Plane Wake Research Articles

Key Factors in Determining the Magnitude of Vorticity in Turbulent Plane Wakes

We examine the effects of local turbulence Reynolds number Rλ and inflow conditions on the magnitude of vorticity in plane turbulent wakes. Measurements of the spanwise component (ω3) of the fluctuating vorticity vector ω = ω1i + ω2j + ω3k (here the subscripts 1, 2 and 3 denote the streamwise, lateral and spanwise directions, respectively) are made in turbulent wakes of a screen and a circular cylinder. Lateral distributions of ω3* (normalized) in general depend on both Rλ and inflow conditions. In the developing region, as the downstream distance x1 increases, ω3* increases significantly in the screen wake but decreases slightly in the cylinder wake. Far downstream in the self-preserving region, ω3* increases linearly with Rλ while it no longer varies with x1 and depends weakly on the influence of inflow conditions. Our analysis suggests that these findings from the measurement of ω3* should apply for the magnitude of ω.

Influence of confinement on temporal stability of plane jets and wakes

It has been recently shown [M. P. Juniper, “The effect of confinement on the stability of two-dimensional shear flows,” J. Fluid Mech. 565, 171 (2006)] that the absolute instability of a two-dimensional jet/wake was favored when confined in the normal direction by two flat plates. In this paper, we confirm the destabilizing influence of confinement using the temporal stability analysis of an idealized Kelvin–Helmholtz flow in a plane channel with surface tension that serves as a model of confined jet/wake flow. We show that there is a region in the wavenumber-confinement parameter plane where the confinement increases the instability of the flow. The joined influence of density ratio and confinement is also studied.

Shallow turbulent wakes behind bed-mounted cylinders in open channels

Flow patterns in the shallow turbulent near-wakes behind bed-mounted cylinders are investigated on smooth and rough beds. A wall wake analysis revealed that the flows in the region away from the bed are similar and well described by the plane wake equation. Wall wake similarity was also observed for turbulent kinetic energy and primary Reynolds stress for moderate to deeply submerged cylinders.Wake analyses in the horizontal plane revealed that similarity of mean velocity profiles across the flow exist in the near-wake region at all elevations for slightly submerged and surface piercing cylinders; and at all elevations below the object height close to moderate and deeply submerged cylinders. For slightly submerged and surface piercing cylinders similarity was observed in the near-wake region between the non-dimensional velocity profiles across the wake and the plane wake equation. However, the similarity of the velocity profiles improves if a modified transverse length scale is used for shallow near-wake flows.

Evaluation of linear and nonlinear convection schemes on multidimensional non‐orthogonal grids with applications to KVLCC2 tanker

AbstractAlmost all evaluations of convection schemes reported in the literature are conducted using simple problems on uniform orthogonal grids; thus, having limited contribution when solving industrial computational fluid dynamics (CFD), where the grids are usually non‐orthogonal with distortions. Herein, several convection schemes are assessed in uniform and distorted non‐orthogonal grids with emphasis on industrial applications. Linear and nonlinear (TVD) convection schemes are assessed on analytical benchmarks in both uniform and distorted grids. To evaluate the performance of the schemes, four error metrics are used: dissipation, phase and L1 errors, and the schemes' effective order of accuracy. Qualitative and quantitative deterioration of these error metrics as a function of the grid distortion metrics are investigated, and rigorous verifications are performed. Recommendations for effective use of the convection schemes based on the range of grid aspect ratio (AR), expansion ratio (ER) and skewness (Q) are included. A ship hydrodynamics case is studied, involving a Reynolds averaged Navier–Stokes simulation of a bare‐hull KVLCC2 tanker using linear and nonlinear convection schemes coupled with isotropic and anisotropic Reynolds‐stress (ARS) turbulence models using CFDShip‐Iowa v4. Predictions of local velocities and turbulent quantities from the midships to the nominal wake plane are compared with experimental fluid dynamics (EFD), and rigorous verification and validation analyses for integral forces and moments are performed for 0° and 12° drift angles. Best predictions are observed when coupling a second‐order TVD scheme with the anisotropic turbulence model. Further improvements are observed in terms of prediction of the vortical structures for 30° drift when using TVD2S‐ARS coupled with DES. Copyright © 2009 John Wiley & Sons, Ltd.

LES/FDF simulation of particle dispersion in a gas-particle two phase plane wake flow

A filtered density function (FDF) transport equation was derived for the fluid velocity seen by the particles in gas-particle two-phase flow. An LES/FDF simulation of a two-phase plane wake flow was carried out. The simulation results were compared with both the experimental photograph and the simulation results without using the FDF model, and proved that the LES/FDF model can clearly improve the spatial dispersion of the particle phase.

The effect of surface tension on the stability of unconfined and confined planar jets and wakes

In this theoretical study, a linear spatio-temporal analysis is performed on unconfined and confined inviscid jet/wake flows with surface tension in order to determine convective/absolute instability criteria. There is a single mode that is due to surface tension and many modes that are due to the jet/wake column. In the unconfined case, the full impulse response is considered in the entire outer flow. On the one hand, the surface tension mode propagates slowly in the cross-stream direction but dominates at the front and back of the wavepacket. On the other hand, the jet/wake column modes propagate more quickly in the cross-stream direction and therefore define the boundaries of the central region of the wavepacket. The flow is particularly unstable when these modes interact. For unconfined flows, it is found that at low and intermediate surface tensions the flow can be more absolutely unstable than that without surface tension but at high surface tensions the flow is stabilized. The effect of confinement has previously been studied but not with the inclusion of surface tension. Confinement and surface tension combined cause the transition from convective to absolute instability to occur even with significant coflow. This effect is examined over an infinite domain of density ratios and confinement.

Reynolds number effects on wavelet components of self-preserving turbulent structures

The effect of the Reynolds number on the wavelet-decomposed turbulent structures in a self-preserving plane wake has been investigated for Re_{theta} (based on the free-stream velocity and momentum thickness, theta, of the wake) =1350 and 4600. Measurements were made at x/theta (x is the streamwise distance downstream of the cylinder) =580 for the circular cylinder using two orthogonal arrays of 16 X wires, eight in the (x,y) plane, and eight in the (x,z) plane. A wavelet multiresolution technique is used to analyze the measured hot-wire data. This technique decomposes turbulence structures into a number of components based on their central frequencies, which are linked with the turbulence scales. Sectional streamlines and vorticity contours at the same central frequency, i.e., the comparable scales of turbulent structures, are examined and compared between the two Reynolds numbers. Discernible differences are observed in the turbulent structures of relatively large to intermediate scales. The differences are further quantified in terms of contributions from the turbulent structures of different scales to the Reynolds stresses, vorticity variance, and probability density functions of the fluctuating velocities. The large-scale structure contributes most to the Reynolds stresses and this contribution drops for the higher Re_{theta}.

Numerical study on heavy rigid particle motion in a plane wake flow by spectral element method

AbstractExperimental particle dispersion patterns in a plane wake flow at a high Reynolds number have been predicted numerically by discrete vortex method (Phys. Fluids A 1992; 4:2244–2251; Int. J. Multiphase Flow 2000; 26:1583–1607). To address the particle motion at a moderate Reynolds number, spectral element method is employed to provide an instantaneous wake flow field for particle dynamics equations, which are solved to make a detail classification of the patterns in relation to the Stokes and Froude numbers. It is found that particle motion features only depend on the Stokes number at a high Froude number and depend on both numbers at a low Froude number. A ratio of the Stokes number to squared Froude number is introduced and threshold values of this parameter are evaluated that delineate the different regions of particle behavior. The parameter describes approximately the gravitational settling velocity divided by the characteristic velocity of wake flow. In order to present effects of particle density but preserve rigid sphere, hollow sphere particle dynamics in the plane wake flow is investigated. The evolution of hollow particle motion patterns for the increase of equivalent particle density corresponds to that of solid particle motion patterns for the decrease of particle size. Although the thresholds change a little, the parameter can still make a good qualitative classification of particle motion patterns as the inner diameter changes. Copyright © 2008 John Wiley & Sons, Ltd.

Two-point similarity in temporally evolving plane wakes

The governing equations for the two-point correlations of the turbulent fluctuating velocity in the temporally evolving wake were analysed to determine whether they could have equilibrium similarity solutions. It was found that these equations could have such solutions for a finite-Reynolds-number wake, where the two-point velocity correlations could be written as a product of a time-dependent scale and a function dependent only on similarity variables. It is therefore possible to collapse the two-point measures of all the scales of motions in the temporally evolving wake using a single set of similarity variables. As in an earlier single-point analysis, it was found that the governing equations for the equilibrium similarity solutions could not be reduced to a form that was independent of a growth-rate dependent parameter. Thus, there is not a single ‘universal’ solution that describes the state of the large-scale structures, so that the large-scale structures in the far field may depend on how the flow is generated.The predictions of the similarity analysis were compared to the data from two direct numerical simulations of the temporally evolving wakes examined previously. It was found that the two-point velocity spectra of these temporally evolving wakes collapsed reasonably well over the entire range of scales when they were scaled in the manner deduced from the equilibrium similarity analysis. Thus, actual flows do seem to evolve in a manner consistent with the equilibrium similarity solutions.

Effects of initial conditions on wavelet-decomposed structures in a turbulent far-wake

The effect of initial conditions on a self-preserving plane wake has been investigated for two wake generators, i.e. a circular cylinder and a screen of 50% solidity. Measurements were made at x / θ ( x is the streamwise distance downstream of the cylinder and θ is the momentum thickness of the wake) = 580 for the circular cylinder and 830 for the screen using 8 X-wires in the plane of mean shear. A vector wavelet multi-resolution technique is applied for analysis of the hot-wire data. This technique decomposes turbulent structures into a number of components based on their central frequencies, which are linked with the turbulence scales. Sectional streamlines, vorticity contours and probability density functions of the fluctuating velocities at the same central frequency, i.e. the comparable scales of turbulent structures, are examined and compared between the two generators. Discernible differences are observed in the turbulent structures of large-down to intermediate-scales. The differences are further quantified in terms of contributions from the turbulent structures of different scales to the Reynolds stresses and vorticity variance. The vorticity behavior of the screen wake is different from that of the circular cylinder. The difference exhibits an extraordinary similarity to that in the near wake.

Integration of infinite chain of transport equations for cumulants

AbstractThe method of compatible differential constraints is applied to integrate an infinite chain of transport equations for cumulants. The dynamics of a stratified momentumless turbulent planar wake is considered as an example. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Scaling Properties and Wave Interactions in Confined Supersonic Turbulent Bluff-Body Wakes

Results are presented from an experimental study that compares the large-scale structure, mean velocities, and scaling properties of confined, supersonic, planar, turbulent, bluff-body wakes at M∞ ≈ 2 and 3 with corresponding results from incompressible wakes. The large-scale structure of incompressible planar turbulent wakes is recovered at downstream locations x where the local relative Mach number Mr(x) has decreased to about 0.4. Interactions between these large-scale structures and reflected expansion waves from the near-field recompression region lead to substantial changes in the scaling properties of the flow. Wave interactions with the flow create local self-induced forcing as the large-scale structures pass through the reflected expansion waves. In the M∞ ≈ 2 wake, a subsonic upstream path exists from the first wave interaction point that allows upstream propagation of the wave-induced forcing, and consequently the measured wake scaling constants match values reported from forced incompressible wakes. In the M∞ ≈ 3 wake, no subsonic upstream path exists from the first interaction point, and as a consequence the measured scaling constants match values reported from unforced incompressible wakes. Downstream of each wave interaction point, the self-induced forcing leads to an increase in the wake growth rate and in the entrainment of free-stream fluid. The wake subsequently detrains this fluid and returns to its original growth rate. This local forcing mechanism and its effects on the wake scaling properties repeat at each downstream location where the reflected expansion waves intersect the wake.

Laboratory observations of inner surf and swash-zone hydrodynamics on a steep slope

Laboratory data of free surface elevation and fluid velocities (cross-shore and vertical velocities) were obtained using a laser-Doppler velocimeter (LDV) for the case of a periodic wave plunging over an impermeable, steep (1:10) slope with a fixed bottom roughness (nominal diameter=2.2 mm). The measurements were conducted over 14 cross-shore locations from the outer surf zone to the swash zone with approximately 10 vertical points at each location. The vertical resolution included measurements in the bottom boundary layer and some points above the trough level. Various hydrodynamic quantities, including ensemble-averaged turbulent kinetic energy (TKE) (〈 k〉) and turbulence intensity (〈 u′ 2〉 and 〈 w′ 2〉), were estimated to better understand inner surf and swash zone hydrodynamics induced by strongly plunging waves. In the outer region of the bottom boundary layer, the ratio of the time-averaged horizontal turbulence intensity ( u ′ 2 ¯ ) to the time-averaged vertical turbulence intensity ( w ′ 2 ¯ ) agreed well with the case of plane wake turbulence ( u ′ 2 ¯ / w ′ 2 ¯ = 1.31 ) . The average ratio near the bottom was larger ( u ′ 2 ¯ / w ′ 2 ¯ = 5.91 ) which is closer to flows typified by the inner region of a boundary layer ( u ′ 2 ¯ / w ′ 2 ¯ = 6.20 ) . In the surf zone, the ensemble-averaged horizontal fluid velocity (〈 u〉) near the bottom leads in phase compared to 〈 u〉 in the upper layer, and 〈 u〉 at the bore front exceeds the theoretical wave celerity. At the impinge point, a strong return flow occurred ( u min = - 50.98 cm / s ) which was greater in magnitude than the onshore directed velocity ( u max = 40.09 cm / s ) . In the surf zone, 〈 k〉 was largest just below trough level with a forward shift in phase of the peak intensity. The turbulence energy was mostly dissipated after the passage of the crest at this location. 〈 u〉 near the bottom was leading in phase with a sharp vertical gradient in 〈 u〉 indicating that boundary layer processes may have been important. In the swash zone, the vertical gradient of 〈 u〉 was relatively small compared to the vertical gradient of 〈 u〉 in the surf zone and may be due to the effect that the strong downrush had on turbulent mixing. Finally, the time-averaged estimate of TKE ( 〈 k 〉 ¯ ) was vertically uniform over the inner surf and swash zone.

Acoustic Wave Generation in a Compressible Wake

The compressible Navier-Stokes equations are numerically solved to study the acoustic generation mechanism associated with the evolution of the structure in a compressible plane wake undergoing transition to turbulence. High-order compact finite difference schemes are used for spatial derivatives and a 4th-order Runge-Kutta scheme is employed for time advancement. Navier-Stokes characteristic boundary conditions are used in the vertical direction and periodic boundary conditions in the streamwise and spanwise directions. Three-dimensional structures of the wake are studied by means of temporally evolving plane wakes forced with a combination of unstable modes obtained from linear stability theory using a mapped Fourier method for the viscous compressible equations. Forcing with a pair of oblique subharmonic unstable modes yields streamwise/vertical counter- rotating vortices in the saddle region. As the streamwise/vertical vortices evolve outside, their self-induction causes inclined braidlike structures to form in the wake, which are similar to observations in the experimental supersonic flat wake transition. The oblique subharmonic unstable modes also cause the spanwise variations in the core that lead to roller distortion. Acoustic waves of plane wakes are generated when two-dimensional rollup structures appear and rotate in such wakes. Near-field sound wave pressure decreases downstream due to the three-dimensional evolution of the wake.

Experimental Investigation of Unstably Stratified Buoyant Wakes

A water channel has been used as a statistically steady experiment to investigate the development of a buoyant plane wake. Parallel streams of hot and cold water are initially separated by a splitter plate and are oriented to create an unstable stratification. At the end of the splitter plate, the two streams are allowed to mix and a buoyancy-driven mixing layer develops. The continuous, unstable stratification inside the developing mixing layer provides the necessary environment to study the buoyant wake. Downstream a cylinder was placed at the center of the mixing layer. As a result the dynamic flows of the plane wake and buoyancy-driven mixing layer interact. Particle image velocimetry and a high-resolution thermocouple system have been used to measure the response of the plane wake to buoyancy driven turbulence. Velocity and density measurements are used as a basis from which we describe the transition, and return to equilibrium, of the buoyancy-driven mixing layer. Visual observation of the wake does not show the usual vortex street associated with a cylinder wake, but the effect of the wake is apparent in the measured vertical velocity fluctuations. An expected peak in velocity fluctuations in the wake is found, however the decay of vertical velocity fluctuations occurs at a reduced rate due to vertical momentum transport into the wake region from buoyancy-driven turbulence. Therefore for wakes where buoyancy is driving the motion, a remarkably fast recovery of a buoyancy-driven Rayleigh-Taylor mixing in the wake region is found.

Compatible differential constraints to an infinite chain of transport equations for cumulants

We study an infinite chain of transport equations for cumulants which appears in modeling the dynamics of a momentumless turbulent planar wake. The method of compatible differential constraints for formulating its integrability properties is applied. The compatibility conditions obtained make it possible to realize a reduction of the original infinite chain of transport equations and to present an algorithm for calculating cumulants of arbitrary order.

Unsteady RANS simulation of the ship forward speed diffraction problem

The forward speed diffraction problem for a surface ship is analyzed numerically, using a RANS approach with a single-phase level set method to compute the free surface and a blended k– ε/ k– ω model for the turbulent viscosity. Simulations were run for a DTMB 5512 model under head incident waves at two speeds and two wavelengths with the same wave amplitude ( a = 0.006 L, with L the ship length). The medium speed case ( Fr = 0.28) with long wavelength incident waves ( λ = 1.5 L) behaves linearly and has been extensively compared against available experimental data for resistance and heave forces and pitching moment, unsteady free surface elevations, and unsteady velocity fields at the nominal wake plane ( x/ L = 0.935). Quantitative verification and validation was performed for this case by running three grids and three time steps with refinement ratio of 2 and the flow field analyzed in detail. The behavior of the boundary layer is analyzed to explain the origin of large first harmonic amplitudes on the axial velocity observed both experimentally and numerically. The high speed case ( Fr = 0.41) with short wavelength incident waves ( λ = 0.5 L) exhibits non-linear behavior on the forces and moment with a strong second harmonic component and an unsteady breaking bow wave. The second harmonic has been reproduced by the CFD computations and the breaking wave predicted. Analysis of the flow indicates that the breaking wave could be responsible for the non-linear behavior on the forces and moment.

A Methodology for Simulating Compressible Turbulent Flows

A flow simulation Methodology (FSM) is presented for computing the time-dependent behavior of complex compressible turbulent flows. The development of FSM was initiated in close collaboration with C. Speziale (then at Boston University). The objective of FSM is to provide the proper amount of turbulence modeling for the unresolved scales while directly computing the largest scales. The strategy is implemented by using state-of-the-art turbulence models (as developed for Reynolds averaged Navier-Stokes (RANS)) and scaling of the model terms with a “contribution function.” The contribution function is dependent on the local and instantaneous “physical” resolution in the computation. This physical resolution is determined during the actual simulation by comparing the size of the smallest relevant scales to the local grid size used in the computation. The contribution function is designed such that it provides no modeling if the computation is locally well resolved so that it approaches direct numerical simulations (DNS) in the fine-grid limit and such that it provides modeling of all scales in the coarse-grid limit and thus approaches a RANS calculation. In between these resolution limits, the contribution function adjusts the necessary modeling for the unresolved scales while the larger (resolved) scales are computed as in large eddy simulation (LES). However, FSM is distinctly different from LES in that it allows for a consistent transition between RANS, LES, and DNS within the same simulation depending on the local flow behavior and “physical” resolution. As a consequence, FSM should require considerably fewer grid points for a given calculation than would be necessary for a LES. This conjecture is substantiated by employing FSM to calculate the flow over a backward-facing step and a plane wake behind a bluff body, both at low Mach number, and supersonic axisymmetric wakes. These examples were chosen such that they expose, on the one hand, the inherent difficulties of simulating (physically) complex flows, and, on the other hand, demonstrate the potential of the FSM approach for simulations of turbulent compressible flows for complex geometries.

Turbulent plane wakes subjected to successive strains

Six direct numerical simulations of turbulent time-evolving strained plane wakes have been examined to investigate the response of a wake to successive irrotational plane strains of opposite sign. The orientation of the applied strain field has been selected so that the flow is the time-developing analogue of a spatially developing wake evolving in the presence of either a or an streamwise pressure gradient. The magnitude of the applied strain rate a is constant in time t until the total strain e(sup at) reaches about four. At this point, a new simulation is begun with the sign of the applied strain being reversed (the original simulation is continued as well). When the total strain is reduced back to its original value of one, yet another simulation is begun with the sign of the strain being reversed again back to its original sign. This process is done for both initially favourable and initially adverse strains, providing simulations for each of these strain types from three different initial conditions. The evolution of the wake mean velocity deficit and width is found to be very similar for all the strained cases, with both measures rapidly achieving exponential growth at the rate associated with the cross-stream expansive strain e(sup at). In the favourably strained cases, the wake widths approach a constant and the velocity deficits ultimately decay rapidly as e(sup -2at). Although all three of these cases do exhibit the same asymptotic exponential behaviour, the time required to achieve this is longer for the cases that have been previously strained (by at approx. equals 1). These simulations confirm the generality of the conclusions drawn in Rogers (2002) regarding the response of plane wakes to strain. The evolution of strained wakes is not consistent with the predictions of classical self-similar analysis; a more general equilibrium similarity solution is required to describe the results. At least for the cases considered here, the wake Reynolds number and the ratio of the turbulent kinetic energy to the square of the wake mean velocity deficit are determined nearly entirely by the total strain. For these measures the order in which the strains are applied does not matter and the changes brought about by the strain are nearly reversible. The wake mean velocity deficit and width, on the other hand, differ by about a factor of three when the total strain returns to one, depending on whether the wake was first favourably or adversely strained. The strain history is important for predicting the evolution of these quantities.

Sailing Boat Performance Prediction

The inviscid, steady, hydro-aerodynamic flow around a sailing boat fully equipped with sails and crew was investigated using two different solvers. The hydrodynamic problem was solved based on Green functions for the linearised wave resistance problem, using a simple plane wake model. The aerodynamic problem is fully non-linear with the sails wakes discretised by means of vortex carrying particles. The boat speed was obtained as a result of the balance between hydro- and aerodynamic forces. Only heel and drift angles were free. The balance problem was solved for two cases: crew position or heel angle prescribed.