Characterisation of rough-wall drag in compressible turbulent boundary layers

This study investigates the applicability of incompressible roughness parameters in compressible turbulent boundary layers across a wide range of Mach and Reynolds numbers, revealing that while velocity transformations have limited impact, the fully rough regime exhibits a Mach-number-dependent shift that is best addressed by a temperature-ratio-based correction factor, thereby highlighting the need for custom rough-wall transformations.

The Gist

The Big Picture: Why Does a Rough Plane Drag?

Imagine you are driving a car on a highway. If the road is perfectly smooth asphalt, the car glides easily. But if the road is covered in gravel, the car has to push through all those rocks, slowing it down. This "slowing down" is called drag.

Now, imagine that car is a supersonic jet flying at the speed of sound (or faster). The air around it isn't just a gentle breeze; it's a chaotic, compressible soup that behaves very differently than the air on a calm day.

This paper asks a simple but tricky question: If we know how much drag a rough surface creates at slow speeds, can we use that same number to predict drag at supersonic speeds?

The Problem: The "Magic" Roughness Number

In the world of slow-moving (incompressible) air, engineers have a "magic number" called $k_s$ (equivalent sand-grain roughness). Think of this like a universal translator for roughness.

• If you have a bumpy surface, you can measure it and say, "This is as rough as 5 grains of sand."
• Once you know it's "5 grains," you can plug that number into a famous formula (from a guy named Nikuradse in 1933) to predict exactly how much the car will slow down.

The Catch: This formula works great for slow cars. But for supersonic jets, the air gets hot, squishes together, and creates shockwaves (like sonic booms) when it hits the bumps. The old formula doesn't account for these "shockwaves" or the heat.

The researchers wanted to know: Can we still use the "5 grains of sand" number for a supersonic jet, or do we need a new translator?

The Experiment: The Wind Tunnel Rollercoaster

To find out, the team built a giant wind tunnel at the University of the Bundeswehr Munich. They didn't just test one speed; they tested a whole range:

1. Slow speed (subsonic, like a commercial airliner).
2. Fast speed (transonic, near the speed of sound).
3. Super fast (supersonic, up to Mach 2.9, which is nearly three times the speed of sound).

They tested two types of "rough" surfaces:

• P60 Sandpaper: Like fine sandpaper (small bumps).
• P24 Sandpaper: Like coarse sandpaper (big, jagged bumps).

They measured how the air flowed over these surfaces using high-speed cameras (PIV) that act like a super-fast movie camera, taking thousands of pictures of tiny particles in the air to see how they move.

The Discovery: The "Shockwave Tax"

Here is what they found, using a simple analogy:

Imagine the rough surface is a toll booth on a highway.

• At slow speeds: The toll booth just slows you down a little bit because you have to drive over the bumps. The "cost" (drag) depends only on how big the bumps are.
• At supersonic speeds: The air hits the bumps so hard it creates a shockwave (a mini sonic boom). It's like the toll booth suddenly has a giant, invisible wall in front of it that you have to crash through. This adds a "shockwave tax" to the drag.

The researchers found that if you just use the old "sand-grain" number, your predictions are wrong at high speeds. The drag is higher than expected because of this extra "shockwave tax."

The Solution: Three New "Translators"

The team tried three different ways to fix the old formula so it works for supersonic speeds. Think of these as trying to find the right lens to look through so the picture makes sense.

1. The "Same Sand" Approach: They tried to find a matching "slow-speed" experiment for every "fast-speed" test.

   • Result: It was hit-or-miss. Sometimes it worked, sometimes it didn't. It was like trying to match a puzzle piece from a different box.

2. The "Viscosity" Approach: They tried to adjust the "sand-grain" number based on how thick the air gets when it heats up (viscosity).

   • Result: This worked well for their specific sandpaper tests, but it failed when they tried to apply it to data from other scientists' experiments. It was too specific.

3. The "Temperature" Approach (The Winner): They realized the key was the temperature difference between the air far away from the wall and the wall itself.

   • When air moves fast, it heats up. The wall stays cooler (or gets hot, depending on the setup). This temperature gap changes how the air behaves.
   • They created a new correction factor based on this temperature gap.
   • Result: This worked best! When they applied this "temperature lens," all their data (and even data from other scientists' papers) collapsed onto the same smooth curve. It meant they could finally use the old "sand-grain" numbers to predict drag at supersonic speeds, as long as they applied this temperature correction.

Why This Matters

This is a huge step forward for aerospace engineering.

• For Engineers: It means they can design faster, more efficient planes and missiles. They can now predict how much fuel a rough surface (like ice, dust, or damage from debris) will cost them at high speeds.
• For the Future: The authors admit their method is still a bit of a "hack" (an empirical fix). They used a formula originally designed for smooth walls and tweaked it for rough walls.
• The Next Step: They hope future research will create a brand-new, custom formula specifically for rough walls that accounts for the shockwaves hitting the bumps directly, rather than just guessing with a correction factor.

The Bottom Line

Rough surfaces on supersonic planes create extra drag because of shockwaves, not just because they are bumpy. You can't just use the old "slow-speed" math. However, by looking at how hot the air gets compared to the wall, we can now fix the math and predict the drag much more accurately. It's like realizing that to drive fast on a bumpy road, you don't just need a better suspension; you need to account for the heat the engine generates.

Technical Summary

1. Problem Statement

In compressible turbulent boundary layers (TBLs) over rough surfaces, drag is typically characterized by first applying a velocity transformation to account for compressibility, then inferring the momentum deficit ($\Delta U^+$) and the equivalent sand-grain roughness ($k_s$).

• The Core Issue: Current practice often derives $k_s$ from measurements at a single Mach number ($M$) and Reynolds number ($Re$), forcing the data to collapse onto the incompressible Nikuradse (1933) relation.
• The Question: If a rough surface has a known $k_s$ in incompressible flow, under what conditions can this value be used to predict drag in compressible flows?
• Gap in Knowledge: Existing frameworks do not account for wave drag generated by shock waves forming over roughness elements in supersonic flow. Furthermore, there is a lack of experimental data covering independent variations of $M$ and $Re$ for realistic roughness types in the compressible regime.

2. Methodology

The authors conducted a series of experiments in the trisonic wind tunnel at the University of the Bundeswehr Munich.

• Test Surfaces: Two types of sandpaper roughness were used: P60-grit and P24-grit. Surface topologies were scanned to determine physical roughness heights ($k$) and statistical parameters.
• Flow Conditions:
  • Mach Number ($M$): Spanning subsonic ($M=0.3$), transonic ($M=0.8$), and supersonic regimes ($M=2.0, 2.9$).
  • Reynolds Number ($Re_\tau$): Ranging from $7,427$ to $30,292$.
  • Independent Variation: A specific sweep was conducted at constant $M=2$ by varying stagnation pressure to achieve independent $Re_\tau$ variation.
• Measurements:
  • Velocity: Planar 2D2C Particle Image Velocimetry (PIV) was used to measure streamwise and wall-normal velocity profiles.
  • Temperature: Direct wall temperature ($T_w$) was not measured due to facility limitations. Instead, $T_w$ and temperature profiles were estimated assuming adiabatic walls using Crocco's relation and Sutherland's formula.
  • Wall Shear Stress (WSS): Direct WSS measurement was not possible on rough walls. WSS and the Hama roughness function ($\Delta U^+$) were estimated using a modified Clauser fitting method applied to the transformed velocity profiles.
• Transformations: Five different compressibility transformations (Howarth, Van Driest, Brun, Trettel & Larsson, Volpiani) were tested to convert compressible velocity profiles ($U, y$) into incompressible equivalents ($U_I, y_I$).

3. Key Contributions

1. Experimental Dataset: Provided a comprehensive experimental dataset for rough-wall TBLs covering a wide range of $M$ (0.3–2.9) and high $Re_\tau$ ($7.4k$–$30.3k$), including an independent $Re$ sweep at constant $M$.
2. Validation of Framework: Tested the applicability of existing incompressible drag characterization frameworks (Nikuradse/Hama) to compressible flows using various velocity transformations.
3. Empirical Scaling Factors: Proposed and evaluated three distinct scaling/correction factors to bridge the gap between incompressible $k_s$ and compressible drag:
   • Equivalent Incompressible $k_s$: Matching compressible cases to incompressible studies based on roughness type and $\delta/k$.
   • Viscosity-Scaled Roughness ($k^*$): Scaling roughness height by the ratio of freestream to wall kinematic viscosity ($k/\nu_\infty^+$).
   • Momentum Deficit Correction ($\sqrt{1/F_c}$): Applying a correction factor derived from smooth-wall skin friction ($F_c$) to the momentum deficit $\Delta U^+$.

4. Key Results

• Velocity Transformation Insensitivity: The Hama roughness function ($\Delta U^+$) was found to be largely insensitive to the choice of velocity transformation (VD, TL, VO, etc.). The primary effect of compressibility appears as a shift in the log-law intercept rather than a change in the profile shape relative to the transformation.
• Mach Number Dependence: In the fully rough regime, the log-law intercept ($B - B_{FR}$) exhibits a Mach-number-dependent shift. As $M$ increases into the supersonic regime, the intercept increases, indicating additional drag (wave drag) not captured by incompressible models.
• Evaluation of Scaling Factors:
  • Equivalent $k_s$: Using an equivalent incompressible $k_s$ (matched by geometry) failed to consistently collapse data across different Mach numbers, particularly at $M=2.9$.
  • Viscosity Scaling ($k^*$): Scaling the roughness height by the freestream viscosity ratio ($k^* = k/\nu_\infty^+$) successfully collapsed the current experimental data to the Nikuradse line but performed poorly on historical datasets.
  • Momentum Deficit Correction ($\sqrt{1/F_c}$): This factor, applied to $\Delta U^+$, provided the most consistent improvement across both the current experimental data and a broad range of previous studies (spanning sandpaper, cubes, and mesh roughnesses). It successfully collapsed the data onto the Nikuradse relation within $\pm 10\%$ error for most cases.
• Wave Drag Evidence: Schlieren imaging confirmed the presence of persistent bow shocks over roughness elements in supersonic flow, extending into the freestream. This confirms the existence of wave drag, which current smooth-wall transformations do not account for.

5. Significance and Future Work

• Practical Application: The study demonstrates that while incompressible $k_s$ values cannot be directly transferred to compressible flows without modification, a simple empirical correction factor ($\sqrt{1/F_c}$) based on the freestream-to-wall temperature ratio ($T_\infty/T_w$) can effectively predict drag across a wide range of conditions.
• Limitations: The current framework relies on smooth-wall compressibility transformations and empirical corrections. The study highlights that the "wave drag" component is currently treated as a residual error rather than being physically modeled.
• Future Directions:
  • Development of custom transformations specifically for rough-wall TBLs that account for roughness geometry and shock interactions.
  • Direct measurement of wall shear stress and wall temperature on rough surfaces to reduce reliance on fitting methods and adiabatic assumptions.
  • Investigation into the physical mechanisms of wave drag generation by specific roughness topologies.

In conclusion, the paper establishes that while the standard incompressible framework is insufficient for compressible rough-wall flows, the drag can be characterized with reasonable accuracy using a temperature-ratio-based correction factor, paving the way for more physics-based models in future research.