Direct numerical simulation of trailing vortices in homogeneous turbulence

Abstract

Among the various meteorological parameters acting on the lifespan of the wake vortices, the atmospheric turbulence is one of major importance. Direct numerical simulations are one of the best tools to study turbulence even if the Reynolds number used in the computations is far bellow the Reynolds number of the phenomenon. Respecting as much as possible the scales of the real flow field, namely the integral length scale of atmospheric turbulence and the r.m.s. velocity, phenomena explaining the decay and the lifetime of trailing vortices are studied. It is shown in this paper that the interaction between the vortices and the ambient turbulence creates ring of vorticity through a tridimensional mechanism. The first effect of these rings is to generate axial velocities as great than velocities induced by the vortices. Furthermore, the vortices are curved and the vortex pair could become unstable. Nomenclat ure d0 = initial spacing between vortices. Kd = wavenumber of the maximum in dissipation spectrum Ke = wavennmber of the maximum in energy spectrum Le = length scale associated to Kf rc = core radius Re = Reynolds number t = time, f * = trc/V0 Tc = turbulence turn-over time TV = vortex time scale a' = r.m.s. turbulent velocity V0 = maximal vortex velocity x,y,z = coordinates x = horizontal cross-plane direction y = vertical cross-plane direction z — axial direction e = turbulence dissipation rate A = integral length of turbulence v — kinematic viscosity fl! = vorticity vector Q* = |Q| rc/V0 = l/2rot({7) •Researcher, email: rissoQiraft.enseeiht.fr ^Research Engineer, Member AIAA, email: corjonQcerfacs.fr. *Research Engineer, Member AIAA, email: stoesselOifp.fr. Copyright ©1996 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. INTRODUCTION Acting for the DGAC (Direction Generale de PAviation Civile), the STNA (Service Technique de la Navigation Aerienne) has defined objectives to study wake vortices. The first goal is to decrease the delays at landing and take off with an advisory system for ATC (Air Traffic Control) based on meteorological parameters forecasting. This system would allow to reduce momentarily the separations to specified Radar minima. In the long term, an automatic system optimizing separations between aircraft would be integrated in an advanced ATC system and more realistic categorization criteria would be defined than the MTOW (Maximum Take-Off Weight). Both these systems require to take into account the effects of atmospheric turbulence on the lifespan of the wake vortices. For the more simple system only turbulence classes are necessary while accurate forecasting are needed for the optimal system. It was decided to develop first semi-empirical models which allow to estimate the formation of the trailing vortices, to simply take into account the effect of meteorological conditions on the vortices behavior and to study the interaction between a vortex and an aircraft. This engineering model was presented in a former paper and is called VORTEX. From an operational point of view, it is necessary to develop simple models which could be used in real time. But these models require a large amount of validation because many of their parameters are difficult to fix. Therefore it was decided to develop not only operational models but also more realistic tools based on a Navier-Stokes solver. For certain phenomena important in the wake vortex proccess, these solvers allow cheaper validation than experiments and could provide crucial informations to improve for example the modeling of the vortex decay process. The ambient air turbulence is one of the major effects influencing the decay of the trailing vortices. The operational model uses the results of Donaldson and Bilanin which relate exponentionally the decay of the vortices and the r.m.s. velocity of the turbulence (see equation 10). The first aim of this work is to study this relationship by comparing the results of direct numerical simulations made with NTMIX3D and the simple model. Secondly, it is important to understand what is exactly the process of trailing vortices decay. Several authors have shown that the most important, vortex-decay mechanism works from outside of the vortex to the center. NUMERICAL TOOL The present NTMIX3D code is solving 3D unsteady compressible Navier-Stokes equations with multi-species, transport and chemistry. Obviously, for this study, multispecies and chemistry capabilities are not used. A highly accurate finite difference method over a cartesian grid is used to fully handle most of the different scales in the 3D flow: spatial derivatives are computed with a 6 order compact scheme (Fade scheme) which has the property of being low-dispersive and non-dissipative. These schemes even if they are quite expensive are allowing some quasispectral accuracy. Time advancement is achieved with a 3 order Runge-Kutta method. The code is supporting accurately many different type of boundary conditions thanks to the use of the NSCBC method . In particular, nonreflecting open boundary conditions as well as adiabatic or isothermal solid walls are implemented. The cost in CPU and memory of a 3D simulation is so high that it requires the use of MPP computers to be affordable. Thus, the code has been written directly in parallel with a Message Passing Paradigm. Specific domain decomposition methods have been developped for the parallel treatment of compact schemes and are allowing the availability of a highly efficient version of the code on a wide range of distributed memory parallel computers such as IBM SP2, Fujitsu VPP500 and Cray T3D. The simulation presented in the following has required 10 hours 20 min on a IBM SP2 (32 ns/GridPoint/Timestep).