2 Problem formulation

This study considers the flow past a circular cylinder with diameter $D$ aligned on the $z$ direction in a 3-D $(35\times 20\times L_{z})D$ rectangular domain, where $L_{z}$ is the non-dimensional span. To study the span effect on the turbulence nature of the wake, the following cases have been considered: $L_{z}=10,\unicode[STIX]{x03C0},1,0.5,0.25,0.1$ as well as a fully 2-D case.

A Reynolds number of $Re=10\,000$ is selected. We define the Reynolds number as,

(2.1) $$\begin{eqnarray}Re=\frac{UD}{\unicode[STIX]{x1D708}},\end{eqnarray}$$

where $U$ is the scaling velocity and $\unicode[STIX]{x1D708}$ is the kinematic viscosity of the fluid. Using this Reynolds number we ensure that the flow is well within the turbulent regime encompassing very different spatial and time scales.

2.1 Governing equations and numerical method

The incompressible viscous fluid motion is described by the continuity equation and the non-dimensional Navier–Stokes momentum equations,

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0, & \displaystyle\end{eqnarray}$$

(2.3) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\boldsymbol{u}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}=-\unicode[STIX]{x1D735}p+Re^{-1}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}, & \displaystyle\end{eqnarray}$$

where $\boldsymbol{u}(\boldsymbol{x},t)=(u,v,w)$ is the velocity vector field, $p(\boldsymbol{x},t)$ is the pressure field, $t$  is the time and $\boldsymbol{x}=(x,y,z)$ is the spatial vector. The initial condition is defined as $\boldsymbol{u}(\boldsymbol{x},0)=(1,0,0)$ in the fluid. The boundary conditions are: a uniform velocity profile on the inlet boundary, a natural convection condition on the outlet boundary, a no-penetration slip condition on the upper and lower boundaries, a periodic condition on the spanwise direction boundaries and a no-slip velocity condition on the cylinder. Periodic boundary conditions on the constricting planes are used as in previous studies on the cross-over of 2-D and 3-D turbulence for obstacle-free flows such as Smith et al. (Reference Smith, Chasnov and Waleffe1996), Celani et al. (Reference Celani, Musacchio and Vincenzi2010) and Biancofiore (Reference Biancofiore2014). This choice is made to avoid the artificial high intensity turbulence enhancements and the deterioration of the 2-D behaviour of the flow when a no-penetration condition is enforced on the constricting planes, as noted on Biancofiore, Gallaire & Pasquetti (Reference Biancofiore, Gallaire and Pasquetti2012) for obstacle-free turbulent wakes. Other studies such as Bao et al. (Reference Bao, Palacios, Graham and Sherwin2016) and Bao et al. (Reference Bao, Zhu, Huan, Wang, Zhou, Han, Palacios, Graham and Sherwin2019) also use a periodic spanwise condition for thick strips modelling long circular cylinders.

The governing equations are numerically solved in a discrete rectilinear grid (details in § 2.2). We couple the discrete velocity and pressure fields with the pressure-Poisson equation. The boundary data immersion method (BDIM) from Weymouth & Yue (Reference Weymouth and Yue2011) has been used in our in-house implicit large eddy simulation (ILES) solver. In short, the BDIM is an immersed boundary (IB) method which maps the governing equations of both the fluid motion and the solid motion into an interface providing a smooth transition of the system variables between the mediums. In this way, the system of fluid and body equations can be solved with a single meta-equation discretised on a rectilinear staggered grid. The BDIM has been extensively validated including cases of flow around bluff bodies such as Maertens & Weymouth (Reference Maertens and Weymouth2015) and Schulmeister et al. (Reference Schulmeister, Dahl, Weymouth and Triantafyllou2017), where the latter also provides a detailed explanation of the numerical method. Our in-house code is second-order accurate in space (using the quadratic upstream interpolation for convective kinematics, a.k.a. QUICK, scheme) and second-order accurate in time (using a predictor–corrector algorithm). The implicit turbulence modelling derives from a flux-limited QUICK treatment of the convective terms, equivalent to optimum finite-volume schemes (for a review on ILES see Adams & Hickel (Reference Adams and Hickel2009)). Hendrickson et al. (Reference Hendrickson, Weymouth, Yue and Yue Yu2019) have validated this ILES approach for intermediate Reynolds numbers similar to the $Re=10\,000$ used in the current work.

Figure 2.

Computational domain sketch non-dimensionalised with the cylinder diameter $D$ . A very fine Cartesian grid domain (depicted in discontinuous lines) surrounds the cylinder and the close and mid-wake regions. A stretched rectilinear grid (depicted in solid lines) transitions from the Cartesian grid to the boundaries.

3 Results and discussion

The flow field is displayed in figure 3 in terms of the instantaneous vorticity component $\unicode[STIX]{x1D714}_{z}$ as the span is varied. The most striking feature is how the coherence of the Kármán vortices increases as the span is reduced. However, even in highly anisotropic geometries such as $L_{z}=0.25$ , small-scale 3-D structures are generated from the cylinder wall.

An important result is that the two-dimensionalisation of these structures is faster (in the sense that it occurs closer to the cylinder) as the domain is constricted because of the geometry constriction and the natural rotation of the Kármán vortices. The combination of these two mechanisms as a two-dimensionalisation method is also found in Smith et al. (Reference Smith, Chasnov and Waleffe1996) and Xia et al. (Reference Xia, Byrne, Falkovich and Shats2011). For the $L_{z}\geqslant 1$ cases, the 3-D small-scales structures detaching from the cylinder wall are not two-dimensionalised as rapidly. In fact, it can be appreciated that only the far wake region of the $L_{z}=1$ case displays a coherent Kármán vortex. This means that less anisotropic geometries promote a direct TKE cascade on the wake so that the 3-D dissipative structures are still sustained far from the cylinder.

Figure 3.

Instantaneous vorticity $\unicode[STIX]{x1D714}_{z}$ (red is positive, blue is negative) at the $z=L_{z}/2$ plane for: (a)  $L_{z}=0$ , (b)  $L_{z}=0.1$ , (c)  $L_{z}=0.25$ , (d)  $L_{z}=0.5$ , (e)  $L_{z}=1$ , (f)  $L_{z}=\unicode[STIX]{x03C0}$ .

Whether the wake turbulence dynamics is two-dimensional or three-dimensional is better captured on the TKE spectra, which can be directly compared to classic turbulence theory. For this, the Taylor’s hypothesis is considered and the temporal spectrum at different points of the wake is calculated. Figure 4(a,b,c) shows the temporal power spectrum (PS) of the $v$ velocity component at $(x,y)=(2,0.8),(4,0.8),(8,0.8)$ . The Welch method (using a time signal of 500 units split in 6 parts with 75 % overlapping) has been employed to compute the spectra at 8 different points along the span for each $(x,y)$ point. These spectra are then averaged resulting in a single spectrum for each case.

First, note that all of the spectra display a peak around the 0.2 non-dimensional frequency corresponding to the non-dimensional Strouhal number, $S_{t}=f_{s}D/U$ , where $f_{s}$ is the vortex-shedding frequency. A smaller harmonic peak around 0.4 non-dimensional frequency is also found for the spectra at $(x,y)=(2,0.8),(4,0.8)$ . Second, the spectrum at the closest analysed point to the cylinder (figure 4 a) displays a 3-D turbulence behaviour with a $-5/3$ decay rate with the exception of the pure 2-D and the $L_{z}=0.1$ cases. For the latter cases, a decaying slope around $-11/3$ is found. These 2-D flow spectra are steeper than the $-3$ rate predicted by the classical 2-D turbulence theory (Kraichnan Reference Kraichnan1967) because of the interaction of the coherent large-scale Kármán vortices, in agreement with the finds of Dritschel et al. (Reference Dritschel, Scott, Macaskill, Gottwald and Tran2008) and Biancofiore (Reference Biancofiore2014). The filamentary vorticity (filaments of vorticity around the coherent vortices) is likely to be destroyed by the interaction 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 effect which limits the range of scales of incoherent filamentary vorticity (Gilbert Reference Gilbert1988). On the other hand, as a $-5/3$ decay rate is captured for the $L_{z}=0.25$ case, it can be argued that 3-D turbulence is being generated from the cylinder wall even for highly reduced spans.

Figure 4.

(a–c) Vertical velocity component temporal power spectrum at different $(x,y)$ locations on the wake: $(a)=(2,0.8)$ , $(b)=(4,0.8)$ , $(c)=(8,0.8)$ . The PS lines of each case are shifted a factor of 10 for clarity and the vertical axis ticks correspond to the $L_{z}=\unicode[STIX]{x03C0}$ case. The dashed lines have a $-11/3$ slope and the dotted lines have a $-5/3$ slope. (d–f) Two-point correlations along $z$ at the same $(x,y)$ locations as (a–c) respectively. The correlation value for a given $d$ corresponds to the averaged value of the multiple correlations of pairs of points separated a distance $d$ along the span.

The $L_{z}=0.25$ and $L_{z}=0.5$ cases feature a decay rate that transitions from $-5/3$ to $-11/3$ as the spectra are computed further downstream from the cylinder. In particular, both cases have $-5/3$ slopes in figure 4(a) and $-11/3$ in figure 4(c), quantifying the turbulent two-dimensionalisation in the rotating wake. Also note the coexistence of both 2-D and 3-D turbulence features for the $L_{z}=0.25,0.5$ and $L_{z}=1$ cases in figures 4(b) and 4(c) respectively. The low-frequency structures behave mostly two-dimensionally (decay rate between $-3$ and $-11/3$ , resulting from the presence of less coherent 2-D structures than the pure 2-D case) up to a certain point where a $-5/3$ rate is briefly recovered. Hence, high-frequency structures interact in a 3-D fashion while low-frequency structures interact two-dimensionally. This finding is in agreement with Smith et al. (Reference Smith, Chasnov and Waleffe1996) and Celani et al. (Reference Celani, Musacchio and Vincenzi2010).

Additionally, two-point correlations along the span have been analysed at the same $(x,y)$ locations as the PS plots (figure 4 d–f). Given a distance $d$ along $z$ ranging from 0 to $L_{z}/2$ , the two-point correlation is calculated with the temporal signals of the vertical velocity component at multiple pairs of points (namely $v_{1}$ and $v_{2}$ ) as follows,

(3.1) $$\begin{eqnarray}\langle v_{1}(\boldsymbol{x},t),v_{2}(\boldsymbol{x}+\boldsymbol{r},t)\rangle =\frac{\text{cov}(v_{1},v_{2})}{\sqrt{\text{cov}(v_{1},v_{1})\text{cov}(v_{2},v_{2})}},\end{eqnarray}$$

where the distance vector is defined as $\boldsymbol{r}=(0,0,d)$ . The multiple correlation coefficients for a given $d$ are then averaged corresponding to a data point in the plots.

Very close to the cylinder (figure 4 d), the correlation coefficient quickly decreases with increasing $d$ for $L_{z}>0.1$ . This indicates the presence of 3-D structures near the body as also noted on the velocity spectrum plot counterpart. Also, the decrease is more pronounced as the span increases. For $L_{z}=\unicode[STIX]{x03C0}$ , it is worth noting a local correlation coefficient maximum around $d\approx 0.9$ . This distance approximately corresponds to the Mode B instability wavelength $(\unicode[STIX]{x1D706}_{z})$ . 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 Reference Chyu and Rockwell1996), the correlation increase is not as significant as at lower $Re$ . Still, $L_{z}=\unicode[STIX]{x03C0}$ is the only case displaying such phenomena because of the spanwise boundary conditions’ periodicity, which only allow the instability to develop if $L_{z}>\unicode[STIX]{x1D706}_{z}$ (the $L_{z}=1$ case might be too critical to display such phenomena considering also its intermittent nature). The correlation coefficient increases when calculated further downstream as shown in figures 4(e) and 4(f), evidencing again the wake two-dimensionalisation.

Figure 5.

TKE spatial plots. The TKE is computed from the normal Reynolds stresses and averaged on the vertical direction: (a) total TKE, (b) ratio of the spanwise component $\overline{w^{\prime }w^{\prime }}$ over the total TKE.

In summary, the transition to a 2-D wake is found at a certain point along the wake on all the cases with $L_{z}\leqslant 1$ . The combination of the large-scale rotation from the Kármán vortices plus the geometry constriction is mainly responsible for this phenomenon. The main physical mechanism differing among the compared cases is the ability of the flow to develop Mode B-like 3-D structures in the wake as a result of a sufficiently long span. The current $Re$ regime is characterised by a transition to turbulent flow at the shear layer (i.e. the TrSL2 regime) as reported in Bloor (Reference Bloor1964) and Kourta et al. (Reference Kourta, Boisson, Chassaing and Ha Minh1987). We argue that when the span is too short for the Mode B instability to develop and thereby sustain these 3-D turbulent structures, the stratification effect of the Kármán vortices leads to the more coherent and energised wake seen in figure 4(d–f).

Next, the TKE along the $x$ direction, the Lumley triangle of turbulence and the ratio between the vortex-stretching and advection terms are examined to further support the observed phenomena. The TKE is defined as,

(3.2) $$\begin{eqnarray}\text{TKE}={\textstyle \frac{1}{2}}(\overline{u^{\prime }u^{\prime }}+\overline{v^{\prime }v^{\prime }}+\overline{w^{\prime }w^{\prime }}),\end{eqnarray}$$

where $\overline{\cdot }$ denotes a time average plus a spanwise average and the subscript $\cdot ^{\prime }$ denotes a fluctuating quantity such as $a^{\prime }=a-\overline{a}$ . The six components of the Reynolds stress tensor $\overline{u_{i}^{\prime }u_{j}^{\prime }}$ have been computed using the following relation,

(3.3) $$\begin{eqnarray}\overline{a^{\prime }b^{\prime }}=\overline{ab}-\overline{a}\overline{b}.\end{eqnarray}$$

Figure 5 shows the streamwise spatial distribution of the TKE averaged along the $y$ direction from $-L_{y}/2$ to $L_{y}/2$ (noted as $\cdot |_{y}$ ). From a general point of view, it can be observed that the total TKE (figure 5 a) peaks right after the recirculation region noting that the latter increases slightly with the span. Also, the total TKE increases as the span is reduced because of the 2-D vortex-merging processes that generate larger and more energised vortical structures. The contribution of the spanwise normal stress to the total TKE increases with the span as shown in figure 5(b). Also, a decay of $\overline{w^{\prime }w^{\prime }}$ right after being generated from the cylinder wall can be noted and it becomes faster as the span is constricted.

The effect of the span on the TKE compared to the lift coefficient root mean square (r.m.s.) value $(\overline{C}_{L})$ displays a quasi-linear relation as shown in figure 6. Note that the drop on both values is more sensitive to the span constriction at the range where both 2-D and 3-D turbulence dynamics co-exist, i.e. $0.25\leqslant L_{z}\leqslant 1$ . On the other hand, a very small change of both values can be appreciated when 3-D turbulence fully dominates the wake, i.e. $L_{z}\geqslant \unicode[STIX]{x03C0}$ . Furthermore, as shown in figure 5(a), highly constricted domains yield large values of TKE because of the energised 2-D large-scale vortical structures present at the wake. Even when small-scale 3-D structures are present in the close wake region and coexist with the large 2-D structures, the values of both the TKE and the $\overline{C}_{L}$ spike.

Figure 6.

Effect of the span constriction on the TKE and the $\overline{C}_{L}$ . The latter is calculated as $\overline{C}_{L}=|2F_{y}/(\unicode[STIX]{x1D70C}U^{2}DL_{z})|_{rms}$ , where $F_{y}$ is the vertical lift force and $\unicode[STIX]{x1D70C}$ is the constant fluid density.

Figure 7.

(a) $C_{L}$ and $C_{D}$ temporal signals. The latter is calculated as $C_{D}=2F_{x}/(\unicode[STIX]{x1D70C}U^{2}DL_{z})$ and the former as detailed in figure 6 (without the r.m.s. norm). (b) Upper $\unicode[STIX]{x1D703}_{u}$ and lower $\unicode[STIX]{x1D703}_{l}$ separation angles temporal signal. The separation angle is calculated from the front stagnation point of the cylinder to the free-shear layer separation point (vorticity practically 0 at the wall). (c) Correlation between the lift coefficient and the difference between $\unicode[STIX]{x1D703}_{u}$ and $360-\unicode[STIX]{x1D703}_{l}$ . Left: $L_{z}=\unicode[STIX]{x03C0}$ . Right: $L_{z}=0.5$ .

In figure 7, the temporal signals of the lift and drag coefficients as well as the upper and lower separation angles of the free-shear layer are displayed for the $L_{z}=0.5$ and the $L_{z}=\unicode[STIX]{x03C0}$ cases. The oscillation of the separation points increases as the span is constricted inducing larger forces on the cylinder. A very high correlation of the upper and lower separation points angle with the lift coefficient is shown as well. Again, the lack of the Mode B instability on the constricted cases yields more coherent 2-D vortical structures in the near wake region. Therefore, the small-scale 3-D structures which normally dissipate most of the kinetic energy in 3-D turbulence are not present leading to high-intensity vortices. This can be quantified by the enstrophy ( $\unicode[STIX]{x1D6FA}$ ) of the span-averaged spanwise vorticity in the near wake region ${\mathcal{D}}$ ( $x\in [0.55,2.1]$ , $y\in [-0.8,0.8]$ ), which is time averaged for the same temporal length as the body forces signals as follows,

(3.4) $$\begin{eqnarray}\unicode[STIX]{x1D6FA}=\int _{{\mathcal{D}}}\frac{1}{t_{2}-t_{1}}\int _{t_{1}}^{t_{2}}\frac{1}{L_{z}}\int _{0}^{L_{z}}\unicode[STIX]{x1D714}_{z}^{2}\,\text{d}z\,\text{d}t\,\text{d}{\mathcal{D}}.\end{eqnarray}$$

It can be observed in table 1 that the enstrophy increases as the span is reduced, which can be understood as an increase of the rotational energy of the flow.

Table 1.

Near body enstrophy as defined in (3.4) for the different span cases. $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x03C0}}$ is the enstrophy of the $L_{z}=\unicode[STIX]{x03C0}$ case.

The highly energised coherent vortices forming at the near wake region for the constricted cases induce a larger convective force on the free-shear layer. This translates to the large oscillations observed in figure 7 and, ultimately, to the forces induced on the cylinder. With this, it can be argued that the coherent 2-D structures have a greater impact on the forces induced to the cylinder than the 3-D small-scale structures when both are present.

Lumley & Newman (Reference Lumley and Newman1977) proposed the Lumley triangle of turbulence which provides a way to classify the anisotropic state of turbulence. The anisotropy property of the Reynolds stress tensor can be extracted with,

(3.5) $$\begin{eqnarray}\unicode[STIX]{x1D623}_{ij}=\frac{\overline{u_{i}^{\prime }u_{j}^{\prime }}}{\overline{u_{k}^{\prime }u_{k}^{\prime }}}-\frac{1}{3}\unicode[STIX]{x1D6FF}_{ij},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}$ is the Kronecker delta and $\unicode[STIX]{x1D623}_{ij}$ is the anisotropic Reynolds stress tensor (which evidently vanishes for isotropic turbulence). This dimensionless and traceless tensor has two non-zero invariants, $II=-\unicode[STIX]{x1D623}_{ij}\unicode[STIX]{x1D623}_{ji}/2$ and $III=\unicode[STIX]{x1D623}_{ij}\unicode[STIX]{x1D623}_{jk}\unicode[STIX]{x1D623}_{ki}/3$ . These invariants are often rewritten as $\unicode[STIX]{x1D702}^{2}=-II/3$ and $\unicode[STIX]{x1D709}^{3}=III/2$ to better appreciate the nonlinear behaviour of the trajectories of return to isotropy of homogeneous turbulence (Choi & Lumley Reference Choi and Lumley2001). It has been shown that all the possible turbulence states are mapped within the triangle (Lumley & Newman Reference Lumley and Newman1977; Lumley Reference Lumley1978).

The different states of turbulence are classified in the triangle as follows: the top right elbow indicates a one-dimensional (or one component) state with a single non-zero eigenvalue (the eigenvalues can be understood in physical terms as the normal stresses in the principal axes of the anisotropic Reynolds stress tensor). The top left elbow indicates a 2-D isotropic state where one eigenvalue vanishes and the two remaining are equal. The top curve connecting the elbows indicates a 2-D turbulence state where one eigenvalue vanishes and the addition of the two remaining eigenvalues is constant. The left and right straight lines correspond to a negative or positive $(\unicode[STIX]{x1D709})$ axisymmetric state since one eigenvalue 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 3-D isotropic turbulence since all of the anisotropic tensor eigenvalues vanish.

Figure 8.

Lumley’s triangle constructed across the wake width at: (a)  $x=2$ , (b)  $x=4$ , (c)  $x=6$ , (d)  $x=8$ .

Figure 8 displays the Lumley triangle constructed across the wake width at different $x$ locations. Hence, every point in the triangle corresponds to a point in the domain and the collection of points for each case represents the collection of vertically aligned points in the domain at different $x$ locations on the wake. It can be appreciated that all of the points are located within the triangle, therefore all of computed Reynolds stresses are realisable (i.e. have positive and real eigenvalues). In general, most of the cases transition from one state to another state of turbulence across the wake width. Only for the almost 2-D case $(L_{z}=0.1)$ , the state of turbulence is always 2-D (or two-component), since $\overline{w^{\prime }w^{\prime }}$ is effectively negligible everywhere.

As the triangle is constructed further downstream on the wake, the trajectories (or collection of points) of the $L_{z}=0.25$ and $L_{z}=0.5$ cases move closer to the 2-D turbulence state location (upper curve). This emphasises once again the wake two-dimensionalisation caused by the large-scale vortical structures on cases with critical span which contain a cross-over of 2-D and 3-D turbulence, as shown in previous results. In contrast, the $L_{z}=1$ and $L_{z}=\unicode[STIX]{x03C0}$ cases remain approximately at the same $\unicode[STIX]{x1D702}$ region showing that the two-dimensionalisation is not as effective.

Additionally, the trajectories present a negative axisymmetric almost two-component state for the locations far from the wake centreline, i.e. the location close to $y=\pm L_{y}/2$ , since one of the normal turbulent stresses $\overline{w^{\prime }w^{\prime }}$ is smaller than the others. On the centreline, $\overline{v^{\prime }v^{\prime }}$ is larger than the other stresses, causing a shift to the positive axisymmetric state.

A comparison of the different cases at the same $x$ location also shows a more noticeable 2-D turbulence state as the span is constricted. This difference is less noticeable close to the cylinder (figure 8 a), where even the $L_{z}=0.5$ case trajectory resembles the $L_{z}=\unicode[STIX]{x03C0}$ case. Again, this evidences that 3-D turbulence is present close to the wall even in considerably constricted cases.

Consider now the vorticity transport equation (VTE) which can be written as,

(3.6) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D74E}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D74E}=\unicode[STIX]{x1D74E}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}+Re^{-1}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D74E},\end{eqnarray}$$

where $\unicode[STIX]{x1D74E}(\boldsymbol{x},t)=(\unicode[STIX]{x1D714}_{x},\unicode[STIX]{x1D714}_{y},\unicode[STIX]{x1D714}_{z})$ is the vorticity vector field defined as $\unicode[STIX]{x1D74E}=\unicode[STIX]{x1D735}\times \boldsymbol{u}$ . The vortex-stretching term, $\unicode[STIX]{x1D74E}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}$ , is often pointed to as the term responsible for the direct energy cascade of the TKE. The stretching of a vortex tube causes a reduction on its diameter while increasing the rotation speed of the vortex by conservation of angular momentum. This term vanishes in the 2-D formulation of the VTE since the stretch of the vortex tube is perpendicular to the plane of rotation. From this mathematical and physical difference, different turbulence dynamics is captured in 2-D or 3-D computations and, therefore, 3-D 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 in the vertical direction. In figure 9(a), it can be observed that the vortex-stretching term decays faster while 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 $R$ in figure 9(b) shows that the vortex-advecting term decays faster (because of the wake momentum deficit) than the vortex-stretching term along the streamwise direction up to $x=6$ where the ratio is kept constant. It can also be observed that the vortex-stretching term becomes as important as the vortex-advecting term with increasing span.

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 mean vortex-advecting term averaged along the vertical direction.