Turbulent Topology Optimization

• Turbulent topology optimization is a design framework that employs continuum design variables and advanced turbulence models to optimize fluid and thermal structures under high Reynolds conditions.
• It integrates rigorous RANS equations with modified k–ε/k–ω models and Brinkman penalization to accurately capture wall-bounded turbulence and porous media effects.
• Algorithmic strategies like adjoint-based gradients, density filtering, and neural surrogates enable efficient exploration of high-dimensional design spaces, leading to significant reductions in drag, energy loss, and heat transfer nonuniformity.

Turbulent topology optimization is the systematic design of fluidic, thermal, or fluid-structure layouts under high Reynolds number (Re ≫ 1) flow conditions using continuum-based design variables and rigorous turbulence modeling. This field seeks to identify optimal spatial distributions of fluid, solid, and/or porous material to minimize objective functionals (e.g., drag, pressure loss, energy dissipation, or heat/mass transfer nonuniformity) subject to the governing partial differential equations (PDEs) of incompressible or compressible turbulent flows. Optimization is performed over high-dimensional design spaces and must account for wall-bounded turbulence, anisotropic transport, and geometric manufacturability. Unlike laminar-flow topology optimization, the turbulent regime presents unique challenges in physics modeling, numerical stability, and gradient accuracy.

1. Governing Physics and Mathematical Formulation

Topology optimization in turbulent flows is fundamentally governed by the Reynolds-Averaged Navier–Stokes (RANS) equations coupled to closure relations for turbulent viscosity, typically k–ε or k–ω models, and extended as needed by heat or mass transport equations for conjugate problems (Philippi et al., 2015, Wu et al., 2022, Bayat et al., 5 Jan 2026). The design variable in density-based or porosity-based methods is a spatially varying field (e.g., $\gamma(\mathbf{x})$ or $\varepsilon(\mathbf{x})$) interpolating between fluid ($\gamma=1$) and solid ($\gamma=0$) states. The solid phase introduces a Darcy–Brinkman penalization in the momentum equations by means of a spatially dependent inverse permeability $\alpha(\gamma)$, which must be appropriately scaled for high Re:

$$\alpha(\gamma) = \alpha_{\max}\,\frac{1-\gamma}{1+q_\alpha\,\gamma}$$

with $\alpha_{\max} \gg 1$ s$^{-1}$ set to ensure near-zero velocity in putative solid (Wu et al., 2022, Bayat et al., 5 Jan 2026). At high Reynolds number, precise scaling of $\alpha_{\max}$ is critical to avoid spurious leakage and to mimic impermeable walls (Wu et al., 2022). In thermofluidic applications, turbulent heat transfer is addressed by the energy equation with turbulent (eddy) diffusivity, $D_t = \nu_t / Pr_t$ (Kii et al., 3 Oct 2025).

A generic optimization problem is then: $$\begin{array}{ll} \min\limits_{\gamma(\mathbf{x})} & J[\gamma,\,u,\,p,\,k,\varepsilon,\ldots] \ \text{s.t.} & \text{RANS} + \text{turbulence closure} + \text{Brinkman penalization}\ & \text{volume fraction constraint}\ & 0 \le \gamma(\mathbf{x}) \le 1 \end{array}$$ where $J$ is the objective functional determined by the application (e.g., total pressure loss, mixing variance, mechanical dissipation) (Philippi et al., 2015, Wu et al., 2022, Kou et al., 6 Mar 2025, Kii et al., 3 Oct 2025). Sensitivities of $J$ with respect to $\gamma$ are computed either via continuous/adjoint or discrete/automatic differentiation, with special care to include or approximate the effect of turbulent viscosity in the sensitivity chain (Philippi et al., 2015, Bayat et al., 5 Jan 2026).

2. Turbulence Modeling within Topology Optimization

Turbulence modeling in topology optimization requires augmenting standard closure models to account for the presence of Brinkman-penalized (porous or solid) regions and their effects on local eddy viscosity. Approaches include:

• Modified k–ε/k–ω models with Darcy sink terms: The turbulent kinetic energy and dissipation equations acquire additional penalization proportional to the local Brinkman term, e.g., $-c_k \alpha(\gamma) k$ and $-c_\varepsilon \alpha(\gamma) \varepsilon$, to enforce rapid decay of turbulence quantities in solid regions (Wu et al., 2022).
• Porosity-dependent eddy viscosity: In methods such as SPAM (Philippi et al., 2015), the turbulent viscosity in porous regions is scaled by the local porosity, $\nu_t^p = \varepsilon \nu_t$, under the “frozen turbulence” assumption (no derivative of $\nu_t$ with respect to the flow field in the adjoint equations).
• Implicit wall-function approaches: Recently, wall effects in density-based topology optimization are robustly enforced using implicit, volumetric wall-penalty formulations derived from the gradient of the design variable, enabling accurate boundary layer resolution on diffuse, non-body-fitted meshes (Bayat et al., 5 Jan 2026).
• Wall-distance computations for SST models: For advanced models like SST k–ω in diffuse interfaces, wall-distance is computed via a penalized Poisson or p-Poisson equation recognizing Brinkman solids (Wu et al., 2022).

These extensions are essential for producing valid near-wall physics, boundary layers, and drag predictions at Re ≫ 10³. Without them, density-based approaches tend to overly diffuse boundary layers and mispredict optimal topologies (Bayat et al., 5 Jan 2026).

3. Algorithmic Approaches and Computational Strategies

Several algorithmic frameworks for turbulent topology optimization have been developed:

• Adjoint-based gradient methods: For PDE-constrained optimization in high dimension, the adjoint method provides scalable and accurate sensitivity information (Philippi et al., 2015, Wu et al., 2022, Bayat et al., 5 Jan 2026). The Lagrangian is constructed from the state equations (RANS+turbulence) and the objective, and its derivative with respect to $\gamma$ yields the update direction.
• Density filtering and projection: To ensure mesh-independence and manufacturability, the design variable is commonly filtered (e.g., via Helmholtz PDE filter) and projected (e.g., Heaviside, sigmoid) to sharpen 0–1 boundaries and enforce minimum feature sizes (Philippi et al., 2015, Bayat et al., 5 Jan 2026).
• Explicit-boundary, binary methods: For multiphysics FSI with turbulence, optimization grids of binary variables are mapped via smoothing and trimming to body-fitted CAD/FEA geometries, allowing wall-function boundary layers and high-order turbulence models to be accurately imposed in COMSOL or similar solvers (Picelli et al., 2023).
• Neural and evolutionary frameworks: For computationally intensive or multiobjective scenarios, deep neural representations (“neural topology”) (Kou et al., 6 Mar 2025), neural operator surrogates, and data-driven evolutionary algorithms employing Wasserstein-distance barycenter crossover (Kii et al., 3 Oct 2025) accelerate exploration of topological design spaces. These approaches leverage pretraining, active learning, and multifidelity models for data efficiency and scalability.

Typical solution workflows involve outer optimization (gradient-based or EA), repeated forward/adjoint PDE solves (or neural operator surrogates), periodic thresholding to enforce discreteness, and possibly post-processing for CAD compatibility.

4. Benchmark Applications and Quantitative Results

Turbulent topology optimization has been validated and applied across a variety of canonical benchmark problems:

• Channel flow with obstacles and dissipative elements: In (Philippi et al., 2015), the SPAM method demonstrates 10–20% reduction in mechanical energy loss in a turbulent channel (Re = $10^5$) containing roughness elements, with further improvement upon careful porosity-to-solid projection and filtering.
• Pipe bends, U-bends, Tesla valves: The implicit-wall-function RANS k–ε framework (Bayat et al., 5 Jan 2026) yields pressure-drop reductions of 50% or more relative to conventional diffusion-based methods, recovers velocity and turbulence profiles matching explicit body-fitted mesh simulations, and obviates the need for fine near-wall meshes at $Re\sim10^5-2\times10^5$.
• Aerodynamic shape optimization: Modified LSKE/SST models with proper Darcy penalization (including wall-distance computation) in (Wu et al., 2022) achieve drag reductions up to 56% (moderate Re) for airfoil-like profiles, and substantial dissipation reduction for turbomachinery-relevant rotating domains.
• Multiobjective turbulent mass/heat transfer: In (Kou et al., 6 Mar 2025, Kii et al., 3 Oct 2025), neural and EA frameworks optimize flow channels to improve mass transfer uniformity by 37% in experiment (versus smooth channels) and simultaneously advance the Pareto frontier for heat transfer and pressure loss, discovering nontrivial morphologies that blend features of different parents via Wasserstein crossover.

Empirical benchmarks highlight the necessity of precise turbulence modeling, wall treatment, and robust regularization for physically valid and manufacturable outcomes.

5. Innovations in Sensitivity Analysis and Wall Modeling

A central technical challenge is the accurate and efficient evaluation of sensitivities of flow objectives with respect to diffuse design variables, particularly in the presence of expensive turbulence closures and non-body-fitted walls.

• Frozen turbulence adjoint: The SPAM methodology (Philippi et al., 2015) simplifies sensitivity computation by freezing turbulent quantities in the adjoint (i.e., decoupling eddy viscosity from variations in the velocity during sensitivity evaluation), enabling closed-form expressions for $\partial \mathcal{L}/\partial \varepsilon$.
• Automatic differentiation: Modern frameworks often employ automatic differentiation (AD) for discrete adjoint sensitivity of the overall objective, as implemented in OpenFOAM/CoDiPack (Wu et al., 2022) and COMSOL (Bayat et al., 5 Jan 2026, Picelli et al., 2023).
• Volume-penalty wall-functions: The implicit wall-function method (Bayat et al., 5 Jan 2026) introduces a wall intensity field derived from design-variable gradients, enforcing the correct logarithmic-law behavior in diffuse penalty zones while maintaining automatic-differentiation compatibility and adjoint accuracy.
• Treatment of wall distance: Modified SST k–ω models employ Poisson or p-Poisson equations for the wall distance field within Brinkman-penalized solids, permitting variable and complex topologies without manual grid annotation (Wu et al., 2022).

These innovations balance computational tractability with the need for adjoint or gradient fidelity in the fully turbulent regime.

6. Challenges, Limitations, and Future Directions

Despite substantial progress, turbulent topology optimization remains challenged by several factors:

• Turbulence model accuracy: Most frameworks are built on steady RANS (k–ε, SST k–ω), which cannot resolve unsteady separation or complex transitional physics. Recent data-driven (neural operator) surrogates remain limited in extrapolation to novel geometries or boundary conditions (Kou et al., 6 Mar 2025, Kii et al., 3 Oct 2025).
• Wall effects and manufacturability: Conventional density-based TO without wall-function corrections fails at high Re, necessitating implicit or explicit wall treatments, or neural surrogates trained on wall-resolving solutions (Bayat et al., 5 Jan 2026).
• Computational cost: High-fidelity adjoint solvers and black-box HF evaluations under multifidelity frameworks are computationally expensive; surrogate-based and EA-based approaches partially address this at the cost of interpretability.
• Extendibility and multiphysics: Full-3D, time-dependent (URANS/LES) TO, incorporation of multiphysics (e.g., FSI with turbulence (Picelli et al., 2023), heat and mass transfer), uncertainty quantification, and geometric regularization remain open avenues for future research.

Anticipated advancements include hybrid surrogate–physics-informed neural operators, enhanced feature-size control during projection/filtering, and scalable parallel solvers for transient turbulent TO.

7. Summary Table of Key Approaches

Approach / Paper	Turbulence Model	Wall Treatment	Sensitivity Technique	Application Domains
SPAM (Philippi et al., 2015)	RANS k–ε (frozen)	Porosity-driven walls	Analytical adjoint, frozen ν_t	Channel drag/dissipation
Neural TO (Kou et al., 6 Mar 2025)	RANS k–ε (2D)	Pretrained Op surrogate	Backpropagation/autodiff	Mass transfer/mixing
FSI-TO (Picelli et al., 2023)	RANS k–ε	Explicit boundary + wall-fn	Semi-automatic diff/ILP	Fluid-structure stress
EA/Wasserstein (Kii et al., 3 Oct 2025)	RANS k–ε	Body-fitted via threshold	No gradients (EA, NSGA-II)	Turbulent heat transfer
Modified RANS (Wu et al., 2022)	LSKE, SST w/ α-sink	Modified wall-dist (SST)	Discrete adjoint/AD	Aerodynamics, turbomachinery
Implicit Wall (Bayat et al., 5 Jan 2026)	RANS k–ε	Volumetric penalty wall-f	Full adjoint/automatic diff	Bends, valves (Re up to 2e5)

In sum, turbulent topology optimization is a rapidly maturing field leveraging advanced PDE-constrained optimization, turbulence closure modeling, and modern computational techniques to deliver high-performance, physically-consistent, and manufacturable fluidic designs in high-Reynolds-number environments. Continued improvements in turbulence modeling, wall representation, and optimization algorithms are expected to further expand its industrial and scientific impact.