# FlowHaptics: Mid-Air Haptic Representation of Liquid Flow
## 3 Methodology
### Fluid Flow Simulation  
The magnitude of the force that the skin receives from flowing water is proportional to the square of the velocity. Moreover, since the pressure distribution and force change  
Figure 2: The sensation of fluid flow is a combination of the drag force and skin friction. We added additional air tubes on the left and right to emphasize skin friction on both sides of the finger.  
with time, we should be able to change the value for the velocity of the flowing water in real time.  
The methods to describe the motion of a fluid can be divided into the Euler method [19] and the Lagrange method [20]. The Euler method is a method in which a space is divided into regular grids, and the physical quantity of the fluid is updated every time step, while each regular grid has its own physical quantity such as the flow velocity. In the Lagrange method, a fluid is expressed as a set of particles or moving grids, and the position of each particle/grid is tracked every time step. Since we would like to estimate the velocity in real time, we selected the Euler method approach. Though there are several Computational Fluid Dynamics (CFD) simulation software based on the Euler method and the Navier-Stokes equation, such as OpenFOAM [21] or ANSYSCFX [22], this method is computationally expensive and cannot be performed in real time for velocity estimation.  
In this research, we used FluidNet [4], which is a machine-learning-accelerated fluid simulation software using the Euler method to estimate the flow velocity of a fluid at the antinode and both sides of the finger.  
The input data were derived using a 2D finger cross-section model to generate the training fluid data for the simulation. The output data were the real-time estimated fluid data. In this research, in order to generalize the prediction of the airflow in any environment, we used a pretrained model from FluidNet, which was derived from 50 standard models from the NTU 3D Model Database [23] using random scaling, rotation, and translation of the patterns in the simulation. As a result, the pretrained model can perform the general prediction of the fluid flow for any model. The fluid simulation with one finger and with three fingers is presented in Figures 3a and 4a, respectively.  
Since we expected our application to be able to estimate the flow in real time based on different finger configurations, we utilized a machine-learning-based approach over the traditional physics simulation approach. The outcome from the simulation was not perfectly symmetric because the flow from FluidNet is an estimated version of the flow from the physics simulation approach. This was a tradeoff between the speed and accuracy of the machine-learning-based flow estimation approach. Furthermore, the FluidNet model we used also included wavelet turbulence when predicting the flow so that the flow result was similar to the outdoor environment where air disturbance is expected to occur. As a result, we could obtain the maximum and minimum values of the velocity of the flow on the left, right, and center parts of each finger. Based on the normalized value of the estimated flow velocity, the air jet flow was produced.  
In FluidNet's 2D simulation, the virtual space is pixelized, and information on the flow velocity, volume, and pressure of the fluid is held for each pixel. The simulation was performed on a PC with Ubuntu 16.04 LTS with a GTX 1080 Nvidia Graphic Card. After  
Figure 3: (**a**) Fluid simulation for 1 finger and (**b**) the velocity measured in the fluid simulation on the left, center, and right parts of the middle finger.  
training, the fluid simulation could be adapted based on the participants' interaction in real time.