Modification of the k-omega0 model for roughness

This paper extends the k-omega0 turbulence model to account for surface roughness by introducing an effective origin that links log-layer offsets to equivalent sandgrain roughness, while deriving a virtual origin formula that ensures consistency with the fully rough flow limit.

The Gist

The Big Picture: Why Do We Care About Roughness?

Imagine you are sliding down a playground slide.

• The Smooth Slide: If the slide is perfectly smooth, you glide down fast and predictably.
• The Rough Slide: If the slide is covered in sandpaper or pebbles, you slow down, get bumped around, and your path changes.

In the world of engineering (like designing airplanes or cars), scientists use computer models to predict how air or water flows over surfaces. These models work great for smooth surfaces. But real life is rarely smooth. Even a surface that looks smooth to the eye (like a metal wing) can be "rough" to the air molecules if the air is moving fast enough.

The problem? Standard computer models get confused by roughness. They assume the air stops moving right at the surface (like a smooth slide). But on a rough surface, the air actually keeps swirling and moving right down into the "pebbles," creating extra drag and turbulence.

The Solution: The "Ghost Floor" (Effective Origin)

The authors, Paul Durbin and Zifei Yin, propose a clever trick to fix the computer model. Instead of trying to model every single tiny pebble on the surface (which would take a supercomputer forever), they introduce a concept called an "Effective Origin" or a "Ghost Floor."

The Analogy:
Imagine you are measuring the height of a person standing on a pile of rocks.

• The Old Way: You try to measure from the very bottom of the rocks. It's messy, and the rocks are uneven.
• The New Way: You pretend the person is standing on a flat, invisible floor that is floating above the rocks. You measure their height from this "Ghost Floor" instead.

In the paper, they shift the starting point of their math. Instead of saying the air stops at the physical wall ($y=0$), they say the air stops at a "virtual" wall ($y = -\ell$) that is slightly below the surface.

• Smooth Surface: The Ghost Floor is right on top of the wall.
• Rough Surface: The Ghost Floor drops down into the roughness. The "rougher" the surface, the deeper the Ghost Floor goes.

By moving this starting point, the computer model can automatically "feel" the roughness without needing to see the actual pebbles.

The Two "Rules of the Road" (Boundary Conditions)

The paper tests two different ways to set up this "Ghost Floor" math. Think of these as two different rulebooks for how the air behaves right at the edge of the wall.

1. Rulebook A (The Quiet Start): This assumes the turbulence is zero right at the wall. It's like saying, "If you are standing on the rocks, you aren't moving yet."
   • Result: This works well, but the math gets a little tricky near the wall.
2. Rulebook B (The Busy Start): This assumes the air is already swirling a bit because of the roughness. It's like saying, "Even if you are on the rocks, the wind is already blowing you around."
   • Result: This turns out to be more accurate for very rough surfaces (fully rough), because it acknowledges that roughness creates its own turbulence immediately.

The "Calibration" (Fitting the Puzzle)

How do they know how deep to put the "Ghost Floor"? They can't just guess.

They used a famous set of data from a scientist named Nikuradse (who measured flow over sand grains a long time ago). The authors ran their computer simulations, adjusted the depth of the "Ghost Floor" until the results matched the real-world data, and then created a lookup chart.

The Analogy:
Think of it like tuning a guitar.

• You have a string (the roughness height).
• You have a tuner (the computer model).
• You turn the peg (the "Effective Origin") until the note (the airflow speed) matches the perfect pitch (the real-world experiment).
• Once they know exactly how much to turn the peg for a specific string, they write it down in a formula. Now, for any new rough surface, they just look up the formula, set the "Ghost Floor," and the model works.

What Did They Find?

1. It Works: Their new method successfully predicts how air slows down and swirls over rough surfaces.
2. The "Fully Rough" Limit: When a surface is very rough (like a gravel road), the air flow behaves in a specific, predictable way. The authors proved that their "Ghost Floor" math naturally leads to this correct behavior without needing special, complicated fixes.
3. Separation: They tested the model on a ramp where the air might separate (peel away from the surface). They found that rough surfaces make the air separate sooner than smooth surfaces. Their model correctly predicted this, showing that roughness acts like a "brake" on the airflow, causing it to detach earlier.

The Takeaway

This paper is about making computer simulations smarter about messy, real-world surfaces.

Instead of trying to build a 3D model of every bump and scratch on a surface (which is too hard), the authors invented a mathematical "magic trick." They simply lowered the floor in their equations. This small shift allows the computer to understand that the surface is rough, predicting the extra drag and turbulence accurately, all while keeping the math simple enough to run on standard computers.

It's like realizing you don't need to count every grain of sand on a beach to know it's rough; you just need to know that the water level effectively starts a few inches lower than the sand itself.

Technical Summary

1. Problem Statement

Surface roughness significantly impacts high Reynolds number turbulent flows, often altering mean flow characteristics even when geometric irregularities are small. In Reynolds-Averaged Navier-Stokes (RANS) modeling, roughness is typically characterized by an equivalent sandgrain roughness ($r$), which correlates to an offset in the logarithmic velocity layer ($\Delta U^+$).

The primary challenge addressed in this paper is adapting the $k-\omega_0$ turbulence model to account for rough surfaces.

• Context: The standard $k-\omega$ model requires a singular solution for $\omega$ near the wall ($\omega \propto y^{-2}$). The $k-\omega_0$ model was originally developed to avoid this singularity by solving for a regular variable $\omega_0$ and transforming it to $\omega$.
• Gap: Existing roughness treatments in $k-\omega$ models often rely on boundary conditions for $\omega$ or $k$ that may not seamlessly integrate with the $k-\omega_0$ formulation's specific transformation logic. The authors aim to incorporate roughness into the $k-\omega_0$ framework using an effective origin approach, ensuring consistency with the fully rough limit and the standard log-law.

2. Methodology

The authors propose a method where surface roughness is modeled by introducing an effective origin ($\ell$) into the model's coordinate system and boundary conditions.

A. Theoretical Framework

• Transformation: The standard transformation relating the regular solution $\omega_0$ to the singular solution $\omega$ is modified to include the effective origin $\ell$:
  $$\omega = \omega_0 + \frac{6\nu}{C_{\omega 2}(y+\ell)^2} e^{-AX}$$
  where $X$ is a function of $\omega_0$ and $y$. This modification alters the near-wall behavior of $\omega$ from $1/y^2$ to $1/(y+\ell)^2$, effectively shifting the singularity inside the wall.
• Boundary Conditions: The paper investigates two distinct boundary condition approaches for $\omega_0$:
  1. $\omega_0(0) = 0$: A simpler implementation where $\omega_0$ vanishes at the wall.
  2. $\omega_0 \propto \tau_w/\mu$: A boundary condition derived from the source term of the $\omega_0$ equation, representing the wall shear stress.
• Turbulent Kinetic Energy ($k$): Under rough conditions, $k$ does not vanish at the wall. A new boundary condition for $k(0)$ is imposed, interpolating between zero (smooth) and a fully rough value based on $\ell$.

B. Calibration Procedure

The model is calibrated using fully developed channel flow simulations at a friction Reynolds number $Re_\tau = 20,000$.

1. Simulation: The model is solved with varying values of the effective origin $\ell$.
2. Log-Layer Offset: The resulting velocity profiles are analyzed to compute the log-layer displacement ($\Delta U^+$) using an averaging integral over the log-region ($80 < y^+ < 0.2\delta^+$).
3. Correlation: The calculated $\Delta U^+$ is mapped to the equivalent sandgrain roughness $r^+$ using the empirical correlation by Ligrani and Moffat (fitting Nikuradse's data).
4. Result: This process generates a calibration curve relating the model parameter $\ell^+$ to the physical roughness $r^+$.

3. Key Contributions

• Extension of $k-\omega_0$ to Roughness: The paper successfully extends the $k-\omega_0$ model (originally designed for numerical robustness) to handle rough surfaces without requiring singular solutions.
• Effective Origin Formulation: It formalizes the use of an effective origin $\ell$ not just as a geometric shift, but as a hydrodynamic property that prevents eddy viscosity from vanishing at the wall, thereby sustaining turbulence generation near rough surfaces.
• Derivation of Virtual Origin: The authors derive an analytical formula for the virtual origin ($y_0$) of the fully rough log-law based on the model parameters. This bridges the gap between the turbulence model's internal parameters and the empirical log-law offset.
• Dual Boundary Condition Analysis: The study provides specific calibration curves and formulas for both the $\omega_0(0)=0$ and the shear-stress-based boundary conditions, offering flexibility for different implementation needs.

4. Results

• Calibration Curves: The authors established polynomial relationships between $\ell^+$ and $r^+$ for both boundary condition variants.
  • For $\omega_0(0)=0$: Fully rough limit ($r^+ > 101$) corresponds to $\ell^+ \approx 3.9$.
  • For Shear Stress BC: Fully rough limit ($r^+ > 97.8$) corresponds to $\ell^+ \approx 8.9$.
• Flow Profiles:
  • $\omega$ and $k$: The model correctly predicts that roughness reduces the singularity of $\omega$ near the wall and elevates $k$ from zero to a constant value in the fully rough regime.
  • Eddy Viscosity: While negligible on linear scales, the modification significantly alters eddy viscosity in the near-wall region on log-log scales, which is critical for the velocity profile.
  • Velocity Displacement: The model successfully reproduces the downward displacement of the log-layer ($\Delta U^+$) as roughness increases.
• Validation Cases:
  • Channel Flow: The model matches DNS data for channels with one smooth and one rough wall (2-D ribs), capturing the asymmetry in the velocity profile.
  • Separated Flow: The model was tested on a contoured ramp with roughness (Song and Eaton experiment). It correctly predicted that roughness thickens the boundary layer, promoting flow separation on a ramp that would remain attached if smooth.

5. Significance

• Physical Insight: The paper clarifies that the "effective origin" is a hydrodynamic construct determined by turbulent mixing rather than a strict geometric average of surface asperities. It effectively represents the mean line through the roughness elements where the turbulence dynamics change.
• Robustness: By utilizing the $k-\omega_0$ formulation, the method avoids the numerical difficulties associated with the $1/y^2$ singularity of the standard $k-\omega$ model, making it more robust for complex geometries involving roughness.
• Predictive Capability: The derived formula for the virtual origin in the fully rough limit ensures the model is consistent with classical fluid dynamics theory, allowing for accurate predictions of drag and separation in high-Reynolds-number rough-wall flows.
• Practical Application: The calibration curves allow engineers to input a known equivalent sandgrain roughness ($r^+$) and directly determine the necessary model parameter ($\ell^+$) for RANS simulations, facilitating the modeling of rough surfaces in aerospace and industrial applications.