Subcritical route to turbulence via the Orr mechanism in a quasi-two-dimensional boundary layer

A subcritical route to turbulence via purely quasi-two-dimensional mechanisms, for a quasi-two-dimensional system composed of an isolated exponential boundary layer, is numerically investigated. Exponential boundary layers are highly stable and are expected to form on the walls of liquid metal coolant ducts within magnetic confinement fusion reactors. Subcritical transitions were detected only at weakly subcritical Reynolds numbers (at most $\approx 70$% below critical). Furthermore, the likelihood of transition was very sensitive to both the perturbation structure and initial energy. Only the quasi-two-dimensional Tollmien–Schlichting wave disturbance, attained by either linear or nonlinear optimization, was able to initiate the transition process, by means of the Orr mechanism. The lower initial energy bound sufficient to trigger transition was found to be independent of the domain length. However, longer domains were able to increase the upper energy bound, via the merging of repetitions of the Tollmien–Schlichting wave. This broadens the range of initial energies able to exhibit transitional behavior. Although the eventual relaminarization of all turbulent states was observed, this was also greatly delayed in longer domains. The maximum nonlinear gains achieved were orders of magnitude larger than the maximum linear gains (with the same initial perturbations), regardless if the initial energy was above or below the lower energy bound. Nonlinearity provided a second stage of energy growth by an arching of the conventional Tollmien–Schlichting wave structure. A streamwise independent structure, able to efficiently store perturbation energy, also formed.

Figure 1

A state space representation of the problem. Four cases are considered, two with initial energies $E_0$ just below and above the minimum initial energy sufficient to cross separatrix 1 ($E_{\mathrm{D}}$) and two with $E_0$ just above and below the maximum initial energy sufficient to cross separatrix 2 ($E_{\mathrm{D},2}$). An initial energy $E_{\mathrm{D}}<E_0<E_{\mathrm{D},2}$ either crosses separatrix 1 (red curve crosses solid dark green line) or avoids crossing separatrix 2 (blue curve eventually avoids solid light green line) to transition to turbulence. Eventually the turbulent state relaminarizes.

Figure 2

Schematic diagram of the sidewall domain with a characteristic length of the Shercliff boundary layer height ${\delta }_{\mathrm{S}}$. The thick horizontal line represents an impermeable no-slip boundary. The dotted line represents a stress-free parallel flow condition. The vertical dashed lines represent a periodicity constraint on velocity and fluctuating pressure. A uniform magnetic field is directed into the page. The out-of-plane Hartmann walls (the sources of linear friction) are not drawn.

Figure 3

Linear transient growth of an exponential boundary layer as a function of $r_{\mathrm{c}}={\text{Re}}_{\mathrm{S}}/{\text{Re}}_{\mathrm{S},\mathrm{crit}}$. (a) Growth optimized over initial field, wave number and time interval. Present data (squares) are compared against Q2D duct results from [27] (circles). The arrow indicates increasing $H$ through 1, 3, 10, 100, and 1000. With increasing $H$, the duct results [27] approach the isolated exponential boundary layer results (this work). (b) Optimal wave number. (c) Optimal time interval.

Figure 4

Optimized $\widehat{v}$-velocity fields. (a) $r_{\mathrm{c}}=0.0146$, ${\alpha }_{\mathrm{opt}}=0.7071$. (b) $r_{\mathrm{c}}=0.146$, ${\alpha }_{\mathrm{opt}}=0.5586$. Simulations computed with $L_y=28.28$ and images clipped at $y=10$. Solid lines (red flooding) positive; dotted lines (blue flooding) negative.

Figure 5

Comparison between linear and nonlinear optimals for various initial energies $E_0=\int {\widehat{u}}^2+{\widehat{v}}^{2}d\mathrm{\Omega }/\int U^{2}d\mathrm{\Omega }$ at $r_{\mathrm{c}}=0.293$. (a) Difference in the maximum linear growth obtained with the linear optimal (LOP) and maximum nonlinear growth with the nonlinear optimal (NLOP), for three domain lengths, and difference in the linear growth of the LOP and the nonlinear growth of the LOP scaled to $E_0$ ($n=1$ only). (b) Comparison between the nonlinear growth of the NLOP and nonlinear growth of the LOP scaled to $E_0$ ($n=1$). The linear growth of the LOP is $G_{\max }=55.9876$.

Figure 6

(a, b) temporal and (c, d) spatial resolution testing of the nonlinear evolution of linear optimals, for various initial energies $E_0$. (a & c) $r_{\mathrm{c}}=0.293$. (b & d) $r_{\mathrm{c}}=0.585$. The smaller polynomial order (value annotated for each curve), or larger time step (see legend), is denoted by a long dashed line for each $E_0$. $n=1$ unless otherwise stated. A black long dashed line represents the linear evolution.

Figure 7

(a) The lower delineation energy as a function of $r_{\mathrm{c}}={\text{Re}}_{\mathrm{S}}/{\text{Re}}_{\mathrm{S},\mathrm{crit}}$ ($n=1$ domain). The dot-dashed line roughly approximates the maximum $r_{\mathrm{c}}$ for which the delineation energy is undefined. (b) Energy time histories at $r_{\mathrm{c}}=0.585$, varying $E_0$. Light red curves with $E_0<E_{\mathrm{D}}$ have a secondary local maximum at best. The orange arrow indicates the switch from local maximum to inflection point, and the lowest initial energy (dashed dark green curve; $E_{\mathrm{D}}$) sufficient to cross separatrix 1. All green curves transition to turbulence. The largest initial energy that avoids crossing separatrix 2 ($E_{\mathrm{D},2}$) is also dashed. Light blue curves with $E_0>E_{\mathrm{D},2}$, which are briefly chaotic, all cross separatrix 2, with the purple arrow indicating the switch back from an inflection point to a local maximum. All curves are rescaled to start at unity to aid visualization, and the linear curve is denoted with a black long dashed line. At $r_{\mathrm{c}}=0.585$, $G_{\max }=89.9630$, while the maximum gain at $E_0=E_{\mathrm{D}}$ exceeds ${10}^3$. (c) Same results as (b), except depicted as a 3D surface, to accentuate the discontinuous changes at the separatrices.

Figure 8

Linearized evolution at $r_{\mathrm{c}}=0.293$, $L_y=28.28$; $\widehat{v}$-velocity contours. Solid lines (red flooding) positive; dotted lines (blue flooding) negative.

Figure 9

Nonlinear evolution at $r_{\mathrm{c}}=0.293$, $L_y=28.28$, $E_0=1.10\times {10}^{-5}$; $\widehat{v}$-velocity contours. Solid lines (red flooding) positive; dotted lines (blue flooding) negative.

Figure 10

(a) An example of the arched TS wave depicted by the $\widehat{v}$-velocity contour lines (solid positive; dotted negative), at $r_{\mathrm{c}}=0.585$, $E_0=2.69187\times {10}^{-6}>E_{\mathrm{D}}$, $t=2.121\times {10}^3$. The underlying backbone of the arch is highlighted by overlaying the high-pass-filtered vorticity ${\widehat{\omega }}_z$, where streamwise Fourier coefficients of modes $\kappa \le 3$ have been removed. (b) An example of the conventional TS wave from the linear transient growth analysis, at $r_{\mathrm{c}}=0.585$, $t=77.78$.

Figure 11

Energy growth at $r_{\mathrm{c}}=0.439$, $E_0=3.869\times {10}^{-6}>E_{\mathrm{D}}$, $n=1$. (a) $E=(1/2)\int {\widehat{u}}^2+{\widehat{v}}^{2}d\mathrm{\Omega }$. (b) $E_{\mathrm{v}}=(1/2)\int {\widehat{v}}^{2}d\mathrm{\Omega }$. At $r_{\mathrm{c}}=0.439$, $G_{\max }=73.9706,$ and $E_{\mathrm{D}}=3.853\times {10}^{-6}$. All curves are rescaled to unit initial energy. The linear evolution is shown as a black long dashed line.

Figure 12

Temporal evolution at $r_{\mathrm{c}}=0.439$, $L_y=28.28$, $n=1$, with $E_0=3.869\times {10}^{-6}>E_{\mathrm{D}}$. (a) Streamwise perturbation $\widehat{u}=u-U$. (b). Wall-normal perturbation $\widehat{v}=v$. Solid lines (red flooding) positive; dotted lines (blue flooding) negative.

Figure 13

Energy time histories at $r_{\mathrm{c}}=0.293$, varying the initial energy and domain length via repetitions $n$ of $l_{x,\mathrm{opt}}$. (a) $E=(1/2)\int {\widehat{u}}^2+{\widehat{v}}^{2}d\mathrm{\Omega }$. (b) $E_{\mathrm{v}}=(1/2)\int {\widehat{v}}^{2}d\mathrm{\Omega }$. Additional nonlinear growth is provided for even multiples of $n$, for all initial energies tested at $r_{\mathrm{c}}=0.293$, via pairwise coalescence of TS wave repetitions. All curves are rescaled to unit initial energy. The linear curves are presented with black long dashed lines. At $r_{\mathrm{c}}=0.293$, $G_{\max }=55.9876$.

Figure 14

Temporal evolution at $r_{\mathrm{c}}=0.293$, $L_y=28.28$, $n=2$, $E_0=5.48\times {10}^{-5}$; $\widehat{v}$-velocity contours. Solid lines (red flooding) positive; dotted lines (blue flooding) negative. This case decays in an $n=1$ domain, but undergoes second-stage growth in an $n=2$ domain because it restructures to an arched TS wave after the coalescence of the two individual perturbation repetitions.

Figure 15

Energy time histories, varying the initial energy and domain length via repetitions $n$ of $l_{x,\mathrm{opt}}$. (a) $r_{\mathrm{c}}=0.585$, $G_{\max }=89.9630$, $E_{\mathrm{D}}=2.6919\times {10}^{-6}$, maximum nonlinear gain observed for $E_0>E_{\mathrm{D}}$ is $\approx 4\times {10}^3$ ($n=2$). (b) $r_{\mathrm{c}}=1.463$, $G_{\max }=166.4092$, $E_{\mathrm{D}}=1.2096\times {10}^{-6}$, maximum nonlinear gain observed for $E_0>E_{\mathrm{D}}$ is $\approx 2\times {10}^4$ ($n=2$). All curves are rescaled to unit initial energy. $E_0<E_{\mathrm{D}}$ are unable to take advantage of the extra domain length, and still rapidly decay.

Figure 16

Contours of $\widehat{v}$-velocity at $r_{\mathrm{c}}=0.585$, $E_0=1.43\times {10}^{-5}$, $L_y=28.28$ at $t\approx 2.8\times {10}^3$. (a) $n=3$. (b) $n=4$. Solid lines (red flooding) positive; dotted lines (blue flooding) negative. Although the $n=3$ and $n=4$ cases coalesce, without the TS wave having an arched appearance, they decay monotonically.