solid and hybrid rocket motors is further analyzed following its complete presentation in a companion article. 1 The development of this solution can be significant as it demonstrates that swirling flows observed in several experimental studies but unaccounted for in the Taylor-Culick flowfield may indeed be described by an alternative flow representation that displays more realistic features than the Taylor-Culick model itself. Previously, it was shown that this new swirl-augmented motion and the ubiquitous Taylor-Culick profile may both be derived as special cases of the Bragg-Hawthorne equation. Building upon this development, we show in this work that at least two families of approximate solutions may be further achieved using the Lagrangian optimization technique; the spinoff models will have the property of identically satisfying the boundary conditions prescribed for the Taylor-Culick problem while exhibiting different levels of kinetic energy. In this study, the energy and entropy contents of the developed solutions are evaluated and discussed. The new formulations pave the way to several prospective studies in combustion and hydrodynamic stability research where the accurate specification of the mean flow plays an important role. Nomenclature a = chamber radius r , z = normalized radial and axial coordinate, / r a , / z a u = normalized velocity ( , , ) / r z w u u u U θ c U = headwall injection velocity, (0, 0) z u c u = normalized injection velocity, / c w U U w U = sidewall injection velocity, ( , ) r u a z − ρ = density Subscripts and Symbols c = centerline property h = headwall property r , z = radial and axial component or partial derivative w = sidewall property X = overbars denote dimensional variables
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