A Time Marching Study of Slender Wing Rock

M.R.Allen and K.J.BadcockDepartment of Aerospace Engineering, University of Glasgow, Glasgow G12 8QQ, U.K.

Aero Report 0320

Abstract

This report is the first six monthly progress report for the European Office of Aerospace Re-search and Development contract FA8655-03-1-3044, which started in June, 2003. The aim ofthe project is to develop a direct prediction of wing rock boundaries using Hopf Bifurcation tech-niques previously demonstrated for the prediction of aeroelastic stability boundaries. An essentialpreliminary for this work is to have a time marching capability for this problem. This has beenestablished and is documented in this report. A preliminary version of the time marching code hasbeen released to the US Air Force and a second version, with updated output features necessaryfor analysis by Proper Orthogonal Decomposition, will be installed during a visit to Dayton byMark Allan in January, 2004. The next stage of the work is to formulate and code the bifurcationbased predictions.

It has previously been shown that Euler simulations adequately model the vortex dynamicsassociated with slender wing rock. The effect of varying parameters such as angle of attack,sweep angle, mass ratio, and sideslip angle has been examined. Whilst varying the mass ratio andsideslip angle alters the characteristics of the wing rock motion (ie. reduced frequency and meanroll angle respectively), they cannot be used to eliminate wing rock. It has been found that for asimple wing (ie. no control surfaces) the onset of wing rock is entirely dependent on the angle ofattack and sweep angle of the wing. By varying these two parameters a wing rock boundary hasbeen generated. A starting point for the planned work on the direct prediction of wing rock onsetthrough Hopf Bifurcation analysis has been established.

1

2

1 Nomenclature

b Wing span

Cl Rolling moment coefficient ( lqSb

)

Clc Rolling moment coefficient based on

cr ( lρU2c3

r

)

CP Pressure coefficient

cr Root chord

k Kinetic energy of turbulent fluctuations

per unit mass

l Rolling moment

Pk Limited production of k

P uk Unlimited production of k

q Dynamic pressure

r Ratio of magnitude of rate of strain

and vorticity tensors

Re Reynolds number

S Wing area

t Time (s)

α Angle of attack

β∗ Closure coefficient

Γ Circulation

η Spanwise coordinate/local span

φ0 Initial roll angle

φ Instantaneous roll angle

ρ Freestream air density

ω Specific dissipation rate

τ Non-dimensional time

3

2 Introduction

The desire for increased speed and agility has led to aircraft designs with increasing wing sweep and

the addition of highly swept leading edge extensions. A common dynamic phenomenon of slender

aircraft flying at high angle of attack is known as wing rock. A classic example of slender wing rock

occurred on the Handley Page 115 research aircraft which was designed to study the aerodynamic

and handling qualities of slender aircraft at low speed [1]. At angles of attack above 20o the aircraft

first experienced wing rock, with the maximum roll amplitude experienced being around 40o at 30o

angle of attack. This motion was suppressed by reducing the angle of attack or applying an input

to the aileron. Aircraft which experience such motions tend to have highly swept surfaces or have

long slender forebodies which produce vortical flow at high angleis of attack. At some critical angle

of attack the aircraft can experience a roll oscillation which grows in amplitude until a limit cycle

oscillation is reached. A loss in roll damping is usually associated with wing rock. Although wing

rock is a complicated motion which involves several degrees of freedom, the primary motion is a

rolling motion around the aircraft longitudinal axis. This motivates experimental and computational

studies involving only one degree-of-freedom rolling motions to adequately reproduce the dynamics.

Wing rock of slender delta wings has been studied experimentally over the past two to three

decades and several review papers have been published within the last decade [2] [3] [4] [5] with

wing rock being a main topic. It should also be noted that wing rock can occur for nonslender delta

wings [] and even rectangular wings of low aspect ratio (less than 0.5) [6].

Levin and Katz [7] performed an experimental study of wing rock with 76o and 80o sweep delta

wings with the wing mounted on a sting-balance. It was observed that only the 80o sweep wing

would undergo self-induced roll oscillations for the given experimental conditions (Reynolds number,

bearing friction and wing moment of inertia), therefore it was concluded that the wing aspect ratio

must be less than 1 for wing rock. The Reynolds number based on the root chord was 5 × 105 and

wing rock was observed for an angle of attack of 20o. However to obtain wing rock at this incidence,

oscillations were started at an angle of attack lowered to 20o. It was not possible for the wing to self

induce wing rock when at a fixed angle of attack of 20o. The presence of vortex breakdown over the

wing was found to limit the amplitude of the LCO. During the free-to-roll motion a loss in the wing

4

average lift was observed relative to the static lift for the same angle of attack. It was also observed

that side forces on the model were high, indicating that the presence of wing rock will have a strong

influence on free flight models. Increasing the wind tunnel speed (Reynolds number) increased the

amplitude of the LCO but the reduced frequency of oscillations remains almost unchanged.

Katz and Levin [8] performed an experimental study of a delta wing / canard configuration. The

canard has an aspect ratio of 0.7 with the wing having an aspect ratio of 1. The Reynolds number

based on the wing’s root chord was 3 × 106 and the mechanical friction was minimal (less than 0.2

g-cm). Unlike the results of Levin and Katz [7] where wing rock was not observed for this identical

wing, with the reduced mechanical friction in the experimental setup wing rock occurred. The effect

of the canard was observed to increase the effective leading edge sweep of the configuration. As

such, with the increased effective sweep, the 75o wing / canard configuration has an enlarged wing

rock envelope. Complex vortex interactions were present (since two vortices were produced by the

canard and two by the wing) and nonsymmetric oscillations occurred.

Arena and Nelson [9] undertook a series of comprehensive studies on an 80o sweep delta wing

undergoing wing rock. An air bearing spindle was developed for the free-to-roll tests that allowed

an isolation of the applied torques due to the flowfield. This implies that the mechanical friction

coefficient is effectively zero. Motion history plots were obtained, as well as static and dynamic flow

visualisation of vortex position and breakdown location, static surface flow visualisation, steady and

unsteady surface pressure distributions. It was observed that there was a rate dependent hysteresis in

vortex location due to a time lag in the motion. This time lag was found to produce a dynamically

unstable rolling moment able to sustain the wing rock motion.

Rolling motion about the longitudinal axis of a delta wing has been computed using CFD by

several researchers [10] [11] [12] [13]. Recently a comprehensive numerical study of wing rock was

conducted by Saad [14] using a three degree-of-freedom flight mechanics model for a generic fighter

aircraft configuration (forebody, 65o leading edge sweep, and vertical fin). Roll, sideslip and vertical

degrees of freedom were allowed. Including the sideslip degree of freedom was found to delay onset

of wing rock and reduce the wing rock amplitude.

This report presents the results of time-accurate CFD simulations of the wing rock of an 80o sweep

5

delta wing. This work will provide the basis for the application of a bifurcation analysis of slender

wing rock. Several possible bifurcation parameters have been assessed with two being selected for

the future bifurcation work.

3 Flow Solver

All simulations described in this paper were performed using the University of Glasgow PMB3D (Par-

allel Multi-Block 3D) RANS solver. A full discussion of the code is given in reference [15]. PMB3D

uses a cell centered finite volume technique to solve the Euler and Reynolds Averaged Navier-Stokes

(RANS) equations. The diffusive terms are discretised using a central differencing scheme, and the

convective terms are discretised using Osher’s approximate Riemann solver with MUSCL interpola-

tion. Steady flow calculations proceed in two parts, initially running an explicit scheme, then switch-

ing to an implicit scheme to obtain quicker convergence. The linear system arising at each implicit

step is solved using a Krylov subspace method. The pre-conditioning is based on Block Incomplete

Lower-Upper BILU(0) factorisation which is decoupled across blocks. For time-accurate simulations,

Jameson’s pseudo-time (dual-time stepping) formulation [16] is applied, with the steady state solver

used to calculate the flow steady states on each physical time step.

For the RANS simulation the two equation k-ω turbulence model is used for closure. It is well

known that most linear two-equation turbulence models over-predict the eddy viscosity within vortex

cores, thus causing too much diffusion of vorticity [17]. This weakens the vortices and can eliminate

secondary separations, especially at low angles of attack where the vortices are weakest. The modifi-

cation suggested by Brandsma et al. [18] for the production of turbulent kinetic energy was therefore

applied to the standard k-ω model of Wilcox [19] to reduce the eddy-viscosity in vortex cores as

Pk = min{P uk , (2.0 + 2.0min{0, r − 1})ρβ∗kω}. (1)

Here P uk is the unlimited production of k, P u

ω is the unlimited production of ω, and r is the ratio

of the magnitude of the rate-of-strain and vorticity tensors. When turbulent kinetic energy is over

predicted in the vortex core, it will be limited to a value relative to the dissipation in that region. This

modification was found to improve predictions compared with the standard k-ω turbulence model and

is used in all simulations presented.

6

4 Free-to-roll model

The non-dimensional one degree-of-freedom roll model implemented is given by

φττ = µClc (2)

where

µ =ρc5

r

Ixx

(3)

and

Clc =l

ρU2∞

c3r

(4)

In equation 2, Clc is the rolling moment coefficient based on the root chord cubed.

The one degree-of-freedom model was coupled to the PMB3D solver by evaluating the flight

mechanics model in the pseudo time stepping loop of the dual time stepping scheme of Jameson

[16]. In this way, the flight mechanics model converges with the flow solution minimising sequencing

errors. Clearly the only variable driving the rolling motion is the rolling moment coefficient which is

updated at each pseudo time step. The most recent update for the rolling moment Clc is used in the

evaluation of the roll angle and roll rate at the following pseudo time step. The implicit integration

scheme is

qn+1,k = qn +∆t

2(Rn+1,k + Rn) (5)

where

q = (φτ , φ)T (6)

and

R = (µClc, φτ )T . (7)

The mesh is rigidly rotated according to the the roll angles computed.

5 Test Case

There is a considerable amount of published experimental data for 80o sweep delta wings. Hence this

sweep angle was selected for the current study. The geometry of the wing was identical to that used

by Arena and Nelson [9]. The wing has a flat upper and lower surface with a 45o windward bevel and

7

a root chord of 0.4222m. The moment of inertia for this wing was given as Ixx = 0.00125 Kg-m2. The

experiment was performed at a Reynolds number of 1.5 × 105. For almost all simulations described

in this paper, an inviscid flow was assumed (with the exception of the RANS simulations described

later where the Reynolds number was matched to experiment) with a freestream Mach number of 0.2.

Since PMB3D is a compressible flow solver, a Mach number of 0.2 was used to avoid convergence

difficulties at very low Mach number.

Since in the CFD simulations all values must be non-dimensional, the freestream air density is

used to non-dimensionalise the moment of inertia of the wing. The freestream air density in experi-

ment is unavailable therefore the freestream air density was assumed to be 1.23Kgm−3 (sea-level ISA

conditions). As can be seen from equation 2, the non-dimensionalisation of the moment of inertia (or

the value of freestream density used) will influence the non-dimensional angular acceleration of the

wing.

6 Verification and Validation

6.1 Euler simulations

Before simulating wing rock the accuracy of steady solutions was verified with a grid refinement

study. A fine grid was created with approximately 1.6 million grid points. From this grid two levels

were extracted by removing every second grid point in each direction. Steady state solutions were

computed for each grid with the residual being reduced 6 orders of magnitude. The test case chosen

for validation purposes was the wing at 30o angle of attack and rolled +10o.

The upper surface pressure distributions from all three grid levels are shown in figures 1 to 3 for

the chordwise stations of 30%cr, 60%cr, and 90%cr. The surface pressures at 60%cr are also com-

pared with those obtained in the experiments of Arena [20]. It should be noted that the experimental

results indicated laminar flow on the upper surface of the wing which has the effect of moving the pri-

mary vortices inboard and upwards off the surface of the wing, which displaces inboard and reduces

the primary vortex suction footprint. This should be kept in mind when considering wing rock since

wing rock can be attributed to hysteretic vortex movement [9]. Examination of the surface pressure

distributions from each grid indicates that the solutions are not grid converged. However Euler sim-

8

η

-Cp

-1 -0.5 0 0.5 1

-0.5

0

0.5

1

1.5

2

2.5Fine GridMedium GridCoarse Grid

Figure 1: Grid refinement study - Surface pressure distributions at 30%cr

ulations of delta wing flows are known to be highly sensitive to grid density [21] and therefore the

current results are as expected. Despite the requirement to chose an appropriate level of grid density,

it is well known that Euler simulations can accurately predict the dynamic response of leading edge

vortices. Comparing the surface pressure distributions at 60%cr, there is a reasonably good agreement

between the solution from the medium grid and experiment.

In figures 1 to 3 the port side of the wing is given by a positive η coordinate. Since the wing

is rolled port side down, the wing is experiencing a lower effective angle of attack and a negative

sideslip angle. As such the port vortex is stronger (lower effective sweep) than the starboard vortex

which induces a restoring rolling moment. Comparing the suction peaks it can be seen that the port

vortex increases in strength at a higher rate than the starboard vortex with increasing grid density. It

would therefore appear that as the grid is refined the flow is more sensitive to sideslip effects. This

may be due to the fact that since the starboard vortex is weaker than the port vortex, the effect of grid

refinement (reduction in vorticity dissipation) may be less than that observed for the stronger port

vortex.

9

η

-Cp

-1 -0.5 0 0.5 1

-0.5

0

0.5

1

1.5

2

2.5Fine GridMedium GridCoarse GridExperiment

Figure 2: Grid refinement study - Surface pressure distributions at 60%cr

η

-Cp

-1 -0.5 0 0.5 1

-0.5

0

0.5

1

1.5

2

2.5Fine GridMedium GridCoarse Grid

Figure 3: Grid refinement study - Surface pressure distributions at 90%cr

10

φ [o]

Cl

-50 0 50-0.1

-0.05

0

0.05

0.1 Medium GridExperiment

Figure 4: Comparison of rolling moments through a steady wing rock cycle - CFD and experiment

The increase in pressure difference between the port and starboard sides as the grid is refined will

induce a stronger restoring moment which is likely to produce a stronger wing rock response. This is

seen in figure 5 where clearly the largest amplitude wing rock is predicted by the fine grid (followed

by the medium and coarse grids). The wing rock amplitudes for the fine, medium, and coarse grids

are 70o, 50o, and 15o, with the wing rock amplitude observed in experiment being 40o. Due to the

low vorticity dissipation of the fine grid the wing rock amplitude predicted is unrealistically large,

and due to the large vorticity dissipation associated with the coarse grid the wing rock amplitude is

unrealistically low. The variations in wing rock reponse with grid refinement are as expected based

on the surface pressure distributions shown in figures 1 to 3.

A comparison of the rolling moments from the medium grid and experiment through one complete

steady wing rock cycle is shown in figure 4. Despite the maximum roll angle being exceeded in the

Euler solution it is clear that there is reasonably good agreement with experiment, with the rolling

moment distribution and magnitudes being predicted well. As discussed by Arena and Nelson [9]

there is a clockwise loop where energy is added to the system, and two anti-clockwise damping lobes

11

τ

φ[o ]

200 400 600 800-80

-60

-40

-20

0

20

40

60

80

Fine Grid, ∆τ=0.1875Medium Grid, ∆τ=0.1875Coarse Grid, ∆τ=0.1875

Figure 5: Grid refinement study - Wing rock histories

when energy is dissipated. Comparing the width of the loops where energy is added, clearly much

more energy is being added to the system in the Euler simulations in comparison to experiment.

It should also be noted, however, that the damping lobes are also larger in the Euler solution in

comparison to experiment. Given the difference in wing rock amplitude between the CFD solutions

and experiment, it is likely that the ratio of energy addition to extraction may not be so well predicted.

Based on comparisons of the static upper surface pressure distributions and the amplitude of the

wing rock response with experiment, it can be concluded that the most realistic simulations are ob-

tained with the medium grid. It should be noted that even with the medium level grid the wing rock

amplitude is 50o. This is higher than the experimentally observed amplitude of 40o. However it was

observed by Levin and Katz [7] that as Reynolds number was increased the wing rock amplitude in-

creases, therefore it is perhaps unsurprising that an Euler solution will exhibit a more energetic wing

rock response. Comparing the computed reduced frequency of motion with the medium grid with the

experiments of Arena [9], the reduced frequency (fcr/U∞) from the medium grid is 0.036 and the

reduced frequency from experiment is 0.039.

12

τ

φ[o ]

0 50 100 150-40

-20

0

20

40Medium Grid, ∆τ=0.1875Medium Grid, ∆τ=0.09375Medium Grid, ∆τ=0.046875

Figure 6: Time step refinement study - Wing rock histories

In order to verify the temporal accuracy of the solutions, a time step refinement study was con-

ducted. As can be seen from figure 6, the non-dimensional time step of 0.1875 provides an adequate

temporal resolution for the current wing rock studies. This equates to around 120 time steps per cycle.

For forced motion studies as few as 50 time steps per cycle are sufficient for temporal convergence,

however since in free-to-roll studies errors amplify with time (where auto-rotation was observed to

occur when the time step was too large), a smaller time step is required to accurately predict the wing

rock behaviour.

6.2 RANS simulations

As shown in the preceeding section Euler simulations exhibit a larger amplitude wing rock response

in comparison to experiment. This is perhaps unsurprising given that the location of the primary

vortices is more outboards and closer to the surface. This results in stronger suction peaks on the

wing surface as seen in figures 1 to 3. To investigate the effect of Reynolds number on the solution, a

RANS simulation was conducted.

13

τ

φ[o ]

0 100 200 300

-50

-40

-30

-20

-10

0

10

20

30

40

50

EulerRANS

Figure 7: Effect of modelling - Comparison between RANS and Euler solutions for α = 30o

The upper surface pressure distributions at 60% cr from the Euler and RANS solutions are com-

pared with experiment in figure 8. The RANS solution was performed on a grid with a high resolution

in the vortical and boundary layer regions. The effect of viscosity can be seen as a drop in the primary

vortex suction peaks, which is due to the presence of a secondary separation shifting the vortex loca-

tion. As such the RANS suction peaks and locations compare very well with experiment. It should

be noted that the surface flow in experiment is laminar, however fully turbulent flow was assumed in

the computations. Therefore, as expected, the secondary vortex suction is larger in experiment than

that predicted in the RANS solutions.

Roll angle histories for Euler and RANS simulations of the 80o sweep delta wing at 30o angle

of attack are shown in figure 7. Comparing first the wing rock amplitudes, the amplitudes from

the RANS solution, Euler solution, and experiment are 35o, 50o, and 40o respectively. Clearly the

effect of Reynolds number is to reduce the wing rock amplitude. This is as expected based on the

previous discussion. With the higher more outboard suction peaks from the Euler solutions there is a

much larger wing rock response in comparison to the RANS solutions, where the suction peaks are

14

η

-Cp

-1 -0.5 0 0.5 1

-0.5

0

0.5

1

1.5

2

2.5Fine GridMedium GridCoarse GridExperimentRANS

Figure 8: Effect of modelling on upper surface pressure - Comparison between RANS and Eulersolutions

15

τ

φ[o ]

0 100 200 300-16

-12

-8

-4

0

4

8

12

16α = 15o

α = 20o

Figure 9: Wing rock roll angle histories - α = 15o and 20o

lower and more inboard. Comparing the period of the wing rock cycle we can see that the Euler and

RANS solutions produce similar results. As such it can be concluded that the Euler solutions predict

qualitatively the correct vortex dynamics and therefore a realistic wing rock response.

7 Parametric studies

In all the simulations described in this section, the time step was chosen such that there were at least

120 time steps per cycle (which has been shown to provide a sufficient temporal resolution). The wing

rock motion was also initiated by starting all solutions from a positive roll angle of 10o and computing

the wing response.

7.1 Effect of varying angle of attack

Simulations of free-to-roll motion at 15o and 20o angle of attack were conducted. The roll angle

histories for the first several oscillations of both simulations are shown in figure 9. Clearly at 15o

angle of attack the solution is dynamically stable (ie. the amplitude of the oscillations decreases

16

ln (µ)

ln(f

c r/U)

-4 -3 -2 -1 0 1 2 3-7

-6

-5

-4

-3

-2

∞

Figure 10: Effect of increasing ρC5r

Ion the reduced frequency of wing rock, and the rate of oscillation

amplitude growth - α = 30o

due to aerodynamic damping). However at 20o angle of attack, the initial roll angle of 10o initiates

a wing rock response with the amplitude of the motion increasing with time. The onset of wing

rock observed in experiment was 22o [9]. The computed result therefore agrees reasonably well with

experimentally observed results, though it is possible that the Euler model of the flow may shift the

wing rock boundary towards lower angle of attack.

7.2 Effect of increasing mass ratio

As we can see from equation 2, the non-dimensional angular acceleration φττ is dependent on the

mass ratio µ (which varies with model size and freestream air density) and the aerodynamic rolling

moment.

The results of a parametric study where the mass ratio µ was varied is presented in figure 10.

It should be noted that for all cases computed wing rock was observed. µ was varied from values

consistent with experimental studies up to and beyond full aircraft scales. The natural logarithm of

17

τ

φ[o ]

0 100 200 300 400 500-60

-50

-40

-30

-20

-10

0

10

20

30

40

50

60

70

β = -5o

β = +5o

β = +10o

Figure 11: Wing rock roll angle histories with varying sideslip angles - α = 30o

reduced frequency of wing rock for each case is plotted against ln(µ). As can be seen in figure 10,

when the results are plotted on logarithmically a linear variation is observed. As such we can easily

derive a simple empirical relationship between µ and reduced frequency. The empirical relationship

for the variation of reduced frequency with µ.

fCr

U∞

=1

75.0

õ (8)

From equation 8, it is clear that as the mass ratio µ decreases (ie as model / aircraft size increases)

the reduced frequency of motion decreases rapidly. Given that the effect of increasing the model size

and hence moment of inertia is to simply increase the period of the oscillations it is unlikely that

further increases in µ would prevent wing rock. In fact based on the empirical relationship, wing rock

can only be prevented by increasing the moment of inertia so much that the aircraft does not move

around its longitudinal axis.

18

7.3 Effect of sideslip

The effect of varying the sideslip angle of the wing is shown if figure 11. It is clear from figure 11 that

as the sideslip angle is varied, although wing rock occurs in all cases the oscillations occur around a

different mean roll angle. It must be recalled, however, that changing the sideslip angle changes the

direction of the freestream velocity vector, and as such it can be expected that the wing may oscillate

around a new mean roll angle. It should also be recalled that as mentioned previously, all simulations

start from an initial positive roll angle of +10o. With a positive sideslip angle of 5o and a positive roll

angle of 10o (port side down), the wing is at a lower effective roll angle. This is clear from the fact

that the oscillations for a sideslip angle of +5o are much lower to start with in comparison to the other

solutions, eventually building up to a limit cycle oscillation. As the positive sideslip angle increases

further, the effective roll angle reduces, becoming negative for a sideslip angle of +10o. This is clear

from the fact that the direction of the restoring moment turns the wing in the opposite sense for a

sideslip angle of +10o, in comparison to the other solutions. Finally if the sideslip of the wing is

negative the effective roll angle increases, and as such, the wing starts to oscillate with a much larger

amplitude.

7.4 Effect of sweep angle

It is well known from experiment that decreasing sweep angle can prevent wing rock from occurring

[5]. Therefore sweep angle can be used as a bifurcation parameter. As such a parametric study was

conducted by varying both bifurcation parameters (angle of attack and sweep angle of the wing).

Mesh density was kept constant, as was the moment of inertia of the model. Figure 12 shows the

wing rock boundary computed by time marching Euler solutions. As can be seen from figure 12 as

sweep angle is decreased the angle of attack at which wing rock occurs increases. This is in agreement

with experiment [23]. Although not presented here, at an angle of attack where wing rock is present

for various sweep angles, as the sweep angle of the wing is decreased the amplitude of the wing rock

oscillation decreases.

19

Sweep angle [o]

Ang

leof

atta

ck[o ]

74 75 76 77 78 79 80 81141516171819202122232425262728293031 No wing rock

Wing rock

Figure 12: Wing rock boundary

8 Concluding Remarks

Computations of slender wing rock have been presented. Time marching Euler simulations have

been shown to adequately predict the vortex dynamics associated with slender wing rock. Various

parameters has been examined in order to establish whether or not wing rock can be eliminated. It

has been found that the occurrance of wing rock can only be prevented by varying either the sweep or

angle of attack of the wing. A wing rock boundary has been generated which can be used to validate

future work using bifurcation methods to predict the onset of slender wing rock.

9 Acknowledgements

This work has been funded by the European Office of Aerospace Research and Development under

contract FA8655-03-1-3044.

20

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22

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