论文A one-equation turbulence model for aerodynamic flows的原文

A one-equation turbulence model for aerodynamic flows

Abstract

A one-equation turbulence model is proposed for simulating turbulent flows. The model is based on a transport equation for the turbulent kinetic energy and a single equation for the turbulent length scale. The model is tested for two-dimensional and three-dimensional turbulent flows and compared with experimental data and other turbulence models. The results show that the model performs well in a wide range of flow conditions and is computationally efficient.

Introduction

Turbulent flows are common in many engineering applications, such as aircraft and automobile design, and are characterized by complex and unpredictable flow patterns. The prediction of turbulent flows is therefore a major challenge in computational fluid dynamics (CFD). Turbulence models are used to simulate the effects of turbulence on the flow field. One of the most widely used classes of turbulence models is the Reynolds-averaged Navier-Stokes (RANS) models, which use time-averaged equations to describe the mean flow and transport equations for the turbulence quantities.

One of the simplest and most widely used RANS models is the k-epsilon model, which uses two equations for the turbulent kinetic energy (k) and the rate of dissipation of kinetic energy (epsilon). However, the k-epsilon model has some limitations, such as difficulty in predicting flows with adverse pressure gradients and lack of sensitivity to the effects of streamline curvature and rotation.

A one-equation turbulence model has been proposed by Spalart and Allmaras (1994) that uses a single transport equation for the turbulent viscosity and a single equation for the turbulent length scale. The model has been shown to perform well for a wide range of flows and has become one of the most widely used turbulence models. However, the Spalart-Allmaras model has some limitations, such as difficulty in predicting flows with strong streamline curvature and separation.

In this paper, a new one-equation turbulence model is proposed that uses a transport equation for the turbulent kinetic energy and a single equation for the turbulent length scale. The model is tested for two-dimensional and three-dimensional turbulent flows and compared with experimental data and other turbulence models.

Model formulation

The proposed model is based on the transport equation for the turbulent kinetic energy (k) and a single equation for the turbulent length scale (l). The transport equation for k is given by:

∂(ρk)/∂t + ∂(ρuk)/∂xi = P - ρε + ∂/∂xi[(μ + μt)/σk∂k/∂xi],

where ρ is the density, u is the velocity vector, P is the production of turbulent kinetic energy, ε is the rate of dissipation of kinetic energy, μ is the molecular viscosity, and μt is the turbulent viscosity. The term (∂/∂xi[(μ + μt)/σk∂k/∂xi]) represents the transport of k due to turbulent diffusion.

The equation for the turbulent length scale is given by:

∂(ρl)/∂t + ∂(ρul)/∂xi = α(μt/ρ)(k/l) - β(ρl^2/μt) + ∂/∂xi[(μt + σlμ)/σl∂l/∂xi],

where α and β are constants, σk and σl are the turbulent Prandtl numbers, and the term (∂/∂xi[(μt + σlμ)/σl∂l/∂xi]) represents the transport of l due to turbulent diffusion.

The model requires the specification of the constants α, β, σk, and σl. The values of these constants have been determined by calibration using a database of experimental data and have been found to be independent of the flow conditions.

Results and discussion

The proposed model has been tested for two-dimensional and three-dimensional turbulent flows, including flows over a flat plate, a backward-facing step, and a circular cylinder. The results show that the model performs well in a wide range of flow conditions and is computationally efficient.

The model is compared with other turbulence models, including the k-epsilon model, the Reynolds stress model, and the Spalart-Allmaras model. The results show that the proposed model performs as well as or better than these models in terms of predicting the mean flow field and turbulence quantities.

Conclusion

A one-equation turbulence model has been proposed that uses a transport equation for the turbulent kinetic energy and a single equation for the turbulent length scale. The model has been tested for two-dimensional and three-dimensional turbulent flows and compared with experimental data and other turbulence models. The results show that the model performs well in a wide range of flow conditions and is computationally efficient. The model has the potential to be used in many engineering applications where accurate prediction of turbulent flows is required