Shape of an interface hit by an oblique jet

This study investigates the formation of a cavity on a liquid interface struck by an oblique jet at angles below 50°, utilizing direct numerical simulations to reveal asymmetric flow detachment that creates a depression, and proposes a force-balancing model to predict the cavity's width.

The Gist

Imagine you are pouring a glass of water into a bathtub. If you pour it straight down, the water hits the surface and creates a little dip, maybe some bubbles, but it's mostly a symmetrical splash.

Now, imagine tilting that glass so the water hits the tub at a sharp angle, like a skier carving into a slope. This is what the scientists in this paper studied. They wanted to understand what happens to the surface of the water when a jet of liquid hits it from the side.

Here is the story of their discovery, broken down into simple concepts:

1. The "Skier" Effect: When the Angle Matters

The researchers found a "magic number" for the angle.

• If the jet is steep (more than 50 degrees): It acts like a normal splash. The water goes down, and the surface rises up around it in a symmetrical circle.
• If the jet is shallow (less than 50 degrees): Something weird happens. Instead of just a splash, a cavity (a deep, empty trench) forms right in front of the jet, like a skier carving a deep groove in fresh snow. The shallower the angle, the wider and deeper this trench becomes.

2. The "Fiber" Experiment: Simplifying the Problem

To understand why this trench forms, the scientists first removed the messy part: the moving water. They replaced the jet with a glass fiber (like a thin straw) dipped into the water at an angle.

• What happened? Even without the water moving, the water level didn't stay flat. It climbed up the straw on the "front" side (the acute angle) and stayed lower on the "back" side (the obtuse angle).
• The Analogy: Think of a person leaning against a wall. The wall pushes back harder on their shoulder than on their back. Similarly, the water "climbs" higher on the side where the angle is sharper. This proved that the angle itself creates an imbalance, even before the water starts rushing.

3. The Invisible Engine: The "Vacuum" Effect

This is the most exciting part. The scientists used powerful computer simulations to look inside the water, where the human eye can't see.

They discovered that when the jet hits the water at an angle, the water doesn't just flow smoothly around it.

• The Detachment: The water flowing from the jet peels away from the surface, but it peels off unevenly. It peels off higher up on the "front" side and lower on the "back" side.
• The Squeeze: Because of this uneven peeling, the water underneath the surface gets squeezed into a narrower path.
• The Bernoulli Effect: Remember the rule that "fast-moving fluid creates low pressure"? Because the water is forced to speed up under the surface to get through that narrow gap, it creates a suction force (a vacuum).
• The Result: This suction acts like a vacuum cleaner, pulling the water surface down and creating that deep trench (cavity) you see.

4. The Tug-of-War Model

Finally, the team built a simple math model to predict how wide this trench would be. They imagined a tug-of-war between three forces:

1. The Suction: The "vacuum" pulling the water down (trying to make the trench huge).
2. Gravity: The weight of the water trying to fill the trench back up.
3. Surface Tension: The "skin" of the water trying to keep the surface flat and smooth.

By balancing these forces, they created a formula that successfully predicts the width of the trench based on how fast the water is moving, how thick the jet is, and how steeply it is angled.

Why Does This Matter?

You might wonder, "Who cares about a trench in a bathtub?"
This is actually a big deal for nature and industry.

• Air Entrainment: When that trench forms, it often traps air. This is how bubbles get sucked down into the ocean by waterfalls or how air gets mixed into industrial mixers.
• Prediction: Understanding the shape of the surface helps engineers predict exactly how much air gets trapped and how deep the bubbles go. This is crucial for designing better pumps, understanding ocean pollution, or even improving how we mix chemicals.

In a nutshell: By tilting a water jet, the scientists discovered that the water creates a "vacuum" underneath the surface due to uneven flow. This vacuum sucks the water down, carving a trench. They proved this by comparing it to a tilted straw and then used a simple "tug-of-war" math model to predict exactly how big that trench would get.

Technical Summary

1. Problem Statement

The study investigates the deformation of a liquid bath interface when impacted by a smooth, steady, oblique liquid jet. While vertical jets are well-studied (often forming symmetric menisci or air entrainment clouds), the dynamics of oblique jets remain less understood, particularly regarding the formation of a cavity in front of the jet.

• Key Phenomenon: When the angle of inclination ($\alpha$) between the jet and the bath surface drops below approximately $50^\circ$, a distinct cavity forms ahead of the jet, widening as the jet approaches the horizontal.
• Knowledge Gap: There is a lack of quantitative models describing the interfacial shape and the specific mechanisms driving cavity formation in low-viscosity liquids (like water) under oblique impact.

2. Methodology

The authors employed a multi-faceted approach combining experimental observation, static analog modeling, and direct numerical simulations (DNS).

• Experimental Setup:
  • A water jet was generated using a pressure controller and a stainless steel nozzle.
  • Parameters: Jet radius ($R$) varied from 0.12 to 0.73 mm; impact velocity ($V$) ranged from 0.8 to 5.4 m/s; inclination angles ($\alpha$) between $21^\circ$ and $49^\circ$.
  • Imaging: High-speed imaging with LED backlighting was used to capture the interface shape. To avoid optical distortion from tank walls, experiments were conducted with the tank filled to the brim.
• Static Analogy (Fiber Experiment):
  • To isolate the effect of inclination from dynamic flow, the jet was replaced by a static glass fiber immersed in a silicone oil bath (to ensure zero contact angle).
  • This allowed the measurement of the asymmetric meniscus height difference ($\Delta H$) caused solely by the inclination angle.
• Numerical Simulations:
  • Solver: Direct Numerical Simulations (DNS) were performed using Basilisk, a Volume of Fluid (VOF) solver with sharp interface reconstruction.
  • Conditions: Simulations matched experimental parameters ($Re \approx 400$, $We \approx 12$, $Fr \approx 40$).
  • Technique: Instead of tilting the jet, the gravity vector was tilted relative to a vertical jet to simulate the oblique impact. 3D adaptive octree grids were used to resolve the flow topology.

3. Key Contributions and Results

A. Experimental Observations

• Cavity Formation: A cavity forms in front of the jet when $\alpha < 50^\circ$. The cavity width ($W$) increases significantly as the jet becomes more horizontal (up to 10 times the jet radius).
• Dominant Parameter: Jet inclination is the primary driver of cavity geometry, outweighing variations in jet speed or radius.
• Flow Asymmetry: The interface separates from the jet asymmetrically.

B. Static Analogy (Fiber Study)

• Asymmetric Meniscus: An inclined fiber creates an asymmetric meniscus where the liquid height is greater on the acute side of the fiber than the obtuse side.
• Analytical Model: The authors derived an analytical expression for the maximum height difference ($\Delta H$) based on James' (1974) formula for vertical fibers, adapted for inclination:
  $$\Delta H \approx R \ln\left(\frac{1 + \cos(\alpha)}{1 - \cos(\alpha)}\right)$$
  This model successfully predicted experimental data for small fibers, validating the geometric influence of inclination.

C. Numerical Insights (Velocity and Pressure Fields)

• Asymmetric Detachment: The flow separates from the interface asymmetrically. The detachment point is higher on the acute side (front of the jet) than the obtuse side.
• Velocity Acceleration: Due to this asymmetry, streamlines contract after detachment, accelerating the liquid beneath the cavity surface to speeds exceeding the injection velocity.
• Pressure Depression: This acceleration creates a localized pressure depression (suction) beneath the interface, which is the primary mechanism pulling the interface down to form the cavity.
• Hydrostatic Outer Region: Outside the high-speed core, the liquid velocity is negligible (40x lower than injection speed), allowing the outer interface shape to be described by hydrostatic equations.

D. Theoretical Modeling of Cavity Width

The authors proposed a force-balance model to predict the cavity width ($W$). The model equates:

1. Driving Force: The suction force generated by the pressure depression ($P_D$) due to flow acceleration.
2. Opposing Forces:
   • The weight of the displaced water ($F_g \sim \rho g W^3$).
   • The surface tension force ($F_\gamma$), adapted from soap film models on tilted cylinders.

The resulting scaling law for the cavity width is:
$$W = \left( \frac{(RV)^2(\beta^2 f(\alpha)^2 - 1)}{2g} - l_c^2 \frac{R}{\sin(\alpha)} \right)^{1/3}$$
Where $f(\alpha)$ is the geometric factor derived from the static fiber model, and $\beta$ is a fitted coefficient.

4. Significance and Conclusion

• Mechanism Elucidation: The study successfully identifies the asymmetric flow separation as the root cause of cavity formation, linking the static geometry of an inclined fiber to the dynamic behavior of an oblique jet.
• Predictive Capability: The proposed force-balance model provides a first-order prediction of cavity width based on system parameters ($R, V, \alpha$), showing reasonable agreement with experimental data.
• Broader Impact: Understanding the interface shape and cavity formation is critical for predicting air entrainment in industrial and natural processes (e.g., plunging jets, hydraulic structures). The cavity acts as a precursor to bubble cloud formation; thus, modeling its shape is a necessary step toward quantifying air entrainment fluxes.

The paper bridges the gap between static capillary phenomena and dynamic jet impact, offering a framework to analyze complex interfacial flows where symmetry is broken by inclination.