A Review of Numerical Simulation and Analytical Calculation Methods for the Study of Flow over a Wedge

Abstract

The study of flow over a wedge holds immense significance in diverse domains, encompassing aerodynamics and heat transfer applications. This research paper presents a comprehensive investigation of essential flow parameters, including pressure, temperature, velocity, and shock wave characteristics. Moreover, it offers a comparative analysis between analytical and numerical approaches for studying flow over a wedge. Theoretical analyses involve calculating vital parameters like pressure ratio and temperature ratio using 1-D inviscid compressible flow equations coupled with conservation laws for energy, mass, and momentum. On the other hand, the numerical investigation employs Computational Fluid Dynamics (CFD) techniques, specifically the ANSYS Fluent CFD Code, to calculate a wide range of parameters and visualise the flow characteristics for the wedge geometry. In contrast, the ANSYS Fluent simulation employs 2-D inviscid compressible equations to conduct a detailed study of flow behaviour past the wedge geometry. Throughout the paper, the key flow parameters are meticulously analysed, taking into account both analytical and numerical results. The comparisons between these two methodologies enable a comprehensive understanding of the advantages and limitations of each approach. Additionally, ANSYS Fluent CFD simulations facilitate the visualization of intricate flow patterns around the wedge, providing valuable insights into the fluid dynamics of the system. This research paper contributes to the knowledge base by presenting a comprehensive review of numerical simulation and analytical calculation methods for the study of flow over a wedge. The insights gained from this analysis can aid researchers and engineers in selecting appropriate methodologies for specific flow scenarios, thus enhancing the efficiency and accuracy of future studies in this domain.

Introduction

I. NOMENCLATURE

θ    Wedge Deflection Angle

β    Wave Angle

M   Mach Number

p    Pressure

T   Temperature

γ    Specific heat ratio = 1.4 (for Air)

μ    Mach Angle

Subscript

1     Before the Shock wave in the Region 1

N1  Normal Component in the Region 1

2     After the Shock wave in the Region 2

N2  Normal Component in the Region 2

For simulation, the boundary conditions need to be specified at the various edges of the domain. In ANSYS Fluent, the "Pressure Far Field" boundary condition is used to simulate conditions at a region far away from the computational domain where the flow is approaching the boundary with a constant pressure. This boundary condition is often employed in external flow simulations where the exact pressure distribution is not known but the flow is assumed to be approaching the domain with a specific pressure value. The Pressure Far Field boundary condition helps simulate open boundary conditions effectively by specifying a reference pressure and inflow direction, allowing accurate representation of the inflow behaviour without the need for detailed information about the upstream conditions.

3)  Solver Setup

The solver setup for flow over a wedge demands careful consideration of the specific characteristics of the problem, such as Mach number, angle of attack, and boundary conditions. A well-configured solver setup not only ensures accurate simulation results but also contributes to a deeper understanding of the complex flow phenomena associated with wedge geometries. In ANSYS Fluent, the first step is to define the general setting for the density- or pressure-based solver, as both models have their own characteristics that can influence the result of simulation. Pressure-based solvers are mostly used for incompressible and weak compressible flows, while density-based solvers are mostly used for compressible flows [14].

After that, a specific turbulence model needs to be chosen for a better and more accurate result. Here, the inviscid viscous model is used, and the air properties are changed to ideal gas because during the higher Mach number, the air behaves like ideal gas because the air molecules do not have time to interact with each other due to high kinetic energy, so for the higher Mach number, the air properties have to be changed to ideal gas [15].

The boundary condition of pressure far field is applied because the region is assumed to be free stream. In the solution method, under spatial discretization select the second-order upwind scheme as the first-order upwind scheme does not accurately capture sharp gradients or shock waves as it uses the value of the variable at the cell face to estimate the value of the variable at the cell centre, while the second-order upwind scheme uses a weighted average of the values of the variable at the cell face and the cell centre to estimate the value of the variable at the cell centre. This can lead to more accurate results for flows with sharp gradients [16].

In the initialization, standard initialization was applied and computed form the far field, and the number of iterations changed to 4000 for the calculation.

After that, the flow parameters such as Mach, pressure, and Temperature contours are visualised, as shown in Figs. 8, 9, and 10, respectively, and the results are analysed in the next section.

Conclusion

In conclusion, this research project has produced a thorough examination of the complexities surrounding flow across a wedge, successfully merging the results of numerical simulation with analytical calculations. Using the computational power of ANSYS Fluent, the study meticulously simulated the complex dynamics of supersonic flow interactions with a Mach number of 3 and a 15? wedge-shaped structure. A wide range of flow parameters were subjected to meticulous examination and rigorous comparison with theoretically derived solutions sourced from oblique shock wave relationships. The striking agreement between the results of the numerical simulations and the analytical derivations highlights the effectiveness of both representational approaches for complex flow phenomena. The modelling technique clearly showed how oblique shock waves during supersonic encounters with the wedge had a transformative effect on flow characteristics. The numerical calculations showed this: the post-shock Mach number showed a distinct decline to 2.25 from the initial free stream Mach number of 3, as shown by the numerical simulations. This sharp compression caused by the oblique shock phenomenon is responsible for the simultaneous rise in temperature and pressure. This study supports the ability of numerical simulation to render precise forecasts while expertly capturing complex flow discontinuities. The comprehensive comparison of many parameters in Table 2\'s tabular portrayal of these results strengthens the validity of both techniques. This thorough comparison study demonstrates the reliability of numerical simulation as a tool for understanding flow dynamics, especially the complex behaviour brought on by oblique shock waves. In conclusion, the symbiotic integration of numerical simulation and analytical computation, as highlighted in this study, confirms their combined effectiveness in explaining the complex details of flow through a wedge. The knowledge gained here helps us better understand these flow phenomena and paves the way for future research into similar phenomena in a variety of engineering fields.

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Copyright

Copyright © 2023 Chintan R. Prajapati, Aniket P. Donga, Akshar D. Patel, Pranav R. Babariya. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.