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This study assessed the boundary layer flow of micropolar fluid induced by the Riga plate. The significance of the flow is conducted from a rotating frame of perspective to produce consistency for enhanced thermodynamics across the Riga plate for fluid flow. The Riga plate is a familiar operator consisting of stably deployed magnetic and electrostatic elements that produce Lorentz force in an alternative way that substantially reduces with distance from the Riga plate. The unsteady, partially differentiated, three-dimensional expressions are reduced to two impartial dimensions (τ , η). For steady-state responses, (τ = 1) , Glerikin quantization is performed with MATLAB software for finite element simulation. For the resulting set of transformed equations, approximative series remedies are attained as convergent. The fluid temperature increased as a result of the rotational parameter, the Hartmann number, and the unstable parameter. The modifications to the Hartmann number (M h) and the material parameter (Λ) decelerate the primary flow velocity in the boundary layer region, while the secondary flow velocity accelerates. The greater modified Hartmann number (M h) and material parameters have a decreasing impact on the skin friction in the primary direction and an increasing impact on the magnitude of the coefficient in the secondary direction. An excellent agreement between the ongoing and previously existing results establishes the validity of the current findings. [ABSTRACT FROM AUTHOR]

Purpose: Water jet propulsion is widely used in various military and civilian fields due to its advantages of simple structure and high propulsion efficiency. The process of mooring involves utilizing specially designed equipment to secure a ship at a designated berth. During the process of water jet propulsion, the single propeller operates within a complex and turbulent three-dimensional flow. Hence, studying the coupling between the water jet propeller and the hull is critical to comprehending the characteristics of the device and the distribution of the flow field in detail. Design/methodology/approach: Firstly, we conducted computational fluid dynamics (CFD)-based self-propulsion calculations to evaluate the interaction between the hull and the propeller. We subsequently analyzed the propeller's performance and the forces acting on the hull to understand how the presence or absence of the hull influenced the water jet propeller. Finally, we performed calculations and analysis of the cavitation characteristics of the coupling between the hull and the water jet propeller, considering different rotational speeds and water depths at the bottom of the pool. Findings: The study demonstrated that the presence of the hull boundary layer under the hull-propeller coupling condition led to reduced uniformity of propeller inlet flow and lower efficiency of the propulsion pump. However, it also increased the bias toward low-flow conditions. Additionally, increasing the impeller speed led to a gradual increase in the cavitation volume within the water jet propeller, resulting in a gradual decrease in the propeller's performance. Originality/value: This research provides the technical support required for effective design and operation of water jet propulsion systems. This paper involves studying and analyzing the performance and flow field of the coupling between the hull and propeller under mooring conditions with a specified hull model. [ABSTRACT FROM AUTHOR]

Discussed in this investigation is the three-dimensional gravity-driven flow of a thin fluid layer down an incline and over topography. A new three-dimensional integral-boundary-layer (IBL) model is proposed to describe the flow. Numerical simulations are presented and comparisons with lubrication and two-dimensional IBL models are discussed for both steady and unsteady flows. Subcritical and supercritical cases are considered, along with symmetrical and asymmetrical bottom topographies. Also, good agreement was found with experiments. [ABSTRACT FROM AUTHOR]

A comprehensive quantitative analysis of a high-loading transonic centrifugal compressor is performed under the design point. The reason for secondary flow within the impeller is examined under the approximation of boundary layer theory. Comparing the amplitudes of different terms of the Navier–Stokes (N–S) equations reveals that the acceleration that creates secondary flow within the shear layer is caused primarily by the Coriolis force difference, which has often been neglected in previous research. Migration flow on the wall surface further leads to the formation of three-dimensional secondary flow structures that occupy the entire blade passage. The vortex structure inside the impeller is visualized, and the contribution of the different vortices to the wake is clarified. The wake is ultimately composed of high-loss fluid that includes migration flow from the hub to the tip on the blade suction surface, leakage flow, and recirculating flow at the impeller outlet. By decomposing the N–S equations, it is clarified that the pressure distribution in the blade passage is mostly determined by the acceleration caused by the three-dimensional secondary flow structure, especially the momentum convection term. Therefore, the relationship between the secondary flow and the impeller loading is established. Furthermore, the natural frequency of the unsteady fluctuation of the secondary flow inside the impeller is discovered by unsteady numerical simulation, revealing the evolution of vortex structures. [ABSTRACT FROM AUTHOR]

We have considered the unsteady effects on the three-dimensional laminar boundary layer flow of a viscous and incompressible fluid over a wedge surface. The outer unsteady flows are assumed proportional to the power of the distance from the leading boundary layer. The model is described by coupled nonlinear third-order differential equations in which streamwise pressure gradient β , the three-dimensionality parameter α and unsteady parameter k predict the boundary layer flow behaviour. We use the Keller-box method, which is based on an implicit finite difference scheme, for the full nonlinear system. To complement Keller-box results, the system of equations is linearized about the boundary layer edge and the corresponding asymptotic solutions, based on eigensystem analysis, are obtained in the form of Kummer's functions. The results thus obtained from both methods are found to be in good agreement for wall-shear stress and velocity profiles. Both approaches show that the solutions to the model exist only in the range - 1 < α < ∞ . The velocity profiles show that the flow is essentially divided into the near-field and far-field. The velocity profiles involve over- and undershoots for some range of the unsteady parameter. Further, it is noticed that the absolute value of wall-shear stresses increases for increasing β for all k. The displacement thicknesses are also obtained for different parameters which shows the opposite nature of wall-shear stresses. [ABSTRACT FROM AUTHOR]

Computational blood flow studies are providing an increasingly important complement to clinical experiments in 21st century biomedical engineering. Motivated by probing deeper into this topic, a theoretical and numerical study is presented of the flow induced by an impulsive acceleration to steady state hemodynamics in a curved tube which is investigated as a boundary layer developing with time from the curved entrance to a straight tube (blood vessel). The transient processes are simulated with a finite volume method solution of the Navier–Stokes equations. The rapid growth of the boundary layer with the core flow is captured in the curved entrance, along the tube to an axisymmetric flow in the downstream. Secondary flow patterns, centrifugal pressures and total head contours are correlated with longitudinal velocity distributions across various sections. It is observed that the entrance zone is controlled by uniform inlet velocity and centrifugal forces. The high pressure drop in the onset flow is associated with strong acceleration which is comparable to generating systolic pressures. The simulations further indicate that a sustained increment in volumetric flow rate is necessary to maintain the pressure wave in the aorta. Furthermore, the velocity distributions are shown to approach Hagen–Poiseuille flow in the downstream zone. The complex hemodynamic characteristics are visualized effectively with computational simulations and the study demonstrates the excellent ability of this approach in elaborating critical flow details in aortic hemodynamics. [ABSTRACT FROM AUTHOR]

Recent attempts to employ deep learning technology for the super-resolution (SR) reconstruction of turbulence have focused chiefly on reconstructing two-dimensional (2D) slices of the three-dimensional (3D) flow fields. However, tomographic particle image velocimetry (Tomo-PIV) measurements yield 3D velocity fields. The resolutions of 3D velocity field data along three directions cannot be simultaneously enhanced using a 2D SR reconstruction model. Additionally, a 3D velocity flow field structure cannot be accurately reconstructed using a 2D model. Therefore, we present a 3D flow field SR reconstruction method (FSR-3D) based on a back-projection network. The FSR-3D model incorporates a multiscale convolutional residual block and an up-and-down projection module and establishes the mapping relationship between low-resolution features and high-resolution (HR) features through an iterative mechanism. The proposed model is validated using two experiments. First, the original velocity field is reconstructed from downsampled forced isotropic turbulence data. Second, the downsampled direct numerical simulation (DNS) data of a turbulent channel flow are used to reconstruct the HR flow field to verify the reconstruction capability of the FSR-3D model with respect to the flow field of the boundary layer. Third, Tomo-PIV measurements of the wake flow behind a circular cylinder are used to further verify the generalization ability of the model and the reconstruction effect of the measured data. The experimental results demonstrate that the proposed FSR-3D model can accurately reconstruct the HR 3D velocity field. At the same time, the properties of the reconstruction results, such as their vortex structure, kinetic energy spectrum, and Reynolds stress, are closer to the DNS results than those of the 2D model. The reconstruction results of the Tomo-PIV measurement data show that the FSR-3D model has good generalization ability. [ABSTRACT FROM AUTHOR]

By using the validated numerical method to simulate the annular shrouded stator with the inter-stage seal cavity, the stator aerodynamic performance related to the cavity leakage flow was examined. The current study also indicated the interactions between the leakage flow and the mainstream. Discussions on the developments of secondary flow movement and hub corner separation were conducted, and the performances at different incoming boundary layer shapes were assessed using both the total pressure loss coefficient and the entropy-based loss coefficient. The results indicate that the cavity leakage flow creates a new vortex close to the leading edge of blade and is crucial to the passage vortex development and the concentrated shedding vortex size. At the same time, the cavity leakage flow weakens the transverse deflection of flow near the end wall and strengthens the three-dimensional flow effect. The two loss coefficients for the shrouded stator with seal cavity change little by thickening the boundary layer. The proper boundary layer skew as well as the interacted cavity leakage flow have stronger resistance to the deflection of the passage vortex and the transverse pressure gradient. [ABSTRACT FROM AUTHOR]

Many turbulence theories in use today are based on two-dimensional equilibrium flows and have limitations when applied to three-dimensional flows. A three-dimensional law of the wall would help to improve simulation fidelity, but while several versions have been proposed, none have been widely accepted. In this study, the three-dimensional attached boundary layer flow over the windward side of the BeVERLI (Benchmark Validation Experiments for RANS/LES Investigations) Hill bump model was measured using near-wall laser Doppler velocimetry in the Virginia Tech Stability Wind Tunnel to study the mean flow and turbulence structure. These mean velocity measurements are compared with the predictions of the proposed three-dimensional (3D) law of the wall of van den Berg [A three-dimensional law of the wall for turbulent shear flows. J Fluid Mech. 1975;70(1):149–160.], which incorporates pressure gradients and inertial effects but assumes alignment of the mean flow gradient and shear-stress angles, and to the sublayer momentum equations, which are exact in the limit of wall-normal x 2 + → 0. In regions with mild stress/strain misalignment, the van den Berg model compares favourably with the experimental data up to a maximum of x 2 + ≈ 800 , and the sublayer momentum relationship compares favourably with the experimental data in the linear sublayer. [ABSTRACT FROM AUTHOR]

This work examines the non-Newtonian Cassonnanofluid's three-dimensional flow and heat and mass transmission properties over a Riga plate. The Buongiorno nanofluid model, which is included in the present model, includes thermo-migration and random movement of nanoparticles. It also took into account the Cattaneo–Christov double flux processes in the mass and heat equations. The non-Newtonian Casson fluid model and the boundary layer approximation are included in the modeling of nonlinear partial differential systems. The homotopy technique was used to analytically solve the system's governing equations. To examine the impact of dimensionless parameters on velocities, concentrations, temperatures, local Nusselt number, skin friction, and local Sherwood number, a parametric analysis was carried out. The velocity profile is augmented in this study as the size of the modified Hartmann number increases. The greater thermal radiative enhances the heat transport rate. When the mass relaxation parameter is used, the mass flux values start to decrease. [ABSTRACT FROM AUTHOR]

The unsteady three-dimensional separated flow on a wall induced by a square protrusion (approximately twice the local boundary layer thickness in width and height), and its control by means of passive suction through holes, is investigated using wind tunnel experiments at Mach 2.87. The baseline flow without any control was characterized and compared against the cases with bleed. A bow-shaped separation line on the wall with a mid-span separation length of 5.57 δ from protrusion face was traced from the oil-flow visualization. The averaged pressure distribution surveyed using static pressure ports placed on the wall has mapped plateau, high-pressure, and low-pressure regions in the separated flow, distinctive to three-dimensional interactions. Ten control configurations were tested with suction holes placed along mid-span in the different pressure regions. Significant spanwise "Mean Reduction in Separation Length" of up to 0.93 δ was observed from oil-flow visualization. A comparison of observations from various control configurations suggested that bleeding the flow from the high-pressure region could delay the separation and reduce the bubble size in general. Furthermore, time-resolved Schlieren visualizations have confirmed reduction in both "mid-span separation length" and "shock-intermittent-region" with the introduction of suction in high-pressure region. Fourier and Proper Orthogonal Decomposition analysis done on the Schlieren data has confirmed the presence of low-frequency separation-shock oscillations at Strouhal Numbers of order 10 − 2 , both with and without control. Furthermore, the amplitudes of separation-shock oscillations in the spectrum were reduced with the introduction of suction simultaneously from two holes placed in high- and low-pressure regions. [ABSTRACT FROM AUTHOR]

We perform three-dimensional particle resolved direct numerical simulations of the flow past a non-spherical obstacle by a Finite Volume cut-cell method, a sub-class of non-body-conforming methods that provides a sharp description of the boundary, which is strictly mass and momentum conservative and can be easily extended to adaptive grids. The present research work discusses the effect of corner rounding and the incidence angle for a range of Reynolds numbers for which the flow exhibits a steady-state behavior. The obstacle is placed in a large cubic domain that properly models an unbounded domain. Hierarchically refined Cartesian meshes are used where the obstacle resides at the finest level of the mesh hierarchy, thus ensuring that the resolution of the boundary layer and the wake of the obstacle is highly accurate, along with significantly reducing the number of grid cells and the computing time. Specifically, we characterize the drag force and the main features of the flow past a bluff obstacle transitioning in shape from spherical to cuboidal through a superquadric geometrical representation. A superquadric representation is suitable for our study since it preserves geometric isometry, and our analysis, thus, focusses on non-sphericity caused by the level of curvature. We investigate a range of Re from 10 to 150, which spans the flow from attached to symmetric and separated past five different obstacle shapes, with the corner radius of the curvature of r / a = 2 / ζ i = 1 , 2 / 2.5 , 2 / 4 , 2 / 8 and 0 placed at incidence angles of α = 0 ° , 15 ° , 30 ° , and 45 ° with respect to the streamwise direction. In general, our results show that the obstacle bluffness increases with α and ζi and this increase is more prominent at higher Re. Higher drag forces are a consequence of either higher viscous forces for more streamlined bodies and in less inertial regimes or higher pressure forces for more bluff bodies and in highly inertial regimes, depending on how the corners are contributing to the frontal and lateral surface areas. [ABSTRACT FROM AUTHOR]

The abyssal overturning circulation is thought to be primarily driven by small-scale turbulent mixing. Diagnosed water-mass transformations are dominated by rough topography "hotspots," where the bottom enhancement of mixing causes the diffusive buoyancy flux to diverge, driving widespread downwelling in the interior—only to be overwhelmed by an even stronger upwelling in a thin bottom boundary layer (BBL). These water-mass transformations are significantly underestimated by one-dimensional (1D) sloping boundary layer solutions, suggesting the importance of three-dimensional physics. Here, we use a hierarchy of models to generalize this 1D boundary layer approach to three-dimensional eddying flows over realistically rough topography. When applied to the Mid-Atlantic Ridge in the Brazil Basin, the idealized simulation results are roughly consistent with available observations. Integral buoyancy budgets isolate the physical processes that contribute to realistically strong BBL upwelling. The downward diffusion of buoyancy is primarily balanced by upwelling along the sloping canyon sidewalls and the surrounding abyssal hills. These flows are strengthened by the restratifying effects of submesoscale baroclinic eddies and by the blocking of along-ridge thermal wind within the canyon. Major topographic sills block along-thalweg flows from restratifying the canyon trough, resulting in the continual erosion of the trough's stratification. We propose simple modifications to the 1D boundary layer model that approximate each of these three-dimensional effects. These results provide local dynamical insights into mixing-driven abyssal overturning, but a complete theory will also require the nonlocal coupling to the basin-scale circulation. [ABSTRACT FROM AUTHOR]

This paper investigates the impact of uniform suction and magnetohydrodynamics on several nanofluids over a decelerating rotating disk. It deals with the typical Von Karman flow pumping but in the presence of water-based nanofluids comprising the fraction of the nanoparticle volume of copper, copper oxide, silver, alumina and titania instead of the regular Newtonian fluid. The influences of the base fluid's nature and the nanoparticle content in the base fluid are considered. After similarity transformations analogous to the traditional Von Karman flow, the leading equations are resolved numerically using the bvp4c MATLAB solver, which is a renowned software. The effect of various parameters on the velocity field is explained graphically. The momentum boundary layer for all types of nanofluids becomes thinner and more prominent for large values of unsteadiness parameter ( α ^ ). With small suction parameter values, the magnetic field M has an evident impact on the axial flow, while its effect is negligible with large suction parameter values. [ABSTRACT FROM AUTHOR]

The dynamically adaptive WAVETRISK-OCEAN global model is used to solve one- and two-layer shallow water ocean models of wind-driven western boundary current (WBC) turbulence. When the submesoscale is resolved, both the one-layer simulation and the barotropic mode of the two-layer simulations have an energy spectrum with a power law of −3, while the baroclinic mode has a power law of −5/3 to −2 for a Munk boundary layer. This is consistent with the theoretical prediction for the power laws of the barotropic and baroclinic (buoyancy variance) cascades in surface quasigeostrophic turbulence. The baroclinic mode has about 20% of the energy of the barotropic mode in this case. When a Munk–Stommel boundary layer dominates, both the baroclinic and barotropic modes have a power law of −3. Local energy spectrum analysis reveals that the midlatitude and equatorial jets have different energy spectra and contribute differently to the global energy spectrum. We have therefore shown that adding a single baroclinic mode qualitatively changes WBC turbulence, introducing an energy spectrum component typical of what occurs in stratified three-dimensional ocean flows. This suggests that the first baroclinic mode may be primarily responsible for the submesoscale turbulence energy spectrum of the oceans. Adding more vertical layers, and therefore more baroclinic modes, could strengthen the first baroclinic mode, producing a dual cascade spectrum (−5/3, −3) or (−3, −5/3) similar to that predicted by quasigeostrophic and surface quasigeostrophic models, respectively. Significance Statement: This research investigates how wind energy is transferred from the largest ocean scales (thousands of kilometers) to the small turbulence scales (a few kilometers or less). We do this by using an idealized model that includes the simplest representation of density stratification. Our main finding is that this simple model captures an essential feature of the energy transfer process. Future work will compare our results to those obtained using ocean models with more realistic stratifications. [ABSTRACT FROM AUTHOR]

Laminar flow control plays a vital role in reducing drag resulting enhancement in the vehicle power by a significant amount. Theoretical and experimental studies have proven that the transition from laminar to turbulent flow (causing the drag coefficient to enhance) may be delayed/prevented by the suction of the fluid from the boundary layer to the wall. The purpose of this work is to study the effects of periodic suction on the unsteady 3-dimensional fluctuating flow of a second grade incompressible fluid flowing laminarly over a horizontal porous infinite plate. The plate is subjected to a sinusoidal transverse suction velocity while the free stream velocity oscillates in time about a constant mean. The flow turns out to be 3-dimensional because of transverse direction of the periodic suction velocity. By the series expansion method, analytic expressions for transient velocity, skin friction components, and pressure are attained. Impact of nondimensional parameters evolving in the mathematical model of the problem on these physical quantities are visualized graphically and discussed analytically. The velocity component u is found to be rising with a growth in suction parameter α. The pressure is noted to be growing with a growth in Reynolds number R e. Further, due to suction at the plate and transient free stream velocity, the pressure increases near the plate and then reaches to a steady value far-off the plate. The skin friction along the main flow is noticed to be decreasing with the rise in α for 0.5 ≤ R e ≤ 1.4 , however, an exponential rise is observed for R e > 1.4. The skin friction along the cross flow is noted to be enhancing for the rising values of suction parameter and elastic parameter. The present results have excellent agreement with previous published results in the limiting sense. This study is beneficial in designing and manufacturing laminar flow control system to enhance the vehicle power requirement. [ABSTRACT FROM AUTHOR]

This numerical work is pertinent to exploring the effect of variation in momentum flux ratios, and geometrical parameters on discharge coefficient (Cd) of single and multiple row anti-screech hole arrangements which resembles as that of a conventional afterburner. The three-dimensional fluid flow domains of both single hole and multiple row hole configurations have been modelled in ANSYS ICEM CFD 20.0 tool and discretized with hexahedral elements to capture the velocity boundary layers more accurately. Flow physics between core and bypass flow through the anti-screech holes are obtained by solving the governing equations, namely the continuity, momentum in three directions and energy equations through the CFD code. Turbulence is captured by Reynolds averaged Navier–Stokes (RANS) two equation model, namely shear stress transport (SST) k-ω mode of closure. It is observed that Cd value increases in direct proportion to the diameter of the hole in case of single row hole configuration. An interesting fact of higher Cd values have been observed for trailing holes in case of multiple row hole configuration compared to the initial holes. The contours representing the static pressure and velocity along the midplane of the domains have been presented and analysed for exploring more insight into complex physics. The effect of hole diameter at lower and higher momentum flux values have been explored in this study with respect to the operating density, and it has been reported that Cd values for the initial holes are lesser than trailing holes in case of multiple row configuration. [ABSTRACT FROM AUTHOR]

The present article assesses the capability of the partially averaged Navier-Stokes (PANS) method to reproduce accurately the self-sustained shock oscillations, also known as transonic buffet, occurring on airfoils and wings at transonic regime under certain conditions of Mach number and angle of attack. The test case under analysis is an OAT15A unswept wing at Mach number M ∞ = 0.73 and Reynolds number Re c = 3 × 10 6. The three-dimensional flow is studied by accounting for the wind tunnel walls adopted in the experiments of Jacquin et al. [1] in the simulations. The computations on a large-span, confined configuration reveal a strong three-dimensionality of the flow both before and after the buffet onset. Attention is paid to the comparison with unsteady Reynolds-averaged Navier Stokes (URANS) results, to show the benefits of PANS in resolving flow unsteadiness at different flow resolutions, especially on affordable CFD grids, at limited additional cost. In this context, the role of the mesh metrics and the local turbulence level in the formulation of the model is described, as well as the relation of this latter with the spatiotemporal discretization used for the numerical simulations. The aim is to extend the use of PANS and obtain accurate predictions of flow cases involving shock-wave boundary layer interactions without expensive approaches. [ABSTRACT FROM AUTHOR]

High-performance compressor cascades are crucial components of aero-engine compressor stages and require wide operational ranges for optimal efficiency. This study examines the aerodynamic effects of implementing 0.5 mm high backward-facing steps (BFS) on the suction side of transonic compressor cascades. Numerical analyses were conducted to assess the influence of step position under specific operating conditions. It was concluded that placing the step in the middle of the cascade is most conducive to improving performance. Subsequent evaluation delineates the flow patterns and modifications to shock wave structures, especially when the step is implanted within the post-leading-edge acceleration zone. Additionally, further comparisons were made on cascade characteristics, channel separation flow dynamics, and blade load distribution at varying incidence angles when incorporating a central step. The adjusted cascade expanded the functional incidence angle from 2.6° to 3.8°, a substantial rise of 46 %. The step primarily affects the flow within a 60 % to 80 % axial chord length sector. Despite engendering a localized recirculation zone, the step notably aids in relocating the end-wall boundary layer, prompting the low-momentum fluid from the root to move spanwise, thereby reducing the channel separation vortex intensity. This leads to a noticeable decrease in blockage factor near the end-wall at the outlet across divergent incidence angles, with reductions of 5 % and 3.4 %, respectively. Overall, these findings propose that situating the BFS proximate to the mid-cascade separation zone enhances the operational capacity of the compressor cascade. The research underscores BFS as a potential asset in broadening the working range for compressor cascades within aeronautical applications. [ABSTRACT FROM AUTHOR]

Supersonic flow over three-dimensional bodies protruding out of the turbulent boundary layer was investigated by means of experiments and numerical computations. A parametric study was performed by varying the shape and dimensions of the protuberance, as well as the freestream Mach numbers (1.5, 2, 2.5, 2.89, and 3.5). Surface streak line visualization, surface pressure measurements, and time-resolved Schlieren visualization were employed along with Reynolds-averaged Navier–Stokes computations to elicit the complex flow features such as the separation line, shock pattern, and the horseshoe vortex, which greatly influence the flow dynamics in the separated region. The rise in surface pressure at mid-span due to separation (plateau pressure) was dependent only on the incoming flow parameters and independent of protuberance geometry. The two-dimensional free interaction theory, applied for normal shock-induced separation, closely predicts the mid-span plateau pressure. Although protuberances are of varying shapes and dimensions, the inviscid bow shock (obtained from Euler computations) provided generalized scales, whose effects on shock boundary layer interactions are analyzed. The radius of curvature of the inviscid shock on the wall plane at the nose, which is theoretically related to the local second derivative (along the shock) of pressure jump, was found to be a determining parameter of mid-span separation length ( L sep ). Since the spanwise distance of the sonic point on the inviscid shock was found to be strongly correlated with its nose radius of curvature, it follows that the "strong" portion of the inviscid bow shock fixes the mid-span separation location. These observations concerning mid-span plateau pressure, and the role of strong shock portion in fixing mid-span separation, suggest that the L sep shall be predicted from a modification of the scaling laws for the length of plateau pressure region in two-dimensional shock boundary layer interaction, with the inclusion of a spanwise relieving effect. A correlation is obtained relating the L sep with various incoming flow parameters and inviscid shock nose radius. The mid-span vortex core position was found to be linearly related to the L sep . The radius of curvature of the separation shock is, however, found to be influenced by the entire inviscid shock, including the "weak" portion. [ABSTRACT FROM AUTHOR]

Field inversion and machine learning are implemented in this study to describe three-dimensional (3D) separation flow around an axisymmetric hill and augment the Spalart–Allmaras (SA) model. The discrete adjoint method is used to solve the field inversion problem, and an artificial neural network is used as the machine learning model. A validation process for field inversion is proposed to adjust the hyperparameters and obtain a physically acceptable solution. The field inversion result shows that the non-equilibrium turbulence effects in the boundary layer upstream of the mean separation line and in the separating shear layer dominate the flow structure in the 3D separating flow, which agrees with prior physical knowledge. However, the effect of turbulence anisotropy on the mean flow appears to be limited. Two approaches are proposed and implemented in the machine learning stage to overcome the problem of sample imbalance while reducing the computational cost during training. The results are all satisfactory, which proves the effectiveness of the proposed approaches. [ABSTRACT FROM AUTHOR]

The three-dimensional flow of a circular synthetic jet interacting with a turbulent crossflow is investigated with unsteady Reynold-Averaged Navier–Stokes simulations. The effects of jet momentum are examined using three blowing ratios ( C B = 0.32 , 0.67 , and 1.10) at constant actuation frequency (f = 300 Hz), approach crossflow Reynolds number, R e θ = 900 , and boundary layer thickness, δ / d = 7.25 , where d is the jet diameter. The results showed that the expelled jet is accompanied by a reverse flow region on the downstream side which undergoes alternating expansion and contraction during the jet cycle. The size of the reverse flow region and depth of penetration increased with increasing jet momentum. For the low momentum jet, the expelled flow structure evolved into a hairpin vortex which significantly enhanced the wall shear stress in the spanwise direction and near the jet exit. The higher momentum jets, on the other hand, exhibited strong vortex loops around the expelled jet column that transitioned into trailing vortex pairs and a tilted vortex ring further downstream. Along the wall, both horseshoe vortex upstream of the jet exit and tertiary vortices downstream were found attached to the wall. The strong tertiary vortices promoted downwash of fluids which significantly enhanced the wall shear stress along the symmetry plane for the medium and high momentum jets. The impact of the tertiary vortices increased with the jet momentum, offering greater potential for flow separation control. [ABSTRACT FROM AUTHOR]

Flow in the three-dimensional boundary layer on a yawed plate of finite length from which a gas is injection along the normal is considered in the case of the regime of strong viscous-inviscid interaction. In order to investigate the upstream disturbance propagation, the flow functions are expanded in four-term series in the neighborhood of the leading edge of plate under the assumption that the base pressure which depends on the transverse coordinate is given on the plate trailing edge. It is shown that these expansions contain an undefined function and its first and second derivatives with respect to the transverse coordinate. The corresponding boundary-value problems are formulated and numerically solved and the eigenvalues are found. It is shown that the possibility of downstream influence becomes significantly higher with increase in both the yawing angle and the intensity of gas injection. The influence of the constitutive parameters on the flow characteristics in the three-dimensional boundary layer is investigated. [ABSTRACT FROM AUTHOR]

This work presents a data-driven approach to improve the predictive accuracy of the Reynolds stresses for flows over an axisymmetric body of revolution. Sparse experimental data of velocity and Reynolds normal stress is used to learn a nonlinear eddy viscosity model represented by neural networks with the ensemble Kalman method. It was recently proposed in Zhang et al., (2023) to enhance the predictive capability of the model by introducing a neural network-based model correction in the turbulent kinetic energy transport equation. Here this newly developed technique is applied for the first time to three-dimensional engineering flows in which the complete set of tensor functions are used. The results show that the method can learn a model with good predictive capability in both velocity and turbulence kinetic energy. The predictive improvement is achieved by suppressing turbulence production in the boundary layer and wake-shear layer with the added correction term in the turbulent kinetic energy transport equation. Moreover, the learned model can improve the prediction of Reynolds normal stresses by capturing the Reynolds stress anisotropy with nonlinear tensor bases. The necessity of combining the nonlinear tensor functions and the correction in turbulence transport equations is highlighted for accurate prediction of the Reynolds stress. • Applies the ensemble Kalman method to marine engineering flows for the first time • Learns neural network turbulence models capturing TKE and Reynolds stress anisotropy • Uses sparse data of velocity and Reynolds normal stress to improve model predictions • Highlights necessity of combining nonlinear tensor functions and TKE corrections [ABSTRACT FROM AUTHOR]

This study presents investigations on the internal flow behaviors and mechanisms of the ventilated partial cavity created by air ventilation behind a backward-facing step. Both planar particle image velocimetry experiments and numerical simulations have been conducted over a range of freestream velocities ( U ∞ ) for varying ventilation rates ( Q As ). The experimental and numerical internal flows are reasonably concordance, revealing three distinct internal flow regions: the ventilation region, the entrained internal boundary layer region, and the reversed region sandwiched in-between. The three-dimensional internal flow structures and the internal pressure gradients are numerically revealed. The internal recirculation vortex in the ventilation region is found to be induced by the stream-wise adverse pressure gradients, while the transverse airflow near the closure of the cavity is attributed to the presentence of the span-wise pressure gradient. Based on the internal velocity and air flux profiles, and the corresponding internal boundary layer thickness, the entrained and reversed air fluxes are revealed to initially increase, attain a maximum, and then decrease along the cavity. For the increment of U ∞ , both entrained and reversed air fluxes increase. For the increment of Q As , the entrained air flux rises, while the reversed air flux declines. Such feature is further verified and explained by applying the Couette flow model with adverse pressure gradients, which provides a reasonable estimation of internal flow profiles and offers a theoretical explanation on the change of internal air flux due to the variation of U ∞ and Q As . [ABSTRACT FROM AUTHOR]

The fluid flow in a precessing cylinder is investigated numerically with focus on the Ekman boundary layers in the strongly forced regime. Not surprisingly, in that regime, we find deviations from the linear theory due to significant modifications of the base flow in terms of an axisymmetric geostrophic mode whose rotation is opposite to that of the container. The transition of the bulk flow from a three-dimensional non-axisymmetric base flow to a geostrophic axisymmetric pattern is reflected in the scaling of both the sidewall boundary layers and the Ekman boundary layers on top and bottom of the cylinder. In our simulations, the Ekman layers surpass the threshold of the first instability (class A) and show an increase in the thickness together with a marked vertical flow advection inside the boundary layer in a limited range of the forcing magnitude. However, due to numerical restrictions in our simulations, which limit the range of achievable Ekman numbers, no developed boundary layer turbulence is found. An estimation by extrapolation shows that, for this purpose, Ekman numbers smaller by a factor of two have to be achieved. [ABSTRACT FROM AUTHOR]

A problem of the flow around a rectangular wing under the transonic buffet conditions in a wind tunnel is investigated numerically and experimentally. It is shown that a three-dimensional flow with a "mushroom" structure of the surface streamlines on the wing surface is formed instead of a close-to-two-dimensional flow in the case with self-excited oscillations along the streamwise coordinate (buffet). For the purpose of buffet control, an ejector is proposed, which provides boundary layer suction from the upper surface of the wing and jet ejection at the trailing edge of the wing. The spectral analysis of the experimental pressure oscillations on the upper surface of the wing shows that the proposed device effectively suppresses pressure oscillations at the buffet frequency. [ABSTRACT FROM AUTHOR]

A parallel plate differential mobility analyzer (DMA) having 100 independent current collectors is calibrated to relate the axial distances Ln between the inlet slit and the detector position to the particle mobility Z at given voltage difference V and sheath gas flow rate Q. Calibrating species are tetraheptylammonium bromide clusters (THABr) and polyethylene glycol (PEG35k, 5 nm in diameter), generated by a bipolar electrospray source, and purified in a cylindrical DMA. Gaussian fitting of the raw discrete mobility spectra in the form of ion current In versus collector position Ln, In(Ln), yield the mean value Lo of the collector position maximizing the signal for a given ion. The many (Z,V,Lo) triads obtained at given Q from many different DMA voltages and standard mobilities collapse into a single 1/(ZiVj) vs Lo curve when slight adjustments are made to the Zi. For different flow rates, Q/(ZiVj) vs. Lo curves collapse also, as long as the peaks are moderately narrow. However, for sufficiently small Q/Z, the THABr cluster peaks become broad, and the curves Q/(ZiVj) vs. Lo cease to collapse precisely. In contrast, the data for PEG show that this behavior is not a low-Q (Reynolds number) effect from the growth of the two lateral boundary layers, but is rather due to the broad and non-Gaussian peak shapes obtained at low Q or high Z. The calibration is accordingly unaffected by the Reynolds number. This simplicity was unexpected, given the three-dimensional flow in this DMA with growing lateral boundary layers. Copyright © 2020 American Association for Aerosol Research [ABSTRACT FROM AUTHOR]

Purpose: This paper addresses the three-dimensional flow of viscous nanofluid bounded by two plates. The lower plate stretches while the upper plate remains stationary. The fluid is electrically conducting in the presence of an applied magnetic field. In addition, the Hall, ion slip and Joule heating effects are retained. Governing equations for the considered physical happening are modeled under the phenomenon of boundary layer analysis. Design/methodology/approach: Both analytical and numerical solutions for the resulting nonlinear system are derived. Numerical solutions have been presented by using bvp4c and NDSolve techniques. The homotopy analysis method is utilized for the development of convergent analytical solutions. A comparative study for the presented solutions is made. An excellent agreement between analytical and numerical solutions is noticed. Findings: The dimensionless velocities, temperature and concentration are examined physically by two-dimensional plots, stream plot and tabular values. It is observed that Hall and ion slip parameters reduce the velocity field and temperature profile increases for the mounting values of the Eckert number. Originality/value: This manuscript contains the novel contents which comprise the Hall and ion slip effects for the transportation of heat and mass for the flow of viscous nanofluid. [ABSTRACT FROM AUTHOR]

In this study, the ground effect over a wing without/with a flap at low Reynolds numbers was investigated experimentally. Three-dimensional flow fields over and behind a flapped wing and single (without flap) wing in/out of ground effect at several clearances and Re numbers (0.75 × 105, 1 × 105 and 1.5 × 105) were investigated using three different experimental methods. From flow visualization with smoke technique, a succession of recorded photographs allowed to reveal the interactions of flow and ground. The changes caused by the employment of trailing-edge flap and ground effect were visualized and compared with data obtained from Constant Temperature Anemometer (CTA). According to force measurement results, the flapped wing near the ground conduced to the increase of lift and drag forces for all clearances. From smoke wire results, the separation zone above the flap appears and contributes to a higher turbulence level in the wake region. The leading-edge vortices, tip vortices behind the wing interact with the ground boundary layer and the flow coming out from the section between the trailing edge of the wing and the ground. For this, dominant vortex frequencies were detected using the spectral analysis from velocity measurements at the wake region of the wing. Further, the ground effect enabled the control of the laminar separation bubble (LSB) on the wing at low angles of attack in low Reynolds flows, the bubble gets smaller as clearance increased; however, the ground effect is not effective in controlling the LSB at high angles of attack. As the Reynolds number raised, improvement of ground effect on the wing increased. When all these results were considered, the flap should not be brought too close to the ground to benefit from the ground effect. The flapped configuration with/without ground effect presented the best performance at low angles of attack. [ABSTRACT FROM AUTHOR]

This research work presents the entropy analysis of the three-dimensional bioconvection flow of nanofluid over a linearly moving plate in the presence of magnetic effects. Nanoparticles and microorganisms have been considered in the fluid, with the magnetic field acting in the transverse direction to the plate. The viscous dissipation term is also considered in the energy equation. The partial differential equations (PDEs) of the boundary layer region of this problem are transformed with suitable non-similarity transformations into a system of nonlinear PDEs. Afterward, the local non-similarity method via the BVP4c MATLAB algorithm with a second level of truncation delivers a solution of the problem. The effect of important parameters such as magnetic field, Eckert number, thermophoresis, Prandtl number, Peclet number, Brownian motion, Schmidt number, Brinkman number, Lewis number, slip factor on temperature distribution, concentration of nanofluid, density of motile microorganisms, and entropy generation are shown in the graphical profiles. [ABSTRACT FROM AUTHOR]

Boundary layer theory is used to show that, at large Reynolds numbers, the three-dimensional Navier–Stokes equations can be rewritten in a form with diffusion velocity that was previously known for the cases of two-dimensional and axisymmetric flows. Relying on this hypothesis, a closed system of equations that is a development of a similar model for the indicated special cases is derived to describe fluid flows in the Lagrangian approach. Simultaneously, a number of mathematical issues are investigated. The existence of an integral representation for the velocity field with integrals with respect to Lagrangian coordinates is proved by analyzing the equations of motion of selected Lagrangian particles and applying the theory of ordinary differential equations with parameters. An equation describing the vorticity flux from the body surface is derived. [ABSTRACT FROM AUTHOR]

Laminar-turbulent transition and corner separation play a critical role in the aerodynamics of the compressor and are quite sensitive to the changes of flow conditions and external disturbances. However, a deep understanding of such fine flow phenomena poses a great challenge for turbulent methods and computer resources. In order to clarify the impacts of incoming flow states on the three-dimensional transitional flow in a compressor cascade, we construct a parallel Large-Eddy-Simulation (LES) methodology and apply it to a full-span compressor cascade. Both the turbulent and laminar incoming endwall boundary layers are considered at a free-stream turbulence level of 4%, which is typical in the multistage axial-flow compressor environment. The parallel performance of the MPI (Message Passing Interface) model and hybrid MPI-OpenMP (Open Multi-Processing) model is particularly analyzed at a parallel scale of 10 000 CPU (Central Processing Unit) cores. The parallel performance test shows that the efficiency of the MPI model is evidently higher than that of the hybrid MPI-OpenMP model. The LES results indicate that the incoming laminar endwall boundary layer results in a more remarkable reduction in the blade loading near the endwall and a larger total pressure loss than the turbulent one. The incoming endwall boundary layer state shows a significant impact on the evolution process of the endwall turbulence and a small impact on the corner separation and the suction-surface transition. This study demonstrates the ability of the parallel LES method to capture complex transitional-flow structures in compressor cascades and its potential application to the deeper understandings of compressor aerodynamics. [ABSTRACT FROM AUTHOR]

The consequences of regularizing the Sawyer–Eliassen equation to calculate the stream function for the axisymmetric secondary circulation of a tropical cyclone are explored. Regularization is an ad hoc procedure in which the coefficients of the equation are suitably modified to replace negative values of the discriminant by small positive values, thereby ensuring that the equation is globally elliptic. The consequences of the procedure may be understood in terms of the analogue behaviour of a stretched membrane subject to a particular force distribution. Several regularization procedures are assessed by comparing the azimuthally averaged radial flow from a three‐dimensional numerical simulation of a tropical cyclone with that from an axisymmetric balance calculation of the Sawyer–Eliassen equation, forced by diabatic and frictional terms diagnosed from the simulation. The comparison shows that the largest challenge for regularization occurs in regions of inertial instability, especially when the diagnosed forcing overlaps with such regions. In the example shown, the diagnosed balanced flow is sensitive to the particular regularization procedure and none of the procedures examined give a flow that is structurally and quantitatively close to that obtained from the numerical solution in and near the region of regularization. The flow in regions of large vertical shear that are common in the lower part of the boundary layer is less sensitive to the regularization procedure, even though such regions are ones in which there is (frictional) forcing. Nevertheless, there are comparatively large differences between the low‐level inflow in the azimuthally averaged numerical solution and the axisymmetric balance solution. These differences can be attributed to the intrinsic lack of balance in the boundary layer. This finding, together with the issues associated with regularization, is further confirmation that balance dynamics is unable to adequately capture the flow in the boundary layer, contrary to recent claims. [ABSTRACT FROM AUTHOR]

A similarity solution of a three-dimensional boundary layer is investigated. The outer flow is given by U=(−xz, −yz, z²), corresponding to an axisymmetric poloidal circulation with constant potential vorticity. This flow is an exact solution of the Navier–Stokes. A wall is introduced at y=0 along which a boundary layer develops towards x=0. We show that a similarity reduction to a system of ODEs is possible. Two distinct solutions are found, one of them through numerical path-continuation, and their stability is investigated. A second three-dimensional solution is also identified for two-dimensional outer flow. The solutions are generalized for outer flows scaling with different powers of z and similar results are found. This behaviour is related to the non-uniqueness of the Falkner–Skan flows in a three-dimensional sense, with a transverse wall-jet. [ABSTRACT FROM AUTHOR]

In this study, a novel theoretical model for three-dimensional (3D) laminar magnetohydrodynamic (MHD) flow of a non-Newtonian second-grade fluid along a plate of semi-infinite length is developed based on slightly sinusoidal transverse suction velocity. The suction velocity involves a steady distribution with a low superimposed perpendicularly varying dispersion. The strength of the uniform magnetic field is incorporated in the normal direction to the wall. The variable suction transforms the fluidic problem into a 3D flow problem because of variable suction velocity in the normal direction to the plane wall. The proposed mathematical modeling and its dynamical analysis are prescribed for the boundary layer flow keeping the magnetic effects without taking into consideration the induced magnetic field. An analytical perturbation technique is employed for the series solutions of the system of ordinary differential equations of velocity profile and pressure. Graphical illustrations are used to analyze the behavior of different proficient parameters of interest. The magnetic parameter is responsible for accelerating the main flow velocity, while controlling the cross flow velocities. [ABSTRACT FROM AUTHOR]

A brief review of the literature for the flow around two surface-mounted finite-height cylinders in a staggered configuration illustrates some of the additional complexity associated with changes in the cylinder aspect ratio (AR), the effects of downwash, the three-dimensional structure of the wake, and the presence of additional vortex structures. To date, however, few extensive systematic studies of centre-to-centre pitch ratio (P / D), incidence angle (α), aspect ratio, boundary layer thickness on the ground plane (δ / D) and Reynolds number (Re) have been reported for two finite cylinders. Some new measurements of the mean drag and lift coefficients (C D and C L) and the Strouhal numbers (St) for two staggered finite cylinders, for AR = 9, 7, 5, and 3, P / D = 1.125–4.5, and α = 0 ∘ –90°, at Re = 6.5 × 104 and δ ∕ D = 1. 4 , illustrate the moderating effect of the three-dimensional flow on the wind loading and vortex shedding behaviour compared to the flow around two infinite cylinders. These new results suggest some changes to the flow pattern boundaries and uncover some new trends in the C D , C L and St data with P / D and/or α ; there is a need for new flow visualization, flow field measurements, and numerical simulations to advance physical interpretation. • Brief literature review for flow around two surface-mounted finite-height cylinders. • New measurements of C D , C L , St for finite cylinders in a staggered configuration. • Effects of aspect ratio, pitch ratio, and incidence angle are presented. [ABSTRACT FROM AUTHOR]

The three dimensional free convection boundary layer flow near a stagnation point region is embedded in viscous nanofluid with the effect of g-jitter is studied in this paper. Copper (Cu) and aluminium oxide (Al2O3) types of water base nanofluid are chosen with the constant Prandtl number, Pr=6.2. Based on Tiwari-Das nanofluid model, the boundary layer equation used is converted into a non-dimensional form by adopting non- dimensional variables and is solved numerically by engaging an implicit finite-difference scheme known as Keller-box method. Behaviors of fluid flow such as skin friction and Nusset number are studied by the controlled parameters including oscillation frequency, amplitude of gravity modulation and nanoparticles volume fraction. The reduced skin friction and Nusset number are presented graphically and discussed for different values of principal curvatures ratio at the nodal point. The numerical results shows that, in- crement occurs in the values of Nusset number with the presence of solid nanoparticles together with the values of the skin friction. It is worth mentioning that for the plane stagnation point there is an absence of reduced skin friction along the y-direction where as for axisymmetric stagnation point, the reduced skin friction for both directions are the same. As nanoparticles volume fraction increased, the skin friction increased as well as the Nusset number. The results, indicated that skin frictions of copper are found higher than aluminium oxide. [ABSTRACT FROM AUTHOR]

Transition to turbulence is known as a factor that impacts the performance of high-lift devices, but only a few numerical results that include transition are available in the literature. We use the Langtry-Menter γ − R e θ transition model to study the influence of transition to turbulence in global aerodynamic coefficients and flow topology in three-dimensional high-lift configurations. The influence of turbulence inflow variables is also addressed. Numerical results are compared with previous results obtained with the Shear Stress Transport (SST) turbulence model, which does not account for transition effects. The numerical simulations are performed using configuration "one" from the 1st AIAA CFD High-Lift Prediction Workshop. Experimental data from the NASA Langley 14 by 22-ft Subsonic Wind Tunnel provide the global aerodynamic information for the verification of our numerical results. It is observed that the inclusion of transition to turbulence impacts the flow topology, as well as the aerodynamic coefficients. [ABSTRACT FROM AUTHOR]

Numerical simulations of three-dimensional flow of a gas mixture with chemical reactions over a flat thermally destructing material surface are presented. To create an effective numerical model, two ways of determining the heat removal are considered, and a comparison with the analytical solution obtained within the frame of the boundary layer approximation is represented. As an example, the numerical simulations of three-dimensional flow in combustion chamber of hybrid rocket are performed. The temperature maps and molar fraction of fuel within the chamber for different times are shown. • Numerical simulations of 3D flow of a gas mixture over a flat surface are presented. • An effective numerical model based on the analytical solution is created. • Two ways of determining the heat removal are considered. • The 3D numerical simulations in combustion chamber of hybrid rocket are performed. [ABSTRACT FROM AUTHOR]

We present three-dimensional simulations of sedimentation of suspended solids in inclined vessels using a Eulerian-Lagrangian model developed specifically for the study of solid-liquid mixtures. The simulation results reveal the three-dimensional vortical flow structures that follow the occurrence of flow instability and induce strong mixing between the particle-laden and clear-fluid layers. We show that the tubes exhibit significant spanwise variation due to the presence of wall boundary layers and are subsequently transformed into three-dimensional turbulent structures. The vortices and turbulence result in significant resuspension from the particle-laden layer and weaken the convective effect that enhances sedimentation. We show that for the present flow condition and particle size, particle inertia plays an important role in particle resuspension. Analysis of the release of potential energy is presented to examine the sedimentation efficiency under different flow conditions. We show that the time history for a typical case can be divided into development, convection, and deposition stages. The analysis for cases with different particle sizes shows that the faster settling associated with larger particles stabilizes the flow by reducing the interfacial shear while settling. From simulation results using various angles of inclination, we can find the optimal angle of inclination to achieve the greatest efficiency in settling enhancement. Finally, critical modeling aspects, including the importance of different forcing terms and two- or three-dimensional considerations, are discussed. Unlabelled Image • Simulations reveal 3D vertical flow structures that follow flow instability. • In unstable conditions, particle inertia plays an important role in resuspension. • Fast settling of larger particles stabilizes the flow by reducing the shear. • The optimal angle of inclination to give the greatest efficiency is obtained. [ABSTRACT FROM AUTHOR]

Abstract To study the three-dimensional flow structure of an incident shock/boundary layer interaction, a M2.7 nozzle model with a ramp was tested based on the background-oriented schlieren (BOS) technique. After comparing the results of CFD, shear-sensitive liquid crystal (SSLC), and high-speed schlieren, we conclude that BOS can capture the internal structure of the flow field and that its results are consistent with those of other methods in which the relative error can be controlled to 7%. In this research, a new type of experiment with several light paths was used with BOS technology. Three-dimensional reconstruction of incident oblique shock (primarily the width of Mach stem of the shock wave) and quantitative results of the flow field can be obtained successfully, and BOS shows great potential for the measurement of flow fields. Highlights • Length of three-dimensional Mach stem is quantitatively measured. • Background oriented schlieren technique can capture internal flow structure. • Rotation angle of optical path had a significant effect on the measurement error. [ABSTRACT FROM AUTHOR]

The elementary task is to calculate the growth rates of disturbances when the eN method in transition prediction is performed. However, there is no unified knowledge to determine the growth rates of disturbances in three-dimensional (3D) flows. In this paper, we study the relation among the wave parameters of the disturbance in boundary layers in which the imaginary parts of wave parameters are far smaller than the real parts. The generalized growth rate (GGR) in the direction of group velocity is introduced, and the conservation relation of GGR is strictly deduced in theory. This conservation relation manifests that the GGR only depends on the real parts of wave parameters instead of the imaginary parts. Numerical validations for GGR conservation are also provided in the cases of first/second modes and crossflow modes. The application of GGR to the eN method in 3D flows is discussed, and the puzzle of determining growth rates in 3D flows is clarified. A convenient method is also proposed to calculate growth rates of disturbances in 3D flows. Good agreement between this convenient method and existing methods is found except the condition that the angle between the group velocity direction and the x-direction is close to 90° which can be easily avoided in practical application. [ABSTRACT FROM AUTHOR]