Rational & Lagrangian LES Parametrization

• Rational and Lagrangian Parametrization are two modeling approaches in LES that define subfilter stresses and achieve algebraic equivalence in two-dimensional flows.
• The Rational LES model employs a Padé approximation for the filter kernel to damp high wavenumbers and ensure improved numerical stability.
• The LANS-α model modifies the kinetic energy via Hamilton’s principle, regularizing turbulence while preserving key physical properties in large eddy simulations.

Rational and Lagrangian Parametrization refers to two distinct, but in some regimes equivalent, mathematical approaches to modeling subfilter stresses in turbulent flows, specifically within the Large Eddy Simulation (LES) framework. The “Rational LES” model employs a sub-diagonal Padé approximation for the filter kernel to obtain better control over small scales and restore numerical stability, while the “Lagrangian-averaged Navier-Stokes-α” (LANS-α) model regularizes turbulence equations through a modification of the kinetic energy in Hamilton’s principle. In two spatial dimensions, these methodologies have been shown to be algebraically identical.

1. Large-Scale Filtering and Subfilter Stress in LES

Large Eddy Simulation (LES) resolves only large-scale components of turbulent flows, modeling unresolved (subfilter or subgrid) stresses. For a velocity field $u(x,t)$ with $\nabla\cdot u=0$, a low-pass filter $G$ of width $\Delta$ defines the resolved (“large-scale”) velocity: $\bar{u}(x,t) = \int G(x-x')\,u(x',t)\,dx'$ By linearity,

$$\overline{u \otimes u} = \int G(x-x')\bigl[u(x')\otimes u(x')\bigr]dx'$$

The unclosed subfilter stress tensor is then: $$\tau(x,t) = \overline{u \otimes u} - \bar{u} \otimes \bar{u}$$ The filtered momentum equations (neglecting body forces and viscosity) are: $$\partial_t\bar{u} + \bar{u}\cdot\nabla\bar{u} + \nabla \cdot \tau + \nabla \bar{p} = 0,\quad \nabla\cdot\bar{u} = 0$$

2. Taylor and Rational Approximations for Subfilter Stresses

The classical nonlinear-gradient model arises by expanding $u$ in a Taylor series around $x$, assuming smoothness and a filter with finite second moment: $$u_i(x') = u_i(x) + r_k \partial_k u_i(x) + O(r^2),\quad r = x' - x$$ Filtering and keeping leading order yields: $$\tau_{ij} \approx c^2 \partial_k\bar{u}_i\,\partial_k\bar{u}_j$$ Setting $c^2 = \Delta^2$, the tensor form is: $$\tau_{ij}^{\text{(NG)}} = \Delta^2 \partial_k\bar{u}_i \partial_k\bar{u}_j$$ This model is not sign-definite and may lack sufficient dissipation, prone to local backscatter and finite-time blow-up when used in isolation.

Alternatively, in Fourier space, the filter kernel can be rationally approximated using the sub-diagonal Padé form: $G(k) \approx \frac{1}{1 + (\alpha k)^2}$ This corresponds in real space to the Helmholtz filter: $\bar{u} = (I - \alpha^2 \Delta)^{-1} u$ The Rational LES momentum equation includes a modeled subfilter-stress divergence: $$\nabla\cdot\tau^{(R)} = 2\alpha^2 (I - \alpha^2 \Delta)^{-1} \nabla\cdot[\nabla u_l \nabla u_l] + O(\alpha^4)$$ where $u_l = \bar{u}$ or its filtered version.

Approach	Filter Kernel Approximation	Filtered Variable
Nonlinear Gradient	$G(k) \approx 1 - (\alpha k)^2$	$\bar{u}$
Rational LES	$G(k) \approx 1/[1 + (\alpha k)^2]$	$\bar{u} = (I-\alpha^2\Delta)^{-1}u$

This rational form damps high wavenumbers more than the Taylor series, yielding improved well-posedness and numerical stability.

3. Lagrangian-Averaged Navier–Stokes-α (LANS-α) Model

The LANS-α methodology modifies the variational principle underlying the Navier–Stokes equations by replacing the standard kinetic energy

$E = \frac{1}{2}\int |u|^2 dx$

with a regularized form,

$$E_\alpha = \frac{1}{2} \int \left[|u|^2 + \alpha^2 |\nabla u|^2\right] dx$$

The resulting Euler–Poincaré (Kelvin–theorem) process leads to the α-model momentum equation: $$\partial_t v + (u \cdot \nabla)v + (\nabla u)^T \cdot v + \nabla p = 0, \quad v := u - \alpha^2 \Delta u,\,\, \nabla\cdot u = 0$$ Equivalently,

$$\partial_t u + u_l\cdot\nabla u + \nabla p = \alpha^2 [(\nabla u_l)^T : \nabla\nabla u_l] + \text{(viscosity + forcing)}$$

where $u_l = (I - \alpha^2 \Delta)^{-1} u$. In two-dimensional vorticity form: $$\partial_t \omega + u_l \cdot \nabla \omega = \text{forcing} + \text{dissipation}$$

4. Algebraic Equivalence in Two Dimensions

In two dimensions, the scalar vorticity $\omega = \partial_x u_y - \partial_y u_x$ is used. The Rational LES vorticity equation is: $$\partial_t \omega_l + u_l \cdot \nabla \omega_l = -2\alpha^2 (I-\alpha^2\Delta)^{-1}[\nabla u_l^T : \nabla\nabla\omega_l] + F_l + D_l$$ Applying $(I-\alpha^2\Delta)$ and redefining $\omega = (I-\alpha^2\Delta)\omega_l$, the resulting evolution is: $$\partial_t \omega + u_l \cdot \nabla \omega = F + D + \delta M$$ where the mismatch

$$\delta M = u_l \cdot \nabla[(I-\alpha^2\Delta)\omega_l] - (I-\alpha^2\Delta)(u_l \cdot \nabla\omega_l) - 2\alpha^2 [\nabla u_l^T : \nabla\nabla\omega_l]$$

Using the identity for incompressible 2D flow,

$$\Delta(u_l \cdot \nabla \omega_l) = 2 \nabla u_l^T : \nabla\nabla\omega_l + u_l \cdot \nabla(\Delta\omega_l)$$

it is found that $\delta M = 0$ identically. Thus, the time evolution of vorticity is exactly governed by the LANS-α equation, and the two models are algebraically identical in two dimensions.

5. Physical Properties, Stability, and Computational Considerations

The nonlinear-gradient model’s lack of sign-definiteness leads to potential instabilities and insufficient dissipation. In contrast, both the Rational LES and the LANS-α models recover global well-posedness and numerical stability in two dimensions, with established theorems on existence and uniqueness. This improved behavior results from enhanced control of high-wavenumber modes by the Helmholtz operator inversion.

Computationally, both approaches require inversion of $(I - \alpha^2 \Delta)$, usually implemented via the Fourier transform (for periodic domains) or multigrid methods (for physical space domains). The extra computational cost relative to bare-filter LES is modest.

Regarding physical principles, LANS-α preserves a modified Kelvin circulation theorem and exhibits a variational (Hamiltonian-like) structure, whereas the Rational LES is motivated directly by subfilter stress modeling and closure approximation in the LES context.

6. Model Selection and Applicability

In two dimensions, model choice can be based on interpretational or implementation preferences since the Rational LES and LANS-α are algebraically identical. Either model may be preferable depending on the desired exploitation of Hamiltonian structure (LANS-α) or the focus on subfilter-stress closure (Rational LES).

In three dimensions, the equivalence breaks down due to the presence of vortex-stretching effects that are not equivalently accounted for in both frameworks. The appropriate model must be selected based on numerical performance, ease of implementation, and the requirement for structure preservation versus closure accuracy. This distinction is critical in applications where physical fidelity at three-dimensional subfilter scales is necessary.

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Main source: "The Equivalence of the Lagrangian-Averaged Navier-Stokes-α Model and the Rational LES model in Two Dimensions" (Nadiga et al., 2012)