Enhanced Delayed Detached-Eddy Simulations

• EDDES are advanced hybrid turbulence models that blend RANS and LES techniques to resolve critical shear-layer instabilities and separation phenomena.
• They introduce adaptive subgrid length scales, like Δ_SLA, to reduce eddy viscosity in thin shear layers, significantly improving predictions of lift, drag, and reattachment.
• Variants such as IDDES-SLA and RSM-EDDES offer specialized benefits for complex flows, extending applicability to scenarios like iced wings and rotor wakes.

Enhanced Delayed Detached-Eddy Simulations (EDDES) encompass a class of hybrid RANS–LES approaches designed specifically to address the limitations of traditional improved delayed detached-eddy simulation (IDDES) in separated, anisotropic free-shear flows. EDDES formulations introduce refined subgrid length scales and adaptive hybridization mechanisms to resolve critical shear-layer instabilities (e.g., Kelvin–Helmholtz) and enable realistic separation/reattachment processes on practical grids. Notable variants include shear-layer–adapted IDDES (IDDES-SLA) and Reynolds-stress background model EDDES (RSM-EDDES), each offering enhanced predictive capability and extended applicability for complex turbulent flows.

1. Theoretical Formulation and Key Innovations

The foundational structure of EDDES arises from modifications to the hybrid RANS–LES length scale governing the eddy-viscosity and switching behavior in the SST-IDDES framework. The turbulent kinetic energy transport equation

$$\partial_t(\rho k) + \partial_j(\rho u_j k) = P_k - \beta^* \rho k \omega + \partial_j \left[ (\mu + \sigma_k \mu_t) \partial_j k \right]$$

maintains its conventional form, but the pivotal innovation is the hybridization of the relevant turbulent length scale: $$\ell_{\text{hybrid}} = f_d \cdot \ell_{\text{RANS}} + (1-f_d) \cdot \ell_{\text{LES}}$$ with $\ell_{\text{RANS}}=k/\omega$ and, in the EDDES context, $$\ell_{\text{LES}} = C_{\text{DES}} \cdot \min(\Delta_{\text{wall}}, \Delta_{\text{SLA}})$$.

The shear-layer–adapted scale $\Delta_{\text{SLA}}$ is defined as: $$\Delta_{\text{SLA}} = \Delta_\omega \cdot F_{\text{KH}}(VTM)$$ where $\Delta_\omega$ is the maximal edge projection perpendicular to local vorticity, and $F_{\text{KH}}(VTM)$ is a damping function of the vortex-tilting measurement (VTM), which suppresses the modeled length scale in thin, planar shear layers. This locally geometric and tensorial construct automatically reduces eddy viscosity ($\mu_t$) in separated shear layers, facilitating the development of small-scale instabilities.

In the RSM-EDDES approach, the hybridization is performed on the Reynolds-stress tensor, combining differential Reynolds-stress modeling for RANS regions with a Boussinesq eddy-viscosity closure in LES regions. The hybrid length scale for this variant is governed by: $$l_{\text{hyb}} = \tilde f_d\,(1+f_e)\,l_{\text{RANS}} +(1-\tilde f_d)\,l_{\text{LES}}$$ with delay/blending functions $f_d$, wall-modeling switches, and an elevating function $f_e$ to minimize log-layer mismatch.

2. Integration Into Hybrid RANS–LES Frameworks

Both IDDES-SLA and RSM-EDDES integrate seamlessly with standard IDDES infrastructure. In the SLA variant, only the definition of the subgrid length scale changes; all other components (blending function, destruction terms, eddy-viscosity formulas, flux-blending numerics) remain as in the SST-IDDES baseline.

In RSM-EDDES, the LES region closure transitions from full Reynolds-stress modeling to a Boussinesq form: $$\tau^F_{ij, \text{LES}} = 2 \mu_t \tilde S_{ij} - \frac{2}{3} \mu_t \frac{\partial \tilde v_k}{\partial x_k} \delta_{ij} - \frac{2}{3} \bar\rho k \delta_{ij}$$ with $\mu_t = \bar\rho \tilde k / \omega$ and hybrid weighting either by continuous delay function or user-defined binary mask. In all variants, wall regions employ the standard near-wall model and mesh restrictions to avoid log-layer mismatch; LES modes prevail away from walls or in strong free-shear.

3. Physical and Numerical Effects

EDDES’s main impact is the accurate representation of separated shear-layer physics. For the iced NACA 0012 wing at $Re=1.5 \times 10^6$, $M=0.2$, and $\text{AoA}=8^\circ$ (Xiao et al., 2022):

• Mean aerodynamic coefficients: EDDES recovers experimental values for lift, drag, and moment coefficients, notably correcting ∼15% drag overprediction and nose-down moment observed in standard IDDES. For example, $C_D = 0.099$ (EDDES) vs $0.108$ (IDDES), experiment $0.10$.
• Separation and reattachment: EDDES accurately predicts both onset and recovery within 5% of experiment along the wing span, whereas IDDES overextends reattachment regions by up to 60%.
• Instantaneous structure: Q-criterion visualizations reveal that EDDES produces small-scale, two-dimensional Kelvin–Helmholtz rollers, which pair and break down naturally into hairpin vortices. Standard IDDES, by contrast, yields “overcoherent” large spanwise rollers and fails to resolve primary instabilities.
• Spectral content: EDDES resolves clear Strouhal number peaks for both rollup and pairing events in the separating shear layer, matching measured values ($St_{\delta\omega} = 0.17$--$0.21$), while IDDES fails to capture these spectral features.
• Eddy viscosity dynamics: The SLA-driven collapse of the local length scale ($\Delta_{\text{SLA}} \sim 10^{-3} c$) reduces modeled viscosity by orders of magnitude in developing shear layers, which is essential to enable rollup and reattachment.
• Generalizability: The $\Delta_{\text{SLA}}$ construct is agnostic to global flow features and can be embedded into any hybrid RANS–LES or wall-modeled LES scheme encountering thin free-shear separation.

In RSM-EDDES (Herr et al., 2023), canonical validation (channel, isotropic turbulence, flat-plate boundary layers) confirms that the energy spectrum, mean velocity, and shear-stress profiles conform with DNS and empirical data when employing eddy-viscosity subgrid models in the LES region and recommended tuning parameters ($c_{\rm DES}=0.65$, $c_l=5$, $c_t=1.87$). The differential RSM subgrid model, by itself, is under-dissipative on low-dissipation numerics, causing energy pile-up and requiring local switch to scalar eddy viscosity.

4. Calibration, Validation, and Limitations

EDDES approaches have undergone systematic calibration and validation:

• Aerodynamics and spectra: EDDES produces integrated coefficient statistics, surface pressure distributions, and power spectral density plots (Kelvin–Helmholtz rollup, vortex shedding) that align with experimental results for separated aerofoils with ice-induced horns. For recirculation region length, reattachment, and surface pressure recovery, EDDES accuracy is within 5% of data.
• Turbulent channel and isotropic turbulence: RSM-EDDES variants recover Reynolds stress, mean velocity, and skin friction within ±2% of DNS for periodic channel flow, conditional on proper subgrid closure.
• Flat-plate boundary layer: Both RSM-EDDES and SST-IDDES yield near-constant skin friction downstream of inflow turbulence conditioning regions, deviating from the slow decay of the empirical Coles–Fernholz law. The persistent offset in wall-modeled LES appears insensitive to RANS–LES blending details and is linked to the synthetic turbulence generator (STG).

A notable limitation is that the purely differential RSM subgrid model is insufficiently dissipative in LES regions with low-numerical-dissipation schemes, causing excessive small-scale energy. The hybrid LES closure, incorporating a Boussinesq eddy viscosity, effectively restores proper LES decay. Embedded wall-modeled LES solutions in flat-plate flows show unavoidable skin-friction offset primarily driven by STG–WMLES coupling, not by RSM/EDDES blending.

5. Comparison of EDDES Variants

EDDES Variant	Subgrid Model in LES Region	Main Benefit
IDDES-SLA (Xiao et al., 2022)	SST $k$–$\omega$ + ΔSLA	Accurate shear instability capture
RSM-EDDES (Herr et al., 2023)	Boussinesq eddy viscosity	Anisotropy resolution; robust wall modeling
Pure RSM subgrid	Differential RSM	Insufficient dissipation on low-dissipation numerics

The SLA-based approach is particularly suited for complex, anisotropic separated flows (e.g., iced wings, rotor wakes), while RSM-EDDES targets applications where Reynolds-stress anisotropy is critical in the RANS region, with a practical transition to LES subgrid modeling for resolved regions.

6. Applicability and Generalization

The geometric and tensorial nature of the shear-layer–adapted length scale ($\Delta_{\text{SLA}}$) renders EDDES formulations broadly applicable to scenarios featuring thin, strongly anisotropic free-shear layers. These include flows around bluffed bodies, open cavities, separated ducts, and ice-affected airfoils. The EDDES methodology extends to any hybrid RANS–LES or wall-modeled LES when the accurate recovery of mean loads, unsteady force spectra, and vortex topology under separated-flow conditions are required.

A plausible implication is that the adoption of local, physically responsive length scales such as $\Delta_{\text{SLA}}$, coupled with appropriate subgrid closures, will be critical for next-generation turbulence models to predict key separation-driven performance and unsteady-response metrics across a wide spectrum of engineering flows. Further refinement in synthetic turbulence injection and interface conditioning between wall-modeled LES and RANS regions is likely necessary to correct skin-friction artifacts in embedded WMLES applications.