Parameters of airplane and atmosphere.
Aircraft wake vortex is a pair of intensive counter‐rotating airflow generated by a flying aircraft. Wake vortex is one of the most dangerous hazards in aviation because it may cause a following aircraft to roll out of control, particularly during the taking off and landing phases. The real‐time detection of wake vortex is a frontier scientific problem emerging from many fields like aviation safety and atmospheric physics, and the dynamics and scattering characteristics of it remain as key problems to develop corresponding detection technologies. This chapter aims at presenting a simulation scheme for the dynamics of wake vortex under different weather conditions. For wake vortex generated in clear air, changes of the atmospheric dielectric constant produced by the density variation and water vapour variation are analysed; for wake vortex generated in rainy condition, the raindrop distribution in the wake vortex is also analysed. Both of them are essential for further analysing the scattering characteristics and developing new detection algorithms.
- wake vortex
- clear air
- wet weather
Wake vortex is an inevitable physical phenomenon that exists in the rear zone of a flying aircraft, which rotates intensively and has a complex structure. The wake vortex generated by a large aircraft could be very hazardous to aviation safety since it might cause a following aircraft to roll out of control, particularly during the departure and landing phases.
In Air Traffic Management (ATM) field, International Civil Aviation Organization (ICAO) established a series of flight interval rules. These rules can ensure the flight safety in most time, but they are too conservative. In order to reach a good balance between avoiding the encountering hazard of wake vortex and increasing the transport capacity of airports, much attention has been paid on the real‐time monitoring and detection of wake vortex in the past decades. Some major ATM programmes like Single European Sky ATM Research (SESAR) and Next Generation Air Transportation System, USA (NextGen) have also launched many projects on this topic, and the representative research institutes include Thales, Office National d'Etudes et de Recherches Aerospatiales (ONERA), Deutsches Zentrum für Luft- und Raumfahrt e.V. (DLR), Université catholique de Louvain (UCL), National Aeronautics and Space Administration (NASA), Federal Aviation Administration (FAA), Lincoln Lab, Boeing, and so on. In all these studies, the characteristics, detection technology and parameter retrieval are the key issues, and the characteristics of wake vortex serve as the basis for the rest studies.
In aviation safety, we mainly concern the clear air condition and wet weather condition. According to the scattering theory, the scattering of wake vortex in clear air is mainly determined by the fluctuation of dielectric constant inside the wake; while under wet condition, the key scattering factor becomes the massive number of precipitation droplets carried by the velocity of wake vortex. In this chapter, we present simulation schemes for the dielectric constant distribution and droplet distribution of wake vortex. The distributions are caused by the dynamics of wake vortex and serve as the physical basis for scattering analysis.
First, we study the dielectric constant distribution of wake vortex generated in clear air.
2. Dielectric constant of wake vortex generated in clear air
2.1. Two key parameters for determining the dielectric constant of wake vortex
The relative dielectric constant of atmosphere () can be well depicted by the following expression :
where ,,are the pressure (pa), absolute temperature (K), and water vapour content (kg/kg), respectively. Generally, the second term in the square bracket is much smaller than 1, so the variation in dielectric constant between wake vortex and ambient air can be approximated as follows when the Taylor expansion is taken into account:
In the expression, the parameters with subscript “a” refer to the ambient parameters and those without “a” refer to the wake vortex's parameters.
As is known, an isentropic process is a process in which there is neither heat exchange nor any friction effect . Typically, the wake vortex of a subsonic airplane can be assumed as an isentropic flow, and the thermodynamic parameters at different points along a streamline can be written as follows:
where is the adiabatic coefficient for air. Consequently, the variation in dielectric constant is transformed to
Here we have denoted , and the effects due to density and water vapour are mixed in the term . In order to separate the two factors, this term is transformed to
with being the water vapour variation between the local wake and the ambient air. Since the variations in density and water vapour for wake vortex are very small, say and , the dielectric constant variation can be approximated as follows when the Taylor expansion is adopted:
In this expression there are two undetermined parameters, and . They are separated into two different terms:
The first term is determined by the density variation :
and the second term is determined by the water vapour variation :
In this manner, the key of modelling the dielectric constant is to determine the two parameters, and .
2.2. Effect of density variation on the dielectric constant
2.2.1. Velocity field of wake vortex
When the stationary phase of a subsonic wake vortex is taken into account, the dynamics can be well characterized by the steady Lamb momentum equation :
where is the total velocity of wake vortex, and is the vorticity. In this expression, the velocity and density are separated, which makes it convenient to work out the thermodynamics parameters according to the velocity field.
Typically, when no cross‐wind is considered, a stable‐stage wake is composed of two contour rotating vortices of the same strength, and they descend at a velocity . In this manner, in a coordinate system descending with the vortex cores, the wake vortex is steady, and the velocity for a given point can be written as (see Figure 1):
where and are the velocities deduced by the left and right vortices, and is the descending velocity. In Figure 1, the parameter is the circulation which defines the strength of the wake vortex.
In the expression, the deduced velocity of each vortex can be presented by existing velocity profile models, such as Rankine model, Lamb‐Oseen model, and so on. Among them, the Rankine model is a widely used one, and the corresponding tangential velocity () follows :
where is the distance of a given point to the vortex centre, the vortex core radius, and the vortex spacing. As shown in Figure 2, the velocity outside the vortex core is irrotational, while the inside part is with a uniform vorticity:
The descending velocity can be derived from the deduced velocity of a vortex core:
2.2.2. Stream function of wake vortex
When the velocity components are determined, the vorticity is finally obtained as
In this expression, we have considered , and is an identification function for the left vortex core region () and right one ():
As a result, we have
where and are the velocity components in and directions, and is the stream function:
In this manner, the Lamb momentum equation for wake vortex is rewritten as
Furthermore, integrating the above equation from a point to infinity gives
where we have considered , and the total stream function is
with the four terms on the right-hand side being the up‐wash flow at infinity, the stream function due to the left vortex and right vortex, and a constant.
For a Rankine vortex, the stream function follows
Then and can be obtained by replacing the circulation with and , respectively.
Also, the constant is chosen to meet when is on the vortex core boundary:
Due to the impact between two vortices, the constant has a very slight variation along the vortex core boundary. In practice, the average of along one vortex core boundary is chosen as the constant.
As a result, combination of the total stream function and velocity gives the distribution of integral as shown in Eq. (20), which is then used to calculate the density distribution of wake vortex.
2.2.3. Determination of dielectric constant due to density variation
Substituting the isentropic relationship (3) into the item on the right hand of Eq. (20) gives
which can then be transformed as
For a wake vortex, the integral has the same magnitude as (generally not larger than 1000). On the other hand, and are, respectively, on the magnitude of 1 and . Therefore, the second term in Eq. (25) is much smaller than 1, and the Taylor expansion gives:
Consequently, the effects of dielectric constant can be rewritten as
2.3. Effect of water vapour variation on the dielectric constant
Generally, the atmospheric water vapour inside the wake vortex can be modelled as a passive scalar, which is convected by the wake vortex velocity field and is governed by the convection-diffusion equation :
where is the water vapour concentration, is the diffusion coefficient for an air/water vapour system, and is the velocity filed of a wake.
In fluid dynamics, the Péclet number is a dimensionless number indicating the rates of convection and diffusion of a flow :
where is the characteristic length, is the velocity, and is the mass diffusion coefficient. Generally, a flow is convection-dominated, if the Péclet number is large. In this study, the diffusion coefficient for air () is very small, which leads to a big Péclet number and the flow is convection dominated. In this manner, neglecting the impact of diffusion leads to a simplified governing equation:
As is known, the initial water vapour gradient is very important to characterize the equation. Here the stratified model is adopted in this work
with and being the offset and gradient of water vapour content, respectively. Substituting the initial condition into the governing equation gives
with being the water vapour variation. At the same time, the initial condition of Eq. (33) becomes ; this is coincident with the physical image that the water vapour variation is initially zero.
Eq. (33) is a hyperbola differential equation, which is often numerically difficult to solve. In the present work, the leapfrog scheme is adopted to solve the target equation, and good convergence and stability are achieved. The scheme is as follows:
with and being the velocity components in and direction, respectively. In the process, the Von Neumann method leads to the following stable condition :
where is the time step, is the minimum grid spacing, and is the maximum velocity in the wake vortex.
In addition, non‐uniform grids and symmetric condition are used to reduce computational cost. On the one hand, the velocity distribution shows that the flow in and around the vortex core is relatively complex and the flow far from the vortex core is slowly variational. Typically, sparse grids are adequate to characterize the slowly variational zones, but complex zones require dense grids. In this manner, non‐uniform grid scheme can be adopted to reduce the total number of grids. On the other hand, the wake vortex is symmetric, so only half of wake vortex needs to be computed. Overall much computational cost can be saved through the above scheme.
In this above simulation, the moving coordinate system is also used. The dielectric constant distribution can be transformed into the stationary coordinate system if the following transform is used:
where is the descending velocity, is the evolutional time, and are the coordinates in the moving coordinate system and stationary coordinate system, respectively.
2.4. Numerical examples
Here we give several numerical examples for the dielectric constant distribution due to different effects. The parameters of airplane and atmosphere are as shown in Table 1.
|Airplane mass||250,000 kg|
|Ambient pressure||100,000 pa|
|Ambient absolute temperature||288 K|
|Water vapour content offset||0.018 kg/kg|
|Water vapour content gradient||kg/(m kg)|
According to the parameters, the distribution of can be worked out, and the intensity of dielectric constant due to density variation can be obtained as shown in Figure 3. It is observed that in the vortex cores are much larger than that outside, so the vortex core could present a big contribution to the scattering of wake vortex.
The dielectric constant due to water vapour variation () can be obtained when the given partial equation (33) is solved, and Figure 4 presents at . It is observed that the convection effect of water vapour results in a non‐uniform laminar structure in and around the wake vortex cores, and these structures could be good contributor to the scattering in high frequency.
Consequently, the sum of Figures 3 and 4 gives the total distribution of dielectric constant (see Figure 5). Comparing the magnitude of and that of shows that is much less than , and dominates the magnitude of . However, and have different structures; they make different contribution to the scattering in different frequency bands. Typically, the scattering of clear air wake vortex at a frequency lower than 100 MHz is mainly determined by the density variation ; otherwise, the water vapour variation makes the major contribution.
2.5. Extrapolation of dielectric constant distribution
The extrapolation includes two parts: one is relate to the density distribution and another is related to the water vapour distribution.
2.5.1. Extrapolation of dielectric constant related to density distribution
As shown in Eq. (27), the key of density distribution is the integral ; this could be obtained with the normalized parameters.
First, if the space is normalized by vortex separation , say , the Rankine model can be normalized as
with being the normalized velocity:
Other velocity profile models have similar expressions. Substituting the normalized velocity into the integral Eq. (20) gives
Where the normalized integral is only related to the velocity model, and , .
For a stably flying airplane, the lift force balances the weight, which leads to the following initial circulation expression:
Consequently, substituting the circulation into the integral gives
The dielectric constant related to density distribution is then rewritten as
Since the normalized integral is only related the velocity model, the following relationship can be obtained when the different airplane parameters and air conditions are introduced:
with and .
2.5.2. Extrapolation of dielectric constant related to water vapour distribution
The key of water vapour distribution is to solve the partial differentiation equation (33).
Define some related normalized parameters as follows: ,,,, where and . Consequently, the target partial differentiation equation is rewritten as
If the variable is solved from above equation, the dielectric constant related to water vapour distribution becomes
In this manner, the dielectric constant for different airplane and air parameters is extrapolated as
where , , , and . For stably flying airplane, we have , and then the relationship becomes
This is a very simple relationship.
With the combination of two extrapolation formulae, the dielectric constant distribution is then determined. This can save a lot of computation cost when different airplane wake vortices are to be analysed.
Another condition we always experience is wet weather condition (fog, rain, and snow). Here we mainly concern the rainy condition since it is the most common situation around airports.
3. Wake vortex generated in rainy weather
In still air, the raindrops fall vertically towards the ground and reach a constant terminal falling velocity. When an aircraft takes off or lands in rainy weather, the raindrops will be inevitably involved in the aircraft wake vortices. Raindrops’ motion in wake vortex is modified by the vortex flow. This modification of the trajectories of the raindrops may induce changes in raindrops’ number concentration and velocity distribution in wake vortices, therefore results in changes in the recorded radar signal. This section presents a modelling scheme for raindrops’ motion and distribution within the wake vortex.
3.1. Parameterization of raindrops
In still air, a falling raindrop reaches its terminal fall velocity with the equilibrium between the inertial force and the drag force acting on it . A widely used exponential expression between (m/s) and the diameter (mm) is given by 
where = 9.65 m/s, = 10.3 m/s, = 0.6 m/s, and is a density ratio correction factor adjusting deviation of the terminal fall velocity due to the air density change with the fall altitude.
Drop size distributions (DSD) have been widely used by radar meteorologists as they are directly related to radar reflectivity . In the following analysis, a suitable model to describe the size distribution of the rainfall in Europe is adopted 
where =64500(m-3 mm-3) with R (mm/h) being the considered rain rate, (), represents the number of raindrops of the diameter D per unit volume per unit diameter class interval.
3.2. The motion equation of raindrops in wake vortices
Typically, the diameter of raindrops disperses between 0.5 and 4 mm. Usually their Stokes number in wake vortex flow is approximate to 1, which makes their motion trajectories significantly differ from the streamlines of total velocity field. To obtain those trajectories, the motion equation of the raindrops is studied .
When a raindrop enters into the wake vortex flow, its movement is governed by
where t is the time, a is the acceleration of the raindrop, is the fluid drag force acting on the raindrop, is its mass, and g is the gravity acceleration. For a raindrop moving with velocity in the fluid whose velocity field is , if its diameter ranges from 0.5 to 4 mm, the drag force can be approximately considered in the Newton regime  and given by
where denotes raindrops’ position, is the relative velocity between the vortex flow and the raindrop, and is the fluid drag coefficient. The impact of air density variations in the vortex flow on can be neglected because the density of raindrops is much larger than that of air. Thus, for a raindrop of diameter D is derived by the equilibrium equation of its weight and the drag force when falling at terminal falling velocity in still air:
The instantaneous position and velocity of raindrops can be obtained from the above equations, but it is not easy to obtain a simple and closed expression. Here, a fourth‐order four variables Runge‐Kutta algorithm is proposed to solve the equation of motion .
3.3. Examples of trajectories of raindrops in wake vortices
In still air, the raindrop falls along a vertical trajectory to the ground. In presence of wake vortices, the trajectory of a raindrop is depending on its diameter and the location where it enters into the wake vortex flow. Circulation is a very important parameter to characterize wake vortex since it describes the strength of wake vortex. For raindrops moving in the vortex flow, their motion characteristics, that is, trajectory and velocity also largely depend on the vortex circulation. For simplicity, only the impacted part of the wake vortex region is shown in Figure 6, where four sets of trajectories are illustrated for raindrops of diameters 0.5, 1.0, 2.0, and 4.0 mm. Each set of trajectories corresponds to a given circulation value of the wake vortices. For a given circulation, the corresponding trajectory and velocity of raindrops in the wake vortices are unique. Comparisons between different trajectories give the following conclusions: (1) the smallest raindrops are much more sensitive to the vortex circulation and (2) the motion characteristic of raindrops in wake vortices is representative of the vortex strength.
3.4. Raindrops’ distribution in wake vortices
A Doppler radar will be very sensitive to raindrops’ motion and possibly enable the detection of wake vortices in rain. To better understand the impact of wake vortices on the raindrops’ motion, it is necessary to develop the methodology to quantitatively analyse the raindrops’ distribution in wake vortices.
3.4.1. Raindrops’ number concentration
The box counting method is adopted here to quantitatively compute raindrops’ distribution in wake vortices. For simplification, we consider the situation where the raindrops are falling into the two dimensional rectangular region of wake vortex in stable phase. Before entering into the wake vortex region, the raindrops are falling in still air with the constant terminal falling velocity, and they are named as “initial raindrops”. The number density of initial raindrops of diameter D (mm) is assumed to be (). In the wake vortex region, the raindrops’ trajectory and velocity are changed and they are denoted as “disturbed raindrops”. The number density of disturbed raindrops is assumed to be (), where (x, y) are the coordinates in the wake vortex region. Obviously, for a given wake vortex pair, depends on both the diameters of raindrops and their locations in wake vortex. In order to better illustrate the influence of wake vortices on the raindrops’ distribution, the raindrops’ relative number concentration at a point (x, y) is defined as
where depicts the change in raindrops’ concentration induced by the wake vortex. If , the concentration of raindrops is enhanced, otherwise it is reduced.
In order to obtain the quantitative estimation of , the wake vortex region is divided into grid boxes with equal size. The size of each grid box in xy plane are Δx and Δy, respectively. Above the wake vortex region, there are grid boxes where the initial raindrops of diameter D are homogeneously distributed, and the number of initial raindrops in each grid box is recorded as . At each time step, their positions and velocities are updated by computing the equation of motion. If some of the initial raindrops enter into the wake vortex region, the same number of new initial raindrops is added to the first row of the grid boxes, say, the grid boxes above the wake vortex region. When all the raindrops released at initial time arrive at the bottom of wake vortex region, the number of disturbed raindrops in each grid box in wake vortex region is counted. Thus, can be approximated by the ratio of the number of disturbed raindrops in a grid box centred at (x, y) to the number of initial raindrops in a grid box above the wake vortex region, that is
Obviously, the estimation accuracy of depends on the choice of the grid box size: Δx and Δy, and the number of initial raindrops in each grid box above the wake vortex region: .
In Figure 7, the raindrops’ number concentration in wake vortices is illustrated. The simulation parameters are listed in Table 2. It can be noticed that in the wake vortex region, there are two columns where the raindrops’ concentration is very small, even to zero. These two columns appear symmetrically below the two vortex cores and the distance between them in (a) is much wider than the others. Between these two columns, there are two narrow regions where the number concentration of raindrops is enhanced. For the raindrops of 1 and 2 mm of diameter, the number concentration value exceeds 8 in some grid boxes.
|Aircraft wingspan||60.30 m|
|Aircraft maximum landing weight||259,000 kg|
|Aircraft landing speed||290 km/h|
|Grid box size||1 m×1 m|
|Raindrops’ diameter||0.5 mm, 1.0 mm, 2.0 mm, and 4.0 mm|
3.4.2. Raindrops’ velocity distribution
Besides the number concentration, the raindrops’ velocity distribution in each grid box is of great interest. If the grid box size used for the box counting method is small enough and the number of raindrops in each grid box is large enough, the velocity components of the raindrops in one grid box can be thought to obey Gaussian distributions. The mean value and variance of the velocity of the raindrops in each grid box are computed. If the number concentration of the grid box is zero, the raindrops’ velocity in this grid box is set to 0.
In Figures 8 and 9, the raindrops’ horizontal and vertical velocity distribution in wake vortices are illustrated, respectively. From Figure 8, it is interesting to find that the raindrops’ horizontal velocity field is similar to the wake vortex velocity field. From Figure 9, it is interesting to find that between the two vortex cores, the raindrops’ vertical velocity is speeded up. At the same time, the standard deviation of the raindrops’ velocity distribution in a grid box is sufficiently low to consider it as constant within each grid box. In fact, in wake vortex, the raindrops’ motion is affected by the vortex flow; the raindrops’ velocity field is not the same as the superimposition of the vortex flow velocity and raindrops’ terminal velocity, but it is representative of the wake vortex velocity characteristics.