Evolution Of Coherent Structures Research Articles

Evolution of coherent structures in the flow around a circular pier with a developing scour hole: A numerical study

Large-eddy simulations (LES) were used to simulate the flow features around a circular pier with evolving scour hole. The bathymetry corresponding to four stages of scour hole evolution, published in the previous literature, was used in the flow domain to recreate the flow at a particular stage of scour hole development. The growth of coherent structures and their interactions in a developing scour hole were investigated. The formation of secondary vortices inside the scour hole and their interactions with the primary horseshoe vortex (HV) made the HV system highly dynamic. High-level pressure fluctuations were observed inside the core of HV. The simulation captured the bimodal aperiodic oscillations in the HV at all stages of scour hole development. The mechanisms responsible for the formation of scour hole behind the pier were discussed. High levels of mean and instantaneous bed shear stress values were observed beneath the horseshoe vortex. These values were found to have decreased with an increase in the scour hole depth. The LES also captured the detachment of vortices from the legs of HV and their convection in the pier downstream.

Convective modes reveal the incoherence of the Southern Polar Vortex

The Southern Polar Vortex (SPV) is prominent over Antarctica in the Austral winter, and typically associated with a region of low temperature, low ozone concentration, negative potential vorticity, and polar stratospheric clouds. Seasonal and unexpected changes in the SPV have a profound influence on global weather. A methodology which identifies the SPV’s coherence and breakup using only wind and pressure data is developed and validated against temperature, ozone and potential vorticity data. The process identifies “convective modes”, each with an assigned “coherence” value, which form building blocks for the observed spatial variation of the SPV. Analysis and interpretation are presented for 4 years with quite different known behavior of the SPV: 1999 (a relatively standard year), 2002 (when the SPV split into two), 2019 (an atmospheric warming year which led to an early dissipation in the SPV), and the most recent year 2022 (which was influenced by submarine volcano eruptions and a prolonged La Niña event). In decomposing convective effects into modes with quantifiable coherence, this study solidifies connections between wind velocities and atmospheric variables while providing new tools to study the evolution of coherent structures and signal the occurrence of atypical geophysical events.

Turbulence structure of the Rayleigh–Bénard convection using liquid CO2 as working fluid

We studied the evolution of flow structures and large-scale circulations (LSC) in Rayleigh–Bénard convection (RBC) using liquid carbon dioxide as the working medium. In this experiment, a transparent sapphire pressure vessel with observable internal flow was designed, and different temperature differences were applied between the upper and the lower surfaces of the fluid to obtain different Rayleigh numbers (Ra). We employed proper orthogonal decomposition and reconstruction to extract internal flow structures from the shadowgraphy images. We used optical flow techniques to acquire the velocity field of the flow, and we reconstructed the temperature field inside the supercritical fluid using the relationship between shadowgraphy images and refractive index. It is clearly observed that the RBC begins to produce different flow structures under a small temperature difference of 0.4 °C. As the number of Ra increases, the number and the speed of plumes increase, and the morphology of plumes gradually becomes elongated. When Ra exceeds a certain critical value, an LSC structure appears in the flow field, and the plumes translate laterally with the large-scale circulation, and the disorder of the vortex structure in the central flow region increases significantly. Three typical flow structures were observed: (1) single plume, (2) thermal boundary layer traveling waves, and (3) Rayleigh–Taylor instability waves. We believe that the traveling wave structure is the precursor to the single plume. The temperature field analysis of the three structures was carried out, and the velocity of the typical plume was calculated by the optical flow method. It was found that LSC transitioned from oval to square shape with the increase in Ra, and the internal plume Reynolds number slowly increased with the increase in Ra. By the in-depth study of the thermal turbulence characteristics and the coherent structure evolution law of RBC, this paper provides experimental support for revealing the mechanism of enhanced heat transfer in energy system with a liquid CO2 working fluid.

Multi-scale proper orthogonal decomposition (mPOD) analysis of vortex evolution and viscous dissipation in a circular-cylinder wake controlled by parallel symmetric jets

This paper describes a novel flow control scheme over circular-cylinder wakes using parallel symmetric jets. The effects of different jet momentum coefficients (Cm) at θ = ±30° on vortex evolution and viscous dissipation in the circular-cylinder wake are experimentally investigated using high-speed particle image velocimetry (PIV) in an open-jet wind tunnel at a Reynolds number of 1.01 × 104. Multiscale proper orthogonal decomposition (mPOD) is used to analyze the evolution of coherent structures and establish the relationship between flow structures and viscous dissipation. At Cm ≤ 0.0089, the vortex formation length is elongated and the scale and intensity of the vortex are suppressed. Small-scale jet vortices are then generated by the interaction between the jet shear layers at Cm = 0.0174, whereupon the turbulent kinetic energy attains a minimum and the Reynolds shear stress in the wake is basically eliminated. At higher values of Cm, band-like jet oscillations appear in the wake, increasing the turbulence intensity. The inhibition effect of the jets on the wake vortex decreases the viscous dissipation in the wake, with optimal viscous dissipation control obtained at Cm = 0.0089. When Cm > 0.0089, the coupling between the jets increases the viscous dissipation of small-scale, high-frequency turbulent structures, and this effect gradually becomes more obvious with increasing Cm.

Large-Eddy simulations on interaction flow structures of pulsed jets in a crossflow

Numerical investigations of pulsed jets in a crossflow are carried out using large-eddy simulation. The flow mechanism and coherent structure evolution of pulsed jets with different frequencies (f = 20 Hz, 50 Hz, and 100 Hz) are analyzed and compared with the steady-state jet. The proper orthogonal decomposition (POD) method is used to analyze the effect of different vortex structures in the flow field, which further elaborates the influence mechanism of frequency on pulse jets in the crossflow. The results show that under the same velocity ratio, the normal penetration depth of pulsed jets to the crossflow are greater than that of the steady-state jet. Due to the pulsating effect, the pulsed jets will form a large-scale shear layer vortex, which strengthens the turbulence pulsation and promotes the mixing between jets and the crossflow. As the pulse frequency increases, the scale of the vortex rings in the downstream shear layer will become smaller but the number will increase, and fine vortex clusters will be formed around them. At f = 50 Hz, the counter-rotating vortex pair (CVP) and the jet front shear layer vortex have a greater influence in the flow field, followed by the downstream shear layer vortex, and the wake vortex has the least influence in the flow field. At f = 20 Hz and 100 Hz, the wake vortex has a greater influence in the flow field than that at f = 50 Hz.

Four-way coupled modelling of swirling particle-laden flow in Methane-central coaxial jets

A novelty particle subgrid scale model is established to model the effect of gas flow on particle motion behaviors and a four-way coupling method to describe the particle-particle collisions by granular temperature equation is also developed. Menthane-central jets of particle-laden swirling flow in coaxial chamber are modeled by a multiphase second-order moment model and numerically predicted by the large-eddy simulation. Simulation results of two-phase hydrodynamics are in good agreement with experimental measurement. Evolution of coherent structure, recirculation region, periodic vortices formation, breakup, coalescence and pair, stronger mixing are captured and analyzed successfully. Compared to gas-annular flow, it is benefit for the stabilization of flame due to the strongly radial spread of particle dispersions and mixing momentum exchanges. The core-annular flow structures of particle concentration are observed, and instantaneous particle velocity are complied with the normal distribution. At shear layers of core-jetting regions of X/R=0.56, correlations of axial-tangential gas and particle fluctuation velocities are approximately 3.0 and 6.0 times larger than those of far downstream flows.

Formation and evolution of coherent structures in 3D strongly turbulent magnetized plasmas

We review the current literature on the formation of coherent structures (CoSs) in strongly turbulent 3D magnetized plasmas. CoSs [Current Sheets (CS), magnetic filaments, large amplitude magnetic disturbances, vortices, and shocklets] appear intermittently inside a turbulent plasma and are collectively the locus of magnetic energy transfer (dissipation) into particle kinetic energy, leading to heating and/or acceleration of the latter. CoSs and especially CSs are also evolving and fragmenting, becoming locally the source of new clusters of CoSs. Strong turbulence can be generated by the nonlinear coupling of large amplitude unstable plasma modes, by the explosive reorganization of large-scale magnetic fields, or by the fragmentation of CoSs. A small fraction of CSs inside a strongly turbulent plasma will end up reconnecting. Magnetic Reconnection (MR) is one of the potential forms of energy dissipation of a turbulent plasma. Analyzing the evolution of CSs and MR in isolation from the surrounding CoSs and plasma flows may be convenient for 2D numerical studies, but it is far from a realistic modeling of 3D astrophysical, space, and laboratory environments, where strong turbulence can be exited, such as in the solar wind, the solar atmosphere, solar flares and Coronal Mass Ejections, large-scale space and astrophysical shocks, the magnetosheath, the magnetotail, astrophysical jets, and Edge Localized Modes in confined laboratory plasmas (tokamaks).

Comparative study on gas-particle transport characteristics subjected to the central and annular coaxial jets

The gas-particle transport characteristics subjected to the central and annular coaxial jets were investigated using particle image velocimetry (PIV) and optical fiber probe technologies. The powder transport behaviors, the distributions of particle velocity, concentration, fluxes, and turbulence properties, as well as the spatial variations of vortex structures and vorticity, were measured with various jet velocities. The results showed that the injection patterns significantly influence the transport characteristics of the gas-particle two-phase coaxial jets. Compared with the annular coaxial jets, the powder jet subjected to the central coaxial jets exhibits better momentum transfer and particle dispersion characteristics, leading to the acceleration and fluctuation in the axial and radial velocities of particles. Correspondingly, the vortex structures formed in the shear layer of the annular coaxial jets do not undergo pairing, coalescing, and breaking up into small-scale eddies as rapidly as the central coaxial jets, which significantly decreases the entrainment and turbulence intensity. The combing interactions of the recirculating entrainment and the wall effect result in the particle accumulation near the wall ( r / R = 0.83– 1). Although the uneven distribution of particles was observed in both coaxial jets, the radial profiles of particle concentration are more uniform in the central coaxial jets than in the annular coaxial jets. • The injection patterns showed a significant effect on powder transport behaviors. • The turbulence properties and the evolution of coherent structures were captured. • The central coaxial jet exhibited better momentum transfer and particle dispersion. • The near-wall accumulation of particles increases with gas jet velocity.

Effect of coolant pulsation on film cooling performance on flat plate

• Large eddy simulation was used for investigating square-wave pulsating film cooling on flat plate. • The experiment of pulsating film cooling was carried out. • Evolution of coherent structures in pulsating film cooling was analyzed. Combined with infrared thermography tests, large eddy simulation was used to study the square-wave pulsating film cooling on flat plate at the time-averaged blowing ratios of 0.5 and 1.5. The effects of two pulsating parameters including frequency and amplitude were discussed in detail. At low blowing ratio, the introduction of coolant pulsation leads to the decrease of cooling effectiveness, and the effect of pulsating amplitude is much more pronounced than frequency. At high blowing ratio, cooling effectiveness can be enhanced by coolant pulsation at low amplitude, but reduced at high amplitude. As the typical vortex structure in pulsating film cooling, the starting vortex is formed as the jet is turned on. The interaction between hairpin vortex and starting vortex shows that, at high pulsating amplitude, the starting vortex can help coolant detach from the wall, and has an adverse influence on cooling performance. The statistical characteristics and power spectrum density of instantaneous velocity were investigated. It was found that the introduction of coolant pulsation can result in the increases of eddy energy and turbulent kinetic energy in the near filed region, but this effect becomes weaker and weaker as the streamwise distance increases.

Mechanical energy loss and Rayleigh-Taylor instability in free discharge of vertical sharp-edged orifices

Orifice discharge is a classical hydrodynamic phenomenon. The experiments of liquid free jets through vertical sharp-edged orifices were conducted on the influences of orifice geometry and liquid depth. The flow mechanism indicates that the higher energy loss of thin-walled orifice is caused by the inspired air annulus inside orifice channel which will lead to the Rayleigh-Taylor instability. The theoretical derivation further demonstrated that the consecutive collisions between main stream and surface waves can cause significant energy dissipation. Besides, a kind of transition orifice was found for the first time: the increase of liquid depth can induce a sudden switch from the thick-walled orifice mechanism to the thin-walled one. The presentation in this work on the formation and development of coherent structures, the migration and bursting of vortexes, as well as the interaction between vortexes and free interface, is universal for revealing the essential features of multi-phase flow in form friction.

Effect of various structures on the shell-side flow over coil tube bundles

In response to the increasing number of applications and safety considerations for coil tube bundles, an investigation on the influence of structural parameters (the pitch diameter ratio between adjacent tube layers, a, the pitch diameter ratio in the same layer, b, and the helix angle, α) on the flow in the shell side over simplified three-layer bundles was carried out using detached-eddy simulation (DES), which was verified by previous studies. In this study, the local flow structures were discussed by observing a zoomed-in view of the gaps, which revealed that irregularly distributed vortices appeared. The spiral wound structure made the flow structures asymmetric and increased the lateral velocity between tubes of the same layer. By applying power spectral density (PSD) and continuous wavelet transform (CWT) functions, the time–frequency data of velocity fluctuations were analyzed to obtain the flow and evolution of coherent structures over time, revealing vortex shedding frequency bands and Strouhal numbers (St). In addition, multimodal Gaussian distribution approximations (MGDA) were employed to indicate that the shell-side flow was composed of multiple flow modes.

Modal analysis of tip vortex cavitation with an insight on how vortex roll-up enhances its instability

The instability mechanism of tip vortex cavitation (TVC) is investigated in the present paper, which is found to be significantly influenced by the roll-up of secondary vortices. The three-dimensional proper orthogonal decomposition (3D-POD) analysis is utilized to identify the coherent structures around the TVC, and a modal turbulence transport theory is introduced to further measure the modal contribution to the TVC instability. According to the dynamic evolution of coherent structure identified by the 3D-POD analysis, the fluctuating component is gradually introduced into the tip vortex together with the roll-up of secondary vortices. Moreover, the dynamic evolution of coherent structure is always accompanied by a significant distribution of modal turbulence kinetic energy (TKE), implying that the TVC instability is promoted along with the merging of secondary vortices into the tip vortex. A deeper understanding is gained with the assistance of the modal turbulence transport theory. It is found that the modal Reynolds stress introduced by the vortex roll-up is the trigger for the local TVC instability, because it can significantly enhance the energy input from mean flow to fluctuating flow.

The evolution of coherent vortical structures in increasingly turbulent stratified shear layers

We study the morphology of Eulerian vortical structures and their interaction with density interfaces in increasingly turbulent stably stratified shear layers. We analyse the three-dimensional, simultaneous velocity and density fields obtained in the stratified inclined duct laboratory experiment (SID). We track, across 15 datasets, the evolution of coherent structures from pre-turbulent Holmboe waves, through intermittent turbulence, to full turbulence and mixing. We use the rortex–shear decomposition of the local vorticity vectors into a rortex vector capturing rigid-body rotation and a shear vector. We describe the morphology of ubiquitous hairpin-like vortical structures (revealed by the rortex), similar to those commonly observed in boundary-layer turbulence. These are born as relatively weak vortices around the strong three-dimensional shearing structures of confined Holmboe waves, and gradually strengthen and deform under increasing turbulence, transforming into pairs of upward- and downward-pointing hairpins propagating in opposite directions on the top and bottom edge of the shear layer. The pair of legs for each hairpin are counter-rotating and entrain fluid laterally and vertically, whereas their arched-up ‘heads’, which are transverse vortices, entrain fluid vertically. We then elucidate how this large-scale vortex morphology stirs and mixes the density field. Essentially, vortices located at the sharp density interface on either edge of the mixing layer (mostly hairpin heads) engulf blobs of unmixed fluid into the mixing layer, whereas vortices inside the mixing layer (mostly hairpin legs) further stir it, generating strong, small-scale shear, enhancing mixing. These findings provide new insights into the role of turbulent coherent structures in shear-driven stratified mixing.

Vertical propagation of submeso and coherent structure in a tall and dense amazon forest in different stability conditions. PART II: Coherent structures analysis

Observations of the vertical structure of the turbulent flow in different stability regimes above and within the Amazon Forest at the Amazon Tall Tower Observatory (ATTO) site have been presented in Part I. Here, the influence of stability on the inception and development of coherent structures is investigated. According to the mixing-layer analogy the coherent vortices that dominate the turbulent flows at the canopy-atmosphere interface are generated by hydrodynamical instabilities triggered by an inflection in the vertical profile of the mean wind speed at or near the canopy top. The coherent motions time, Tpeak and separation length scale, Λ, depend on the shear length scale, Ls, close to the canopy top. The present analysis studies the characteristics of Ls, Tpeak and Λ in the five stability regimes defined in Part I.The behaviour of Ls with stability is evaluated and parameterized. Ls increases with decreasing stability, presenting two asymptotes for large unstable and stable stratification and a linear behaviour close to neutral stratification. Coherent structures and their timescale are detected with an original method based on the autocorrelation functions of 5-min subsets of turbulent quantities. The vertical time scale is larger in neutral conditions and decreases for both increasing and decreasing stability. At the canopy top the separation length scale presents a linear dependence on Ls, whose slope is maximum in neutral conditions and decreases departing from neutrality. An original parameterization of the dependence of this slope on h/L, where h is the canopy height and L is the Obukhov length, is shown. Combining the parameterizations, the dependence of the separation length scale is finally presented. Λ has a maximum for slightly unstable conditions, linearly decreases towards neutral and weakly stable stratifications and then it tends to zero for very intense unstable and stable stratifications.

Research on coherent structure in 4-rod bundle lattice based on CFD

Coherent structures exist only when the ratio of pitch to rod diameter (P/D) in the rod bundle lattice is less than 1.1. In order to produce coherent structures in large P/D lattices, roughness elements are arranged on the outer surface of the rod bundle. This method can enhance the heat transfer capability of the rod bundle lattice. The whole process of the generation and development of coherent structures are analyzed. The Reynolds stress model (RSM) is used for the numerical calculation of the flow and heat transfer of the working fluid in a 4-rod bundle lattice with a P/D of 1.25. The whole process of the generation and development of the coherent structures in the 4-rod bundle lattice is shown. The influence of different micro-rib size (e) on the coherent structures and the flow and heat transfer of the rod bundle lattice is analyzed. The interaction between the fluid channels in the 4-rod bundle lattice is also analyzed. The results show that with the arrangement of the micro-ribs in the large P/D rod bundle lattice, coherent structures are produced, which enhances the flow mixing between the channels of the 4-rod bundle lattice. The flow and heat transfer characteristics of the rod bundle lattice have been greatly changed, especially when P/(D + 2e) = 1.02, the heat transfer capacity of the rod bundle lattice increases by 44.1% while the flow resistance increases by 3.4 times. This research can provide theoretical support for the design and improvement of reactor fuel elements, heat exchangers and steam generators.

Evolution of instability modes and spreading characteristics of a planar jet flow under long-wave excitations

Switching phenomenon and parametric resonance have been identified in near field flows subjected to high-frequency controlled perturbations. This study reports the correlation between switching phenomenon, parametric resonance, and spreading characteristics caused by the instability modes after the end of the potential core of a planar jet (far field). Two periodically oscillating strips were used to control perturbations in free shear layers and three different long-wave excitations (i.e., symmetric, antisymmetric, and one-sided) were imposed at the excitation frequency (fe). The excitation frequency was fixed relative to the preferred mode frequency (fp) at fe = fp/4. To investigate the evolution of coherent structures, frequency spectral analysis and phase correlation were applied to hot-wire anemometer measurements. Under symmetric excitation, the instability mode of vortical structures is observed twice at mode switching, retaining the in-phase pattern. Jet puffing occurs when the cross correlation R12(0) is positive at the far field. These features cause vortex shedding to remain symmetric and suppresses the flapping motion of the jet. However, out-of-phase behavior reoccurs under antisymmetric and one-sided excitations and the cross correlation R12(0) becomes a negative after the end of the potential core. As a result, flapping motion is enhanced and the jet spreads wider after the end of the potential core.

Vortex Shedding Frequency Scaling of Coherent Structures Induced by a Plasma Actuator

The spatial–temporal evolution of coherent structures generated by an asymmetrical dielectric barrier discharge (DBD) plasma actuator is an intriguing unsteady process with profound implications for revealing the controlling mechanism of the plasma actuator and promoting the control effect of the DBD plasma actuator at high wind speed and high Reynolds number. In this study, the induced coherent structures of the plasma actuator are studied systematically with a particle image velocimetry for a range of force coefficients (). Results indicate that the vortices that shed periodically from the downstream edge of the exposed electrode are a result of the amplification of natural disturbances in the jet shear layer. During later evolution, the adjacent vortices pair up to form a new vortex, leading to a reduction of the dominant frequency in the velocity spectra. Further downstream, the coherent vortical structures decay considerably, correlating with the shear-layer transition process in the plasma wall jet. The vortex shedding frequency is observed to scale with the force coefficient according to , which is of importance to the selection of the dominant frequency by adjusting the actuation parameters of the plasma actuator based on the characteristic frequency of the controlled flowfield and improving the control effect of the plasma actuator. With increasing force coefficient, the propagation speed of the roll-up vortices increases linearly, whereas the streamwise vortex spacing decreases monotonically. It is of great significance that the nondimensional Strouhal number based on the vortex shedding frequency, vortex propagation speed, and streamwise vortex spacing is proposed and varies slightly between 0.045 and 0.065, for all the tested cases, which lays a foundation for promoting the numerical simulation model of the plasma actuator and revealing the controlling mechanism of the flow control using the plasma actuator.

Experimental investigation of flow around a 45∘ oriented cube for Reynolds numbers between 500 and 50 000

Particle imaging velocimetry (PIV) measurements were performed for flow around an oriented cube, to study the wake at various Reynolds numbers. The wake shows numerous small eddies and the velocity profiles have multiple peaks. At higher Reynolds number the streamwise to transverse root mean square velocity ratio ${U}_{r\phantom{\rule{0}{0ex}}m\phantom{\rule{0}{0ex}}s}/{V}_{r\phantom{\rule{0}{0ex}}m\phantom{\rule{0}{0ex}}s}$ ~ 1 indicates homogenizing and better mixing ability of an oriented cube as compared to a normal cube. Proper orthogonal decomposition (POD) has been used to examine the energy content of the flow and the evolution of coherent structure.

The role of the “monopole” instability in the evolution of two-dimensional turbulent free shear layers

The role of instability in the growth of a two-dimensional, temporally evolving, “turbulent” free shear layer is analyzed using vortex-gas simulations that condense all dynamics into the kinematics of the Biot–Savart relation. The initial evolution of perturbations in a constant vorticity layer is found to be in accurate agreement with the linear stability theory of Rayleigh. There is then a stage of non-universal evolution of coherent structures that is closely approximated not by Rayleigh stability theory, but by the Karman–Rubach–Lamb linear instability of monopoles, until the neighboring coherent structures merge. After several mergers, the layer evolves eventually to a self-preserving reverse cascade, characterized by a universal spread rate found by Suryanarayanan, Narasimha, and Hari Dass [“Free turbulent shear layer in a point vortex gas as a problem in nonequilibrium statistical mechanics,” Phys. Rev. E 89, 013009 (2014)] and a universal value of the ratio of dominant spacing of structures (Λf) to the layer thickness (δω). In this universal, self-preserving state, the local amplification of perturbation amplitudes is accurately predicted by Rayleigh theory for the locally existing “base” flow. The model of Morris, Giridharan, and Lilley [“On the turbulent mixing of compressible free shear layers,” Proc. R. Soc. London, Ser. A 431, 219–243 (1990)], which computes the growth of the layer by balancing the energy lost by the mean flow with the energy gain of the perturbation modes (computed from an application of Rayleigh theory), is shown, however, to provide a non-universal asymptotic state with initial condition dependent spread rate and spectra. The reason is that the predictions of the Rayleigh instability, for a flow regime with coherent structures, are valid only at the special value of Λf/δω achieved in the universal self-preserving state.

Investigations into transient wakes behind a custom airfoil undergoing pitching motion

The wake dynamics behind a sinusoidally pitching custom designed symmetrical airfoil is analyzed computationally. The effect of pitching frequency and pitching amplitude is investigated independently at a constant Reynolds number. The wake patterns are found to exhibit a transition from von Karman to reverse von Karman vortex street and from symmetric reverse von Karman vortex street to asymmetric deflected wake as established. For laminar flow under investigation, the contours of time-varying velocity fluctuations are presented. The mean profiles of velocity and the fluctuations at different locations in the wake show that the thrust wakes are generated at the expense of energy of the surrounding fluid. The temporal behavior of the flow field is analyzed using the phase plot. The time series plots of force coefficients show a periodic loading on the airfoil. Proper orthogonal decomposition (POD) analysis of drag, neutral and thrust wakes displayed the evolution of coherent structures in the flow field. The evolution of these structures portray the transitions in the wake of the airfoil. The merging of coherent structures in an asymmetric deflected wake turns out to be an efficient way of identifying the location of deflection.