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Go to the source code of this file.
Namespaces | |
| SPE | |
| Namespace containing all functions and data classes for this solver. | |
Functions | |
| PetscInt | SPE::set_A_and_B_OSS_zi (SPE &data, const PetscInt &zi=0) |
| set A and B matrix for Orr-Sommerfeld equations for zi plane Note that the wavelike ansatz is \(u=\hat{u}(y,z) \exp (i (-\omega t + \alpha x))\) \[ \begin{aligned} (-i \omega - \frac{1}{Re}(\partial_y^2 + \partial_z^2))\hat{u} + U' \hat{v} &= \alpha (-i \hat{P} - \frac{i}{Re}(\partial_y \hat{v} + \partial_z \hat{w}) - iU \hat{u})\\ (-i \omega - \frac{1}{Re}(\partial_y^2 + \partial_z^2))\hat{v} + \partial_y \hat{P} &= \alpha (- \frac{1}{Re}(\alpha \hat{v}) - iU \hat{v})\\ (\alpha \hat{v}) &= \alpha \cdot \hat{v}\\ (-i \omega - \frac{1}{Re}(\partial_y^2 + \partial_z^2))\hat{w} + \partial_z \hat{P} &= \alpha (- \frac{1}{Re}(\alpha \hat{w}) - iU \hat{w})\\ (\alpha \hat{w}) &= \alpha \cdot \hat{w}\\ \partial_y \hat{v} + \partial_z \hat{w} &= \alpha (-i \hat{u})\\ \end{aligned}\] Where we have \[ \begin{aligned} \mathbf{z}_i= \begin{bmatrix} \hat{u}\\ \alpha \hat{v}\\ \hat{v}\\ \alpha \hat{w}\\ \hat{w}\\ \hat{P} \end{bmatrix}_{z=z(i)} \textrm{ and } & \mathbf{q} = \begin{bmatrix} \mathbf{z}_1\\ \mathbf{z}_2\\ \vdots \\ \mathbf{z}_{nz - 1}\\ \mathbf{z}_{nz}\\ \end{bmatrix}\\ \end{aligned}\] To solve the spatial eigenvalue problem \(\mathbf{\mathcal{A}q} = \alpha \mathbf{\mathcal{B}q}\) More... | |
1.8.11
