Search NASA⌕ Search

SEARCH · Search NASA

Results for “Vortex structure”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 163 records · Page 9
All sourcesNASA NTRSDOE OSTI
NASA NTRS2002

Reduction of Base Drag on Launch Vehicles

Current reentry vehicle designs exhibit large amounts of base drag due to large base areas. These large base areas can arise from the integration of the propulsion system (X-33) or control surface placement (X-38). Large base drag limits the vehicle's cross-range capability and causes a large glide-slope angle. Fortunately, there appears to be a possible means of lowering the base drag on these vehicles. Based on early work on the subsonic aerodynamics of lifting bodies, it appears that the addition of small amounts of viscous fore-body drag can produce a significant reduction in base drag. Recent work suggests that this phenomenon also occurs in the transonic and supersonic flight regimes. This report summarizes a study designed to demonstrate the reduction of base drag through the addition of fore-body viscous drag. The present study has focused on the measurement of viscous fore-body drag and the demonstration of the relationship between fore-body viscous drag and base drag at Reynolds Numbers up to 2.5 x 10(exp 6). The results of the present work do not conclusively demonstrate that viscous fore-body drag reduces base drag. The apparent contradictory results of the present study are attributed to the different geometry used in the present study. However, the results suggest that the increased boundary layer thickness at separation caused by larger fore-body viscous drag somehow affects the vortex structure in the wake thereby reducing the base drag. More research is required to confirm this postulated mechanism.

Naughton, Jonathan W.↗
NASA NTRS2002

Pitched and Yawed Circular Jets in Cross-Flow

Results from an experimental investigation of flow field generated by pitched and yawed jets discharging from a flat plate into a cross-flow are presented. The circular jet was pitched at alpha = 20 degrees and 45 degrees and yawed between beta = 0 degrees and 90 degrees in increments of 15 degrees. The measurements were performed with two X-wires providing all three components of velocity and turbulence intensity. These data were obtained at downstream locations of x = 3, 5, 10 and 20, where the distance x normalized by the jet diameter, is measured from the center of the orifice. Data for all configurations were acquired at a momentum-flux ratio J = 8. Additionally, for selected angles and locations, surveys were conducted for J = 1.5, 4, and 20. As expected, the jet penetration is found to be higher at larger alpha. With increasing beta the jet spreads more. The rate of reduction of peak streamwise vorticity, with the downstream distance is significantly lessened at higher alpha but is found to be practically independent of alpha. Thus, at the farthest measurement station x = 20, omega(sub xmax) is about five times larger for beta = 0 degrees compared to the levels at beta = 0 degrees. Streamwise velocity within the jet-vortex structure is found to depend on the parameter J. At J = 1.5 and 4, 'wake-like' velocity profiles are observed. In comparison, a 'jet-like' overshoot is present at higher J.

Milanovic, Ivana M.↗
NASA NTRS2002

The Distribution of Ozone in the Early Stages of Polar Vortex Development

Previous analysis has shown that the distribution of O3 at high northern latitudes in the lower-to-middle stratosphere at the beginning of the winter season, 1999-2000 has a characteristic distribution, which is consistent between in situ and satellite measurements [Kawa et al., The Interaction Between Dynamics and Chemistry of Ozone in the Set-up Phase of the Northern Hemisphere Polar Vortex, submitted manuscript, 2001 ]. Initial O3 profiles in the vortex are similar to each other and are quite different from outside the vortex at the same latitude and also from a zonal mean climatology. In the vortex, O3 is nearly constant from 500 to above 800 K with a value at 3 ppmv +/- approx.10%. Values outside the vortex are up to a factor of 2 higher and increase significantly with potential temperature. The seasonal time series of POAM data shows that relatively low O3 mixing ratios, which characterize the vortex in late fall, are already present at high latitudes at the end of summer in September before the vortex circulation sets up. This suggests a possible feedback role between O3 chemistry and the formation of the vortex, which is dominated by the seasonal radiation balance. Here we show that these characteristic O3 distributions are consistent from year to year and between the hemispheres. We will attempt to determine whether variations in fall vortex O3 are related in any way to O3 abundances and vortex structure later during winter and into spring.

Kawa, S. R.↗
NASA NTRS2003

Comparing Model Ozone Loss during the SOLVE and SOLVE-2 Winters

Model simulations have been used to analyze the factors influencing ozone loss during the 1999-2000 and 2002-2003 js. For both winters, the evolution of the Arctic vortex from November to April has been simulated using a trajectory-based microphysical and photochemical model. Extensive PSC formation and strong ozone depletion are evident in both winters. However, the ozone loss begins earlier in the 2002-2003 winter, with significant ozone depletion by early January. Analysis of the model results shows that during December 2002 not only cold temperatures but also the vortex structure was critical, allowing PSC-processed air parcels to experience significant solar exposure. The resultant ozone loss can be differentiated from ozone loss that occurs in the springtime, in particular because of the continued exposure to PSCs. For example, chlorine reactivation by the PSCs causes ozone loss to be insensitive to denitrification. Therefore, diagnosing the extent of ozone loss early in the winter is critical In understanding the overall winter-long ozone depletion.

Drdla, K.↗
NASA NTRS2004

Physical Analysis and Scaling of a Jet and Vortex Actuator

Our previous studies have shown that the Jet and Vortex Actuator generates free-jet, wall-jet, and near- wall vortex flow fields. That is, the actuator can be operated in different modes by simply varying the driving frequency and/or amplitude. For this study, variations are made in the actuator plate and wide-slot widths and sine/asymmetrical actuator plate input forcing (drivers) to further study the actuator induced flow fields. Laser sheet flow visualization, particle- image velocimetry, and laser velocimetry are used to measure and characterize the actuator induced flow fields. Laser velocimetry measurements indicate that the vortex strength increases with the driver repetition rate for a fixed actuator geometry (wide slot and plate width). For a given driver repetition rate, the vortex strength increases as the plate width decreases provided the wide-slot to plate-width ratio is fixed. Using an asymmetric plate driver, a stronger vortex is generated for the same actuator geometry and a given driver repetition rate. The nondimensional scaling provides the approximate ranges for operating the actuator in the free jet, wall jet, or vortex flow regimes. Finally, phase-locked velocity measurements from particle image velocimetry indicate that the vortex structure is stationary, confirming previous computations. Both the computations and the particle image velocimetry measurements (expectantly) show unsteadiness near the wide-slot opening, which is indicative of mass ejection from the actuator.

Lachowicz, Jason T.↗
NASA NTRS2001

Instability Analysis of a Low-Density Gas Jet Injected into a High-Density Gas

The objective of this study was to determine the effects of buoyancy on the absolute instability of low-density gas jets injected into high-density gas mediums. Most of the existing analyses of low-density gas jets injected into a high-density ambient have been carried out neglecting effects of gravity. In order to investigate the influence of gravity on the near-injector development of the flow, a linear temporal stability analysis and a spatio-temporal stability analysis of a low-density round jet injected into a high-density ambient gas were performed. The flow was assumed to be isothermal and locally parallel; viscous and diffusive effects were ignored. The variables were represented as the sum of the mean value and a normal-mode small disturbance. An ordinary differential equation governing the amplitude of the pressure disturbance was derived. The velocity and density profiles in the shear layer, and the Froude number (signifying the effects of gravity) were the three important parameters in this equation. Together with the boundary conditions, an eigenvalue problem was formulated. Assuming that the velocity and density profiles in the shear layer to be represented by hyperbolic tangent functions, the eigenvalue problem was solved for various values of Froude number. The temporal growth rates and the phase velocity of the disturbances were obtained. It was found that the presence of variable density within the shear layer resulted in an increase in the temporal amplification rate of the disturbances and an increase in the range of unstable frequencies, accompanied by a reduction in the phase velocities of the disturbances. Also, the temporal growth rates of the disturbances were increased as the Froude number was reduced (i.e. gravitational effects increased), indicating the destabilizing role played by gravity. The spatio-temporal stability analysis was performed to determine the nature of the absolute instability of the jet. The roles of the density ratio, Froude number, Schmidt number, and the lateral shift between the density and velocity profiles on the jet s absolute instability were determined. Comparisons of the results with previous experimental studies show good agreement when the effects of these variables are combined together. Thus, the combination of these variables determines how absolutely unstable the jet will be. Experiments were carried out to observe the qualitative differences between a round low-density gas jet injected into a high-density gas (helium jet injected into air) and a round constant density jet (air jet injected into air). Flow visualizations and velocity measurements in the near-injector region of the helium jet show more mixing and spreading of the helium jet than the air jet. The vortex structures develop and contribute to the jet spreading causing the helium jet to oscillate.

Lawson, Anthony Layiwola↗
NASA NTRS2003

Mechanisms of Rotating Instability in Axial Compressors Investigated

Rotating instability is a phenomenon that occurs in the tip flow region of axial compressor stages during stable operation. It can be observed in highly staggered rotors with significant tip clearance and is strongest at high-load operating points where the characteristic levels off. In this condition, the single-stage fan under investigation radiates an audible, whistling tone, and wall pressure spectra in the vicinity of the rotor disk exhibit nonrotational components. The graph shows the spectrum of static pressure at a point on the endwall near the leading edge. A hump appears at roughly half of the blade passing frequency (BPF) and is characteristic of rotating instability. A computational model was developed at the NASA Glenn Research Center to investigate the mechanism behind this phenomenon. A three-dimensional steady Navier-Stokes code that has been successfully tested for a wide range of turbomachinery flows was modified to execute a time-accurate simulation of the full annulus of the compressor. At the inlet of the computational domain, the total pressure, total temperature, and two velocity components are specified. Since no unsteady measurements of static pressure or other flow variables were available downstream of the rotor, circumferentially averaged static pressure was specified on the shroud at the outlet of the computational domain. A three-dimensional view of the vortex from the numerical model is shown. Particle traces released near the leading edge tip have rolled up to illustrate the tip clearance vortex. Flow near the trailing edge is pushed forward by the axially reversed flow. It then interacts with the tip clearance flow and the incoming flow and results in the rotating instability vortex, the core of which is illustrated by total pressure shading on planes located successively downstream. The rotating instability vortex is formed periodically midway between the blades and moves toward the pressure side of the passage. The unsteady behavior of this vortex structure is the main mechanism of the rotating instability is shown. The numerical model can be used to detect any possible occurrence of rotating instability when the tip clearance increases during engine service.

Hah, Chunill↗
NASA NTRS2006

Aerodynamic Comparison of Hyper-Elliptic Cambered Span (HECS) Wings with Conventional Configurations

An experimental study was conducted to examine the aerodynamic and flow field characteristics of hyper-elliptic cambered span (HECS) wings and compare results with more conventional configurations used for induced drag reduction. Previous preliminary studies, indicating improved L/D characteristics when compared to an elliptical planform prompted this more detailed experimental investigation. Balance data were acquired on a series of swept and un-swept HECS wings, a baseline elliptic planform, two winglet designs and a raked tip configuration. Seven-hole probe wake surveys were also conducted downstream of a number of the configurations. Wind tunnel results indicated aerodynamic performance levels of all but one of the HECS wings exceeded that of the other configurations. The flow field data surveys indicate the HECS configurations displaced the tip vortex farther outboard of the wing than the Baseline configuration. Minimum drag was observed on the raked tip configuration and it was noted that the winglet wake lacked the cohesive vortex structure present in the wakes of the other configurations.

Lazos, Barry S.↗
NASA NTRS2007

Experimental Study of Transitional Flow Behavior in a Simulated Low Pressure Turbine

A detailed investigation of the flow physics occurring on the suction side of a simulated Low Pressure Turbine (LPT) blade was performed. A contoured upper wall was designed to simulate the pressure distribution of an actual LPT airfoil onto a flat lower plate. The experiments were carried out for the Reynolds numbers of 35,000, 70,000, 100,000, and 250,000 with four levels of freestream turbulence ranging from 1 to 4 percent. For the three lower Reynolds numbers, the boundary layer on the flat plate was separated and formed a bubble. The size of laminar separation bubble was measured to be inversely proportional to the freestream turbulence levels and Reynolds numbers. However, no separation was observed for the Re = 250,000 case. The transition on a separated flow was found to proceed through the formation of turbulent spots in the free shear layer as evidenced in the intermittency profiles for Re = 35,000, 70,000, and 100,000. Spectral data show no evidence of Kelvin-Helmholtz of Tollmien-Schlichting instability waves in the free shear layer over a separation bubble (bypass transition). However, the flow visualization revealed the large vortex structures just outside of the bubble and their development to turbulent flow for Re = 50,000, which is similar to that in the free shear layer (separated-flow transition). Therefore, it is fair to say that the bypass and separated-flow transition modes coexist in the transition flows over the separation bubble of certain conditions. Transition onset and end locations and length determined from intermittency profiles decreased as Reynolds number and freestream turbulence levels increase.

Sohn, Ki Hyeon↗
NASA NTRS2008

Computational Investigation of the NASA Cascade Cyclonic Separation Device

Devices designed to replace the absent buoyancy separation mechanism within a microgravity environment are of considerable interest to NASA as the functionality of many spacecraft systems are dependent on the proper sequestration of interpenetrating gas and liquid phases. Inasmuch, a full multifluid Euler-Euler computational fluid dynamics investigation has been undertaken to evaluate the performance characteristics of one such device, the Cascade Cyclonic Separator, across a full range of inlet volumetric quality with combined volumetric injection rates varying from 1 L/min to 20 L/min. These simulations have delimited the general modes of operation of this class of devices and have proven able to describe the complicated vortex structure and induced pressure gradients that arise. The computational work has furthermore been utilized to analyze design modifications that enhance the overall performance of these devices. The promising results indicate that proper CFD modeling may be successfully used as a tool for microgravity separator design.

Hoyt, Nathaniel C.↗
NASA NTRS2007

An Application of Overset Grids to Payload/Fairing Three-Dimensional Internal Flow CFD Analysis

The application of overset grids to the computational fluid dynamics analysis of three-dimensional internal flow in the payload/fairing of an expendable launch vehicle is described. In conjunction with the overset grid system, the flowfield in the payload/fairing configuration is obtained with the aid of OVERFLOW Navier-Stokes code. The solution exhibits a highly three dimensional complex flowfield with swirl, separation, and vortices. Some of the computed flow features are compared with the measured Laser-Doppler Velocimetry (LDV) data on a 1/5th scale model of the payload/fairing configuration. The counter-rotating vortex structures and the location of the saddle point predicted by the CFD analysis are in general agreement with the LDV data. Comparisons of the computed (CFD) velocity profiles on horizontal and vertical lines in the LDV measurement plane in the faring nose region show reasonable agreement with the LDV data.

Kandula, Max↗
NASA NTRS2018

Data Mining for Vortices on the Earth's Magnetosphere - Algorithm Application for Detection and Analysis

Unsteady processes in the solar wind– magnetosphere interaction, such as vortices developed at the magnetopause boundary by the Kelvin–Helmholtz instability, may contribute to the process of mass, momentum and energy transfer into the Earth’s magnetosphere. The research described in this paper validates an algorithm to automatically detect and characterize vortices based on velocity data from simulations. The vortex identification algorithm (VIA) systematically searches the 3-D velocity fields to identify critical points where the magnitude of the velocity vector vanishes. The velocity gradient tensor is computed and its invariants are used to assess vortex structure in the flow field. We use the Community Coordinated Modeling Center (CCMC) Runs on Request capability to create a series of model runs initialized from the conditions observed by the Cluster mission in the Hwang et al. (2011) analysis of Kelvin–Helmholtz vortices observed during southward interplanetary magnetic field (IMF) conditions. We analyze further the properties of the vortices found in the runs, including the velocity changes within their motion across the magnetosheath. We also demonstrate the potential of our tool to identify and characterize other transient features (e.g., flux transfer events, FTEs) with vortical internal structures. We find that the vortices are associated with flows on the magnetosheath side of the magnetopause that reach speeds greater than the solar wind speed at the bow shock.

Collado-Vega, Yaireska M.↗
NASA NTRS2019

The Intermediate Wake of a Thick Flat Plate with a Circular Trailing Edge

The intermediate wake region of a thick flat plate with a circular trailing edge (TE) is investigated with a direct numerical simulation (DNS). The upper and lower separating boundary layers are both turbulent and are statistically identical; the resulting wake is symmetric in the mean. Earlier research dealt with the near/very-near wake of the same plate (x/D < 13.0, x is the streamwise distance from the center of the circular TE and D is the plate-thickness/TE-diameter). In the present investigation the emphasis is on the evolution of shed-vortex structure and turbulence intensity distributions with increasing x; the focus is on the region 20.0 < x/D < 40.0. Profile similarity in wake velocity statistics is explored.

Rai, Man Mohan↗
NASA NTRS2021

The Near and Intermediate Wake of a Thick Flat Plate with a Circular Trailing Edge

The near and intermediate wake regions of a thick flat plate with a circular trailing edge (TE) are investigated with a direct numerical simulation (DNS). The focus is on the region x/D < 43.0(D is the diameter of the TE). The upper and lower separating boundary layers are both turbulent and are statistically identical; the resulting wake is symmetric in the mean. The present study is a continuation of an earlier investigation (Case AX). The earlier research dealt with shed vortex structure, in particular the persistence of the coherence of these vortices with increasing streamwise distance (except for infrequent, random disruption/rupture). In addition, across-centerline movement (ACM)of shed vortices, that is prevalent past x/D = 20.0 in a cylinder case at a comparable Reynolds number, was found to be essentially absent in Case AX. In the present investigation, an underlying cause for the occasional disruption/rupture of shed vortices is explored. Strong interactions between subsequent shed vortices in the trailing edge region is found to be an important contributor to this phenomenon. The distributions of phase-averaged streamwise and cross-stream components of intensity and the corresponding budget terms in the near wake are compared with those obtained in the intermediate wake here. In addition, the main features of the distribution of the phase-averaged velocity pressure-gradient term are compared with those of the production term, for the two intensity components. Except for a change in sign, the important features in the two budget terms are qualitatively similar for both components.

Direct Numerical Simulations of Flat Plate Wake↗
NASA NTRS2021

Characterization of Chevron Nozzle Performance

The flow fields from a set of 50.8 mm diameter circular nozzles were characterized using both streamwise and cross-stream 3-component Stereo Particle Image Velocimetry (SPIV). The test matrix of nozzles included 15 different chevron nozzle designs and one baseline, circularly symmetric nozzle. Nozzle jet operating conditions ranged from acoustic Mach numbers of 0.9 to 1.5, with static temperature ratios ranging between 0.84 and 2.7. Detailed surveys of the single jet flows were performed to capture three-dimensional features of the turbulent exhaust jet evolution. Cross-flow planar measurements were obtained at twelve axial locations, ranging from 0.1 to 20 nozzle diameters downstream of the nozzle exit planes. Streamwise measurements, along the jet centerlines, were obtained at ten partially overlapping downstream locations, providing complete axial surveys over a region extending beyond 20 nozzle diameters downstream of the nozzle exit planes. In both optical configurations, the measurement planes were sized to completely capture the fully turbulent jet shear layer growth. The measured three-dimensional mean and turbulent velocity fields, along with computed second order statistics including axial vorticity and turbulent kinetic energy, were evaluated for all test points. Well-defined streamwise vortex structures in the jet shear layers were measured and reported.

Jet Noise↗
NASA NTRSDate not supplied

USM3D-ME Contributions to the 5th AIAA High Lift Prediction Workshop

This paper presents the results of Reynolds-averaged Navier-Stokes (RANS) simulations conducted by NASA’s flow solver, mixed-element USM3D (USM3D-ME), for the 5th AIAA High-Lift Prediction Workshop. As part of the Fixed-Grid RANS Technology Focus Group (TFG), these simulations were performed to assess the accuracy and efficiency of the USM3D-ME solutions in predicting high-lift flows. The High-Lift Common Research Model (CRM-HL) served as the primary geometry. Several CRM-HL configurations were used for three case studies: a verification study (Case 1), a configuration buildup study (Case 2), and a Reynolds-number variation study (Case 3). Overall, USM3D-ME RANS results aligned with the solutions selected by the Fixed-Grid RANS TFG and available wind tunnel data Simulations for Cases 1 and Configuration 2.1 achieved machine-zero residual convergence, with aerodynamic coefficients converging to steady-state values. However, Configurations 2.2-2.4 and Case 3 encountered iterative- and grid-convergence challenges, particularly at high angles of attack. Compared with the experimental data available for Configurations 2.2-2.4, close agreement was demonstrated at low angles of attack. However, for angles of attack approaching the maximum lift conditions, the predicted lift coefficient and pitching moment deviated from experimental values. The drag-coefficient predictions were in a relatively good agreement, however, slight overpredictions were observed at the highest angle of attack corresponding to the maximum-lift condition. Although iterative convergence for Configurations 2.2-2.4 at high angles of attack remains a persistent challenge, averaging aerodynamic coefficients over the last 5000 iterations yielded satisfactory agreement with the available wind tunnel experimental data. During the workshop, the lack of iterative convergence was attributed to the vortex structures emanating from the slat brackets. To investigate this issue further, post-workshop simulations were conducted on Configuration 2.2. In one study, RANS simulations were performed on a simplified geometry with the slat brackets removed. The second study focused on performing URANS simulations on the original Configuration 2.2 geometry. Preliminary results from both studies are presented and compared with wind tunnel data for Configuration 2.2. Consistent with the findings of other participants in the Fixed-Grid RANS TFG, this study emphasizes the necessity for further exploration and advancement in RANS technology for predicting high-lift flows.

CFD↗
NASA NTRSDate not supplied

USM3D-ME Contributions to the 5th AIAA High Lift Prediction Workshop

This paper presents the results of Reynolds-averaged Navier-Stokes (RANS) simulations conducted by the NASA flow solver, mixed-element USM3D (USM3D-ME), for the 5th AIAA High-Lift Prediction Workshop. As part of the Fixed-Grid RANS Technology Focus Group (TFG), these simulations were performed to assess the accuracy and efficiency of the USM3D-ME solutions in predicting high-lift flows. The High-Lift Common Research Model (CRM-HL) served as the primary geometry. Several CRM-HL configurations were used for three case studies: a verification study (Case 1), a configuration buildup study (Case 2), and a Reynolds-number variation study (Case 3). Overall, USM3D-ME RANS results aligned with the solutions selected by the Fixed-Grid RANS TFG and available wind tunnel data. Simulations for Case 1 and Configuration 2.1 achieved machine-zero residual convergence, with aerodynamic coefficients converging to steady-state values. However, Configurations 2.2-2.4 and Case 3 encountered iterative- and grid-convergence challenges, particularly at high angles of attack. Compared with the experimental data available for Configurations 2.2-2.4, close agreement was demonstrated at low angles of attack. However, for angles of attack approaching the maximum lift conditions, the predicted lift coefficient and pitching moment deviated from experimental values. The drag-coefficient predictions were in relatively good agreement, however, slight overpredictions were observed at the highest angle of attack corresponding to the maximum-lift condition. Although iterative convergence for Configurations 2.2-2.4 at high angles of attack remains a persistent challenge, averaging aerodynamic coefficients over the last 5000 iterations yielded satisfactory agreement with the available wind tunnel experimental data. During the workshop, the lack of iterative convergence was attributed to the vortex structures emanating from the slat brackets. To investigate this issue further, post-workshop simulations were conducted on Configuration 2.2. In one study, RANS simulations were performed on a simplified geometry with the slat brackets removed. The second study focused on performing unsteady RANS (URANS) simulations on the original Configuration 2.2 geometry. Preliminary results from both studies are presented and compared with wind tunnel data for Configuration 2.2. Consistent with the findings of other participants in the Fixed-Grid RANS TFG, this study emphasizes the necessity for further exploration and advancement in RANS technology for predicting high-lift flows.

Aerodynamics↗
NASA NTRS1983

A Numerical Investigation of the Multiple Vortex Phenomenon: Vortex Number and Structure as a Function of Swirl Ratio

A numerical investigation of the multiple vortex phenomenon (MVP) for tornado like flows is conducted to determine the conditions for when a vortex becomes unstable and divides into smaller subsidiary vortices, as well as to determine the structure of these vortices. A three dimensional numerical model developed by Rotunno (1983) is utilized which has been demonstrated to successfully simulate MVP with properties observed both in natural as well as laboratory tornado like vortices. The MVP is generated for several swirl ratio conditions in order to determine the number of vortices generated for those flow configurations. These results are then compared to experimental measurements to validate the numerical model. The number of vortices produced is consistent with observational results made in the Purdue tornado vortex chamber. Furthermore, horizontal and vertical cross sections are taken through the vortices to determine the structure of MVP. Preliminary results indicate that tangential velocities within these smaller asymmetric vortices increase by 20% over values observed in a single axisymmetric vortex at the same swirl ratio.

Smith, D. R.↗