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

The statistical properties of the self-sustained turbulent wake downstream of a long, high-aspect-ratio, laminar separation bubble are studied. The primary focus is on the relaxation of turbulent wake to a zero pressure gradient (ZPG) turbulent boundary layer, in terms of mean flow, Reynolds stresses, and turbulent kinetic energy (TKE) budget. The bubble is initiated by the strong adverse pressure gradient (APG) induced by a large-amplitude, laboratory-scale internal solitary wave of depression propagating against an oncoming barotropic current. The high-resolution large eddy simulation data generated in Part I of this two-part article is used for analysis. It is shown that the wake development in the early stage, following the breakdown of coherent vortices toward the trailing region of the separation bubble, possesses a statistical character similar to that of a self-similar plane mixing layer. The mean streamwise velocity profile relaxes to that of a ZPG turbulent boundary layer at 15 water column depths form the wave trough. This relaxation distance normalized by the bubble length is equal to 5 which is shorter than reported by Alam and Sandham's short bubble [J. Fluid Mech. 403, 223 (2000)] that is induced by a stronger APG. In regions further downstream of the bubble, distributions of the Reynolds stresses resemble those of channel flow. The relaxation of the TKE budget to that of a ZPG turbulent boundary layer is slower far away from the wall and faster near the wall, as compared to the relaxation of the mean flow to that of a ZPG turbulent boundary layer. Large bed stresses beneath the shed vortices and persistent near-bed turbulence offer an insight into frequently observed sediment resuspension events in coastal seas.

Figure 1

Schematic diagram of physical model. The frame of reference is fixed to the wave propagating leftward (negative $x$ direction). 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.

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 categorized 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 flow reattachment point.

Figure 3

(a) Streamwise variations of the time-averaged pressure coefficient along the bottom boundary and (b) the corresponding dimensionless pressure gradient along the bottom, $H(d{C}_p/dx)$. The streamwise location of the flow separation point is marked by the symbol $S$, and that of the mean flow reattachment point is marked by $R$, along with vertical dashed lines for both.

Figure 4

(a) Boundary layer thickness $\delta$, displacement thickness ${\delta }_{*}$, and momentum thickness ${\delta }_{\theta }$; (b) displacement thickness Reynolds number ${\mathrm{Re}}_{*}$ and momentum thickness Reynolds number ${\mathrm{Re}}_{\theta }$; (c) shape factor ${\delta }_{*}/\theta$; and (d) skin friction coefficient $c_f$ together with the Ludwieg-Tillman correlation in dashed curve, all as functions of the streamwise coordinate $x$. Vertical dashed lines indicate the separation point ($S$) and reattachment point ($R$). The wavelength is ${\lambda }_w=2.33H$.

Figure 5

Streamwise velocity profile in wall-scaled coordinates for different streamwise locations together with Murlis et al. [21] at ${\mathrm{Re}}_{\theta }=1900$ ($\square$), viscous-wall law $u^{+}=z^{+}$ (black solid for $z^{+}<10$), and log law of the wall $u^{+}=\frac{1}{0.41}log{z}^{+}+5.2$ (black solid for $z^{+}>10$). $u^{+}\equiv (\overline{u}-c)/u_{\tau }$, where $u_{\tau }$ is the local friction velocity defined as $u_{\tau }\equiv \sqrt{{\tau }_b/{\rho }_0}$. $z^{+}\equiv z{u}_{\tau }/\nu$.

Figure 6

Spatial distribution of Reynolds stresses. All the stresses are scaled by the maximum along-bed stress $u_{{\tau }_{\max }}^2$. (a) $\overline{u^{\prime }{u}^{\prime }}$: fifteen contour levels from 2.0 (blue) to 30.0 (red), equally spaced. (b) $\overline{v^{\prime }{v}^{\prime }}$: fifteen contour levels from 0.333 (blue) to 5.0 (red), equally spaced. (c) $\overline{w^{\prime }{w}^{\prime }}$: fifteen contour levels from 1.67 (blue) to 25 (red), equally spaced. (d) $-\overline{u^{\prime }{w}^{\prime }}$: eleven contour levels from 0.001 (blue) to 2.5 (red), equally spaced. $x$ coordinates of separation ($S$) and reattachment ($R$) are indicated by dashed lines.

Figure 7

Wall-normal distribution of Reynolds stresses and turbulence kinetic energy $0.5{\overline{u_i^{\prime }{u}_i^{\prime }}}^{+}$ (in dashed line) for different streamwise locations (a) $x=7H\approx 3{\lambda }_w$, (b) $x=9H\approx 3.9{\lambda }_w$, (c) $x=11H\approx 4.3{\lambda }_w$, and (d) $x=17H\approx 7.3{\lambda }_w$. Dot-dashed lines in (d) are corresponding stress values computed for ZPG flat-plate boundary layer by Spalart [22] for ${\mathrm{Re}}_{\theta }=1410$. All values are scaled by streamwise local value of $u_{\tau }^2\equiv {\tau }_b/{\rho }_0$. Note the different vertical axis limits across panels.

Figure 8

Wall-normal distribution of wall-scaled Reynolds stresses at $x=17H\approx 7.3{\lambda }_w$ presented in (a) viscous scale and (b) boundary layer scale. Dashed line represents the turbulence kinetic energy $0.5{\overline{u_i^{\prime }{u}_i^{\prime }}}^{+}$. Dot-dashed line represents DNS data for a channel flow obtained by Moser et al. [23] for ${\mathrm{Re}}_{\tau }(\equiv u_{\tau }\delta /\nu )=590$.

Figure 9

Wall-normal distribution of wall-scaled Reynolds stresses presented over viscous-wall scale $z^{+}$ for the viscous-wall region. Dashed line represents the turbulence kinetic energy $0.5{\overline{u_i^{\prime }{u}_i^{\prime }}}^{+}$. (a) $x=7H\approx 3{\lambda }_w$, (b) $x=9H\approx 3.9{\lambda }_w$, (c) $x=11H\approx 4.3{\lambda }_w$, and (d) $x=17H\approx 7.3{\lambda }_w$. Dot-dashed line in (d) is Spalart [22] for ${\mathrm{Re}}_{\theta }=1410$. Note the different vertical axis limits across panels.

Figure 10

Wall-scaled TKE budget terms in the viscous-wall region. (a) $x=7H\approx 3{\lambda }_w$, (b) $x=9H\approx 3.9{\lambda }_w$, (c) $x=11H\approx 4.3{\lambda }_w$, and (d) $x=17H\approx 7.3{\lambda }_w$. In all cases the advection term is essentially zero. Black lines with $+$ (with the same line styles defined in the legend) in (c) and (d) are Spalart [22] for ${\mathrm{Re}}_{\theta }=1410$. Positive values (gain) and negative values (loss). Note the different vertical axis limits across panels.

Figure 11

Boundary-layer-scaled turbulent kinetic energy budget terms away from the wall. (a) $x=7H\approx 3{\lambda }_w$, (b) $x=9H\approx 3.9{\lambda }_w$, (c) $x=11H\approx 4.3{\lambda }_w$, and (d) $x=17H\approx 7.3{\lambda }_w$. Black lines with $+$ (production, dissipation, and turbulent diffusion, with the same line styles defined in the legend) in (c) and (d) are Spalart [22] for ${\mathrm{Re}}_{\theta }=1410$. Positive values (gain) and negative values (loss). Note the different vertical axis limits across panels.

Figure 12

Instantaneous spanwise-averaged bed shear stress coefficient $c_{f0}=2{\tau }_b/{\rho }_0{c}_{w0}^2$ along the streamwise axis, together with critical shear stress for sediment mobilization for grain sizes 100, 200, 300, and $400\mu \mathrm{m}$ in dashed lines.

Figure 13

(a) Wall-normal turbulence intensities $\sqrt{\overline{w^{\prime 2}}}/c_{w0}$ as functions of wall-scaled wall-normal coordinate $z^{+}$ at different streamwise locations $x/H=\{4,9,17\}$ together with the particle settling velocities $w_s$ (horizontal solid lines) for particle diameters $d=100,200,300$, and $400\mu \mathrm{m}$. The vertical axis is log scaled. The region $z^{+}<5$, as indicated by the vertical dashed line, is the viscous sublayer. (b) The same quantities ($\sqrt{\overline{w^{\prime 2}}}/c_{w0}$) are presented in linear scale as functions of the global wall-normal coordinate $z/H$.