This chapter, titled "Dynamics and Scaling," by Lawrence Sirovich, delves into the dynamics of coherent structures in turbulent flows. The author assumes a complete set of vector eigenfunctions \(\{V_n\}\) of the correlation matrix operator \(\mathbf{K}\), which fit the boundary conditions and satisfy the continuity equation for incompressible flows. The flow \(\mathbf{v}\) is expanded in this set, and the Navier-Stokes equations are projected onto these eigenfunctions to obtain ordinary differential equations for the coefficients \(a_k\). The Galerkin approximation is discussed, where the infinite system is truncated to a finite one, yielding an approximate solution. The ordering of eigenfunctions by eigenvalues is emphasized to maximize the retained energy fraction. The ambiguity in the expansion of total flow quantities versus fluctuations is addressed, suggesting that only the fluctuations should be expanded. The effect of changes in physical parameters, such as Reynolds number, on the eigenfunctions is discussed, noting that the calculated eigenfunctions may still be useful over a range of parameter space. The chapter then focuses on channel flow, detailing the geometry and governing equations. It explores different solution strategies, including expanding the mean flow and using the known mean flow to fit the fluctuations. The approach is extended to handle off-reference values of the Reynolds number, using iterative procedures to update the coherent structures. The discussion section highlights the importance of data compression using eigenfunction representations, demonstrating significant savings in storage. The final remarks emphasize the efficiency of eigenfunctions as a basis set for describing flows and their role in reducing the dimensionality of the attractor. The chapter concludes with a note on recent related work and acknowledges the contributions of several individuals and organizations.This chapter, titled "Dynamics and Scaling," by Lawrence Sirovich, delves into the dynamics of coherent structures in turbulent flows. The author assumes a complete set of vector eigenfunctions \(\{V_n\}\) of the correlation matrix operator \(\mathbf{K}\), which fit the boundary conditions and satisfy the continuity equation for incompressible flows. The flow \(\mathbf{v}\) is expanded in this set, and the Navier-Stokes equations are projected onto these eigenfunctions to obtain ordinary differential equations for the coefficients \(a_k\). The Galerkin approximation is discussed, where the infinite system is truncated to a finite one, yielding an approximate solution. The ordering of eigenfunctions by eigenvalues is emphasized to maximize the retained energy fraction. The ambiguity in the expansion of total flow quantities versus fluctuations is addressed, suggesting that only the fluctuations should be expanded. The effect of changes in physical parameters, such as Reynolds number, on the eigenfunctions is discussed, noting that the calculated eigenfunctions may still be useful over a range of parameter space. The chapter then focuses on channel flow, detailing the geometry and governing equations. It explores different solution strategies, including expanding the mean flow and using the known mean flow to fit the fluctuations. The approach is extended to handle off-reference values of the Reynolds number, using iterative procedures to update the coherent structures. The discussion section highlights the importance of data compression using eigenfunction representations, demonstrating significant savings in storage. The final remarks emphasize the efficiency of eigenfunctions as a basis set for describing flows and their role in reducing the dimensionality of the attractor. The chapter concludes with a note on recent related work and acknowledges the contributions of several individuals and organizations.