Brian A. Maicke

On the rotational compressible Taylor flow in injection-driven porous chambers

This work considers the compressible flow field established in a rectangular porous channel. Our treatment is based on a Rayleigh-Janzen perturbation applied to the inviscid steady two-dimensional isentropic flow equations. Closed-form expressions are then derived for the main properties of interest. Our analytical results are verified via numerical simulation, with laminar and turbulent models, and with available experimental data. They are also compared to existing one-dimensional theory and to a previous numerical pseudo-one-dimensional approach. Our analysis captures the steepening of the velocity profiles that has been reported in several studies using either computational or experimental approaches. Finally, explicit criteria are presented to quantify the effects of compressibility in two-dimensional injection-driven chambers such as those used to model slab rocket motors.

A constant shear stress core flow model of the bidirectional vortex

In this paper, we discuss the merits of two models for the swirl velocity in the core of a confined bidirectional vortex. The first is piecewise, Rankine-like, based on a combined-vortex representation. It stems from the notion that a uniform shear stress distribution may be assumed in the inner vortex region of a cyclone, especially at high Reynolds numbers. Thereafter, direct integration of the shear stress enables us to retrieve an expression for the swirl velocity that overcomes the inviscid singularity at the centreline. The second model consists of a modified asymptotic solution to the problem obtained directly from the Navier–Stokes equations. Both solutions we present transition smoothly to the outer, free-vortex approximation at some intermediate position in the chamber. This position is deduced from available experimental data to the extent of providing an accurate swirl velocity distribution throughout the chamber. By scaling the constant shear radius to the core layer thickness, the constant of proportionality is readily calculated using the method of least squares. Interestingly, the constant of proportionality is found to be invariant at several vortex Reynolds numbers, thus helping to achieve closure. The combined-vortex representation is validated against a large body of experimental measurements and through comparisons to a laminar core model that is enhanced through the use of an eddy viscosity. Other heuristic schemes are discussed and the two most suitable models to capture realistic flow behaviour at high vortex Reynolds numbers are identified. Our two models are first derived analytically and then anchored on the available experimental measurements.

Current State of High Speed Propulsion: Gaps, Obstacles, and Technological Challenges in Hypersonic Applications

The object of this study is to canvas the literature for the purpose of identifying and compiling a list of Gaps, Obstacles, and Technological Challenges in Hypersonic Applications (GOTCHA). The significance of GOTCHA related deficiencies is discussed along with potential solutions, promising approaches, and feasible remedies that may be considered by engineers in pursuit of next generation hypersonic vehicle designs and optimizations. Based on the synthesis of several modern surveys and public reports, a cohesive list is formed consisting of widely accepted areas needing improvement that fall under several general categories. These include: aerodynamics, propulsion, materials, analytical modeling, CFD modeling, and education in high speed flow physics. New methods and lines of research inquiries are suggested such as the homotopy-based analysis (HAM) for the treatment of strong nonlinearities, the use of improved turbulence models and unstructured grids in numerical simulations, the need for accessible validation data, and the refinement of mission objectives for Hypersonic Airbreathing Propulsion (HAP).

On the Compressible Bidirectional Vortex. Part 1: A Bragg-Hawthorne Stream Function Formulation

The Bragg-Hawthorne equation, also named Squire-Long, takes advantage of a requirement that the stagnation pressure head, H, and angular momentum, B, may be generally related to the stream function, . This reduces the Navier-Stokes equations to a single stream function representation, wherein H and B may be specified based on the application of interest. In this paper, the Bragg-Hawthorne equation (BHE) is extended in the context of a steady, inviscid, and compressible fluid. Then given an assortment of suitable assumptions, our approach leads to a pair of partial di erential equations that must be treated simultaneously. Rather than solve the ensuing coupled equations numerically, we opt for a reduced-order model by implementing the Rayleigh-Janzen expansion method in which the square of the reference Mach number is employed as a perturbation parameter. The resulting linearized equations are retrievable to an arbitrary level of precision, thus leading to the establishment of a reliable, compressible BHE framework. From a practical standpoint, we find that a first-order correction is su cient to capture the bulk compressible contribution in most physical settings. In the second part of this paper series, the procedure is tested to derive a compressible solution for the linear Beltramian model of the bidirectional vortex; this motion corresponds to a swirl-driven cyclonic flowfield with an axially reversing character that proves to be of particular interest to the development of an innovative, self-cooled, liquid rocket engine concept. It can therefore be seen that, although the application of interest remains focused on the bidirectional vortex motion, the framework that we o er retains su cient generality to be useful in handling a wide range of axisymmetric problems, especially those that may be conveniently expressed in polar-cylindrical coordinates.

On the Compressible Bidirectional Vortex. Part 2: A Beltramian Flowfield Approximation

In the first part of this paper series, a compressible Bragg-Hawthorne framework is developed in the form of a density-stream function formulation that may be applied to a wide class of steady, axisymmetric flow problems. In this sequel, the procedure is implemented with the aim of retrieving an approximate compressible solution to describe the helical motion observed in a cyclonic, bidirectional vortex chamber. Our approach centers on sequentially solving the linearized density-stream function relations to the extent of producing a closed-form expression for the compressible, swirl-dominated motion. This e ort begins by conceiving a judicious set of boundary conditions that may be paired with the Bragg-Hawthorne procedure established previously. Then using a Rayleigh-Janzen expansion of the resulting system of partial di erential equations, a solution is achieved at the leading and first orders in the injection Mach number squared. At the leading order, the incompressible approximation of the linear Beltramian flowfield is recovered, with compressibility e ects relegated to the first order. In moving to higher orders, our approach requires the numerical integration of several groups of Bessel functions. These are specified individually as special functions that enable us to retain the analytical character of the first-order correction. Among significant findings predicted by this model, an appreciable steepening of the axial velocity profile is captured, thus mirroring a similar mechanism observed in solid rocket motors (SRMs). Furthermore, the mantle location, which is ordinarily committed to a single radial location in the absence of compressibility, gains an axial dependence that is reminiscent of the radial shifting of mantles reported in some experimental trials and numerical simulations. This sensitivity becomes more pronounced at higher injection Mach numbers and ratios of specific heats. The sensitivity of the solution to variations in , the inflow swirl parameter, is also investigated. We find that increasing leads to relative growth in both the incompressible and compressible velocities in the axial and radial directions. Conversely, at small values of , i.e. when the axial and radial velocities are overwhelmingly dominated by the tangential motion, the compressible solution approaches the leading-order result. Albeit counter-intuitive at first, imparting a progressively larger swirl component stands to promote the axisymmetric distribution of flowfield properties, and these include an implicit resistance to compression in the tangential direction, lest axisymmetry is violated. As for the density, its largest excursions occur near the centerline, and these become more appreciable at higher Mach numbers and ratios of specific heats. From a broader perspective, this study not only provides a viable approximation to the linear Beltramian motion associated with the classic cyclonic flowfield, it also o ers a proof-of-concept of the procedure introduced by the authors in their companion article. Specifically, the present analysis confirms the validity of the newly established compressible Bragg-Hawthorne framework in the treatment of swirl-driven and other axisymmetric fluid motions.

Explicit Inversion of Stodola’s Area-Mach Number Equation

Stodola’s area-Mach number relation is one of the most widely used expressions in compressible flow analysis. From academe to aeropropulsion, it has found utility in the design and performance characterization of numerous propulsion systems; these include rockets, gas turbines, microcombustors, and microthrusters. In this study, we derive a closed-form approximation for the inverted and more commonly used solution relating performance directly to the nozzle area ratio. The inverted expression provides a computationally efficient alternative to solutions based on traditional lookup tables or root finding. Here, both subsonic and supersonic Mach numbers are obtained explicitly as a function of the area ratio and the ratio of specific heats. The corresponding recursive formulations enable us to specify the desired solution to any level of precision. In closing, a dual verification is achieved using a computational fluid dynamics simulation of a typical nozzle and through Bosley’s formal approach. The latter is intended to confirm the truncation error entailed in our approximations. In this process, both asymptotic and numerical solutions are compared for the Mach number and temperature distributions throughout the nozzle.

On the Compressible Bidirectional Vortex

The purpose of this paper is to develop a theoretical solution that describes the compressible bidirectional vortex. Similar studies by the authors have extended the Taylor and Culick profiles to incorporate the effects of compressibility in porous channels and tubes. Our study is prompted by the need to better understand the flow behavior at high speed in swirl-driven thrust chambers in which a reversing cyclonic motion is established. Such chambers have the advantage of promoting mixing, efficiency, and internal wall cooling. This is accomplished by confining combustion to an inner vortex tube that remains separated from the chamber walls by an outer stream of swirling, low temperature oxidizer. Our closed-form analytical solution is based on steady, rotational, axisymmetric, compressible, and inviscid flow conditions. It is constructed using a Rayleigh-Janzen expansion in the injection Mach number. At the outset, the compressible axial and radial velocities are captured along with the mantle movement at various Mach numbers and vortex Reynolds numbers. In view of the underlying assumption of axisymmetry, all properties are held constant about the chamber axis. We find that, so long as this condition is maintained, the swirl velocity remains invariant in the tangential direction.

Evaluation of CFD Codes for Hypersonic Flow Modeling

The intent of this study is to evaluate existing gaps in computational fluid dynamics (CFD) technology with an emphasis on hypersonic flow modeling. The importance of validation is highlighted, especially with respect to the dearth of available hypersonic flow data. Modeling challenges relevant to hypersonic air-breathing propulsion, such as multi-phase flow and turbulent mixing, are discussed and areas for continued research are identified. Two specialized codes (DPLR and VULCAN) and two general purpose solvers (CFD++ and FLUENT) are also reviewed for use in hypersonic applications.

Microwave system and methods for combined heating and radiometric sensing for blood perfusion measurement of tissue

A combination of microwave heating and radiometry provides a promising means of the noninvasive measurement of blood perfusion. In this paper, a microwave system, along with a perfusion mimicking setup, for perfusion/flow measurement is described. This paper also presents for the first time i) considerations for achieving high sensitivity of temperature response versus flow, ii) radiometric measurement of temperature decay at the presence of flow, iii) a proper choice of the perfusion phantom to mimic the tissue permittivity, and iv) insights on the proper choice of heating and radiometry frequencies from the near-field antenna beam and bioheat transfer considerations.

Inversion of the Fundamental Thermodynamic Equations for Isentropic Nozzle Flow Analysis

The isentropic flow equations relating the thermodynamic pressures, temperatures, and densities to their stagnation properties are solved in terms of the area expansion and specific heat ratios. These fundamental thermofluid relations are inverted asymptotically and presented to arbitrary order. Both subsonic and supersonic branches of the possible solutions are systematically identified and exacted. Furthermore, for each branch of solutions, two types of recursive approximations are provided: a property-specific formulation and a more general, universal representation that encompasses all three properties under consideration. In the case of the subsonic branch, the asymptotic series expansion is shown to be recoverable from Bürmann’s theorem of classical analysis. Bosley’s technique is then applied to verify the theoretical truncation order in each approximation. The final expressions enable us to estimate the pressure, temperature, and density for arbitrary area expansion and specific heat ratios with no intermediate Mach number calculation or iteration. The analytical framework is described in sufficient detail to facilitate its portability to other nonlinear and highly transcendental relations where closed-form solutions may be desirable. [DOI: 10.1115/1.4003963]