High-Reynolds-number turbulent boundary layers under adverse pressure gradients. Part 2. A composite mean velocity profile

This paper presents a robust, three-parameter composite mean velocity profile for high-Reynolds-number turbulent boundary layers under adverse pressure gradients, which incorporates history effects and velocity overshoot to enable reliable estimation of flow parameters and demonstrates that the von Kármán coefficient converges to an invariant value of approximately 0.39 at sufficiently high Reynolds numbers.

The Gist

Imagine you are a river flowing smoothly over a flat, smooth riverbed. The water near the bottom moves slowly because of friction, while the water at the top rushes along. This is a "Zero Pressure Gradient" flow—it's calm, predictable, and follows a well-known set of rules.

Now, imagine that river suddenly hits a steep hill or a narrowing canyon. The water has to fight against gravity and the narrowing space. It slows down, swirls, and the flow near the bottom gets messy and turbulent. In physics, we call this an Adverse Pressure Gradient (APG). It's the "bad weather" of fluid dynamics.

This paper is about creating a new, better map for this messy, turbulent river.

The Problem: The Old Map Was Outdated

For decades, scientists have used a standard "composite map" (a mathematical formula) to describe how water moves in these turbulent layers. Think of this map like a generic GPS route. It works perfectly for a calm, straight highway (the smooth river).

However, when the river hits that hill (the adverse pressure gradient), the old GPS starts to fail. It gets confused about:

1. The History: Did the river just come from a steep drop, or has it been climbing slowly for miles? The old map didn't care about the journey, only the current spot.
2. The Speed Bumps: Near the riverbed, the water sometimes speeds up unexpectedly (a "velocity overshoot") before slowing down again. The old map missed this bump.
3. The Edge: The map didn't know exactly where the river ends and the open air begins, especially when the flow is stretched out.

The Solution: A New, "Smart" GPS

The authors of this paper (Zarei, Lozier, Deshpande, and Marusic) have built a new, upgraded GPS specifically for these difficult, turbulent conditions. They took an existing model and added three special "features" to make it robust:

1. The "Memory Lane" Feature (History Effects)

Imagine two hikers standing at the same spot on a mountain.

• Hiker A just walked up a gentle slope.
• Hiker B just sprinted up a steep cliff.
  Even though they are at the same height, Hiker B is out of breath and moving differently than Hiker A.

The old map treated them as identical. The new map introduces a "History Parameter." It asks, "Where did you come from?" This allows the map to account for the fact that the water's current behavior is shaped by the pressure gradients it experienced upstream. It distinguishes between a river that just started slowing down and one that has been struggling for miles.

2. The "Speed Bump" Detector (Velocity Overshoot)

In the old map, the water speed near the bottom was expected to rise smoothly. But in reality, under these tough conditions, the water near the wall sometimes gets a sudden burst of speed—a "speed bump"—before settling into the normal flow.

The new map includes a special "Overshoot Function." It's like adding a little hump to the road on the map to match the real terrain. This ensures the model captures that extra burst of energy near the wall, which is crucial for calculating things like drag (how much the water "sticks" to the surface).

3. The "Stretchy Ruler" (Wake Region)

The top part of the river (the "wake") is where the flow is most chaotic. The old map used a rigid ruler to measure this area. But when the pressure gradient changes, this part of the river stretches or squishes.

The new map uses a "Stretchy Ruler." It introduces a parameter that allows the map to expand or contract the top section of the flow to match reality. This makes the map accurate even when the river is being pulled apart by strong forces.

Why Does This Matter? (The Real-World Impact)

You might ask, "Why do we need a better map for a river?" Here is why this matters for the real world:

• Saving Fuel: Airplanes and ships are essentially "flying" or "swimming" through these turbulent layers. If we can predict the friction (drag) more accurately, we can design planes and ships that use less fuel.
• The "Invisible" Measurements: Sometimes, in a wind tunnel or a real ocean, it's impossible to measure the exact speed of the water right at the surface (the "friction velocity") or the exact height of the turbulent layer. It's like trying to measure the temperature of a flame without touching it.
  • The new map acts like a Sherlock Holmes. By looking at the shape of the flow above the surface, the map can mathematically deduce (estimate) the hidden values at the bottom. This is a huge tool for engineers who can't always put sensors everywhere.
• The Universal Constant: The study found something surprising. No matter how strong the pressure gradient is, if the Reynolds number (a measure of how turbulent the flow is) gets high enough, the "friction rule" (the von Kármán coefficient) settles on a single, unchanging value. It's like finding that no matter how chaotic the storm gets, the wind eventually follows a specific, predictable rhythm.

The Bottom Line

This paper is about upgrading the "instruction manual" for turbulent fluids. By adding memory of the past, detecting speed bumps, and using a stretchy ruler, the authors have created a tool that works for a much wider range of real-world conditions.

It turns a messy, unpredictable flow into something we can understand, measure, and predict with much higher confidence. This helps engineers build better, more efficient vehicles and helps scientists understand the fundamental rules of how fluids move.

Technical Summary

1. Problem Statement

Turbulent Boundary Layers (TBLs) subjected to Adverse Pressure Gradients (APGs) are critical in engineering applications (e.g., aircraft wings, turbomachinery) but remain difficult to model accurately. While the mean velocity profile is fundamental for predicting skin-friction drag and developing turbulence models, existing analytical formulations face significant limitations:

• Inadequacy of Classical Models: Traditional "law of the wall" and "law of the wake" formulations (e.g., Coles, 1956) and even the two-parameter composite profile by Nickels (2004) fail to capture the complex behavior of high-Reynolds-number ($Re_\tau$) APG flows.
• Pressure Gradient (PG) History Effects: Previous studies (including Part 1 of this series) demonstrated that upstream PG history significantly alters the mean velocity profile, particularly in the overlap and wake regions. Existing models rely on local parameters (like the Clauser parameter $\beta$ or viscous-scaled $p_x^+$) and cannot account for this cumulative history.
• Measurement Limitations: In high-$Re_\tau$ experiments, direct measurement of friction velocity ($U_\tau$) and boundary-layer thickness ($\delta$) is often difficult or impossible. There is a need for a robust analytical tool to estimate these quantities and evaluate derived metrics (like the indicator function) from sparse velocity data.

2. Methodology

The authors developed a three-parameter composite mean velocity profile by extending the framework of Nickels (2004). The methodology involved:

• Dataset Compilation: A comprehensive database of APG TBLs was assembled, spanning over three decades of experimental and numerical data (DNS, LES, and wind tunnel experiments). This included datasets with varying Reynolds numbers ($Re_\tau \approx 400 - 24,000$), local pressure gradients ($\beta$), and distinct PG history conditions.
• Formulation Modification: The composite profile $U^+ = U^+_{inner} + U^+_{overlap} + U^+_{wake}$ was modified in three key ways:
  1. Inner Region: A velocity-overshoot function (Gaussian-type) was added to the inner layer expression to capture the buffer region overshoot observed in high-$Re_\tau$ flows, which previous models missed.
  2. Overlap Region: The relationship between the sublayer thickness ($z_c^+$) and the pressure gradient was reformulated. Instead of relying solely on the local viscous-scaled parameter $p_x^+$, the authors introduced an effective pressure gradient parameter ($\beta_{eff} = \beta \cdot C_{H_i}$). This links the local PG and the upstream PG history to the shift in the logarithmic region.
  3. Wake Region: The wake function was reformulated using an independent, physically motivated definition of boundary-layer thickness ($\delta$) based on the skewness of velocity fluctuations (Lozier et al., 2025). A new wake-stretching parameter ($C_{H_w}$) was introduced to account for the horizontal stretching of the wake profile caused by PG history.
• Fitting Procedure: A nonlinear least-squares curve-fitting algorithm was applied to the compiled datasets to optimize the three free parameters:
  • $z_c^+$: Sublayer thickness (governs the overlap shift).
  • $C_{H_i}$: Inner/overlap history factor (relates to $\beta_{eff}$).
  • $C_{H_w}$: Wake history factor (relates to wake stretching).
  • $\Pi$: Wake strength parameter (Coles' wake parameter).

3. Key Contributions

• Three-Parameter Composite Profile: The paper presents a unified analytical formulation (Equations 3.5–3.7) that accurately describes mean velocity profiles across a wide range of $Re_\tau$ and APG conditions, including strong PG history effects.
• Quantification of PG History: The study introduces a framework to quantify PG history effects using two distinct parameters:
  • $C_{H_i}$: Characterizes the "history-enhanced" or "history-damped" shifting of the overlap region.
  • $C_{H_w}$: Characterizes the stretching of the wake region.
  • This allows researchers to distinguish between flows with matched local conditions but different upstream histories.
• Effective Pressure Gradient ($\beta_{eff}$): The concept of $\beta_{eff}$ is proposed to unify the influence of local and upstream pressure gradients on the inner/overlap regions, providing a single metric to predict the additive constant $B$ (or $z_c^+$) of the log-law.
• Practical Estimation Tools: The formulation enables the estimation of $U_\tau$ and $\delta$ in datasets where these are not directly measured, provided the flow is "well-behaved" (minimal history effects, $C_{H_i} \approx C_{H_w} \approx 1$).
• Analytical Differentiation: The analytical nature of the profile allows for the precise calculation of the indicator function ($z^+ dU^+/dz^+$) and the identification of inflection points, which are difficult to obtain from noisy experimental data.

4. Results

• Accuracy: The new composite profile provides excellent agreement with experimental and numerical data across all regions (viscous sublayer, buffer, overlap, and wake) for both Zero Pressure Gradient (ZPG) and APG flows.
• Universality of $\kappa$: Analysis of the fitted parameters reveals that at sufficiently high Reynolds numbers ($Re_\tau \gtrsim 10,000$), the von Kármán coefficient ($\kappa$) approaches an invariant value of $\kappa \approx 0.39$, independent of the pressure gradient strength or history. This supports the attached eddy hypothesis.
• Logarithmic Extent: The study predicts the extent of the logarithmic region (in decades of $z^+$). It shows that while increasing $\beta$ reduces the logarithmic extent, high $Re_\tau$ can compensate for this, maintaining a significant overlap region even in moderate APGs.
• History Effects:
  • Flows with "history-enhanced" shifting ($C_{H_i} > 1$) exhibit a larger downward shift in the overlap region than predicted by local $\beta$ alone.
  • Flows with "history-damped" shifting ($C_{H_i} < 1$) show a smaller shift.
  • The wake parameter $C_{H_w}$ deviates from unity in flows with significant upstream PG events, indicating wake stretching.
• Validation: The method successfully estimated $U_\tau$ and $\delta$ for the high-$Re_\tau$ dataset from Part 1, matching direct measurements within experimental uncertainty. It also successfully re-analyzed historical datasets (e.g., Marusic & Perry, 1995) where near-wall data was missing.

5. Significance

This work provides a robust, physically interpretable framework for analyzing and modeling high-Reynolds-number APG turbulent boundary layers. Its significance lies in:

1. Bridging the Gap: It resolves the discrepancy between local pressure gradient models and the observed cumulative effects of PG history, offering a unified description for "well-behaved" and complex APG flows.
2. Engineering Utility: It offers a practical tool for engineers to estimate critical boundary layer parameters ($U_\tau, \delta$) and evaluate turbulence models in scenarios where direct measurement is infeasible.
3. Theoretical Insight: By confirming the invariance of $\kappa \approx 0.39$ at high $Re_\tau$ and quantifying the distinct mechanisms of PG history on the inner vs. wake regions, it advances the fundamental understanding of wall-bounded turbulence.
4. Future Guidance: The formulation provides criteria for selecting experimental parameters to ensure sufficient logarithmic scaling in future high-$Re_\tau$ APG studies.

In conclusion, this paper extends the legacy of Nickels (2004) by incorporating modern physical insights and high-fidelity data, resulting in a composite profile that is both mathematically rigorous and practically applicable to complex turbulent flows.