At every crank angle position, we record
N independent velocity fields during successive engine cycles, the so-called snapshots (
U
n
). Each snapshot consists of
m 2D velocity vectors. Phase-dependent POD, as the name suggests, calculates the POD modes for every specific CAD. The components
u and
v in
x and
y directions of each velocity vector (
u,
v) for each measured velocity field are stored in a velocity matrix,
U, in which each column represents one measured velocity field taken at the same crank angle, hence
$$ {\bf U}=[U^{1}U^{2} \ldots U^{N}]= \left( \begin{array}{llll} u_{1}^{1}u_{1}^{2} &\ldots &u_{1}^{N}\\ \vdots &\vdots & &\vdots\\ u_{m}^{1}&u_{m}^{2} &\ldots &u_{m}^{N}\\ v_{1}^{1}&v_1^2 &\ldots &v_{1}^{N}\\ \vdots &\vdots & &\vdots\\ v_{m}^{1}&v_{m}^{2} &\ldots &v_{m}^{N}\\ \end{array} \right). $$
(1)
Using the velocity matrix, the
N ×
N space correlation matrix, or autocovariance matrix, is defined as:
$$ {\rm R}={\bf U}^{\rm T}{\bf U}, $$
(2)
for which the eigenvalue problem can be written as
$$ {\rm RA}=\Uplambda {\rm A}. $$
(3)
The eigenvalues (λ) from the eigenvalue array
\(\Uplambda\) are then sorted in decreasing order:
\(\lambda_1>\lambda_2>\ldots>\lambda_N\). The eigenvectors (A
i
) are sorted in the same order as the eigenvalues and stored as a matrix used to define the POD mode matrix (
\(\Upphi\)) which is normalized:
$$ \Upphi=\frac{A {\bf U}}{\|A {\bf U}\|}. $$
(4)
The POD coefficients (
a) for a specific snapshot are determined by projecting the velocity field of the snapshot onto the POD modes
\(\Upphi\):
$$ a^n_i=\Upphi^{i} U^{n}. $$
(5)
Using the POD coefficients and the POD modes, a snapshot (
n) can be reconstructed (subscript
r) using:
$$ U^n_r=\sum^{i_{max}}_{i=1} a_i^{n} \Upphi^{i}. $$
(6)
When all POD mode contributions are included (
i
max =
N), the snapshot is fully reconstructed.