In this paper, we propose a dimensionless numerical mesh-free model for the simulation of the compressible isothermal viscous flows. The novelty of this work consists to formulate the Navier-Stokes equations under a dimensionless form and to solve them by a high order mesh-free algorithm to simulate the compressible fluid flows. This algorithm combines a classical implicit Euler scheme, a high order continuation with the Moving Least Squares (MLS) and a homotopy transformation. The MLS approximation and implicit Euler scheme are used respectively for the spatial and temporal discretizations of dimensionless Navier-Stokes equations. The homotopy transformation serves to introduce in dimensionless Navier Stokes equations an arbitrary operator and a parameter without physical dimension. The obtained equations are solved by a high order continuation. The performance of the presented model is tested on the standard benchmark lid-driven cavity problem. Then, the Mach and Reynolds numbers effect is discussed. The obtained results are compared with those of the Finite Difference Method (FDM) coupled with an explicit Runge-Kutta (R-K) scheme and those of literature. This comparison reveals that the results of the dimensionless model are obtained with a less expensive CPU time compared to that of the other algorithms.

在本文中，我们提出了一个无因次数值无网格模型来模拟可压缩等温粘性流。这项工作的新颖性在于以无量纲形式表示Navier-Stokes方程，并通过高阶无网格算法模拟可压缩流体流动来求解它们。该算法结合了经典的隐式Euler方案，带有移动最小二乘法（MLS）的高阶连续和同伦变换。MLS近似和隐式Euler方案分别用于无量纲Navier-Stokes方程的空间和时间离散化。同伦变换用于在无量纲的Navier Stokes方程中引入任意运算符和无物理尺寸的参数。所获得的方程通过高阶连续求解。在标准基准盖驱动腔问题上测试了该模型的性能。然后，讨论了马赫数和雷诺数效应。将获得的结果与有限差分法（FDM）结合显式Runge-Kutta（RK）方案的结果进行比较，并与文献进行比较。这种比较表明，与其他算法相比，无量纲模型的结果使用较少的CPU时间获得。