Overshoot-resolved transition modeling based on field inversion and symbolic regression

TL;DR

This work tackles the persistent overshoot in high-speed transitional flows by marrying field inversion with symbolic regression (FISR) to produce an interpretable, overshoot-capable augmentation for a high-speed transition model. EnKF-based field inversion identifies a spatial correction β(𝑥) tied to the intermittency field γ, while PySR-derived expressions translate β(𝑥) into a compact analytical form with explicit physical meaning. The resulting SR-augmented model reproduces skin-friction and heat-transfer overshoot across flat plates and cone configurations without compromising transition onset or length, and it introduces protections to avoid spurious laminar corrections. The approach reveals a physically grounded mechanism: turbulent Mach number elevates γ, which strengthens TKE production and distorts velocity/temperature profiles to generate overshoot, with generalization to low-speed regimes demonstrated and a path toward a concise, universal correction discussed.

Abstract

Overshoot of high-speed transitional skin-friction and heat-transfer values over their fully turbulent levels is well documented by numerous direct numerical simulations (DNS) and experimental studies. However, this high-speed-specific overshoot phenomenon remains a longstanding challenge in Reynolds-averaged Navier-Stokes (RANS) transition models. In this paper, field inversion and symbolic regression (FISR) methodologies are adopted to explore a generalizable and interpretable augmentation for resolving the missing overshoot characteristic. Specifically, field inversion is implemented on our previous high-speed-improved $k$-$ω$-$γ$-$\widetilde{Re}_{θ\rm{t}}$ transition-turbulence model. Then symbolic regression is employed to derive an analytical map from RANS mean flow variables to the pre-defined and inferred corrective field $β(\mathbf{x})$. Results manifest that the excavated expression faithfully reproduces the overshoot phenomena of transition region over various test cases while does not corrupt model behavior in transition location and length. Based on its transparent functional form, mechanistic investigations are conducted to illustrate the underlying logic for accurate capture of overshoot phenomenon. In addition, importance of protect function in $β(\mathbf{x})$, feasibility of a more concise expression for $β(\mathbf{x})$, and reliable performance of $β(\mathbf{x})$ in low-speed transitional flows are emphasized.

Figures (26)

• Figure 1: Model parameter sensitivity analysis for adiabatic flat plate at $Ma_\infty=2.25, Re_\infty=2.50\times10^7$: (a) $c_{a1}$, (b) $c_{e1}$, (c) $c_{a2}$, (d) $c_{e2}$, (e) $\sigma_\gamma$, (f) $c_{\theta \rm{t}}$, and (g) $\sigma_{\theta \rm{t}}$. Here, legends "Original Model" and "Improved Model" refer to the results of low-speed model Langtry-2009-AIAA and our high-speed improved transition-turbulence model WuLei-2026-IJHMT, respectively, while "Samples" stands for results from individual ensemble members.
• Figure 2: Zoom-in view of intermittency factor $\gamma$ contours with (a) $c_{e1}=0.5$, (b) $c_{e1}=1.0$, and (c) $c_{e1}=1.5$ for adiabatic flat plate at $Ma_\infty=2.25, Re_\infty=2.50\times10^7$.
• Figure 3: Distributions of skin-friction coefficient $C_f$ for an adiabatic flat plate at $Ma_{\infty}=2.25, Re_{\infty}=2.50\times10^7$ obtained at different EnKF epochs. Legend "Samples Average" denotes the mean among all ensemble samples. DNS Flat-Plate-DNS2.25 and turbulent skin-friction estimated via the reference temperature method (RTM) WuLei-2026-IJHMT are shown for comparison. All other legends are consistent with those in Fig. \ref{['fig:Dependence-parameter']}.
• Figure 4: Intermittency factor fields of an adiabatic flat plate at $Ma_{\infty}=2.25, Re_{\infty}=2.50\times10^7$ near the transition region from (a) improved model and (b) first ensemble member of EnKF at tenth epoch.
• Figure 5: Distributions of heat-transfer coefficient $h/h_{\rm{ref}}$ for an isothermal sharp cone at $Ma_{\infty}=6.00, Re_{\infty}=2.03\times10^7$ obtained at different EnKF epochs. Legend "EXP" refers to the experimental results of Horvath. All other legends are consistent with those in Fig. \ref{['fig:F1-EnKF-Cftau']}.