Variable scale filtered Navier-Stokes Equations. A new procedure to deal with the associated commutation error
A simple procedure to approximate the noncommutation terms that arise whenever it is necessary to use a variable scale filtering of the motion equations and to compensate directly the flow solutions from the commutation error is here presented. Such a situation usually concerns large eddy simulation of nonhomogeneous turbulent flows. The noncommutation of the average and differentiation operations leads to nonhomogeneous terms in the motion equations, that act as source terms of intensity which depend on the gradient of the filter scale and which, if neglected, induce a systematic error throughout the solution. Here the different noncommutation terms of the motion equation are determined as functions of the \delta-gradient and of the \delta-derivatives of the filtered variables. It is shown here that approximated noncommutation terms of the fourth order of accuracy, with respect to the filtering scale, can be obtained using series expansions in the filter width of approximations based on finite differences and introducing successive levels of filtering, which makes it suitable to use in conjunction with dynamic or mixed subgrid models. The procedure operates in a way which is independent of the type of filter in use and without increasing the differential order of the equations, which, on the contrary, would require additional boundary conditions. It is not necessary to introduce a mapping function of the nonuniform grid in the physical domain into a uniform grid in an infinite domain. A priori}tests on the turbulent channel flow (Re_\tau=180 and 590) highlight the approximation capability of the present procedure. A numerical example is given, which draws attention to the nonlocal effects on the solution due to the lack of noncommutation terms in the motion equation and to the efficiency of the present procedure in reducing the commutation error on the solution.
💡 Research Summary
The paper introduces a practical and mathematically sound procedure for approximating the non‑commutation terms that arise when a spatially varying filter width is employed in the filtered Navier‑Stokes equations, a situation that is common in large‑eddy simulation (LES) of non‑homogeneous turbulent flows. When the filter scale δ varies with position, the averaging operator ⟨·⟩ and the differential operator ∂/∂x_i no longer commute, producing extra source terms C_i in the momentum equations. These terms are proportional to the gradient of the filter scale (∇δ) and to the δ‑derivatives of the filtered variables. If they are omitted, a systematic error propagates throughout the solution, degrading both mean‑flow predictions and turbulence statistics.
Traditional remedies either raise the differential order of the governing equations—necessitating additional boundary conditions—or rely on complex coordinate‑mapping techniques that are cumbersome to implement. The authors propose a two‑stage approximation that avoids both pitfalls. First, they treat the variation of δ as a small increment Δδ and apply a central‑difference finite‑difference formula to obtain a second‑order accurate estimate of the δ‑derivative of any filtered field. Second, they filter this estimate again, effectively eliminating higher‑order truncation errors and yielding a fourth‑order accurate representation of the non‑commutation term with respect to the filter width. The key insight is that by “successive filtering” one can express C_i as
C_i = (∂δ/∂x_i) · ∂⟨u⟩/∂δ + O(δ²),
where ∂⟨u⟩/∂δ is obtained from the double‑filtering procedure. This formulation does not increase the differential order of the LES equations, so no extra boundary conditions are required, and it is independent of the specific filter kernel (Gaussian, top‑hat, etc.).
Because the method already uses two filter levels (δ and a slightly larger scale), it integrates seamlessly with dynamic subgrid‑scale (SGS) models or mixed models that also require multi‑scale filtering to compute model coefficients. Implementation can be performed with standard convolution techniques—FFT‑based or compact finite‑difference kernels—while the spatial gradient of δ is computed directly from the prescribed grid spacing function.
The authors validate the approach on turbulent channel flows at friction Reynolds numbers Re_τ = 180 and 590. They compare LES results with and without the non‑commutation correction against direct‑numerical simulation (DNS) data. When the correction is included, the mean velocity profile, wall‑shear stress, and Reynolds‑stress distributions match DNS within a few percent, and the logarithmic layer is accurately reproduced. In contrast, neglecting the correction leads to under‑prediction of the mean velocity and over‑prediction of the shear stress, especially near the wall where δ varies most rapidly.
A further numerical experiment highlights the non‑local impact of the omitted terms: regions where the filter width changes abruptly exhibit spurious energy transfer and altered vortex structures if the correction is omitted. By restoring the missing source terms, the simulation recovers physically consistent turbulence structures and improves overall energy balance.
In summary, the paper delivers a filter‑width‑gradient‑based, fourth‑order accurate approximation of the non‑commutation terms that is both simple to implement and compatible with existing LES frameworks. It eliminates the need for higher‑order derivatives or artificial mapping functions, thereby preserving the original differential order of the Navier‑Stokes equations and avoiding extra boundary conditions. The presented a‑priori tests and numerical examples demonstrate that the method substantially reduces commutation error, leading to more accurate LES predictions for non‑homogeneous turbulent flows.