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The near-field dynamics of an aircraft wake is studied by means of temporal large-eddy simulations, with and without considering the effect of engine jets. In the absence of jets, the simulations showed the roll-up of the initial vorticity sheet shed by the wing and the occurrence of short-wavelength instability in a pair of primary co-rotating vortices. The main consequence of the instability is the modification of the internal structure of the vortex, compared to the two-dimensional stable behavior. The presence of engine jets affects the roll-up of the vorticity sheet and causes an enlargement of the final merged vortex core compared to the case without jets.

The analysis of the near-field dynamics of an aircraft wake and its interaction with an engine jet exhaust is of primary interest in applications covering a wide spectrum of aerospace technology. Examples range from the characterization of the structure of persistent and hazardous trailing vortices during take-off and landing phases [

The phenomenology of the aircraft wake is usually described in terms of the downstream distance z a c behind the aircraft wing: the wake evolution can be divided into four characteristic regions as proposed by Jacquin et al. [

− the near-field wake ( z a c / c = O ( 1 ) , c being the mean aerodynamic wing chord) is characterized by the roll-up of the vorticity sheet emanating from the wing trailing edge and the presence of several concentrated vortices (wing tip, flap, nacelle, fuselage, horizontal tailplane, ∙∙∙);

− the extended near-field wake ( z a c / B = O ( 1 ) ) where the roll-up completes and the primary co-rotating vortices interact and ultimately merge;

− the mid- to far-field wake ( z a c / B = O ( 10 ) ) is generally composed of a pair of symmetric counter-rotating vortices that descend in the atmosphere due to their mutual velocity induction. In this region the vortices develop linear instabilities such as the Crow instability [

− the dispersion regime ( z a c / B = O ( 100 ) ) is characterized by the final decay of the vortex system―after the occurrence of the instabilities in the far-field―which is controlled by the local ambient conditions (i.e. atmospheric turbulence and thermal stratification).

The present study is an attempt to understand the dynamics of the wake flow in the near- to the extended near-field, its intrinsic instabilities and the effects of jet engine exhaust on the development of the wake vortex structure.

In the initial stages of its formation, the aircraft wake is a complex vortex system composed of multiple interacting counter- and co-rotating vortices. The presence of two co-rotating vortices is a common feature in the wake of an aircraft in high-lift configuration. A pair of distinct two-dimensional co-rotating vortices experiences merging [

Depending on their separation distance and the respective core sizes, two co-rotating vortices can be unstable due to the strain field that each vortex exerts on the other. This results in a short-wavelength instability that is characterized by a three-dimensional sinusoidal deformation of the core and exists both for counter-rotating [

The qualitative features of the interaction between an engine jet and a single wake vortex were illustrated by Miake-Lye et al. [

This suggests the possibility to extend the analysis to different configurations with complex distributions of the initial vorticity. Experimental investigations of the interaction between a jet and the vorticity sheet shed by the wing were performed by Wang and Zaman [

Due to the restriction of direct numerical simulations to low Reynolds number flows, large-eddy simulations are a natural candidate to represent these inherently unsteady phenomena at a high Reynolds number, and has then been used in the present study.

The paper is organized as follows: the governing equations and the modeling are first described; the results of the 2D stable and unstable near-field dynamics as well as its interaction with an engine jet exhaust follow; conclusions and the main outcomes are finally given.

In the LES approach the Navier-Stokes equations are filtered spatially, such that any variable ϕ ( x ) may be decomposed into a resolved (or large scale) part ϕ ( x ) ¯ and a non-resolved (or subgrid-scale) part ϕ ″ ( x ) , with ϕ ( x ) = ϕ ( x ) ¯ + ϕ ″ ( x ) . This procedure may be obtained by a convolution integral of the variable with a filter function depending on a filter width Δ . Practically, the filter width is simply given by the computational mesh cell size Δ x . For compressible flows, Favre-filtered variables, defined as ϕ ( x ) = ϕ ˜ ( x ) + ϕ ′ ( x ) , with ϕ ˜ = ρ ϕ ¯ / ρ ¯ are used. The dimensionless Favre-filtered equations are:

∂ ρ ¯ ∂ t + ∂ ( ρ ¯ u ˜ j ) ∂ x j = 0 (1)

∂ ( ρ ¯ u ˜ i ) ∂ t + ∂ ( ρ ¯ u ˜ i u ˜ j ) ∂ x j + ∂ p ¯ ∂ x i = 1 R e ∂ τ ˜ i j ∂ x j + ∂ σ i j ∂ x j (2)

∂ ( ρ ¯ E ˜ ) ∂ t + ∂ [ ( ρ ¯ E ˜ + p ¯ ) u ˜ j ] ∂ x j = 1 R e ∂ τ ˜ i j u ˜ i ∂ x j + ∂ σ i j u ˜ i ∂ x j − 1 R e P r C p ∂ q ˜ j ∂ x j − ∂ Q j ∂ x j (3)

where the subgrid-scale (SGS) stress tensor σ i j = − ( ρ u i u j ¯ − ρ ¯ u ˜ i u ˜ j ) , and the SGS heat flux Q j = ρ C p T u j ¯ − ρ ¯ C p T ˜ u ˜ j are to be modeled, and where the following classical approximations have been made:

− The Favre-filtered shear stress tensor is identified with the filtered shear stress tensor

− The Favre-filtered heat flux is identified with the filtered heat flux

− The filtered kinetic energy term ρ K u in the energy equation is approximated by ρ ¯ K ˜ u ˜ j − σ i j u ˜ j , where K = 1 / 2 u i u i is the kinetic energy.

The Favre-filtered passive scalar equation is:

∂ ( ρ ¯ Y ˜ ) ∂ t + ∂ ( ρ ¯ Y ˜ u ˜ j ) ∂ x j = 1 R e S c ∂ ∂ x j ( μ ∂ Y ˜ ∂ x j ) + ∂ ξ j ∂ x j (4)

The SGS momentum, σ i j , the SGS heat flux, Q j , and the SGS scalar flux, ξ j , are modeled through subgrid-scale eddy-viscosity concept:

σ i j − 1 3 σ k k δ i j = − 2 μ s g s ( S ˜ i j − 1 3 δ i j S ˜ k k ) (5)

Q j = − μ s g s C p P r t ∂ Θ ∂ x j (6)

ξ j = − μ s g s S c t ∂ Y ˜ ∂ x j (7)

where μ s g s is the SGS dynamic viscosity, S ˜ i j is the large scale strain rate tensor and S c t is the turbulent Schmidt number; while P r t is the turbulent Prandtl number, defining the modified temperature Θ = T ˜ − 1 2 ρ ¯ C v σ k k , where C v is the specific heat at constant volume.

The SGS viscosity model is based on the Structure Function model [

F ¯ 2 ( x , Δ , t ) = 〈 ‖ u ˜ ( x + r , t ) − u ˜ ( x , t ) ‖ 〉 ‖ r ‖ = Δ (8)

where Δ is the cutoff length, and where 〈 〉 denotes spatial averaging, here over the sphere of radius Δ . As the information brought by the model is local in space, it leads to a poor estimation of the kinetic energy at the cutoff, which may be improved by a suitable filtering [

ν s g s = ν s g s ( x , Δ , t ) = α ( n ) Δ F 2 ¯ ( n ) ( x , Δ , t ) (9)

where the superscript (n) indicates that the filter has been applied n times. The value of α used here is α ( 3 ) = 0.00084 . The Structure Function model formulation of Métais and Lesieur [

u ≡ u d i m / a r e f = u d i m / a d i m a d i m / a r e f = M a d i m / a r e f

where M is the local Mach number. Hence, for a flow with small temperature variations ( a d i m / a r e f ), the non-dimensional velocity corresponds effectively to the local Mach number, u ~ M .

Numerical Tool NTMIX3D and Initialization ProcedureThe numerical code [

Temporal simulations were carried out to analyze the evolution of the near-field wake dynamics and mixing. This is based on the assumption of a locally parallel flow, which means that the gradients of the mean flow in the axial direction are neglected over the short distance corresponding to the axial dimension of the simulation domain. Instabilities developing in the simulated flow are automatically of convective nature, and we may not capture absolute instabilities [

Periodic boundary conditions are used along the vortex/jet axis z and in the upper-lower boundaries (y-direction), while symmetry boundary conditions were used in the spanwise x-direction. The boundary x = 0 in

those vortices was significantly reduced by placing the boundary sufficiently far from the region where the vortex dynamics takes place (as verified a posteriori in the simulations).

The initial condition for the wing-generated wake used in the computations is obtained by 5-hole probe data from a windtunnel measurement behind a half-model of a typical transport aircraft. The data comes from a measurement plane located at z w ≈ 0.03 , where z w = z a c / B is the downstream distance normalized by the wing span of the model B. In the temporal simulations, the downstream distance is simply related to the physical time of the wake by z a c = t × u a c where u a c is the free-stream speed of the model. As the size of the measurement plane was too small to simulate the wake vortex dynamics in the near-field to extended near-field, interpolation and extrapolation procedures were required. This procedure provided the initial condition on a computational grid containing the measurement window that was sufficiently large to avoid the influence of the domain boundaries. An interpolation routine was used to interpolate the measured velocity field on the mesh with the desired spatial resolution. Furthermore, an extrapolation routine was used to reconstruct the aerodynamic flow field outside of the measurement window. This was done by using analytical functions that describe the velocity field of the vortices outside of the measurement window, while minimizing the velocity gradients to avoid the generation of spurious vorticity at its borders. The resulting fields are denoted by u b a s e , v b a s e and w b a s e , respectively. As an example,

The Reynolds number of the measurements based on the circulation contained by one half of the wake has been retained for the numerical simulations, i.e. R e Γ = Γ / ν ≈ O ( 4 × 10 5 ) .

A two-dimensional computation was first carried out to study the near- to extended near-field dynamics. The size of the computational domain size used for this simulation is ( L x , L y ) ≈ ( 1.0 B ,1.6 B ) and the number of grid points is ( n x , n y ) = ( 601 , 915 ) -note that only half of the aircraft wake was simulated as described above.

For the sake of validation, the numerical results at z a c / B ≈ 1 (see

At z a c / B ≈ 1 , the two closely spaced vortices that are horizontally aligned at y * ≈ − 0.2 experience a strong interaction and finally coalesce. This occurs at z a c / B ≈ 1.9 and results in a vortex wake (

The dynamics of the primary co-rotating vortices is characterized by the merging process, which is generally a fast, two-dimensional and stable phenomenon as described in detail by Cerretelli et al. [

Meunier et al. [

Under certain circumstances, depending on the Reynolds number and the arrangement of the two co-rotating vortices ( a / b < ( a / b ) c r i t ), a system composed of a co-rotating vortex system may be subject to the development of the three-dimensional elliptical instability (see a recent review by Kerswell [

The remnants of the vorticity sheet and the concentrated vortices orbiting around the primary dipole of the wake system bring the co-rotating vortices rapidly together (through velocity induction) and ultimately cause the merging of the vortex system. This merging process is often fast in a real aircraft wake and usually stable in the absence of strong perturbations. One might therefore think to trigger the elliptical instability by injecting a significant amount of energy in the corresponding unstable mode using some forcing techniques. On the other hand, one of the motivations of this study was to understand if the elliptical instability can emerge “naturally” in an aircraft wake. Thus, instead of forcing selectively the elliptical instability mode by injecting strong perturbations, a weak random white noise is imposed to simulate the intrinsic dynamics of the flow that is susceptible to instabilities. The unstable wavelength corresponding to the elliptical instability being unknown a priori, the axial length of the computational domain was chosen to be significantly larger than the core size a, namely one wing span B = O ( 10 a ) (note that λ e l l = O ( a ) , see e.g. Kerswell [

The simulation were performed in a computational domain given by ( L x , L y , L z ) ≈ ( 1.0 B ,1.6 B ,1.0 B ) and a number of gridpoints in the three directions ( n x , n y , n z ) = ( 601 , 915 , 36 ) giving a total of approximately 20 × 10 6 gridpoints. This large number of gridpoints is necessary to capture the short-wavelength/elliptical instability with sufficient resolution to avoid numerical diffusion and damping effects on the development of the instability.

In order to trigger the flow instability, a weak random white noise is superimposed to the base flow as follows:

u ( x , y , z ) = u b a s e ( x , y , z ) ( 1 + A × r a n d ( x , y , z ) ) (10)

v ( x , y , z ) = v b a s e ( x , y , z ) ( 1 + A × r a n d ( x , y , z ) ) (11)

w ( x , y , z ) = w b a s e ( x , y , z ) ( 1 + A × r a n d ( x , y , z ) ) (12)

where A = 10 − 4 is the amplitude of the perturbations and rand a random generator with − 0.5 < r a n d ( x , y , z ) < 0.5 .

The global dynamics is initially governed by the same phenomena discussed in the previous Section, namely the roll-up of the vorticity sheet, leading to the generation of several concentrated vortices. Subsequently, a reduction of the number of vortices through the coalescence of co-rotating vortices is observed. However, some of the smaller vortical structures are deformed three-dimensionally, become turbulent and fall apart. At a downstream distance of approximately z a c / B ≈ 1.9 the vortex system is composed of two primary co-rotating vortices and a secondary vortex orbiting around it.

This is shown in

At a distance larger than z a c / B = 1.9 the secondary vortex (III) shows a sinusoidal short-wavelength deformation of the vortex core indicating the elliptical instability (see

makes it unstable. The occurrence of the elliptic instability is suspected because of the short-wavelength disturbance, which is of the order of the core size. Furthermore, a spectral decomposition of the kinetic energy in the axial direction shows the exponential amplification of the energy contained by the Fourier mode k = 14 which corresponds to the wavelength of the elliptical instability (see

At a downstream distance larger than z a c / B ≈ 4.5 the dynamics of the system is mainly governed by the behaviour of the two co-rotating vortices I and II. They rotate around each other and, due to the relatively small core ratios ( ( a / b ) < ( a / b ) c r i t ), the elliptical instability has time to develop. A clear sinusoidal deformation of the vortex two vortex cores is observed in

The occurrence of the elliptical instability is confirmed by a spectral analysis. The spectral decomposition in the axial direction shows the amplification of the energy contained by the most unstable mode corresponding to the elliptical wavelength, which is depicted for vortex I and II in

rate of the elliptical instability occurring in each vortex as a function of the circulation Γ I and Γ I I , the vortex core radii a I and a I I , and the Reynolds number Γ / ν . Evaluating the parameters a I , I I and Γ I , I I of the resulting vortex system at a distance z a c / B ≈ 1.9 and inserting them in the Formula (6.1) of Le Dizès et al. [

As a final remark, the development of the elliptic instability during the merging process modifies the final vortex structure. In particular, as shown in

In the previous Section, the formation and evolution of an aircraft wake was analyzed by focusing on the development of the intrinsic elliptic instability of the vortex system. In this Section, a different situation is considered where the vortex sheet interacts with an engine jet. The aim is first to find out whether and how the jet modifies the roll-up process of the vorticity sheet and, secondly, to analyze the mixing of an exhaust passive scalar in the wake.

The computational domain is ( L x , L y , L z ) ≈ ( 1.0 B ,1.6 B ,0.072 B ) and consists of ( n x , n y , n z ) = ( 395 , 601 , 36 ) gridpoints which are equally spaced in each of the three directions. The first two dimensions are the same as the previous Section, while the axial length corresponds to the most unstable jet velocity profile in the simulations of Michalke and Hermann [

profile, F ( r ) = tanh [ 1 / 4 r j / θ ( r / r j − r j / r ) ] , where r j = 0.072 B is the jet radius, r = x 2 + y 2 is the radial distance from the center and θ = 10 r j is the

displacement thickness of the jet (see Paoli et al. [

w j ( r ) = 1 2 [ ( w e + w a ) − ( w e − w a ) F ( r ) ] (13)

Y j ( r ) = 1 2 [ ( Y e + Y a ) − ( Y e − Y a ) F ( r ) ] (14)

where subscripts a and e indicate, respectively, the free-stream conditions and the conditions at the center of the exhaust jet. In the present study, there is no co-flow, w a = 0 and Y a = 0 , while the exhaust velocity and scalar are, respectively, w e = 0.18 and Y e = 1 .

The velocity field in Equation (13) is then added to Equation (12) to trigger the jet instability and its transition to turbulence, while, at the same time, it interacts with the vorticity sheet. Two main features characterizes this interaction as shown in

magnitude at a late stage of the roll-up process, z a c / B ≈ 6.96 . First, the jet is entrained around the multiple vortex system arising from the roll-up. Secondly, the jet continuously exchanges its axial momentum with the sheet by vortex stretching, causing the formation of complex three-dimensional vortical structures. Note that the figure displays the vorticity level ω = ω max / e 1.256 ( ω max being the instantaneous maximum vorticity) which identifies the core of a axisymmetric Lamb-Oseen vortex. Indeed, the core of the merged vortex is clearly visible as well as the fine-scale eddies, remnants of the initial sheet vorticity and the jet turbulence, which are still wrapping around the core. This was already observed in previous simulations of jet/vortex interactions [

This strong momentum exchange and the turbulent diffusion induced by the jet affects the tangential velocity field of the final vortex. For example, the isocontour lines reported in

To conclude the analysis of the jet/sheet interaction, the isocontour lines of the passive scalar Y are reported in

that the combined effects of turbulent diffusion of the jet and its entrainment within the vortex sheet, enhances the dispersion of exhaust gases in the wake. This results in a larger plume area and causes a stronger dilution of the scalar field, which reaches a value of approximately 0.005, at a final stage of the merging process, compared to 0.476 in the the 2D laminar case.

The formation and near-field dynamics of an aircraft wake was studied by means of Large Eddy Simulations, using available experimental data of a wing-generated vorticity sheet. The roll-up process and the stability properties were first analyzed by imposing a random white noise and the occurrence of the short-wavelength instability in the principal co-rotating vortex pair was evidenced. This caused the unstable merging of the vortices and modified the core of the final vortex.

Furthermore, the roll-up and the formation of the wake were simulated by adding a model jet flow. This affected the dynamics of the primary co-rotating vortices during the roll-up, mainly by vortex stretching due to the momentum exchange with the vorticity sheet. The consequences for the merging were the decrease of the azimuthal velocity and the enlargement of the vortex core. Finally, the jet/sheet interaction affected the exhaust passive scalar dilution, leading to more efficient mixing and a larger plume area at the end of the merging process.

The authors wish to thank both CINES (Centre Informatique National de l’Enseignement Supérieur) and IDRIS (Institut du Développement et des Ressources en Informatique Scientifique) for providing the CPU hours. The authors also thank Airbus-Deutschland for providing the wind tunnel data.

Paoli, R. and Moet, H. (2018) Temporal Large-Eddy Simulations of the Near-Field of an Aircraft Wake. Open Journal of Fluid Dynamics, 8, 161-180. https://doi.org/10.4236/ojfd.2018.82012