Advances in Simulations of Incompressible Liquids
By Jennifer Garland, Applied Physics, 2021
This article was originally published as part of Issue 35: Motion.
The Navier-Stokes equations are a set of differential equations that directly arise from Newton’s Second Law and are used to simulate the motion of fluids. Differential equations, commonly seen in electrical circuits, population change, and heat transfer problems, contain derivatives and are used to model changes over time. Computers can be used to implement numerical methods which approximate solutions to these equations. However, a switch from continuous to discrete is necessary because they can only process discrete data. The resulting fluid motion models calculate values for velocity and pressure at every point in space and time, and, for example, can recreate how fluids move from high to low pressure.
Researchers in computational fluid dynamics are attempting to create new methods of simulation for incompressible liquids using the Navier-Stokes equations. Incompressible liquids, such as water or brake fluid, have constant density under different pressures. While easier to model than compressible fluids, problems arise in situations with complex geometries. New techniques are needed to improve existing approaches that tend to be expensive, time-consuming, and error-prone.
The simulation very accurately predicts laminar flow, where fluids travel smoothly in layers, with approximately one hundred times less error than previous simulations.
Recently, a new scheme for solving the Navier-Stokes equations for incompressible liquids in two dimensions was created by a collaboration between researchers at the Università di Bari, the National Research Saratov State University, and the Joint Institute for Nuclear Research. The simulation very accurately predicts laminar flow, where fluids travel smoothly in layers, with approximately one hundred times less error than previous simulations. Laminar flow is simpler to model than turbulent flow, which requires supercomputers due to the presence of vortices like those seen in whirling liquids. Nonetheless, any simulation used for particle interactions, including laminar flow, needs a lot of computing power, especially since many particles are needed to model the smooth surface of a liquid.
One researcher from the Russia-Italy team sums up the significance of their precise method of numerical solution, stating, “other schemes that are known and studied in our paper have substantially larger errors that grow rather quickly with time, while our error is very small and grows very slowly, it is almost a constant…We compared it to standard numerical methods by other authors and saw that our scheme is better.” After additional comparisons with exact results, the next steps are to expand the method to 3D motion.
Improved fluid simulations have the potential to bleed over into endless applications. Atmospheric flow and oceanic motion models are important for accurately predicting the weather and natural disasters. In some cases, even air can be treated as an incompressible fluid, leading to advances in rocketry, airplanes, and vehicles. In addition, fluids are notoriously hard to animate, so better mathematical approximations will enhance visuals used for entertainment as well as practical purposes. Luckily, the future points towards increasingly accurate models of our seemingly unpredictable world.
App. Math. And Comp. (2017). Doi: 10.1016/j.amc.2017.06.037