Minimal Seed for Relaminarization in Boundary Layers

• Minimal seed for relaminarization is the optimal, energy-minimal disturbance in shear flows that sits on the laminar–turbulent boundary, offering precise control thresholds.
• It employs nonlinear optimization and adjoint methods to compute the critical energy threshold, using dynamic rescaling and localization in a Blasius boundary layer.
• The framework informs flow-control strategies by quantifying the vulnerability of the laminar state and guiding both active and passive methods to delay transition.

A minimal seed for relaminarization is defined as the optimal, fully localized disturbance of minimal energy in a shear flow that precisely sits on the laminar–turbulent separatrix: it is the critical initial perturbation that is energetic enough to neither quickly decay (relaminarize) nor immediately transition to sustained turbulence at subcritical Reynolds numbers. This concept is central to quantifying thresholds for transition in spatially developing boundary layers, particularly within the framework of adjoint-based nonlinear optimization techniques. The minimal seed encapsulates both localization and optimality—being a compact disturbance whose evolution maximally exploits nonlinear amplification mechanisms—providing fundamental insights into the controllability and vulnerability of transitional and relaminarizing flows (Vavaliaris et al., 2019).

1. Formulation and Dynamic Rescaling

The theoretical framework is developed for the Blasius boundary layer, with the incompressible base flow given by

$$\mathbf{U}(x,y)=\bigl(U_B(x,y),\,V_B(x,y),\,0\bigr),$$

satisfying

$$U_B\,\partial_x U_B + V_B\,\partial_y U_B = \nu\,\partial_{yy} U_B, \quad \partial_x U_B + \partial_y V_B = 0,$$

where $U_B \rightarrow U_\infty$, $V_B \rightarrow 0$ as $y\rightarrow \infty$, and no-slip at $y=0$. Perturbations are superimposed such that $$\mathbf{u}_{\mathrm{tot}} = \mathbf{U} + \mathbf{u}$$, $p_\mathrm{tot} = P + p$, and the perturbation dynamics are governed by

$$\partial_t \mathbf{u} + (\mathbf{U}\!\cdot\!\nabla)\mathbf{u} + (\mathbf{u}\!\cdot\!\nabla)\mathbf{U} + (\mathbf{u}\!\cdot\!\nabla)\mathbf{u} = -\nabla p + \nu \nabla^2 \mathbf{u}, \quad \nabla \cdot \mathbf{u}=0.$$

Non-dimensionalization is done using the inflow displacement thickness $\delta_0^*$, freestream velocity $$U_B\,\partial_x U_B + V_B\,\partial_y U_B = \nu\,\partial_{yy} U_B, \quad \partial_x U_B + \partial_y V_B = 0,$$0, and time $$U_B\,\partial_x U_B + V_B\,\partial_y U_B = \nu\,\partial_{yy} U_B, \quad \partial_x U_B + \partial_y V_B = 0,$$1, leading to the specification of an inflow Reynolds number $$U_B\,\partial_x U_B + V_B\,\partial_y U_B = \nu\,\partial_{yy} U_B, \quad \partial_x U_B + \partial_y V_B = 0,$$2. As the boundary layer grows downstream, a dynamic rescaling transforms to similarity variables $$U_B\,\partial_x U_B + V_B\,\partial_y U_B = \nu\,\partial_{yy} U_B, \quad \partial_x U_B + \partial_y V_B = 0,$$3, with

$$U_B\,\partial_x U_B + V_B\,\partial_y U_B = \nu\,\partial_{yy} U_B, \quad \partial_x U_B + \partial_y V_B = 0,$$4

and

$$U_B\,\partial_x U_B + V_B\,\partial_y U_B = \nu\,\partial_{yy} U_B, \quad \partial_x U_B + \partial_y V_B = 0,$$5

which reconciles the expanding boundary-layer with a quasi-parallel formulation suitable for adjoint-based optimization (Vavaliaris et al., 2019).

2. Nonlinear Optimization and Variational Framework

The minimal seed is identified through a variational optimization in the nonlinear state space. Let $$U_B\,\partial_x U_B + V_B\,\partial_y U_B = \nu\,\partial_{yy} U_B, \quad \partial_x U_B + \partial_y V_B = 0,$$6 be the initial perturbation, constrained to have fixed kinetic energy

$$U_B\,\partial_x U_B + V_B\,\partial_y U_B = \nu\,\partial_{yy} U_B, \quad \partial_x U_B + \partial_y V_B = 0,$$7

which is then evolved under the nonlinear Navier–Stokes equations. The objective is to maximize the kinetic energy at a specified target time $$U_B\,\partial_x U_B + V_B\,\partial_y U_B = \nu\,\partial_{yy} U_B, \quad \partial_x U_B + \partial_y V_B = 0,$$8,

$$U_B\,\partial_x U_B + V_B\,\partial_y U_B = \nu\,\partial_{yy} U_B, \quad \partial_x U_B + \partial_y V_B = 0,$$9

A Lagrangian is constructed with Lagrange multipliers $U_B \rightarrow U_\infty$0 for the equations of motion, $U_B \rightarrow U_\infty$1 for incompressibility, and $U_B \rightarrow U_\infty$2 for the fixed energy constraint: $U_B \rightarrow U_\infty$3 where

$U_B \rightarrow U_\infty$4

Stationarity gives rise to the direct (primal) and adjoint equations, corresponding to evolution of $U_B \rightarrow U_\infty$5 and backward evolution of $U_B \rightarrow U_\infty$6, respectively. The boundary and terminal conditions are set to ensure no-slip at $U_B \rightarrow U_\infty$7, stress-free outflow, zero perturbation at inflow, and periodicity in $U_B \rightarrow U_\infty$8 for the direct; adjoint boundary conditions are the formal transpose of these operations. The gradient with respect to the initial condition,

$U_B \rightarrow U_\infty$9

is used in a descent-rotation algorithm (with explicit norm constraint) to iteratively converge to the critical optimal disturbance (Vavaliaris et al., 2019).

3. Critical Energy Threshold and Minimal Seed Computation

Computation of the minimal seed involves identification of the critical energy $V_B \rightarrow 0$0 that just separates relaminarization from transition. This is achieved in two stages:

• Bisection Method: Initial bracketing energies $V_B \rightarrow 0$1 (below threshold) and $V_B \rightarrow 0$2 (above threshold) are chosen. Interpolation and re-optimization yield a new candidate $V_B \rightarrow 0$3, and its nonlinear evolution determines on which side of the transition the system lies. Iteration proceeds until $V_B \rightarrow 0$4.
• Edge-Tracking Refinement: To achieve extreme precision, edge trajectories are tracked: two states, one relaminarizing, one turbulent, are evolved; their energy gap is repeatedly halved, refining $V_B \rightarrow 0$5 to machine accuracy.

The resulting initial field at $V_B \rightarrow 0$6 is the minimal seed $V_B \rightarrow 0$7—the lowest-energy, fully nonlinear perturbation that sits exactly at the laminar–turbulent boundary (Vavaliaris et al., 2019).

4. Structure, Localization, and Dynamical Portrait

The minimal seed is sharply localized in all three spatial dimensions. It consists of a compact "packet" of streamwise-elongated vortical perturbations, with initial tilting of streamwise vortices against the base flow shear (Orr mechanism), followed by amplification via the lift-up of high- and low-speed streaks. These streaks and vortices constitute the most dangerous configuration for triggering early transition in the Blasius boundary layer.

Spatial localization is critical: the minimal seed for the spatially developing boundary layer forms at a specific streamwise location, at which the local Reynolds number $V_B \rightarrow 0$8. The critical energy at inflow $V_B \rightarrow 0$9 was numerically found to be

$y\rightarrow \infty$0

This localization and scaling behavior notably contrasts with results in parallel shear flows, where power-law scaling $y\rightarrow \infty$1 with $y\rightarrow \infty$2 is typical—implying that boundary-layer-specific analyses are necessary (Vavaliaris et al., 2019).

In a dynamically rescaled phase-space portrait, plotting variables such as

$y\rightarrow \infty$3

reveals that the minimal-seed trajectory approaches the same attracting "edge state" manifold (as previously described by Beneitez et al. 2019), but does so more efficiently and from lower energy (Vavaliaris et al., 2019).

5. Implications for Relaminarization and Flow Control

The minimal seed provides a precise diagnostic of flow controllability: it is the least energetic disturbance that neither triggers nor suppresses transition. An infinitesimal perturbation toward the laminar side in the phase space will cause the flow to completely relaminarize. Conversely, a perturbation toward turbulence will result in subcritical transition. Thus, comprehensive characterization of $y\rightarrow \infty$4 is directly relevant for designing both active and passive flow-control strategies, offering a rigorous mechanism for identifying the "most effective" means of delaying or reversing boundary-layer transition. The minimal seed framework quantifies the vulnerability of the laminar state and determines the disturbance-energy threshold that control strategies must robustly suppress to ensure relaminarization (Vavaliaris et al., 2019).

6. Numerical Implementation and Research Context

The adjoint optimization and minimal seed computation are carried out using the spectral element code Nek5000 with gradient-rotation and checkpointing for memory efficiency, and edge-tracking is performed using the code SIMSON. The technique builds upon prior foundational works by Cherubini et al. (2011a, 2011b), Duguet et al. (2012), and Beneitez et al. (2019), but extends these methods to spatially developing boundary layers with full rescaling and precise variable localization. Full implementation details, boundary condition transposes, and adjoint code structure are reported by Vavaliaris et al. (2020) (Vavaliaris et al., 2019).

7. Connections and Further Directions

Understanding minimal seeds yields insight into subcritical transition mechanisms in boundary layers, links optimal disturbance theory to nonlinear phase space geometry, and offers a framework in which both relaminarization and transition can be viewed dynamically as outcomes of phase space trajectories near the edge manifold. This paradigm informs both theoretical hydrodynamic stability analysis and practical flow-control applications, and serves as a basis for future work on minimal acts of relaminarization and resilience of laminar/turbulent states to complex, localized disturbances (Vavaliaris et al., 2019).