3.1. Numerical set-up

The simulations are performed in a periodic box of volume $V_e = L_e^3 = (4b_0)^3$ and using a pseudospectral code with phase shift dealiasing (Canuto et al. Reference Canuto, Yousuff Hussaini, Quarteroni and Zang2006). First, a forced homogeneous isotropic turbulence (HIT) simulation is performed to obtain a converged turbulence field. Then, a model 2VS is added to this precursor field. We consider vortices with a core size of $r_c/b_0 = 0.050$, as in Bricteux et al. (Reference Bricteux, Duponcheel, De Visscher and Winckelmans2016), and we first use the BH model for the initial circulation distribution within each vortex. The grid size used for that pseudospectral LES is $256^3$ (thus a grid spacing $\varDelta =b_0/64$); thanks to the spectral properties of the code, this is equivalent to a $512^3$ simulation when using our fourth-order finite difference code in § 4.

We do not claim that the dynamics of all small scales inside the vortex cores are here fully resolved; that would require a finer grid, and likely also running the simulation in direct numerical simulation mode: something that would be extremely costly at the high Reynolds number considered here. The present study focuses on obtaining a turbulent state of two counter-rotating vortices, and using LES (thus necessarily under-resolved for the turbulence, with subgrid scales (SGSs) not captured). The turbulence will be obtained for radii greater than $r_c$ and up to roughly $b_0/2$, where the turbulent vorticity of one vortex will then strongly interact with that of the other one, making the total circulation of each vortex slowly decrease. The turbulent part of the vortices is the part that we focus on.

We also stress that the Crow instability, that has a slow growth rate, and which is maximum for the wavelength $\lambda _x\simeq 8 b_0$, will not develop here; this is on purpose since we aim to obtain a fully developed T-2VS that, in reality, would form within less than one $t_0$ behind a wing, through the rollup of the near wake; in such a short time, the Crow instability does not have time to develop. We refer to De Visscher et al. (Reference De Visscher, Bricteux and Winckelmans2013a) for a previous investigation performed using a 2VS subjected to various levels of atmospheric turbulence and stratification, with a periodic box of $(8b_0)^3$ and using the same pseudospectral LES code, with the same core size and grid resolution, and using BH vortices for the initial condition. Those simulations focused on capturing the evolution of the wake up to its complete demise, and also on developing a simplified operational model. Simulations of wake demise due to the Crow instability in atmospheric turbulence, and also combined with stratification effects, were also presented in Misaka et al. (Reference Misaka, Holzäpfel and Gerz2015).

As to the energy dissipation rate, $\epsilon _0$ (units of $\textrm {{m}}^2\ \textrm {{s}}^{-3}$), of the HIT perturbation field used here, it must be characterised relatively to the 2VS global length and velocity scales; hence using the dimensionless $\epsilon _0^\ast = (\epsilon _0 b_0)/W_0^3$. We obtain that $\epsilon _0^\ast = 2.42 \times 10^{-4}$; this is indeed a case of very weak turbulence.

The initial kinetic energy of the mean cross-flow velocity field, noted $(V,W)$, is

(3.1)\begin{equation} E_{2D,0} = \int_{S_e} \frac{\left(V^2 + W^2\right)}{2}\,{\rm d}S, \end{equation}

where $S_e = 4b_0 \times 4b_0$. Its units are $\textrm {{m}}^4\ \textrm {{s}}^{-2}$. We obtain, in dimensionless form, $E_{2D,0}/(W_0b_0)^2 = 15.38$. The kinetic energy of the total initial flow field (HIT field added to the BH-2VS, thus with $(u,v,w)=(u', V+v', W+w')$) is, after longitudinally averaging in order to compare with $E_{2D,0}$,

(3.2)\begin{equation} E_0 = \frac{1}{L_e} \int_{V_e} \frac{\left(u^2 + v^2 + w^2\right)}{2} \,{\rm d}V = E_{2D,0} + E_{f,0}. \end{equation}

We obtain $E_0/(W_0b_0)^2 = 15.44$. Thus, the kinetic energy associated with the added fluctuations amount to only approximately 0.4 % of $E_0$. Nevertheless, and as we will see later, this very weak HIT field added to the 2VS will suffice to lead, after a long transient, to a fully developed turbulent 2VS.

For completeness, we also report the energy of the same BH-2VS when considered in unbounded space. Since the core size is small, this energy can be estimated accurately by using the formulae developed in de Bruin & Winckelmans (Reference de Bruin and Winckelmans2005),

(3.3)\begin{equation} E_{2D,0}^\infty = \frac{\varGamma_0^2}{2{\rm \pi}}\left(\log\left(\frac{b_0}{r_c}\right)-C\right) \quad\Rightarrow\quad \frac{E_{2D,0}^\infty}{(W_0b_0)^2} = 2{\rm \pi}\left(\log\left(\frac{b_0}{r_c}\right)-C\right) , \end{equation}

where $C=\frac 12$ when using BH vortices (note that $C=-0.05617$ when using LO vortices). We obtain $E_{2D,0}^\infty /(W_0b_0)^2 = 15.68$. Thus only a 2 % difference with the value for the 2VS put in a periodic domain; comforting us that the size $L_e = 4 b_0$ of the periodic computational domain used here is large enough for the purpose of our study. The descent velocity $W_0$ of the 2VS in the periodic domain is also measured and compared with that in an unbounded domain; it is a bit slower, by roughly 1.3 %.

Finally, we also note that the present BH-2VS has roughly the same energy as that of the near wake vortex sheet that would emanate from a wing with an elliptical loading; indeed, the lifting line theory gives $E_{2D,{ellip}}=\varGamma _0^2\,({{\rm \pi} }/{8})$, and thus $E_{2D,{ellip}}/(W_0b_0)^2 = {{\rm \pi} ^3}/{2} = 15.50$. As to the LO-2VS with the same $r_c/b_0=0.050$, its energy is significantly higher: $19.18$, which would correspond to a wing that is significantly more outboard loaded than one with elliptical loading. To have a LO-2VS that has the same energy as that of the present BH-2VS would require us to use $r_c/b_0\simeq 0.087$.

3.3. Sensitivity analysis and circulation distribution of the T-2VS

To investigate if the obtained equilibrium T-2VS is fairly generic, a sensitivity analysis is further performed. Case 0 is the ‘baseline simulation’ discussed so far, using BH model vortices as initial conditions and perturbed using a very weak HIT. Case 1 corresponds to using the more realistic PW model. The energy of that PW-2VS is then roughly the same as that of the baseline (approximately 3 % higher). Case 2 corresponds to using the LO model, hence with an energy significantly higher than that of the baseline (22 % higher). Finally, Case 3 corresponds to the BH-2VS perturbed using an HIT with $10 \times$ higher dissipation rate than for the other cases; yet still corresponding to weak HIT).

Figure 3 displays the evolution of the dissipation rate for the various cases. We see that, regardless of the initial circulation distribution, the turbulent equilibrium reached is essentially the same, with a similar value of $\mathcal {E}$ when at equilibrium (compare Cases 0, 1 and 2). Also, and as expected, the stronger HIT forcing leads to a shorter transient to obtain a similar turbulent equilibrium (compare Cases 0 and 3).

The obtained circulation distributions at turbulent equilibrium (here measured one time unit after the time of maximum energy dissipation rate) are also reported in figure 4. For each case, we display the distribution measured for each of the vortices (left-hand and right-hand), and each scaled using its measured core size and total circulation. This is done on purpose, in order to also highlight the natural variations that can be observed in the results, even in the present well-controlled simulation set-up (results that are also affected by the limitations of the procedure described in § 3.2 for measuring the circulation distribution and the core size).

Figure 4.

Circulation distribution of the obtained left-hand and right-hand vortices in the T-2VS, as measured at turbulent equilibrium (and each scaled using its measured $r_c$ and $\varGamma _{max}$): (a) Case 0 (solid), Case 1 (dashed–dotted), Case 2 (dashed); (b) Case 0 (solid), Case 3 (dashed).

We see that the obtained distributions are essentially the same for all cases, except Case 2 which as a steeper growth for small $r$. Recall that this case was started using a 2VS with significantly more energy than that of the other cases, due to the LO vortices having more circulation within $r_c$. Hence, the obtained T-2VS also has more energy and vortices with a steeper growth of the circulation profile for small $r$.

We clearly observe a two-scale behaviour in each of the obtained circulation distributions: a Gaussian inner part (up to $r\lesssim 0.8r_c$) with a rapid growth of the circulation, and an outer part for $r\gtrsim 4r_c$ (that is $r\gtrsim 0.2b_0$) with a slow growth, and in accordance with the two-scale models (P and PW), see figure 5 for the vortices of the baseline (Case 0) T-2VS. It is not possible to define a precise circulation distribution for the obtained vortices, except by taking the average on both vortices. Indeed, there is already some difference between the left-hand and right-hand vortex distributions when each is scaled using its own measured core size. This difference can be as large as the difference between two cases (compare Cases 0 and 3). As expected, the circulation profiles obtained when using the BH (Case 0) or PW (Case 1) models for the initial 2VS are very close; they will be retained for wings with elliptical loading, or close to it. That obtained when using the LO model (Case 3) could be appropriate for wings that are highly outboard loaded.

Figure 5.

Analysis of the outer and inner scalings for the circulation distributions: T-2VS of Case 0 (solid); partially completed rollup ($t/t_0\simeq 1$) of a wake started using five-hole probe measurements of the velocity field just behind a large A340-type generic model in landing configuration (dash–dot; using simulation data by UCLouvain in the AWIATOR project). The straight lines indicate the slope $2$ and the slope $q=0.75$ (here $\beta _o \simeq 6.0$).

We stress again that the present simulations cannot be used to predict the core size of rolled up vortices in real wakes: the grid resolution used here, together with the dealiased pseudospectral method, only allowed to simulate vortices with $r_c \simeq 0.05b_0$.

We also speculate that the two-scale behaviour observed in the circulation distribution could be representative of the distribution for real wakes developing behind a wing at high Reynolds number, or aircraft, after full completion of the rollup. The core size $r_c$ would, however, be smaller than here. For instance, simulations were carried out in the AWIATOR project of the EU using the same numerical methodology as here, but started using five-hole probe measurements of the 3-D velocity field just behind a large A340-type generic model in landing configuration. The Reynolds number was $Re_\varGamma \simeq 3.6 \times 10^5$ and the grid resolution used was $\varDelta =b_w/150$, as we needed to capture the fine details of the measured near wake. Various configurations were simulated, including a baseline configuration with both inner and outer flaps deflected at $26^{\circ }$. The simulation was carried up to $t/t_0=1$; by then, the half-plane circulation was still conserved, thus the vortices were not aged yet. The wake was such that $b_0/b_w\simeq 0.81$. The core size of the vortices was obtained as $r_c/b_0 \simeq 0.038\unicode{x2013} 0.040$ (it was already established by $t/t_0 \simeq 0.3$ and it remained constant after that). This simulation was, however, not run up to a full completion of the rollup (i.e. where the half-plane circulation starts to decrease), which would here take up to $2t_0$, as the near wake vorticity field of a large aircraft in landing is much more complex than that in cruise. Using, nevertheless, the circulation profile measured at $t/t_0=1$, we obtain that the outer part with the slow growth is also in accordance with the P or PW model, see figure 5. It even appears to be valid for $r\gtrsim 2.5r_c$ (that is $r\gtrsim 0.1b_0$). This wake is, however, not fully rolled up, hence the span of the outer region could still change.

3.4. Definition of circulation diagnostics

Using the measured vortex circulation distribution $\varGamma (r)$, we define the ‘total circulation’ that is associated with that vortex, $\varGamma _{tot}$. The usual way to define it is to use $\varGamma _{max}$, the maximum of the measured $\varGamma (r)$ distribution. Another way to estimate the total circulation is to only consider the vorticity contained within the circle of radius $r=b_0/2$; we use the notation $\varGamma _{b_0/2}$. For cases where the vorticity is widespread, as for the obtained T-2VS, $\varGamma _{b_0/2}$ is a bit lower than $\varGamma _{max}$ (by approximately 1.5 %–2 %).

For the baseline case, the total circulation was conserved up to $t/t_0\simeq 3.5$. The circulation decay that happens later on will be presented and discussed later, in § 4, when comparing with the case where the same T-2VS interacts with the ground.

Another global diagnostic, commonly used in the LIDAR (light detection and ranging) community, is the so-called ‘$\varGamma _{5-15}$ circulation’, which is defined as

(3.4)\begin{equation} \varGamma_{5-15}= \frac{1}{b_w/6}\int_{b_w/12}^{b_w/4} \varGamma(r)\, {\rm d}r . \end{equation}

This definition is connected to processing algorithms of LIDAR data relating to aircraft wakes. Originally, the wake produced by heavy aircraft of wingspan $b_w \simeq 60$ m were considered, also in landing configuration (thus with wings more inboard loaded than the elliptical loading). The ‘5–15’ used in the subscript then refers to the range of distances to the vortex centre, ranging from $b_w/12=5$ m to $b_w/4=15$ m. The motivation of this average is that LIDARs cannot resolve well the velocities within the vortex core ($r < 5$ m) and that the 15 m distance is well within $b_0/2$, even for such aircraft in landing. We note that, for medium aircraft ($b_w\simeq 35$ m), the 15 m distance reaches beyond $b_0/2$, but not by much; at least when the wing has a loading close to elliptical loading: the region where the neighbouring vortex reduces the circulation measurement is then usually not reached with the 15 m distance. Nevertheless, (3.4) is the generalisation of the definition for a wing of general span $b_w$; it is also the definition used here, as our simulation did not assume any specific wingspan. Finally, to connect $b_0$ to $b_w$, we here will further assume that the wing that produced the wake has an elliptical loading, and we thus use $b_0/b_w ={\rm \pi} /4$ in what follows.

A last global diagnostic is $\varGamma _{15}$ which, following the generalisation above, is taken as $\varGamma (r= b_w/4)$.

The diagnostics defined above will be all be presented and discussed later, in § 4, when comparing with the case where the T-2VS further interacts with the ground.

4.1. Numerical set-up

This simulation is performed using our fourth-order finite difference code: it uses the discretisation schemes of Vasilyev (Reference Vasilyev2000) on a staggered mesh, and it is energy conserving up to fourth order (Morinishi et al. Reference Morinishi, Lund, Vasilyev and Moin1998; Vasilyev Reference Vasilyev2000). The SGS model used here is the multiscale variant of the wall-adapting local eddy-viscosity (WALE) model, see Bricteux, Duponcheel & Winckelmans (Reference Bricteux, Duponcheel and Winckelmans2009). This model solely acts on the high-pass filtered LES field (hence its multiscale character); moreover, its SGS viscosity is computed using the WALE scaling of Nicoud & Ducros (Reference Nicoud and Ducros1999) but with the high-pass filtered LES field. This model has been shown to exhibit the proper near-wall behaviour (essential for wall-resolved simulations) and to also handle vortex flows (by not dissipating artificially the vortex cores). It was already applied to the study of wake vortices put in ground effect and with a turbulent crosswind by Bricteux et al. (Reference Bricteux, Duponcheel, De Visscher and Winckelmans2016).

The size of the computational domain is $L_x = 4 b_0 \times L_y = 8 b_0 \times L_z = 4 b_0$. No slip is imposed at the wall $z=0$, and slip is used at $z= L_z$. Periodic boundary conditions are used in $x$ and $y$. The mesh is $768 \times 1280 \times 1536$ grid points (thus roughly $1.5\times 10^9$). It is uniform in $x$ and $y$ (with $\Delta x/b_0 =1/192$ and $\Delta y/b_0 =1/160$) and stretched in $z$, with $\Delta z_{min}/b_0 \simeq 6.0\times 10^{-4}$ and $\Delta z_{max}/b_0 \simeq 4.7\times 10^{-3}$. The resolutions in $x$ and $y$ are thus, respectively, $3 \times$ and $2.5 \times$ those used in the pseudospectral code; which is more than what is required to obtain the same accuracy (a factor of 2 would have sufficed). As we want to also capture well the streaks that will develop at the ground when the T-2VS interacts with it, the resolution in $x$ was taken finer than what is required for the OGE phase.

To insert the T-2VS field in the new numerical domain, we use the streamfunction $\boldsymbol {\psi }$ (i.e. $\boldsymbol {u} = \boldsymbol {\nabla }\times \boldsymbol {\psi }$) obtained by solving the Poisson equation ($\nabla ^2\boldsymbol {\psi } = -\boldsymbol {\omega }$ with $\boldsymbol {\omega } = \boldsymbol {\nabla }\times \boldsymbol {u}$ the vorticity field). The vorticity field of the T-2VS OGE is interpolated to the new grid where the domains intersect, and is set to zero elsewhere. The Poisson equation is solved using slip-wall boundary conditions (i.e. $\psi _x = \psi _y = {\partial \psi _z}/{\partial z} = 0$). The obtained velocity field is then used as the initial condition for the present simulation with the no-slip condition imposed at the ground, hence thin boundary layers develop there. We stress that this process is here quite smooth, thanks to the high release altitude of the T-2VS.

The maximum wall friction is then reached at $t/t_0 \simeq 2.5$, which occurs a bit earlier than when the vortices reach their lowest altitude. The mesh is then such that $\Delta x_{max}^+ \simeq 31$, $\Delta y_{max}^+ \simeq 37$ and $z_{1_{max}}^+ \simeq 1.8$; which are sufficient for wall-resolved simulations (see Sagaut Reference Sagaut2006). The simulation is run up to $t/t_0$ a bit larger than $5$.

As a basis for comparison, another wall-resolved simulation is performed using the BH-2VS as initial conditions. It is perturbed using a random velocity excitation (white noise) whose amplitude is set to 0.1 % of the maximum azimuthal velocity of the vortices, $u_\theta (r_c)$, which also corresponds to roughly $1.0$ % of $W_0$. The small perturbations will trigger 3-D instabilities when the 2VS interacts with the ground, which will also lead, after rebound, to a transition towards a turbulent vortex flow. That simulation is run up to $t/t_0$ a bit smaller than $5.9$; because it was less turbulent initially, we knew that its decay would be a bit delayed, hence we ran it longer.

Finally, an OGE simulation of the T-2VS is also run using the same code, thus again in the periodic domain of size $(4 b_0)^3$ but now using $512^3$ grid points (i.e. using a resolution twice finer than with the pseudospectral code so as to have the same level of accuracy); those results will be compared with those of the IGE simulation.

4.2. Analysis of the behaviour IGE

The temporal evolutions of both the T-2VS and the BH-2VS are displayed in figure 6, using a volume rendering of the vorticity magnitude field, and in figure 7 using the longitudinally averaged axial vorticity field (which corresponds to a Reynolds-averaged view of that field).

Figure 6.

Volume rendering of the vorticity magnitude at $t/t_0=1$, $2$, $3$, $4$, $5$: T-2VS (a,c,e,g,i); BH-2VS (b,d,f,h,j).

Figure 7.

Mean axial vorticity, $\langle \omega _x\rangle b_0^2/\varGamma _0$, at $t/t_0=1$, $2$, $3$, $4$, $5$: T-2VS (a,c,e,g,i); BH-2VS (b,d,f,h,j).

In the BH-2VS case, the global behaviour IGE is artificial, as there is no physical turbulence present in the initial condition (only some white noise used to trigger instabilities). The boundary layer is seen to separate in a laminar and quasi-2-D fashion, as also confirmed when examining the wall vorticity in figure 8. Each separated shear layer rolls up to form a coherent secondary vortex. That vortex then quickly becomes unstable through short-wave instabilities, and then turbulent, see figures 6 and 7 at $t/t_0 = 3$ and $4$.

Figure 8.

Axial vorticity at the wall, $\omega _x\, b_0^2/\varGamma _0$, at $t/t_0=2$ and $3$: T-2VS (a,b); BH-2VS (c,d).

The situation is thus different from that in Stephan et al. (Reference Stephan, Holzäpfel and Misaka2013), Holzäpfel et al. (Reference Holzäpfel, Tchipev and Stephan2016) and Bricteux et al. (Reference Bricteux, Duponcheel, De Visscher and Winckelmans2016), where the initial condition was made of non-turbulent model vortices put in a turbulent boundary layer flow. In the present case, the transition to turbulence is solely due to short-wave instabilities that develop in the secondary vorticity, quickly after the separation of the boundary layer. Similarly to the lower Reynolds cases, the turbulent secondary vortices start to orbit the primary vortices and they interact with them. This interaction starts with azimuthal vortex filaments from the secondary vortices which are stretched and wrapped around the primary vortices, see figure 6. This is quickly followed by turbulent mixing, which finally results, around $t/t_0 = 5$, in the primary vortices having become turbulent (except in the core region), and still being fed continuously by opposite sign vorticity from the ongoing boundary layers separation.

The T-2VS is seen to behave quite differently from the BH-2VS. The vorticity of the T-2VS fills all the Rankine oval (see figure 7), which, outside of the vortex cores, is mainly populated by elongated azimuthal vortices (figure 6). There is also some vorticity in the wake of the oval. Even in the NGE phase, the boundary layers forming at the wall are perturbed by the axially inhomogeneous induced velocity of the T-2VS and are 3-D from the onset, as can be observed in figure 6 at $t/t_0 = 1$. Furthermore, as the vortices further descend IGE, the outer turbulent structures almost reach the wall and directly interact with the boundary layers (figures 6 and 7 at $t/t_0 = 2$). The traces of these structures are the transverse streaks which can be observed in the wall vorticity (figure 8). We note that these streaks have some similarities with those observed by Bricteux et al. (Reference Bricteux, Duponcheel, De Visscher and Winckelmans2016) in their simulation of a BH-2VS at $Re_\varGamma = 2.0\times 10^4$ that was released IGE within a fully turbulent crosswind. This tends to confirm their interpretation that the disturbances of the developing boundary layers are mostly due to the pre-existing turbulent structures of the flow (in their case, solely from the wind) that are being stretched by the vortices during their descent. In the present case, the turbulent structures of the flow are solely those of the T-2VS and the Reynolds number is also much higher.

The direct interaction of the T-2VS and the boundary layers significantly modifies the separation, which is also strongly 3-D. The separated secondary vorticity is then much less coherent (it does not roll up to form a coherent vortex), and it directly interacts with the turbulence of the primary vorticity (figures 6 and 7). This, in turn, significantly alters the rebound of the primary vortices, which is seen to be less intense than that of the BH-2VS at $t/t_0 = 5$. Consequently, the turbulent primary vortices are also farther apart from each other.

For completeness, and also to highlight the fine grid resolution used near the wall and in the vortex cores, we provide profiles of the longitudinally averaged velocity components at $t/t_0=1.9$ and $3.0$, see figure 9. For each case, the profiles were measured on a vertical line passing through the primary vortex centre.

Figure 9.

Profiles of the longitudinally averaged velocity component $V$ for the T-2VS: $t/t_0=1.9$ and $y/b_0=0.78$ (a,b); $t/t_0=3.0$ and $y/b_0=1.46$ (c,d), with (b,d) corresponding to a zoom near the ground. The dots correspond to the location of the grid points.

4.3. Transport and decay of the vortices IGE

In order to analyse the transport and decay of the primary vortices, their centres are tracked in a fashion similar to that of Bricteux et al. (Reference Bricteux, Duponcheel, De Visscher and Winckelmans2016). The circulation distribution $\varGamma (r, x)$ is obtained in each cross-plane and then averaged over all planes to obtain $\varGamma (r)$. From this distribution, the circulation diagnostics are obtained: $\varGamma _{max}$, $\varGamma _{b_0/2}$, $\varGamma _{5-15}$ and $\varGamma _{15}$ (the reported value being the average of the two vortices).

The transport of the vortices is described using the time evolution of the vortex spacing and height (figure 10) and the trajectory of the vortex centres in the $y\unicode{x2013} z$ plane (figure 11). As already discussed in § 4.2, the rebound of the T-2VS is less pronounced than that of the BH-2VS; the maximum height after rebound is also smaller, and the vortex spacing is larger (the minimal height being, however, similar, with $h_{min} \simeq 0.55 b_0$). The shape of the trajectories is thus different. For the BH-2VS, the trajectory exhibits the typical ‘loop’, as also observed at lower Reynolds numbers (Türk et al. Reference Türk, Coors and Jacob1999; Stephan et al. Reference Stephan, Holzäpfel and Misaka2013), and which stops when the primary and the secondary vortices mix. The T-2VS exhibits a more progressive and smoother rebound, without loop, which is also similar to what is observed experimentally for aircraft wake vortices (Holzäpfel & Steen Reference Holzäpfel and Steen2007; De Visscher, Lonfils & Winckelmans Reference De Visscher, Lonfils and Winckelmans2013b).

Figure 10.

Evolution of the vortex height $h/b_0$ and of the vortex spacing $b/b_0$: T-2VS (thick solid) and BH-2VS (dashed). The evolutions for the T-2VS evolving OGE is also shown for comparison (thin solid).

Figure 11.

Trajectory of the vortices: T-2VS (thick solid) and BH-2VS (thin solid). The symbols are spaced by one $t_0$. The inviscid hyperbolic trajectory is also shown (dotted).

The decay of the T-2VS, as measured using the circulation diagnostics, is shown in figure 12. A two-phase decay, typical of vortices IGE (Holzäpfel & Steen Reference Holzäpfel and Steen2007; De Visscher et al. Reference De Visscher, Lonfils and Winckelmans2013b), is clearly visible: an initial slow decay phase, followed by a fast decay phase due to the interaction with the ground. The evolution of $\varGamma _{max}$ is seen to be more irregular than that of $\varGamma _{b_0/2}$. This diagnostic is also more sensitive to the deformation of the vortex pair and to the presence of opposite sign vorticity within the boundary layer and/or orbiting the primary vortex (see figure 7). We see that it first decays slowly (and remains roughly 1 % above $\varGamma _{b_0/2}$, as was expected for $t/t_0 \leq 1$ which is still OGE), then exhibits an intermediate faster decay around $t/t_0 \simeq 2$ and finally coincides with $\varGamma _{b_0/2}$ beyond $t/t_0 \simeq 2.5$. The $\varGamma _{max}$ diagnostic is thus not the best to use as a measure of the total vortex circulation for wake vortices going IGE. From now on, the $\varGamma _{b_0/2}$ diagnostic will be preferred.

Figure 12.

Evolution of the various vortex circulation diagnostics for the T-2VS: $\varGamma _{max}$ (thin dash); $\varGamma _{b_0/2}$ (thick solid); $\varGamma _{15}$ (thin solid); $\varGamma _{5-15}$ (thick dash).

For completeness, and since it is of interest to the LIDAR community, the $\varGamma _{5-15}$ diagnostic is also provided in figure 13.

Figure 13.

Evolution of $\varGamma _{5-15}$: T-2VS (thick solid) and BH-2VS (dashed). The evolution of $\varGamma _{5-15}$ for the T-2VS evolving OGE is also shown (thin solid).

Finally, the ‘half-plane circulation but excluding the boundary layer’ is also used in order to estimate the total amount of free vorticity,

(4.1)$$\begin{gather} \varGamma^+_{HP|z>z_\delta} = \int_{y>0} \int_{z>z_\delta} \langle\omega_x\rangle \,{\rm d}S, \end{gather}$$

(4.2)$$\begin{gather}\varGamma^-_{HP|z>z_\delta} = \int_{y<0} \int_{z>z_\delta} \langle\omega_x\rangle \,{\rm d}S. \end{gather}$$

As for the other diagnostics, the value reported in figure 14 is the average of the two. This quantity has a weak dependence on the choice of $z_\delta$, provided that it is large enough to exclude the boundary layer attached vorticity, yet small enough to detect the detached vorticity – we used, as a trade-off, $z_\delta = 0.10b_0$. The quantity $\varGamma _0 - \varGamma _{HP|z> 0.1b_0}$ thus quantifies approximately the amount of opposite sign and separated vorticity that is available to interact with the primary vortex, and that is responsible for its fast decay phase. We note that it is not immediately correlated with the fast decay of $\varGamma _{b_0/2}$ as it takes some time for this secondary free vorticity to strongly interact with the primary vortex and affect it significantly within the region $r \leq b_0/2$.

Figure 14.

Evolution of the vortex circulation $\varGamma _{b_0/2}$ (top curves) and of the separated secondary vorticity $\varGamma _0 - \varGamma _{HP|z>0.1b_0}$ (bottom curves): T-2VS (thick solid) and BH-2VS (dashed). The evolution of $\varGamma _{b_0/2}$ for the T-2VS evolving OGE is also shown (thin solid).

For the BH-2VS, both $\varGamma _{b_0/2}$ (figure 14) and $\varGamma _{5-15}$ (figure 13) are almost constant up to the start of the fast decay phase. The start of that phase also varies depending on the considered diagnostic: at $t/t_0 \simeq 2.7$ for $\varGamma _{b_0/2}$, but only at $t/t_0 \simeq 3.7$ for $\varGamma _{5-15}$; thus a quite large time difference of $\simeq 1.0 t_0$, which reflects the long time required for the turbulent mixing of the secondary vorticity from the vortex outer region to its inner region.

Contrary to the BH-2VS, the T-2VS exhibits a clear ‘two-phase’ decay, with a slow decay during the first phase, and for both circulation diagnostics. The fast decay phase also starts earlier: at $t/t_0 \simeq 3.0$ for $\varGamma _{b_0/2}$, and at $t/t_0 \simeq 3.4$ for $\varGamma _{5-15}$, thus a time difference of only $\simeq 0.4 t_0$.

We also see in figure 14 that the slow decay of $\varGamma _{b_0/2}$ for the T-2VS IGE is similar to that obtained for its simulation OGE up to $t/t_0\simeq 2.5$; this confirms that the slow decay phase corresponds to the natural turbulent decay when OGE (of course before any fast decay phase eventually occurs much later, due to the development of the long-wavelength Crow instability).

The differences in the decay behaviours of the T-2VS and BH-2VS are further understood by examining, in figure 15, the time evolution of the circulation distributions $\varGamma (r)$, from which both $\varGamma _{b_0/2}$ and $\varGamma _{5-15}$ were derived. Since the BH-2VS is initially laminar, there is almost no vorticity diffusion between the two vortices. Consequently, $\varGamma (r)$ remains the same for $r/b_0 \lesssim 0.6$ up to $t/t_0 \simeq 3.0$, hence, $\varGamma _{b_0/2}$ and $\varGamma _{5-15}$ remain constant. The opposite sign, ground generated, vorticity is responsible for the decrease of $\varGamma (r)$ at large radii. When the primary vortex strongly interacts with that secondary vorticity, the maximum of $\varGamma (r)$ sharply decreases and its shape is also modified: this corresponds to the start of the fast decay phase.

Figure 15.

Vortex circulation distribution at (a) $t/t_0=0$ (black), $1$ (blue) and $2$ (red); (b) $t/t_0 = 3$ (black), $4$ (blue) and $5$ (red), for the T-2VS (thick solid) and for the BH-2VS (thin dashed).

For the T-2VS OGE, and then NGE, turbulent exchanges between the vortices occur from the start, which translates into a decrease of the $\varGamma (r)$ distribution at all radii, while its shape is preserved for $r/b_0 \lesssim 0.6$ up to $t/t_0 \simeq 2$. This explains the slow decay of $\varGamma _{b_0/2}$ and $\varGamma _{5-15}$ during that first phase. Similarly to the BH-2VS, the fast decay phase of the T-2VS is due to the turbulent mixing of the separated and turbulent secondary vorticity with the primary vortices. However, as seen in figure 14, the separation of the boundary layer vorticity starts earlier, and with a lower rate of opposite sign vorticity injection. The fast decay phase thus starts earlier and it is more progressive. Those significant differences highlight the importance of using a realistic turbulent 2VS field as initial condition for the present investigations.

Finally, the decay rate of the vortices in the late phase is found to be quite similar for both cases, yet still a bit higher for the T-2VS (as is seen when comparing the slopes of the top curves in figure 14 at late times). We also see, in figure 15, that the BH-2VS global decay IGE at $t/t_0=5$ is less than that of the T-2VS IGE at the same time; it compares better to that at $t/t_0=4$. To compare with the T-2VS IGE at $t/t_0=5$, it is thus better to consider the results of the BH-2VS IGE at the end of the simulation, $t/t_0 \simeq 5.9$. Both cases then correspond to turbulent vortices that are aged similarly, and that slide slowly along the wall and interact with it, with secondary vorticity being continuously incorporated into them (recall figures 6 and 7), at a roughly similar rate (see figure 14). The circulation distributions scaled using the instantaneous vortex circulation and core size are then also found to be similar, see figure 16. We have thus obtained some sort of other turbulent equilibrium: that of a turbulent vortex sliding slowly along the ground and continuously interacting with it (and no longer interacting with the other vortex, as too far from it).

Figure 16.

Circulation distribution at late times: T-2VS at $t/t_0=5$ (thick solid) and BH-2VS at $t/t_0=5.9$ (dashed). The circulation distribution of the T-2VS OGE is also recalled for comparison (thin solid).