Document [original]

Vol.:(0123456789)

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CEAS Aeronautical Journal (2022) 13:763–778

https://doi.org/10.1007/s13272-022-00587-1

ORIGINAL PAPER

Simplified vortex methods tomodel wake vortex roll‑up inreal‑time

simulations forfuel‑saving formation flight

HenrikSpark1 · RobertLuckner1

Received: 18 August 2021 / Revised: 25 April 2022 / Accepted: 28 April 2022 / Published online: 29 June 2022

Abstract

One possibility for reducing fuel consumption is to fly in the upwind field of the wake vortex generated by an aircraft that

is flying ahead. Migratory birds use this principle. Manually flying an aircraft at the point of optimal fuel reduction is not

suited for routine flight operations as the pilot workload is excessively high. Hence, an autopilot function has to carry out this

task. For designing the autopilot, a flight mechanical simulation with a wake vortex velocity model is required that has the

ability to calculate the vortex-induced velocity fields. This paper contributes to the choice of a real-time simulation method

for modelling vortex-induced velocities that the wake vortex of a leading aircraft generates and that the trailing aircraft shall

use during fuel-saving formation flight.Two different wake vortex velocity models are introduced and compared during

steady, horizontal flight. One model is based on the Lifting Line Method (LLM) and the other on the unsteady Vortex Lat-

tice Method (VLM). Both models are able to calculate the wake vortex roll-up phase for arbitrary lift distributions, whereas

the commonly used Single Horseshoe Vortex Model (SHVM) ignores the near-field roll up. The differences in the induced

upwind distribution and vortex filament position are analysed for coarse spatial and temporal discretisation that the real-time

constraint requires. Despite the more stringent simplifications of LLM, both methods yield similar filament positions and

similar velocity fields for the same discretisation of the lifting surfaces. Finally, the influence of the discretisation parameters

is discussed and parameter values are recommended for using VLM and LLM in real-time flight simulations.

Keywords Wake vortex modelling· Fuel saving formation flight· Real-time flight simulation· Vortex Lattice Method·

Lifting Line Method

List of symbols

b Wing span [m]

Vortex pair spacing [m]

Lift coefficient [–]

d𝐥

Discrete vortex filament segment [m]

H Altitude [m]

l(y) Chord length [m]

Core radius [m]

𝐫

Distance vector [m]

Vortex spacing parameter [m]

V Airspeed [m/s]

𝛼

Angle of attack [

◦

]

Circulation [

m2∕s

]

Γ0

Root circulation [

m2∕s

]

𝜌

Air density [

kg∕m3

]

Subscript/abbreviation

a Aerodynamic coordinate system

f Body coordinate system

ind Induced

W Wing

HTP Horizontal tailplane

TE Trailing edge

1 Introduction

Aviation has to noticeably reduce its environmental impact.

Innovative airframe and engine concepts as well as sus-

tainable aviation fuels shall contribute [1]. Also aircraft

operations shall be part of the solution [2]. Even though not

explicitly mentioned in Refs. [1, 2], it is long known that for-

mation flying techniques, inspired by migratory birds, have

the potential to significantly save energy. Transport aircraft

* Henrik Spark

[email protected]

1 Chair ofFlight Mechanics, Flight Control andAeroelasticity,

Institute ofAeronautics andAstronautics, Technical

University ofBerlin, Marchstr. 12, 10587Berlin,

Deutschland

764 H.Spark, R.Luckner

1 3

can use this technique as well. The Applied Vehicle Tech-

nology (AVT) Panel of the NATO Science and Technology

Organization (STO) has published a comprehensive over-

view on the state of the art of formation flight and the related

research activities including an extensive reference compi-

lation [3]. The overview addresses the history of energy-

saving formation flight, the physical principles, related tech-

nologies and systems, as well as operational aspects.

In a formation, the following aircraft can reduce its thrust

and hence its fuel consumption, when flying in the upwind

field of a wake vortex that a leading aircraft generates. So,

the trailing aircraft takes advantage of wake energy that

otherwise is left unused. The position for maximum fuel

savings in a vortex-induced velocity field is called sweet-

spot. For flying in the sweet-spot, the trailing aircraft has to

control its position relative to the leader. During long haul

flights, where the sweet-spot position has to be maintained

over many hours, formation flight needs to be automated.

This automation requires new autopilot functions. For the

development of such autopilot functions, a flight mechanical

model with a realistic wake vortex velocity model is needed

to simulate the vortex-induced forces and moments. Vortex

methods, as comprehensively described in Ref. [4], are used

for generation of vortex velocity fields. The research at the

TU Berlin aims at increasing the realism of formation flight

simulations by means of high-fidelity, real-time methods.

The objective of this paper is to evaluate the fidelity of real-

time methods for the calculation of the vortex velocity field.

Wieselberger [6] was the first who described the funda-

mental physics of formation flight by the vortex pair behind

a lifting surface. He calculated the upwind field by Biot

Savart’s Law. Schlichting [7] explained the resulting energy

savings by applying Aerodynamic Wing Theory. Hummel

[8] extended this approach to inhomogeneous formations.

In 1986, Beukenberg and Hummel demonstrated the theo-

retically explained energy savings by flight tests for the first

time [9, 10]. Further flight tests with different aircraft types

confirmed fuel savings up to 18%, for example in Ref. [11].

A simple vortex model that is commonly used for simula-

tions of formation flight and wake vortex encounters is the

Single Horseshoe Vortex Model (SHVM). It approximates

the wake vortex system by a pair of two straight vortices, see

Refs. [5] and [12]. This vortex model represents a rolled-up

vortex. It is valid from 15 up to 150 wing spans behind an

aircraft. Hence, it is not well suited to model the near field of

the wake that extends up to 15 wing spans behind the vortex

generating aircraft. As formation flight can be in within the

near field, modelling the vortex roll-up increases realism.

Sarpkaya [13] comprehensively reviews the computa-

tional methods that are developed in fluid mechanics to sim-

ulate and describe the characteristics of three-dimensional,

unsteady vortical flows. The application of those methods

to real-time flight simulator investigations is not straight

forward, as the real-time requirements on the computational

performance significantly differ from accuracy requirements

for fluid dynamic applications.

Fuel-saving formation flights are assumed to be carried

out when atmospheric turbulence is so low that decay and

deformation of the wake can be neglected for the envisioned

separation distances. However, to investigate cockpit pro-

cedures and pilot workload in real-time flight simulators,

vortex models are necessary for distances, where the vortex

roll-up is not completed (10 wing spans are 0.5 NM for an

80 m wing span aircraft like Airbus A380). Whereas the

computational-cheap SHVM can be used for formations with

separations above 15 wing spans, more accurate methods are

needed for closer distances.

For the investigation of vortex effects on a trailing air-

craft, very high accurate methods, like Large Eddy Simula-

tions (LES), have been applied, e.g. by Bieniek etal. [14].

There, the vortex flow field was pre-computed for a cer-

tain distance behind the generating aircraft and stored in

a “box”, from which the wind field can be used. If LES is

not available, LLM can be used in a similar manner—with

some loss of realism. However, such simulation techniques

require steady flight conditions. If the formation is manoeu-

vring, e.g. changing altitude or course, a pragmatic solution

to stay with the “box” technique by adapting the box loca-

tion and orientation in space was proposed by Kaden [15].

A superior approach would be to use methods, like LLM or

VLM that can online address the impact of the lift changes

of the vortex-generating aircraft and the impact on the wing.

However, in real-time flight simulations, available CPU per-

formance constrains the discretisation granularity. The moti-

vation for this paper was to prepare the online approach by

comparing LLM and VLM vortex models using very coarse

discretisation.

Research on vortex methods for wake vortex roll-up simu-

lations dates back to the 1930s [16]. Numerical problems

resulting from this approach and methods permitting their

resolution have coined a research area, see Refs. [17–19].

In addition to summing up the state of the art, Devoria and

Mohseni [19] describe the possibility of chaotic vortex tra-

jectories in roll-up simulations. As a solution to this prob-

lem, extensions to the roll-up methods are recommended.

The extensions can, for example, rediscretise the vortex

elements, see Ref. [20], or introduce specific procedures to

regulate the distortion of the Langrangian methods, see Refs.

[21, 22].

This paper compares two well-known methods for the

calculation of the vortex velocity field: the Lifting Line

Method (LLM) and an unsteady Vortex Lattice Method

(VLM). Extensions of the LLM or VLM, as proposed by

Refs. [20–22], are not considered here. The focus is on the

comparison of the unaugmented LLM and VLM calculation

schemes and their results. Only the core radius function is

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Simplified vortex methods tomodel wake vortex roll‑up inreal‑time simulations forfuel‑saving…

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introduced as smoothing parameter. The results of the LLM

and VLM formulations are compared to the SHVM.

LLM can be used for all distances behind aircraft. Kaden

implemented the LLM code that is used at TU Berlin and

in his work. He compared it to the simple SHVM, see Refs.

[5, 15]. A set of parameters for the LLM was optimised to

match the vortex-induced velocities in the far field, where

the roll-up is complete, with SHVM. The optimised param-

eters comprise the number of vortex filaments, the integra-

tion step size

Δt

, the core radius

and its regularisation

function. The calculation consists of two main steps: first, an

offline computation of the vortex particle field, and second,

the subsequent calculation of the vortex-induced velocities

at the follower’s position. Using simplifying assumptions,

the LLM’s second step, important for the formation flight

simulations, has real-time capability and is therefore cur-

rently used at TU Berlin.

Also at TU Berlin, Loftfield developed an unsteady Vor-

tex Lattice Method (VLM) for flight mechanical investiga-

tions at separated air flow as part of the project MoSS [23].

The unsteady VLM enables the calculation of wake vortex

roll-up behind manoeuvring aircraft. Less simplifications

in the VLM yield more realism compared to the LLM. But,

the VLM is computational expensive and the present for-

mulation is not suited for real-time application. Here, the

unsteady VLM is used. It is applied to a steady, horizontal

cruise flight scenario. A one kilometre long wake vortex

wind field is generated by an offline simulation. For the same

scenario, the wind field is calculated by the LLM. This is

achieved by flying the aircraft through planes and capturing

the wake properties at their locations, see Fig.1. Comparing

the results of both methods, the effect of the simplifications

made in the LLM is investigated.

For the comparison, the filament positions of the wake

vortices are analysed. The computed wind field data, result-

ing from all filament positions and circulations, are used

to identify the wake vortex axes. The axes represent the

wake vortex position. This position has to be known for

fuel-efficient formation flight. In addition to the position-

related comparison, the upwind areas in the wake vorti-

ces are compared. The vertical wind velocity maximum,

hence the maximum tangential velocity near the vortex

core, are related to the vortex-induced rolling moments, as

Bieniek has shown in Ref. [24]. Because the induced rolling

moments are important for the simulation of the following

aircraft, the vertical wind velocity maxima are used as a

substitute in our comparison.

The paper starts with the description of both wake vortex

methods LLM and VLM and the resulting models in Sect.2

addressing assumptions, differences, advantages and disad-

vantages. To enable a comparison, Sect.3 explains the meth-

odology to set up the models for a steady, horizontal flight

scenario. Section4 explains how the wake vortex axes are

identified. The identification method is used in Sect.5 for the

comparison of VLM and LLM results. Finally, an analysis of

the parameter set for this comparison is conducted in Sect.6.

2 Vortex filament methods

This section describes the methodologies behind LLM,

VLM and SHVM. SHVM is the reference as it is commonly

used in wake vortex simulations. The assumptions made by

the methods, the differences, advantages and disadvantages

of the LLM and VLM are explained.

LLM and VLM are Lagrangian methods, as described

in Refs. [4, 25, 26], tracing particles in a velocity field. The

particles are the edges of vortex filaments, which are shed

from the aircraft’s lifting surfaces. Using the Biot–Savart

law, see Sect.2.4, each filament induces a velocity on every

particle and therefore on every filament. The aircraft’s lift-

ing surfaces are modelled by vortex filaments, too. These

bound filaments also induce velocities, generating the flow

field around the lifting surfaces and influencing the wake

vortex roll-up. The mutual induced velocity of all vortex

filaments leads to a roll-up of the free filaments in the wake,

Fig. 1 Concept of the wake vortex roll-up simulation using grid planes according to Ref. [5]

766 H.Spark, R.Luckner

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generating a rotational velocity field behind the left and right

wing of a leading aircraft. The induced velocity from the

opposing side induces the typical descent of the wake vortex.

In addition to calculating the induced velocity by the bound

and free filaments at the vortex filaments’ edges, the induced

velocity can be evaluated at arbitrary points to compute the

induced velocity fields.

Evaluation planes are placed behind an aircraft to calcu-

late and visualise the velocities induced by the vortex sheet

in grids. The evaluation planes can be arbitrarily placed.

They consist of a grid of particles. At the particle positions,

the local velocity is evaluated using the same operations as

for the free vortex filaments in the wake roll-up calculation.

The induced 3D velocity field is calculated from multiple

evaluation planes stacked in longitudinal

-direction.

Often, an elliptical lift distribution on the wing and tail-

plane is assumed for LLMs, see Refs. [5, 12, 27]. In this

paper, the regional airliner VFW 614 that is modelled at

TU Berlin’s research flight simulator is used as example.

Detailed geometric and aerodynamic data are available,

such as the exact geometry, wing twist and airfoil profiles.

Using those data, the elliptical distribution that was used

in previous projects is replaced by a lift distribution that is

calculated by the VLM. Thus, the wind field computation

starts with the VLM calculating the circulation at the lifting

surfaces (at the bound filaments). The following steps for the

wind field generation are equivalent in the VLM and LLM.

These steps are: computing the wake vortex roll-up, calculat-

ing the induced velocities at all particles in the evaluation

planes and saving the 3D wind field data.

2.1 Single horseshoe vortex model

The Single Horseshoe Vortex Model (SHVM) satisfyingly

represents the wake vortex velocity field after roll-up and it

is commonly utilised for that purpose, see Ref. [5]. Here, it

is used as reference for the LLM and VLM.

The SHVM consists of a bounded vortex at the wing

(its influence is commonly ignored) and two straight, free

counter-rotating vortices that extend behind the generating

aircraft to infinity. The circulation of the two straight vor-

tex lines is determined by the Kutta–Joukowski law using

the actual parameters of the leading aircraft. For a certain

vortex age, vortex decay can be considered by reducing the

circulation. In [5], Kaden defined the SHVM parameters

for cruise flight (weight of 17.4 tons, altitude of 6400 m

and airspeed of

140 m∕s

) for the VFW 614, assuming an

elliptical circulation distribution. The vortex spacing is

b0=

𝜋

∕4b=16.89 m

, circulation

Γ=114.42 m2∕s

, and

global core radius

rc=0.45 b=0.9675 m

In the following, the global core radius of the SHVM has

to be distinguished from the local core radius used to com-

pute the velocities induced by the discrete vortex filaments

in LLM and VLM. Figure2 shows the resulting induced

vertical velocities for the elliptical circulation distribution

and for the distribution that is calculated by the VLM (for

discretisation 1, see Sect.3). In the VLM calculation, which

is used here, the values for aircraft mass, altitude and the

airspeed are the same as in [5] and yield

Γ=137.78 m2∕s

and a vortex spacing

b0=13.88 m

. The global core radius

is the same.

2.2 Lifting line method

The LLM calculates the roll-up in several steps that are

described in [5]. The wake vortex roll-up is calculated using

multiple calculation planes for the wake vortex roll-up cal-

culation, whereas the evaluation planes describe the 3D

wind field and define the vortex-induced wind field for an

aircraft flying through the planes. Both types of planes are

stacked in negative

-direction behind the aircraft in flight

direction, see Fig.1. The LLM represents lifting surfaces by

lifting lines. These lines hold a certain circulation distribu-

tion

Γ(y)

. The local circulation generally increases from a

lifting surface’s tip to its root. Following the Kutta–Jouk-

owski theorem, the lift is proportional to the circulation

air density

𝜌

and airspeed V [28]

The axis of the lifting lines are bound to the aircraft’s quarter

chord of the aerodynamic surfaces. Here, the bound vortex

filaments originate. The continuous circulation distribu-

tion

Γ(y)

is replaced by the sum of the discrete circulations

ΔΓi

of the filaments with stepwise constant strength, see

Fig.3. Each discrete step in the bound circulation distribu-

tion produces a free vortex filament of the circulation

ΔΓi

see [29]. In other words, the discrete changes in the lifting

(2.1)

L=𝜌VΓ.

Fig. 2 Induced velocities of the SHVM with the VLM’s circulation

compared to the classical SHVM

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line’s bound circulation determine the circulation of the free

wake vortex filaments.

The origin of the aerodynamic coordinate system (index

a) lies in the wing’s aerodynamic mean chord. The calcu-

lation planes span in

- and

-direction. The roll-up of

the free vortex filaments is calculated for each plane. The

time step size

Δt

is defined by the distance between those

planes,

Δt=Δxa∕V

. One can either assume the aircraft

to travel through the planes or assume that the planes are

following the aircraft in flight. For the previously intro-

duced evaluation planes, both points of view are equiva-

lent as the resultant wake age at each evaluation plane is

the same both ways when flying with a constant speed V.

The calculation planes are iteratively solved in the LLM

to calculate the wake shape. So, visualising the calculation

planes behind the aircraft and extending the wake shape in

each time step to the next gate is preferred. Another cal-

culation plane is added each time step to extend the wake

shape. The iterative calculation starts from the position of

the lifting lines and employs the forward Euler integration

method. In each calculation plane, the vortex filaments are

assumed to be straight and aligned with the

-axis. By

this approximation, the induced velocity in

-direction

vanishes and the roll-up is simplified, see Ref. [5]. So,

the LLM only uses the preceding calculation plane that

contains the last position of the free filaments to compute

the next position of the free filaments. The influence of

the bound filaments is considered, but it vanishes down-

stream. By using this approximation instead of calculating

the induced velocities from streaklines, the algorithm time

complexity of the calculation decreases.

As a result of this scheme, the runtime of the LLM scales

linear with the number of computation planes. Thus, it scales

linear with the wake vortex length. The runtime scales quad-

ratically with the number of vortex filaments, as all vortex

filaments interact. However, a disadvantage of LLM is, that

the lifting line’s circulation distribution has to be prescribed,

considering the necessary total lift of the aircraft. This can

either be done by assuming an elliptical distribution or, as in

this paper, by calculating the distribution for example with

the VLM.

2.3 Vortex lattice method

The unsteady Vortex Lattice Method (VLM) uses a lat-

tice of vortex filaments to represent the lifting surfaces and

the wake, as Fig.4 shows. Each lattice consists of indi-

vidual vortex rings, each vortex ring comprises four vortex

filaments.

Panels model the aircraft’s lifting surface geometry, as

described in Sect.3. Based on the panel position and the

convention to place the vortex rings’ leading segments at

the panels’ quarter chord [28], the vortex lattice geometry

is derived accordingly, see Fig.4. Each vortex ring has the

same oriented circulation

Γij

(circulation of panel i at sec-

tion j) for all four filaments of the ring, see Ref. [28]. The

vortex rings placed on the panel geometry are bound to that

geometry. Their circulation

Γij

is calculated so a no-flow-

through condition is reached at the collocation points. The

collocation points are placed at the centre of the vortex rings,

therefore at three quarter of the panels chord. The condition

that no flow passes through surfaces causes a tangential air-

flow over the lifting surfaces.

The free vortex rings representing the wake are gener-

ated at the lifting surfaces’ trailing edges and move with

their individual local velocity. Obeying the Helmholtz

theorems and the conservation of circulation [30], the cir-

culation of the free vortex rings is bound to the filaments

bound vortex filaments

free vortex filaments

ΔΓ2ΔΓ1

ΔΓ1

ΔΓ2

Fig. 3 Schematic of free vortex filaments origination at unswept lift-

ing lines

panel geometry

Γ1,1

bound vortex corner point

free vortex corner point

XXX

collocation point

Γ2,1 Γ3,1 Γw,1,1 Γw,2,1 Γw,3,1 Γw,4,1

Δx

Δy =Δt V

Fig. 4 Vortex lattice formed by four filaments per vortex ring,

adapted from [28] and [23]

768 H.Spark, R.Luckner

1 3

forming the rings and the free rings circulation is constant

over time. In each VLM calculation time step, the induced

velocities at the free vortex ring corner points are evalu-

ated and the points are moved according to the induced

velocity and airspeed. This procedure yields the wake vor-

tex roll-up in the VLM.

The circulation

Γij

of the vortex filaments at the trailing

edges of wing and HTP is related to the lift of the respec-

tive section, as can be seen in Fig.4. As each vortex ring

circulation is oriented clockwise, the resultant circulation at

the intersection of rings must be calculated. This resultant

circulation is used to calculate the lift and pitching moment

in the VLM. As a consequence of this principle, the circula-

tion of resultant vortex filaments with spanwise orientation

adds up downstream. The bound circulation at the trailing

edge,

Γ3,1

in Fig.4, is related to the circulation and the lift

of wing Sect.1.

The trailing-edge Kutta condition

ΓTE =0

, is satisfied by

generating a free wake vortex ring with the same circulation,

for example

Γ3,1 =Γ

w,1,1

in Fig.4, so they are cancelling

each other out [28].

The same superposition principle for the vortex ring cir-

culation applies in the other direction, too. This results in

chordwise oriented filaments reflecting the spanwise change

in circulation. Particularly, the circulation of those filaments

in the first wake row defines the spanwise change in total

circulation. This relation is in analogy to the circulation of

free filaments in the LLM. The spanwise distance between

filaments at the trailing edge is

Δy

and the separation

Δx

of the free vortex rings is defined by the time step size

Δt

Δx=Δt⋅V

The VLM’s roll-up calculation is build upon less restric-

tive assumptions than the LLM’s. For instance, free vortex

filaments are not coupled to predefined computation planes.

The wake vortex roll-up is calculated by evaluating the

induced velocity from all vortex filaments at all vortex ring

corner points (wake particles). This results in a quadratic

time complexity for the VLM algorithm with respect to the

number of bound vortex filaments at the wing and tailplane

in spanwise direction and to the number of the free vor-

tex filaments that depends on the simulated vortex lengths.

Compared to the VLM, LLM has linear algorithm time com-

plexity with respect to the wake length. Besides the quad-

ratic scaling with wake length, the VLM evaluates all vortex

rings while calculating each wake particle’s movement in

contrast to LLM that evaluates only the latest evaluation

plane and the bound vortices. Consequently, computations

with the VLM are significantly slower. Another disadvan-

tage is that the simulated wake shape has to be longer than

the length for which the wake vortex induced velocities are

calculated. Otherwise, a starting vortex might be included

in the results that leads to induced velocities not resembling

the wake vortex velocity field.

The more complex VLM calculation has advantages:

the method derives the circulation distribution from the

geometry and actual flight condition as it simulates the

tangential flow condition at the lifting surfaces. So, the

roll-up calculation can be unsteady because the trailing-

edge condition is evaluated at each time step. In this way

it is possible to include the effect of changing lift distri-

butions during manoeuvres in the circulation of the wake

vortex filaments and eventually in the wake vortex roll-up.

2.4 Induced velocities

In both methods, the induced velocities are calculated

using the Biot–Savart law. The calculation is based on

the vortex filament circulations and the distance to a point

P, defined by the position vector

𝐱P

at which the induced

velocity

𝐕ind

shall be determined as Fig.5 shows. The

velocity induced by a segment j of the filament i at point

P is

with the distance vector

𝐫

and the circulation

Γi

of the vortex

filament

d𝐥

As stated, for the LLM each vortex filament

d𝐥

is a part

of the lifting lines or a line normal to the active calcula-

tion plane. In the VLM, each vortex ring comprises four

filaments

d𝐥

and all rings are recalculated in every time

step. The different application of the Biot–Savart law in

both methods results in more calculations per time step

for the VLM.

(2.2)

𝐕ind,i,j(𝐱P)= 1

4𝜋

Γi

d𝐥

i,j

×𝐫

i,j

i, j

ri, j

Filament i

Segment

j - 1

j + 1

j + 2

Fig. 5 Geometry definitions used for the calculation of the induced

velocity at point P

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Simplified vortex methods tomodel wake vortex roll‑up inreal‑time simulations forfuel‑saving…

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2.5 Core radius

Equation2.2 is valid for an inviscid potential vortex and

results in a singularity at a radius

r→0

. To compensate

that singularity a regularisation function is introduced.

As the goal is to compare the methods, both methods use

the same function to regularise the induced velocity in

the vicinity of a vortex filament, the low-order algebraic

regularisation. A review summary of 2D and 3D regu-

larisations for the Biot–Savart equation is given by Ref.

[31]. The transcriptions of Eq. (2.2) used in this paper for

computation that include regularisation in the Biot–Savart

formula are given in [5] for the LLM and for the VLM in

[28].

Regularisation functions introduce a viscid core of

radius

around the vortex filaments. From a vortex fila-

ment to a distance of

, the induced velocities increase.

At distances larger than

, the induced velocities decrease

and approach the velocities induced by a potential vortex.

The choice of

influences the result of the wake roll-up

calculations and thus the resulting vortex-induced flow

field [5]. The smaller

is chosen, the nearer to the vortex

axis the transition to the velocity of the potential vortex

happens, thus generating higher induced peak velocities.

The wake vortex effect on the trailing aircraft is calcu-

lated in two steps: (1) the calculation of the vortex particle

field during vortex roll-up, which is currently done offline

and (2) the subsequent calculation of the vortex-induced

velocities in real-time. The maximum upwind velocity and

an upwind area after roll-up are essential for formation

flight simulations. The comparison requires that LLM and

VLM yield an upwind area outside the core and after roll-

up that is as similar as possible to the SHVM—despite the

relatively coarse gridding that has to be used to comply

with computing-time constraints. To preserve the rela-

tionship between the wake vortex roll-up and the vortex-

induced velocities, the same local core radius must be used

for both calculation steps. Kaden investigated the impact

of the core size on the roll-up for core radii between

0.015b and 0.035b in Ref. [5] and selected

rc=0.025 b

for his investigations. Here,

rc=0.02 b

is used, see Sect.3.

3 Geometry andparameter definition

The circulation distribution in the VLM results from the

lifting surfaces geometry and the no-flow-through condi-

tion, as described in Sect.2.3. The lifting surface geom-

etry is modelled by panels and the aircraft is trimmed for

steady horizontal flight as defined in Table2.

The geometry and profile data of the wing, horizon-

tal and vertical tailplane of the VFW 614 are extracted

from the documentation of this short haul airliner. The

lifting surfaces geometry is modelled in OpenVSP [32]

and exported to Matlab. Three models with different dis-

cretisation are generated, D1, D2, and D3, see Table1. D1,

D2 and D3 are used in the VLM computations. D1 with

the highest resolution is shown in Fig.6—for the Open-

VSP representation—and in Fig.7 for the Matlab model

used for the VLM computations. Additionally to D1, D2

Table 1 Discretisation of the VFW 614’s lifting surfaces

Discretisation

Ny,W

Ny,HTP

Ky,W

D1 32 16 0.78 357

D2 8 4 3.125 357

D3 4 2 6.25 357

DF 128 64 0.19 10000

Fig. 6 OpenVSP representation of D1

Fig. 7 Matlab representation of D1

770 H.Spark, R.Luckner

1 3

and D3, a finer discretisation for the LLM (DF) has been

generated by interpolation of D1.

As Fig.7 shows, the fuselage is neglected in the VLM

calculations. Since the flight condition is a steady and hori-

zontal flight without sideslip or rudder input and therefore

symmetrical, no forces act on the vertical stabiliser. So, the

circulation of the vertical stabiliser is negligible. That is no

serious restriction, as the rudder is seldomly used in cruise

in airline operations. Therefore, it is not relevant for the

investigated formation flight scenario.

In Ref. [5], Kaden initialised the LLM with an elliptical

lift distribution. He used a circulation of

Γ0,W

=104.94 m

∕

for the wing and

Γ0,HTP

=22.63 m

∕

for the horizontal tail-

plane (HTP). The aircraft’s centre of gravity was in an aft

position to achieve the same lift distribution on wing and

tail plane as in Kaden’s study [5]. Then, the HTP with zero

elevator deflection was used to trim the pitching moment.

This trim condition is used for all three discretisation

cases that are applied to compare VLM and LLM. The pro-

cess starts with loading the geometries D1 to D3 into the

VLM computer program and calculating the roll-up with the

parameters of Table2.

The VLM program calculates the circulation of the vortex

filaments at the trailing edges. This circulation corresponds

to the total lift distribution over the lifting surfaces. Figure8

shows the circulation distribution for the three discretisation

cases. The geometry yields a non-elliptical lift distribution.

The root circulation at the HTP is close to the given refer-

ence values [5] for the elliptical loading in this flight condi-

tion, whereas the wing root circulation is higher as a result

of the non-elliptical loading.

The differences between the SHVM resulting from the

actual circulation distribution on the wing and HTP (D1)

and the classical SHVM with an elliptical distribution and

only considering the wing is shown in Fig.2.

The SHVM from the actual circulation distribution

induces higher velocities and a smaller vortex spacing due

to higher wing root circulation and the positive lift on the

HTP. Whereas the VLM computes the spanwise circulation

distribution from the panel geometry, the circulation dis-

tribution along the lifting lines has to be predefined for the

LLM. To achieve comparable conditions for the VLM and

LLM for the three discretisation cases, the LLM is initialised

with the VLM’s circulation distribution of the corresponding

discretisation. Thus, the VLM defines the spanwise distribu-

tion of the LLM’s circulation.

The chordwise positioning of the lifting lines is inves-

tigated as follows. For three potential positions, LLM and

VLM results are compared with the wake vortex filament

positions as criterion. Position 1 is a straight line parallel

to the y-axis at

x=1∕4

as shown in Fig.3. It approximates

the lifting surfaces with a line while neglecting the sweep.

Position 2 also is at the 1/4 line but in addition the sweep is

modelled. Position 3 of the lifting line is at the swept lifting

surfaces’ trailing edges. The last option results in the best fit

of the VLM’s and LLM’s wake filament positions. For that

reason it is selected to chordwise position the lifting lines.

Thus, by adapting the lifting lines position and using the

VLM’s circulation distribution, the parameters of the LLM

are tuned for the comparison of the roll-up.

Another criterion for the parameter selection is to pro-

duce wind velocities in the far field comparable to that

of the SHVM with the actual circulation distribution in

Fig.2. References [22] and [33] state that the cores of

the vortex filaments should be overlapping throughout

the simulation. The overlap ratio

is the ratio between

vortex spacing and core radius. It can be approximated by

Ky=b∕(rc

⋅

, where N stands for the number of span-

wise vortex segments along the wing. A core radius was

selected, for which all discretisation cases satisfied the

criteria of maximum upwind velocity and upwind area

best, yielding

rc=0.02 b=0.43 m

. Only D1 and DF have

Table 2 Settings for the simulation of the flight scenario

Wind field length 1000 m

≈7.14

s flight time

Δt

0.02 s

→Δx=2.8 m

Δt

(DF) 0.0007 s

→Δx=0.1 m

Regularisation function Low-order algebraic

0.43 m

0.43

𝛼

3.85 deg

𝜌

0.6309

kg∕m3

VTAS

140 m∕s

6400 m

m17400 kg

Evaluation grid resolution

0.1 m

Fig. 8 Circulation distribution of the VFW 614 as computed by the

VLM. The HTP’s lift spans in the range [

−4.5 m;4.5 m

]

771

Simplified vortex methods tomodel wake vortex roll‑up inreal‑time simulations forfuel‑saving…

1 3

overlap (

). In x-direction, D1, D2 and D3 do not

achieve an overlap, as

Kx=Δx∕rc=2.8 m∕0.43 m =6.5

This unfavourable ratio is used under the assumption that

the impact on the accuracy is low, if steady or almost

steady flight conditions are investigated. For the refer-

ence case DF (

Δx=0.1 m

Δt=0.0007 s

), the overlap

ratio

Kx=Δx∕rc=0.1 m∕0.43 m =0.23

is adequate. The

escape of filaments from the spiral can be avoided by reduc-

ing the step size and thus the distance each filament trav-

els in between time steps. Reducing the time step size also

increases the vortex core overlap.

4 Vortex identification method

Introducing regularisation to the potential vortex solution

in the Biot–Savart equation leads to rotation in the velocity

field, see Sect.2.5. This rotation is bound to the proximity

of the vortex cores. Overlapping cores result in common

rotational centre [34]. As the cores of the neighbouring fila-

ments are overlapping during wake vortex roll-up, a com-

mon rotational centre evolves during vortex roll-up. The

rotation in the merging cores allows the identification of the

wake vortex axes.

Given a monotonously increasing circulation from the

tips to the middle of the lifting surfaces, all free vortices

on the same side of the aircraft rotate in the same direc-

tion. The rolled-up wake vortex core positions for the left

and right can thus be identified by calculating the position

of two (positive and negative) extrema in the vorticity, see

also Ref. [35]. The vorticity is calculated as the curl of the

- and

- velocity field in the evaluation planes, neglect-

ing a small u-component in the VLM. The determination of

the cores of the rolled-up wake vortices over all evaluation

planes yields the trajectory of the wake vortex axes. Further-

more, the wake vortex span can be calculated. Gerz etal.

[35] determine the distance between the rolled-up cores by

b0=sv

⋅

, where

For an elliptical lift distribution,

equals

𝜋∕4

. The root

circulation

Γ0

at the middle of a lifting surface

is calculated from the mass m to be lifted and the gravita-

tional acceleration g. The root circulation

Γ0

is equal to the

lifting surfaces contribution to the total circulation of the

wake vortex. Equations (4.1) and (4.2) determine how lift,

rolled-up vortex core span and the total circulation interact.

(4.1)

v=2

b∫b

Γy

Γ0

(4.2)

𝜌s

By rearranging Eq. (4.1) and using

b0=sv

⋅

, the half of the

rolled-up vortex core span becomes

Using the discrete circulation distribution readout (

Γi

at i-th

filament y-position

yFil,i

) from the VLM and discretising the

integral 4.3 into the Riemann sum, the expected spanwise

-position of the rolled-up cores becomes

5 Comparison ofLLM andVLM results

This section describes the methods for comparison of LLM

and VLM and the results. The comparison is structured

as follows. First, the position of all free vortex filaments,

and the position of the identified rolled-up vortex cores for

LLM and VLM (see Sect.4) are compared in Sect.5.1. The

expected rolled-up vortex core span

, calculated by Eq.

(4.4), is also compared. The expected value of

repre-

sents a vortex pair spacing equivalent to the spacing of the

SHVM’s two free vortices. In the case of an often assumed

elliptical circulation distribution,

becomes

𝜋∕4b

. In the

present study,

is identified from the results of the roll-up

with a non-elliptical distribution, as explained in Sect.4.

Second, the upwind distribution and the maximum verti-

cal velocity are used to substitute the tangential wind veloc-

ity analysis as a measurement for the severity of induced

rolling moments, and compared for LLM and VLM in

Sect.5.2. The upwind distribution is also compared to that

of the SHVM, see Fig.2.

5.1 Comparison offilament position andvortex

axes position

The differences between the VLM and LLM results are com-

pared by evaluating the position of all free vortex filaments.

For each time step, the y- and z-position of the i-th filament

in the LLM is compared to the i-th filament of the VLM.

This procedure is applied to each lifting surface.

As the flight condition is symmetric, only comparing one

side of the wake shape (characterised by the filament posi-

tions) is sufficient. Figure9 illustrates this comparison for

D2, ten spans downstream—equivalent to about 1.5 s behind

the leading aircraft.

On the left side of Fig.9, only the filament positions

from the LLM are visualised and on the right those from

the VLM.

(4.3)

=∫

Γy

(4.4)

∑

i=1

ΓiyFil,i

772 H.Spark, R.Luckner

1 3

The mean value and the standard deviation of the dif-

ference in the vortex filament position are illustrated in

Figs.10, 11 and 12.

For n given filaments per lifting surface side, the mean

value

𝜇

of the spanwise filament position difference

between LLM and VLM at a filament age t is

The standard deviation of the spanwise positions

𝜎

is cal-

culated by

The same formulae apply for the analysis of the z-locations.

For the finest resolution of the aircraft’s geometry, D1,

the standard deviation

𝜎

increases for filaments older than

one second as can be seen in Fig.10. For D2 and D3, this

phenomenon starts later and is less pronounced, see Figs.11

and 12. In the mean deviations

𝜇

, a large increase cannot

be observed. The mean deviations are starting at values up

to 0.4m for the vertical distance of the HTP filaments but

are becoming small at large distances behind the aircraft

(

<0.1

m) respectively for older filaments. This leads to the

conclusion that the mean filament position does not greatly

differ for VLM and LLM calculations.

The standard deviation

𝜎

in y- and z-direction increases

with time, as differences in filament positions cause different

induced velocities and hence different filament positions in

the next time step. Also, finer discretisations cause higher

(5.1)

𝜇

(t)=

∑

i=1

(yFil,i,LLM(t)−yFil,i,VLM(t))

(5.2)

𝜎

(t)=

√

∑

i=1

yFil,i,LLM(t)−yFil,i,VLM(t)−𝜇(t)

Fig. 9 Comparison of the vortex filament positions for D2 for a vor-

tex age of

1.5 s

02468

0.5

1.5

2.5

y:Wing

z:Wing

y:HTP

z:HTP

02468

-0.5

-0.4

-0.3

-0.2

-0.1

0.1

0.2

0.3

y:Wing

z:Wing

y:HTP

z:HTP

Fig. 10 Mean and standard filament position deviation for D1

02468

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

y:Wing

z:Wing

y:HTP

z:HTP

02468

-0.4

-0.3

-0.2

-0.1

0.1

0.2

y:Wing

z:Wing

y:HTP

z:HTP

Fig. 11 Mean and standard filament position deviation for D2

02468

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

y:Wing

z:Wing

y:HTP

z:HTP

02468

-0.4

-0.3

-0.2

-0.1

0.1

0.2

y:Wing

z:Wing

y:HTP

z:HTP

Fig. 12 Mean and standard filament position deviation for D3

773

Simplified vortex methods tomodel wake vortex roll‑up inreal‑time simulations forfuel‑saving…

1 3

𝜎

-values, as smaller distances between the filaments lead

to higher induced velocities at the neighbouring filaments.

But, as Sect.6 shows, finer discretisations do not lead to

significant differences in the upwind profiles. So the stand-

ard deviation of the vortex filaments is not a good metric for

comparing the fidelity of the methods.

Next, the positions of the wake vortex cores are ana-

lysed, defined by the identified wake axis trajectory. The

axes are determined with the vortex identification method

that is explained in Sect.4. The comparison is carried out

for D1, that resulted in the highest standard deviation

𝜎

for

the individual filaments. Figure13 shows how well the wake

trajectories of both methods are matching.

The cores originate from the wingtips at

ya=±10.75

m

and form the vortex axes with the typical descent.

Also, the expected convergence to a spanwise distance

is visible. Table3 lists the

-positions of the vortex axes

at the end of the simulated wind field that is one kilometer

behind the aircraft, equivalent to a vortex age and travel time

of the aircraft of 7.14 s.

Additionally, the expected value of

b0∕2

is calculated by

Eq. (4.4) for (a) only the wing’s circulation distribution and

(b) the combined circulation distribution of wing and HTP.

For the latter case, Eq. (4.4) is evaluated for all positions

and circulations of bound vortex filaments of wing and HTP

together, adding up the root circulations

Γ0,w

and

Γ0,HTP

. For

D1, the expected end value for

b0∕2

6.94 m

and its evolu-

tion is visualised in Fig.14 together with half vortex span

that would result from an elliptical distribution (

8.45 m

A helical behaviour of the vortex axes that Figs.13 and 14

are showing for the cases D1 and DF also exists for the cases

D2 and D3. This motion continues throughout the simula-

tion, so the listed values for the end positions are not steady

state values. The results thus depend on the

-position, in

contrast to the SHVM where the spacing of the vortex pair

is fixed. Between LLM and VLM, the differences in the end

positions are small. D1 and D2 sufficiently well agree with

expected values for the combined circulation distribution of

wing and HTP, see Table3, D3 does not. In Fig.14, mark-

ers are placed on the wake vortex axes one second apart.

After 15b, equivalent to

2.3 s

, the roll-up is assumed to be

completed. The axes are close to the expected

6.94 m

at this time. However, they differ after the

2.3 s

, rotating

about a centre that lies less than

left of the

6.94 m

line.

5.2 Comparison ofinduced wind fields

Now, the calculated vertical wind fields of both methods

and the vertical wind field of the SHVM are compared. The

comparison is carried out at two positions,

xa=10 b

and

km behind the aircraft. The first position reflects the

minimal distance between aircraft in the recent formation

flight simulations at TU Berlin [15]. The second distance

marks the end of the computed wind field. The position at 1

km is farther downstream than 40 spans, which is a typical

distance in formation flight simulations in Ref. [15].

The wind fields are evaluated by an analysis of the upwind

difference in the

-plane and of the vertical velocity at

the local height

of the identified wake vortex axes.

Because the wind field is symmetrical to the

-plane,

only the starboard wind field (

) is evaluated. Figure15

shows the upwind at a distance of 10b behind the lead-

ing aircraft, corresponding to a separation of 1.5 s between

leader and follower. Figure16 compares the upwind veloci-

ties at the distance of 1 km that corresponds to a flight time

of 7.14 s. The right half of the leading aircraft illustrates the

scale in Figs.15 and 16.

The wake vortex parameters of the identified core loca-

tion and the maximum vertical wind velocity calculated by

LLM and VLM are similar as can be seen by the zero cross-

ings of the vertical wind velocity and the adjacent maxima

on the right side of Figs.15 and 16.

The results differ for the three discretisation cases. This

means, the calculations are sensitive regarding the choice

of spanwise discretisation. The chosen parameter set, see

Table2, is suboptimal as—against expectation and based on

a coincidentally favourable position of the strongest wingtip

vortices—only the coarsest discretisation, D3, achieves an

upwind distribution comparable to the SHVM reference, and

only for

xa=10 b

, see Fig.15. The reference distribution

in Fig.2 has no local minima and maxima. Such a smooth

profile is not generated with the used parameter set.

The contour plots on the left of Figs.15 and 16 show

that the local differences between VLM and LLM in the

plane increase with growing distance behind the leader

aircraft, in other words for a higher vortex age. The devia-

tions also become more pronounced for finer discretisa-

tions, for example D1 in Fig.16. The deviation contours

have their peak values where the wake vortex axes are

located, as a comparison with the wake trajectories in

Figs.13 and 14 shows. In Fig.16, the maximal deviations

are vertically located approximately at

=12

m for all

discretisation cases. This location corresponds to the lat-

eral and vertical wake position in Fig.14 at the end of the

computed field, where the vortex age is

t=7.14

Despite not precisely fitting the SHVM’s vertical wind

distribution, the curves on the right side of Figs.15 and

16 exhibit a smooth positive upwind distribution outside

the vortex cores. This area is important for formation flight

simulations. LLM and VLM calculate similar upwind

distributions. So, the maximal vertical velocities that are

related to the rolling moment and related to the energy

usable in formation flight are also similar. Furthermore,

it is expected that the similarity between LLM and VLM

also holds for distances larger than 1 km.

774 H.Spark, R.Luckner

1 3

6 Discussion ofdiscretisation parameters

Section5 compared LLM and VLM for steady horizontal

flight. This flight condition is relevant for energy-saving for-

mation flight. At large, VLM and LLM yield similar results

for the three discretisation cases (D1, D2 and D3). As the

VLM is computationally runtime-expensive, the discretisa-

tion parameters are set to relatively coarse values. The values

are significantly coarser than in Kaden’s LLM study [5].

=1 km

, the vortex has rolled up. This should be

visible in the induced vertical velocity. However, none of the

three discretisation cases has reached the reference veloci-

ties of the SHVM. That is an effect of the discretisation.

Although the vortices are rolled-up, that means the vortex

filaments have approached to a common axis, the conver-

gence is not sufficient to attain the expected core radius with

the corresponding peak velocities. The persistent helical

deformation causes the difference in the vortex axis loca-

tion. Figure17 shows the expected induced vertical veloc-

ity of a rolled-up vortex computed with SHVM (low-order

algebraic regularisation) and with the LLM for D1 and the

finer discretisation case DF from Sect.3. The number of

bound vortices in DF (128 filaments for the half wing and

64 for the half HTP) is four times higher than for D1 and the

time step size

Δt

is roughly 28 times smaller. Reference [5]

used the same discretisation. The induced wind velocities

are only computed with LLM as the VLM results are similar.

The comparison shows that the discretisation causes differ-

ences. The differences are larger for D2 and D3—not shown

here. This is not surprising as the overlap ratio in spanwise

direction

is poor for D2 and D3. Therefore, D3 and D2

are not further investigated. But the D1 results deviate as

well. Obviously,

Δy

and

Δt

are still too large. However, the

computer performance requirements for real-time computa-

tion prohibit a finer discretisation. This raises the question:

are the selected parameters for D1 (

Δy

Δx

resp.

Δt

and

the regularisation function) adequate for the task?

To evaluate the suitability of D1 for the formation flight

application and to analyse the impact of the discretisation

step size, DF is used as reference. In order to assess the

impact of

and the regularisation function, an additional

discretisation case was generated with the same

Δy

and

Δt

as DF but with

rc=0.015 b

and the Gauss regularisation. It

is labelled DFmod.

Figure18 compares the induced vertical velocity in span-

wise direction for D1, DF, DFmod and the SHVM 10 wing

spans behind (

xa=10 b

). The vertical wind velocities out-

side the cores (

10 m

), which are relevant for formation

flight, are similar in all cases. Therefore, a finer discretisa-

tion than in D1 would not significantly increase the quality

of the velocity model. Surprisingly, this can be achieved

with a coarse

Δt

The vortex core locations of D1 and DF lie close together

=8m

. They differ by approximately

Δya=1m

from

the theoretical values that the SHVM uses. The vortex roll-

up is not completed. Within the core (

7.5 m

8.5 m

Fig. 13 Identified vortex axes for D1

Fig. 14 Identified vortex axes for D1 and F1 with markers every sec-

ond

Table 3 Comparison of the half vortex span

b0∕2

at the end of the

simulated wind field and the results of Eq. (4.4)

Discretisation

(b0∕2)1 km

b0∕2

wing and

HTP

b0∕2

wing only

D1 LLM: 7.60 m

VLM: 7.62 m

6.94 m 7.44 m

D2 LLM: 6.82 m

VLM: 6.87 m

6.98 m 7.49 m

D3 LLM: 4.14 m

VLM: 4.15 m

7.02 m 7.61 m

775

Simplified vortex methods tomodel wake vortex roll‑up inreal‑time simulations forfuel‑saving…

1 3

Fig. 15 Upwind comparison for

distance

10 b

. View from

behind the leader aircraft, its contour

is overlaid in grey

Fig. 16 Upwind comparison for dis-

tance

km. View from behind

the leader aircraft, its contour is

overlaid in grey

776 H.Spark, R.Luckner

1 3

the induced wind velocities are smoother for DF than for

D1. However, that effect seems to be less important for for-

mation flight simulations than the difference in amplitudes

(D1:

6.8 m∕s

and

−5.5 m∕s

, SHVM:

13 m∕s

and

−9.9 m∕s

The difference can be reduced by selecting a regularisation

function that provides higher maximum tangential veloci-

ties. Using a Gaussian instead of the low-order algebraic

regularisation function in the LLM increases the maxima by

approximately 20%, which is close to the theoretical value

of the SHVM. As the following aircraft in formation flight

shall avoid the region near the core, where the induced roll-

ing moment is excessive, the size of the maxima is only

relevant for failure case investigations, when this region is

unintentionally penetrated. But, a Gaussian regularisation,

as well as a high-order algebraic regularisation, can improve

the convergence with increasing number of vortex filaments

compared to the low-order algebraic regularisation, see Ref.

[31]. If required, the LLM results can be further tuned to bet-

ter fit a reference vertical velocity function. Kaden [5] stud-

ied for discretisation DF how variations of the core radius

between 0.015 and 0.035b impact the induced vertical

velocities.

Heintsch [36] analysed the influence of the step size on

vortex filament location and on the induced velocities for

a forward Euler method that is also used for the numeri-

cal integration in this study. He states, that beginning at a

step size above

Δt=0.01 s

deviations from the typical and

expected spiral form of the vortex roll-up appear. This ren-

ders the used step size

Δt=0.02 s

for D1, D2 and D3 too

large. For DF,

Δt=0.0007 s

is appropriate. Heintsch also

analysed the impact of the selected integration method by

comparing the Euler integration method to second-order

Adam–Bashford and fourth-order Runge–Kutta for identical

time step sizes. As the results of all methods were similar,

Heintsch favoured the Euler method because it is the sim-

plest one. We used the Euler method in this work without

investigating more sophisticated integration methods.

7 Conclusions

This paper contributes to the choice of a real-time simula-

tion method for modelling vortex-induced velocities that are

generated by the wake vortex of a leading aircraft and that

the trailing aircraft shall use for energy saving in formation

flight. Two methods are compared, LLM and VLM. Both

are more sophisticated than the simple SHVM that is often

used to represent the vortex mid field (> 15 wing spans)

but that is not well suited to model the vortex roll-up in the

near field.

The comparison is focused on characteristics that are

important for fuel-saving formation flight. Hence, the loca-

tion of the vortex axes and the vortex-induced upwind veloc-

ities are compared. They influence the sweet-spot position

and the potential energy savings.

The LLM requires the circulation distribution over the

wing for the definition of the lifting lines. If it is not avail-

able, an elliptical distribution is often used. The VLM

calculates the circulation distribution from the lifting sur-

face geometry and the airflow condition. So, it is possible

to adapt it to different wing geometries and varying flight

conditions. For the comparison, the circulation distribution

was computed with the VLM. This circulation distribution

significantly differs from the often-assumed elliptical distri-

bution that only considers the main wing. Thus, calculation

of the actual distribution increases the realism, especially

if the HTP contributes to the overall lift. The circulation

distribution calculated by VLM was also used by the LLM

in order to eliminate the influence of different distributions.

Fig. 17 Induced vertical velocity in span wise direction for D1, DF

and SHVM at

1 km

Fig. 18 Induced vertical velocity in span wise direction for D1, DF,

DFmod and SHVM at

10 b

777

Simplified vortex methods tomodel wake vortex roll‑up inreal‑time simulations forfuel‑saving…

1 3

Both methods use parameters (core radius, discretisa-

tion of wing and tailplane, time step size) that are tuned

to achieve a velocity field after roll-up that resembles the

velocity field of a SHVM. The selected discretisation param-

eters and the time step size are coarse to allow simulation

with both methods on a standard computer. Despite using

coarser parameters than in Refs. [18] and [36], the upwind

field in the area outside of the cores, which is important for

formation flight simulation, has a smooth shape. The limits

for the discretisation which still allow achieving the required

accuracy for formation flight simulation are strongly coupled

to the time step size and the core radius value. Discretisa-

tion D1 with 32 filaments on the half of the wing leads to

a comparable updraft distribution as the four times higher

discretisation DF that additionally uses a 28 times finer inte-

gration step size

Δt

The analysis leads to the recommendation to reduce the

time step size from

0.02 s

below

0.01 s

. A selected discre-

tisation can be optimised to better fit a reference velocity

profile. This fitting can be achieved by changing the regu-

larisation function and the local vortex core size.

Vortex roll-up computations with LLM and VLM show

similar results for the wake vortex axes and for the maximum

vertical wind velocity. Deviations are locally observed for

the position of the free vortex filaments and for the wind

velocities. They increase with finer discretisation as the core

radius and the integration time step are kept constant. The

mean deviations for the filaments position are very small

(

<0.4

m), standard deviations reach 2 m and deviations in

local vertical velocities near the wake cores reach 2 m/s.

However, the distance differences are small in relation to

the relative distance between the aircraft and the velocity

variations will be smoothened as the trailing aircraft flies

parallel to the vortex axes. Thus, it can be assumed that the

simplifications for the LLM do not significantly impact the

accuracy of the velocity field simulation for steady flight—

when compared with the more accurate VLM results.

As the LLM is less computationally expensive than the

VLM and delivers similar results for straight trajectories, TU

Berlin uses the LLM for vortex field calculations in real-time

simulations where the leading aircraft is in a steady cruise

flight condition. Even real-time simulations of formation

flight manoeuvres have been performed with LLM. Wake

vortex velocity fields are simulated along curved trajectories

using coordinate transformations as explained in Ref. [15].

However, only VLM can directly model the wake vortex

roll-up in manoeuvres as stated in [37]. For real-time simu-

lations of the roll-up, the calculation runtime needs to be

significantly reduced.

Future work should consider methods as they are

described in [19] and [20], for example to use multiple levels

of vortex filament discretisation downstream and thus reduc-

ing the number of filaments, to reduce the computation time

for both, VLM and LLM, and to achieve finer discretisation

for a given computational performance.

Acknowledgements This publication is based on a German version

originally published for the annual congress of the Deutsche Gesells-

chaft für Luft- und Raumfahrt (DGLR) in 2020 [37]. This research was

performed with internal funding of TU Berlin. The authors would like

to acknowledge the help of André Kaden and Kai Loftfield.

Funding Open Access funding enabled and organized by Projekt

DEAL.

Open Access This article is licensed under a Creative Commons Attri-

bution 4.0 International License, which permits use, sharing, adapta-

tion, distribution and reproduction in any medium or format, as long

as you give appropriate credit to the original author(s) and the source,

provide a link to the Creative Commons licence, and indicate if changes

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the article's Creative Commons licence and your intended use is not

permitted by statutory regulation or exceeds the permitted use, you will

need to obtain permission directly from the copyright holder. To view a

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