Compressible turbulent boundary layers over two-dimensional square-rib roughness

This study utilizes direct numerical simulations at Mach 2.5 to investigate compressible turbulent boundary layers over square-rib roughness under adiabatic and cooled conditions, proposing a fitting-based optimization for virtual origin determination, demonstrating the superiority of the GFM transformation over van Driest, and formulating a modified rough-wall Generalized Reynolds Analogy to address the breakdown of classical analogies caused by the disparity between momentum drag and heat transfer mechanisms.

The Gist

Imagine you are flying a supersonic jet. The air rushing over the wings is moving at 2.5 times the speed of sound. In this high-speed world, the air isn't just a fluid; it's a hot, chaotic mess of friction and heat. Engineers need to predict exactly how this air behaves to design planes that don't melt or lose control.

Usually, they rely on "smooth wall" rules—mathematical shortcuts that work perfectly if the plane's skin is polished glass. But real planes aren't perfect. They have rivets, seams, and thermal protection tiles that act like tiny bumps. This paper investigates what happens when you combine high-speed air, rough surfaces, and extreme heat transfer (like a wall that is either super hot or super cold).

Here is the story of their findings, explained through everyday analogies.

1. The Setup: The "Turbulent Highway"

The researchers built a super-accurate computer simulation (like a digital wind tunnel) to watch air flow over a surface covered in square bars (like a row of tiny, evenly spaced speed bumps). They tested two scenarios:

• The Hot Wall: The surface is insulated (adiabatic), letting the air get hot from friction.
• The Cold Wall: The surface is actively cooled, sucking heat out of the air.

2. The Dynamic Problem: Finding the "Ground"

When air flows over a rough surface, it doesn't start sliding from the actual physical surface. It starts sliding from a "virtual" height above the bumps because the air gets trapped in the gaps between them.

• The Old Rule (The Zero-Moment Method): Traditionally, engineers tried to find this virtual height by calculating the "center of gravity" of the drag forces. It's like trying to find the balance point of a seesaw.
• The Discovery: For these specific square bumps, the old rule failed. It pointed to a height that made the math look weird and broken (like an "S" shape instead of a straight line).
• The New Solution: The team invented a "fitting" method. Instead of guessing the physics, they adjusted the virtual height until the data looked like a perfect, smooth highway. They found that the air actually behaves as if it's sliding on a floor much lower than the old rule suggested.

3. The Velocity Transformation: The "Magic Glasses"

In high-speed aerodynamics, the air gets hot and expands, which messes up the standard speed measurements. To fix this, scientists use "transformations" (like special glasses) to translate the chaotic, hot, fast air into a calm, slow, incompressible language that we understand.

• The Old Glasses (Van Driest): These worked great for smooth walls but got foggy when the wall was rough and cold.
• The New Glasses (GFM Transformation): The team found that a newer set of "glasses" (the Griffin–Fu–Moin transformation) worked perfectly. When they put these on, the messy data from the rough, cold wall suddenly looked exactly like the data from a smooth wall. It's as if the roughness and the coldness were just optical illusions that the new glasses could see through.

4. The Thermodynamic Problem: The "One-Way Street"

Here is the most surprising part. In physics, there is a classic rule called the Reynolds Analogy. It says: "If you know how the air drags (friction), you automatically know how it transfers heat." It assumes momentum (pushing) and heat (warming) are best friends who always stick together.

• The Breakup: The researchers found that on a rough wall, this friendship breaks up.
  • Why? The square bars create a massive "form drag" (like a parachute catching wind). This is a mechanical push. But heat doesn't work that way. Heat just conducts through the air; it doesn't get "pushed" by the shape of the bars in the same way.
  • The Result: The old rule (Generalized Reynolds Analogy) failed miserably, especially on the cold wall. It was like trying to predict the temperature of a car engine just by looking at how hard the tires are gripping the road—the two things stopped talking to each other.

5. The Fix: The "Slip Plane"

To fix the broken heat rule, the team proposed a clever workaround. They imagined a "Slip Plane"—a magical, invisible floor located slightly above the rough bumps.

• The Analogy: Imagine trying to measure the speed of a river flowing over rocks. The water swirls wildly near the rocks. Instead of trying to measure the chaos right at the rocks, you pretend there is a smooth, invisible bridge a few feet above the rocks. You measure the flow there, where the water is calm and straight.
• The Result: By using this "Slip Plane" concept, they could rebuild the math. Suddenly, the relationship between speed and temperature made sense again, even on the rough, cold wall.

6. The Fluctuations: The "Storm"

Finally, they looked at the tiny, chaotic swirls (turbulence) in the air.

• Near the Wall: The roughness and cooling created a chaotic storm zone where the old rules didn't work.
• Farther Out: Once you move away from the wall (into the "outer layer"), the air calms down. The researchers found that a refined version of the old rules (Refined Strong Reynolds Analogy) works perfectly here. It's like the storm only exists in the basement; once you go to the top floor, the house is quiet and predictable.

Summary

This paper is a story of fixing broken rules for a messy world.

1. Old rules for finding where the air starts moving failed on rough surfaces.
2. New math (GFM transformation) fixed the speed measurements.
3. Old heat rules failed because roughness breaks the link between friction and heat.
4. A new "Slip Plane" trick fixed the heat rules.

The takeaway? When designing high-speed vehicles with rough surfaces and extreme temperatures, you can't just use the old textbooks. You need these new, smarter tools to understand how the air really behaves.

Technical Summary

1. Problem Statement

The study addresses the complex interaction between surface roughness and wall heat transfer in compressible turbulent boundary layers (CTBL) at high Mach numbers ($Ma = 2.5$). While theoretical frameworks exist for smooth walls (e.g., van Driest transformation, Generalized Reynolds Analogy) and for rough walls under incompressible conditions, significant gaps remain when these factors are combined. Specifically:

• Dynamic Challenge: Existing methods for determining the "zero-plane displacement" (virtual origin) for 2D cavity-type roughness often fail to restore the logarithmic law of the wall. Furthermore, it is unclear if outer-layer similarity holds for 2D roughness under diabatic (heat-transferring) conditions.
• Thermodynamic Challenge: The classical Reynolds Analogy assumes a coupling between momentum and heat transfer that breaks down over rough walls due to the physical asymmetry between form drag (pressure drag) and heat conduction. The effective turbulent Prandtl number ($Pr_e$) is known to deviate from unity, invalidating standard models like the Generalized Reynolds Analogy (GRA).
• Fluctuation Challenge: The applicability of Strong Reynolds Analogy (SRA) variants to predict temperature and velocity fluctuation intensities in the presence of strong roughness and wall cooling is uncertain.

2. Methodology

The authors performed Direct Numerical Simulations (DNS) using a high-fidelity finite-difference solver (OpenCFD).

• Flow Configuration:
  • Mach Number: $M_\infty = 2.5$.
  • Roughness: Two-dimensional transverse square bars with a pitch-to-height ratio $\lambda_x/k = 8$ and a roughness height $k^+ \approx 35$.
  • Thermal Conditions: Four cases were simulated: Smooth/Rough walls under Adiabatic ($T_w/T_r = 1.0$) and Cold-wall ($T_w/T_r = 0.5$) conditions.
  • Inflow Generation: A dual-domain strategy was used, coupling an auxiliary Compressible Equivalent Boundary Layer (CEBL) to generate realistic turbulent inflow for the main spatially developing CTBL domain.
• Numerical Techniques:
  • Immersed Boundary Method (IBM): A sharp-interface ghost-cell IBM was employed to resolve the complex geometry of the square ribs.
  • Statistical Framework: A double-averaging method (triple decomposition) was used to separate time-averaged means, spatial fluctuations (due to roughness), and turbulent fluctuations.
  • Grid Resolution: High-resolution grids were used, with local refinement near the roughness elements, ensuring $y^+$ resolution suitable for DNS.

3. Key Contributions

The paper introduces several novel methodological and theoretical advancements:

1. Optimization-Based Zero-Plane Displacement: The authors demonstrate that the conventional zero-moment method (Jackson, 1981) fails to yield a consistent virtual origin for 2D cavity roughness, resulting in non-logarithmic velocity profiles. They propose a fitting-based optimization procedure to determine the kinematic virtual origin that maximizes the extent of the logarithmic region.
2. Validation of GFM Transformation: The study rigorously evaluates the Griffin–Fu–Moin (GFM) transformation against the classical van Driest (VD) transformation. It confirms that GFM successfully restores outer-layer similarity for velocity defects in rough-wall compressible flows with wall heat transfer, whereas VD fails under cold-wall conditions.
3. Modified Rough-Wall Generalized Reynolds Analogy (rGRA): The authors identify that the classical GRA fails because the effective turbulent Prandtl number ($Pr_e$) deviates severely from unity within the roughness sublayer. They formulate a rGRA by introducing an equivalent slip-plane or reference-point boundary condition. This bypasses the near-wall thermal heterogeneity, allowing for accurate reconstruction of the temperature-velocity relationship.
4. Assessment of SRA Variants: A critical evaluation of Strong Reynolds Analogy models (GSRA, HSRA, and RSRA) is provided, identifying the specific regions where near-wall modulation by roughness and cooling disrupts fluctuation coupling.

4. Key Results

• Flow Structure & Drag:

  • Roughness induces strong leading-edge oblique shocks and recirculation zones.
  • Form drag dominates, constituting $\approx 109\%$ of the total drag (with negative skin friction in the cavity).
  • Cold-wall conditions lead to a more rapid boundary layer expansion and higher frictional drag compared to adiabatic cases.

• Mean Velocity Profiles:

  • Virtual Origin: The kinematically optimized zero-plane displacement ($d^+$) is significantly lower ($d^+/k^+ \approx 0.15\text{--}0.30$) than the zero-moment prediction ($d_m/k \approx 0.60\text{--}0.64$).
  • Outer-Layer Similarity: Under the GFM transformation, the velocity defect profiles for both adiabatic and cold rough walls collapse onto the smooth-wall baseline in the outer layer ($y/\delta > 0.4$). The VD transformation fails to achieve this collapse for cold walls.
  • Roughness Function ($\Delta U^+$): The cold-wall roughness function remains close to incompressible references, while the adiabatic case shows a significantly higher $\Delta U^+$.

• Thermodynamics (Temperature-Velocity Coupling):

  • Breakdown of Classical GRA: The classical GRA fails to predict temperature profiles in the roughness sublayer, particularly for cold walls, due to the severe deviation of $Pr_e$ from unity.
  • Success of rGRA: By anchoring the rGRA at reference heights above the roughness crest ($y \ge 2k$), the model accurately reconstructs the temperature-velocity relationship. This confirms that the thermal field requires a larger wall-normal distance to achieve spatial homogeneity compared to the velocity field.
  • Effective Prandtl Number: $Pr_e$ drops to $\approx 0.8$ in the roughness sublayer for adiabatic cases and deviates even more for cold walls, invalidating the $Pr_e \approx 1$ assumption of classical theories.

• Fluctuation Intensities:

  • The Refined SRA (RSRA) maintains predictive accuracy for fluctuation intensities in the outer layer ($y > 0.5\delta$) despite near-wall disturbances.
  • Near-wall deviations are significant for all SRA variants due to the combined effects of roughness and wall cooling, requiring larger wall-normal distances for the analogies to recover.

5. Significance

This study provides a comprehensive physical understanding of how 2D roughness and severe wall heat transfer jointly alter compressible turbulent boundary layers. Its significance lies in:

• Correcting Theoretical Assumptions: It challenges the universal applicability of the zero-moment method for 2D roughness and the $Pr_e \approx 1$ assumption in rough-wall sublayers.
• Practical Modeling: The proposed rGRA and the validation of the GFM transformation offer robust tools for engineers and researchers to model high-speed aerodynamic heating and skin friction on rough surfaces (e.g., hypersonic vehicles with thermal protection systems).
• Fundamental Insight: The work highlights the distinct spatial recovery rates of hydrodynamic and thermal fields over rough surfaces, a critical factor often overlooked in simplified aerothermodynamic models.