Self-sustained instability, transition, and turbulence induced by a long separation bubble in the footprint of an internal solitary wave. I. Flow topology

The development of a separated bottom boundary layer in the footprint of a large-amplitude internal solitary wave of depression, propagating against an oncoming barotropic current, is examined in detail using high-resolution implicit large eddy simulation. The wave is supported by a continuous two-layer stratification. The Reynolds number based on the water column height is $1.6\times {10}^5$. This numerical simulation is the first to reproduce the self-sustained three-dimensional vortex shedding, resultant transition, and turbulence under an ISW, which have long been hypothesized to occur in field experiments. No artificial noise is inserted into the flow domain. Part I of this study focuses on a structural description of the sequence of flow regimes developing from a wave-induced, long, high-aspect-ratio, laminar separation bubble. Three illuminating topological features are identified. (a) The spatial development of the self-sustained turbulence is composed of three transitional stages: (i) spontaneous excitation of a global instability in the separation bubble that emanates trailing vortices, (ii) vortex breakup and degeneration into turbulent clouds, and (iii) relaxation to a spatially developing turbulent boundary layer. (b) In the separation bubble, there exists a three-dimensional linear global oscillator, which is primarily excited by the two-dimensional absolute instability of the separated shear layer. This global mode possesses a transverse coherent structure. The transverse perturbation subsequently excites an elliptic instability mode inside the shed vortex, resulting in an axial distortion of the vortex core. (c) A shortwave secondary instability is excited in the form of a series of coherent streamwise vortex streaks that wrap around each shed vortex, leading to rapid break up and burst of the vortex.

Figure 1

Schematic diagram of physical model. The frame of reference is fixed to the wave propagating leftward (negative $x$ direction). $L_z$ is a reduced height used for computational domain. In the wave-fixed reference frame, in addition to the background barotropic current, one experiences a uniform current $c$ oriented in the left-to-right $x$ direction. For the sake of visual clarity, in any subsequent figures showing the along-wave velocity, this uniform contribution is omitted.

Figure 2

Spatial development of instantaneous spanwise perturbation vorticity ${\omega }_y{\delta }_s/c_{w0}$ over $1.5<x/H<19$ split equally. Each window has a length of $1.5{\lambda }_w$. The flow domain is characterized roughly into the three regimes: (i) two- and three-dimensional instability and transition: $2.5<x/H<6$ ($1.1<x/{\lambda }_w<2.6$); (ii) vortex breakup and formation of turbulent clouds: $6<x/H<13$ ($2.6<x/{\lambda }_w<5.6$); (iii) developing turbulent boundary layer: $x/H>13$ ($x/{\lambda }_w>5.6$). The symbol $R$ together with an arrow in the top panel indicate the mean reattachment point.

Figure 3

Spatial development of instantaneous spanwise perturbation vorticity ${\omega }_y{\delta }_s/c_{w0}$ in the transition region over $3<x/H<7.5$ equally split. Each window has a length of $0.64{\lambda }_w$. Isosurfaces of ${\omega }_y{\delta }_s/c_{w0}=3.0$ (in red) and ${\omega }_y{\delta }_s/c_{w0}=-3.0$ (in blue) are shown with a contoured background. All coordinates are scaled by $H$. Vortex pairing can be seen at $x/H\approx 3.5$. Vortex bending through elliptic instability can be seen at $x/H\approx 5$ and $x/H\approx 5.75$.

Figure 4

Contours of instantaneous transverse $v$ velocity scaled by $c_{w0}$ sampled with several instantaneous streamlines at two different downstream locations on the midspan plane: (a) near the maximum reverse flow region. The lower half streamlines flow upstream (to the left) and upper half ones flow downstream; (b) near the aft end of the separation bubble. Boxes in the the top panel delineate the location of panels (a) and (b) within the separation bubble.

Figure 5

Contours of instantaneous streamwise vorticity ${\omega }_x{\delta }_s/c_{w0}$ near the maximum reverse flow region: (a) cross section at the midspan plane (this section corresponds to the region (a) in Fig. 4), (b) horizontal slice along the primary vorticity layer ($z=0.02H$), and (c) horizontal slice along the bottom boundary.

Figure 6

Isosurface of instantaneous streamwise vorticity ${\omega }_x{\delta }_s/c_0$ tracking the same vortex (a) $t=t_0$ with ${\omega }_x{\delta }_s/c_{w0}=-0.03$ (blue) and ${\omega }_x{\delta }_s/c_{w0}=0.03$ (red), (b) $t=t_0+9.9{\delta }_s/c_{w0}$ with ${\omega }_x{\delta }_s/c_0=-0.06$ (blue) and ${\omega }_x{\delta }_s/c_{w0}=0.06$ (red), (c) $t=t_0+13.2{\delta }_s/c_{w0}$ with ${\omega }_x{\delta }_s/c_{w0}=-0.5$ (blue) and ${\omega }_x{\delta }_s/c_{w0}=0.5$ (red), and (d) $t=t_0+17.1{\delta }_s/c_{w0}$ with ${\omega }_x{\delta }_s/c_{w0}=-2.0$ (blue) and ${\omega }_x{\delta }_s/c_{w0}=2.0$ (red). All coordinates are scaled by $H$.

Figure 7

${\lambda }_2$ representation of Fig. 6. Isosurface of ${\lambda }_2=-0.5$ in all cases: (a) $t=t_0$, (b) $t=t_0+9.9{\delta }_s/c_{w0}$, (c) $t=t_0+13.2{\delta }_s/c_{w0}$, and (d) $t=t_0+17.1{\delta }_s/c_{w0}$. All coordinates are scaled by $H$.

Figure 8

Spatial development of turbulent coherent structures over selected windows (top) $7.5<x/H<9$, (middle) $10.8<x/H<12.3$, and (bottom) $15.8<x/H<17.3$ (each window has a length of $0.64{\lambda }_w$). Isosurface of ${\lambda }_2=-0.5$ on which the contour of $w/c_{w0}$ is rendered. All coordinates are scaled by $H$. Pairs of positive and negative $w$ velocities indicate distinct “memory” of vortices shed from the separated shear layer in the upstream (top and middle). Such distinct memory of parent vortices is blurred further downstream (bottom).

Figure 9

(a) Typical time series of wall-normal velocity $w$ sampled at $x=3H(\approx 1.3{\lambda }_w)$, $x=9H(\approx 3.9{\lambda }_w)$, and $x=17H(\approx 7.3{\lambda }_w)$, all at $z=0.025H(\approx 0.183{\delta }_s)$ at the midspan. The signals are equally separated vertically by one half unit. (b) Single-point frequency spectra of turbulent kinetic energy for fluctuation signals sampled at the same locations presented in (a). The spectrum for $x=9H$ is shifted by twofold upward, the spectrum for $x=17H$ is shifted by fourfold upward, all relative to one for $x=3H$. Reference lines with slopes of $-1$, $-5/3$, and $-7$ are included. Vertical dashed line indicates the vortex shedding frequency.

Figure 10

Time-averaged one-dimensional turbulent kinetic energy spectra at $z/{\delta }_s=\{0.015,0.033,0.15,0.37,0.73\}$ together with $-5/3$ slope. (a) Streamwise spectra. All cases are separated by threefold in vertical direction starting from the case $z/{\delta }_s=0.015$. (b) Spanwise spectra. All cases are separated by 2.5-fold starting from the case $z/{\delta }_s=0.015$.