Large Eddy Simulations (LES)

• Large Eddy Simulation (LES) is a numerical technique that resolves dominant turbulent motions while modeling smaller eddies via spatial filtering and subgrid-scale closures.
• LES employs various methodologies such as Smagorinsky, WALE, and renormalization-group models to capture turbulence in engineering, astrophysical, and environmental applications.
• LES validation involves benchmarking against DNS and experiments by comparing energy spectra, fluxes, and resolution effects to ensure accurate turbulence simulation.

Large Eddy Simulation (LES) is a numerical approach for simulating turbulent flows with high Reynolds numbers in which the large turbulent motions are resolved directly, while the effect of smaller, unresolved eddies—the subgrid scales (SGS)—is modeled through closure terms introduced by spatial filtering of the governing equations. In compressible settings, Favre filtering is standard; in implicit large eddy simulations (ILES), energy is dissipated through numerical diffusivities, whereas explicit SGS models are applied in LES to compute energy dissipation, mixing, and related transfers due to numerically unresolved turbulent eddies (Schmidt-Brückner, 8 Sep 2025, Schmidt, 2014).

1. Scale decomposition and filtered equations

The mathematical foundation of LES is a scale decomposition by means of a linear spatial filter. In the astrophysical reviews, the filtered field is written as

$$\langle q\rangle_G(\mathbf{x}) = \int G(\mathbf{x}-\mathbf{x}') q(\mathbf{x}',t)\, d^3x',$$

with filter length typically identified with the grid scale $\Delta$ (Schmidt-Brückner, 8 Sep 2025, Schmidt, 2014). For compressible flows, density-weighted Favre filtering is used,

$$\tilde{u}_i = \frac{\langle \rho u_i \rangle}{\langle \rho \rangle},$$

because filtering the compressible Navier-Stokes equations yields equations with extra subgrid-scale terms that arise from nonlinearities (Schmidt-Brückner, 8 Sep 2025).

The filtered momentum equation contains the SGS stress tensor,

$$\tau_{ij} = -\langle \rho u_i u_j \rangle + \frac{\langle \rho u_i \rangle \langle \rho u_j \rangle}{\langle \rho \rangle},$$

and its trace defines the SGS turbulent kinetic energy through

$\langle\rho\rangle K := -\frac{1}{2}\tau_{ii}.$

This is the classical closure problem: the filtered equations are not closed because the SGS terms depend on unresolved fluctuations (Schmidt-Brückner, 8 Sep 2025, Schmidt, 2014).

The same logic extends beyond neutral-fluid Navier-Stokes turbulence. For cosmological applications, the filtered equations are formulated in co-moving coordinates, with additional source terms associated with expansion and gravitational potential fluctuations (Schmidt-Brückner, 8 Sep 2025, Schmidt, 2014). The decomposition is also extended to magnetohydrodynamics, where filtered momentum and induction equations generate SGS magnetic stresses and SGS electromotive forces (Schmidt-Brückner, 8 Sep 2025). In gyrokinetics, filtering is performed in Fourier space in the perpendicular directions, and the resolved distribution function obeys

$$\partial_t \overline{f}_{ki}=L[\overline{f}_{ki}] + N[\overline{f}_{ki},\overline{f}_{ki}] + D[\overline{f}_{ki}] + \overline{T},$$

with $\overline{T}$ encoding the influence of unresolved modes on the resolved scales (Morel et al., 2011).

2. Subgrid-scale closures and model families

A large fraction of LES methodology is organized around how the SGS terms are represented. The most commonly used models in astrophysics are the Smagorinsky model, the hydrodynamical SGS turbulence energy equation model, and the non-linear structural model for both non-relativistic and relativistic MHD (Schmidt-Brückner, 8 Sep 2025). In compressible jet LES, the Smagorinsky closure is written as

$$\mu_{sgs} = \overline{\rho}(C_s\Delta)^2 \left(2\check{S}_{ij}\check{S}_{ij}\right)^{1/2},$$

with $C_s=0.18$ and $Pr_{sgs}=0.6$ in the cited solver (Junqueira-Junior et al., 2022). The SGS turbulence energy equation model instead evolves $\Delta$0 dynamically, with modeled production, dissipation, and turbulent diffusion (Schmidt-Brückner, 8 Sep 2025, Schmidt, 2014).

Several closures are designed to improve the behavior of eddy viscosity in specific regimes. The WALE model uses both the strain tensor and the rotation tensor, unlike the classical Smagorinsky model that considers only the strain portion of the velocity gradient tensor, and thereby yields the correct near-wall scaling $\Delta$1 (Alam et al., 2017). In high-order DG LES for a Formula 1 front wing, the explicit formulation uses the Vreman model, which adapts to laminar, transitional and turbulent regimes, whereas the implicit formulation relies on the discretization scheme rather than an explicit SGS model (Ntoukas et al., 2024). In urban atmospheric boundary layer LES over roughness, the Vreman model is also used for SGS viscosity within a high-order, low-dissipation incompressible solver (De, 15 Feb 2025).

Renormalization-group-based closures represent a different line of development. In decaying homogeneous and isotropic turbulence, the total LES viscosity is

$\Delta$2

with $\Delta$3 and $\Delta$4 (Vashishtha et al., 2017). The same renormalized viscosity is used in LES of turbulent thermal convection, with $\Delta$5 and turbulent Prandtl number set to unity (Vashishtha et al., 2018).

Structural and filter-based models pursue closure without a simple eddy-viscosity ansatz. The non-linear structural model in astrophysical MHD follows a gradient expansion such as

$\Delta$6

and analogous expressions for magnetic SGS stress and EMF (Schmidt-Brückner, 8 Sep 2025). A different proposal applies the filter not to the Navier-Stokes equations, but to generalized Navier-Stokes equations with nonlinear viscosity, and then closes the SGS tensor through the Clark approximation (Rodríguez et al., 2015). In explicit filtering LES, no added SGS terms are introduced at all; instead, the governing equations are integrated and a low-pass filter is applied after each time step, so that the high-wavenumber part acts as a spectral buffer (Mathew, 2016).

3. Numerical realizations and computational frameworks

LES is implemented across a wide range of numerical discretizations. Pseudo-spectral realizations appear in decaying homogeneous and isotropic turbulence using the code Tarang and in turbulent thermal convection using a sharp spectral filter at the cutoff wavenumber (Vashishtha et al., 2017, Vashishtha et al., 2018). Gyrokinetic LES is implemented in the GENE code by reducing the resolution of the simulation grid in the perpendicular directions while keeping the other phase-space directions fixed (Morel et al., 2011).

High-order finite-difference and discontinuous-Galerkin realizations are prominent in compressible and external-aerodynamic applications. A structured finite difference LES solver for supersonic jets employs second-order accurate centered differencing with Turkel-Vatsa artificial dissipation and a 5-stage, 2nd-order accurate Runge-Kutta scheme (Junqueira-Junior et al., 2022). Supersonic jet resolution studies with the FLEXI framework use a nodal discontinuous Galerkin method, Roe scheme with entropy fix, FV sub-cell shock capturing, and five-stage, fourth-order explicit Runge-Kutta time integration (Abreu et al., 2023). High-order DGSEM also underpins implicit LES of a low-pressure turbine cascade and explicit/implicit LES comparisons in Horses3D, where the Kennedy-Gruber entropy stable formulation with Gauss-Lobatto points enables larger time steps in under-resolved simulations (Morsbach et al., 2023, Ntoukas et al., 2024).

Finite-element and finite-volume variants broaden the computational landscape. Industrial-scale LES with complex geometries is achieved by combining adaptive octree meshes with immersogeometric analysis, the variational multiscale method, adaptive quadrature, and weak imposition of Dirichlet boundary conditions using Nitsche’s method (Saurabh et al., 2020). For quasi-geostrophic equations, a segregated three-step algorithm is combined with a computationally efficient Finite Volume method and a nonlinear differential low-pass filter acting on potential vorticity (Girfoglio et al., 2022).

Mesh-free and stochastic formulations show that LES is not restricted to Eulerian grid solvers. In bubbly flows and breaking waves, the liquid phase is advanced with a semi-implicit isothermally compressible SPH framework and coupled to a discrete Lagrangian bubble model through exchanges of volume and momentum, bubble breakup and entrainment closures, bubble-induced turbulence, and free-surface interaction models (King et al., 2022). A random LES method integrates the technology of random vortex method with spatial filtering and the integral representation theorem for parabolic equations, then computes filtered velocities through Monte-Carlo approximation (Guo et al., 2024). Lagrangian LES via physics-informed machine learning replaces the fixed mesh by interacting particles whose pairwise dynamics, smoothing kernel, and SGS effects are learned from DNS Lagrangian data under differentiable programming (Tian et al., 2022).

4. Validation, diagnostics, and resolution assessment

A recurring feature of LES research is the use of diagnostics that probe both large-scale statistics and scale-by-scale transfer. In renormalized-viscosity LES of decaying homogeneous and isotropic turbulence, coarse-grid LES on $\Delta$7, $\Delta$8, and $\Delta$9 was compared with DNS on $$\tilde{u}_i = \frac{\langle \rho u_i \rangle}{\langle \rho \rangle},$$0; the reported agreement covers the temporal evolution of turbulence kinetic energy $$\tilde{u}_i = \frac{\langle \rho u_i \rangle}{\langle \rho \rangle},$$1, kinetic energy spectrum $$\tilde{u}_i = \frac{\langle \rho u_i \rangle}{\langle \rho \rangle},$$2, kinetic energy flux $$\tilde{u}_i = \frac{\langle \rho u_i \rangle}{\langle \rho \rangle},$$3, and isosurfaces of velocity magnitude, while lower resolutions show a bottleneck effect near the dissipation wavenumber (Vashishtha et al., 2017). In thermal convection, LES on $$\tilde{u}_i = \frac{\langle \rho u_i \rangle}{\langle \rho \rangle},$$4 agrees with DNS on $$\tilde{u}_i = \frac{\langle \rho u_i \rangle}{\langle \rho \rangle},$$5 for the evolution of kinetic energy and entropy, spectra and fluxes of velocity and temperature fields, and isosurfaces of temperature (Vashishtha et al., 2018).

For under-resolved shock-driven mixing, numerical dissipation and physical dissipation are assessed through a posteriori diagnostics. The proposed effective viscosity and effective species diffusivity framework is agnostic to the LES model, numerics and the physical Reynolds number of the simulation, and leads to effective Reynolds and Schmidt numbers that clarify when the solution behaves as if at lower effective Reynolds number (Olson et al., 2014). In supersonic jets, hp-refinement studies on 50 to 410 million DOFs show that mean velocity results and RMS values of velocity fluctuations improve with increasing resolution; the potential core length error decreases from 30% to 13% and then to 5.5% across the reported refinements (Abreu et al., 2023).

Urban-flow validation places particular emphasis on combined statistical and spectral tests. In wall-modeled LES over urban roughness, mean-velocity profiles, wake regions, and rooftop acceleration are captured with excellent agreement in the streamwise velocity component; FAC2 routinely exceeds 0.7–1.0, and spectral analysis verifies that LES accurately resolves the turbulent energy cascade over approximately two frequency decades, reproducing the Kolmogorov $$\tilde{u}_i = \frac{\langle \rho u_i \rangle}{\langle \rho \rangle},$$6 slope in the pre-multiplied spectra below and above the urban canopy (De, 15 Feb 2025). In gyrokinetic turbulence, validation is based on the electron density fluctuation spectrum, free energy spectra, nonlinear transfer spectra, fluxes, shell-to-shell transfers, and locality functions; the LES recovers spectral shape and power-law behavior that under-resolved simulations flatten artificially (Navarro et al., 2013). In the turbine cascade, realistic synthetic inflow conditions allow discrepancies between LES and RANS to be attributed to physical modelling problems, and excellent agreement with the experiment is obtained in blade loading at midspan and wake total pressure losses at midspan and over the channel height (Morsbach et al., 2023).

5. Major application domains

Canonical turbulence remains a central test bed. Decaying homogeneous and isotropic turbulence has been used to evaluate parameter-free RG closures and dynamic energy-flux-based viscosity updates (Vashishtha et al., 2017). Turbulent thermal convection extends the same renormalized approach by adding thermal diffusivity and shows the capability of LES to simulate thermal convection at very high Rayleigh numbers, with reported calculations at $$\tilde{u}_i = \frac{\langle \rho u_i \rangle}{\langle \rho \rangle},$$7 on a $$\tilde{u}_i = \frac{\langle \rho u_i \rangle}{\langle \rho \rangle},$$8 grid (Vashishtha et al., 2018).

Compressible jet flows constitute a major application area because they combine turbulence, shocks, and acoustics. LES of perfectly expanded supersonic jets at $$\tilde{u}_i = \frac{\langle \rho u_i \rangle}{\langle \rho \rangle},$$9 and $$\tau_{ij} = -\langle \rho u_i u_j \rangle + \frac{\langle \rho u_i \rangle \langle \rho u_j \rangle}{\langle \rho \rangle},$$0 is used for aeroacoustic analyses of rocket plumes and as input to the Ffowcs Williams and Hawkings approach (Junqueira-Junior et al., 2022). Resolution studies for a perfectly expanded free jet with exit Mach number 1.4 examine how mesh and polynomial refinement affect mean and fluctuating velocity predictions, shock-cell structure, and mixing-layer development (Abreu et al., 2023).

External and industrial aerodynamics motivate high-order and large-scale implementations. In a Formula 1 front wing section, both explicit LES with Vreman SGS viscosity and implicit LES with an entropy stable DG split form agree well with Nektar++, while the implicit LES better captures transition and allows for larger time steps at a similar cost per iteration (Ntoukas et al., 2024). Adaptive-octree immersogeometric LES accurately computes the drag coefficient of a sphere across Reynolds numbers $$\tau_{ij} = -\langle \rho u_i u_j \rangle + \frac{\langle \rho u_i \rangle \langle \rho u_j \rangle}{\langle \rho \rangle},$$1–$$\tau_{ij} = -\langle \rho u_i u_j \rangle + \frac{\langle \rho u_i \rangle \langle \rho u_j \rangle}{\langle \rho \rangle},$$2, encompassing the drag crisis regime, and simulates flow across a semi-truck to investigate platooning, including a reported 32% reduction in $$\tau_{ij} = -\langle \rho u_i u_j \rangle + \frac{\langle \rho u_i \rangle \langle \rho u_j \rangle}{\langle \rho \rangle},$$3 for the trailing vehicle (Saurabh et al., 2020). In a low-pressure turbine cascade, implicit LES resolves separation-induced transition, secondary vortical structures, and anisotropy fields that RANS models fail to reproduce satisfactorily (Morsbach et al., 2023).

Urban and environmental flows illustrate the interaction between turbulence modelling and geometric complexity. A canopy stress method represents the loss of pressure and the sink of momentum due to interaction between eddies and roughness elements without a mesh conforming to urban roughness elements, and is reported to predict vertical profiles of mean and variance as well as the temporal intermittency of coherent structures (Alam et al., 2017). Wall-modeled LES over arrays of square prisms and the Michel-Stadt geometry uses an online precursor simulation to supply realistic turbulent inflow and reproduces mean flow, turbulent kinetic energy at most locations, and spectral signatures within and above the canopy (De, 15 Feb 2025).

LES also extends to non-Navier-Stokes and multiphase systems. In Richtmyer-Meshkov instability, LES and DNS are used to study the interaction of grid resolution, Reynolds number and numerical method on mixing (Olson et al., 2014). In gyrokinetic plasma turbulence, LES reproduces free-energy spectra for ion-temperature-gradient-driven turbulence and, in later work, electron density fluctuation spectra across regimes driven by different microinstabilities (Morel et al., 2011, Navarro et al., 2013). In astrophysics, LES methodology is applied to cosmological structure formation, turbulent combustion in thermonuclear supernovae, star formation and feedback in galaxies, magnetic field amplification in binary neutron star mergers, and explicit modelling of turbulent diffusion of metals in mesh-free codes (Schmidt-Brückner, 8 Sep 2025, Schmidt, 2014). In bubbly flows and breaking waves, the LES-SPH framework predicts the Hinze scale, bubble size distribution, and growth rate of the entrained bubble population (King et al., 2022). For barotropic quasi-geostrophic double-gyre flow, a nonlinear differential low-pass filter recovers the four-gyre pattern of the time-averaged stream function even with extremely coarse meshes (Girfoglio et al., 2022).

6. Limits, misconceptions, and current directions

LES performance depends strongly on the relation between SGS modelling, numerical dissipation, and the target flow class. The renormalized-viscosity model is derived under the assumption of homogeneity and isotropy, and the cited work cautions that it may not be immediately applicable for anisotropic, inhomogeneous, or wall-bounded flows (Vashishtha et al., 2017). In thermal convection, the reported $$\tau_{ij} = -\langle \rho u_i u_j \rangle + \frac{\langle \rho u_i \rangle \langle \rho u_j \rangle}{\langle \rho \rangle},$$4 scaling for $$\tau_{ij} = -\langle \rho u_i u_j \rangle + \frac{\langle \rho u_i \rangle \langle \rho u_j \rangle}{\langle \rho \rangle},$$5 is attributed to the lack of SGS modeling for boundary layers (Vashishtha et al., 2018). In urban atmospheric boundary layers, wall-models struggle in regions of strong separation, contributing to discrepancies in turbulent kinetic energy near the ground (De, 15 Feb 2025). In Richtmyer-Meshkov instability, under coarse grids numerical dissipation dominates and the solution behaves as if at lower effective Reynolds number (Olson et al., 2014).

A common misconception is that LES accuracy is controlled only by the formal SGS closure. The cited literature shows that numerical method, quadrature, filtering strategy, inflow generation, and wall treatment are equally consequential. In the Formula 1 front wing study, the implicit LES owes its behavior to the Kennedy-Gruber entropy stable formulation and Gauss-Lobatto collocation, while the explicit LES depends on the Vreman eddy viscosity and Gauss-point integration (Ntoukas et al., 2024). In supersonic jets, the largest remaining discrepancies close to the jet lip are attributed not to insufficient resolution but to the use of a flat inflow velocity profile (Abreu et al., 2023). In the turbine cascade, realistic Fourier-series-based synthetic turbulence generation is essential for obtaining a validated LES reference (Morsbach et al., 2023).

Another misconception is that explicit SGS models always dominate the under-resolved dynamics. In many astrophysical codes, significant numerical diffusion means that the additional effect of SGS models is generally small, although convergence with resolution increases in some cases and explicit modelling can be essential for magnetic field amplification or turbulent metal diffusion (Schmidt-Brückner, 8 Sep 2025). This suggests that the practical distinction between LES, ILES, and numerics-aware closure design remains problem-dependent rather than doctrinal.

Current directions broaden the LES concept rather than narrow it. Explicit filtering interprets SGS action as a spectral buffer (Mathew, 2016). Random LES bypasses classical SGS stress modelling by filtering the stochastic integral representation of incompressible viscous flow (Guo et al., 2024). Lagrangian LES via physics-informed machine learning learns pairwise particle interactions, smoothing kernels, and loss functions that constrain both Eulerian and Lagrangian turbulence statistics (Tian et al., 2022). Taken together, these developments suggest that LES is best understood as a scale-separation framework whose concrete realization can be pseudo-spectral, finite-volume, DG, finite-element, SPH, stochastic, or machine-learned, provided that the resolved-to-unresolved transfer is controlled and validated against appropriate statistics, spectra, and fluxes.