Combined dynamic-kinematic validation of droplet-wall impact modeling

This paper introduces a combined dynamic-kinematic validation framework and a novel $(\beta_{max}, Ca_{char})$ diagram to demonstrate that relying solely on maximum spreading diameter is insufficient for accurate droplet impact modeling, advocating instead for a hybrid contact angle model that better captures both geometric spreading limits and internal kinematic receding dynamics.

The Gist

Imagine you drop a single raindrop onto a windshield. What happens next is a tiny, high-speed drama: the drop splats, spreads out like a pancake, and then sometimes shrinks back or bounces away. Scientists have been trying to build computer programs to predict exactly how this happens, which is crucial for everything from 3D printing to making better car windshields that don't ice over.

However, there's a problem with how scientists have been testing these computer programs.

The "Pizza Dough" Problem

For years, researchers checked if their computer simulations were working correctly by looking at just one thing: How wide did the "pancake" get at its widest point?

Think of it like judging a pizza chef. If you only look at the final diameter of the pizza, you might think the chef is a genius. But what if the chef rolled the dough out perfectly wide, but then the sauce was running off the side, or the dough was tearing apart? You missed the process.

This paper argues that just measuring the final size (the "maximum spreading diameter") isn't enough. You need to watch the dance—how the liquid moves inside, how fast it spreads, and how it behaves when it tries to pull back together.

The Two "Rules" of the Dance

To simulate this, the scientists used two different mathematical "rulebooks" to describe how the edge of the droplet (the contact line) behaves:

1. The "Generalized HVT Law" (The Geometry Expert): This rulebook is great at predicting how wide the drop will get. It's like a mapmaker who knows exactly how far a river will flood. However, when it comes to the speed and flow of the water inside the river, this mapmaker gets a bit confused. It predicts the water speeding up when it should be slowing down, like a car that keeps accelerating even after the driver hits the brakes.
2. The "Hoffman Function" (The Kinematics Expert): This rulebook is better at understanding the movement. It knows how the liquid flows and slows down realistically. It's like a traffic engineer who understands the flow of cars perfectly. But, it's not quite as precise at predicting the exact final width of the flooded area.

The Solution: The "Hybrid Driver"

The researchers realized that relying on just one rulebook was like hiring a driver who only knows how to turn the steering wheel but doesn't know how to use the brakes, or vice versa.

They created a Combined Model:

• When the drop is spreading out (the "splat"): They used the "Geometry Expert" (HVT Law) because it's great at predicting the width.
• When the drop starts to shrink back (the "recede"): They switched to the "Kinematics Expert" (Hoffman Function) because it correctly predicts how the liquid slows down and stops without doing weird, impossible things.

The New "Scorecard"

The paper introduces a new way to grade these simulations. Instead of just looking at the final size, they now look at:

• The Shape: Did it spread to the right size?
• The Flow: Did the liquid inside move at the right speed?
• The Stop: Did it stop moving at the right time?

They even created a new "map" (a diagram) that links the size of the drop to the internal speed of the liquid. It's like saying, "If you see a drop of this specific size, you can guess exactly how fast the water inside is swirling."

Why Does This Matter?

If you are designing a spray paint system, a medical inkjet printer, or a cooling system for a hot engine, you don't just want the paint to cover the right area. You need to know how it gets there. If the simulation gets the physics wrong, your real-world product might fail.

In short: This paper says, "Don't just check the final score; watch the whole game." By combining the best parts of two different mathematical models, they created a simulation that is not only accurate in size but also physically realistic in how the liquid moves.

Technical Summary

1. Problem Statement

Droplet impact on solid substrates is critical for applications ranging from spray coating and inkjet printing to anti-icing and 3D bioprinting. While Computational Fluid Dynamics (CFD) is widely used to simulate these processes, current validation methodologies are often insufficient.

• The Limitation: Most studies validate models solely based on the maximum spreading diameter ($\beta_{max}$).
• The Gap: Relying only on geometric parameters fails to capture the internal flow dynamics (kinematics). A model might accurately predict the final spread size but fail to reproduce the correct velocity fields, contact line behavior, or receding dynamics.
• Objective: The authors aim to develop a combined dynamic-kinematic validation method that evaluates both geometric outcomes (spreading diameter) and kinematic characteristics (internal velocity fields, contact line velocity) to ensure physical fidelity.

2. Methodology

Experimental Setup

• Fluid: Water-glycerol mixture (60 wt% glycerol) with titanium dioxide particles for PIV seeding.
• Substrate: Sapphire glass.
• Conditions: Impact Weber numbers ($We_{imp}$) of 20, 80, and 250, corresponding to impact velocities of 0.63, 1.17, and 2.08 m/s.
• Measurement: High-speed videography for geometry and Particle Image Velocimetry (PIV) for internal velocity fields at a height of 250 $\pm$ 50 $\mu$m above the substrate.

Numerical Simulation

• Solver: Ansys Fluent 2023 R1 using a 2D axisymmetric Volume of Fluid (VOF) model.
• Governing Equations: Incompressible Navier-Stokes equations.
• Mesh: Adaptive Mesh Refinement (AMR) based on VOF field gradients, with a convergence study selecting 5 adaptation levels.
• Dynamic Contact Angle (DCA) Models: Two models were implemented via User-Defined Functions (UDF):
  1. Generalized Hoffman-Voinov-Tanner (HVT) Law: $\theta_D^3 - \theta_0^3 = \kappa Ca$.
  2. Hoffman Function: A non-linear relationship between contact angle and capillary number ($Ca$) for both advancing and receding modes.

Data Processing & Validation Strategy

• Preprocessing: A custom Python/SciPy pipeline was developed to preprocess CFD data for fair comparison with PIV. This included:
  • Linear interpolation onto a uniform mesh.
  • FFT-based Gaussian filtering to remove high-wavenumber noise.
  • Cubic spline interpolation near the droplet rim (to correct for PIV optical distortion errors).
  • Weighted averaging of radial velocities.
• Metrics: Validation was performed against:
  • Maximum spreading diameter ($D_{max}$).
  • Average radial velocity ($v_{avg}$) during spreading and receding phases.
  • Droplet spreading velocity ($U_{spr}$).
  • Mean internal flow velocity ($U_{mean}$).

3. Key Contributions

1. Critique of Geometric-Only Validation: The study demonstrates that a model can achieve high accuracy in predicting maximum spreading diameter ($\le$7% error) while simultaneously failing to reproduce correct receding dynamics (e.g., predicting non-physical negative spreading velocities or incorrect internal acceleration).
2. Development of a Combined DCA Model: The authors propose a hybrid model that leverages the strengths of both tested approaches:
   • Spreading Phase ($Ca > 0$): Uses the Generalized HVT Law (with $\kappa = 8.78$) for superior accuracy in predicting maximum spread and mean internal velocity.
   • Receding Phase ($Ca < 0$): Uses the Hoffman Function to ensure physically consistent contact line behavior and accurate receding velocity trends.
3. New Dimensionless Diagram ($\beta_{max}$ vs. $C_{achar}$): The authors introduce a diagram relating the maximum spreading factor ($\beta_{max}$) to a characteristic capillary number ($C_{achar}$) defined by the mean internal horizontal velocity rather than impact velocity. This links geometric outcomes directly to internal flow dissipation.

4. Results

Comparison of Individual Models

• Generalized HVT Law:
  • Pros: Excellent agreement with experimental maximum spreading diameter (error $\le$ 7%) and mean internal velocity during the spreading phase.
  • Cons: Fails during the receding phase. It predicts unrealistic dynamics, including a monotonic increase in velocity and negative spreading velocities (contact line moving backward) even after the contact line has pinned. The median absolute error (MedAE) in radial velocity during receding was up to 3 times higher than the Hoffman model.
• Hoffman Function-Based Model:
  • Pros: Captures the correct kinematic trends, particularly the deceleration and pinning of the contact line during receding. It provides the most consistent radial velocity profiles.
  • Cons: Slightly higher error in maximum spreading diameter compared to the optimized HVT law (6.8%–12.5%).

Performance of the Combined DCA Model

The proposed combined model successfully unified the advantages of both:

• Geometric Accuracy: Maintained the low error ($\le$ 7%) in maximum spreading diameter characteristic of the HVT law.
• Kinematic Accuracy: Achieved the lowest MedAE in radial velocity during the receding phase (outperforming both individual models).
• Physical Consistency: Correctly predicted the transition to contact line pinning ($U_{spr} \to 0$) without non-physical negative velocities.
• Internal Flow: The relative error for mean internal velocity ($U_{mean}$) was comparable to the best-performing individual model (Generalized HVT) for most cases.

Scaling Laws

• The study confirmed a power-law relationship between maximum spreading diameter and mean internal velocity ($D_{max} \propto U_{mean}^b$, where $b \approx 0.5$).
• The $(\beta_{max}, C_{achar})$ diagram revealed distinct regimes:
  • Inertia-dominated: Increasing impact velocity increases both $\beta_{max}$ and $C_{achar}$.
  • Viscosity-dominated: Increasing viscosity (at fixed velocity) decreases $\beta_{max}$ while increasing $C_{achar}$, highlighting the suppression of spreading by viscous dissipation.

5. Significance and Conclusion

• Validation Paradigm Shift: The paper argues that validating droplet impact models requires a dual-metric approach. Relying solely on maximum spreading diameter is methodologically weak because it can mask fundamental errors in the hydrodynamics of the receding phase.
• Improved Modeling: The proposed Combined DCA model offers a robust solution for industrial applications (spraying, cooling, printing) where both the final wetted area and the transient flow dynamics are critical.
• Future Outlook: The introduction of the $(\beta_{max}, C_{achar})$ diagram suggests a pathway to estimate internal kinematic properties based on observable geometric contact-line data, provided sufficient experimental data is available.

In summary, this work establishes that rigorous validation of droplet impact must integrate geometric metrics (spreading diameter) with kinematic metrics (internal velocity fields and contact line dynamics) to ensure the physical fidelity of CFD simulations.