Large-scale characteristics of stratified wake turbulence at varying Reynolds number

We analyze a large-eddy simulation data set of wakes of a towed sphere of diameter $D$ at speed $U$ in a uniformly stratified Boussinesq fluid with buoyancy frequency $N$ and kinematic viscosity $\nu$. These temporally evolving wakes are simulated using a spectral multidomain penalty-method-based incompressible Navier-Stokes solver for $\text{Fr}\equiv 2U/ND\in \{4,16,64\}$ and $\text{Re}\equiv UD/\nu \in \{5\times {10}^3,{10}^5,4\times {10}^5\}$, enabling a systematic examination of stratified wakes at three different values of $\text{Re}$ sufficiently separated in magnitude. As such, particular attention is paid to the effects of varying $\text{Re}$ on the evolution of large-scale characteristics of stratified wake turbulence. We examine the evolution of horizontal and vertical integral length scales (${\ell }_h$ and ${\ell }_v$), horizontal and vertical fluctuation velocities ($\mathcal{U}$ and $\mathcal{W}$), local vertical shear, as well as the resulting dimensionless parameters based on the above quantities. In particular, the vertical turbulent Froude number ${\text{Fr}}_v^{★}\equiv 2\pi \mathcal{U}/N{\ell }_v$ is found to be of order unity, a signature of the dynamics in the strongly stratified regime where shear instabilities develop between anisotropic flow layers. The horizontal turbulent Reynolds number ${\text{Re}}_h\equiv \mathcal{U}{\ell }_h/\nu$ stays approximately constant in time and the horizontal turbulent Froude number ${\text{Fr}}_h\equiv \mathcal{U}/N{\ell }_h$ decays in time as ${\left(Nt\right)}^{-1}$, consistent with scaling analysis of freely decaying turbulence. We characterize the transitions between distinct stratified flow regimes and examine the effects of body-based parameters $\text{Re}$ and $\text{Fr}$ on these transitions. The transition from the weakly to the strongly stratified regime, which is marked by ${\text{Fr}}_v^{★}$ decaying to unity, occurs when ${\text{Fr}}_h\simeq O\left(0.01\right)$. We further show that the initial value of ${\text{Re}}_h$ at which the flow completes the above transition scales as $\text{Re}{\text{Fr}}^{-2/3}$, which provides a way to predict the possibility of accessing the strongly stratified regime for a wake of given $\text{Re}$ and $\text{Fr}$. The analysis reported here constitutes an attempt to obtain the predictive capability of stratified wake turbulence in terms of Reynolds number $\text{Re}$, applying select elements of strongly stratified turbulence theory, so far typically utilized for homogeneous turbulence, to a canonical inhomogeneous turbulent free-shear flow.

Figure 1

Parameter space [24] based on turbulent horizontal Reynolds and Froude numbers ${\text{Re}}_h$ and ${\text{Fr}}_h$, respectively, and the proposed regime classification for stratified flows. Gray dashed lines delineate the proposed transitional boundaries between adjacent regimes.

Figure 2

Computational domain for implicit large-eddy simulation of a temporally evolving, stratified towed-sphere wake [9, 10, 16, 18, 19]. The centerline of the wake is at $(y,z)=(0,0)$. The effect of the towed sphere is not computed explicitly by the Navier-Stokes solver but rather introduced as a complex two-stage turbulent wake initialization procedure [10]. The sphere is assumed to be towed along the $x$ axis, which is the only statistically homogeneous direction in these simulations. The dimensions of the computational domain in this schematic are not drawn to scale.

Figure 3

Subdomain distributions for all three Reynolds numbers on the $z/D\in [-6,6]$ interval. The black horizontal lines delineate subdomain interfaces with the local Gauss-Lobatto-Legendre grid points omitted for clarity. The total number of grid points for $z/D\in [-6,6]$ is given by ${\widehat{N}}_z=M(\widehat{N}+1)+1$, i.e., as reported in Table 1. Here $M$ is the number of subdomains, and $\widehat{N}$ is the order of polynomial approximation.

Figure 4

Colormaps of (a)–(c) vertical vorticity ${\omega }_z(x,y)$ fields at $Nt=120$ sampled at the $Oxy$ horizontal midplane ($z=0$) for simulations R5F4, R100F4, and R400F4, respectively (from left to right) and (d)–(f) spanwise vorticity ${\omega }_y(x,z)$ fields for the same instances sampled at the $Oxz$ vertical midplane ($y=0$). The sphere travels from left to right. The length of the visualization window is (a)–(c) $\frac{80}{3}D$ in $x$ and $20D$ and (d)–(f) $\frac{40}{3}D$ in $x$ and $4D$ in $z$. The colorbar limits are (a)–(c) $\pm 0.09(U/D)$, (d) $\pm 0.15(U/D)$, and (e) and (f) $\pm 0.6(U/D)$.

Figure 5

Colormaps of spanwise vorticity ${\omega }_y(x,z)$ fields at $Nt=40$, 80, and 120, respectively (from left to right) sampled at the $Oxz$ vertical midplane ($y=0$) for simulations (a) R5F16, (b) R100F16, and (c) R400F16, respectively. The length of the visualization window is $\frac{40}{3}D$ in $x$ and $4D$ in $z$. The colorbar limits are (a) $\pm 0.09(U/D)$ and (b) and (c) $\pm 0.36(U/D)$. The median value of the local gradient Richardson number ${\text{Ri}}_{g,\text{loc}}$ at each time instance is marked in the bottom right corner of each panel.

Figure 6

Time evolution of (a) and (c) vertical turbulence length scale ${\ell }_v$ and (b) and (d) horizontal turbulence length scale ${\ell }_h$, both normalized by the sphere diameter $D$. Results are shown for (a) and (b) $\text{Re}=5\times {10}^3$ (R5) and (c) and (d) $\text{Re}={10}^5$ and $4\times {10}^5$ (R100 and R400, respectively). The line legends are shown in (a) and (c). The line types and colors distinguish various $\text{Fr}$ values, and increasing thickness of lines corresponds to a larger value of $\text{Re}$, a scheme that is followed throughout this paper.

Figure 7

(a) and (c) Inverse cyclical vertical Froude number ${\text{Fr}}_v^{★-1}\equiv {\ell }_v{N}^{★}/\mathcal{U}$ as a function of time. (b) and (d) Test of the viscous scaling for length scales ${\ell }_v/{\ell }_h\sim {\text{Re}}_h^{-1/2}$. The circles in (a) and (c) mark the unity crossing time of ${\text{Fr}}_v^{*}$ and the crosses in (b) and (d) mark the unity crossing time of buoyancy Reynolds number $\mathcal{R}$ (Fig. 11). The lightly shaded region in (c) corresponds to $1<{\text{Fr}}_v^{★-1}<2$, a characteristic dynamical signature of the strongly stratified regime.

Figure 8

Time series of (a) $\mathcal{U}/U$ and (b) $\mathcal{W}/\mathcal{U}$, where $\mathcal{U}$ and $\mathcal{W}$ are the horizontal and vertical fluctuation velocities, respectively, and $U$ is the tow speed. (c) Plot of the ratio $\mathcal{W}/\mathcal{U}$ against the length-scale ratio ${\ell }_v/{\ell }_h$. Again, the circle on each curve marks the unity crossing time of ${\text{Fr}}_v^{*}$ and the cross marks the unity crossing time of buoyancy Reynolds number $\mathcal{R}$, a convention that is followed in all subsequent figures. The lines in (a) are identical to those defined in the legends of the other panels.

Figure 9

Time series of (a) the cyclical vertical Froude number ${\text{Fr}}_v^{★}$ and (b) the median local gradient Richardson number ${\text{Ri}}_{g,\text{loc}}$.

Figure 10

Variation of horizontal turbulence Reynolds and Froude numbers ${\text{Re}}_h$ and ${\text{Fr}}_h$, respectively, with the dimensionless time $Nt$. The shaded region is (b) corresponds to $0.021<{\text{Fr}}_h<0.035$, the range of ${\text{Fr}}_h$ values at which the wakes enter the strongly stratified regime.

Figure 11

Trajectories followed by stratified wakes in the parameter space formed by ${\text{Re}}_h$ and ${\text{Fr}}_h^{-1}$ [24]. Circles mark the times at which the vertical Froude number ${\text{Fr}}_v^{★}$ becomes unity, i.e., ${\text{Fr}}_v^{★†}\equiv 1$, and crosses mark when the buoyancy Reynolds number $\mathcal{R}$ becomes unity, i.e., ${\mathcal{R}}^{‡}\equiv 1$. The shaded region corresponds to $29<{\text{Fr}}_h^{-1}<48$, the range of ${\text{Fr}}_h^{-1}$ values at which the wakes enter the strongly stratified regime.

Figure 12

Transitional horizontal turbulent Reynolds number ${\text{Re}}_h^{†}$ as the flow enters the strongly stratified regime and its dependence on the wake's body-based parameters $\text{Re}$ and $\text{Fr}$. The horizontal line corresponds to the lower bound of ${\text{Re}}_h^{†}$ in order for the wake to access the strongly stratified regime, according to Eq. (17). The vertical line corresponds to the threshold value of $\text{Re}{\text{Fr}}^{-2/3}$ for wakes accessing the strongly stratified regime, according to Eq. (22).

Figure 13

Transfer function $\widehat{G}$ as a function of (a) $k_{x}D$ and (b) $k_{x}h$ for simulations at $\text{Re}\in \{5\times {10}^3,{10}^5,4\times {10}^5\}$. The dimensionless cutoff wave number $k_{c}D$ (as measured by the wave number at which $\widehat{G}$ drops to 0.99) is 19.8, 26.2, and 53.0 for the three $\text{Re}$ values, respectively, corresponding to $\mathrm{\Delta }/D$ ratios of 0.16, 0.12, and 0.059. The value of $k_{c}h$ is 2.06, 1.38, and 1.38 for the three $\text{Re}$ values, corresponding to $\mathrm{\Delta }/h$ ratios of 1.5, 2.3, and 2.3.

Figure 14

Sample compensated energy spectrum of the streamwise velocity $u$ for various $Nt$ values from the R400F4 wake. The vertical dashed line indicates the cutoff wave number $k_{c}D\simeq 53$ beyond which the numerical solutions are directly affected by the Fourier filter (Fig. 13). The spectrum is first sampled pointwise for $(y,z)$ locations within the wake core defined by Eq. (C3) and then averaged over all $(y,z)$ locations sampled to yield the wake-averaged spectrum.