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All sourcesNASA NTRSDOE OSTI
NASA NTRS1992

Bounded energy states in homogeneous turbulent shear flow - An alternative view

The equilibrium structure of homogeneous turbulent shear flow is investigated from a theoretical standpoint. Existing turbulence models, in apparent agreement with physical and numerical experiments, predict an unbounded exponential time growth of the turbulent kinetic energy and dissipation rate; only the anisotropy tensor and turbulent time scale reach a structural equilibrium. It is shown that if a residual vortex stretching term is maintained in the dissipation rate transport equation, then there can exist equilibrium solutions, with bounded energy states, where the turbulence production is balanced by its dissipation. Illustrative calculations are presented for a k-epsilon model modified to account for net vortex stretching.

Bernard, P. S.↗
NASA NTRS1974

Geodesic synchrotron radiation in the Kerr geometry by the method of asymptotically factorized Green's functions

The scalar, electromagnetic, and gravitational geodesic-synchrotron-radiation (GSR) spectra are determined for the case of a test particle moving on a highly relativistic circular orbit about a rotating (Kerr) black hole. It is found that the spectral shape depends only weakly on the value of the angular-momentum parameter (a/M) of the black hole, but the total radiated power drops unexpectedly for a value of at least 0.95 and vanishes as the value approaches unity. A spin-dependent factor (involving the inner product of the polarization of a radiated quantum with the source) is isolated to explain the dependence of the spectral shape on the spin of the radiated field. Although the scalar wave equation is solved by separation of variables, this procedure is avoided for the vector and tensor cases by postulating a sum-over-states expansion for the Green's function similar to that found to hold in the scalar case. The terms in this sum, significant for GSR, can then be evaluated in the geometric-optics approximation without requiring the use of vector or tensor spherical harmonics.

Chrzanowski, P. L.↗
NASA NTRS1999

On the Alignment of Strain, Vorticity and Scalar Gradient in Turbulent, Buoyant, Nonpremixed Flames

The alignment of vorticity and scalar gradient with the eigendirections of the rate of strain tensor is investigated in turbulent buoyant nonpremixed horizontal and vertical flames. The uniqueness of a buoyant nonpremixed flame is that it contains regions with distinct alignment characteristics. The strain-enstrophy angle Psi is used to identify these regions. Examination of the vorticity field and the vorticity production in these different regions indicates that Psi and consequently the alignment properties near the flame surface identified by the mixture fraction band F approximately equals F(sub st) differ from those in the fuel region, F > F(sub st) and the oxidizer region, F < F(sub st). The F approximately equals F(sub st) band shows strain-dominance resulting in vorticity/alpha alignment while F > F(sub st) (and F < F(sub st) for the vertical flame) band(s) show(s) vorticity/beta alignment. The implication of this result is that the scalar dissipation, epsilon(sub F), attains its maximum value always near F approximately equals F(sub st). These results are also discussed within the framework of recent dynamical results [Galanti et al., Nonlinearity 10, 1675 (1997)] suggesting that the Navier-Stokes equations evolved towards an attracting solution. It is shown that the properties of such an attracting solution are also consistent with our results of buoyant turbulent nonpremixed flames.

Boratav, O. N.↗
NASA NTRS1993

The radiated noise from isotropic turbulence revisited

The noise radiated from isotropic turbulence at low Mach numbers and high Reynolds numbers, as derived by Proudman (1952), was the first application of Lighthill's Theory of Aerodynamic Noise to a complete flow field. The theory presented by Proudman involves the assumption of the neglect of retarded time differences and so replaces the second-order retarded-time and space covariance of Lighthill's stress tensor, Tij, and in particular its second time derivative, by the equivalent simultaneous covariance. This assumption is a valid approximation in the derivation of the second partial derivative of Tij/derivative of t exp 2 covariance at low Mach numbers, but is not justified when that covariance is reduced to the sum of products of the time derivatives of equivalent second-order velocity covariances as required when Gaussian statistics are assumed. The present paper removes these assumptions and finds that although the changes in the analysis are substantial, the change in the numerical result for the total acoustic power is small. The present paper also considers an alternative analysis which does not neglect retarded times. It makes use of the Lighthill relationship, whereby the fourth-order Tij retarded-time covariance is evaluated from the square of similar second order covariance, which is assumed known. In this derivation, no statistical assumptions are involved. This result, using distributions for the second-order space-time velocity squared covariance based on the Direct Numerical Simulation (DNS) results of both Sarkar and Hussaini(1993) and Dubois(1993), is compared with the re-evaluation of Proudman's original model. These results are then compared with the sound power derived from a phenomenological model based on simple approximations to the retarded-time/space covariance of Txx. Finally, the recent numerical solutions of Sarkar and Hussaini(1993) for the acoustic power are compared with the results obtained from the analytic solutions.

Lilley, Geoffrey M.↗
NASA NTRS1994

Experimental study of the flow field inside a whirling annular seal

The flow field inside a whirling annular seal has been measured using a 3-D Laser Doppler Anemometer (LDA) system. The seal investigated has a clearance of 1.27 mm, a length of 37.3 mm and is mounted on a drive shaft with a 50 percent eccentricity ratio. This results in the rotor whirling at the same speed as the shaft rotation (whirl ratio = 1.0). The seal is operated at a Reynolds number of 12000 and a Taylor number of 6300 (3600 rpm). The 3-D LDA system is equipped with a rotary encoding system which is used to produce phase averaged measurements of the entire mean velocity vector field and Reynolds stress tensor field from 0.13 mm upstream to 0.13 mm downstream of the seal. The mean velocity field reveals a highly three-dimensional flow field with large radial velocities near the inlet of the seal as well as a recirculation zone on the rotor surface. The location of maximum mean axial velocity migrates from the pressure side of the rotor at the inlet to the suction side at the exit. Turbulence production is a maximum near the seal inlet as indicated by the rapid increase of the turbulence kinetic energy (kappa). However, turbulence production and dissipation attain equilibrium fairly quickly with kappa remaining relatively constant over the last half of the seal.

Morrison, Gerald L.↗
NASA NTRS1992

Experimental study of the flow field inside a whirling annular seal

The flow field inside a whirling annular seal was measured using a 3-D Laser Doppler Anemometer (LDA) system. The seal investigated has a clearance of 1.27 mm, a length of 37.3 mm, and is mounted on a drive shaft with a 50 percent eccentricity ratio. This results in the rotor whirling at the same speed as the shaft rotation (whirl ratio = 1.0). The seal is operated at Reynolds number of 12,000 and a Taylor number of 6,300 (3,600 rpm). The 3-D LDA system is equipped with a rotary encoding system which is used to produce phase averaged measurements of the entire mean velocity vector field and Reynolds stress tensor field from 0.13 mm upstream to 0.13 mm downstream of the seal. The mean velocity field reveals a highly three dimensional flow field with large radial velocities near the inlet of the seal as well as a recirculation zone on the rotor surface. The location of maximum mean axial velocity migrates from the pressure side of the rotor at the inlet to the suction side at turbulence kinetic energy. However, turbulence production and dissipation attain equilibrium fairly quickly with remaining relatively constant over the last half of the seal.

Morrison, Gerald L.↗
NASA NTRS1996

Second moment closure analysis of the backstep flow database

A Second Moment Closure computation (SMC) is compared in detail with the Direct Numerical Simulation (DNS) data of Le and Moin for the backstep flow at Re = 5,000 in an attempt to understand why the intensity of the backflow and, consequently, the friction coefficient in the recirculation bubble are severely underestimated. The data show that this recirculation bubble is far from being laminar except in the very near wall layer. A novel 'differential a priori' procedure was used, in which the full transport equation for one isolated component of the Reynolds stress tensor was solved using DNS data as input. Conclusions are then different from what would have been deduced by comparing a full simulation to a DNS. One cause of discrepancy was traced back to insufficient transfer of energy to the normal stress by pressure strain, but was not cured. A significant finding, confirmed by the DNS data in the core region of a channel flow, is that the coefficient that controls destruction of dissipation, C epsilon(sub 2), should be decreased by a factor of 2 when production is vanishing. This is also the case in the recirculation bubble, and a new formulation has cured 25% of the backflow discrepancy.

Parneix, S.↗
NASA NTRS1987

The role and behavior of spin in gravitational physics

A self-consistent method of introducing spin into any Lagrangian based theory of gravitation was developed. The metric variation of the Lagrangian in the theory leads to an improved energy-momentum tensor which represents the source term in the gravitational field equations. The goal of the research is the construction of a theory general enough to be used to investigate spin effects in astrophysical objects and cosmology, and also to serve as a basis for discussion of the theoretical ideas tested by the NASA Gyroscope Experiment (aboard Gravity Probe B). Specific accomplishments in the following areas are summarized: the inclusion of electromagnetism into the variational principle for spinning matter, formulation of a self-consistent theory for the case of a fluid in which particle production processes occur, and the derivation of the Raychaudhuri equation in the case of spinning matter.

Ray, John R.↗
NASA NTRS2000

Turbulence Model Predictions of Strongly Curved Flow in a U-Duct

The ability of three types of turbulence models to accurately predict the effects of curvature on the flow in a U-duct is studied. An explicit algebraic stress model performs slightly better than one- or two-equation linear eddy viscosity models, although it is necessary to fully account for the variation of the production-to-dissipation-rate ratio in the algebraic stress model formulation. In their original formulations, none of these turbulence models fully captures the suppressed turbulence near the convex wall, whereas a full Reynolds stress model does. Some of the underlying assumptions used in the development of algebraic stress models are investigated and compared with the computed flowfield from the full Reynolds stress model. Through this analysis, the assumption of Reynolds stress anisotropy equilibrium used in the algebraic stress model formulation is found to be incorrect in regions of strong curvature. By the accounting for the local variation of the principal axes of the strain rate tensor, the explicit algebraic stress model correctly predicts the suppressed turbulence in the outer part of the boundary layer near the convex wall.

Rumsey, Christopher L.↗
NASA NTRS1987

An investigation of the flow characteristics in the blade endwall corner region

Studies were undertaken to determine the structure of the flow in the blade end wall corner region simulated by attaching two uncambered airfoils on either side of a flat plate with a semicircular leading edge. Detailed measurements of the corner flow were obtained with conventional pressure probes, hot wire anemometry, and flow visualization. The mean velocity profiles and six components of the Reynolds stress tensor were obtained with an inclined single sensor hot wire probe whereas power spectra were obtained with a single sensor oriented normal to the flow. Three streamwise vortices were identified based on the surface streamlines, distortion of total pressure profiles, and variation of mean velocity components in the corner. A horseshoe vortex formed near the leading edge of the airfoil. Within a short distance downstream, a corner vortex was detected between the horseshoe vortex and the surfaces forming the corner. A third vortex was formed at the rear portion of the corner between the corner vortex and the surface of the flat plate. Turbulent shear stress and production of turbulence are negligibly small. A region of negative turbulent shear stress was also observed near the region of low turbulence intensity from the vicinity of the flat plate.

Hazarika, Birinchi K.↗
NASA NTRS1979

Sound production due to large-scale coherent structures

The acoustic pressure fluctuations due to large-scale finite amplitude disturbances in a free turbulent shear flow are calculated. The flow is decomposed into three component scales; the mean motion, the large-scale wave-like disturbance, and the small-scale random turbulence. The effect of the large-scale structure on the flow is isolated by applying both a spatial and phase average on the governing differential equations and by initially taking the small-scale turbulence to be in energetic equilibrium with the mean flow. The subsequent temporal evolution of the flow is computed from global energetic rate equations for the different component scales. Lighthill's theory is then applied to the region with the flowfield as the source and an observer located outside the flowfield in a region of uniform velocity. Since the time history of all flow variables is known, a minimum of simplifying assumptions for the Lighthill stress tensor is required, including no far-field approximations. A phase average is used to isolate the pressure fluctuations due to the large-scale structure, and also to isolate the dynamic process responsible. Variation of mean square pressure with distance from the source is computed to determine the acoustic far-field location and decay rate, and, in addition, spectra at various acoustic field locations are computed and analyzed. Also included are the effects of varying the growth and decay of the large-scale disturbance on the sound produced.

Gatski, T. B.↗
NASA NTRS2004

Taking Measure of the Milky Way

We intend to use SIM to make definitive measurements of fundamental structural and dynamical parameters of the Milky Way. The important niche in dynamical parameter space afforded by SIM can be exploited to resolve, with unprecedented precision, a number of classical problems of Galactic astronomy. In addition, we have developed new tests of the Galactic mass distribution specifically designed for data with the special properties of SIM products. Our proposed suite of experiments will utilize the SIM Astrometric Grid as well as complementary observations of star clusters and other strategically-selected, distant "test particles" for a definitive characterization of the major components (bulge, disk, halo, satellite system) of the Milky Way. Specifically, our goals will be: 1) The determination of two fundamental parameters that play a central role in virtually every problem in Galactic astronomy, namely (a) the solar distance to the center of the Milky Way, R(sub 0); (b) the solar angular velocity around the Galactic: center, omega(sub 0). 2) The measurement of fundamental dynamical properties of the Milky Way, among them (a) the pattern speed of the central bar (b) the rotation field and velocity-dispersion tensor in the disk (c) the kinematics (mean rotational velocity and velocity dispersion tensor) of the halo as a function of position 3. The definition of the mass distribution of the Galaxy, which is dominated by the presence of dark matter. We intend to measure (a) the relative contribution of the disk and halo to the gravitational potential (b) the local volume and surface mass density of the disk (c) the shape, mass and extent of the dark halo of the Milky Way out to 250 kpc.

Majewski, Steven R.↗
NASA NTRS1997

The Spectral and Statistical Properties of Turbulence Generated by a Vortex/Blade-Tip Interaction

The perpendicular interaction of a streamwise vortex with the tip of a lifting blade was studied in incompressible flow to provide information useful to the accurate prediction of helicopter rotor noise and the understanding of vortex dominated turbulent flows. The vortex passed 0.3 chord lengths to the suction side of the blade tip, providing a weak interaction. Single and two-point turbulence measurements were made using sub-miniature four sensor hot-wire probes 15 chord lengths downstream of the blade trailing edge; revealing the mean velocity and Reynolds stress tensor distributions of the turbulence, as well as its spanwise length scales as a function of frequency. The single point measurements show the flow downstream of the blade to be dominated by the interaction of the original tip vortex and the vortex shed by the blade. These vortices rotate about each other under their mutual induction, winding up the turbulent wakes of the blades. This interaction between the vortices appears to be the source of new turbulence in their cores and in the region between them. This turbulence appears to be responsible for some decay in the core of the original vortex, not seen when the blade is removed. The region between the vortices is not only a region of comparatively large stresses, but also one of intense turbulence production. Velocity autospectra measured near its center suggests the presence quasi-periodic large eddies with axes roughly parallel to a line joining the vortex cores. Detailed two-point measurements were made on a series of spanwise cuts through the flow so as to reveal the turbulence scales as they would be seen along the span of an intersecting airfoil. The measurements were made over a range of probe separations that enabled them to be analyzed not only in terms of coherence and phase spectra but also in terms of wave-number frequency (kappa-omega) spectra, computed by transforming the measured cross-spectra with respect to the spanwise separation of the probes. These data clearly show the influence of the coherent eddies in the spiral wake and the turbulent region between the cores. These eddies produce distinct peaks in the upwash velocity kappa-omega spectra, and strong anisotropy manifested both in the decay of the kappa-omega spectrum at larger wave-numbers and in differences between the kappa-omega spectra of different components. None of these features are represented in the von Karman spectrum for isotropic turbulence that is often used in broadband noise computations. Wave-number frequency spectra measured in the cores appear to show some evidence that the turbulence outside sets tip core waves, as has previously been hypothesized. These spectra also provide for the first time a truly objective method for distinguishing velocity fluctuations produced by core wandering from other motions.

Devenport, William J.↗
NASA NTRS2011

Magnetospheric Multiscale (MMS) Mission Attitude Ground System Design

This paper describes the attitude ground system (AGS) design to be used for support of the Magnetospheric MultiScale (MMS) mission. The AGS exists as one component of the mission operations control center. It has responsibility for validating the onboard attitude and accelerometer bias estimates, calibrating the attitude sensors and the spacecraft inertia tensor, and generating a definitive attitude history for use by the science teams. NASA's Goddard Space Flight Center (GSFC) in Greenbelt, Maryland is responsible for developing the MMS spacecraft, for the overall management of the MMS mission, and for mission operations. MMS is scheduled for launch in 2014 for a planned two-year mission. The MMS mission consists of four identical spacecraft flying in a tetrahedral formation in an eccentric Earth orbit. The relatively tight formation, ranging from 10 to 400 km, will provide coordinated observations giving insight into small-scale magnetic field reconnection processes. By varying the size of the tetrahedron and the orbital semi-major axis and eccentricity, and making use of the changing solar phase, this geometry allows for the study of both bow shock and magnetotail plasma physics, including acceleration, reconnection, and turbulence. The mission divides into two phases for science; these phases will have orbit dimensions of l.2xl2 Earth radii in the first phase and l.2x25 Earth radii in the second in order to study the dayside magnetopause and the nightside magnetotail, respectively. The orbital periods are roughly one day and three days for the two mission phases. Each of the four MMS spacecraft will be spin stabilized at 3 revolutions per minute (rpm), with the spin axis oriented near the ecliptic north pole but tipped approximately 2.5 deg towards the Sun line. The main body of each spacecraft will be an eight-sided platform with diameter of 3.4 m and height of 1.2 m. Several booms are attached to this central core: two axial booms of 14.9 m length, two radial magnetometer booms of 5 m length, and four radial -wire booms of 60 m length. Attitude and orbit control will use a set of axial and radial thrusters. A four-head star tracker and a slit-type digital Sun sensor (DSS) provide input for attitude determination. In addition, an accelerometer will be· used for closed-loop orbit maneuver control. The primary AGS product will be a daily definitive attitude history. Due to power limitations; the star tracker and accelerometer data will not be available at all times. However, tracker data from at least 10 percent of each orbit and continuous DSS data will be provided. An extended Kalman filter (EKF) will be used to estimate the three-axis attitude (i.e., spin axis orientation and spin phase) and rotation rate for all times when the tracker data is valid. For other times, the attitude is generated by assuming a constant angular momentum vector in the inertial frame. The DSS sun pulse will provide a timing signal to maintain an accurate spin phase. There will be times when the Sun is occulted and DSS data is not available. If this occurs at the start or end of a definitive attitude product, then the spin phase will be extrapolated using the mean rate determined by the EKF.

Sedlak, Joseph E.↗
NASA NTRS2010

Magnetospheric Multiscale (MMS) Mission Attitude Ground System Design

This paper describes the attitude ground system (AGS) design to be used for support of the Magnetospheric MultiScale (MMS) mission. The AGS exists as one component of the mission operations control center. It has responsibility for validating the onboard attitude and accelerometer bias estimates, calibrating the attitude sensors and the spacecraft inertia tensor, and generating a definitive attitude history for use by the science teams. NASA's Goddard Space Flight Center (GSFC) in Greenbelt, Maryland is responsible for developing the MMS spacecraft, for the overall management of the MMS mission, and for mission operations. MMS is scheduled for launch in 2014 for a planned two-year mission. The MMS mission consists of four identical spacecraft flying in a tetrahedral formation in an eccentric Earth orbit. The relatively tight formation, ranging from 10 to 400 km, will provide coordinated observations giving insight into small-scale magnetic field reconnection processes. By varying the size of the tetrahedron and the orbital semi-major axis and eccentricity, and making use of the changing solar phase, this geometry allows for the study of both bow shock and magnetotail plasma physics, including acceleration, reconnection, and turbulence. The mission divides into two phases for science; these phases will have orbit dimensions of 1.2 x 12 Earth radii in the first phase and 1.2x25 Earth radii in the second in order to study the dayside magnetopause and the nightside magnetotail, respectively. The orbital periods are roughly one day and three days for the two mission phases. Each of the four MMS spacecraft will be spin stabilized at 3 revolutions per minute (rpm), with the spin axis oriented near the ecliptic north pole but tipped approximately 2.5 deg towards the Sun line. The main body of each spacecraft will be an eight-sided platform with diameter of 3.4 m and height of 1.2 m. Several booms are attached to this central core: two axial booms of 14.9 m length, two radial magnetometer booms of 5 m length, and four radial wire booms of 60 m length. Attitude and orbit control will use a set of axial and radial thrusters. A four-head star tracker and a slit-type digital Sun sensor (DSS) provide input for attitude determination. In addition, an accelerometer will be used for closed-loop orbit maneuver control. The primary AGS product will be a daily definitive attitude history. Due to power limitations, the star tracker and accelerometer data will not be available at all times. However, tracker data from at least 10 percent of each orbit and continuous DSS data will be provided. An extended Kalman filter (EKF) will be used to estimate the three-axis attitude (i.e., spin axis orientation and spin phase) and rotation rate for all times when the tracker data is valid. For other times, the attitude is generated by assuming a constant angular momentum vector in the inertial frame. The DSS sun pulse will provide a timing signal to maintain an accurate spin phase. There will be times when the Sun is occulted and DSS data is not available. If this occurs at the start or end of a definitive attitude product, then the spin phase will be extrapolated using the mean rate determined by the EKF.

Sedlak, Joseph E.↗
NASA NTRS1998

Incompressible Turbulent Wing-Body Junction Flow

The overall objective of this study is to contribute to the optimized design of fan bypass systems in advanced turbofan engines. Increasing the engine bypass ratios have provided a major boost in engine performance improvement over the last fifty years. An engine with high bypass ratio (11-16:1) such as the Advanced Ducted Propulsion (ADP) is being developed and is expected to provide an additional 25% improvement in overall efficiency over the early turbofans. Such significant improvements in overall efficiency would reduce the cost per seat mile, which is a major government and Industry challenge for the 21th century. The research is part of the Advanced Subsonic Technology (AST) program that involves a NASA, U.S. Industry and FAA partnership with the goal of a safe and highly productive global air transportation system. The immediate objective of the study is to perform numerical simulation of duct-strut interactions to elucidate the loss mechanisms associated with this configuration that is typical of advanced turbofan engines such as ADP. However, at present experimental data for a duct-strut configuration are not available. Thus, as a first step a wing-body junction flow would be studied and is the specific objective of the present study. At the outset it is to be recognized that while duct-strut interaction flow is similar to that of wing-body junction flows, there are some differences owing to the presence of a wall at both ends of the strut. Likewise, some differences are due to the sheared inflow (as opposed to a uniform inflow) velocity profile. It is however expected that some features of a wing-body junction flow would persist. Next, some of the salient aspects of the complex flow near a wing-body junction, as revealed by various studies reported in the literature will be reviewed. One of the principle characteristics of the juncture flow, is the presence of the mean flow components in a plane perpendicular to the direction of the oncoming free-stream flow. The lateral curvature of the wing/strat causes the oncoming turbulent layer to skew about am axis (x-axis) parallel to the plane (xz-plane) of the mean shear. This is the principle mechanism for the generation of secondary flow. Such skew-induced secondary flows are slow to be attenuated by Reynolds stresses. Additional contribution to the generation of secondary flow comes from anisotropies in Reynolds stresses. Upstream of the strut, the mean-vorticity is directed span wise (along the y-direction). The presence of secondary flow in the vicinity of the strut causes the vorticity to stretch around the obstacle in a horse-shoe shape, with each leg having a vorticity of the opposite sense. The blockage effect of the strut imposes a severe adverse pressure gradient on the oncoming turbulent shear layer, causing boundary layer separation ahead of the leading edge, resulting in a vortex that rolls up and flows downstream into the juncture region. The separation vortices trailing in the wake of the wing can alter the lift or drag characteristics of the surfaces downstream of the wing-body juncture. Likewise, on submarines, the wake flow behind the appendage can degrade the performance of the propeller located downstream. The complex nature of this flow is caused by the presence of all six components of Reynolds stresses. Devenport and Simpson report that in the vicinity of the horse-shoe vortex there is intense recirculation with turbulent stresses being much larger than those normally observed in turbulent flows. These features contribute to making this flow a challenge to predict numerically. Some of the past studies provide useful insights into this flow that would guide our numerical efforts. In measurements reported by Shabaka and Bradshaw, the eddy viscosity tensor is seen to be non-isotropic and has negative components in certain regions. In an effort to evaluate the closure assumptions of various turbulence models, Devenport and Simpson used their own extensive measurements in juncture flows around the nose of a wing-body junction. Measured values of mean-velocity and/or turbulence kinetic energy was used to predict the magnitude of the shear stress vector. Algebraic stress models performed the best followed by Cebeci-Smith eddy viscosity model. The flow is reported to be dominated by a pressure field produced by the wing and the velocity field generated by the horseshoe vortex that is wrapped around the junction between the wing and wall. Kubendran et al. conclude from an experimental study that the shape of leading edge of the wing as characterized by its slenderness ratio is a major factor in determining the flow fields in the juncture region. The more thinner the leading edge of the juncture, the weaker the horseshoe vortex is. Also, with a slender leading edge, the secondary flow in the juncture would be due mainly to the cross-stream gradients of Reynolds stresses rather than due to a lateral skewing of the shear layer.

Krishnamurthy, R.↗