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For those who have been involved in compressible flow research for a while, the term “solutions” may be a bit of a buzz word. A few people even exclusively use it as shorthand to describe any of the numerical methods that are available for modelling incompressible flow, including Boussinesq’s method and Prandtl’s method. In terms of fluid dynamics, however, solutions refers to both the solutions to equations and the solution of problems. In this blog post we shall argue that using numerical methods is important because they provide an accurate – or nearly so – approximation to incompressible flow at free stream conditions. Numerical methods are simply models of equations. Boussinesq’s model for incompressible flow is an approximation to the Navier-Stokes equations, which are themselves incredibly accurate for all but the most hostile environments. However, numerical methods are not limited to incompressible flow, as they have been used with great success by computational fluid dynamics (CFD) practitioners to model compressible flows. For that reason, this blog post will focus on numerical methods that can accurately model compressible fluids over a range of Mach numbers. The first numerical method that will be covered is called gas turbine mixing layer simulation (GTMLS). It was developed by Dr. Harald Wilhelmi, professor of fluid dynamics in the Department of Mechanical Engineering at the University of Wisconsin-Madison, in collaboration with NASA. The gas turbine mixing layer simulation methodology was originally developed to simulate the mixing layer in a gas turbine engine. It has since been used to model compressible flows in many different applications, including jet engines, rocket engines, and spacecraft. As shown by Drs. Wilhelmi and Young in their 2008 paper “Gas Turbine Mixing Layer Simulation”, use of this method enables accurate modelling of compressible flow at much higher Mach numbers than other methods can achieve. This has resulted in the numerical approximation of the compressible flow equation, namely: where "ρ" is the density, "v" is the mean local velocity, and "P". The first term on the right hand side represents fluid kinetic energy flux. The second term represents potential energy flux. However, as we will see later, this method was not successful at modelling compressible flows with high Mach numbers. The first problem that needs to be solved in order to calculate this expression accurately is to obtain a solution for flow that includes a moving compressible flow along with a stationary compression region. The purpose of the moving compressible flow is to establish a slope in the pressure-density function, which will enable subsequent calculations. This is achieved by solving the coupled incompressible flow problem. The incompressible equations are used to obtain approximate solutions, but these should be supplemented by additional terms that are required for this purpose. One of these additional terms includes pressure gradient due to compressibility. The other important component needed in order to obtain an accurate solution for incompressible flow is the use of the Navier-Stokes equations along with empirical correction factors. These correction factors enable us to obtain a value for "P" at any given Mach number, corrected for compressibility effects. eccc085e13
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