Planar Velocity Fields Research Articles

Planar Lorentz invariant velocities with a wave equation application

In this paper we determine the functional form of those planar velocity fields for which the associated system of two ordinary differential equations are automatically invariant under a Lorentz transformation. For planar motion we determine first order partial differential equations for the velocity components u(x,y,t) and w(x,y,t) in the x− and y−directions respectively and their general solutions in terms of two arbitrary functions. These partial differential equations and the associated partial differential relations connecting energy and momentum are fully compatible with the Lorentz invariant energy–momentum relations and appear not to have been given previously in the literature. For a particular special relativistic model, one example is given involving similarity solutions of the wave equation. An interesting special case gives rise to a family of particle paths which are characterized by a single arbitrary function, and for which the magnitude of their velocities is the speed of light. This is indicative of the abundant possibilities existing in the “fast-lane”.

Three dimensional flows beneath a thin layer of 2D turbulence induced by Faraday waves

Faraday waves occur on a fluid being subject to vertical shaking. Although it is well known that form and shape of the wave pattern depend on driving amplitude and frequency, only recent studies discovered the existence of a horizontal velocity field at the surface, called Faraday flow. This flow exhibits attributes of two-dimensional turbulence and is replicated in this study. Despite the increasing attention towards the inverse energy flux in the Faraday flow and other not strictly two-dimensional (2D) systems, little is known about the velocity fields developing beneath the fluid surface. In this study, planar velocity fields are measured by means of particle image velocimetry with high spatio-temporal resolution on the water surface and at different depths below it. A sudden drop in velocity and turbulent kinetic energy is observed at half a Faraday wavelength below the surface revealing that the surface flow is the main source of turbulent fluid motion. The flow structures below the surface comprise much larger spatial scales than those on the surface leading to very long-tailed temporal and spatial velocity (auto-) correlation functions. The three-dimensionality of the flow is estimated by the compressibility, which increases strongly with depth while the divergence changes its appearance from intermittent and single events to a large scale pattern resembling 2D cut-planes of convection rolls. Our findings demonstrate that the overall fluid flow beneath the surface is highly three-dimensional and that an inverse cascade and aspects of a confined 2D turbulence can coexist with a three-dimensional flow.Graphic abstract

Frameworks for investigation of nonlinear dynamics: Experimental study of the turbulent jet

Analysis methods that have been developed in the field of nonlinear dynamics have provided valuable insights into the physics of turbulent flows, although their application to open flows is less well explored. The nonlinear dynamics of a turbulent jet with a low-to-moderate Reynolds number is investigated by using the single-trajectory framework and ensemble framework. We have used Lyapunov exponents to calculate the spectra of scaling indices of the attractor. First, we evaluated the frameworks on two theoretical models, one with a stationary attractor (Lorenz-63) and the other with time-varying characteristics (Lorenz-84). Theoretical studies showed that in dynamical systems with a stable attractor, both frameworks estimated the same largest Lyapunov exponent. The ensemble framework enables us to resolve the unsteady characteristics of a time-varying strange attractor. Second, we applied both frameworks to time-resolved planar velocity fields in a turbulent jet at local Reynolds numbers (Reδ) of 3000 and 5000. Time-resolved particle image velocimetry was utilized to measure streamwise and transverse velocity components. Results support the presence of a low-dimensional attractor in the reconstructed phase space with a chaotic characteristic. Despite considerable changes in the dynamics for the higher Reynolds number case, the system’s fractal dimension did not change significantly. We have used Lagrangian Coherent Structures (LCSs) to study the relationship between changes in the Lyapunov exponent with flow topological features. Results suggest that holes in the stable LCSs provide a path for the entrainment of the coflow, which is shown to be one of the main contributors to high Lyapunov exponents.

Streamwise evolution of drag reduced turbulent boundary layer with polymer solutions

The effect of local polymer solution injection on the evolution of a flat-plate turbulent boundary layer has been investigated experimentally. Polyethylene oxide (PEO) solutions were injected through an inclined slot. The influence of polymer injection on boundary layer development downstream of the slot is assessed at three different concentrations (100 ppm, 500 ppm, and 1000 ppm) using planar velocity field and concentration measurements. A local drag reduction (DR) of up to 60% was obtained in the vicinity of the slot. A systematic change observed in the inverse of the von Kármán constant (k = 1/κ) with an increase in DR is used to define the sub-regimes of the high DR regime, and a linear relation between k and DR is shown to persist over a wide range of Reynolds numbers. The analysis of combined velocity and concentration measurements provides added insight into the associated changes in the boundary layer characteristics and the underlying flow physics.

Dual-plane stereo-astigmatism: a novel method to determine the full velocity gradient tensor in planar domain

A novel approach, which utilizes astigmatism-based depth codification and discrete dual-plane illumination, is developed to provide information on the full velocity-gradient tensor in pair of planar domains by means of stereoscopic particle image velocimetry (stereo-PIV). The technique is accordingly referred to as dual-plane stereo-astigmatism (DPSA). In contrast to conventional stereo-PIV, DPSA provides the determination of the desired out-of-plane gradient. The principle of the DPSA approach relies on the joint recording of two, consecutive, planar measurement domains and the subsequent allocation of particles based on their particle image shapes. For the identification of particles, a correlation-based particle identification approach (CPI), which addresses non-overlapping particles, is introduced. With regard to dense particle fields, a method for the iterative particle reconstruction (IPR) is adapted for DPSA. For displacement analysis, a modified PIV evaluation strategy is introduced, which utilizes derived information of particle locations. The DPSA approach is tested experimentally and synthetically by applying it to spray-induced flow measurement and synthetic images generated from DNS data of a turbulent boundary flow. Separate planar velocity fields are obtained by means of PIV and PTV evaluation. A performance analysis is carried out to assess the influence of noise, particle density and particle image size on the procedure of particle identification and allocation. The CPI technique provides adequate results for low to medium particle densities and further features a high robustness regarding particle identification with respect to background noise, optical aberrations and non-uniform particle images. The adapted IPR method, on the other hand, shows viable results for particle densities of up to 0.1 ppp.

Flow pattern analysis in a rotating scraped surface plate heat exchanger

The flow pattern inside a scraped surface plate heat exchanger used for ice slurry production is studied both experimentally and numerically. Isothermal planar velocity fields are reported in the range 2.2 · 104⩽Rerot⩽1.7 · 105 using a phase averaged PIV technique. Unsteady RANS modelling of the flow, using the rotating frame methodology is used to obtain a time-averaged description of the flow structure, which was successfully validated against the PIV data. The flow structure is characterized by the presence of a roll-up vortex generated by the scraper blade, which is the main factor for both the radial and vertical displacements of the flow. Increasing the Rerot number leads to a more uniform flow, reducing the presence of stagnant zones. However, the main pattern structure does not change significantly when increasing the scraping velocity.

Stokes-layer formation under absence of moving parts—A novel oscillatory plasma actuator design for turbulent drag reduction

A novel plasma actuator concept is proposed to mimic the effect of spanwise wall oscillations without mechanically moving parts, where four groups of electrodes and three independently operated high-voltage power supplies maintain a pulsatile dielectric barrier discharge (DBD) array. Time-resolved planar velocity fields are obtained with high-speed particle image velocimetry (PIV) in proximity of the discharge zones for quiescent ambient conditions. Resulting flow topologies and wall-normal velocity profiles indicate the Stokes-layer-like flow formation, which is elevated above the wall due to the no-slip condition. The underlying body forces are derived from the PIV data to provide further insight into cause-effect relations between pulsatile discharge and oscillatory flow. The momentum transfer domain is found to be only interrupted with the width of the exposed electrode, which is an important step toward homogeneous virtual wall oscillations. A comparison with earlier studies by Gatti et al. [“Experimental assessment of spanwise-oscillating dielectric electroactive surfaces for turbulent drag reduction in an air channel flow,” Exp. Fluids 56, 110 (2015)] leads to the hypothesis that DBD-based turbulent drag reduction might be a competing alternative to conventional active and passive shear-layer formation strategies, where the adjustability of both oscillation frequency and velocity amplitude might cover a wide range of Reynolds numbers.

Digital particle image velocimetry study on parameter influence on the behavior of impinging synthetic jets

This study conducts an experimental investigation of the effects of the stroke length and the Reynolds number on synthetic jet vortex rings impinging onto a solid wall. A two-dimensional time-resolved particle image velocimetry system is employed to measure the planar velocity field across the jet centerline. All the experiments have been performed at a constant orifice-to-wall distance (H0/D0 = 8.0), whereas three different stroke lengths (L0/D0 = 1.8, 3.6, and 7.2) and two Reynolds numbers (Resj = 111 and 333) are selected for comparison. Time-mean characteristics are first presented to reveal the basic flow feature of an impinging synthetic jet, and phase-averaged vorticity fields provide the information of the vortex evolutions during the interaction between synthetic jet vortex rings and the wall, i.e., the unsteady behavior of the impinging synthetic jet. With the help of the quantitative particle image velocimetry data, instantaneous wall pressure and wall shear stress distributions are evaluated simultaneously to link the dynamic vortical events in the vicinity of the wall with the wall fields. Finally, a proper orthogonal decomposition analysis is applied to extract dominant coherent structures in the flow field of the impinging synthetic jet and to highlight the combined effects of the stroke length and the Reynolds number. It is found that the stroke length effect on an impinging synthetic jet is mainly reflected in the vortex ring coherence before impacting the wall, whereas the effect of the Reynolds number on the flow behavior for a small stroke length is more significant than that for a large one. In particular, the previously impinged vortex ring is beneficial in suppressing the flow separation as well as the development of the secondary vortex ring. This validates the substantial difference between the impingement of consecutive vortex rings of a synthetic jet and that of a single vortex ring.

Estimating velocity in Gasoline Direct Injection sprays using statistical pattern imaging velocimetry

Statistical pattern imaging velocimetry (SPIV) is a new technique for the estimation of the planar velocity field from the high-speed videos. SPIV utilizes an ensemble of either backlit or side lit videos to obtain full planar velocities in sprays and flames. Unlike conventional particle imaging velocimetry, statistical pattern imaging velocimetry does not require well-resolved images of particles within turbulent flows. Instead, the technique relies of patterns formed by coherent structures in the flow. Therefore, SPIV is well suited for the estimating planar velocities in sprays and turbulent flames, both of which have well-defined patterns embedded in the flow videos. The implementation of the SPIV technique is relatively quite straightforward since high-speed videos can be readily obtained either in a laboratory or production floor setting. The biggest challenge for the SPIV techniques is that the procedure is computationally expensive even with an ordinary mega-pixel camera. To improve the computation speed, a successive partitioning scheme was employed. In addition, to improve spatial resolution to subpixel dimensions, a weighted central averaging scheme was used. With these two enhancements, the SPIV method was used to obtain planar radial and axial velocities in a spray emanating from a GDI injector. Sprays from GDI injectors are very dense (with obscuration levels close to the injector being greater than 99%), and velocity measurements are difficult. However, further away from the nozzle, a Phase Doppler Anemometer can be used to obtain velocity measurements. The velocities obtained using these two methods showed reasonable agreement.

On the spatial organization of hairpin packets in a turbulent boundary layer at low-to-moderate Reynolds number

The present study is devoted to characterizing the coherent organization of vortical structures, which can be fitted into the paradigm of the hairpin-packet model, in the streamwise–wall-normal plane of a canonical turbulent boundary layer at $Re_{\unicode[STIX]{x1D70F}}=377{-}1093$. Proper orthogonal decomposition (POD) of the planar velocity fields measured via two-dimensional particle image velocimetry, together with a spatio-temporal coherence analysis, shows that the first four leading-order POD modes share both geometric similarity and dynamic coherence and jointly depict the downstream convection of the large-scale Q2/Q4 events, which can be regarded as the low-order imprints of the hairpin packets. A simple low-order indicator is then proposed to extract the inclined interfaces of the hairpin packets, based on which a two-point conditional correlation analysis forms a statistical picture of the spatial organization of multiple prograde vortices aligned along the interface within one packet. A saturation of the self-similar growth of the streamwise gap between two neighbouring vortices is seen. This implies a detachment of the hairpin packets from the inner layer. Both the detachment height and the saturated streamwise spacing are found to scale as $Re_{\unicode[STIX]{x1D70F}}^{1/2}$.

Ultrasonic measurements of the bulk flow field in foams.

The flow field of moving foams is relevant for basic research and for the optimization of industrial processes such as froth flotation. However, no adequate measurement technique exists for the local velocity distribution inside the foam bulk. We have investigated the ultrasound Doppler velocimetry (UDV), providing the first two-dimensional, non-invasive velocity measurement technique with an adequate spatial (10mm) and temporal resolution (2.5Hz) that is applicable to medium scale foam flows. The measurement object is dry aqueous foam flowing upward in a rectangular channel. An array of ultrasound transducers is mounted within the channel, sending pulses along the main flow axis, and receiving echoes from the foam bulk. This results in a temporally and spatially resolved, planar velocity field up to a measurement depth of 200mm, which is approximately one order of magnitude larger than those of optical techniques. A comparison with optical reference measurements of the surface velocity of the foam allows to validate the UDV results. At 2.5Hz frame rate an uncertainty below 15 percent and an axial spatial resolution better than 10mm is found. Therefore, UDV is a suitable tool for monitoring of industrial processes as well as the scientific investigation of three-dimensional foam flows on medium scales.

Assessment of various turbulence models (ELES, SAS, URANS and RANS) for predicting the aerodynamics of freight train container wagons

This study presents an assessment of the capabilities of various turbulence modelling approaches —ELES, SAS, URANS and RANS—to predict the aerodynamic flow around a double-stacked freight wagon, both in isolation and within a train. The numerical predictions are compared with experimental measurements at the same Reynolds number to determine the accuracy of each model. Specifically, aerodynamic drag, front and rear surface pressures, planar velocity fields and skin friction lines are validated against the wind tunnel results. In particular, predictions from the ELES and SAS models show good agreement with the wind tunnel data, both qualitatively and quantitatively. Indeed, ELES predicts both the range and distribution of the rear-face surface pressure very closely, indicating that the separated flow is also likely to be well predicted. Both SAS and ELES predict the pressure drag of the multi-wagon configuration to within 2% of the experimental value. However, the steady RANS model predicts the trends in pressure drag in line with the experiments as the front and rear gaps are varied, even though individual drag predictions are considerably worse. Overall, the set of results establishes the benefits and deficiencies of using particular turbulence models to capture different aspects of freight train aerodynamics.

Flow topology of a container train wagon subjected to varying local loading configurations

How the positioning and length of a container placed on an arbitrary train wagon in an otherwise fully loaded train affects the local aerodynamics, and consequently the contribution to drag, is examined here. Results from scale-model wind-tunnel tests undertaken at a Reynolds number of 0.3 × 106 for a combination of 49 upstream and downstream gap spacings (Gf,Gr) are presented. Surface flow topology, pressure profiles and planar velocity fields are measured. Gf dominated the drag variations, with Gr only causing a secondary effect. The greatest drag reduction potential is found between gaps size of 1.77W and 3.23W, where W represents the wagon width. Over the range of Gf and Gr investigated, a number of distinct physical mechanisms were observed. These affect the separation size and the nature of boundary layer enveloping the wagon, which have a direct impact on the entrainment and shedding frequency of the wake.

Tumbling of a Brownian particle in an extensional flow

The phenomenon of tumbling of microscopic objects is commonly associated with shear flows. We address the question of whether tumbling can also occur in stretching-dominated flows. To answer this, we study the dynamics of a semi-flexible trumbbell in a planar extensional velocity field. We show that the trumbbell undergoes a random tumbling-through-folding motion. The probability distribution of long tumbling times is exponential with a time scale exponentially increasing with the Weissenberg number.

Estimation and optimization of loss-of-pair uncertainties based on PIV correlation functions

The uncertainty quantification of particle image velocimetry (PIV) measurements is still an open problem, and to date, no consensus exists about the best suited approach. When the spatial resolution is not appropriate, the largest uncertainties are usually caused by flow gradients. But also the amount of loss-of-pairs due to out-of-plane flow motion and insufficient light-sheet overlap causes strong uncertainties in real experiments. In this paper, we show how the amount of loss-of-pairs can be quantified using the volume of the correlation function normalized by the volume of the autocorrelation function. The findings are an important step toward a reliable uncertainty estimation of instantaneous planar velocity fields computed from PIV and stereo-PIV data. Another important consequence of the analysis is that the results allow for the optimization of PIV and stereo-PIV setups in view of minimizing the total error. In particular, it is shown that the best results (concerning the relative uncertainty) can be achieved if the out-of-plane loss-of-correlation is smaller than one (F o ). The only exception is the case where the out-of-plane motion is exactly zero. The predictions are confirmed experimentally in the last part of the paper.

Planar near-nozzle velocity measurements during a single high-pressure fuel injection

In order to reduce the fuel consumption and exhaust emissions of modern Diesel engines, the high-pressure fuel injections have to be optimized. This requires continuous, time-resolved measurements of the fuel velocity distribution during multiple complete injection cycles, which can provide a deeper understanding of the injection process. However, fuel velocity measurements at high-pressure injection nozzles are a challenging task due to the high velocities of up to 300 m/s, the short injection durations in the $$\upmu\text{s}$$ range and the high fuel droplet density especially near the nozzle exit. In order to solve these challenges, a fast imaging Doppler global velocimeter with laser frequency modulation (2D-FM-DGV) incorporating a high-speed camera is presented. As a result, continuous planar velocity field measurements are performed with a measurement rate of 200 kHz in the near-nozzle region of a high-pressure Diesel injection. The injection system is operated under atmospheric surrounding conditions with injection pressures up to 1400 bar thereby reaching fuel velocities up to 380 m/s. The measurements over multiple entire injection cycles resolved the spatio-temporal fluctuations of the fuel velocity, which occur especially for low injection pressures. Furthermore, a sudden setback of the velocity at the beginning of the injection is identified for various injection pressures. In conclusion, the fast measurement system enables the investigation of the complete temporal behavior of single injection cycles or a series of it. Since this eliminates the necessity of phase-locked measurements, the proposed measurement approach provides new insights for the analysis of high-pressure injections regarding unsteady phenomena.

Comparison of vortex identification criteria for planar velocity fields in wall turbulence

This study derives and compares vortex identification methods for detecting vortices in planar velocity fields. Two-dimensional (2D) forms of the commonly used Δ, Q, λci, and λ2 criteria are derived in detail based on the 2D counterpart of the full velocity gradient tensor. These four criteria are compared mathematically and experimentally in the case of using zero thresholds. The results show that while all methods are capable of extracting strong vortices, their efficiencies in identifying weaker vortices are not necessarily the same. The Δ and λci criteria impose the least requirements on the identified structures and extract the most number of vortices, and the λ2 criterion is the most restrictive one and tends to discard the weakest vortices. However, non-zero thresholds are generally necessary for applying vortex identification criteria in real turbulent flows, and normalizing the vortex indicators with their root mean squares is needed to enable the selection of universal threshold for vortices residing at different wall-normal positions in wall turbulence. The introduction of threshold makes the four vortex identification criteria equally efficacious, and equivalent thresholds are proposed to facilitate quantitative comparison of results based on different criteria in wall turbulence.

Some nonlinear stability results for a magnetic induction equation with Hall, ion-slip and shear effects: necessary and sufficient condition

We study the nonlinear stability, with respect to axisymmetric perturbations, of the solution magnetic field to the induction equation for a weakly ionized gas, subjected to an assigned planar velocity field which, in a special case, keeps it in proximity of a gravitational center. In other cases, this velocity field can generate hyperbolic trajectories. Whatever, assuming the presence of Hall and ion-slip effects, we will try to determine how the geometric and kinematic characteristics of the gas stream affect the stability/instability of the magnetic field. Then, we obtain a necessary and sufficient stability condition and estimate the radius of attraction.

Real-time planar flow velocity measurements using an optical flow algorithm implemented on GPU

This paper presents a high-speed implementation of an optical flow algorithm which computes in real-time planar velocity fields in an experimental flow. Real-time computations of the flow velocity field allow the experimentalist to have instantaneous access to quantitative features of the flow. This can be very useful in many situations: fast evaluation of the performances and characteristics of a new setup, design optimization, easier and faster parametric studies, etc. It can also be used as a visual sensor for an input in closed-loop flow control experiments where fast estimation of the state of the flow is needed. The algorithm is implemented on a graphics processor unit. The accuracy of the computation is demonstrated. Computation speed and scalability of the processing are highlighted along with guidelines for further improvements. The system architecture is flexible, scalable and can be adapted on the fly in order to process higher resolutions or achieve higher precision. The setup is applied on a backward-facing step flow in a hydrodynamic channel. For validation purposes, classical particle image velocimetry (PIV) is used to compare with instantaneous optical flow measurements. The important flow characteristics like the dynamics of the recirculation bubble, computed in real time for the first time, are well recovered. The accuracy of real-time optical flow measurements is comparable to off-line PIV computations.

Evaluation of Lift Formulas Applied to Low-Reynolds-Number Unsteady Flows

Different lift decompositions into the elemental terms are compared based on direct numerical simulations of a flapping flat plate and a flapping rectangular wing at low-Reynolds-number flows. The simple lift formula is given as a useful approximate form that has the vortex force and local acceleration terms. The accuracy of the simple lift formula in lift estimation is quantitatively evaluated in comparison with the general force formulas based on the fully resolved two- and three-dimensional unsteady velocity fields and the planar velocity fields at several spanwise locations in simulated particle-image-velocimetry measurements. In addition, the mathematical connections between the different force formulas are discussed.