Parabolised Coherent Structures (PCS) Method

• Parabolised Coherent Structures (PCS) is a spatial-marching framework that extends vortex–wave interaction theory to predict finite-amplitude wave evolution in complex boundary layers.
• PCS couples nonlinear feedback between mean streak and wave equations, enabling quantitative predictions of wave amplitude and displacement thickness in experimental geometries.
• PCS has been validated against Görtler vortex experiments, demonstrating improved accuracy over linear methods in capturing finite-Reynolds-number and nonparallel effects.

The Parabolised Coherent Structures (PCS) method is a spatial-marching framework for predicting finite-amplitude wave evolution in boundary-layer flows, particularly where vortex–wave interactions and nonparallel effects dominate instability development. Originating in the context of Görtler vortices over concave walls, PCS extends vortex–wave–interaction (VWI) theory by embedding nonlinear feedback into a parabolic system, thereby enabling quantitative prediction of wave amplitude and displacement thickness in experimental geometries (Song et al., 25 Jan 2026).

1. Governing Equations and Asymptotic Decomposition

PCS is formulated from the incompressible Navier–Stokes equations in boundary-layer scalings within cylindrical coordinates $(r^*, \varphi^*, z^*)$ around a concave wall of radius $a^*$, with characteristic freestream speed $U^*_\infty$ and kinematic viscosity $\nu^*$. The boundary-layer thickness is defined as $\delta^* = L^* / \sqrt{Re}$, where $Re = U^*_\infty L^* / \nu^*$. Nondimensionalization introduces scaled variables: $$x = \frac{a^*\,\varphi^*}{\delta^*},\quad y = \frac{r^* - a^*}{\delta^*},\quad z = \frac{z^*}{\delta^*},\quad t = \frac{U^*_\infty}{\delta^*} t^*,$$ with velocities $(u,v,w)=(u_\varphi, u_r, u_z)/U^*_\infty$ and pressure $p = p^* / (\rho^* U^{*2}_\infty)$. The effective Reynolds number is $$R_\delta = U^*_\infty \delta^* / \nu^* = Re^{1/2} \gg 1$$, and the system includes curvature effects via the Görtler number $G = 2 R_\delta^2 / a$, $a = a^*/\delta^*$.

The flow is split into a slow–fast decomposition following VWI theory: the slow coordinate $X = R_\delta^{-1} x$ and fast phase $\theta = R_\delta \int^X \alpha(s)\,ds - \Omega t$, with $\alpha(X)$ as streamwise wavenumber and $\Omega$ the frequency. The velocity field is decomposed as

$$[\mathbf{u},p](X, y, z, \theta) = [\overline{\mathbf{u}}, \overline{p}](X,y,z) + [\widetilde{\mathbf{u}}, \widetilde{p}](X,y,z,\theta),$$

where the mean adopts scalings $(U,\,R_\delta^{-1} V,\,R_\delta^{-1} W)$, and the wave part scales as $O(\epsilon^{1/2} R_\delta^{-1})$ with $\epsilon = R_\delta^{-1/3}$.

2. PCS System: Mean and Wave Equations

The PCS methodology couples parabolised equations for coherent mean structures and nonlinear evolution equations for the waves. The mean (roll–streak) equations form a parabolic system in $X$, neglecting streamwise viscous derivatives but retaining advection and cross-stream diffusion: $$\begin{aligned} &\left[ U\partial_X + V\partial_y + W\partial_z - \partial_y^2 - \partial_z^2 \right] (U, V, W)^\top \ &\quad + (0, \partial_y P + \frac{G}{2} U^2, \partial_z P)^\top = \mathbf{F}(X, y, z), \ &\partial_X U + \partial_y V + \partial_z W = 0. \end{aligned}$$ Here, the forcing $\mathbf{F}$ derives from the divergence of the wave Reynolds stress: $$\mathbf{F} = -\overline{ ( \widetilde{\mathbf{u}} \cdot \widetilde{\nabla} ) \widetilde{\mathbf{u}} }.$$

The wave field $\widetilde{\mathbf{u}}$ evolves through nonlinear equations under a local parallel-flow approximation: $$\begin{aligned} &\left[ \overline{\mathbf{u}} \cdot \widetilde{\nabla} - c \alpha \partial_\theta \right] \widetilde{\mathbf{u}} + (\widetilde{\mathbf{u}} \cdot \nabla ) \overline{\mathbf{u}} + ( \widetilde{\mathbf{u}} \cdot \widetilde{\nabla} ) \widetilde{\mathbf{u}} - \overline{ ( \widetilde{\mathbf{u}} \cdot \widetilde{\nabla} ) \widetilde{\mathbf{u}} } = - \widetilde{\nabla} \widetilde{p} + \frac{1}{R_\delta} \widetilde{\nabla}^2 \widetilde{\mathbf{u}}, \end{aligned}$$ subject to $$\widetilde{\nabla} \cdot \widetilde{\mathbf{u}} = 0$$, where $$\widetilde{\nabla} = (\alpha \partial_\theta, \partial_y, \partial_z)$$ and the phase speed $c = \Omega / \alpha$.

The explicit retention of nonlinear triadic wave–wave interactions in both the mean and wave equations distinguishes PCS from linear secondary instability approaches.

3. Nonlinear Vortex–Wave Coupling

PCS incorporates full nonlinear vortex–wave coupling:

• The mean receives forcing via $$\mathbf{F} = -\overline{ ( \widetilde{\mathbf{u}} \cdot \nabla ) \widetilde{\mathbf{u}} }$$, including all Reynolds stress triads.
• The mean streak $U$ and roll components $(V, W)$ enter the wave equations via advection and linear vortex–wave coupling terms.
• Viscosity $1/R_\delta$ regularizes the wave critical layer, enabling finite-amplitude waves of order $R_\delta^{-7/6}$ to sustain streaks of order unity.

This nonlinear feedback produces self-sustaining coherent structures, embedding the essential mechanisms known from exact coherent structure studies but in an efficient spatial-marching setting.

4. Spatial-Marching and Numerical Algorithm

PCS advances the coupled mean and wave fields along the slow streamwise coordinate $X$ via an implicit spatial-marching algorithm:

• At each step, explicit $X$-derivatives are updated implicitly, akin to searching for time-periodic traveling wave solutions.
• The Newton–Raphson method iteratively solves for the new fields $$\{ \overline{\mathbf{u}}, P, \widetilde{\mathbf{u}}, \widetilde{p} \}$$.
• Spectral discretization is employed: Fourier–Galerkin expansions for $\theta$ and $z$, with typical retention of only the fundamental mode, and Chebyshev collocation in $y$ mapped to cluster nodes near the wall ($y=0$).
• Boundary conditions enforce no-slip for perturbations at the wall, and decay or free-stream at the edge ($y=H$).
• The global wave frequency $\Omega$ is fixed by experiment, with $\alpha(X)$ updated local-to-global such that $\alpha c = \Omega$.

A plausible implication is that PCS provides a scalable computational approach for nonparallel instability problems beyond the scope of traditional linear analyses.

5. Application to Görtler Vortex Instability

PCS is validated against experimental data on Görtler vortices (SB87). Representative parameters are $U^*_\infty = 5\;\mathrm{m/s}$, $a^* = 3.2\;\mathrm{m}$, $\lambda^* = 2.3\;\mathrm{cm}$, $\nu^* = 1.6\times 10^{-5}$, leading to $Re = 3.125\times 10^4$, $R_\delta \approx 177$, $G \approx 11.05$. The spanwise periodicity sets the fundamental $z$ wavenumber.

Initial conditions for PCS derive from linear and nonlinear solutions of the Boundary Region Equations (BRE), with transition from a linearized stage ($x^* = 10$–$30$ cm) via a seeded mode, moving to nonlinear BRE up to $x^* \approx 95$ cm. Linear secondary instability analysis yields a critical point and mode frequency $f^* \approx 170$ Hz.

PCS is launched at the critical point using imperfect bifurcation by imposing a small external forcing in the wave equation near $x^* = 95$ cm. Upstream of this point, pure BRE is solved; downstream, the fully-coupled PCS system is marched forward.

6. Experimental Validation and Quantitative Predictions

PCS predictions for local displacement thickness $\delta^*_{\text{disp}}$ at Görtler-vortex peaks and valleys match experimental measurements to within uncertainty up to $x^* \approx 110$ cm. The root-mean-square wave amplitude is defined as

$$\widetilde{u}_{\rm rms}(X, y, z) = \sqrt{ \frac{1}{2\pi} \int_0^{2\pi} \widetilde{u}^2 d\theta },\quad \widetilde{u}_{\max}(X) = \max_{y, z} \widetilde{u}_{\rm rms}.$$

The growth rate $$\sigma^* = \frac{1}{L^*} \frac{d\ln \widetilde{u}_{\max}}{dX}$$ predicted by PCS closely tracks SB87 data, whereas linear secondary-instability theory substantially overestimates wave amplitude growth.

This suggests that PCS is essential for capturing finite-amplitude wave evolution and its interaction with the mean flow in nonparallel boundary-layer phenomena.

7. Extensions and Broader Significance

PCS embeds the self-sustaining vortex–wave interaction of exact coherent structures into an efficient computational framework. By retaining nonlinear triadic interactions and finite-Reynolds-number viscous regularization of the critical layer, PCS quantitatively predicts finite-amplitude waves and boundary-layer displacement thickness in concave-wall boundary layers. A plausible implication is the extensibility of the PCS approach to other nonparallel instability problems requiring finite-amplitude feedback, including crossflow instabilities and shock/boundary-layer interactions (Song et al., 25 Jan 2026).

The rigorous coupling of nonlinear dynamics, validated against experiment, establishes PCS as a key tool in the quantitative theory of transitional flows in complex geometries.