Yes, it is possible to use POD and DMD to determine the coherent structures of fluid flow at a single time instance. This approach is often referred to as snapshot-based analysis or static mode decomposition.

To apply POD to a single snapshot of the flow over an airfoil, you can collect a set of velocity fields at different spatial locations and time instances, and arrange them into a data matrix X. This matrix can then be decomposed using singular value decomposition (SVD) to obtain the spatial modes and corresponding temporal coefficients. The spatial modes represent the coherent structures of the flow at the given time instance, while the temporal coefficients describe the amplitudes and time evolution of these structures.

Similarly, DMD can also be applied to a single snapshot of the flow to extract the dominant coherent structures. In this case, the data matrix X is partitioned into two matrices, X1 and X2, representing the flow at consecutive time instances. The DMD algorithm computes the eigenvectors and eigenvalues of the matrix product Y = X2 X1+ , where X1+ is the pseudo-inverse of X1. The eigenvectors represent the spatial modes, while the eigenvalues describe their growth or decay rates and frequencies.

It's worth noting that while these snapshot-based approaches can provide useful insight into the coherent structures of the flow at a single time instance, they may not capture the full dynamics and evolution of the flow. For a more comprehensive analysis, it's recommended to use a time series of velocity fields and perform a temporal mode decomposition using techniques such as POD or DMD.

Here is a python script to help you -:

import numpy as np

from scipy.linalg import svd

# Generate sample velocity field data (size N x M)

N = 100 # number of spatial locations

M = 1 # number of time instances (snapshots)

X = np.random.rand(N, M)

# Compute the mean velocity field

mean_X = np.mean(X, axis=1)

# Subtract the mean from the data

X_centered = X - mean_X[:, np.newaxis]

# Compute the SVD of the centered data

U, S, Vt = svd(X_centered, full_matrices=False)

# Compute the POD modes (size N x K)

K = 10 # number of modes to keep

Phi = U[:, :K]

# Compute the DMD modes (size N x K)

X1 = X_centered[:, :-1]

X2 = X_centered[:, 1:]

U2, s, Vh = np.linalg.svd(X1, full_matrices=False)

Sinv = np.diag(1/s)

Atilde = U2.T @ X2 @ Vh.T @ Sinv

Lambda, W = np.linalg.eig(Atilde)

Phi_DMD = X2 @ Vh.T @ Sinv @ W

# Print the first POD and DMD modes

print("First POD mode:\n", Phi[:, 0])

print("First DMD mode:\n", Phi_DMD[:,

Thank you

Naman Bajpai: Since we have only the M=1 column, wouldn't mean_X be equal to X?