Three-dimensional flow structure in highly buoyant jet by scanning stereo PIV combined with POD analysis

The flow characteristics and the structure of highly buoyant jet of low density fluid issuing into a stagnant surrounding of high density fluid is studied by scanning stereo PIV combined with proper orthogonal decomposition (POD) analysis. The experiment is carried out at Froude number of 0.3 and Reynolds number of 200, which satisfies the inflow condition due to the unstable density gradient near the nozzle exit. An increase in the maximum mean velocity occurs and the vertical velocity fluctuation is highly amplified near the nozzle exit, which suggests the influence of inflow due to the unstable density gradient. The POD analysis indicates that the vertical velocity fluctuation is the major source of fluctuating energy contributing to the development of the highly buoyant jet. The examination of the POD modes show that the longitudinal structure of the vertical velocity fluctuation is generated along the jet axis having the opposite sign of velocity fluctuation on both sides of the jet axis. The vertical scale of the POD mode decreases with increasing the mode number and results in the frequent appearance of cross-flow across the buoyant jet. The reconstruction flow from the POD modes indicates that the vortex structure is caused by the highly sheared layer between the upward and downward velocity and the inflow is induced by the vortex structure. The magnitude of the vortex structure seems to be weakened with an increase in the distance from the nozzle and the buoyant jet approaches to an asymptotic state in the further downstream.     

Numerical analysis of characteristic features of shallow and deep cavity in supersonic flow

Flow past open cavities are numerically simulated at a Mach number of 1.5, and Reynolds number, based on initial momentum thickness at the front lip of cavity, of 3333 for variable depths (D) with constant length (L). The dominant frequency of oscillation shows a sudden jump when there is a transition from shallow (L/D > 1) to deep cavity (L/D < 1). The vorticity thickness displays two different growth rates along the length of cavity: (1) initial lower spreading rate, followed by (2) higher spreading rate. The lower spreading rate of shear layer is dictated by the type of cavity (either shallow or deep), while the higher spreading rate is directly related to the amplitude of oscillations. Proper orthogonal decomposition (POD) is implemented to visualise the coherent structures based on their energy content. The first two POD spatial structures in the shallow cavity represent vortex shedding, while in the deep cavity, they comprise vortex pairing interactions as in mixing layer. The higher POD modes contain coherent structures at mixed frequencies. The behaviour of coherent structures associated with a temporal frequency is further investigated using dynamic mode decomposition (DMD). The higher DMD modes confirm the dominance of mixing layer behaviour in the deep cavity.     

Experimental Investigation of Three-Dimensional Vortex Structures Downstream of Vortex Generators Over a Backward-Facing Step

An experimental investigation of vortex generators has been carried out in turbulent backward-facing step (BFS) flow. The Reynolds number, based on a freestream velocity U₀ = 10 m/s and a step height h = 30 mm, was Re_h = 2.0 × 10⁴. Low-profile wedge-type vortex generators (VGs) were implemented on the horizontal surface upstream of the step. High-resolution planar particle image velocimetry (2D-2C PIV) was used to measure the separated shear layer, recirculation region and reattachment area downstream of the BFS in a single field of view. Besides, time-resolved tomographic particle image velocimetry (TR-Tomo-PIV) was also employed to measure the flow flied of the turbulent shear layer downstream of the BFS within a three-dimensional volume of 50 × 50 × 10 mm³ at a sampling frequency of 1 kHz. The flow control result shows that time-averaged reattachment length downstream of the BFS is reduced by 29.1 % due to the application of the VGs. Meanwhile, the Reynolds shear stress downstream of the VGs is considerably increased. Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) have been applied to the 3D velocity vector fields to analyze the complex vortex structures in the spatial and temporal approaches, respectively. A coherent bandwidth of Strouhal number 0.3 < St_h < 0.6 is found in the VG-induced vortices, and moreover, Λ-shaped three-dimensional vortex structures at St_h = 0.37 are revealed in the energy and dynamic approaches complementarily.     

Global stability analysis of axisymmetric boundary layer over a circular cylinder

This paper presents a linear global stability analysis of the incompressible axisymmetric boundary layer on a circular cylinder. The base flow is parallel to the axis of the cylinder at inflow boundary. The pressure gradient is zero in the streamwise direction. The base flow velocity profile is fully non-parallel and non-similar in nature. The boundary layer grows continuously in the spatial directions. Linearized Navier–Stokes (LNS) equations are derived for the disturbance flow quantities in the cylindrical polar coordinates. The LNS equations along with homogeneous boundary conditions forms a generalized eigenvalues problem. Since the base flow is axisymmetric, the disturbances are periodic in azimuthal direction. Chebyshev spectral collocation method and Arnoldi’s iterative algorithm is used for the solution of the general eigenvalues problem. The global temporal modes are computed for the range of Reynolds numbers and different azimuthal wave numbers. The largest imaginary part of the computed eigenmodes is negative, and hence, the flow is temporally stable. The spatial structure of the eigenmodes shows that the disturbance amplitudes grow in size and magnitude while they are moving towards downstream. The global modes of axisymmetric boundary layer are more stable than that of 2D flat-plate boundary layer at low Reynolds number. However, at higher Reynolds number they approach 2D flat-plate boundary layer. Thus, the damping effect of transverse curvature is significant at low Reynolds number. The wave-like nature of the disturbance amplitudes is found in the streamwise direction for the least stable eigenmodes.     

Analysis of time-resolved PIV measurements of a confined turbulent jet using POD and Koopman modes

We present a comparative analysis of proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) computed from experimental data of a turbulent, quasi 2-D, confined jet with co-flow (Re?=?11,500, co-flow ratio inner-to-outer flow ≈2:1). The experimental data come from high-speed 2-D particle image velocimetry. The flow is fully turbulent, and it contains geometry-dependent large-scale coherent structures; thus, it provides an interesting benchmark case for the comparison between POD and DMD. In this work, we address issues related to snapshot selections (1), convergence (2) and the physical interpretation (3) of both POD and DMD modes. We found that the convergence of POD modes follows the criteria of statistical convergence of the autocovariance matrix. For the computation of DMD modes, we suggest a methodology based on two criteria: the analysis of the residuals to optimize the sampling parameters of the snapshots, and a time-shifting procedure that allows us to identify the spurious modes and retain the modes that consistently appear in the spectrum. These modes are found to be the ones with nearly null growth rate. We then present the selected modes, and we discuss the way POD and DMD rank them. POD analysis reveals that the most energetic spatial structures are related to the large-scale oscillation of the inner jet (flapping); from the temporal analysis emerges that these modes are associated with a low-frequency peak at St?=?0.02. At this frequency, DMD identifies a similar mode, where oblique structures from the walls appear together with the flapping mode. The second most energetic group of modes identified is associated with shear-layer oscillations, and to a recirculation zone near the inner jet. Temporal analysis of these modes shows that the flapping of the inner jet might be sustained by the recirculation. In the DMD, the shear-layer modes are separated from the recirculation modes. These have large amplitudes in the DMD. In conclusion, the DMD modes with eigenvalues on the unit circle are found to be similar to the most energetic POD modes, although differences appear due to the fact that DMD isolates structures associated with one frequency only.     

Calculation of Linear Stability of a Stratified Gas–Liquid Flow in an Inclined Plane Channel

Linear stability of liquid and gas counterflows in an inclined channel is considered. The full Navier–Stokes equations for both phases are linearized, and the dynamics of periodic disturbances is determined by means of solving a spectral problem in wide ranges of Reynolds numbers for the liquid and vapor velocity. Two unstable modes are found in the examined ranges: surface mode (corresponding to the Kapitsa waves at small velocities of the gas) and shear mode in the gas phase. The wave length and the phase velocity of neutral disturbances of both modes are calculated as functions of the Reynolds number for the liquid. It is shown that these dependences for the surface mode are significantly affected by the gas velocity.     

Experimental Investigation of Separated Shear Flow under Subharmonic Perturbations over a Backward-Facing Step

Subharmonic-perturbed shear flow downstream of a two-dimensional backward-facing step was experimentally investigated. The Reynolds number was Re_h = 2.0 ×10⁴, based on free-stream velocity and step height. Planar 2D-2C particle image velocimetry was employed to measure the separating and reattaching flow in the horizontal-vertical plane in the center position. The subharmonic perturbations were generated by an oscillating flap which was implemented over the step edge and driven by periodic Ampere force. The subharmonic frequency was 55 Hz as the half of the fundamental frequency of the turbulent shear layer. As a result of the subharmonic perturbations, the size of recirculation region behind the backward-facing step is reduced and the time-averaged reattachment length is 31.0% shorter than that of the natural flow. The evolution of vortices, including vortex roll-up, growth and breakdown process, is analyzed by using phase-averaging, cross-correlation function and proper orthogonal decomposition. It is found that Reynolds shear stress is considerably increased in which the vortices roll up and then break down further downstream. In particular, rapid growth of vortices based on the “step mode” occurs at approximate half of the recirculation region, caused by in interaction between the shear layer and the recirculation region. Furthermore, the coherent structures, which are represented by a phase-correlated POD mode pair, are reconstructed in phases in order to show regular patterns of the subharmonic-perturbed coherent structures.     

Flow field investigation in rotating rib-roughened channel by means of particle image velocimetry

The turbulent velocity field over the rib-roughened wall of an orthogonally rotating channel is investigated by means of two-dimensional particle image velocimetry (PIV). The flow direction is outward, with a bulk Reynolds number of 1.5 × 10⁴ and a rotation number ranging from 0.3 to 0.38. The measurements are obtained along the wall-normal/streamwise plane at mid-span. The PIV system rotates with the channel, allowing to measure directly the relative flow velocity with high spatial resolution. Coriolis forces affect the stability of the boundary layer and free shear layer. Due to the different levels of shear layer entrainment, the reattachment point is moved downstream (upstream) under stabilizing (destabilizing) rotation, with respect to the stationary case. Further increase in rotation number pushes further the reattachment point in stabilizing rotation, but does not change the recirculation length in destabilizing rotation. Turbulent activity is inhibited along the leading wall, both in the boundary layer and in the separated shear layer; the opposite is true along the trailing wall. Coriolis forces affect indirectly the production of turbulent kinetic energy via the Reynolds shear stresses and the mean shear. Two-point correlation is used to characterize the coherent motion of the separated shear layer. Destabilizing rotation is found to promote large-scale coherent motions and accordingly leads to larger integral length scales; on the other hand, the spanwise vortices created in the separating shear layer downstream of the rib are less organized and tend to be disrupted by the three-dimensional turbulence promoted by the rotation. The latter observation is consistent with the distributions of span-wise vortices detected in instantaneous flow realizations.     

Meandering jets in shallow rectangular reservoirs: POD analysis and identification of coherent structures

The effect of the shallowness on meandering jets in a shallow rectangular reservoir is investigated. Four meandering flows were investigated in an experimental shallow rectangular reservoir. Their boundary conditions were chosen to cover a large range of friction numbers (defined with the sudden expansion width). Due to the unsteady characteristics of the flows, a proper orthogonal decomposition (POD) of the fluctuating part of the surface velocity fields measured using Large-Scale Particle Image Velocity was used for discriminating the flow structures responsible for the meandering of the jet. Less than 1 % of the calculated POD modes significantly contribute to the meandering of the jet, and two types of instability are in competition in such a flow configuration. The sinuous mode is the dominant mode in the flow, and it induces the meandering of the flow, while the varicose mode is a source of local mixing and weakly participates to the flow. The fluctuating velocity fields were then reconstructed using the POD modes corresponding to 80 % of the total mean fluctuating kinetic energy, and the coherent structures were identified using the residual vorticity, their centres being localised using a topology algorithm. The trajectories of the structures centres emphasise that at high friction number the coherent structures are small and laterally paired in the near, middle and far fields of the jet, while with decreasing friction number, the structures merge into large horizontal vortices in the far field of the jet, their trajectories showing more variability in space and time. The analysis of the stability regime finally reveals that the sinuous mode is convectively unstable and may become absolutely unstable at the end of the reservoir when the friction number is small.     

Turbulence Filter and POD Analysis for Velocity Fields in Lifted CH4-Air Diffusion Flames

The turbulence filter and the proper orthogonal decomposition (POD)methods are applied to PIV measurements of lifted CH₄-air diffusionflames at three different Reynolds numbers. Properties such as vorticityand strain rate distributions of the decomposed fields are analyzed inorder to assess the physical behavior. The turbulence filter can userelatively less data than the POD method while still providingsignificant insight about the flow field. The energy activation ratesshow the first modes (mean flow) account for 85% of the total energy.Reconstruction of the POD modes reveals that the combination of modesyields a complex fluctuating behavior. The averaged Reynolds stress ofthe mean flow removed parts shows interesting correlation with the meanflow vorticity and strain rate distributions. Selected turbulentproperties are calculated and discussed.     

PIV measurement of separated flow over a blunt plate with different chord-to-thickness ratios

The influence of the chord-to-thickness ratio (c/t) on the spatial characteristics of the separated shear layer over a blunt plate and the leading-edge vortices embedded in the separated shear layer was studied extensively using planar particle image velocimetry (PIV). Three systems corresponding to different shedding modes were chosen for the comparative study: c/t=3, 6 and 9. The Reynolds number based on the plate's thickness (t) was Re_t=1×10³. A gigapixel CCD camera was used to acquire images with a spatial resolution of 0.06t×0.06t in the measurement range of 9.5t×4.5t. Distributions of statistical quantities, such as the streamline pattern, streamwise velocity fluctuation intensity, shear stress and reverse flow intermittency, showed that the separated shear layer in the system with c/t=3 did not reattach to the plate's surface, while the near‐wake behind the trailing edge was highly unstable because the energetic leading-edge vortices were shed into the wake. The separated shear layer of the system with c/t=6 periodically reattached to the plate's surface, which resulted in intensified fluctuations of the near wake behind the trailing edge. In the longest system (c/t=9), the separated shear layer always reattached to the plate's surface far upstream from the trailing edge, which did not induce large fluctuations of the near wake. Furthermore, the proper orthogonal decomposition (POD) was extensively employed to filter the original velocity fields spatially to identify the large-scale vortices immersed in the separated shear layer easily. The distribution of the v-v correlation coefficients of the spatially filtered flow fields reflected the organized large-scale vortices in the three systems. The number of alternations of the positive and negative correlation coefficients across the flow field were determined to be 1, 2 and 3 for the systems with c/t=3, 6 and 9, respectively; this is in agreement with the shedding mode of each system. The distribution of the swirling strength of the separated shear layer accurately determined the positions and structures of the large-scale vortices formed above the plate surface.     

An improved algorithm for the shallow water equations model reduction: Dynamic Mode Decomposition vs POD

We propose an improved framework for dynamic mode decomposition (DMD) of 2‐D flows for problems originating from meteorology when a large time step acts like a filter in obtaining the significant Koopman modes, therefore, the classic DMD method is not effective. This study is motivated by the need to further clarify the connection between Koopman modes and proper orthogonal decomposition (POD) dynamic modes. We apply DMD and POD to derive reduced order models (ROM) of the shallow water equations. Key innovations for the DMD‐based ROM introduced in this paper are the use of the Moore–Penrose pseudoinverse in the DMD computation that produced an accurate result and a novel selection method for the DMD modes and associated amplitudes and Ritz values. A quantitative comparison of the spatial modes computed from the two decompositions is performed, and a rigorous error analysis for the ROM models obtained is presented. Copyright © 2015 John Wiley & Sons, Ltd.     

Transient growth mechanisms of low Reynolds number flow over a low-pressure turbine blade

A direct transient growth analysis for three-dimensional perturbations to flow past a periodic array of T-106/300 low-pressure turbine fan blades is presented. The methodology is based on a singular value decomposition of the flow evolution operator, linearised about a steady or periodic base flow. This analysis yields the optimal growth modes. Previous work on global mode stability analysis of this flow geometry showed the flow is asymptotically stable, indicating a non-modal explanation of transition may be more appropriate. The present work extends previous investigations into the transient growth around a steady base flow, to higher Reynolds numbers and periodic base flows. It is found that the notable transient growth of the optimal modes suggests a plausible route to transition in comparison to modal growth for this configuration. The spatial extent and localisation of the optimal modes is examined and possible physical triggering mechanisms are discussed. It is found that for longer times and longer spanwise wavelengths, a separation in the shear layer excites the wake mode. For shorter times and spanwise wavelengths, smaller growth associated with excitation of the near wake are observed.     

Particle image velocimetry measurements of a backward-facing step flow

Particle image velocimetry (PIV) measurements were carried out on a backward-facing step flow at a Reynolds number of Re_h=U_Xh/9=4,660 (based on step height and freestream velocity). In-plane velocity, out-of-plane vorticity, Reynolds stress and turbulent kinetic energy production measurements in the x-y and x-z planes of the flow are presented. Proper orthogonal decomposition was performed on both the fluctuating velocity and vorticity fields of the x-y plane PIV data using the method of snapshots. Low-order representations of the instantaneous velocity fields were reconstructed using the velocity modes. These reconstructions provided insight into the contribution that the various length scales make to the spatial distribution of mean and turbulent flow quantities such as Reynolds stress and turbulent kinetic energy production. Large scales are found to contribute to the Reynolds stresses and turbulent kinetic energy production downstream of reattachment, while small scales contribute to the intense Reynolds stresses in the vicinity of reattachment.     

POD investigation of the unsteady turbulent boundary layer developing over porous moving flexible fishing net structure

Particle image velocimetry (PIV) measurements are made to investigate the boundary layer developing over a modeled bottom trawl. The random motion of the fishing net structure as well as the flexibility and the porosity of this structure means that it is not enable to access the main characteristics of such a flow, using classical post-processing mathematical tools. An innovative post-treatment tool based on proper orthogonal decomposition (POD) is then developed to extract the mean velocity flow field from each available PIV instantaneous unsteady velocity field. In order to do so, the whole available velocity database is used to compute POD eigenfunctions and the first POD modes are identified as representing the mean flow field. It is then possible to deduce the mean boundary layer flow field for each position of the fishing net structure during PIV measurements. It is then observed that the mean flow field strongly depends on multiple parameters such as surface curvature, structure porosity, random motion of the structure. Streamwise evolution of classical thicknesses of boundary layer flow are also analyzed. The present work also provides benchmark PIV data of the unsteady flow developing on fishing net porous structures, which helps the progress in unsteady numerical codes for this investigation.     

Three-dimensional magnetic resonance velocimetry measurements of turbulence quantities in complex flow

A magnetic resonance velocimetry (MRV) experimental technique based on magnetic resonance imaging and capable of measuring the turbulent Reynolds stresses in a 3D flow domain is described. Results are presented in backward facing step flow in a square channel with a Reynolds number of 48,000 based on step height and freestream velocity at the step. MRV results are compared to particle image velocimetry (PIV) measurements in the centerplane containing the streamwise and cross-stream axes. MRV and PIV mean velocity measurements show excellent agreement. MRV measurements for Reynolds normal stresses compare to within ±20% of the PIV results while results for the turbulent shear are less accurate.     

Particle-image velocimetry measurements of flow over interacting barchan dunes

Barchan dunes are crescentic planform-shaped dunes that are present in many natural environments, and may occur either in isolation or in groups. This study uses high-resolution particle-image velocimetry (PIV) experiments using fixed-bed models to examine the effects of barchan dune interaction upon the flow field structure. The barchan dune models were created from an idealized contour map, the shape and dimensions of which were based upon previous empirical studies of dune morphology. The experimental setup comprised two, co-axially aligned, barchan dune models that were spaced at different distances apart. In this paper, two volumetric ratios (V _r, upstream dune: downstream dune) of 1.0 and 0.175 were examined. Models were placed in a boundary-layer wind tunnel and flow quantification was achieved via PIV measurements of the mean and turbulent flow field in the streamwise–wall-normal plane, along the centerline of the barchan(s), at an average flow Reynolds number of 59,000. The presence of an upstream barchan dune induces a “sheltering effect” on the flow. Flow on the stoss side of the downstream dune is controlled by the developing internal boundary layer from the upstream dune, as well as by the turbulent flow structures shed from the free shear layer of the upstream dune leeside. At both volumetric ratios, enhanced turbulence is present over the downstream barchan dune leeside, which is proposed to be caused by the interaction of shear layers from the upstream and downstream dunes. Both the size and magnitude of the shear layer formed in the leeside of the upstream dune control this interaction, together with the proximity of this shear layer to the stoss side of the downstream dune. Proper orthogonal decomposition (POD) analysis shows that the distribution of turbulent kinetic energy is shifted to higher modes (i.e., smaller spatial scales) over interacting barchan dunes, which also reflects the role of the leeside free shear layer in dominating the flow field by generation, or redistribution, of TKE to smaller scales.     

Simulations of the unsteady separated flow past a normal flat plate

Well-resolved two-dimensional numerical simulations of the unsteady separated flow past a normal flat plate at low Reynolds numbers have been performed using a fractional step procedure with high-order spatial discretization. A fifth-order upwind-biased scheme is used for the convective terms and the diffusive terms are represented by a fourth-order central difference scheme. The pressure Poisson equation is solved using a direct method based on eigenvalue decomposition of the coefficient matrix. A systematic study of the flow has been conducted with high temporal and spatial resolutions for a series of Reynolds numbers. The interactions of the vortices shed form the shear layers in the near-and far-wake regions are studied. For Reynolds numbers less than 250 the vortices are observed to convect parallel to the freestream. However, at higher Reynolds numbers (500 and 1000), complex interactions including vortex pairing, tearing and deformations are seen to occur in the far-wake region. Values of the drag coefficient and the wake closure length are presented and compared with previous experimental and numerical studies.