Experimental Studies on the Load Characteristics of Low-Speed Droplets Impinging onto Surface

Abstract

Droplet impingement on a wall is a fundamental scientific problem with wide engineering applications. When a droplet impacts the surface of an aircraft, it generates shock waves, airflow disturbances, and splashing phenomena. This not only has a negative impact on the aerodynamic performance and stability of the aircraft but also obstructs the field of view of optical sensors or causes distortion in optical devices. It can also damage the aircraft's structure, thus it’s vital to assess the droplet impact force for flight safety. However, droplets are often treated as rigid spheres for simplicity, but this does not reflect the real physical situation. In this paper, we utilized high-precision force sensors and high-speed imaging technology to experimentally investigate the impact dynamic of droplet impingement on a dry wall. The temporal evolution of force, the associated morphology changes and their relationship during collisions were analyzed systematically, we also elucidated the physical mechanisms underlying flow phenomenon. An unified and accurate mechanical model were established for droplet impingement, providing guidance for related engineering designs.

1 Introduction

The collision between a liquid drop and a dry substrate is a dynamic, complicated physical process and widespread in nature, industrial and agricultural production, therefore it has received extensive attention from scientists. As a common basic scientific problem in many engineering applications, various outcome of drop impact [1,2,3] and complex phenomenon [4,5,6] during collision have been investigated widely. Nowadays we have already acquired a systematic understanding of these phenomenon and underling mechanisms.

The impact force exerted by droplets plays an important role in aerospace, energy, chemical and other technical fields: the take-off and landing of aircraft on water-polluted runways, aircraft/missiles flying in rainy circumstance, the erosion of soil and steam turbine blades. In these fields, the impact force is an essential factor for designing materials and structures and has become one of the most concerned issues for engineers. However, there is a relative scarcity of experimental studies that specifically examine the evolution of impact force, there are still numerous cognitive gaps.

Impact force is can be written as $F = f(\rho ,V_{0} ,D_{0} ,\mu ,\sigma ,t)$, where $\rho ,V_{0} ,D_{0} ,\sigma ,\mu$ is droplet density, impact velocity, droplet diameter, surface tension and dynamic viscosity respectively. A majority of the existing studies have primarily concentrated on analyzing the maximum impact force exerted by liquid drops [7,8,9]. Such measurements revealed that impact forces increase with the increase of droplet diameter, density and velocity, which is in accordance with intuitions. Dimensionless analysis indicates that $\tilde{F} = f(Re,We,\tau )$, where $Re = \rho V_{0} D/\mu ,We = \rho V_{0}^{2} D/\sigma ,$ $\tilde{F} = F/\rho V_{0}^{2} D_{0}^{2} ,\tau = V_{{0}} t/D_{0}$.

Zhang et al. [9] investigated the influence of Re and We on the peak forces systematically for the first time. It has been shown that the dimensionless impact force is hardly influenced by Weber number, whereas the Reynolds number is the key factor that affects impact behaviors, the conclusion were reinforced in subsequent studies [10, 11]. Therefore, impact force characteristics are only related to Reynolds number, namely:

$$\tilde{F} = f(Re,\tau )$$

(1)

Impact regimes can be categorized into two zones by Reynolds numbers: an inertial-dominated and a viscous regime. The value of critical Re for classification is between 200–280 [9,10,11]. When considering finite Reynolds numbers, the viscous effects cannot be disregarded, the force evolution varies with the Reynolds number, however, the normalized peak force collapse into a universe curve at high Re. Experimental results show that the impact force first increases to the peak value in a short time, and then decays to zero slowly during the impact duration. Philippi et al. [12] analyzed the semi-similar flow structure and deduced the normal force of liquid droplet impinging onto the wall at early times, the dimensionless form is given by:

$$\widetilde{F} = \frac{3\sqrt 6 }{2}\sqrt \tau$$

(2)

However, the analytical solution is only valid for a very short time ($\tau < 0.03$), it deviates rapidly from measurements as time increased. In the refined research of Mitchell et al. [10], they developed an empirical model that represents the entire force evolution of droplet impacts for high Re:

$$\tilde{F} = \sqrt {\frac{1000\pi \tau }{{243}}} e^{ - 10\tau /3}$$

(3)

However, the early time dynamics and exponential decay law phase in impact force require a theoretical understanding, as the maximum impact force occurs during the transition and the underlying physical mechanisms are still uncertain. With regard to impact force dynamics in viscous regime, Gordillo et al. [11] analyzed the early impact force characteristics at finite Re via a perturbation method based on Philippi et al. [12], they found that the propagation speed of the self-similar filed increases due to viscous boundary layer in early stages, nevertheless, our understanding of the entire evolution of drop impact forces in viscous regime is still limited and requires further exploration because of the complex nature of the problem. The existence of viscous boundary layer makes it harder to predict impact force in viscous regime theoretically and there is still a lack of model equation to represent the force profiles at finite Re.

In this paper, through analyzing the impact forces and the associated morphology change in impact process across different Reynolds numbers, we verify the interaction of inertial force, viscous force and surface tension and their roles played in force–time history, providing a theoretical support for exponential decay law at long times, and an uniform expression is constructed for the scaling of force profiles of different Reynolds numbers due to the similar physical mechanisms and therefore connect the two impact regimes successfully.

2 Experiments

2.1 Experimental Setup

Schematic diagram of the experimental apparatus is shown in Fig. 1. The experiment setup consists of four main systems: three-axis mobile platform, droplet generator and control device, visualization and force measurement system.

Fig. 1

Experimental apparatus; a Image, and b Schematic

Droplets were produced using a high-precision syringe pump, in which stainless steel needles of varying sizes were attached to control the diameter of the droplets. Droplets were released under gravity and impacted a 2.25 g, 25 mm diameter quartz glass plate, the surface was designed to be larger than the droplets’ maximum spreading diameter to ensure that the droplets did not flow out of the substrate. The time evolution of the droplet morphology during impact was recorded by a high-speed camera (FASTCAM AX200) at 20,000 frames per second. We used a PCB model 209C01 force transducer to measure force profiles. The sensitivity and force revolution of the transducer were 489.2 mV/N and 0.1 mN respectively. The signal acquisition equipment (NIPXLE-4499) was employed and the sample rate was set to 200 kHz.

In the paper, silicone oil with different viscosity (10–1000 cst) and distilled water were used to vary the physical properties of the impacting droplets. The inner and outer diameters of needle were 0.41 mm and 0.72 mm respectively. The fall height was adjusted to change the impact velocity, the silicon oil droplets were kept at 220 mm while water droplets were released from 530 mm approximately, the precise control of x–y plane location made sure of that the droplets fall on the center of the substrate. We synchronized the force evolution and high-speed imaging by triggering the high-speed camera using a photo-interrupter (KEYENCE LV-N10) activated by the falling drops, which enabled us to investigate the kinematics and dynamics of drop impacts simultaneously. Droplet diameter and impact velocity were determined from image analysis using 50 frames up to the last frame before impact. It’s noted that droplets deform as they fall and may not be a perfect spherical shape, droplet equivalent diameter is determined by $D_{0} = \sqrt[3]{{D_{h}^{2} D_{V} }}[3]$${{D_{h}^{2} D_{V} }}$, where $D_{h} ,D_{v}$ denotes droplet horizontal diameter, vertical diameter respectively. The vibration of system resulted from the droplet impact leads to low-amplitude, high-frequency oscillations in force signals, we applied s filter to eliminate small oscillations to enhance its reliability.

The test conditions are listed in Table 1. In the experiment, the temperature of room was controlled 25 °C, we mainly pay attention to the Reynolds number, the advantage of silicon oil lies in its nearly constant surface tension and thus the Weber numbers are kept constant (approximately 36). The accuracy of the experiments was validated by conducting a comparison between the measured impulse and the initial momentum of droplets. Theoretically, we have $\int_{0}^{\infty } {F(t)\;} dt = mV_{0} = \rho \cdot \frac{4}{3}\pi R^{3} V_{0}$, in dimensionless form:

$$\int_{0}^{\infty } {\tilde{F}(\tau )\;} d\tau = \frac{\pi }{6}$$

(4)

Table 1 Test conditions of the present experiments

3 Results and Discussion

3.1 Impact Dynamics

In this section the shape deformation of droplets and the corresponding force profiles are combined to investigate the impact dynamics and kinetics during collision. We mainly analyze the interplay of inertial force, viscous force and surface tension, as well as their varying significance in the impact process.

The normalized profiles are displayed in Fig. 2, the results reveal the existence of a viscous and inertial self-similar regime. Dimensionless peak force decreases rapidly at small Re number, when Re number increases above 200, all force profiles almost collapse into one curve, which has been confirmed by previous studies as well.

Fig. 2

Non-dimensional force profiles; a Viscous regime, and b Self-similar inertial regime

Figure 3 exhibits the corresponding temporal evolution of spreading factor and dimensionless apex height for test conditions, the two parameters are defined as $\beta = d/D_{0}$,$\tilde{h}_{max} = {{h_{max} } \mathord{\left/ {\vphantom {{h_{max} } {D_{0} }}} \right. \kern-0pt} {D_{0} }}$ respectively, where d denotes wetting length. $\tilde{h}_{max}$ is good choice to reflect impact force dynamics since it characterizes the momentum transfer rate between droplets and substrate. The results show that viscosity plays a significant role in preventing droplet deformation at finite Re, the formation of liquid sheet is limited due to high radical shear stress near the initial point of contact, the droplet can be approximated as a rigid ball in the scenario where the liquid viscosity approaches infinite, therefore leading to a shorter impact duration and a greater peak force.

Fig. 3

Non-dimensional morphology change; a Normalized apex height, and b Spreading factor

The impact behavior of droplets is affected by inertial force, viscous force and surface tension, which is a complex dynamic phenomenon. The entire impingement process can be divided into early kinetic phase and later spreading phase. In kinetic phase, inertial force dominates the initial stage of collision, droplet can be regarded as a truncated sphere in early stages, the apex height follows the same trend regardless of the Reynolds number; viscosity is a key factor in the later spreading phase, the central height slows down gradually and approaches boundary thickness, central height spends less time reaching steady state and remains unchanged, although the surface tension is responsible for retraction and spreading behaviors such as maximum spreading factor, it exerts no evident effects on impact forces.

3.2 Impact Force Model

Regardless of Reynolds number, the drop keeps falling at the initial-impact velocity in early phases as if it had not experienced any impact at all and early pre-peak force is proportional to the square root of time, this unusual phenomenon can be attributed to the finite propagation speed of the self-similar physical fields at high Reynolds numbers [12], the existence of the viscous boundary layer makes similar pressure field travels faster than that in the inviscid flow, thus the apex velocity deviates in advance, which induces a faster rise of impact force and a shorter peak time [11].

In the later stages of deformation, the inertial force weakens, impact force can be well approximated with an exponential decay, which is verified by several researchers [10, 13]. Here we provide a theoretical support for the conclusion briefly by analogy. Considerating that the surface tension and viscous force dominate the behaviors of impacting liquid drops at long times, which can be equivalent to a spring and a damper respectively in post-peak stages (see Fig. 4). Furthermore, we have acknowledged that force profiles are independent of We numbers and it has little effect on impact forces. Therefore the influence of surface tension during decline stage can be ignored.

Fig. 4

The scheme of force model at long times

For the spring-damper model, the Newton’s second law can be expressed as: $m\frac{dV}{{dt}} + kV = 0$, where m and k are the mass of the droplet and damping factor respectively. Simple dimensional analyses indicts that $m \sim \rho D_{0}^{3} ,k \sim \mu D_{0}$.The impact force exerts on the wall is considered equal to the damping force, thus we have: $\frac{{d\tilde{F}}}{d\tau } \sim - \frac{{D_{0} k}}{{U_{0} m}}\tilde{F} \sim - \frac{\mu }{{\rho U_{0} D_{0} }}\tilde{F}$. In other words, $\frac{{d\tilde{F}}}{d\tau } = - c\tilde{F}$, where c is a function of Reynolds numbers. At given Re, we have:

$$\int_{{\tilde{F}_{decay} }}^{{\tilde{F}}} {\frac{{d\tilde{F}}}{{\tilde{F}}}} = \int_{{\tau_{decay} }}^{\tau } { - c(Re)d\tau }$$

(5)

$\tau_{decay}$_,$\tilde{F}_{decay}$ _{are recession moment and the corresponding dimensionless impact force respectively,} integrating both sides of the equation by separation of variables:

$$\tilde{F}(\tau ) = \tilde{F}_{decay} * e^{{ - c(\tau - \tau_{decay} )}}$$

(6)

Equation (6) indicts the exponential decay law of impact force at long times successfully, furthermore, it predicts that impact force declines faster at relatively low Re result from the increase of viscosity and therefore the force curves exhibit a shorter non-dimensional time duration. The speculation is in line with measurements, which proves the effectiveness of the analog damper model. In self-similar regime, droplet impact dynamics is almost independent of Re and c gradually approaches constant.

In spite of the fact that a self-similar regime and a viscous regime which can be classified by dimensionless peak force, from the perspective of the impact process, the evolution of impact force exhibits similar trends and there are same physical mechanisms across different Re numbers during collision: early impact force can be scaled as the square root of time and long time exponential decay. Therefore the empirical model equation form proposed in reference [10] used for inertial impact is still applicable to finite Reynolds numbers:

$$\tilde{F}(\tau ) = c_{1} \sqrt \tau e^{{ - c_{2} \tau }}$$

(7)

where the physical meanings represented by $c_{1} ,c_{2}$ can be regarded as the propagation speed of the similar field in the early stage and the decay rate at long times, and the two parameters vary with Re because of the existence of boundary layer.

In early stages, $\tilde{F} \sim c_{1} \sqrt \tau$, the initial-impact theory developed by Gordillo et al. [11] reveals that early impact force can be scaled as $1/\sqrt {Re}$, and gives the coefficient theoretically using a perturbation method, namely:

$$c_{1} = \frac{3\sqrt 6 }{2}(1 + \frac{8\sqrt 6 }{{3\pi^{3/2} }}\frac{1}{{Re^{1/2} }})$$

(8)

The parameter $c_{2}$ can be determined by the fact that the impulse is equal to the initial momentum of the droplet numerically (see Eq. (4)), we then fit the parameter quantitatively as follows:

$$c_{2} = \frac{10}{3}(1 + \frac{0.825}{{Re^{1/2} }})$$

(9)

We can see that the two parameters decrease with the increase of Reynolds number and then gradually tend to constant values, which is in line with our cognition. In order to demonstrate the validity of the force formula to represent the evolution of impact force, the comparison of peak force and corresponding peak time over a wide range of Reynolds numbers are exhibited in Fig. 5, the results reveal that the model equation are in good well with experiments very well when $Re > 10$.

Fig. 5

Comparison of peak force and corresponding peak time over a wide range of Reynolds numbers

4 Conclusion

We investigate the influence of the Reynolds numbers on the impact dynamics of liquid drops impingement and divide our discussion into two parts: the inertia-dominated kinetic phase and long decay regime. An analogous damper model characterizing viscosity-dominant effect at sufficient long times is put forward to explain the exponential decay law successfully.

Based on the fact that impact force is proportional to the square root of time in early stages and obeys exponential decay law at long times regardless of Reynolds number, the differences lies in that it rises more rapid and decays faster at finite Re. The similar physical mechanisms underlying flow phenomenon across different Re are explained and thus allow us to develop an unified semi-empirical force model to represent the entire force evolution for various Reynolds numbers，providing guidance for related engineering designs.