Inviscid Core Research Articles

Hydrodynamic structure of the boundary layers in a rotating cylindrical cavity with radial inflow

A flow model is formulated to investigate the hydrodynamic structure of the boundary layers of incompressible fluid in a rotating cylindrical cavity with steady radial inflow. The model considers mass and momentum transfer coupled between boundary layers and an inviscid core region. Dimensionless equations of motion are solved using integral methods and a space-marching technique. As the fluid moves radially inward, entraining boundary layers develop which can either meet or become non-entraining. Pressure and wall shear stress distributions, as well as velocity profiles predicted by the model, are compared to numerical simulations using the software OpenFOAM. Hydrodynamic structure of the boundary layers is governed by a Reynolds number, Re, a Rossby number, Ro, and the dimensionless radial velocity component at the periphery of the cavity, Uo. Results show that boundary layers merge for Re < < 10 and Ro > > 0.1, and boundary layers become predominantly non-entraining for low Ro, low Re, and high Uo. Results may contribute to improve the design of technology, such as heat exchange devices, and turbomachinery.

Characterization of CO2 flow in a hypersonic impulse facility using DLAS

This work documents diode laser absorption measurements of CO2 flow in the free stream of the Longshot hypersonic impulse facility at Mach numbers ranging from 10 to 12. The diode laser sensor was designed to measure absorption of the P12 (30013) \(\leftarrow\) (00001) transition near 1.6 \(\upmu\)m, which yields relatively weak direct absorption levels (3.5 % per meter at peak Longshot free-stream conditions). Despite this weak absorption, measurements yielded valuable flow property information during the first 20 ms of facility runs. Simultaneous measurements of static temperature, pressure, and velocity were acquired in the inviscid core flow region using a laser wavelength scanning frequency of 600 Hz. The free-stream values obtained from DLAS measurements were then compared to Longshot probe-derived values determined from settling chamber and probe measurements. This comparison enabled an assessment of the traditional method of flow characterization in the facility, which indicated negligible influence from possible vibrational freezing of reservoir gases.

Theoretical Model of Buoyancy-Induced Flow in Rotating Cavities

The Ekman-layer equations, which have previously been solved for isothermal source–sink flow in a rotating cavity, are derived for buoyancy-induced flow. Although the flow in the inviscid core is three-dimensional and unsteady, it is assumed that the flow in the Ekman layers is axisymmetric and steady; and, as for source–sink flow, the average mass flow rate in the Ekman layers is assumed to be invariant with radius. In addition, it is assumed that the flow in the core is adiabatic, and consequently the core temperature increases with radius and with rotational speed. Approximate solutions are obtained for laminar flow, and it is shown that the Nusselt numbers for the rotating disks and the mass flow rate in the Ekman layers are proportional to Grc1/4, where Grc is a Grashof number based on the rotational Reynolds number and the temperature difference between the disk and the core. The equation for the Nusselt numbers, which includes two empirical constants, depends strongly on the radial distribution of the temperature of the disks.

On the effect of surface roughness in transonic high Reynolds number flow through slender channels

AbstractIn the present study transonic weakly three‐dimensional flows at high Reynolds numbers in slender channels, as they are found e.g. in micro nozzles and turbo‐machines of micro‐electromechanical‐ systems (MEMS), are considered. The channel height is taken to be so small that the viscous layers forming at the channel walls start to interact strongly rather than weakly with the predominantly inviscid core flow and, therefore, the classical boundary layer approach fails. The resulting viscous‐inviscid interaction problem is formulated using matched asymptotic expansions and found to be governed by a triple deck structure. As a consequence, the properties of the inviscid core region and the viscous wall layers have to be calculated simultaneously in the interaction region. Weakly three‐dimensional effects caused by surface roughness, upstream propagating flow perturbations and boundary layer separation are discussed. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Approximate modeling of the flow structure in a λ-shaped pseudoshock

An approximate analytical model of the flow structure in a plane λ-shaped pseudoshock consisting of a viscous boundary layer and an inviscid core flow is proposed. It is assumed that the boundary layer edge is a streamline with a specified pressure distribution along the channel and that the flow in the pseudoshock consists of an input segment (Mach reflection of an oblique shock wave) and a sequence of internal segments with an identical structure (shock train). Comparisons with experimental data and results of numerical calculations are performed. It is shown that the model provides a sufficiently accurate description of the pseudoshock flow structure.

Triple-deck analysis of transonic high Reynolds number flow through slender channels.

In this work, laminar transonic weakly three-dimensional flows at high Reynolds numbers in slender channels, as found in microsupersonic nozzles and turbomachines of micro-electro-mechanical systems, are considered. The channel height is taken so small that the viscous wall layers forming at the channel walls start to interact strongly rather than weakly with the inviscid core flow and, therefore, the classical boundary layer approach fails. The resulting viscous-inviscid interaction problem is formulated using matched asymptotic expansions and found to be governed by a triple-deck structure. As a consequence, the properties of the predominantly inviscid core region and the viscous wall layers have to be calculated simultaneously in the interaction region. Weakly three-dimensional effects caused by surface roughness, upstream propagating flow perturbations, boundary layer separation as well as bifurcating solutions are discussed. Representative results for subsonic as well as supersonic conditions are presented, and the importance of these flow phenomena in technical applications as, for example, a means to reduce shock losses through the use of deformed geometry is addressed.

On the steady flow in a rectangular cavity at large Reynolds numbers: A numerical and analytical study

Steady flow in a rectangular cavity at high Reynolds numbers is numerically and analytically investigated. Numerical simulations are reported up to a maximum Reynolds number, Re, value of 15000 for deep cavities and 20000 for shallow cavities using a compact fourth-order accurate central difference scheme and a stream function–vorticity formulation. At high Reynolds numbers, the eddy structure in shallow cavities consists of counter-rotating primary eddies, with each eddy behaving as an inviscid core with uniform vorticity. For deep cavities, the increase in Reynolds number results in the growth and eventually merger of the corner eddies into new primary eddies. Two merger patterns are identified, a symmetric pattern and an asymmetric pattern depending on a local Reynolds number based on the properties of the bottom primary eddy. A cavity with effectively infinite depth, D=10, is also numerically investigated up to a maximum Re value of 10000. Numerical results indicate that for an infinitely deep cavity and at a large Reynolds number, inertia effects would dominate near the upper moving wall, while Stokes flow behavior would dominate away from the moving wall. An overlap region would exist, in which both inertia and viscous effects are of comparable magnitude. Finally, an analytical solution is developed for the steady flow in a rectangular cavity at large Reynolds numbers. Results from the analytical model are compared to numerical solutions obtained from the full Navier–Stokes equations for both one-sided and four-sided driven cavity configurations.

Stability of high-Reynolds-number flow in a collapsible channel

AbstractWe study high-Reynolds-number flow in a two-dimensional collapsible channel in the asymptotic limit of wall deformations confined to the viscous boundary layer. The system is modelled using interactive boundary-layer equations for a Newtonian incompressible fluid coupled to the freely moving elastic wall under constant tension and external pressure. The deformation of the membrane is assumed to have small amplitude and long wavelength, whereas the flow comprises the inviscid core and the viscous boundary layers on both walls coupled to each other and to the membrane deformation. Firstly, by linking the interactive boundary-layer model to the small-amplitude, long-wavelength inviscid analysis, we conclude that the model is valid only when the pressure perturbations are fixed downstream from the wall indentation, contrary to the common assumption of classical boundary-layer theory. Next we explore possible steady states of the system, showing that a unique steady solution exists when the pressure is fixed precisely at the downstream end of the membrane, but there are multiple states possible if the pressure is specified further downstream. We examine the stability of these states by solving the generalized eigenvalue problem for perturbations to the nonlinear steady solutions and also by performing time integration of the full boundary-layer equations. Surprisingly, we find that no self-excited oscillations develop in the collapsible channel systems with finite-amplitude deformations. Instead, for each point in the parameter space, with the exception of points subject to numerical instabilities associated with the boundary-layer equations, exactly one of the steady states is predicted to be stable. We discuss these findings in relation to the results reported previously in the literature.

Three-dimensional gas flow in a rotating cylinder with a retarding cover

The study of rotating flows is of interest due to both the development of the centrifugal method of separation of gas and isotope mixtures and the possibility of astrophysical applications. An analytical nonlinear model for calculating the hydrodynamic characteristics of the viscous incompressible fluid flow in a rotating cylinder in the presence of a retarding cover is presented. The cases of stationary and rotating covers are considered. The analysis is performed on the basis of the system of hydrodynamic Navier-Stokes equations. The flow domain is divided up into the main flow and end boundary layers at the cylinder bottom and at the rotating cover. In its turn, the main flow is divided up into an inviscid quasi-rigid core and a lateral layer within which almost the entire upward circulatory flow is concentrated. The equations of the boundary layers at the end surfaces are analyzed by the approximate Slezkin-Targ method. The solutions in the boundary and lateral layers are “stitched” together with the velocity distribution in the main flow core. The unknown angular velocity ω1 and radial boundary R1 of the core are determined from the balance of the moments of the friction forces acting on the main rotating flow and the continuity condition for the circulatory flow. The experimental and calculated data are compared.

Theory of isobaric pressure exchanger for desalination

Abstract A theory is developed to predict the time of sustained operation of a rotary pressure exchanger used for energy recovery in seawater reverse osmosis system. Based on past experiments for oscillating pipe flows, it is found that the existing plug flow velocity in the ducts is not high enough to induce turbulence in the wall boundary layer. Modeling the time series of the flow velocity in the inviscid core as a periodic series of rectangular pulses, the structure of the laminar momentum boundary layer is first derived. The mass boundary layer induced by the oscillating velocity is then solved in order to obtain the slow diffusion of the averaged brine concentration along the duct. With the result the effective longitudinal diffusivity (dispersivity) is found explicitly for arbitrary Schmidt number. The dispersivity is found to be small due to the small viscosity and mass diffusivity in the very thin boundary layers, however it is still augmented to hundreds times of the molecular diffusivity. For a range of duct and rotor dimensions and rotor frequencies, the time needed for the mixing zone to spread to the ends of the duct is predicted for large Schmidt number appropriate for salt in water. After transient mixing is over, a certain amount of salt leaks steadily into the fresh seawater reentering the membrane. However the leakage is shown to be small due to the small dispersivity.

Shear Layer Time-Delay Correction Using a Non-Intrusive Acoustic Point Source

Microphone array processing algorithms often assume straight-line source-to-observer wave propagation. However, when the microphone array is placed outside an open-jet test section, the presence of the shear layer refracts the acoustic waves and causes the wave propagation times to vary from a free-space model. With a known source location in space, the propagation time delay can be determined using Amiet's theoretical method. In this study, the effects of shear layer refraction are examined using a pulsed laser system to generate a plasma point source in space and time for several different test section flow speeds and configurations. An array of microphones is used to measure the pulse signal, allowing for the use of qualitative beamforming and quantitative timing analysis. Results indicate that Amiet's method properly accounts for planar shear layer refraction time delays within experimental uncertainty. This is true both when the source is in the inviscid core of the open-jet test section, as well as when the source is located in different model wakes of varying complexity. However, the method breaks down where the thin layer assumption fails, such as in the region where the tunnel test section's open jet interacts with the facility jet collector.

A theoretical model of the pressure field arising from asymmetric intraglottal flows applied to a two-mass model of the vocal folds

A theoretical flow solution is presented for predicting the pressure distribution along the vocal fold walls arising from asymmetric flow that forms during the closing phases of speech. The resultant wall jet was analyzed using boundary layer methods in a non-inertial reference frame attached to the moving wall. A solution for the near-wall velocity profiles on the flow wall was developed based on a Falkner-Skan similarity solution and it was demonstrated that the pressure distribution along the flow wall is imposed by the velocity in the inviscid core of the wall jet. The method was validated with experimental velocity data from 7.5 times life-size vocal fold models, acquired for varying flow rates and glottal divergence angles. The solution for the asymmetric pressures was incorporated into a widely used two-mass model of vocal fold oscillation with a coupled acoustical model of sound propagation. Asymmetric pressure loading was found to facilitate glottal closure, which yielded only slightly higher values of maximum flow declination rate and radiated sound, and a small decrease in the slope of the spectral tilt. While the impact on symmetrically tensioned vocal folds was small, results indicate the effect becomes more significant for asymmetrically tensioned vocal folds.

Viscous–inviscid interactions in transonic flows through slender nozzles

Considering the miniaturization trend in technical applications, the need of a slender nozzle theory for such conventional, that is ideal-gas-like, fluids, which accounts for a strong boundary-layer interaction with the core region, arises in quite a natural way as the dimensions of the flow device are successively reduced. Moreover, a number of modern technological processes (e.g. organic Rankine cycles) involve fluids with high molecular complexity, some of which are expected to exhibit embedded regions with negative values of the fundamental derivative in the vapour phase commonly termed Bethe–Zel'dovich–Thompson (BZT) fluids. Linked to it, unconventional Laval nozzle geometries are needed to transform subsonic to supersonic internal flows. In the present paper, the transonic flows through nozzles of short length scales located in a channel of constant cross-section so slender that the flow in the inviscid core region is one-dimensional are considered. Rapid streamwise changes of the flow field caused by the nozzle then lead to a local breakdown of the classical hierarchical boundary-layer approach, which is overcome by the triple-deck concept. Consequently, the properties of the inviscid core and the near-wall (laminar) boundary-layer regions have to be calculated simultaneously. The resulting problem is formulated for both regular (ideal-gas-like) fluids and dense gases. Differences and similarities of the resulting flow pattern with respect to the well-known classical Laval nozzle flow are presented for perfect gases, and the regularizing influence of viscous–inviscid interactions, is examined. Furthermore, the analogous problem is considered for BZT fluids in detail as well. The results indicate that the passage through the sonic point in the inviscid core is strongly affected by the combined influence of nozzle geometry and boundary-layer displacement effects suggesting in turn an inverse Laval nozzle design in order to obtain the desired flow behaviour.

Transonic viscous inviscid interactions in a slender Laval nozzle

AbstractTransonic, high Reynolds number flows through a Laval nozzle, which is so narrow that the classical boundary layer correction can no longer be considered to be an effect of higher order, are considered. As a consequence the properties of the inviscid core and the viscosity dominated boundary layer region can no longer be determined in subsequent steps but have to be calculated simultaneously. The resulting interaction problem for laminar flows in a small nozzle is presented for perfect gases. Representative solutions including the internal structure of pseudo‐shocks forming in the diffuser part of the nozzle and being strongly associated with the chocking phenomenon will be presented. The linear stability of the various flow regimes observed in the nozzle will be discussed. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Lagrangian transport induced by peristaltic pumping in a tube

This is an analytical study, based on equations of motion in Lagrangian form, for the steady Lagrangian fluid transport induced by long peristaltic waves (which can be progressive, purely or partially standing) of small amplitude traveling on the boundary of a flexible tube. The first-order oscillatory viscous flow and the higher-order time-mean Lagrangian drifts (or steady streaming) are obtained as functions of the wave properties. Two cases are considered. Firstly, the wave frequency is slow such that the steady-streaming Reynolds number (Res) is very small and the viscous diffusion is significant across the entire flow region. The time-mean flow can be found in the second-order problem. Secondly, high-frequency pumping is considered such that Res = O(1). Under this condition, the flow domain is divided into a thin Stokes boundary layer near the wall and the inviscid core region. The steady streaming in the core region is to be found in the fourth-order problem. Based on the Lagrangian coordinates, all the solutions are analytically expressed. Results are generated to illustrate the effects of wave properties on the Lagrangian transport. The phenomenon of reflux, a backward time-mean flow, is examined in particular.

Shock regularization in dense gases by viscous–inviscid interactions

Transonic high-Reynolds-number flows through channels which are so narrow that the classical boundary-layer approach fails locally are considered in the presence of a weak stationary normal shock. As a consequence, the properties of the inviscid core and the viscosity-dominated boundary-layer region can no longer be determined in subsequent steps but have to be calculated simultaneously in a small interaction region. Under the requirement that the core-region flow should be considered to be one-dimensional to the leading order the resulting problem of shock–boundary-layer interaction is formulated by the means of matched asymptotic expansions for laminar flows of dense gases (Bethe–Zel'dovich–Thompson, or BZT, fluids). Such fluids have the distinguishing feature that the fundamental derivative of gas dynamics can become negative or even change sign under the thermodynamic conditions to be considered. The regularizing properties of the mechanism of viscous–inviscid interactions on the different anomalous shock forms possible in the flow of dense gases with mixed nonlinearity, namely rarefaction, sonic, double-sonic and split shocks, will be discussed. To this end we show the consistency of the resulting internal-shock profiles because of strong shock–boundary-layer interaction with a generalized shock admissibility criterion formulated for the case of purely inviscid flows. Representative solutions for the internal-shock structures are presented, and the importance of such flow phenomena in technical applications in the near future are shortly discussed by considering estimates of the actual dimensions of the interaction region for a specific representative situation in which the BZT fluid PP10 (C13F22) has been selected.

The compressional and compositional stratifications in Maxwell earth models: the gravitational overturning and the long-period tangential flux

SUMMARY We analyse self-gravitating compressible Maxwell earth models with compressional and compositional stratifications by taking into account the continuous variations of the initial density and of the bulk modulus. We derive the analytical solution in the Laplace domain of an earth model composed of a homogeneous inviscid core and a viscoelastic mantle characterized by a specific Darwin-law density profile. This makes us discover a new class of relaxation modes, the compositional C-modes, of which the previously discussed RT-modes are a subclass. Compositional stratifications present a denumerable set of C-modes situated on the positive and negative real axis depending on the sign of the square of the Brunt–Vaisala frequency ω and converging to the origin of the Laplace domain, while compressional stratifications do not present such modes. If ω2 0 the C-modes are stable, but they affect the tangential perturbation in such a way that the tangential fluid limit diverge. We interpret this perturbation as a long-period tangential flux of material. By analysing the tangential velocities, we show that this flux fades in the fluid limit controlled by the viscosity. Our findings shed new light on the isostatic criterion, which we reformulate in a new way. Successively, by means of a numerical algorithm, we consider earth models including also the continuous variations of the bulk modulus. In this case, the differential system describing the conservation of the momentum and the self-gravitation is not uniformly Lipschitzian in the region of the Laplace domain associated with the inverse compressional relaxations times. This implies a continuous spectrum in the transient region that the normal mode approach cannot deal with. Nevertheless, the buoyancy mode regions is not characterized by a continuous spectrum and the C-modes are still present for the compositional stratifications.

Gain and output power measurements in an electrically excited oxygen–iodine laser with a scaled discharge

Singlet delta oxygen (SDO) yield, small signal gain, and output power have been measured in a scaled electric discharge excited oxygen–iodine laser. Two different types of discharges have been used for SDO generation in O2–He–NO flows at pressures up to 90 Torr, crossed nanosecond pulser/dc sustainer discharge and capacitively coupled transverse RF discharge. The total flow rate through the laser cavity with a 10 cm gain path is approximately 0.5 mole s−1, with steady-state run time at a near-design Mach number of M = 2.9 of up to 5 s. The results demonstrate that SDO yields and flow temperatures obtained using the pulser-sustainer and the RF discharges are close. Gain and static temperature in the supersonic cavity remain nearly constant, γ = 0.10–0.12% cm−1 and T = 125–140 K, over the axial distance of approximately 10 cm. The highest gain measured is 0.122% cm−1 at T = 140 K. Positive gain measured in the supersonic inviscid core extends over approximately one half to one third of the cavity height, with absorption measured in the boundary layers near top and bottom walls of the cavity. Laser power has been measured using (i) two 99.9% mirrors on both sides of the resonator, 2.5 W, and (ii) 99.9% mirror on one side and 99% mirror on the other side, 3.1 W. Gain downstream of the resonator is moderately reduced during lasing (by up to 20–30%) and remains nearly independent of the axial distance, by up to 10 cm. This suggests that only a small fraction of power available for lasing is coupled out, and that additional power may be coupled in a second resonator. Preliminary laser power measurements using two transverse resonators operating at the same time (both using 99.9–99% mirror combinations) demonstrated lasing at both axial locations, with the total power of 3.8 W.

Local Interaction Theory for Laminar Transonic Flows in Slender Channels

AbstractTransonic flows through channels so narrow that the classical boundary layer approach fails are considered. As a consequence the properties of the inviscid core and the viscosity dominated boundary layer region can no longer be determined in subsequent steps but have to be calculated simultaneously. The resulting viscous inviscid interaction problem for weakly three dimensional laminar flows is formulated for perfect gases under the requirement that the channel is sufficiently narrow so that the flow outside the viscous wall layers becomes planar in the leading order approximation. Representative solutions for subsonic as well as for supersonic flows disturbed by three dimensional surface mounted obstacles will be presented. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Multiple inner core wobbles in a simple Earth model with inviscid core

The inner core wobble (ICW) is the chandler wobble of the inner core. Its predicted period for the PREM model is about 7.5 years, based upon the resolution of the Liouville equations of conservation of angular momentum. Here, solving the local equation of conservation of linear momentum with a truncated chain that couples the toroidal and spheroidal displacement fields, the ICW is computed for a model made up of three homogeneous layers: an incompressible liquid outer core and rigid mantle and inner core. Contrary to the angular momentum approach, as implemented up to now, that provides a single ICW, the linear momentum approach shows that the dynamics of the neutrally stratified outer core may generate a family of ICWs with periods ranging from a few dozens to thousands of days. The mode with the largest wobble amplitude in the inner core has a period close to that obtained with the angular momentum approach.