The development of reynolds averaged navier stokes solver for a two dimensional compressible flow problem - M. Elkamel, Hasan Taher (2017)
The computational fluid dynamics represented by fluid dynamic science focuses on the way how to solve the flow problems numerically. The governing equation of fluid motion passing through an object flow can be presented in various forms depending on the assumption imposed to the flow problem in hand. Initially, in solving the flow problem passing through an object such as the flow passing through an aircraft, the flow is incompressible, irrotational, and inviscid flow. Resulting from the initial form of governing equation called the Navier-Stokes equations; the flow can be simplified as the Laplace equation. When the incompressible condition cannot be maintained, the compressibility effects have to be taken into account due to the increasing incoming velocity, while the inviscid and irrotational conditions are still maintained. The Navier-Stokes can be reduced to become a full potential equation. The Navier-Stokes equation becomes the Euler equations by ignoring the viscous effects. If the viscous effects are included, the presence of turbulent flow phenomena creates a small fluctuation to the flow variables resulting in the Navier-Stokes equation to reduce and become a Reynolds-averaged Navier-Stokes (RANS) equation. For instance, these various models of the governing equations had been formulated before the era of computer started. The manner on how to solve the flow problem according to the level of governing equations is based on the achievement of computer technology. In 1960, the aerodynamic problems were solved when the computer capability was limited, which led to the change of the Laplace equation by the method known as the Panel Method. As the computer power became more available, the aerodynamic problems were solved through the full potential equation. Further improvement in computing power made the aircraft designers since 1980 to use Euler equation as the governing equation of motion for the flow problem in hand. Continuous support gained from computer technology development has helped aircraft designers since 1990 by using the RANS equations in solving their flow problems. The success in the use of RANS equations depends on the manner in combining the numerical grid generation and scheme for discretizing the governing equation and turbulence model, which need to be provided in making the RANS equation solvable. In developing the RANS solver, the present research uses the unstructured grid for meshing the flow domain, combined with the Roe’s finite volume scheme for discretizing the RANS equation and Spalart-Allmaras for fulfilling the required turbulent modeling. For the purpose of validation, the result of the developed computer code was compared with the experimental result available in the literature and result through running the Fluent software. The validation was carried out by using airfoil NACA 0012 and RAE 2822. Both two airfoils have the experimental result in terms of distribution pressure coefficient along the airfoil surfaces at different angles of attacks and Mach numbers. The comparison result over these two airfoil models had found that the developed RANS solver was able to produce the results closed to the experimental result, as well as the Fluent software. The developed computer code was applied to further evaluate the aerodynamic airfoil characteristics NACA 4415 and Supercritical Airfoil 26a at various angles of attacks and Mach numbers. For the airfoil NACA 4415, the aerodynamic analysis were carried by treating the flow problem as inviscid flow problems while the other as viscous flow problems. In other words, the flow problems in hand were solved by the Euler and RANS solvers. As for the results of the pressure coefficient distribution along the airfoil surface, there was a significant difference between the result provided by the Euler and RANS solvers. While for the supercritical airfoil, the result of the developed computer code as RANS solver found the position of the shock wave strongly influenced by the angle of attacks as well as the Mach number. Combining Roe’s finite volume scheme, the Spalart-Allmaras turbulent model, and unstructured grid made RANS solver developed successfully. In addition, developing the code for RANS solver simultaneously develops the Euler solver. When viscous term was set up to zero, the RANS solver became Euler solver. Hence, the present work developed both the RANS and Euler solver.