# Influence of free-stream turbulence on the boundary layer stability of a wind turbine airfoil and near wake: PSE operators

T C L Fava, B A Lobo, P A S Nogueira, A P Schaffarczyk, M Breuer, D S Henningson, A Hanifi

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## To cite this version:

T C L Fava, B A Lobo, P A S Nogueira, A P Schaffarczyk, M Breuer, et al.. Influence of free-stream turbulence on the boundary layer stability of a wind turbine airfoil and near wake: PSE operators. 2023. [hal-04075160v2](https://hal.science/hal-04075160v2)

HAL Id: hal-04075160

https://hal.science/hal-04075160v2

Preprint submitted on 15 Jun 2023

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# Influence of free-stream turbulence on the boundary layer stability of a wind turbine airfoil and near wake: PSE operators

T C L Fava¹, B A Lobo², P A S Nogueira³, A P Schaffarczyk², M Breuer⁴, D S Henningson¹, A Hanifi¹

¹KTH Royal Institute of Technology, Dept. of Engineering Mechanics, FLOW and SeRC, SE-100 44 Stockholm, Sweden

²Mechanical Engineering Dept., Kiel University of Applied Sciences, D-24149 Kiel, Germany

³Aerospace Engineering, Monash University, 3800, Clayton, Australia

⁴Professur für Strömungsmechanik, Helmut-Schmidt-Universität Hamburg, D-22043 Hamburg, Germany

E-mail: fava@kth.se

**Abstract.** This document reports the operators of the parabolized stability equations (PSE) model employed in the article "Influence of free-stream turbulence on the boundary layer stability of a wind turbine airfoil and near wake" [1]. This model was first developed in [2].

## 1. Introduction

The following ansatz is used in the current version of the parabolized stability equations (PSE):

$$ q'(x, y, z, t) = \exp \left( i \int_{0}^{x} \alpha(\xi) d\xi - i\omega t \right) \sum_{n=-N_s}^{N_s} \left( \hat{q}_n(x, y, \omega) e^{i(\beta+n\beta_s)z} \right), \quad (1) $$

where $x$, $y$ and $z$ are the three spatial coordinates, $\alpha$ and $\beta$ are the wavenumbers in the $x$ and $z$ directions, $\omega = 2\pi f$ is the angular frequency, $t$ is the time, and $N_s$ is the number of considered harmonics. The base-flow is constituted of an undisturbed part (subscript 0) and a streaky part (subscript s), according to

$$ U(x, y, z) = U_0(x, y) + \mathrm{Re}\{U_s(x, y, z)\}, \quad (2) $$

where

$$ U = [\rho, U, V, W, P]^T, \quad U_0 = [\rho_0, U_0, V_0, W_0, P_0]^T, \quad U_s = [\rho_s, U_s, V_s, W_s, P_s]^T, \quad (3) $$

where $\rho$ is the density, $U$, $V$, and $W$ are the $x$, $y$, and $z$ velocity components, and $P$ is the pressure. $Re\{\cdot\}$ indicates the real part. Notice that, even though the density was used to define the base flow $U$, it was more convenient to rewrite the perturbation in density as a perturbation in specific volume in the final stability equations. The system of equations of the PSE is given by
$$i\mathbf{F} \frac{\partial \tilde{\mathbf{q}}}{\partial x} = -\mathbf{L}\tilde{\mathbf{q}}, \qquad (4)$$

where $\tilde{\mathbf{q}} = [\hat{\mathbf{q}}_{-N_s}, \dots, \hat{\mathbf{q}}_0, \dots, \hat{\mathbf{q}}_{N_s}]$, with $\hat{\mathbf{q}}_i = [\hat{v}_i, \hat{u}_i, \hat{v}_i, \hat{w}_i, \hat{p}_i]^T$, $i = -N_s, \dots, N_s$, where $v, u, v, w$, and $p$ are the perturbations of specific volume, velocities in $x, y$, and $z$, and pressure, respectively. $\gamma$ is the ratio of specific heats, and $Pr = \mu_\infty c_p / \kappa_\infty$ is the Prandtl number. Here, $\mu_\infty$ is the reference dynamic viscosity and $\kappa_\infty$ is the reference thermal conductivity.

The boundary conditions are

$$\hat{\mathbf{u}}_i = 0, \hat{\mathbf{v}}_i = 0, \hat{\mathbf{w}}_i = 0, \hat{\mathbf{T}}_i = 0, i = -N_s, \dots, N_s \qquad (5)$$

at $y = 0$ and $y = y_{max}$. The temperature disturbance $\hat{\mathbf{T}}_i$ can be obtained from $\hat{v}_i$ and $\hat{\mathbf{P}}_i$ via the ideal gas law. The governing equations are non-dimensionalized according to the reference quantities in Table 1:

<table><caption>Table 1: Reference quantities.</caption><thead><tr><th>length</th><th>velocity</th><th>density</th><th>pressure</th><th>temperature</th></tr></thead><tbody><tr><td>L</td><td>a<sub>∞</sub></td><td>ρ<sub>∞</sub></td><td>ρ<sub>∞</sub>a<sub>∞</sub><sup>2</sup></td><td>a<sub>∞</sub><sup>2</sup>/c<sub>p</sub></td></tr></tbody></table>

where $L$ is a generic length scale, $a_\infty$ is the reference speed of sound, $\rho_\infty$ is the reference density, and $c_p$ is the specific heat at constant pressure. The Reynolds number can be defined as $Re = \rho_\infty u_\infty L / \mu_\infty$

The following operators are defined

$$\mathcal{D}_x = \frac{1}{\mathbf{h}_1} \frac{\partial}{\partial x}, \quad \mathcal{D}_y = \frac{\partial}{\partial y}, \quad \mathbf{\Delta} = \mathcal{D}_y^2 + \mathbf{m}_{13} \mathcal{D}_y. \qquad (6)$$

The scale factors and their derivatives are given by

$$\mathbf{h}_i^2 = \sum_{j=1}^{3} \left( \frac{\partial y^j}{\partial x^i} \right)^2, \quad \mathbf{m}_{ij} = \frac{1}{\mathbf{h}_i \mathbf{h}_j} \frac{\partial \mathbf{h}_i}{\partial x^j}, \quad i = 1, \dots, 3, \quad j = 1, \dots, 3, \qquad (7)$$

where $y^j$ are the Cartesian coordinates of the reference system and $x^i = x, y, z$.

The PSE operators in eq. 4 for $N_s = 4$ are given below, where * denotes complex conjugate. The generalization for $N_s > 4$ follows the same pattern and is trivial.

$$\mathbf{L} = -i\omega\mathbf{I} + \mathbf{L}_t - \alpha\mathbf{F} + \mathbf{E}_v + \mathbf{L}_\theta + \mathbf{E}_{vt1} + \mathbf{E}_{vt2} \qquad (8)$$

$$\mathbf{L}_t = \begin{bmatrix} \Psi_0 & \Psi_c & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \Psi_s & \Psi_0 & \Psi_c & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \Psi_s & \Psi_0 & \Psi_c & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \Psi_s & \Psi_0 & \Psi_c & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \Psi_s & \Psi_0 & \Psi_c & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \Psi_s & \Psi_0 & \Psi_c & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \Psi_s & \Psi_0 & \Psi_c & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \Psi_s & \Psi_0 & \Psi_c \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \Psi_s & \Psi_0 \end{bmatrix} \qquad (9)$$

$$\Psi_0 = L_{cn0} + L_{cr0}, \quad \Psi_s = L_{cn_s} + L_{cr_s}, \quad \Psi_c = L_{cn_c} + L_{cr_c} \qquad (10)$$
$$ L_{cn_0} = \begin{bmatrix} \eta & \mathcal{D}_x \rho_0 & \mathcal{D}_y \rho_0 - \rho_0 \mathcal{D}_y & \mathbf{0} & \mathbf{0} \\ \mathcal{D}_x \mathbf{P}_0 & \mathbf{V}_0 \mathcal{D}_y + \mathcal{D}_x \mathbf{U}_0 & \mathcal{D}_y \mathbf{U}_0 & \mathbf{0} & \mathbf{0} \\ \mathcal{D}_y \mathbf{P}_0 & \mathcal{D}_x \mathbf{V}_0 & \mathbf{V}_0 \mathcal{D}_y + \mathcal{D}_y \mathbf{V}_0 & \mathbf{0} & \rho_0 \mathcal{D}_y \\ \mathbf{0} & \mathcal{D}_x \mathbf{W}_0 & \mathcal{D}_y \mathbf{W}_0 & \mathbf{V}_0 \mathcal{D}_y & \mathbf{0} \\ \mathbf{0} & \mathcal{D}_x \mathbf{P}_0 & \mathcal{D}_y \mathbf{P}_0 + \gamma \mathbf{P}_0 \mathcal{D}_y & \mathbf{0} & \xi \end{bmatrix} \quad (11) $$

$$ \begin{aligned} \eta &= \mathbf{V}_0 \mathcal{D}_y - \mathcal{D}_y \mathbf{V}_0 - \mathcal{D}_x \mathbf{U}_0 \\ \xi &= \mathbf{V}_0 \mathcal{D}_y + \gamma \mathcal{D}_y \mathbf{V}_0 + \gamma \mathcal{D}_x \mathbf{U}_0 \end{aligned} \quad (12) $$

$$ L_{cn_s} = \begin{bmatrix} \sigma & \mathcal{D}_x \rho_s & \mathcal{D}_y \rho_s - \rho_s \mathcal{D}_y & i \beta_s \rho_s & \mathbf{0} \\ \mathcal{D}_x \mathbf{P}_s & \mathbf{V}_s \mathcal{D}_y + \mathcal{D}_x \mathbf{U}_s & \mathcal{D}_y \mathbf{U}_s & i \beta_s \mathbf{U}_s & \mathbf{0} \\ \mathcal{D}_y \mathbf{P}_s & \mathcal{D}_x \mathbf{V}_s & \mathbf{V}_s \mathcal{D}_y + \mathcal{D}_y \mathbf{V}_s & i \beta_s \mathbf{V}_s & \rho_s \mathcal{D}_y \\ i \beta_s \mathbf{P}_s & \mathcal{D}_x \mathbf{W}_s & \mathcal{D}_y \mathbf{W}_s & \mathbf{V}_s \mathcal{D}_y + i \beta_s \mathbf{W}_s & \mathbf{0} \\ \mathbf{0} & \mathcal{D}_x \mathbf{P}_s & \mathcal{D}_y \mathbf{P}_s + \gamma \mathbf{P}_s \mathcal{D}_y & i \beta_s \mathbf{P}_s & \tau \end{bmatrix} \quad (13) $$

$$ \begin{aligned} \sigma &= \mathbf{V}_s \mathcal{D}_y - \mathcal{D}_y \mathbf{V}_s - i \beta_s \mathbf{W}_s - \mathcal{D}_x \mathbf{U}_s \\ \tau &= \mathbf{V}_s \mathcal{D}_y + \gamma (\mathcal{D}_y \mathbf{V}_s + i \beta_s \mathbf{W}_s) + \gamma \mathcal{D}_x \mathbf{U}_s \end{aligned} \quad (14) $$

$$ L_{cn_c} = \begin{bmatrix} \Gamma & (\mathcal{D}_x \rho_s)^* & (\mathcal{D}_y \rho_s)^* - \rho_s^* \mathcal{D}_y & -i \beta_s \rho_s^* & \mathbf{0} \\ (\mathcal{D}_x \mathbf{P}_s)^* & \mathbf{V}_s^* \mathcal{D}_y + (\mathcal{D}_x \mathbf{U}_s)^* & (\mathcal{D}_y \mathbf{U}_s)^* & -i \beta_s \mathbf{U}_s^* & \mathbf{0} \\ (\mathcal{D}_y \mathbf{P}_s)^* & (\mathcal{D}_x \mathbf{V}_s)^* & \mathbf{V}_s^* \mathcal{D}_y + (\mathcal{D}_y \mathbf{V}_s)^* & -i \beta_s \mathbf{V}_s^* & \rho_s^* \mathcal{D}_y \\ -i \beta_s \mathbf{P}_s^* & (\mathcal{D}_x \mathbf{W}_s)^* & (\mathcal{D}_y \mathbf{W}_s)^* & \mathbf{V}_s^* \mathcal{D}_y - i \beta_s \mathbf{W}_s^* & \mathbf{0} \\ \mathbf{0} & (\mathcal{D}_x \mathbf{P}_s)^* & (\mathcal{D}_y \mathbf{P}_s)^* + \gamma \mathbf{P}_s^* \mathcal{D}_y & -i \beta_s \mathbf{P}_s^* & \zeta \end{bmatrix} \quad (15) $$

$$ \begin{aligned} \Gamma &= \mathbf{V}_s^* \mathcal{D}_y - (\mathcal{D}_y \mathbf{V}_s)^* + i \beta_s \mathbf{W}_s^* - (\mathcal{D}_x \mathbf{U}_s)^* \\ \zeta &= \mathbf{V}_s^* \mathcal{D}_y + \gamma ((\mathcal{D}_y \mathbf{V}_s)^* - i \beta_s \mathbf{W}_s^*) + \gamma (\mathcal{D}_x \mathbf{U}_s)^* \end{aligned} \quad (16) $$

$$ L_{cr_0} = \begin{bmatrix} -m_{13} \mathbf{V}_0 & 0 & -m_{13} \rho_0 & 0 & 0 \\ 0 & m_{13} \mathbf{V}_0 & m_{13} \mathbf{U}_0 & 0 & 0 \\ 0 & 2m_{13} \mathbf{U}_0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \gamma m_{13} \mathbf{P}_0 & 0 & \gamma m_{13} \mathbf{V}_0 \end{bmatrix} \quad (17) $$

$$ L_{cr_s} = \begin{bmatrix} -m_{13} \mathbf{V}_s & 0 & -m_{13} \rho_s & 0 & 0 \\ 0 & m_{13} \mathbf{V}_s & m_{13} \mathbf{U}_s & 0 & 0 \\ 0 & 2m_{13} \mathbf{U}_s & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \gamma m_{13} \mathbf{P}_s & 0 & \gamma m_{13} \mathbf{V}_s \end{bmatrix} \quad (18) $$
$$ L_{cr_c} = \begin{bmatrix} -m_{13}V_s^* & 0 & -m_{13}\rho_s^* & 0 & 0 \\ 0 & m_{13}V_s^* & m_{13}U_s^* & 0 & 0 \\ 0 & 2m_{13}U_s^* & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \gamma m_{13}P_s & 0 & \gamma m_{13}V_s^* \end{bmatrix} \quad (19) $$

$$ F = -\frac{1}{h_1} \begin{bmatrix} \Psi_0 & \Psi_c & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \Psi_s & \Psi_0 & \Psi_c & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \Psi_s & \Psi_0 & \Psi_c & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \Psi_s & \Psi_0 & \Psi_c & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \Psi_s & \Psi_0 & \Psi_c & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \Psi_s & \Psi_0 & \Psi_c & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \Psi_s & \Psi_0 & \Psi_c & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \Psi_s & \Psi_0 & \Psi_c \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \Psi_s & \Psi_0 \end{bmatrix} \quad (20) $$

$$ \Psi_0 = F_{u_0} + F_{v_0} \qquad \Psi_s = F_{u_s} + F_{v_s} \qquad \Psi_c = F_{u_c} + F_{v_c} \quad (21) $$

$$ F_{u_0} = \begin{bmatrix} iU_0 & -i\rho_0 & 0 & 0 & 0 \\ 0 & iU_0 & 0 & 0 & i\rho_0 \\ 0 & 0 & iU_0 & 0 & 0 \\ 0 & 0 & 0 & iU_0 & 0 \\ 0 & \gamma iP_0 & 0 & 0 & iU_0 \end{bmatrix} \quad (22) $$

$$ F_{u_s} = \begin{bmatrix} iU_s & -i\rho_s & 0 & 0 & 0 \\ 0 & iU_s & 0 & 0 & i\rho_s \\ 0 & 0 & iU_s & 0 & 0 \\ 0 & 0 & 0 & iU_s & 0 \\ 0 & \gamma iP_s & 0 & 0 & iU_s \end{bmatrix} \quad (23) $$

$$ F_{u_{sc}} = \begin{bmatrix} iU_s^* & -i\rho_s^* & 0 & 0 & 0 \\ 0 & iU_s^* & 0 & 0 & i\rho_s^* \\ 0 & 0 & iU_s^* & 0 & 0 \\ 0 & 0 & 0 & iU_s^* & 0 \\ 0 & \gamma iP_s^* & 0 & 0 & iU_s^* \end{bmatrix} \quad (24) $$

$$ F_{v_0} = -\frac{1}{Re} \begin{bmatrix} 0 & 0 & 0 & 0 & 0 \\ 2im_{13}D_x V_0 & 0 & 2im_{13}\rho_0 & 0 & 0 \\ -2im_{13}D_x U_0 & -2im_{13}\rho_0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 2i\gamma/PrD_x P_0 & 0 & 0 & 0 & 2i\gamma/PrD_x\rho_0 \end{bmatrix} \quad (25) $$
$$F_{v_s} = -\frac{1}{Re} \begin{bmatrix} 0 & 0 & 0 & 0 & 0 \\ 2im_{13}\mathcal{D}_x V_s & 0 & 2im_{13}\rho_s & 0 & 0 \\ -2im_{13}\mathcal{D}_x U_s & -2im_{13}\rho_s & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 2i\gamma/Pr\mathcal{D}_x P_s & 0 & 0 & 0 & 2i\gamma/Pr\mathcal{D}_x\rho_s \end{bmatrix} \quad (26)$$

$$F_{v_s c} = -\frac{1}{Re} \begin{bmatrix} 0 & 0 & 0 & 0 & 0 \\ 2im_{13}(\mathcal{D}_x V_s)^* & 0 & 2im_{13}\rho_s^* & 0 & 0 \\ -2im_{13}(\mathcal{D}_x U_s)^* & -2im_{13}\rho_s^* & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 2i\gamma/Pr(\mathcal{D}_x P_s)^* & 0 & 0 & 0 & 2i\gamma/Pr(\mathcal{D}_x\rho_s)^* \end{bmatrix} \quad (27)$$

$$E_v = \begin{bmatrix} \Pi_0 & \Pi_c & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \Pi_s & \Pi_0 & \Pi_c & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \Pi_s & \Pi_0 & \Pi_c & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \Pi_s & \Pi_0 & \Pi_c & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \Pi_s & \Pi_0 & \Pi_c & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \Pi_s & \Pi_0 & \Pi_c & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \Pi_s & \Pi_0 & \Pi_c & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \Pi_s & \Pi_0 & \Pi_0 \end{bmatrix} \quad (28)$$

$$\Pi_0 = E_{v_0} + E_{cr_0} \quad \Pi_s = E_{v_s} + E_{cr_s} \quad \Pi_c = E_{v_c} + E_{cr_c} \quad (29)$$

$$E_{v_0} = -\frac{1}{Re} \begin{bmatrix} 0 & 0 & 0 & 0 & 0 \\ \mathcal{D}_y^2 \mathbf{U}_0 & \rho_0 \Delta & 0 & 0 & 0 \\ \mathcal{D}_y^2 \mathbf{V}_0 & 0 & \rho_0 \Delta & 0 & 0 \\ \mathcal{D}_y^2 \mathbf{W}_0 & 0 & 0 & \rho_0 \Delta & 0 \\ \gamma/Pr(\mathbf{P}_0 \Delta + \mathcal{D}_y^2 \mathbf{P}_0 + 2\mathcal{D}_y \mathbf{P}_0 \mathcal{D}_y) & 0 & 0 & 0 & \Upsilon \end{bmatrix} \quad (30)$$

$$\Upsilon = \gamma/Pr(\rho_0 \Delta + \mathcal{D}_y^2 \rho_0 + 2\mathcal{D}_y \rho_0 \mathcal{D}_y) \quad (31)$$

$$E_{v_s} = -\frac{1}{Re} \begin{bmatrix} 0 & 0 & 0 & 0 & 0 \\ \mathcal{D}_y^2 \mathbf{U}_s - \mathbf{U}_s \beta_s^2 & \rho_s \Delta & 0 & 0 & 0 \\ \mathcal{D}_y^2 \mathbf{V}_s - \mathbf{V}_s \beta_s^2 & 0 & \rho_s \Delta & 0 & 0 \\ \mathcal{D}_y^2 \mathbf{W}_s - \mathbf{W}_s \beta_s^2 & 0 & 0 & \rho_s \Delta & 0 \\ \gamma/Pr(\mathbf{P}_s \Delta + \mathcal{D}_y^2 \mathbf{P}_s - \mathbf{P}_s \beta_s^2 + 2\mathcal{D}_y \mathbf{P}_s \mathcal{D}_y) & 0 & 0 & 0 & \varphi \end{bmatrix} \quad (32)$$

$$\varphi = \gamma/Pr(\rho_s \Delta + \mathcal{D}_y^2 \rho_s - \rho_s \beta_s^2 + 2\mathcal{D}_y \rho_s \mathcal{D}_y) \quad (33)$$

$$E_{v_c} = -\frac{1}{Re} \begin{bmatrix} 0 & 0 & 0 & 0 & 0 \\ (\mathcal{D}_y^2 \mathbf{U}_s)^* - \mathbf{U}_s^* \beta_s^2 & \rho_s^* \Delta & 0 & 0 & 0 \\ (\mathcal{D}_y^2 \mathbf{V}_s)^* - \mathbf{V}_s^* \beta_s^2 & 0 & \rho_s^* \Delta & 0 & 0 \\ \mathcal{D}_y^2 \mathbf{W}_s^* - \mathbf{W}_s^* \beta_s^2 & 0 & 0 & \rho_s^* \Delta & 0 \\ \gamma/Pr(\mathbf{P}_s^* \Delta + (\mathcal{D}_y^2 \mathbf{P}_s)^* - \mathbf{P}_s^* \beta_s^2 + 2(\mathcal{D}_y \mathbf{P}_s)^* \mathcal{D}_y) & 0 & 0 & 0 & \varpi \end{bmatrix} \quad (34)$$
$$ \bar{\varpi} = \gamma/\mathrm{Pr}(\rho_s^* \Delta + (\mathcal{D}_y^2 \rho_s)^* - \rho_s^* \beta_s^2 + 2(\mathcal{D}_y \rho_s)^* \mathcal{D}_y) \quad (35) $$

$$ \mathbf{E}_{cr_0} = -\frac{1}{Re} \begin{bmatrix} 0 & 0 & 0 & 0 & 0 \\ 2\mathbf{m}_{13}\mathcal{D}_x\mathbf{V}_0 & 0 & 0 & 0 & 0 \\ -2\mathbf{m}_{13}\mathcal{D}_x\mathbf{U}_0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \gamma/\mathrm{Pr}(\mathbf{m}_{13}\mathbf{P}_0\mathcal{D}_y + \mathbf{m}_{13}\mathcal{D}_y\mathbf{P}_0) & 0 & 0 & 0 & \gamma/\mathrm{Pr}(\mathbf{m}_{13}\rho_0\mathcal{D}_y + \mathbf{m}_{13}\mathcal{D}_y\rho_0) \end{bmatrix} \quad (36) $$

$$ \mathbf{E}_{cr_s} = -\frac{1}{Re} \begin{bmatrix} 0 & 0 & 0 & 0 & 0 \\ 2\mathbf{m}_{13}\mathcal{D}_x\mathbf{V}_s & 0 & 0 & 0 & 0 \\ -2\mathbf{m}_{13}\mathcal{D}_x\mathbf{U}_s & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \gamma/\mathrm{Pr}(\mathbf{m}_{13}\mathbf{P}_s\mathcal{D}_y + \mathbf{m}_{13}\mathcal{D}_y\mathbf{P}_s) & 0 & 0 & 0 & \gamma/\mathrm{Pr}(\mathbf{m}_{13}\rho_s\mathcal{D}_y + \mathbf{m}_{13}\mathcal{D}_y\rho_s) \end{bmatrix} \quad (37) $$

$$ \mathbf{E}_{cr_c} = -\frac{1}{Re} \begin{bmatrix} 0 & 0 & 0 & 0 & 0 \\ 2\mathbf{m}_{13}(\mathcal{D}_x\mathbf{V}_s)^* & 0 & 0 & 0 & 0 \\ -2\mathbf{m}_{13}(\mathcal{D}_x\mathbf{U}_s)^* & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \gamma/\mathrm{Pr}(\mathbf{m}_{13}\mathbf{P}_s^*\mathcal{D}_y + \mathbf{m}_{13}(\mathcal{D}_y\mathbf{P}_s)^*) & 0 & 0 & 0 & \delta \end{bmatrix} \quad (38) $$

$$ \delta = \gamma/\mathrm{Pr}(\mathbf{m}_{13}\rho_s^*\mathcal{D}_y + \mathbf{m}_{13}(\mathcal{D}_y\rho_s)^*) \quad (39) $$

$$ \bar{\beta}_n = \beta - n\beta_s \quad (40) $$

$$ \mathbf{L}_{\theta} = \begin{bmatrix} \bar{\beta}_{-4}\mathbf{L}_{\theta_0} & \bar{\beta}_{-3}\mathbf{L}_{\theta_c} & 0 & 0 & 0 & 0 & 0 & 0 \\ \bar{\beta}_{-4}\mathbf{L}_{\theta_s} & \bar{\beta}_{-3}\mathbf{L}_{\theta_0} & \bar{\beta}_{-2}\mathbf{L}_{\theta_c} & 0 & 0 & 0 & 0 & 0 \\ 0 & \bar{\beta}_{-3}\mathbf{L}_{\theta_s} & \bar{\beta}_{-2}\mathbf{L}_{\theta_0} & \bar{\beta}_{-1}\mathbf{L}_{\theta_c} & 0 & 0 & 0 & 0 \\ 0 & 0 & \bar{\beta}_{-2}\mathbf{L}_{\theta_s} & \bar{\beta}_{-1}\mathbf{L}_{\theta_0} & \bar{\beta}_0\mathbf{L}_{\theta_c} & 0 & 0 & 0 \\ 0 & 0 & 0 & \bar{\beta}_{-1}\mathbf{L}_{\theta_s} & \bar{\beta}_0\mathbf{L}_{\theta_0} & \bar{\beta}_1\mathbf{L}_{\theta_c} & 0 & 0 \\ 0 & 0 & 0 & 0 & \bar{\beta}_0\mathbf{L}_{\theta_s} & \bar{\beta}_1\mathbf{L}_{\theta_0} & \bar{\beta}_2\mathbf{L}_{\theta_c} & 0 \\ 0 & 0 & 0 & 0 & 0 & \bar{\beta}_1\mathbf{L}_{\theta_s} & \bar{\beta}_2\mathbf{L}_{\theta_0} & \bar{\beta}_3\mathbf{L}_{\theta_c} \\ 0 & 0 & 0 & 0 & 0 & 0 & \bar{\beta}_2\mathbf{L}_{\theta_s} & \bar{\beta}_3\mathbf{L}_{\theta_0} \\ 0 & 0 & 0 & 0 & 0 & 0 & \bar{\beta}_3\mathbf{L}_{\theta_s} & \bar{\beta}_4\mathbf{L}_{\theta_c} \end{bmatrix} \quad (41) $$

$$ \mathbf{L}_{\theta_0} = i \begin{bmatrix} \mathbf{W}_0 & 0 & 0 & -\rho_0 & 0 \\ 0 & \mathbf{W}_0 & 0 & 0 & 0 \\ 0 & 0 & \mathbf{W}_0 & 0 & 0 \\ 0 & 0 & 0 & \mathbf{W}_0 & \rho_0 \\ 0 & 0 & 0 & \gamma\mathbf{P}_0 & \mathbf{W}_0 \end{bmatrix} \quad (42) $$

$$ \mathbf{L}_{\theta_s} = i \begin{bmatrix} \mathbf{W}_s & 0 & 0 & -\rho_s & 0 \\ 0 & \mathbf{W}_s & 0 & 0 & 0 \\ 0 & 0 & \mathbf{W}_s & 0 & 0 \\ 0 & 0 & 0 & \mathbf{W}_s & \rho_s \\ 0 & 0 & 0 & \gamma\mathbf{P}_s & \mathbf{W}_s \end{bmatrix} \quad (43) $$
$$ L_{\theta_c} = i \begin{bmatrix} W_s^* & 0 & 0 & -\rho_s^* & 0 \\ 0 & W_s^* & 0 & 0 & 0 \\ 0 & 0 & W_s^* & 0 & 0 \\ 0 & 0 & 0 & W_s^* & \rho_s^* \\ 0 & 0 & 0 & \gamma P_s^* & W_s^* \end{bmatrix} \quad (44) $$

$$ E_{vt1} = \begin{bmatrix} \bar{\beta}_{-4} \Xi_0 & \bar{\beta}_{-3} \Xi_c & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \bar{\beta}_{-4} \Xi_s & \bar{\beta}_{-3} \Xi_0 & \bar{\beta}_{-2} \Xi_c & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \bar{\beta}_{-3} \Xi_s & \bar{\beta}_{-2} \Xi_0 & \bar{\beta}_{-1} \Xi_c & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \bar{\beta}_{-2} \Xi_s & \bar{\beta}_{-1} \Xi_0 & \bar{\beta}_0 \Xi_c & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \bar{\beta}_{-1} \Xi_s & \bar{\beta}_0 \Xi_0 & \bar{\beta}_1 \Xi_c & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \bar{\beta}_0 \Xi_s & \bar{\beta}_1 \Xi_0 & \bar{\beta}_2 \Xi_c & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \bar{\beta}_1 \Xi_s & \bar{\beta}_2 \Xi_0 & \bar{\beta}_3 \Xi_c & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \bar{\beta}_2 \Xi_s & \bar{\beta}_3 \Xi_0 & \bar{\beta}_4 \Xi_c \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \bar{\beta}_3 \Xi_s & \bar{\beta}_4 \Xi_0 \end{bmatrix} \quad (45) $$

$$ \Xi_0 = 0 \quad (46) $$

$$ \Xi_s = -\frac{1}{Re} \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -2\gamma/Pr\beta_s P_s & 0 & 0 & 0 & -2\gamma/Pr\beta_s\rho_s & 0 & 0 & 0 & 0 \end{bmatrix} \quad (47) $$

$$ \Xi_c = -\frac{1}{Re} \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2\gamma/Pr\beta_s P_s^* & 0 & 0 & 0 & 2\gamma/Pr\beta_s\rho_s^* & 0 & 0 & 0 & 0 \end{bmatrix} \quad (48) $$

$$ E_{vt2} = \begin{bmatrix} \bar{\beta}_{-4}^2 \Lambda_0 & \bar{\beta}_{-3}^2 \Lambda_c & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \bar{\beta}_{-4}^2 \Lambda_s & \bar{\beta}_{-3}^2 \Lambda_0 & \bar{\beta}_{-2}^2 \Lambda_c & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \bar{\beta}_{-3}^2 \Lambda_s & \bar{\beta}_{-2}^2 \Lambda_0 & \bar{\beta}_{-1}^2 \Lambda_c & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \bar{\beta}_{-2}^2 \Lambda_s & \bar{\beta}_{-1}^2 \Lambda_0 & \bar{\beta}_0^2 \Lambda_c & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \bar{\beta}_{-1}^2 \Lambda_s & \bar{\beta}_0^2 \Lambda_0 & \bar{\beta}_1^2 \Lambda_c & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \bar{\beta}_0^2 \Lambda_s & \bar{\beta}_1^2 \Lambda_0 & \bar{\beta}_2^2 \Lambda_c & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \bar{\beta}_1^2 \Lambda_s & \bar{\beta}_2^2 \Lambda_0 & \bar{\beta}_3^2 \Lambda_c & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \bar{\beta}_2^2 \Lambda_s & \bar{\beta}_3^2 \Lambda_0 & \bar{\beta}_4^2 \Lambda_c \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \bar{\beta}_3^2 \Lambda_s & \bar{\beta}_4^2 \Lambda_0 \end{bmatrix} \quad (49) $$
$$ \Lambda_0 = -\frac{1}{Re} \begin{bmatrix} 0 & 0 & 0 & 0 & 0 \\ 0 & -\rho_0 & 0 & 0 & 0 \\ 0 & 0 & -\rho_0 & 0 & 0 \\ 0 & 0 & 0 & -\rho_0 & 0 \\ -\frac{\gamma}{PrP_0} & 0 & 0 & 0 & -\frac{\gamma}{Pr\rho_0} \end{bmatrix} \quad (50) $$

$$ \Lambda_s = -\frac{1}{Re} \begin{bmatrix} 0 & 0 & 0 & 0 & 0 \\ 0 & -\rho_s & 0 & 0 & 0 \\ 0 & 0 & -\rho_s & 0 & 0 \\ 0 & 0 & 0 & -\rho_s & 0 \\ -\frac{\gamma}{PrP_s} & 0 & 0 & 0 & -\frac{\gamma}{Pr\rho_s} \end{bmatrix} \quad (51) $$

$$ \Lambda_c = -\frac{1}{Re} \begin{bmatrix} 0 & 0 & 0 & 0 & 0 \\ 0 & -\rho_s^* & 0 & 0 & 0 \\ 0 & 0 & -\rho_s^* & 0 & 0 \\ 0 & 0 & 0 & -\rho_s^* & 0 \\ -\frac{\gamma}{PrP_s^*} & 0 & 0 & 0 & -\frac{\gamma}{Pr\rho_s^*} \end{bmatrix} \quad (52) $$

## References

[1] Fava T C L et al 2023 J. Phys.: Conf. Ser. 2505 012002

[2] Nogueira P A S et al 2022 Proc. of the 12th Int. Symp. on Turbulence and Shear Flow Phenomena (TSFP12)