A Modal One-Way Navier-Stokes Approach to Modelling Non-Modal Boundary Layer Instabilities

TL;DR

This paper introduces modal One-Way Navier-Stokes (M-OWNS), a stability-analysis framework that combines the computational efficiency of modal approaches (PSE-like structure) with the robustness of OWNS-R by preserving the streamwise pressure gradient term $\partial p/\partial x$ and using a recursive projection operator. M-OWNS stabilises the streamwise march without a minimum step-size constraint, while retaining the ability to capture both modal and non-modal disturbance dynamics. A carefully designed adaptive wavenumber iteration and a set of recursion-parameters $\beta^{\pm}$ enable stable parabolisation across subsonic, transonic, and supersonic boundary layers, including three-dimensional configurations and acoustic receptivity phenomena. Validation across flat-plate, swept-cylinder, and Mach 4.5 boundary layers demonstrates that M-OWNS reproduces modal and non-modal features with comparable fidelity to OWNS-R and LHNS but at substantially reduced computational cost—often by factors of 5–30 or more—thereby enabling efficient practical engineering analyses of receptivity and disturbance evolution in complex flows.

Abstract

This paper presents a method to solve the modal form of the linearised one-way Navier-Stokes (OWNS) equations for investigating disturbance development in developing subsonic and supersonic boundary layers. The modal framework offers significant advantages in robustness and computational efficiency over the conventional non-modal OWNS framework. Notably, we demonstrate that modal OWNS (M-OWNS) retains the capability to capture non-modal disturbance development. Our technique leverages the modal ansatz of parabolised stability equations (PSE) whilst employing the recursion-parameter parabolisation strategy of non-modal OWNS to stabilise the streamwise-marching modal algorithm. A key contribution is that we overcome the minimum streamwise step-size requirement that constrains conventional PSE in capturing short-scale disturbance evolution. We demonstrate through canonical test cases that fine-scale variations in the baseflow can be captured effectively. The M-OWNS procedure permits arbitrarily small streamwisemarching step-sizes whilst maintaining stability. Crucially, we find the algorithm to be more robust than conventional non-modal OWNS, whilst recovering identical modal and non-modal phenomena.

Figures (23)

• Figure 1: Spectral structure of the streamwise spatial marching operator ${\mathbf{M}}$ for compressible boundary layers under the parallel flow ($\beta = 0$, ${W} = 0$). The subsonic spectrum comprises continuous branches of downstream-propagating vorticity/entropy modes $\alpha_{1,2,3}(\eta)$ and acoustic mode $\alpha_{4}(\eta)$, an upstream-propagating acoustic branch $\alpha_{5}(\eta)$, and discrete eigenvalues (circles). Arrows denote the direction of increasing $\eta^2$ along each branch.
• Figure 2: Numerical spectrum of the two-way system $\sigma\left({\mathbf{M}}^\pm\right) = \sigma\left({\mathbf{M}}\right)$ computed with the QZ method, and the parabolised downstream system $\sigma\left({\mathbf{M}}^+\right) = \sigma\left(\mathbf{P}_{N_\beta}{\mathbf{M}}\right)$ for a 2D boundary layer ($M=0.02$, $R=400$, $F=86$, $b=0$, $W=0$) with $N_\beta=30$. Downstream eigenvalues are preserved whilst upstream eigenvalues are eliminated. Blue and red diamonds represent $\beta^+$ and $\beta^-$ recursion parameters, respectively. The green diamonds denote $\beta^*$ parameters, roots of the characteristic polynomial and lie near the $|\mathcal{P}_{N_\beta}|=0.5$ contour. The contours show the norm of $\mathcal{P}_{N_\beta}$, with teal regions (approximately 1) indicating retained modes and pink regions (approximately 0) indicating suppressed modes. Notable features include the acoustic and vorticity/entropy continuous branches, and the discrete T-S wave. The discrete T-S wave must be retained throughout the march, we place a recursion parameter on it as it evolves. In this case $c=1$.
• Figure 3: Recursion selection algorithm for subsonic 2D-boundary layers at $R_\delta=400$ with $b=0$, ${W}=0$ and $F=86$ and variable Mach number. As $M$ increases, the real parts of $\alpha_{4,5}$ become more pronounced.
• Figure 4: Recursion selection algorithm for subsonic boundary layers with flow parameters $R_\delta=400$, $M=0.02$, ${W}=0$ and $F=86$ and $b=0.1,0.2$. As $b$ increases, the acoustic branches separate moving up and down the imaginary axis. This makes the recursion parameter selection process much easier as the $\beta^\pm$ parameters do not have to compete against each other.
• Figure 5: (a) Wavenumber and (b) growth rates for a 2D-disturbance ($b=0$) with $F=86$ in a subsonic boundary layer with variable Mach number. () $M=0.02$, () $M=0.2$, () $M=0.5$, () $M=0.8$. All calculations include PSE with $n_x=110$, M-OWNS with $n_x=110$ and $n_x=1000$, and OWNS at $n_x=1000$. The agreement between the high-resolution and low resolution methods is excellent until the end of the computational march, where the high-resolution computations tend to be attracted (or migrate) towards a vorticity wave with wavenumber $d_1=\alpha_{1,2,3}(0)$.