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In this work, an asymptotic expansion is presented that takes into account a naturally-occurring perturbation parameter in the context of a circular tube with an open-open endpoint configuration. This approach is shown to produce accurate predictions of pressure mode shapes and frequencies for arbitrary temperature distributions that mimic a wide variety of flow heating arrangements including, but not limited to, those associated with a Rijke tube. The underlying formulation consists of two linearly coupled partial differential equations that can be solved simultaneously while using a Green’s function to capture the thermoacoustic pressure. In the present investigation, the strategy leading to an accurate prediction of the unsteady pressure oscillations is fully detailed and then applied to several representative cases. Results pertaining to the pressure oscillations are systematically discussed and compared to other recently developed models in the literature. 

We vary the inflow properties in a finite-volume solver to investigate their effects on the computed cyclonic motion in a right-cylindrical vortex chamber. The latter comprises eight tangential injectors through which steady-state air is introduced under incompressible and inviscid conditions. To minimize cell skewness around injectors, a fine tetrahedral mesh is implemented first and then converted into polyhedral elements, namely, to improve convergence characteristics and precision. Once convergence is achieved, our principal variables are evaluated and compared using a range of inflow parameters. These include the tangential injector speed, count, diameter, and elevation. The resulting computations show that well-resolved numerical simulations can properly predict the forced vortex behavior that dominates in the core region as well as the free vortex tail that prevails radially outwardly, beyond the point of peak tangential speed. It is also shown that augmenting the mass influx by increasing the number of injectors, injector size, or average injection speed, further amplifies the vortex strength and all peak velocities while shifting the mantle radially inwardly. Overall, the axial velocity is found to be the most sensitive to vertical displacements of the injection plane. By raising the injection plane to the top half portion of the chamber, the flow character is markedly altered, and an axially unidirectional vortex is engendered, particularly, with no upward motion or mantle formation. Conversely, the tangential and radial velocities are found to be axially independent and together with the pressure distribution prove to be the least sensitive to injection plane relocations. 

In this work, the Kármán–Pohlhausen (KP) momentum-integral approach based on optimized fourth-order (MX4) polynomial approximations of the velocity and temperature profiles is applied to a classical benchmark problem, namely, that of a cylinder in crossflow with a variable pressure gradient. This enables us to extract closed-form expressions for both hydrodynamic and thermal boundary-layer parameters and then compare the newly found solutions to their counterparts obtained using Pohlhausen's cubic (KP3) and quartic (KP4) polynomials. As usual, the farfield around the cylinder is modeled using potential flow theory and the momentum-integral analysis is paired with Walz's empirical expression for the momentum thickness, which is based on a wide collection of experiments. This procedure permits retrieving explicit relations for the pressure-sensitive KP3, KP4, and MX4 velocity profiles across the boundary layer; one also obtains accurate approximations for the pressure distribution around the cylinder as well as an improved prediction of the separation point, namely, to within 0.87% of the actual location. In this process, refined estimates are produced for several characteristic parameters whose distributions are found to be in favorable agreement with experimental measurements and numerical simulations. These include the disturbance, momentum, and displacement thicknesses as well as the skin friction, pressure, and total drag coefficients. Finally, the thermal analysis is undertaken using both isothermal and isoflux boundary conditions. For each of these cases, closed-form analytical solutions are obtained for the local Nusselt number distribution around the cylinder, and these distributions are found to exhibit noticeably reduced errors relative to their classical values. 

We vary the inflow properties in a finite-volume solver to investigate their effects on the computed cyclonic motion in a right-cylindrical vortex chamber. The latter comprises eight tangential injectors through which steady-state air is introduced under incompressible and inviscid conditions. To minimize cell skewness around injectors, a fine tetrahedral mesh is implemented first and then converted into polyhedral elements, namely, to improve convergence characteristics and precision. Once convergence is achieved, our principal variables are evaluated and compared using a range of inflow parameters. These include the tangential injector speed, count, diameter, and elevation. The resulting computations show that well-resolved numerical simulations can properly predict the forced vortex behavior that dominates in the core region as well as the free vortex tail that prevails radially outwardly, beyond the point of peak tangential speed. It is also shown that augmenting the mass influx by increasing the number of injectors, injector size, or average injection speed further amplifies the vortex strength and all peak velocities while shifting the mantle radially inwardly. Overall, the axial velocity is found to be the most sensitive to vertical displacements of the injection plane. By raising the injection plane to the top half portion of the chamber, the flow character is markedly altered, and an axially unidirectional vortex is engendered, particularly, with no upward motion or mantle formation. Conversely, the tangential and radial velocities are found to be axially independent and together with the pressure distribution prove to be the least sensitive to injection plane relocations. 

This work presents an exact solution of Euler's incompressible equations in the context of a bidirectional vortex evolving inside a conically shaped cyclonic chamber. The corresponding helical flowfield is modeled under inviscid conditions assuming constant angular momentum. By leveraging the axisymmetric nature of the problem, a steady-state solution of the generalized Beltramian type is obtained directly from first principles, namely, from the Bragg–Hawthorne equation in spherical coordinates. The resulting stream function representation enables us to fully describe the ensuing swirl-dominated motion including its fundamental flow characteristics. After identifying an isolated singularity that appears at a cone divergence half-angle of 63.43°, two piecewise formulations are provided that correspond to either fluid injection or extraction at the top section of the conical cyclone. In this process, analytical expressions are readily retrieved for the three velocity components, vorticity, and pressure. Other essential flow indicators, such as the theoretically preferred mantle orientation, the empirically favored locus of zero vertical velocity, the maximum polar and axial velocities, the crossflow velocity, and other such terms, are systematically deduced. Results are validated using limiting process verifications and comparisons to both numerical and experimental measurements. The subtle differences between the present model and a strictly Beltramian flowfield are also highlighted and discussed. The conically cyclonic configuration considered here is relevant to propulsive devices, such as vortex-fired liquid rocket engines with tapered walls; meteorological phenomena, such as tornadoes, dust devils, and fire whirls; and industrial contraptions, such as cyclonic flow separators, collectors, centrifuges, boilers, vacuum cleaners, cement grinders, and so on. 

In this work, an exact inviscid solution is developed for the incompressible Euler equations in the context of a bidirectional, cyclonic flowfield in a right-cylindrical chamber with a hollow core. The presence of a hollow core confines the flow domain to an annular swirling region that extends into a toroid in three-dimensional space. The procedure that we follow is based on the Bragg–Hawthorne framework and a judicious assortment of boundary conditions that correspond to a wall-bounded cyclonic motion with a cylindrical core. At the outset, a self-similar stream function is obtained directly from the Bragg–Hawthorne equation under the premises of steady, axisymmetric, and inviscid conditions. The resulting formulation enables us to describe the bidirectional evolution of the so-called inner and outer vortex motions, including their fundamental properties, such as the interfacial layer known as the mantle; it also unravels compact analytical expressions for the velocity, pressure, and vorticity fields, with particular attention being devoted to their peak values and spatial excursions that accompany successive expansions of the core radius. By way of confirmation, it is shown that removal of the hollow core restores the well-established solution for a fully flowing cylindrical cyclone. Immediate applications of cyclonic flows include liquid and hybrid rocket engines, swirl-driven combustion devices, as well as a multitude of heat exchangers, centrifuges, cyclone separators, and flow separation devices that offer distinct advantages over conventional, non-swirling systems.