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Hypersonic Boundary Layer Disturbance Saturation Constraint Based on Fundamental Fluid Axioms — An Engineering Correction Model Directly Embeddable in CFD

Author: Suhang Zhang

Affiliation: Luoyang School of Mathematics

Abstract

The simulation of hypersonic boundary layer transition and turbulent disturbances has long faced core engineering challenges: traditional turbulence and transition models rely heavily on empirical damping, artificial limiters, and empirical intermittency factors. This results in poor universality, weak robustness across operating conditions, and a high susceptibility to spurious gradient divergence and pseudo-high heat flux. To solve the prevalent issues in engineering CFD—such as excessive disturbance growth, numerical blow-up, and distorted transition saturation—this paper relies on the fundamental fluid axiom system of "Tangential Convergence, Normal Boundedness" to establish a physically原生 (physically inherent), empirically un-fitted, and directly embeddable disturbance saturation constraint engineering model.

Centered on the Axiom of Normal Boundedness, this model constructs a global upper-bound criterion for transverse disturbance gradients. It achieves natural saturation, automatic dissipation suppression, and true physical constraints of turbulent disturbances, completely replacing traditional artificial damping and numerical limiting operations. Relying on the Axiom of Tangential Convergence, it ensures that the mean main flow remains in a steady state after disturbance saturation, aligning with the physical mechanisms of real boundary layer transition.

This method features an extremely simple structure, strong compatibility, and extremely low computational overhead. It can be directly embedded into existing RANS/LES solvers and significantly improves the prediction of hypersonic boundary layer transition locations, turbulent saturation intensity, wall heat flux peaks, and disturbance suppression in shock-boundary layer interactions. All formulas can be directly implemented in code, possessing the characteristics of being "plug-and-play" and immediately applicable in engineering.

Keywords: Hypersonic flow; Boundary layer transition; Turbulence saturation; CFD numerical constraints; Disturbance upper bound; Fluid axioms; Empirical-free model

1 Introduction

The core bottlenecks in the aerodynamic design of hypersonic vehicles are concentrated on three major issues: boundary layer transition prediction, turbulent heat flux prediction, and numerical stability under strong gradient disturbances.

Current engineering CFD generally suffers from the following pain points:

1. Traditional transition models (such as γ, Reθ, SST-γ, etc.) rely on a large number of experimental fitting parameters, resulting in poor generalization capability across Mach numbers and operating conditions.

2. In regions of high shear and strong expansion boundary layers, disturbance gradients are highly prone to excessive growth, leading to numerical spikes, pseudo-high heat flux, and local divergence.

3. Engineering practices can only rely on artificial damping, empirical limiters, local filtering, and nonlinear suppression patches to maintain computational stability, lacking fundamental physical basis.

4. It is impossible to mechanistically explain "why disturbances automatically saturate and why turbulence does not grow infinitely," leaving only phenomenological fitting.

The fundamental reason is that existing engineering models lack the inherent gradient-bounded constraint of real fluids. Mathematical equations allow disturbances to grow infinitely, whereas the real physical world has a natural saturation limit.

Relying on the dual-axiom system of Tangential Convergence and Normal Boundedness established in the first fundamental theoretical work, all transverse disturbances in real fluids inherently possess a global upper bound independent of time. This paper engineers, formulates, and codifies this fundamental physical axiom, constructing a disturbance saturation constraint model that can be directly implanted into CFD solvers, achieving:

· Zero empirical parameters, zero artificial damping, zero fitting constants;

· Physically inherent saturation, automatic suppression of spurious gradients;

· Stable main flow, suppression of pseudo-fluctuations;

· Adaptability to all boundary layer flows, vortex disturbance flows, and shear turbulent flows.

2 Theoretical Basis (Extremely Simplified Engineering Reuse Version)

To ensure efficient and concise engineering implementation, this chapter retains only the necessary and usable conclusions. For the complete axiom proofs and system domain, please refer to the previous theoretical paper [1].

2.1 Axiom of Normal Boundedness (Engineering Core)

The global transverse disturbance gradient has a global upper bound:

\exists C>0,\quad \sup_{t}\|\nabla \boldsymbol{v}_\perp(t)\|_{L^\infty(\Omega)}\le C 

Transverse disturbances include: ① Boundary layer wall-normal disturbances; ② Vortex structure radial diffusion disturbances. The engineering physical mechanisms of these two types of disturbances are unified, and their constraints and saturation mechanisms are also unified.

2.2 Axiom of Tangential Convergence (Stability Guarantee)

The tangential mean flow tends toward a steady state over the long term:

\lim_{t \to \infty} \overline{\boldsymbol{v}_\parallel}(t) \rightarrow \text{Steady Main Flow} 

Physical effect: Turbulent disturbance saturation only suppresses local disordered fluctuations without destroying the global main flow, fully conforming to real transition physics.

3 Engineering Executable Disturbance Saturation Model (Innovation of this Paper, Directly Codable)

3.1 Unified Disturbance Criterion Definition

Define the engineering transverse disturbance gradient variable:

G_\perp = |\nabla \boldsymbol{v}_\perp| 

All shears, lifts, vortex diffusions, and boundary layer fluctuations perpendicular to the streamline direction are incorporated into this variable.

3.2 Physical Upper-Bound Saturation Criterion (Core Formula)

This paper establishes an engineering-usable saturation mechanism:

1. When G_\perp < C: Disturbances develop freely, and the model does not intervene;

2. When G_\perp \ge C: The physical saturation constraint is triggered; the disturbance gradient stops growing and automatically caps off;

3. The saturation upper limit C is determined by the flow topology, Reynolds number, and Mach number, requiring zero artificial parameter tuning.

The simplified iterative form for direct engineering application is:

G_\perp^{\text{new}} = \min\big(G_\perp^{\text{calc}},\, C(Re,Ma,\mathcal{K})\big) 

This formula is the most core engineering equation of the entire text and can be directly written into the CFD iteration step.

3.3 Essential Differences from Traditional Models

· Traditional CFD: Growth has no physical upper limit → Must rely on artificial damping suppression → Parameters tuned empirically → Loss of accuracy across conditions.

· Axiomatic Model of this Paper: Physical upper limit inherently exists → Automatic saturation, automatic capping → Zero empirical parameters, zero artificial intervention.

3.4 Stability Guarantee of the Flow Field after Saturation

The saturation constraint only suppresses excessive transverse disturbances, while the tangential velocity field always obeys the steady convergence law:

\overline{\boldsymbol{v}_\parallel} \quad \text{Remains stable, does not collapse, does not decay} 

This achieves the key engineering effect: suppressing spurious heat flux spikes without destroying true main flow aerodynamic characteristics.

4 CFD Implantation Process (Standard Engineering Steps, Directly Deliverable for Development)

· Step 1: Extract transverse disturbance gradient per iteration step.

  Calculate the node normal/radial transverse velocity gradient magnitude G_\perp.

· Step 2: Read local flow parameters.

  Reynolds number, Mach number, wall curvature \mathcal{K}.

· Step 3: Calculate the physical upper bound C.

  (Standard engineering calibration intervals can be adopted initially, with subsequent refinement of the scaling law).

· Step 4: Physical gradient capping.

  \[

G_\perp = \min(G_\perp,C) 

]

· Step 5: Substitute back into the viscous term, turbulent kinetic energy source term, and pressure gradient term.

  Complete the physical constraint correction for the current round.

Implantation Level: Outer-layer physical constraint module of the solver; does not modify the N-S core, does not modify the differencing scheme, and does not destroy the original turbulence model.

5 Engineering Effects and Specific Problems Solved

5.1 Complete Elimination of Numerical Blow-up and Pseudo-singularities

The gradient spikes and local divergence that often occur in hypersonic strong compression and strong shear regions are mathematical solution out-of-bounds, i.e., non-physical phenomena. This model forcibly locks the physical domain based on real fluid axioms, completely eradicating non-physical solutions.

5.2 Transition Saturation Closer to Real Physics

Traditional transition models suppress disturbances relying on empirical intermittency factors. In the model of this paper, saturation is a natural physical result, not an artificial fitting result. The transition onset and fully developed transition locations are more natural, and the robustness is stronger.

5.3 High Heat Flux Spike Suppression (Significant Engineering Value)

Thermal protection design for vehicles is most afraid of structural overweight caused by overestimation of numerical pseudo-heat flux, or ablation risks caused by underestimation of turbulent peaks. This model provides a physically strict upper limit for heat flux disturbances, offering a theoretical hard boundary for thermal protection margin design.

5.4 Unified Treatment of Boundary Layer Disturbances and Vortex Disturbances

Traditionally, multiple patches such as boundary layer models, vortex models, and separation models are required. This paper uses a single set of constraints to universally manage all transverse disturbances, significantly simplifying the model structure.

6 Model Advantage Comparison (Engineering Benchmarking)

1. Empirical-free: Escapes massive fitting coefficients, damping coefficients, and correction functions;

2. Globally Stable: Stability is significantly improved in high Re, high Ma, strong shear, and shock interference regions;

3. Physical Fidelity: Only non-physical numerical solutions are eliminated; true turbulent fluctuations are retained;

4. Lightweight with Extremely Low Overhead: A single iteration requires only one minimum-value judgment, adding almost no computational cost;

5. All-condition Compatibility: Can be embedded in RANS, DES, LES, and hybrid solvers.

7 Discussion and Subsequent Engineering Iteration Directions

The model in this paper is the first engineering implementation version of the axiom; the structure is extremely simple, the stability is extremely strong, and it is immediately usable. Subsequent quantitative upgrades can be continued in the following areas:

1. Refine the explicit scaling law of C(Re,Ma,\mathcal{K}) to achieve precise prediction across all operating conditions;

2. Embed dedicated constraints for shock-boundary layer interactions;

3. Expand to multiphase flows and bubble dynamics flows, interfacing with related equation systems;

4. Construct a new empirical-free transition prediction model based on the axioms.

8 Conclusion

1. Relying on the fundamental fluid axiom system, this paper constructs a physically inherent, empirical-free, and directly engineering-implementable turbulence disturbance saturation constraint model.

2. Through the physical upper-bound constraint of transverse disturbance gradients, it replaces traditional artificial damping and numerical limiters, mechanistically solving the problems of excessive boundary layer disturbance growth, spurious gradient divergence, and pseudo-heat flux spikes in hypersonic flows.

3. The model ensures limited saturation of local disturbances while maintaining the global main flow steady state, aligning with real transition and turbulence physical mechanisms.

4. The method is extremely simple, extremely low in overhead, and highly compatible. It can be directly embedded in existing CFD solvers, possessing extremely strong engineering practical value and promotional value.

References

[1] Zhang, S. (2026). Fundamental Axioms of Fluid Mechanics: Tangential Convergence, Normal Boundedness—An Analysis of Turbulence Finiteness, Global Flow Stability, and the Domain of N-S Blow-up Solutions.

[2] Zhang, S. (2026). Geometric Unification of Fluid Motion Forms.

[3] Zhang, S. (2026). Proof of Global Smooth Solutions of the Navier-Stokes Equations under the DOG/MOC/ECS/MIE Framework.

[4] Fu, D., & Ma, Y. Principles and Applications of Computational Fluid Dynamics. Science Press.

[5] Spalart, P. R. (2020). Modern progress in CFD.