Simulation the passage of a laminar flow through a local constriction in a pipe

This paper presents a study using discrete simulation methods for problems of spatial dynamics. A cellular automaton model of the gas flow, which was used in the work, is a discrete simulation model. Such methods make it possible to obtain relatively simple software implementations on modern supercomputers. The aim of the work is to check the possibility of using the studied model under conditions of flow in a straight pipe at a constant temperature in a sub-critical mode; under these conditions, the critical Reynolds number is several thousand. The simulation object is a gas flow passing in the space between parallel flat walls. Such a flow corresponds to a two-dimensional case, since the characteristics of a three-dimensional flow between two parallel planes and in a pipe with a circular cross section, which is not considered here, differ only in coefficients. The narrowing or variable distance between the planes in the three-dimensional case corresponds to the variable diameter of the two-dimensional pipe. The objective of the paper is to simulate the flow with the specified boundary conditions using a discrete cellular automaton model and compare it with the known results obtained using continuous models. Such a comparison is necessary in the framework of the study of the possibility of using a discrete model. The desired dependencies are the distribution of velocity and pressure along the direction of flow for various sizes of the constriction and various pressure gradients at the ends of the pipe. The model cover two-dimensional simulated space by hexagonal cells arranged in a regular pattern; each cell has six neighbors. The state of the cell is a vector of six integers denoting the number of discrete model particles with unit mass and unit velocity directed towards one of the six neighboring cells. The cellular automaton operates in synchronous mode using an iterative transition function. An iteration consists of two steps: collision and propagation. During a collision, the particle velocity vectors in each cell are changed, regardless of the states of other cells. During a propagation, each particle moves to one of the neighboring cells in the direction of its velocity vector. The simulation result is the velocity field and the pressure field obtained after the required number of iterations by the averaging method. The averaging neighborhood at each point constituting the fields is the set of cells whose centers are separated from this point no further than some averaging radius. The flow velocity at a point is the average velocity of the particles located in the cells of the averaging neighborhood. The gas pressure at a point is proportional to the concentration of particles located in the cells of the averaging neighborhood centered at this point. A software implementation of the model is made with a set of three modules: a boundary conditions constructor, a simulator, and a post-processor. The constructor in interactive mode allows you to create a cellular array with the required values of the states of each cell and place it in a file that is used by the simulator. The simulator performs a given number of iterations over the original cellular array. It is a CLI parallel program with the ability to run on a computing cluster. Parallelism is implemented by the method of domain decomposition of a cellular array using the MPI library. The post-processor averages the result obtained by the simulator and visualizes the averaged values in the form of velocity and pressure fields built either on the entire cellular array or on its selected area. Computer experiments were carried out with a cellular array of 5000 by 500 cells. The lumen of the constriction of the pipe varied from 100 to 400 cells. The inlet and outlet pressures also varied. To establish a stationary flow regime, 100 thousand iterations were carried out in each of the experiments. After that, the post-processor was launched. To calculate the average particle velocity, the averaging radius was chosen to be equal to twenty cells. To calculate the average concentration of particles, the averaging radius was chosen to be equal to five cells. The velocity and pressure fields of the gas flow in a two-dimensional pipe are obtained. Velocity and pressure distributions along the direction of flow are plotted for various sizes of constriction and various pressure gradients at the ends of the pipe. In the constricted part of the pipe, the flow velocity was maximum in each of the experiments. It can be seen from a comparison of the experiments that the flow velocity has an inverse dependence on the diameter of the constricted part of the pipe, which corresponds to the existing view on the process’ physics. So, the possibility of using the studied model under conditions of flow in a straight pipe at a constant temperature in a sub-critical mode with a critical Reynolds number of the order of several thousand was shown.