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Stability-Based Optimization of Planar Quiet Wind Tunnel Nozzles: Two-Dimensional Design and Three-Dimensional Assessment

This study demonstrates that while two-dimensional stability-based optimization effectively suppresses instabilities on the top wall of planar quiet wind tunnel nozzles, reliable design requires three-dimensional assessment to account for sidewall-induced crossflow disturbances that the planar model fails to capture.

Original authors: Alberto Testa, Pierre Schrooyen, Johan Steelant, Thierry Magin, Guillaume Grossir

Published 2026-07-10

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Alberto Testa, Pierre Schrooyen, Johan Steelant, Thierry Magin, Guillaume Grossir

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to build the ultimate "quiet room" for testing super-fast model airplanes. In this room, the air needs to be perfectly smooth and silent so that the tiny vibrations of the plane itself can be heard, rather than the roar of the wind tunnel's own machinery. This is the dream of a "quiet wind tunnel."

For decades, engineers have tried to design the nozzles (the funnels that speed up the air) by pretending the world is flat. They imagine the air flowing through a long, flat hallway where the only thing that matters is the top and bottom walls. They use powerful computer simulations to tweak the shape of the nozzle and the temperature of the walls, hoping to stop the air from getting "bumpy" (turbulent) before it reaches the test section.

The Big Experiment: Flat vs. Real
In this study, the researchers at the Von Karman Institute and their partners decided to test if this "flat world" idea actually works. They first designed a nozzle using the traditional 2D method, which assumes the air flows perfectly evenly from side to side. They used a clever computer trick called "Bayesian optimization" (think of it as a super-smart robot that tries thousands of shapes and temperatures, learning from each mistake to find the perfect one) to minimize the "bumpiness" of the air.

The Good News: The Top Wall is Happy
When they checked their best design, the results were great for the top wall of the nozzle. The computer simulations showed that by making the nozzle open up very slowly (at a shallow angle of about 2 degrees) and by heating the throat area then cooling it down sharply further down, they could keep the air smooth. In their 2D model, the air stayed calm all the way to the exit, with a "quietness score" (called the N-factor) staying below the danger line of 7.5. The optimized design even managed to keep the air laminar (smooth) for a much longer distance than the old, unoptimized designs.

The Bad News: The Side Walls Have Secrets
But here is the twist. Real nozzles aren't infinite flat hallways; they have side walls, just like a real room has side walls. When the researchers ran a full 3D simulation that included these side walls, the story changed.

Because the nozzle expands vertically (up and down) but not sideways, the air gets squeezed against the side walls. This creates a sneaky, invisible current that swirls along the sides, creating "corner vortices" (tiny tornadoes in the corners). These swirls create a new kind of instability called "crossflow," which the flat 2D model completely missed.

The paper explicitly rules out the idea that the 2D design is perfect for the whole nozzle. While the top wall remained calm, the side walls developed these new, dangerous swirls. In fact, the simulations suggest that on the side walls, the air might start getting bumpy much earlier than expected—around the halfway point of the nozzle (x/L ≈ 0.5)—even with the best design.

The Silver Lining: Heat is a Magic Wand
Here is the most surprising part. Even though the 2D design didn't account for the side walls, the specific temperature pattern they found (hot at the start, then a sharp drop) actually helped the side walls too! It didn't fix the problem completely, but it did reduce the growth of those side-wall swirls. This suggests that controlling the temperature of the walls is a powerful tool that can fight both the "flat" problems and the "3D" problems, even if we didn't know about the 3D problems when we started.

The Verdict
The researchers conclude that while designing based on a flat, 2D view is a great starting point and works well for the top of the nozzle, it is not enough to guarantee a perfectly quiet tunnel. The side walls introduce complex, 3D effects that the simple models can't see. To truly build a reliable quiet wind tunnel, engineers must stop pretending the world is flat and start designing with the side walls in mind, using more advanced tools to catch those hidden corner swirls.

The Numbers
The study focused on a nozzle designed to reach a speed of Mach 3.5 (3.5 times the speed of sound). They aimed for a "quiet Reynolds number" of 20 × 10⁶, a measure of how long the smooth air can last. The old design only managed about 3.3 × 10⁶, but the new optimized design pushed the smooth flow much further, at least on the top wall. The side walls, however, showed that the smooth air might break down around the middle of the nozzle, a limit that the 2D model failed to predict.

In short: The 2D design is a strong foundation, but if you ignore the side walls, you're building a house with a leaky roof you can't see. The solution involves a mix of careful shape-shaping and clever heating, but the full picture requires looking at the whole 3D room.

Technical Summary: Stability-Based Optimization of Planar Quiet Wind Tunnel Nozzles

Problem Statement
The design of quiet supersonic wind tunnels is critical for investigating natural boundary-layer (BL) transition in high-speed flows, as conventional facilities suffer from elevated freestream disturbances that compromise ground-to-flight extrapolation. While axisymmetric nozzles are inherently two-dimensional (2D) in their flow symmetry, planar nozzles possess lateral sidewalls that introduce inherent three-dimensional (3D) effects. Standard design methodologies often rely on 2D assumptions (spanwise-homogeneous flow), neglecting the anisotropic expansion caused by vertical expansion without corresponding spanwise expansion. This simplification generates spanwise pressure gradients and secondary flow structures, potentially inducing instability mechanisms—such as crossflow—that are absent in 2D models. The central problem addressed is whether 2D stability-based optimization strategies remain robust when applied to planar nozzles where 3D sidewall effects are present.

Methodology
The study employs a modular, stability-based optimization framework integrated within a Bayesian Optimization (BO) loop. The process involves three main stages:

Inviscid Contour Generation: The HYPNOZE toolkit uses the Method of Characteristics (MoC) to generate nozzle contours based on geometric parameters (throat curvature, opening angle, straightening length) and operating conditions.
Viscous Boundary-Layer Analysis: The DEKAF solver computes laminar boundary-layer profiles and displacement thickness, providing viscous corrections to the effective geometry.
Stability Assessment: The VESTA toolkit applies Local Linear Stability Theory (LST) to evaluate the amplification of disturbances. The analysis targets first-mode, second-mode, and Görtler instabilities.

The optimization minimizes a cost function defined as the linear sum of N-factors ( $N_{tot} = N_{1st} + N_{2nd} + N_{G\ddot{o}}$ ) to avoid the insensitivity of root-sum-square formulations to weak amplifications. Design variables include the nozzle opening angle ( $\omega$ ), straightening length ( $X_{BC}$ ), and a parameterized wall temperature distribution ( $T_w(x)$ ).

To assess robustness, the optimized 2D designs are re-evaluated using fully 3D Computational Fluid Dynamics (CFD) simulations (using CFD++) that explicitly include lateral sidewalls. Stability analysis on these 3D base flows utilizes both LST and Linear Parabolized Stability Equations (LPSE) to capture non-parallel and non-local effects, particularly along the sidewalls where secondary flows develop.

Key Contributions

2D Optimization Framework: The study demonstrates that a 2D stability-based optimization, combining geometric control and active thermal management, can successfully suppress first-mode and Görtler instabilities along the top wall of a planar nozzle, achieving subcritical amplification levels ( $N_{combined} < 7.5$ ) and extending the quiet flow region.
3D Robustness Assessment: The paper provides a systematic assessment of 2D designs in a 3D environment, revealing that while top-wall stability predictions remain accurate, sidewall regions exhibit distinct instability mechanisms driven by secondary flows.
Thermal Control on 3D Instabilities: A significant finding is that a wall temperature distribution optimized for 2D mechanisms (specifically a sharp temperature drop downstream of the throat) also attenuates crossflow-type amplification along the sidewalls, suggesting a coupling between thermal control and 3D instability mechanisms.
Validity Limits of Stability Theories: The comparison between LST and LPSE delineates the domain of validity for linear stability theories in 3D planar flows. LPSE integration breaks down in regions of strong non-parallelism and rapid streamwise variation (typically downstream of the inflection point), indicating where higher-fidelity approaches are required.

Results

Optimization Outcomes: The optimal configuration (IT048) features a shallow opening angle ( $\approx 2^\circ$ ), a short straightening length, and a specific thermal profile: high temperatures near the throat ($350$ K) to suppress first-mode waves, followed by a sharp drop to $\approx 292$ K to further dampen instabilities, and a slight rise toward the exit. This design maintains laminar flow along the top wall, unlike the reference case which predicts early transition.
Top-Wall Performance: In the 3D CFD simulations, the top-wall stability characteristics (N-factor evolution) of the optimized design remain consistent with 2D predictions, with relative errors below 5%. The 2D framework serves as a reliable first-order approximation for the top wall.
Sidewall Performance: The sidewall boundary layer exhibits strong 3D effects, including corner vortices and transverse velocity components. These induce crossflow-type instabilities that are not captured in the 2D model. While the optimized geometry and thermal control reduce amplification levels compared to the reference case, the N-factor threshold ( $N_{crit} \approx 7.5$ ) is still reached at approximately $x/L \approx 0.5$ along the sidewalls. This suggests that sidewall-induced transition may occur upstream of the intended uniform core region, even for optimized designs.
Mechanism Interaction: The optimized thermal profile effectively decouples the spatial overlap of first-mode and Görtler amplification regions. Furthermore, the thermal control strategy, though derived from 2D physics, provides a secondary benefit by reducing crossflow amplification along the sidewalls.

Significance and Claims
The paper claims that stability-based optimization is an effective and physically interpretable tool for controlling transition on the top wall of planar quiet nozzles. It establishes that while 2D optimization is sufficient for top-wall design, it is insufficient for a complete nozzle assessment because it fails to capture sidewall-induced crossflow instabilities.

The authors conclude that reliable nozzle design for planar configurations requires the explicit consideration of 3D effects. While the current 2D optimization provides a robust baseline, the presence of sidewall transition risks necessitates the integration of higher-fidelity approaches (such as plane-marching PSE or spanwise BiGlobal stability theory) into future design frameworks to fully resolve multidirectional instability dynamics. The study highlights wall temperature control as a versatile design lever that influences both 2D and 3D instability mechanisms, offering a new pathway for mitigating sidewall transition.

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