Onset Of Three-dimensionality Research Articles

Influence of thermal buoyancy on the wake dynamics of a heated square cylinder

Direct numerical simulation of the three-dimensional (3-D) wake transition of a heated square cylinder subjected to horizontal cross-flow is performed in the presence of buoyancy. In order to capture the effects of large-scale heating, a non-Oberbeck–Boussinesq model is utilized, which includes the governing equations for compressible gas flow. All computations are performed at low free stream Mach number $M=0.1$ using air (free stream Prandtl number, $Pr=0.71$ ) as the working fluid. The 3-D instability modes A and B, which correspond to free stream Reynolds numbers of 180 and 250, are observed with longer and shorter spanwise wavelengths, respectively, and the onset of three-dimensionality is triggered at a Reynolds number of 173. In the presence of buoyancy, baroclinic vorticity production in the near-wake plays an important role for streamwise vorticity generation. The chaotic wake of the Mode-A instability bifurcates into periodic and quasiperiodic wakes at various heating levels, expressed by the overheat ratio, $\varepsilon =(T_w-T_\infty )/T_\infty$ , where $T_w$ and $T_\infty$ are the temperature of the cylinder surface and the ambient air, respectively. At low heating ( $\varepsilon =0.2$ ), the 3-D Mode-A instability is suppressed leading to a two-dimensional wake flow. Further increase in heating, again brings back the three-dimensionality in the wake through Mode-E instability. The variation of thermophysical properties and the effective Reynolds number with increase in heating level around the cylinder is examined. It is shown that the effect of thermophysical properties competes with the baroclinic streamwise vorticity generation at higher levels of heating ( $\varepsilon \geqslant 0.4$ ) to control the 3-D modes and wake dynamics.

Effects of wake confinement and buoyancy on three-dimensional flow transitions for a square cylinder near a moving wall

Three-dimensional flow transitions for a square cylinder placed near a moving wall are examined in the presence of buoyancy. Computations are performed for the gap ratio G/D = 0.3, 0.5, and 1.0 for −0.5 ≤ Ri ≤ 0.5 at Re = 150. The flow rate near the cylinder, interactions between the shear-layers near the cylinder and the wall, and the baroclinic production of vorticity play key roles in the onset of three-dimensionality. With a decrease in G/D, the onset of three-dimensionality occurs at a lower value of Re compared to an isolated square cylinder. Both positive (Ri > 0) and negative buoyancy (Ri < 0) assist the onset of three-dimensionality. Mode A, mode B, and mode C of different wavelengths are formed for various values of Ri and G/D. The streamwise baroclinic vorticity is in-phase with the streamwise total vorticity and strengthens three-dimensionality. For negative buoyancy, the flow rate near the cylinder decreases, which reduces the strength of three-dimensionality compared to positive buoyancy. Positive buoyancy enhances the tendency of subcritical transition, while the tendency of supercritical transition increases for negative buoyancy. A stable limit cycle for supercritical transitions and an unstable limit cycle for subcritical transitions are observed. The initiation of three-dimensional transition may occur in the far-wake region, where velocity-vorticity interactions play a significant role. With an increase in Ri, buoyancy enhances enstrophy production, dissipation, and diffusion in the near-wake region. The force coefficients are obtained for different values of Ri.

Onset of three-dimensionality in rapidly rotating turbulent flows

Abstract

Influence of three-dimensional wake transition on heat transfer from a square cylinder near a moving wall

Three-dimensional direct numerical simulations have been performed to study the influence of gap-ratio (G/D) on wake transition and heat transfer from a square cylinder placed near a moving wall. Value of G/D is varied from 0.1 to 4. Three different values of Re = 160, 180 and 200 are considered, which are slightly less than the critical Re for different flow transitions corresponding to a free-stream flow past a square cylinder. With a decrease in G/D, the transition from one flow regime to another takes place at lower values of Re compared to a free-stream flow. At Re = 160, with a decrease in G/D, an onset of three-dimensionality with the presence of vortex dislocations in flow takes place due to an acceleration of flow in the gap. With a further decrease in G/D, dislocations are suppressed and Mode A only is present in the flow. At G/D = 0.4, streamwise vortices of wavelength 2.5D, called Mode S, appear in flow. These vortices show period doubling phenomenon. For G/D ≤ 0.3, three-dimensionality in flow occurs due to interaction of shear-layers formed near the cylinder and moving wall. At low values of G/D, an initial transition occurs from two-dimensional steady state to three-dimensional steady state. Variations in Nusselt number with G/D and Re in the spanwise direction and along the surfaces of the cylinder have been studied. Variation in the global Nusselt number on the cylinder surfaces with G/D can be divided in three parts. For G/D ≥ 0.5, Nu increases with a decrease in G/D and is maximum at G/D ≈ 0.5. For 0.5 ≤ G/D ≤ 0.3, it decreases with G/D and is minimum at G/D ≈ 0.3. For G/D ≤ 0.3, it again increases with a decrease in G/D. Influences of G/D and Re on force coefficients and Strouhal number (St) are examined. With an increase in Re, the drag coefficient increases and decreases at high and low values of G/D, respectively.

Wake transitions and laminar separation bubble in the flow past an Eppler 61 airfoil

The flow past an Eppler 61 airfoil at 10° angle to the free-stream is investigated numerically for 100 ≤ Re ≤ 87 000. Vortex shedding is observed beyond Re ∼ 600 and three-dimensionality sets in at Re ∼ 1280.9 via the mode C instability and hairpin vortex structures that grow with an increase in Re. At larger Re, the shear layer vortices, arising from the instability of the separated shear layer, interact with the flow close to the airfoil and cause it to reattach. A Laminar Separation Bubble (LSB) forms at Re ∼ 20 000 and beyond. The airfoil experiences a very significant increase in the lift and a decrease in the drag at the formation of LSB. Although the flow is three-dimensional, the primary mechanism of the formation of LSB appears to be two-dimensional. The length of the LSB decreases with an increase in Re. The variation of the shear layer and primary wake frequency, with Re, is studied. Both follow the power law. However, the variation before and after the formation of LSB is quite different. Unlike the primary wake frequency, the shear layer frequency suffers a jump at the formation of LSB. Wake formation length (Lf), estimated via the spatial distribution of Reynolds stress, exhibits spanwise periodicity at the onset of three-dimensionality.

Onset of three-dimensionality in supersonic flow over a slender double wedge

An investigation shows that laminar supersonic flow on a 2D compression corner can destabilize to become 3D beyond a critical turn angle. This compressible instability resides in the recirculating flow, producing spiral wall flow patterns postseparation and wall temperature streaks downstream.

Computational analysis of physical mechanisms at the onset of three-dimensionality in the wake of a square cylinder

A high-order spectral element method is used to perform direct numerical simulations of flow past a square cylinder at the transition from the two-dimensional von Kármán vortex street in the wake of the cylinder to three-dimensionality. A Reynolds number range between 100 and 300 has been considered and good agreement with previous numerical and experimental results is obtained. At transition, the spanwise perturbation observed in the cylinder wake occurs before the onset of the streamwise vortex separation, while the former mechanism is commonly shown, in the literature, to originate from the latter by an instability of the cores or interactions of shed vortices in the vicinity of the cylinder. It is shown that the first three-dimensional unstable mode, the mode $A$, originates from an axial stretching of the upstream perturbed vorticity, existing on the braids, due to the strain field created by the spanwise vortices which evolve under a shear instability of the wake when viscous effects are small. On the other hand, the mode $B$ is observed to arise only after spanwise vortices are shed downstream of the cylinder.

Co-existence of A and B modes in the cylinder wake at [formula omitted

We address the issue of co-existence of the modes A and B in the infinite cylinder wake after the onset of three-dimensionality. To distinguish them from each other we use the opposite symmetry of their time Fourier modes obtained by integrating the periodic flow field over one vortex shedding period. As argued in a recent paper, a relatively significant spanwise period of the simulation has to be considered to obtain a realistic simulation of the infinite spanwise geometry. This spanwise period of the simulation is taken equal to 10πd, d standing for the cylinder diameter, which amounts to 8 times the linearly preferred wavelength. In this numerical configuration the 3D state is found fully chaotic and stable down to the Reynolds number (Re) of 170. The obtained asymptotic state is analyzed to evidence the fingerprint of either of the modes A and B by time Fourier analysis. The results show the presence of both modes, which allows us to conclude that the mode B is excited simultaneously with the onset of mode A by non-linear couplings and that the progressive emergence of the mode B, described by Williamson as an “energy transfer” from frequencies associated to mode to those of mode B, observed experimentally to start from Re=220, occurs actually along the whole discontinuous branch of the Williamson Strouhal vs. Reynolds number curve. This explains why a bifurcation giving rise to the mode B has never been observed.

The vortex shedding around four circular cylinders in an in-line square configuration

This paper presents a numerical study of three-dimensional (3D) vortex shedding flow in the wake of four circular cylinders in a square configuration with a constant space-to-diameter ratio of 2. Numerical tests are carried out for Reynolds number (Re) in the range from 100 to 500. Four wake flow regimes are identified at this spacing ratio. Regime 1 ($100 \le {\mathop{\rm Re}\nolimits} \le 220$100≤ Re ≤220) is characterized by the inclination of weak spanwise vortices and weak streamwise vortices, where the wake behind four cylinder array shares similar features to that behind a single cylinder with a large equivalent diameter. It is observed that the onset of three-dimensionality in the wake behind four cylinder array occurs at lower Re than that behind a single cylinder. Regime 2 ($240 \le {\mathop{\rm Re}\nolimits} \le 300$240≤ Re ≤300) is characterized by the appearance of the regular wavy spanwise vortices and rib-shaped streamwise vortices. The wavelength of the spanwise vortices is about 1.2 and the wake flow is similar to the transition mode B of a single cylinder. Regime 3 ($320 \le {\mathop{\rm Re}\nolimits} \le 380$320≤ Re ≤380) is characterized by severe vortex dislocations in the wake of the cylinders and regime 4 ($400 \le {\mathop{\rm Re}\nolimits} \le 500$400≤ Re ≤500) is characterized by the absence of vortex dislocations and the strong streamwise vortices. The flow between the upstream and the downstream cylinders is predominantly two-dimensional in regimes 1, 2, and 3 and becomes 3D in regime 4. Physical mechanisms responsible for different flow regimes are proposed and discussed in details. Significant changes in the root-mean-square force coefficients, wake formation length, and phase angle of the lift coefficients on the downstream cylinders are observed when the flow transits from one regime to another.

Numerical investigation of the subcritical effects at the onset of three-dimensionality in the circular cylinder wake

The presented numerical simulations focus on the strong non-linear effects at the onset of three-dimensionality in the circular cylinder wake. The obtained numerical results shed light on the link between the linear theory and experimental observations. The latter are represented mainly by the Williamson’s Strouhal vs. Reynolds number experimental curve presenting a double discontinuity at the onset of three-dimensionality. In this paper, we show that the subcritical nature of the secondary bifurcation triggering the three-dimensionality considerably weakens the relevance of linear predictions. We first investigate the bi-stability interval of the subcritical bifurcation as a function of the spanwise periodicity of simulations. This allows us to define a non-linear marginal stability curve and to show that it predicts no preferred wavelength. The large spanwise scales and their effect on the simulated flow are then investigated in simulations with a spanwise periodicity of 31.4d and run for subcritical Reynolds numbers Re = 185, 170, and 160. It is shown that the non-linear state is chaotic and that the bi-stability interval in which it co-exists with the 2D parallel vortex shedding extends at least from Re = 170 to the linear secondary instability threshold lying between Re = 188 and Re = 189. Strouhal numbers obtained in simulations with a spanwise period close to 4d (linear preferred wavelength) tend to overestimate the Strouhal number drop. Simulations with a large spanwise period tend to put the Strouhal number closer to experimental values. Below the bi-stability interval, at Re = 160, the wake is observed to settle from the chaotic state to oblique vortex shedding.

Onset of three-dimensionality in electromagnetically driven thin-layer flows

Two-dimensional fluid flow is often approximated in the laboratory with thin electromagnetically forced fluid layers. The faithfulness of such an experimental model must be considered carefully, however, because the physical world is inherently three-dimensional. By adapting an analysis technique developed for oceanographic data, we divide velocity measurements from a thin-layer flow into two components: one that is purely two-dimensional and another that accounts for all out-of-plane flow. We examine the two- and three-dimensional components separately, finding that motion in thin-layer flows is nearly two-dimensional at low Reynolds numbers, but that out-of-plane flow grows quickly above a critical Reynolds number. This onset is likely due to a shear instability.

Unsteady flow around impacting bluff bodies

Abstract The flow resulting from the collision without rebound of generic bluff bodies with a wall in a still viscous fluid is investigated both computationally and experimentally. Emphasis is on the case of a circular cylinder impact (two-dimensional geometry), but comparisons with the flow generated by the impact of a sphere (axisymmetric geometry) are included. For normal cylinder impacts, the two counter-rotating vortices forming behind the body during its motion continue their trajectory towards the wall after the collision, leading to the generation of opposite-signed secondary vorticity at the cylinder and wall surfaces. Secondary vortices forming from this vorticity at higher Reynolds numbers exhibit a short-wavelength three-dimensional instability. Comparison with the sphere impact reveals significant differences in the scales of the vortices after the collision, due to the additional vortex stretching acting in the axisymmetric geometry. This leads to a delay in the onset of three-dimensionality and to a different instability mechanism. Oblique cylinder impacts are also considered. For increasing impact angles, the wall effect is gradually reduced on one side of the cylinder, which favours the roll-up of the secondary vorticity and increases the rebound height of the vortex system.

Modelling blockage effects using a spectral element method

We employ a two-dimensional spectral element method to investigate the effect of various semi-circular blockages on the laminar flow between two flat plates. This simplified geometry has a general relevance to blocked flows, but more academically, also relates to work on the backward-facing step, presenting a qualitatively similar flow with a single, independent step, or blockage, parameter. The parameter space for Reynolds number between 50 and 3000, and blockage ratio ranging from 0.05 to 0.9 is mapped out. The results with blockage ratio of 0.5 compare favourably with previous work on the flow over a backward-facing step, with the analysis suggesting that the onset of three-dimensionality in the flow is likely to be closely matched. Flow phenomena associated with blockage ratios above and below the customary half-blockage are summarised, including smaller than expected flow separation lengths for lower blockages, and vortex shedding and unusual structures within the recirculation zones for higher blockages.

The onset of three-dimensionality in an oscillating flow past a fixed circular cylinder

The onset of three-dimensionality in an oscillating flow past a fixed circular cylinder

Three-dimensional stability analysis of the periodic flow around a circular cylinder

The onset of three-dimensionality in the von Kármán vortex street behind a circular cylinder is investigated by carrying out the first global, nonparallel, three-dimensional stability analysis of the periodic flow. This flow is found to become unstable at a Reynolds number of 170 by a critical, three-dimensional Floquet mode with a spanwise wavelength of 1.8 diam. The spatial structure of this mode indicates that the onset of three-dimensionality is due to a near-wake instability and not caused by a stagnation-line or a boundary-layer instability.

The onset of three-dimensionality and time-dependence in Görtler vortices: neutrally stable wavy modes

Recently Hall & Seddougui (1989) considered the secondary instability of large-amplitude Görtler vortices in a growing boundary layer into a three-dimensional flow with wavy vortex boundaries. They obtained a pair of coupled, linear ordinary differential equations for this instability which constituted an eigenproblem for the wavelength and frequency of this wavy mode. In the course of investigating the nonlinear version of this problem (Seddougui & Bassom 1990), we have found that the numerical work of Hall & Seddougui (1989) is incomplete; this deficiency is rectified here. In particular, we find that many neutrally stable modes are possible; we derive the properties of such modes in a high-wavenumber limit and show that the combination of the results of Hall & Seddougui and our modifications lead to conclusions which are consistent with the available experimental observations.

On the onset of three-dimensionality and time-dependence in Görtler vortices

The secondary instability of large-amplitude Görtler vortices in a growing boundary layer is discussed in the fully nonlinear regime. It is shown that the three-dimensional breakdown to a flow with wavy vortex boundaries, similar to that which occurs in the Taylor vortex problem takes place. However, the instability is confined to the thin shear layers which were shown by Hall & Lakin (1988) to trap the region of vortex activity. The disturbance eigenfunctions decay exponentially away from the centre of these layers, so that the upper and lower shear layers can support independent modes of instability. The structure of the instability, in particular its location and speed of downstream propagation, is found to be entirely consistent with recent experimental results. Furthermore, it is shown that the upper and lower layers support wavy vortex instabilities with quite different frequencies. This result is again consistent with the available experimental observations.

Vortex/separated boundary-layer interactions at transonic Mach numbers

An experimental study has been completed on the effect of a single longitudinal vortex on a separated, transonic, turbulent boundary layer. The vortex was generated by a half-delta wing mounted at the upstream end of an axisymmetric bump model. A flow visualization study was conducted using vapor screen and surface oilflow techniques. In addition surface pressures were measured and mean flow and turbulence data obtained using a two-component laser velocimeter. At precritical Mach numbers, the vortex delayed or eliminated boundary-layer separation on the downwash side and enhanced it on the upwash side, thus converting a nominally twodimensional separation into a three-dimensional one. At the postcritical Mach number, the effect of the vortex was to reduce the size and extent of the shock-induced boundary-layer separation throughout the region of interaction. The boundary-layer turbulence in both cases was found to reorganize accordingly, although in a rather complex manner. The onset of three-dimensi onality in the separation line produced by the vortex resulted in secondary vortices (foci), the sign and number being strongly dependent on the freestream Mach number.