Using Transport Equation Of Subgrid Scale Eddy Viscosity For Large Eddy Simulation And Its Application In Turbulent Flows

Large eddy simulation(LES)can accurately simulate higher Reynolds numbers and more complex turbulent flows on a computational grid with a spatial resolution that is much coarser than that of direct numerical simulations,and obtain more accurate prediction results and more turbulence informations than Reynolds-averaged Navier–Stokes equations method.These characteristics of large eddy simulation are very important for turbulence research and engineering design and application analysis.Therefore,with the development of highperformance computers,LES has become a hot research topic in numerical simulation of turbulence and a means of predicting actual flow.In LES,only large-scale structures with grid-level and critical momentum are directly solved,and small-scale(subgrid-scale)structures dominated by dissipation and irrelevant momentum in turbulence are modeled by the subgrid(SGS)model.Therefore,reasonable and accurate SGS modeling has become the key to LES research.Although various SGS models have been proposed by researchers and have achieved some success,until today there have been no satisfactory models,especially for complex flows with non-equilibrium characteristics,such as laminar-turbulent transitions and flow separation.The research topic of this paper is the basic research on SGS model modeling,especially to improve the prediction accuracy of SGS model for complex flows with nonequilibrium characteristics.For example,in airfoil flow,the calculated grid width varies greatly along the flow lines flowing near the surface of the airfoil.However,most of the existing SGS models do not consider the effect of the mainflow change on the length scale on the SGS stress.In this paper,the effect of this grid width change on the eddy viscosity coefficient of SGS model is theoretically discussed for the first time,and this effect is quantified theoretically.Then,this paper applies the filtered material derivative operator to the eddy viscosity model and implicitly calculates the SGS kinetic energy.After a series of advanced mathematical conversions,a class of SGS model considering the change of the mainflow grid width is derived and presented.This newly developed SGS model is called the transport equation of eddy viscosity model(TEM).This model does not need to use any near-wall damping function and test filtering operation,and theoretically guarantees that the SGS stress has a certain anisotropic property.Further,this article theoretically extends the TEM model and obtains a simplified version suitable for engineering requirements,a more accurate version of the dynamic parameters,and an improved version based on the new characteristic length model.In order to verify the newly proposed TEM model,this paper conducted a prior test and a posteriori test on the planar turbulent channel flow.The test results show that the TEM model is effective;compared with the classic dynamic Smagorinsky model and the traditional one-equation model,the TEM model shows higher prediction accuracy.In particular,in this paper,the LES with turbulent flow in a flat channel under a non-uniform grid is confirmed from the perspective of numerical experiments.The non-uniformity of the grid along the flow direction(even in uniform turbulence)will cause the non-physical behavior of turbulence.It is proved that the TEM model that takes this effect into account has higher prediction accuracy than a general SGS model that does not take this effect into account.Next,this article compared several commonly used SGS models(standard Smagorinsky model,dynamic Smagorinsky model,standard transport equation of SGS kinetic energy model,and dynamic one-equation model)and our developed improved dynamic transport equation of SGS kinetic energy model for a planar diffuser flow.The results showed that the improved dynamic SGS kinetic energy transport equation model has the best predictive performance,followed by the dynamic one-equation model,then followed by the dynamic Smagorinsky model and the standard SGS kinetic energy transfer equation model,and the standard Smagorinsky model performing the worst.At the same time,this paper also compared the prediction performance of the TEM model with the standard SGS kinetic energy transport equation model and the improved dynamic SGS kinetic energy transport equation model.The effectiveness and superiority of the TEM model in simulating turbulent flows with non-equilibrium characteristics were confirmed again.Finally,this paper applied the newly proposed TEM model to the large eddy simulation of the flow around the NACA0012 airfoil(a typical main wing shape of aircraft),and calculated and analyzed the noise around the NACA0012 airfoil based on the aerodynamic results of the large eddy simulation,then the trailing edge of NACA0012 airfoil was serrated,and the noise reduction mechanism of the trailing edge of NACA0012 airfoil was researched.In order to solve the problem of weak compressibility,this paper modified the time advancement method of the conventional incompressible solver.In order to consider the non-equilibrium characteristics of airfoil flow,the newly derived TEM model was incorporated into this weakly compressible scheme.To improve the prediction accuracy of aerodynamic noise,a new noise source model was derived based on the recombination of continuity equation and Navier-Stokes equation.The results of the aerodynamic field around the NACA0012 airfoil showed that it is necessary to consider the compressibility even in the low Mach number flow.This is especially important to properly reproduce the pressure fluctuations and vortex structures around the airfoil.During the transition from the laminar to the turbulent boundary layer,the production and dissipation of SGS kinetic energy has obvious local non-equilibrium characteristics.The TEM model proposed in this paper can accurately reproduce the local non-equilibrium characteristics of SGS kinetic energy.The sound field results around the NACA0012 airfoil showed that: the material derivative of the velocity divergence plays a leading role as the sound source term;the sound source distribution in the acoustic model derived in this paper and the classic sound source models(such as Lighthill model and Powell model)havs the same distribution of sound sources;and the sound pressure level predicted based on the above LES and the sound source model derived in this paper agrees reasonably with the experimental data.This paper also studied the effect of the increase of Mach number on the sound field around the airfoil.The results showed that as the Mach number increases,the position of the peak frequency of the sound pressure level of the aerodynamic noise tends to move from the high frequency region to the low frequency region,and compared with the classic sound source model,the sound source model derived in this paper can more accurately reproduce the impact of this increase in Mach number.The calculation results of NACA0012 airfoil flow with trailing edge sawtooth show that the trailing edge sawtooth hinders the growth of the vortex in the wingspan direction and promotes the development of the vortex in the vicinity of the trailing edge and the wake,which significantly reduces the intensity of pressure fluctuations near the trailing edge.Finally,the effect of reducing trailing edge noise is achieved.