Generally, problems in numerous fields such as fluid dynamics and aerospace are formulated based on partial differential equations. This makes it easy to analyze and solve such problems. Owing to their complexity, several techniques have been developed to simplify the solution of partial differential equations. Presently, smoothed particle hydrodynamics is used to solve these equations owing to its efficiency. For instance, it is a meshfree method thus suitable for situations with interfaces and represents the whole system with a set of particles containing both their chemical and physical properties. Unfortunately, smoothed particle hydrodynamics simulations involve approximations used to obtain the equations governing the dispersed particles that end up inaccurate in most cases due to issues like wavy profiles with a period equivalent to the separation distance between two adjacent particles. To this note, researchers have been looking for alternative corrective approximations capable of eliminating the wavy profiles and enhance the calculation of derivative values.
In a recently published literature, most of the traditionally used formulas have difficulties in achieving accurate first and second order derivatives, particularly for non-uniform particles distribution. Therefore, Taylors series expansion have been identified as an alternative and promising solution to bridge the gap. It is ideal for determining the first order derivative of chemical and physical properties under different particle arrangements.
Recently, Texas A&M University scientists: Mr. Yu Yang and Dr. Sy-bor Wen developed a Taylor series expansion based high accuracy meshfree method. Their main aim was to accurately calculate the first and second derivatives for different particle distribution conditions. Their research work is published in International Journal of Heat and Mass Transfer.
The authors observed that the developed method was capable of accurately providing the first order and second order in Laplacian and gradient calculations of fluid properties respectively. Consequently, it allowed direct simulations of heat transfer and fluid dynamics in incompressible fluids. On the other hand, the boundary conditions were directly determined accurately from the conservation equations at the interfaces of the solid and fluid as compared to the traditional meshfree and smoothed particle hydrodynamics simulations that relied on artificial repulsive boundary conditions.
According to Yu Yang and Sy-bor Wen, the developed method has significantly overcome most of the challenges associated with the traditional meshfree methods. For example, the resulting physical boundary conditions can be effectively used to prevent energy generation and non-physical momentum at the boundaries. It was necessary to investigate the importance of the method of heat transfer and fluid dynamics simulations. The simulation results obtained were similar to the theoretical ones with the small variations attributed to the assumptions used in the traditional methods which are ignored in this case.
Therefore, the study will significantly advance the solution of the partial differential equation not only in heat transfer and fluid dynamics but also in other disciplines as well due to its simplicity and efficiency. Furthermore, particle position rearrangement after a certain number of simulations can be used to correct the minimal variations of the developed Taylor series expansion based high accuracy meshfree method.
Yang, Y., & Wen, S. (2018). A new high accuracy meshfree method to directly simulate fluid dynamics and heat transfer of weakly compressible fluids. International Journal of Heat And Mass Transfer, 123, 25-39.Go To International Journal of Heat And Mass Transfer