Numerical study of aircraft wake vortex evolution near ground in stable atmospheric boundary layer

Mengda LIN,Weixi HUANG,Zhaoshun ZHANG,Chunxiao XU,Guixiang CUI

School of Aerospace,Tsinghua University,Beijing 100084,China

Numerical study of aircraft wake vortex evolution near ground in stable atmospheric boundary layer

Mengda LIN,Weixi HUANG,Zhaoshun ZHANG,Chunxiao XU,Guixiang CUI*

School of Aerospace,Tsinghua University,Beijing 100084,China

Aerodynamics; Aircraft; Aircraft wake vortex; Large eddy simulation; Stable atmosphere boundary layer

The evolutions of aircraft wake vortices near ground in stable atmospheric boundary layer are studied by Large Eddy Simulation(LES).The sensitivity of vortex evolution to the Monin-Obukhov(M-O)scale is studied for the first time.The results indicate that increasing stability leads to longer lifetimes of upwind vortices,while downwind vortices will decay faster due to a stronger crosswind shear under stable conditions.Based on these results,an empirical model of the vortex lifetime as a function of 10-m-high crosswind and the M-O scale is summarized.This model can provide an estimate of the upper boundary of the vortex lifetime according to the realtime crosswind and atmospheric stability.In addition,the lateral translation of vortices is also inspected.The results show that vortices can travel a furthest distance of 722 m in the currently studied parameter range.This result is meaningful to safety analysis of airports that have parallel runways.

1.Introduction

During an approach to an airport,the wake vortex of a leading aircraft poses danger to a following one.Therefore,it is necessary to study the wake vortex behavior in the atmospheric background.Fast-time wake prediction models have been developed by scholars.1–4In the early phase of the approach,the aircraft wake vortex is generated far from the ground,which is the so-called Out-of-Ground-Effect(OGE)condition.The wake vortex will turn into a rapid decay in tens of seconds as a result of the development of linear instability5,6due to the multi-reaction of the vortex pair.Furthermore,the vortex pair will leave the flight path by lateral displacement due to the crosswind,or by descending under the induction of each other,both of which contribute to the safety of the following aircraft.

In the final approach,the wake vortex is generated near the ground.Compared to the OGE condition,the In Ground Effect(IGE)condition requires more concerns.In the ground effect,image vortices will increase the separation of the vortex pair,7which weakens the interaction between the vortex pair and decreases the vortex decay rate.Besides,the vortices will rebound in the ground effect7and come back to the glide slope.These factors increase the wake-encounter risk for the following aircraft.

One of the early IGE wake prediction models was developed by Proctor et al.8They revealed that the vortices turn into a rapid decay soon after the rebound from the ground.In addition,the evolution of vortex circulation after the rebound is described with an exponential model based on Large Eddy Simulation(LES)results.Except for the circulation,the displacement of the vortices was modeled by Robins et al.7In their work,the separation of the vortex pair is modeled by bringing in image vortices below the ground,and the rebound is modeled by introducing secondary vortices.The in fluence of the atmospheric background was taken into consideration by Holzäpfel and Steen.9In their model,the decay rate of vortex circulation is related to the 10-m-height crosswind velocity.The newest model is the Deterministic wake Vortex Model(DVM)developed by De Visscher et al.in 201310,in which Holzäpfel’s model9is used to predict the vortex circulation,and the vortex displacement is predicted by the method of discrete vortices,in which vortex particles are released from the ground to simulate ground induced secondary vortices.11,12

However,the in fluence of atmospheric stability has not been considered in the aforementioned near-ground models.In an unstable boundary layer(or convective boundary layer),vortices will deform and decay rapidly under updraft and downdraft velocities,which is to the bene fit of the following aircraft’s safety.13The influence of a stable boundary layer will be much more complex.On one hand,the stable stratification can suppress the atmospheric turbulence,which will lead to a longer wake vortex lifetime.On the other hand,the strati fication will promote the vortex decay by buoyancy effects.14–16Therefore,it is necessary to study the wake vortex decay in the stable atmospheric boundary layer to improve wake prediction models.

In the current OGE wake vortex prediction models,the B runt–Väisälä(BV)frequency and the Eddy Dissipation Rate(EDR)are used as the parameters describing stability and atmospheric turbulence,respectively.1–3In the IGE condition,these two parameters are not enough because wind shear and ambient turbulence kinetic energy can also influence the wake vortex evolution.However,it is impractical to bring all these atmospheric parameters into a fast prediction model because it is difficult to establish an empirical model with so many independent variables.In this paper,the Monin-Obukhov(M-O)scale L17is chosen as the parameter to measure the atmospheric stability.On one hand,in the IGE condition,vortices are generated in the atmospheric surface layer(below 60 m),where the BV frequency and the EDR are not mutually independent and can be related to the M-O scale.17According to the M-O simulation theory,atmospheric information including wind and potential temperature pro file and the ambient turbulence can be estimated from the 10-meter-height crosswind V10and the M-O scale L.In this way,the number of independent atmospheric parameters is prominently reduced.On the other hand,a model using the M-O scale may be more practical because the M-O scale can be easily estimated from the wind speed and the net radiation index18,19which can be provided by the airport control tower without additional observing facilities.In this paper,LES is applied to the wake vortex decay process in the stable atmospheric boundary layer.The vortex evolution is studied under different V10and L,and simulation results are discussed and modeled.The paper is organized as follows.The numerical method and the physical model are introduced in Section 2.The results are displayed and discussed in Section 3.Finally,main conclusions are summarized in Section 4.

2.Methodology

2.1.Numerical methods

The LES in this paper solves the Boussinesq-approximated Navier-Stokes equations.

where uiand ujare the velocity component and xiand xjare the coordinate,with p the pressure,ρ the density,t the time.ν and νtare the molecular kinematic viscosity and sub-grid scale viscosity,respectively.In the equation of the potential temperature θ,the turbulent Prandtl number Prtis set to 0.72.The Lagrangian dynamic sub-grid model20is used.

These equations are solved numerically on a movable,nonstaggered Cartesian grid which is automatically adapted to the real-time flow field to obtain a higher resolution in a large gradient region such as the vortex cores with a limited total grid number.The self-adaptive algorithm is based on Gnoffo and Nakahashi’s spring analogy method.21,22The Finite Volume Method(FVM)is applied for the spatial discretization and a fourth-order Runge-Kutta integration is used in the time advancement.The details of the numerical method are described in Refs.23,24This method has been proven reliable in both wake vortex simulation18and micro-scale atmospheric boundary layer LES25,26

2.2.Setup of the initial field

The initial field is combined with the atmospheric background and a pair of wake vortices.In the background atmosphere,the pro file of wind V and the potential temperature θ in the surface layer are initialized with the M-O simulation theory27as

where vτand θτare the friction velocity and friction temperature,respectively.L is the M-O scale and θ0=290 K is the reference temperature.κ=0.4 is the Karman constant,and g=9.8 m/s2is the gravity acceleration.z0is the roughness length.At an altitude z>2L,the logarithmic and polynomial approximation suggested by Zilitinkevich28is applied.

The constants C1and C2are set to 5.9 and-0.6 respectively to ensure the continuity of V(or θ)and its vertical gradient at z=2L.H is the boundary layer height,which is estimated by Arya29as

where f=10-4s-1is the Coriolis parameter.When the M-O scale L is large,the stable form of Eq.(7)will overestimate H.30Therefore,the following estimation of H is used in this work:

Above H,the wind speed or potential temperature is set to a constant(if H is smaller than the vertical domain size)as

The initialization of the ambient turbulence is similar to Misaka et al.’s work.31The turbulence field is obtained with Rogallo’s method32to match a modi fied Von Karman spectrum33as

where k is wave number and kkolis the Kolmogorov wave number de fined by(ε/ν3)1/4,in which ε is the EDR.kpis the wave number where the spectrum reaches its peak value,and 1/kpstands for the turbulence integral length.The undetermined variables kpand K0are determined iteratively by a given EDR ε and the Turbulent Kinetic Energy(TKE)kefrom Eq.(10)and the following equations:

For a given 10-m-height wind velocity V10and M-O scale L,the basic parameters can be calculated as

The EDR and TKE are obtained at the flight altitude(the initial altitude of wake vortices)zias30

Eq.(14)has also been used in NASA’s Aircraft Vortex Spacing System(AVOSS)fast-time wake prediction models.30,3

Fig.1 shows the wake vortex roll-up process with the evolution of the normalized axial vorticity|ωx*|=|ωxt0|in the plane perpendicular to the flight direction.In Fig.1,t*is the non-dimensional time normalized with the characteristic time t0=b0/w0,where w0=C0/(2πb0)is the initial descent speed of vortices.The lateral and vertical coordinate y and z are normalized by y*=y/b0and z*=z/b0respectively.After the setup of the atmospheric background,a vortex sheet with a width the same as the wingspan B is generated as shown in Fig.1(a).The total circulation C0is distributed along the vortex sheet according to the elliptically-loaded wing assumption,and then the vortex sheet begins to roll up into a pair of wake vortices separated by b0=(π/4)B under self-induction,which is in agreement with the elliptically-loaded wing assumption(Fig.1(d)).The validation of this vortex generation method is described in detail by Lin et al.34In this paper,the wake vortex of a landing B747-400,a heavy aircraft,is simulated,with B=64.3 m and C0=500 m2/s.The vortex separation is b0=(π/4)B=50.5 m and the initial descent speed of vortices is w0=C0/(2πb0)=1.58 m/s,with the characteristic time t0=b0/w0=32.0 s.

2.3.Setup of simulation cases

Fig.2 shows the setup of the simulation domain.x,y,and z stand for the flight direction(or axial direction of the wake vortex),spanwise direction,and vertical direction,respectively.The domain size is set to Lx×Ly×Lz=8b0×7b0×2b0with the grid number being 360×352×100.A uniform grid space of Δx=0.022b0is applied in the axial direction,while Δy and Δz are set to 0.0074b0near the vortices and 0.052b0away from the vortices.The axial domain size is set to 8b0to recognize the potential long wave instability.5The vortex generation position is shown in Fig.2(b).Periodic boundary condition is applied to the axial and lateral directions,and non-slipping condition is applied to the ground.The top boundary is treated as a first-kind boundary condition where u=0,w=0,and v=v(2b0)according to Eq.(4).The vertical domain size is set to 2b0in balance of reducing the computational work and reducing the in fluence to the vortices from the top boundary Fig.3.

Nine cases with different 10-meter-height crosswind velocitiesand stability parameters L*=L/b0are simulated.In neutral conditions,L*=∞.Therefore,1/L*=b0/L is used instead and 1/L*=0 stands for a neutral condition.The basic parameters of all the cases are listed in Table 1,and the vertical pro files of V and θ are shown in,in which z*=z/b0is the normalized altitude.N is the averaged BV frequency between heights b0/2 and b0and is normalized withand k*=are the normalized EDR and TKE calculated at the height of b0from Eq.(14).As shown in Table 1,N*increases as 1/L*grows.For a givenε*changes slightly with increasing 1/L*,which indicates that the EDR is well correlated withHowever,the TKE k*decreases as the stability increases because the turbulence inte-gral scale is reduced by the stable strati fication.Besides,V(b0),the crosswind speed at the height z=b0appears higher in stable conditions,for the stable strati fication suppresses the vertical turbulent transport of momentum resulting in a higher vertical velocity gradient.In this paper,the headwind(or tailwind)is set to zero,which is conservative for safety because an attendance of the headwind(tailwind)will increase the atmospheric turbulence and accelerate the decay of wake vortices.

Before the cases in Table 1 are simulated,the IDaho Falls(IDF)B-757 Run 9 case35is simulated numerically and compared to the measured data as a verification of the current model.IDF B757 Run9 is one of the experiments carried out by Federal Aviation Administration(FAA)of USA to study the behaviors of B-757 and B-767 wakes.This case is chosen because the real-time atmospheric data including crosswind and potential temperature pro files are recorded in detail.36The measured wind and potential temperature data are shown in Fig.4.The pro files obtained from the current model(Eqs.(4),(5))with L=16.6 m and u*=0.119 are also shown.The largest deviation above 10 m is approximately 0.5 m/s for crosswind and 0.5 K for potential temperature.At an altitude below 10 m,the deviation is larger because the M-O simulation theory is not appropriate in this region.

Table 1 List of cases.

The case is simulated in a domain of Lx×Ly×Lz=240 m×300 m×105 m with the vortex parameters C0=365 m2/s and b0=30 m for a B757,and the vortex generation altitude is 70 m,the same as the measurement.The non dimensional mesh resolution is the same as those in Cases 1–9.In the axial direction,a uniform grid space of Δx=0.022b0=0.66 m is applied,while Δy and Δz are set to 0.0074b0or 0.222 m near the vortices and 0.052b0or 1.56 m away from the vortices.The total mesh point number is 366×326×120.The initial ambient atmosphere including wind and potential temperature field and the atmospheric turbulence are generated with the model introduced in Section 2.2.The modeled pro files rather than the measured data are used in order to examine the rationality of predicting the vortex behavior with a modeled atmospheric background.

3.Results and discussion

3.1.IDF B757 run 9 case

The vortex strength is measured by its circulation,which is calculated as a function of radius as

The r1–r2radius-averaged circulation is de fined as

In this case,the 0–10 m averaged circulation C0–10mis calculated for comparison with the measurement.3The evolution of the simulated circulation is shown in Fig.5 along with the measurement.It can be seen that the Down Wind(DW)vortex decays much faster than the Up Wind(UW)one,and the circulation falls to 100 m2/s in one minute,which is in agreement with the measured data.The result of a finer mesh(Δx=0.015b0=0.45 m,Δy= Δz=0.005b0near the vortex center,and Nx×Ny×Nz=366×326×120)is also shown in Fig.5.It can be seen that the vortex circulation decay rate is slightly reduced by the finer mesh,but the unset time of rapid decay is not affected by the increased resolution,which indicates that the mesh resolution applied in this paper is appropriate.The earlier decay of the DW vortex can be the result of crosswind shear.In Doligalski’s review paper,37it has been pointed out that crosswind can support the formation of a secondary vorticity at the DW vortex and attenuates it at the UW vortex.These secondary vorticities can be seen in the iso-surface of λ2=-1.038(Fig.6),from which a large amount of secondary vorticities can be observed around the DW vortex.The mechanism of this can be explained by the initiate Raleigh instability.When the crosswind pro file reacts to the DW vortex’s tangential velocity pro file,the original circulation pro file C(r)is decreased(Fig.7).The circulation at larger radii will be reduced more than that of smaller radii,resulting in the circulation decreasing as the radius increasing,which will active the initiate Raleigh instability.In Fig.8(a),the evolution of the vortex altitude is compared to the measurement.The simulation reports an earlier rebound of the DW vortex,and a less degree of rebound for the UW vortex.This difference can be due to the deviation of the modeled wind pro file in Fig.4,because the vortex rebound is sensitive to the second derivative of ambient crosswind,and a slight difference in the initial crosswind pro file can lead to a considerable deviation.39,40The lateral displacement is shown in Fig.8(b).The simulation results are in good agreement with the measurement for both vortices.The results of a finer mesh are also shown in Fig.8,which are almost the same as those of the current mesh except that the UW vortex descends faster in the finer mesh(Fig.8(a))due to the slower decay of the vortex strength(Fig.5).

In the current subsection,it is proven feasible to predict the vortex circulation and lateral position with modeled wind and temperature pro files,while the prediction of the vortex altitude is less reliable.

3.2.Results of cases 1–9

3.2.1.Decay of vortex strength

The temporal evolutions of the vortex circulation and altitude in Cases 1–9 are shown in Fig.9.The results are grouped according toand the 5–15 m averaged circulation is calculated to be compatible with existing models.3,4The time is normalized with t0,and the circulation is normalized with C0.In the evolution of the circulation,a two-phase decay can be obviously observed1with a slow decay phase followed by a rapid decay phase.The onset time of the rapid decay can be estimated byapproximately,whereis the normalized time when a vortex reaches its lowest altitude.8

The result shows that a strongereads to a faster decay of the vortex strength,which is reasonable because a strongerusually corresponds to a stronger ambient turbulence(Table 1).In addition,the crosswind leads to an asymmetry of UW and DW vortices.The DW vortex will rebound higher and decays faster than the UW vortex as shown in Fig.10,and this difference increases as the crosswind enhances.

The influence of the stability parameter 1/L*can be revealed by the inter comparison of cases with the sameFor the UW vortices,a larger 1/L*considerably decreases the decay rate and leads to a longer vortex lifetime.This can be explained by the combined action of stronger crosswind, which attenuates the secondary vorticity near the UW vortex,and reduced ambient TKE(Table 1).However,this effect is not obvious in cases withBecause the crosswind shear and the ambient turbulence are both weak,the turbulence generated by the wake vortex itself during the roll-up process or the later evolution will dominate the vortex decay,and the ambient conditions become less important.For the lifetime of DW vortices,the positive correlation with 1/L*is reversed,because increasing 1/L*means a stronger crosswind shear,which has been demonstrated in Section 3.1 to accelerate the decay of DW vortices.Although lower TKE can reduce the vortex decay rate,it can be seen that the crosswind shear plays a leading role.

The influence of buoyancy should also be discussed.As the stability degree increases,the Brunt–Vaisala frequency N*will also increase(Table 1),which can accelerate the vortex decay.15An additional Case 6b is simulated with all conditions the same as those of Case 6except that the vertical gradient of the potential temperature is removed by setting a constant θ.The circulation decay of Case 6b is shown in Fig.9(b).For the upwind vortex,after t*=3,Case 6b starts to deviate from Case 6.For the downwind vortex,the deviation occurs after t*=1.5,which means that the decay rate is underestimated by ignoring the buoyancy effect.However,this deviation is not crucial.This can explain why increasing N*does not reverse the positive correlation between 1/L*and the lifetime of UW vortices.

where A is a constant relative to the equivalent viscosity.The vortex lifetimeis de fined as the time when the vortex first decays to 0.2C0=100 m2/s.The threshold is set to 0.2C0because it is the maximum circulation a medium aircraft such as B737 and A320 can encounter without danger.41Thencan be written as

The onset time of the rapid decayand the vortex lifetimefor both UW and DW vortices are listed in Table 2.The relationship betweenandunder different 1/L*is shown in Fig.11,whereis set negative for DW vortices.

where

where A1is a constant that stands forin a static atmosphere,in which the vortex will decay under only its rollup induced turbulence and the ground effect.A2stands for the asymmetry of UW and DW vortices,and it is obvious that increasing 1/L*will enhance this asymmetry.The coefficient A3is also positively correlated with 1/L*,which means that the vortex lifetime will be more sensitive to the change ofunder stronger stability degrees.In addition,as the vortex lifetime should not be shorter than that in the static atmosphere,Eq.(19)is modified as

Table 2 and for both UW and DW vortices in all cases.

Table 2 and for both UW and DW vortices in all cases.

Case V10*,DW 1 0.5 0.0 9.0 8.5 1.75 1.50 2 0.5 0.5 9.5 6.8 1.75 1.50 3 0.5 1.5 10.6 6.9 1.75 1.50 4 1.0 0.0 5.0 4.5 1.60 1.25 5 1.0 0.5 8.5 4.0 1.60 1.25 6 1.0 1.5 9.8 3.4 1.70 1.25 7 2.0 0.0 3.1 3.0 1.40 1.20 8 2.0 0.5 3.6 2.4 1.40 1.20 9 2.0 1.5 5.1 1.6 2.00 0.75*1/L* T0.2*,UW T0.2*,DW T00*,UW T00

3.2.2.Lateral translation of vortices

Fig.13 shows the lateral movement y*=y/b0of vortices.The curves are cut off at(see Fig.9).Whenis larger,the vortices leave the current runway(y*=0)rapidly and allow the following aircraft approaching at a shorter time separation.The most dangerous situation is observed in Case 1in which the UW vortex does not leave y*=0 until t*=5.2(t=166 s).This is because the UW vortex’s image under the ground induces a backward moving of the UW vortex against the crosswind.This situation improves when 1/L*increases because of a larger(Table 1).

Although the lateral traveling speed is closely related tothe furthest travel distance is not positively correlated withbecause of the variation of the vortex lifetime.The furthest travel distances of the DW vortices range from 6.5 to 8.5b0(330–430 m)for all cases.This range is narrow because the greater traveling speed is offset by the shorter vortex lifetime.The range is much wider for the UW vortices.The furthest traveling distance is observed for the UW vortex in Case 6which is as far as 14.3b0or 722 m.In cases with the crosswindthe vortices have a much shorter lifetime so that they can travel no further than 14.3b0although the traveling speed is faster.That indicates that the wake vortex can pose potential danger to a parallel runway separated less than 722 m away.

4.Conclusions

The evolutions of aircraft wake vortices in near-ground conditions are studied with LES.The influences of crosswind and the M-O length L are investigated.Numerical and physical models are firstly examined by the IDF B757 Run 9 case.Then nine cases with different crosswinds and M-O scales are simulated.The results are summarized as follows.

(1)A faster crosswind speed leads to a shorter lifetime for both UW and DW vortices.Increasing stability(a larger 1/L*)will enlarge the difference between the decay rates of UW and DW vortices.The UW vortices decay slower in stable atmosphere because of the crosswind shear and lower TKE,while the DW vortices decay faster due to a stronger crosswind shear.Although lower TKE can reduce the decay rate for both vortices,the crosswind shear dominates in the stable atmosphere.

(2)Although increasing 1/L*leads to a greater BV frequency,this does not reverse the positive correlation between 1/L*and the UW vortex lifetime.It indicates that the buoyancy effect does not dominate in vortex decay near ground.The vortex decay will be more sensitive to the ambient turbulence and crosswind shear.

(3)The furthest lateral translation distance is observed in a case with strong stability and moderate crosswind.For a stronger crosswind,the vortex decays quickly under strong ambient turbulence,while for a weaker crosswind,the vortex travels too slow to reach a further position.The furthest lateral translation is observed on a UW vortex rather than on a DW one,because UW vortices usually have a longer lifetime.

(4)In addition,the lifetimes for both UW and DW vortices are modeled under neutral and stable conditions.The empirical model shows the vortex life time can be as long as 9t0under weak crosswind and strong stability.It is worth mentioning that some factors that can accelerate vortex decay are not taken into account in this model.For example,the headwind(or tailwind)will increase the ambient turbulence and leads to a shorter vortex lifetime.Besides,the obstacles on the ground may make a vortex decay earlier.Even so,this model provides an upper boundary of the vortex lifetime.

Acknowledgement

The work was supported by Boeing-COMAC Aviation Energy Conservation and Emissions Reduction Technology Center(AECER).

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1 July 2016;revised 12 November 2016;accepted 14 December 2016

Available online 9 September 2017

Ⓒ2017 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is an open access article under the CCBY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

*Corresponding author.

E-mail address:cgx@tsinghua.edu.cn(G.CUI).

Peer review under responsibility of Editorial Committee of CJA.