Span effect on the turbulence nature of flow past a circular cylinder
Bernat Font, Gabriel D. Weymouth, Vinh-Tan Nguyen, Owen R. Tutty
TThis draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics Span effect on the turbulence nature of flowpast a circular cylinder
Bernat Font Garcia , Gabriel D. Weymouth † , Vinh-Tan Nguyen ,and Owen R. Tutty Faculty of Engineering Physical Sciences, University of Southampton, SO17 1BJSouthampton, UK Institute of High Performance Computing, Singapore Agency for Science, Technology andResearch (A*STAR), 138632, Singapore(Received xx; revised xx; accepted xx)
Turbulent flow evolution and energy cascades are significantly different in two-dimensional (2D) and three-dimensional (3D) flows. Studies have investigated thesedifferences in obstacle-free turbulent flows, but solid boundaries have an importantimpact on the cross-over between 3D to 2D turbulence dynamics. In this work, weinvestigate the span effect on the turbulence nature of flow past a circular cylinder at Re = 10000. It is found that even for highly anisotropic geometries, 3D small-scalestructures detach from the walls. Additionally, the natural large-scale rotation of theK´arm´an vortices rapidly two-dimensionalises those structures if the span is 50% ofthe diameter or less. We show this is linked to the span being shorter than the ModeB instability wavelength. The conflicting 3D small-scale structures and 2D K´arm´anvortices result in 2D and 3D turbulence dynamics which can coexist at certain locationsof the wake depending on the domain geometric anisotropy.
1. Introduction
Incompressible viscous flow past two-dimensional (2D) bluff bodies involves com-plex physics such as the well-known von-K´arm´an street phenomenon as well as three-dimensional (3D) wake dynamics as the Reynolds number ( Re ) is increased (Roshko1954; Williamson 1996 a , b ). Due to the two-dimensionality of circular cylinders, someauthors have used the 2D Navier-Stokes equations on multiple planes located along thespan of the cylinder as a simplified model of the three-dimensionality without increasedcomputational cost ( a.k.a. strip theory method). Such strip theory methods are used inoffshore and civil engineering applications to model flow along slender structures wherethe computational cost of fully-resolved simulations is prohibitive, such as marine risers,tow and mooring cable systems, and tall pillars. However, the physics inherent in 2Dsimulations lead to poor 3D predictions (Bao et al. † Email address for correspondence: [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] A ug B. Font Garcia, G. D. Weymouth, V.-T. Nguyen and O. R. Tutty
Figure 1.
Sketch of the TKE cascade distribution across the scales as the span is reduced. Byconstricting the domain, the integral scale is reduced and the energy-containing scales can feedless energy to the inertial and dissipative scales. comprehensive review see Boffetta & Ecke 2012). Studies such as Xiao et al. (2009) haveshown that physical processes such as vortex-thinning and vortex-merging dominate thedynamics of 2D turbulence generating larger and more intense vortical structures theenergy of which piles up at the integral scale for bounded domains.Previous work has studied the transition between 2D and 3D dynamics in obstacle-free turbulent flows in detail. The effect of length scale L z constriction on the TKEdistribution across the scales (or wavenumbers κ ) is sketched in figure 1. By constrictingthe domain, the size of the 3D energy-containing structures (integral scale structures)is reduced. Since the energy-containing structures feed the inertial subrange structuresdown to the 3D small-scale dissipative structures, smaller integral scale structures resultin less energy fed into the inertial subrange structures and, consequently, to the dissipativestructures. On the 2D limit, no 3D dissipative structures are present and, because of thelack of a dissipation mechanism, the turbulent vortical structures can only merge. Thiscreates larger structures promoting an inverse energy cascade as shown in Kraichnan(1967); Leith (1968); Batchelor (1969).Smith et al. (1996) reviewed the aspect ratio depth effect together with a rotation effectof forced turbulence on a L x × L y × L z periodic box. It was found that the turbulencedimensionality of the flow depended not only on the geometry constriction ( A ≡ L z /L x ),but also on the rotation intensity Ω . A critical ratio between the span and the turbulenceforcing scale was revealed below which two-dimensionalisation occurred for non-rotatingcases. Furthermore, it was found that higher rotation rates induced a more significant2D turbulence behaviour and that direct and inverse energy cascades for small andlarge scales can coexist respectively. Celani et al. (2010) found a similar splitting ofthe turbulence cascade for a critical value of the relative forcing on a depth-restrictedperiodic box.These differences between 2D and 3D turbulence dynamics have a significant practicalimpact. For example, the forces induced on a cylinder are larger in magnitude andvariability in 2D systems (Mittal & Balachandar 1995; Norberg 2003). In an attempt todissipate the energised vortical structures and to prevent the vortex-merging dynamics of2D turbulence, models based on the turbulent-viscosity hypothesis ( a.k.a. the Boussinesqhypothesis) are incorporated into 2D strip theory methods. However, these modelsassume that the anisotropic Reynolds stress tensor is proportional to the mean rate- pan effect on the turbulence nature of flow past a circular cylinder et al. (2016) showed how strips witha certain thickness are able develop to 3D turbulence when its span is larger than thewavelength of the Mode B instability of circular cylinders (about one diameter). In fact,this instability creates rib-like streamwise vortical structures along the main K´arm´an 2Dvortices (Noack 1999). Therefore, it can be argued that the two-dimensionalisation of thewake arises from the geometry constriction which prevents the rib-like vortices to developwhen there is not room enough for its natural wavelength. However, the connectionbetween wake and wall turbulence and the persistence of 3D turbulent structures inconstricted span flows has not been fully explored.This work studies the geometry constriction effect on the turbulence nature of a flowpast a circular cylinder at Re = 10000. To do this, a series of simulations ranging from L z = 10 to pure 2D planes have been considered. As discussed above, the inclusion of abody boundary provides an important change to the turbulence production mechanismscompared to previous research and novel information on the transition and cross-overbetween 3D and 2D turbulence for very constricted domains in wall-generated turbulentshear flows. Multiple turbulence statistics are presented for the wide range constrictedwakes, providing new data on the transition from 3D to 2D turbulence.With this intention, we have structured the current paper as follows: § §
3, the turbulence nature of the wake for the different cases is analysed(similarly to Biancofiore (2014)) and discussed with results such as wake visualisation,velocity temporal spectras at different locations, Lumley’s triangle, TKE spatial plotsand vortex-stretching analysis.
2. Problem formulation
This study considers the flow past a circular cylinder with diameter D aligned on the z direction in a 3D (35 × × L z ) D rectangular domain, where L z is the non-dimensionalspan. To study the span effect on the turbulence nature of the wake, the following caseshave been considered: L z = 10 , π, , . , . , . Re = 10000 is selected. We define the Reynolds number as, Re = U Dν , (2.1)where U is the scaling velocity and ν is the kinematic viscosity of the fluid. Usingthis Reynolds number we ensure that the flow is well within the turbulent regimeencompassing very different spatial and time scales.2.1. Governing equations and numerical method
The incompressible viscous fluid motion is described by the continuity equation andthe non-dimensional Navier-Stokes momentum equations, ∇ · u = 0 , (2.2) ∂ t u + u · ∇ u = −∇ p + Re − ∇ u , (2.3)where u ( x , t ) = ( u, v, w ) is the velocity vector field, p ( x , t ) is the pressure field, t is thetime and x = ( x, y, z ) is the spatial vector. The initial condition is defined as u ( x ,
0) =
B. Font Garcia, G. D. Weymouth, V.-T. Nguyen and O. R. Tutty
Figure 2.
Computational domain sketch non-dimensionalised with the cylinder diameter D . Avery fine Cartesian grid domain (depicted in discontinuous lines) surrounds the cylinder and theclose and mid wake regions. A stretched rectilinear grid (depicted in solid lines) transitions fromthe Cartesian grid to the boundaries. (1 , ,
0) in the fluid. The boundary conditions are: a uniform velocity profile on theinlet boundary, a natural convection condition on the outlet boundary, a no-penetrationslip condition on the upper and lower boundaries, a periodic condition on the spanwisedirection boundaries and a no-slip velocity condition on the cylinder. Periodic boundaryconditions on the constricting planes are used as in previous studies on the cross-over of2D and 3D turbulence for obstacle-free flows such as Smith et al. (1996), Celani et al. (2010) and Biancofiore (2014). This choice is made to avoid the artificial high intensityturbulence enhancements and the deterioration of the 2D behaviour of the flow when ano-penetration condition is enforced on the constricting planes, as noted on Biancofiore et al. (2012) for obstacle-free turbulent wakes. Other studies such as Bao et al. (2016)and Bao et al. (2019) also use a periodic spanwise condition for thick strips modellinglong circular cylinders.The governing equations are numerically solved in a discrete rectilinear grid (detailsin § et al. (2017) and Maertens & Weymouth (2015), wherethe latter also provides a detailed explanation of the numerical method. Our in-housecode is second-order accurate in space (using the Quadratic Upstream Interpolation forConvective Kinematics, a.k.a. QUICK, scheme) and second-order accurate in time (usinga predictor-corrector algorithm). The implicit turbulence modelling derives from a flux-limited QUICK treatment of the convective terms, equivalent to optimum finite-volumeschemes (for a review on ILES see Adams & Hickel (2009)). Hendrickson et al. (2019)have validated this ILES approach for intermediate Reynolds numbers similar to the Re = 10000 used in the current work. pan effect on the turbulence nature of flow past a circular cylinder Computational details
The domain is composed of a sufficiently fine Cartesian grid for the close and mid wakeregions defined as ( L x × L y × L z ) D , where L x and L y are the non-dimensional horizontaland vertical lengths respectively. A stretched grid is considered for the regions far fromthe cylinder (see figure 2). A resolution of 90 cells per diameter in all the spatial directionsis chosen for the Cartesian grid subdomain. The resolution in all spatial directions is keptconstant as the span is reduced.The 3D simulations are started from a three-dimensionalised 2D flow snapshot. A timelength of 200 units ( T = tU/D ) is simulated before starting to record the flow statisticsin order to achieve a statistically stationary state of the wake. The flow statistics arethen recorded for a total of 500 T (around 100 wake cycles). A verification and validationof the wake turbulence dynamics of the investigated test case is included in appendix A.Finally, the turbulence statistics of the L z = 10 and L z = π cases are very similar asdisplayed in figure 5a. Hence, only the L z = π results are displayed on the other figuresfor clarity.
3. Results and discussion
The flow field is displayed in figure 3 in terms of the instantaneous vorticity component ω z as the span is varied. The most striking feature is how the coherence of the K´arm´anvortices increases as the span is reduced. However, even in highly-anisotropic geometriessuch as L z = 0 .
25, small-scale 3D structures are generated from the cylinder wall.An important result is that the two-dimensionalisation of these structures is faster (inthe sense that it occurs closer to the cylinder) as the domain is constricted because of thegeometry constriction and the natural rotation of the K´arm´an vortices. The combinationof these two mechanisms as a two-dimensionalisation method is also found in Smith et al. (1996) and Xia et al. (2011). For the L z (cid:62) L z = 1 case displays a coherent K´arm´anvortex. This means that less anisotropic geometries promote a direct TKE cascade onthe wake so that the 3D dissipative structures are still sustained far from the cylinder.Whether the wake turbulence dynamics are 2D or 3D is better captured on the TKEspectra which can be directly compared to classic turbulence theory. For this, the Taylor’shypothesis is considered and the temporal spectra at different points of the wake iscalculated. Figure 4 a,b,c shows the temporal power spectra (PS) of the v velocitycomponent at ( x, y ) = (2 , . , (4 , . , (8 , . x, y ) point. These spectras are thenaveraged resulting in a single spectra for each case.First, note that all of the spectras display a peak around the 0 . S t = f s D/U , where f s is the vortex-shedding frequency. A smaller harmonic peak around 0 . x, y ) = (2 , . , (4 , . − / L z = 0 . − / − et al. (2008) and Biancofiore (2014). The filamentary vorticity B. Font Garcia, G. D. Weymouth, V.-T. Nguyen and O. R. Tutty
Figure 3.
Instantaneous vorticity ω z (red is positive, blue is negative) at the z = L z / a ) L z = 0, ( b ) L z = 0 .
1, ( c ) L z = 0 .
25, ( d ) L z = 0 .
5, ( e ) L z = 1, ( f ) L z = π . (filaments of vorticity around the coherent vortices) is likely to be destroyed by theinteraction of the large-scale vortices rather than viscous effect (specially for high Re flows), thus limiting its range of scales. The coherent vortices induce a spiralling effectwhich limits the range of scales of incoherent filamentary vorticity (Gilbert 1988). On theother hand, as a − / L z = 0 .
25 case, it can be arguedthat 3D turbulence is being generated from the cylinder wall even for highly-reducedspans.The L z = 0 .
25 and L z = 0 . − / − / − / − / L z = 0 . , . L z = 1 cases on figures 4b and 4crespectively. The low-frequency structures behave mostly 2D (decaying rate between − − /
3, resulting from the presence of less coherent 2D structures than the pure2D case) up to a certain point where a − / et al. (1996) and Celani et al. (2010).Additionally, two-point correlations along the span have been analysed at the same( x, y ) locations as the PS plots (figure 4 d,e,f). Given a distance d along z ranging from 0to L z /
2, the two-point correlation is calculated with the temporal signals of the verticalvelocity component at multiple pairs of points (namely v and v ) as follows, (cid:104) v ( x , t ) , v ( x + r , t ) (cid:105) = cov( v , v ) (cid:112) cov( v , v )cov( v , v ) , (3.1) pan effect on the turbulence nature of flow past a circular cylinder y x2 0.8 Figure 4.
Left: vertical velocity component temporal power spectra (PS) at different (( x, y )locations on the wake: ( a ) = (2 , . b ) = (4 , . c ) = (8 , . L z = π case.The dashed lines have a − / − / z at the same ( x, y ) locations as the left figures respectively. The correlationvalue for a given d corresponds to the averaged value of the multiple correlations of pairs ofpoints separated a distance d along the span. B. Font Garcia, G. D. Weymouth, V.-T. Nguyen and O. R. Tutty
Figure 5.
TKE spatial plots. The TKE is computed from the normal Reynolds stresses andaveraged on the vertical direction: ( a ) total TKE, ( b ) ratio of the spanwise component w (cid:48) w (cid:48) overthe total TKE. where the distance vector is defined as r = (0 , , d ). The multiple correlation coefficientsfor a given d are then averaged corresponding to a data point in the plots.Very close to the cylinder (figure 4d), the correlation coefficient quickly decreases withincreasing d for L z > .
1. This indicates the presence of 3D structures near the body asalso noted on the velocity spectra plot counterpart. Also, the decrease is more pronouncedas the span increases. For L z = π , it is worth noting a local correlation coefficientmaximum around d ≈ .
9. This distance approximately corresponds to the Mode Binstability wavelength ( λ z ). Since the rib-like vortices associated with Mode B instability(streamwise and cross-flow vorticity) are not very well defined at this Re regime (Chyu &Rockwell 1996), the correlation increase is not as significant as at lower Re . Still, L z = π is the only case displaying such phenomena because of the spanwise boundary conditionsperiodicity, which only allow the instability to develop if L z > λ z (the L z = 1 case mightbe too critical to display such phenomenon considering also its intermittent nature). Thecorrelation coefficient increases when calculated further downstream as shown in figure4e and 4f, evidencing again the wake two-dimensionalisation.In summary, the transition to a 2D wake is found at a certain point along the wake onall the cases with L z (cid:54)
1. The combination of the large-scale rotation from the K´arm´anvortices plus the geometry constriction are mainly responsible for this phenomenon. Themain physical mechanism differing among the compared cases is the ability of the flow todevelop Mode B-like 3D structures in the wake as a result of a sufficiently long span. Thecurrent Re regime is characterised by a transition to turbulent flow at the shear layer( i.e. the TrSL2 regime) as reported in Bloor (1964) and Kourta et al. (1987). We arguethat when the span is too short for the Mode B instability to develop and thereby sustainthese 3D turbulent structures, the stratification effect of the K´arm´an vortices leads tothe more coherent and energized wake seen in figure 4 d,e,f.Next, the TKE along the x direction, the Lumley’s triangle of turbulence and the ratiobetween the vortex-stretching and advection terms are examined to further support theobserved phenomena. The TKE is defined as, T KE = 12 (cid:0) u (cid:48) u (cid:48) + v (cid:48) v (cid:48) + w (cid:48) w (cid:48) (cid:1) , (3.2) pan effect on the turbulence nature of flow past a circular cylinder .
06 0 .
08 0 .
10 0 .
12 0 . T KE | y )0 . . . . . . . C L π . . . Figure 6.
Effect of the span constriction on the TKE and the C L . The latter is calculated as C L = | F y / ( ρU DL z ) | rms , where F y is the vertical lift force and ρ is the constant fluid density. where · denotes a time average plus a spanwise average and the subscript · (cid:48) denotes afluctuating quantity such as a (cid:48) = a − a . The six components of the Reynolds stress tensor u (cid:48) i u (cid:48) j have been computed using the following relation, a (cid:48) b (cid:48) = ab − ab. (3.3)Figure 5 shows the streamwise spatial distribution of the TKE averaged along the y direction from − L y / L y / ·| y ). From a general point of view, it can beobserved that the total TKE (figure 5a) peaks right after the recirculation region notingthat the latter increases slightly with the span. Also, the total TKE increases as the spanis reduced because of the 2D vortex-merging processes that generate larger and moreenergised vortical structures. The contribution of the spanwise normal stress to the totalTKE increases with the span as shown in figure 5b. Also, a decay of w (cid:48) w (cid:48) right afterbeing generated from the cylinder wall can be noted and it becomes faster as the spanis constricted.The effect of the span on the TKE compared to the lift coefficient r.m.s. value (cid:0) C L (cid:1) displays a quasi-linear relation as shown in figure 6. Note that the drop on both valuesis more sensitive to the span constriction at the range where both 2D and 3D turbulencedynamics co-exist, i.e. . (cid:54) L z (cid:54)
1. On the other hand, a very small change of bothvalues can be appreciated when 3D turbulence fully dominates the wake, i.e. L z (cid:62) π .Furthermore, as shown in figure 5a, highly constricted domains yield large values of TKEbecause of the energised 2D large-scale vortical structures present at the wake. Even whensmall-scale 3D structures are present in the close wake region and coexist with the large2D structures, the values of both the TKE and the C L spike.On figure 7, the temporal signals of the lift and drag coefficients as well as the upperand lower separation angles of the free-shear layer are displayed for the L z = 0 . L z = π cases. The oscillation of the separation points increases as the span is constrictedinducing larger forces on the cylinder. A very high correlation of the upper and lowerseparation points angle with the lift coefficient is shown as well. Again, the lack of theMode B instability on the constricted cases yields more coherent 2D vortical structures in0 B. Font Garcia, G. D. Weymouth, V.-T. Nguyen and O. R. Tutty
Figure 7.
Top: C L and C D temporal signals. The latter is calculated as C D = 2 F x / ( ρU DL z )and the former as detailed in figure 6 (without the r.m.s norm). Middle: Upper θ u and lower θ l separation angles temporal signal. The separation angle is calculated from the front stagnationpoint of the cylinder to the free shear layer separation point (vorticity practically 0 at the wall).Bottom: Correlation between the lift coefficient and the difference between θ u and 360 − θ l . Left: L z = π . Right: L z = 0 . the near wake region. Therefore, the small-scale 3D structures which normally dissipatemost of the kinetic energy in 3D turbulence are not present leading to high-intensityvortices. This can be quantified by the enstrophy ( Ω ) of the span-averaged spanwisevorticity in the near wake region D ( x ∈ [0 . , . , y ∈ [ − . , . Ω = (cid:90) D t − t (cid:90) t t L z (cid:90) L z ω z dz dt d D . (3.4)It can be observed in table 1 that the enstrophy increases as the span is reduced, whichcan be understood as an increase of the rotational energy of the flow.The highly energised coherent vortices forming at the near wake region for the con-stricted cases induce a larger convective force on the free-shear layer. This translates to pan effect on the turbulence nature of flow past a circular cylinder L z π . .
25 0 . Ω/Ω π Table 1.
Near body enstrophy as defined in equation 3.4 for the different span cases. Ω π isthe enstrophy of the L z = π case. the large oscillations observed in figure 7 and, ultimately, to the forces induced on thecylinder. With this, it can be argued that the coherent 2D structures have a greaterimpact on the forces induced to the cylinder than the 3D small-scale structures whenboth are present.Lumley & Newman (1977) proposed the Lumley’s triangle of turbulence which providesa way to classify the anisotropic state of turbulence. The anisotropy property of theReynolds stress tensor can be extracted with, b ij = u (cid:48) i u (cid:48) j u (cid:48) k u (cid:48) k − δ ij , (3.5)where δ is the Kronecker delta and b ij is the anisotropic Reynolds stress tensor (whichevidently vanishes for isotropic turbulence). This dimensionless and traceless tensor hastwo non-zero invariants, II = − b ij b ji / III = b ij b jk b ki /
3. These invariants are oftenrewritten as η = − II/ ξ = III/ ξ ) axisymmetric state since oneeigenvalue is smaller than the other two (which are equal) or greater than the other two(which are equal) respectively. Finally, the (0 ,
0) point indicates 3D isotropic turbulencesince all of the anisotropic tensor eigenvalues vanish.Figure 8 displays the Lumley’s triangle constructed across the wake width at different x locations. Hence, every point in the triangle corresponds to a point in the domain andthe collection of points for each case represents the collection of vertically aligned pointsin the domain at different x locations on the wake. It can be appreciated that all of thepoints are located within the triangle, therefore all of computed Reynolds stresses arerealisable ( i.e. have positive and real eigenvalues). In general, most of the cases transitionfrom one state to another state of turbulence across the wake width. Only for the almost2D case ( L z = 0 . w (cid:48) w (cid:48) is effectively negligible everywhere.As the triangle is constructed further downstream on the wake, the trajectories (orcollection of points) of the L z = 0 .
25 and L z = 0 . L z = 12 B. Font Garcia, G. D. Weymouth, V.-T. Nguyen and O. R. Tutty y x2 y x 4y x 8y x 6
Figure 8.
Lumley’s triangle constructed across the wake width at: ( a ) x = 2, ( b ) x = 4, ( c ) x = 6, ( d ) x = 8. and L z = π cases remain approximately at the same η region showing that the two-dimensionalisation is not as effective.Additionally, the trajectories present a negative axisymmetric almost two-componentstate for the locations far from the wake centreline, i.e. the location close to y = ± L y / w (cid:48) w (cid:48) is smaller than the other ones. Onthe centreline, v (cid:48) v (cid:48) is larger than the other stresses causing a shift to the positiveaxisymmetric state.A comparison of the different cases at the same x location also shows a more noticeable2D turbulence state as the span is constricted. This difference is less noticeable close to thecylinder (figure 8a), where even the L z = 0 . L z = π case.Again, this evidences that 3D turbulence is present close to the wall even in considerablyconstricted cases.Consider now the vorticity transport equation (VTE) which can be written as, ∂ t ω + u · ∇ ω = ω · ∇ u + Re − ∇ ω , (3.6) pan effect on the turbulence nature of flow past a circular cylinder Figure 9. ( a ) Modulus of the mean vortex-stretching term averaged along the vertical direction.( b ) Ratio between the modulus of the mean vortex-stretching term and the modulus of the meanvortex-advecting term averaged along the vertical direction. where ω ( x , t ) = ( ω x , ω y , ω z ) is the vorticity vector field defined as ω = ∇ × u . Thevortex-stretching term, ω · ∇ u , is often pointed as the term responsible for the directenergy cascade of the TKE. The stretching of a vortex tube causes a reduction on itsdiameter while increasing the rotation speed of the vortex by conservation of angularmomentum. This term vanishes in the 2D formulation of the VTE since the stretch ofthe vortex tube is perpendicular to the plane of rotation. From this mathematical andphysical difference, different turbulence dynamics are captured on 2D or 3D computationsand, therefore, 3D turbulence can be directly linked to this term.Figure 9 displays the modulus of the mean vortex-stretching term and its ratio R with the mean vortex-advecting term. These quantities are averaged on the verticaldirection. On figure 9a, it can be observed that the vortex-stretching term decays fasterwhile moving downstream from the cylinder for cases with shorter span. The vortex-stretching decay demonstrates again the two-dimensionalisation of the flow by the large-scale vortices. The ratio R on figure 9b shows that the vortex-advecting term decaysfaster (because of the wake momentum deficit) than the vortex-stretching term along thestreamwise direction up to x = 6 where the ratio is kept constant. It can also be observedthat the vortex-stretching term becomes as important as the vortex-advecting term withincreasing span.
4. Conclusion
The span effect on the turbulence dynamics of a flow past a circular cylinder at Re = 10000 has been investigated using spectras and two-point correlation on differentlocations in the domain, the TKE along the wake, the separation points, the Lumley’striangle of turbulence and the mean vortex-stretching term of the VTE. It has been shownthat 3D turbulence is present even for highly constricted cases (for example L z = 0 . et al. (2016). Since the Mode B4 B. Font Garcia, G. D. Weymouth, V.-T. Nguyen and O. R. Tutty instability helps sustaining the turbulent structures advected from the shear layer, thelack of it prevents large-scale 3D structures to be created and less dissipative structurescan be sustained. In this scenario, 2D turbulence takes over and dominates the wakedynamics creating larger, stronger and more coherent vortices. Ultimately, the coherentand energised vortical structures induce a larger convective force on the free-shear layer.This translates to larger oscillations and, finally, higher forces on the cylinder.The flow turbulence transition from 3D to 2D caused by a geometry constriction foundin this work is in agreement with the physical mechanisms described in the obstacle-freeturbulence work of Smith et al. (1996), Celani et al. (2010) and Biancofiore (2014). In thepresent study with solid boundaries, the main difference is found on the presence of small-scale 3D turbulence even in highly-constricted geometries ( L z = 0 .
25) which leads to acoexistence of 2D and 3D turbulence close to the cylinder wall. This is observed not onlyin the L z = 0 .
25 case but also for the L z = 0 . L z = 1 cases as shown in figures 4b and4c respectively. Note that the crossover between 2D and 3D turbulence dynamics arises indifferent points in the spatial domain depending on the span length. The shorter the span,the closer to the cylinder it takes place evidencing that the wake two-dimensionalisationtransitions at different locations in function of the domain geometric anisotropy.On the other hand, a very rapid two-dimensionalisation is found in the present casesbecause of the natural large-scale rotation motion of the K´arm´an vortices. A large-scalerotation as a mechanism of two-dimensionalisation has been also found in other workssuch as Smith et al. (1996) and Xia et al. (2011). These two mechanisms combined yieldto a rapid transition from the 3D to 2D turbulence dynamics when the span is shorterthan the Mode B instability wavelength. Acknowledgements
The authors acknowledge the use of the IRIDIS High Performance Computing Facility,and associated support services at the University of Southampton, in the completion ofthis work. The authors also acknowledge the support of the Singapore Agency for Science,Technology and Research (A*STAR) Research Attachment Programme (ARAP) as acollaboration between the A*STAR Institute of High Performance Computing (IHPC)and the Faculty of Engineering and the Environment of the University of Southampton.
Appendix A.
A grid convergence study has been conducted in order verify the correct implementa-tion of the governing equations and to show that the selected grid resolution and averagingtime length is suitable for a proper analysis. Three grids with different resolution ( ∆ )designed as shown in figure 2 have been investigated for the L z = π case (see details intable 2). The refinement ratio is √ , .
8) and (8 , . − / pan effect on the turbulence nature of flow past a circular cylinder y x2 0.8 0.8y x 8 0.8y x 8y x2 0.8 0.8y x 8y x2 0.8 Figure 10.
Vertical velocity component temporal power spectra at the close (left) and far (right)wake regions. The coarse, medium and fine grid results are displayed at the first, second and thirdrow respectively. The spectras are shifted a factor of 10 and the vertical axis ticks correspond tothe T = 50 case. The number of the time signal splits selected for the Welch method ( n ) ensuresa minimum of 50 T per split (at least 8 times larger than the lowest frequency of interest, i.e. the shedding frequency which is around 5 T ) and a maximum of 6 splits per total time signal.The dotted line corresponds to a − / B. Font Garcia, G. D. Weymouth, V.-T. Nguyen and O. R. Tutty
Figure 11.
Comparison of the Reynolds shear stress u (cid:48) v (cid:48) between: (a) DNS data from Dong& Karniadakis (2005) and (b) present solver. The contour levels criteria is the same as thereference data: | u (cid:48) v (cid:48) | min = 0 .
03 and ∆u (cid:48) v (cid:48) = 0 .
01. Positive and negative levels are noted withcontinuous and dashed lines respectively.
D/∆ N CG (M) N T (M)64 65.3 109.290 189.5 311.4128 510.6 855.6 Table 2.
Details of the three different grids ranging from coarse (
D/∆ = 64, i.e.
64 cells perdiameter) to fine (
D/∆ = 128) for L z = π . N GC refers to the total number of cells (in millions)for the Cartesian grid subdomain. N T refers to the total number of cells (in millions). region, the coarse grid presents slightly steeper spectras than at the close wake region.This can be caused by an insufficient resolution in span, which would induce a two-dimensionalisation effect. On the other hand, the medium grid still presents convergedspectras for time signals longer than 100T. The fine grid leads to results very similarto the medium grid. Hence, it is shown that using the medium grid with simulationtimes over 100T (500T is used in the results presented in this work) provides statisticallystationary results both in the close and far wake regions.To validate the wake turbulence statistics, the Reynolds shear stress u (cid:48) v (cid:48) has also beenqualitatively analysed. It can be observed in figure 11 that the shear stress predicted bythe present solver is in very good agreement with the Direct Numerical Simulation (DNS)data from Dong & Karniadakis (2005), with just a slight shift of the field structures onthe streamwise direction. All positive and negative regions are correctly captured whichdisplay an antisymmetric pattern with respect to the centreline of the wake. REFERENCESAdams, N. A. & Hickel, S.
Advances in Turbulence XII , pp. 743–750. Berlin, Heidelberg: Springer Berlin Heidelberg.
Bao, Y., Palacios, R., Graham, J. M. R. & Sherwin, S.
J. Comp. Phys. ,1079–1097.
Bao, Y., Zhu, H.B., Huan, P., Wang, R., Zhou, D., Han, Z.L., Palacios, R., Graham, pan effect on the turbulence nature of flow past a circular cylinder M. & Sherwin, S.
J. Fluids and Struct. . Batchelor, G. K.
Phys. Fluids (12), II–233–II–239. Biancofiore, L.
J. Fluid Mech. , 164–179.
Biancofiore, L., Gallaire, F. & Pasquetti, R.
Computers & Fluids , 27–44. Bloor, M. Susan
J. FluidMech. (2), 290–304. Boffetta, G. & Ecke, R. E.
Annu. Rev. Fluid Mech. ,427–451. Celani, A., Musacchio, S. & Vincenzi, D.
Phys. Rev. Lett. (18), 184506.
Choi, K.-S. & Lumley, J. L.
J. FluidMech. , 59–84.
Chyu, C. & Rockwell, D.
J. Fluid Mech. , 117–137.
Dong, S. & Karniadakis, G. E.
J. Fluids and Struct. , 519–531. Dritschel, D. G., Scott, R. K., Macaskill, C., Gottwald, G. A. & Tran, C. V.
Phys. Rev.Lett. , 094501.
Gilbert, A. D.
J. FluidMech. , 475–497.
Hendrickson, K., Weymouth, G. D. & Yue, D. K.-P. Yue Yu, X.
J. Fluid Mech . Kourta, A., Boisson, H. C., Chassaing, P. & Ha Minh, H.
J. Fluid Mech. ,141–161.
Kraichnan, R. H.
Phys. Fluids , 1417–1423. Leith, C. E.
Phys. Fluids ,671–673. Lumley, J. L.
Adv. Appl. Mech. , 123–176. Lumley, J. L. & Newman, G. R.
J. Fluid Mech. , 161–178. Maertens, A. P. & Weymouth, G. D.
Comput. Methods Appl. Mech. Engrg. , 106–129.
Mittal, R. & Balachandar, S.
Phys. Fluids (8). Noack, B. R.
J. Appl. Math. and Mech. , 223–226. Norberg, C.
J. Fluids and Struct. , 57–96. Pope, S. B.
Turbulent Flows . Cambridge University Press.
Roshko, A.
Schulmeister, James C., Dahl, J. M., Weymouth, G. D. & Triantafyllou, M. S.
J. Fluid Mech. , 743–763.
Smith, L. M., Chasnov, J. R. & Waleffe, F.
Phys. Rev. Lett. (12), 2467–2470. Weymouth, G. D. & Yue, D. K. P.
J. Comp. Phys. , 6233–6247.
Williamson, C. H. K. a Three-dimensional wake transition.
J. Fluid Mech. , 345–407. B. Font Garcia, G. D. Weymouth, V.-T. Nguyen and O. R. Tutty
Williamson, C. H. K. b Vortex dynamics in the cylinder wake.
Annu. Rev. Fluid Mech. , 477–539. Xia, H., Byrne, D., Falkovich, G. & Shats, M.
Nature Physics , 321–324. Xiao, Z., Wan, M., Chen, S. & Eyink, G. L.
J. Fluid Mech.619