Homogeneous Flow Approximation Research Articles

Linear impulse response in three-dimensional oscillatory boundary layers formed by periodic modulation of a rotating disk

A numerical investigation is undertaken on the development of linear disturbances in the rotating disk boundary layer, in which a time-periodic modulation is applied to the disk rotation rate. The model gives a prototypical example of a three-dimensional oscillatory boundary layer, by adding a Stokes layer to the von Kármán flow that develops on a steady disk. The study extends the Floquet analysis of Morgan et al. (J. Fluid Mech., vol. 925, 2021, A20), who showed that disk modulation stabilises the stationary convective instabilities found on the steady rotating disk. Using a radial homogeneous flow approximation, whereby the radial dependence of the basic state is ignored, disturbance development is simulated for several modulation settings, with flow conditions matched to both convective and absolute forms of linear instability. Disturbances excited via a stationary periodic wall forcing display behaviour consistent with that found using Floquet theory; time-periodic modulation stabilises the cross-flow instability by reducing the radial growth rate. In addition, convective and absolute instabilities, generated by an impulsive wall forcing, are both stabilised by the introduction of modulation to the disk rotation rate. Modulation establishes a significant reduction in both the temporal growth rate and the disturbance amplitude as it propagates away from the impulse origin. Moreover, greater stabilising control benefits are realised as the modulation amplitude increases.

Effects of partial slip on the local-global linear stability of the infinite rotating disk boundary layer

A numerical investigation is undertaken on the effect of small-scale surface roughness on the local absolute and global stability of the flow due to a rotating disk. Surface roughness is modeled via the imposition of the partial-slip wall boundary condition, with radial and concentric anisotropic roughnesses and isotropic roughness considered. The effect of the partial-slip parameters on the neutral characteristics for absolute instability is presented, while the azimuthal mode numbers required for global linear instability to occur are determined for the genuine inhomogeneous base flow. Predictions for the threshold values for the azimuthal mode numbers needed for globally unstable behavior are also computed by coupling solutions of the Ginzburg–Landau equation with the local linear stability properties obtained using the homogeneous flow approximation. These are found to be in excellent agreement with the exact values realized from the numerical simulations. In general, surface roughness is demonstrated to stabilize the absolute instability and the global linear instabilities.

Global linear instability of rotating-cone boundary layers in a quiescent medium

The global linear stability of the family of infinite rotating-cone boundary layers in an otherwise still fluid is investigated using a velocity-vorticity form of the linearized Navier-Stokes equations. The formulation is separable with respect to the azimuthal direction. Thus, disturbance development is simulated for a single azimuthal mode number. Numerical simulations are conducted for an extensive range of cone half-angles (ψ∈[20∘:80∘]) and stability parameters (Reynolds number, azimuthal mode number), where conditions are taken to be near those specifications necessary for the onset of absolute instability. A localized impulsive wall forcing is implemented that excites disturbances that form wave packets. This allows the disturbance evolution to be traced in the spatial-temporal plane. When a homogeneous flow approximation is utilised that neglects the spatial variation of the basic state, linear perturbations display characteristics consistent with local stability theory. For disturbances to the genuine spatially dependent inhomogeneous flow, global linear instability characterized by a faster than exponential temporal growth arises for azimuthal mode numbers greater than the conditions for critical absolute instability. Furthermore, a reasonable prediction for the azimuthal mode number needed to bring about a change in global behavior is achieved by coupling solutions of the Ginzburg-Landau equation with local stability properties. Thus, the local-global stability behavior is qualitatively similar to that found in the infinite rotating-disk boundary layer and many other globally unstable flows.

Global stability behaviour for the BEK family of rotating boundary layers

Numerical simulations were conducted to investigate the linear global stability behaviour of the Bödewadt, Ekman, von Kármán (BEK) family of flows, for cases where a disc rotates beneath an incompressible fluid that is also rotating. This extends the work reported in recent studies that only considered the rotating-disc boundary layer with a von Kármán configuration, where the fluid that lies above the boundary layer remains stationary. When a homogeneous flow approximation is made, neglecting the radial variation of the basic state, it can be shown that linearised disturbances are susceptible to absolute instability. We shall demonstrate that, despite this prediction of absolute instability, the disturbance development exhibits globally stable behaviour in the BEK boundary layers with a genuine radial inhomogeneity. For configurations where the disc rotation rate is greater than that of the overlying fluid, disturbances propagate radially outwards and there is only a convective form of instability. This replicates the behaviour that had previously been documented when the fluid did not rotate beyond the boundary layer. However, if the fluid rotation rate is taken to exceed that of the disc, then the propagation direction reverses and disturbances grow while convecting radially inwards. Eventually, as they approach regions of smaller radii, where stability is predicted according to the homogeneous flow approximation, the growth rates reduce until decay takes over. Given sufficient time, such disturbances can begin to diminish at every radial location, even those which are positioned outwards from the radius associated with the onset of absolute instability. This leads to the confinement of the disturbance development within a finitely bounded region of the spatial–temporal plane.

Effects of gas density on the structure of liquid jets in still gases

A theoretical and experimental study of the near jet-exit region of nonevaporatin g round liquid jets in still gases is described, emphasizing effects of ambient gas density in the atomization breakup regime where liquid breakup begins right at the jet exit. Mean liquid volume fraction distributions were measured for 9.5-mm-diam water jets in still air at pressures of 1-8 atm. Mixing was strongly affected by the gas/liquid density ratio and the degree of flow development at the jet exit, with the largest gas/liquid density ratio and fully developed turbulent pipe flow yielding the fastest mixing rates. Flow properties were predicted using the locally homogeneous flow approximation, where relative velocities between the phases are assumed to be small in comparison to mean flow velocities. Predictions were in good agreement with measurements, including representation of effects of gas/liquid density ratio and flow development at the jet exit, but only at relatively high mixture fractions. In contrast, separated flow effects caused predictions to overestimate rates of flow development at low mixture fractions.

Temperature and Radiation of Diffusion Flames with Suppression

Abstract Temperature and radiation properties of 1-8 MW heat release rate methane/air jet diffusion flames without and with the addition of liquid water suppressant to the fuel stream are studied. The analysis includes an existing parabolic flow solver, in conjunction with the locally homogeneous flow approximation and the laminar flamelet concept. State relationships for methane + liquid water mixtures are calculated from species concentration measurements for methane flames without water added. The analyses are compared with previously published data. Tests for flames without water addition show the applicability of the analyses to the present flame scales. The results for flames with water addition indicate that finite rates of evaporation and separated flow effects need to be considered for accurate predictions at high water loading.

Structure of the near-injector region of nonevaporating pressure-atomized sprays

The dense-spray region of pressure-atom ized nonevaporating sprays was studied, emphasizing the properties of the multiphase mixing layer that surrounds the liquid core during atomization breakup. The dispersed-phase properties of a large-scale (9.5-mm injector diameter) water jet injected vertically downward in still air were measured using single- and double-pulse holography for both fully developed and slug flow jet exit conditions. The inner portion of the mixing layer contained large irregularly shaped liquid elements and drops, and the proportion of spherical drops increased and drop sizes decreased with increasing radial distance. For present test conditions, the liquid core and the large liquid elements cause mean liquid volume fractions to be high near the axis; however, the gas-containing region was relatively dilute at each instant. Additionally, the velocities of large drops were generally much larger than small drops and predictions based on the locally homogeneous flow approximation, providing direct evidence of significant separated-flow effects in the flow. Finally, the degree of flow development at the jet exit had a substantial effect on the structure of the mixing layer, with increased turbulence levels increasing the number and size of large irregular liquid elements through distortion of the surface of the liquid core—enhancing rates of removal of liquid from the core.

Structure and mixing properties of pressure-atomized sprays

A theoretical and experimental study of the dense-spray region of pressure-atomized nonevaporating sprays is described, emphasizing flows in the wind-induced and atomization breakup regimes. Mean and fluctuating velocities at the injector exit, mean liquid volume fraction distributions, and entrainment rates were measured for large-scale (9.5 and 19.1 mm injector diameters) water jets in still air at atmospheric pressure. It was found that mixing was strongly influenced by the degree of flow development at the injector exit and the breakup regime: fully developed injector flow and atomization breakup yielded the fastest mixing rates. Predictions based on the locally homogeneous flow approximation, where relative velocities between the phases were neglected, gave encouraging predictions of dense-spray properties in the near-injector region for atomization breakup, including representation of flow development effects at the injector exit.

Structure of Turbulent Reacting Gas Jets Submerged in Liquid Metals

Abstract A model for predicting the turbulent, multiphase, diffusion-flame structure of a gaseous oxidizer jet submerged in a molten-metal fuel is described. Axisymmetric flow was considered using the locally homogeneous flow approximation of multiphase flow in conjunction with a well-calibrated k-e-g model of turbulence properties. Effects of buoyancy were considered in the governing equations for mean properties. The model was evaluated using existing data for vertical chlorine jets submerged in dilute-Na/NaCl molten baths at atmospheric pressure. Combustion products for these reactants are condensible; therefore, the portion of the flow containing gas has a finite penetration length—somewhat analogous to flame lengths in gaseous single-phase diffusion flames. Predicted and measured penetration lengths and mean temperatures along the jet axis were in fair agreement using conventional turbulence model constants established for single-phase flows. The predictions indicate a complex flow structure containing several solid and liquid phases due to the large temperature range of the process and the miscibility properties of the reaction products Computations for injection of chlorine into pure sodium showed that while reaction is completed near the injector, the gas phase persists for a considerable penetration length since vapors formed in high temperature portions of the flow must recondense.

Condensation of Submerged Vapor Jets in Subcooled Liquids

A theoretical investigation of turbulent condensing vapor jets submerged in subcooled liquids was conducted. A model of the process was developed employing the locally homogeneous flow approximation of two-phase flow in conjunction with a k-ε-g model of turbulence properties. The model was evaluated using existing data for turbulent condensing water, ethylene glycol and iso-octane vapor jets with liquid to vapor density ratios in the range 324–31,200, initial jet Reynolds numbers greater than 15,000 and negligible effects of buoyancy. Good predictions of vapor penetration lengths were obtained using conventional turbulence model constants established for single phase flows.

Evaluation of a locally homogeneous flow model of spray combustion

A model of spray combustion which employs a second-order turbulence model was developed. The assumption of locally homogeneous flow is made, implying infinitely fast transport rates between the phase. Measurements to test the model were completed for a gaseous n-propane flame and an air atomized n-pentane spray flame, burning in stagnant air at atmospheric pressure. Profiles of mean velocity and temperature, as well as velocity fluctuations and Reynolds stress, were measured in the flames. The predictions for the gas flame were in excellent agreement with the measurements. The predictions for the spray were qualitatively correct, but effects of finite rate interphase transport were evident, resulting in a overstimation of the rate development of the flow. Predictions of spray penetration length at high pressures, including supercritical combustion conditions, were also completed for comparison with earlier measurements. Test conditions involved a pressure atomized n-pentane spray, burning in stagnant air at pressures of 3, 5, and 9 MPa. The comparison between predictions and measurements was fair. This is not a very sensitive test of the model, however, and further high pressure experimental and theoretical results are needed before a satisfactory assessment of the locally homogeneous flow approximation can be made.