NUMERICAL SIMULATION OF THE FREE PITCH OSCILLATIONFOR A RE-ENTRY VEHICLE IN TRANSONIC WIND TUNNEL FLOW

Bodo Reimann

German Aerospace Center (DLR), Institute of Aerodynamics and Flow Technology,Lilienthalplatz 7, 38108 Braunschweig, Germany

bodo.reimann@dlr.de

ABSTRACT

This paper presents the numerical results carried out bythe German Aerospace Center (DLR) in the framework ofthe ESA Technology Research Program (TRP) DampingDerivatives Assessment for Hypersonic Re-Entry Vehi-cles Exhibiting High Angle of Attack (DERIVAS). Aimof the study is the simulation of the free pitching oscilla-tion of a re-entry vehicle in transonic wind tunnel condi-tions. The unsteady numerical simulations with strongcoupling between the flow around the vehicle and itspitch motion show dynamic unstable behaviour of thevehicle in the transonic regime. The presence of themodel support, holding the vehicle in the test section hasa damping effect on the oscillation. Furthermore, the re-sulting pitch damping sum shows a strong dependency onthe amplitude of the pitch oscillation.

Key words: dynamic stability; pitch damping deriva-tives; free oscillation technique; flight mechanics cou-pling; sting effects; CFD; TAU-code; chimera technique.

1. INTRODUCTION

The entry of a flight vehicle in the atmosphere of a planetcomprises a wide range of flow conditions from highspeed chemical reacting flow at high altitudes down tolow speed close to the landing site. Vehicles designedto resist the extreme loads during hypersonic flight of-ten tend to be unstable in the trans- and subsonic flightregime. This unstable dynamic behaviour often requiresthe deployment of a parachute to stabilize the vehicle.However for a safe operation the controllability along theentry trajectory is essential. For this reason numericalsimulations, as well as wind tunnel experiments are usedto predict dynamic stability.

The aim of the study is to compute dynamic pitch damp-ing derivatives of a lifting body type re-entry vehicle intranssonic wind tunnel flow. The simulated cases repre-sent wind tunnel conditions of corresponding measure-ments carried out in the Trisonic Test Section (TMK) at

DLR Koln. Of special interest is the influence of the sup-port sting holding the model in the test section of the tun-nel. For this purpose steady-state computations as wellas unsteady simulations with full coupling between flowand rigid body motion of the vehicle - so-called free mo-tion technique - have been carried out. The simulationshave been performed at the Institute of Aerodynamics andFlow Technology of the DLR in Braunschweig. The usedCFD tool is the DLR TAU-code.

2. NUMERICAL METHOD

2.1. DLR TAU-code

The TAU flow solver [7] uses the finite volume methodto discretize the Euler or Navier-Stokes equations on un-structured grids. Based on this primary grid an edge-based metric called dual-grid is generated in a prepro-cessing step. If multi-grid technique is used, the prepro-cessor also agglomerates coarser levels of the dual-grid.Also domain splitting is done by the preprocessor in caseof parallel computations.

In the CFD (computational fluid dynamics) solver mod-ule inviscid terms are computed employing either asecond-order central scheme or an AUSMDV upwindscheme using linear reconstruction to get second-orderspatial accuracy. Viscous terms are generally computedwith a second-order central scheme. For time integra-tion various explicit Runge-Kutta schemes, as well asan implicite approximate factorization lower-upper sym-metric Gauss-Seidel scheme (LU-SGS) is implemented.For time accurate computations a Jameson-type dual timestepping approach is employed. Additional convergenceacceleration is achieved by explicit residual smoothing.

To simulate turbulent flows the user has the choice be-tween several one- and two-equation turbulence models,a Reynolds stress or a DES model. For the present studythe perfect gas solver is used. For moving grids TAUis written in an arbitrary Eulerian-Lagrangian formula-tion. Chimera technique [2],[3] is an important feature tosimulate configurations with movable parts. The method

handles the data exchange in the overlapping region. Thecurrent implementation of the Chimera technique coverssteady and unsteady simulations for inviscid and viscousflows with multiple moving bodies, and is also availablein parallel mode.

2.2. Flight mechanic coupling

The motion of the rigid body is computed in the RBD(rigid body dynamics) module solving Newtons secondlaw and the Euler equation of rotational dynamics. Thecoupled CFD/RBD problem is solved in a partitionedmanner. A so-called strong coupling scheme is used [6].Strong coupling means that the coupled equations are it-eratively solved within each physical time step by repeat-edly solving the involved disciplines CFD and RBD sep-arately based on the exchanged coupling quatities. Theseare on CFD side aerodynamic loads (forces and mo-ments) and on RBD side the motion state (position andvelocities). Figure 1 shows a operation chart of a strongcoupled simulation.

Figure 1. Scheme of the strong coupling between CFDand RBD.

2.3. Computational domain

As input for the generation of the computational gridsESA provided a CAD file of the geometry of the vehiclebased on the Intermediate Experimental Vehicle (IXV)aeroshape 2.2. Flap deflection is not considered. Minorcorrections to avoid tiny surface panels and to guaran-tee a watertight contour have been introduced by DLR.The CAD description of the wind tunnel sting holdingthe model in the test section has been constructed byDLR. Several details of the mounting mechanism havebeen simplified for the meshing process. The meshingitself has been done with the commercial mesh generat-ing software CENTAUR. Two types of grids have beengenerated, one for the flight configuration which com-prises only the vehicle, and one for the wind tunnel con-figuration which also includes the model support sting.To simulate the rotation of the vehicle around its y-axiswith respect to the sting the method of overlapping grids(Chimera technique) has been applied for the wind tunnelconfiguration. In the experiments the vehicle is attachedto the sting in the center of rotation (CoR) by a cross flex-ure which allows the vehicle to oscillate around the y-axis

Figure 2. Chimera blocks. Shown are the surface andfarfield boundary of the sting mesh (blue) and the surfaceboundary of the vehicle (grey). The farfield boundary ofthe vehicle mesh is not visible.

during the run [1] The cross flexure has not been takeninto account for the simulations. The Chimera mesh con-sist of two blocks (separate grids), one for the vehicleand one for the sting. Figure 2 shows the two parts ofthe mesh. Two additional grids so-called hole grids areneeded to cut a hole for the body in the volumetric grid ofthe other block, respectively. Figure 3 shows the hole def-inition of the Chimera mesh. In order to ensure enoughoverlap for the interpolation between the Chimera grids,and due to the fact that the gap between sting and vehicle

Figure 3. Chimera hole boundaries. The vehicle holedefinition (red) cuts out the vehicle surface from the mesharound the support sting. The sting hole (green) cuts outthe sting surface from the vehicle mesh.

Figure 4. Computational mesh for the wind tunnel con-figuration. Shown is the vehicle mesh (black) the vehi-cle surface (red) and the mesh around the model supportsting (blue). The farfield sphere has a diameter of 3.75 m.

model is very small, only a maximum pitch amplitude of±2◦ can be simulated. The size of the vehicle is 0.125 m.The spherical farfield of the computational domain has adiameter of 3.75 m. Only one half side of the domainis computed exploiting the symmetry of the model. Forthe free flight grid the notch for the support sting on thetop of the vehicle does not exit. The main part of the pri-mary volumetric mesh consist of tetraedras and pyramids.Boundary layers are resolved using a stack of prismaticcells perpendicular to the triangulated viscous walls. Inregions of interest the grid points are clustered to increasethe resolution of the flowfield. A grid convergence studyhas been performed for the free flight case. The findings

Figure 5. Detailed view of the mesh for the wind tunnelconfiguration.

have than been used for the generation of the wind tun-nel mesh, so that the resolution of both final meshes issimilar. The final grid for the free flight has 5.37 millionpoints (20.89 million primary grid cells), with the supportsting included the number of points rises to 17.71 millionpoints (68.02 million primary grid cells). Figures 4 to 6gives an impression of the final primary meshes.

2.4. Test conditions

The freestream conditions of the TMK used for thesimulations are given in Tab. 1. The gas is as-sumed to be calorically perfect air with γ = 1.4 andR = 287 J/(kg K). The wall is modeled as adiabatic noslip boundary. All simulation are carried out with a tur-bulent flow solver with implemented Menter SST k-ωturbulence model [4]. The reference length of the ve-hicle is Lref = 0.125 m. The reference relation area forthe half model is Aref = 2929.68 · 10−6 m2. The cordi-nates of the center of rotation are xCoR = 72.5 · 10−3 m,yCoR = 0.0 m and zCoR = -3.125 · 10−3 m. The momentof inertia for the half model around the y-axis is com-puted with Iyy = 0.978 · 10−3 kg m2. In wind tunnel teststhe angle of attack α is identical to the pitch angle θ Thevehicle .

Table 1. Simulated conditions of the freestream in TMK.

run 18 run19 run 23 run 10

M∞ 0.80 0.95 1.10 2.01Re∞ [×106] 2.82 3.69 4.60 4.39u∞ [m/s] 251.87 291.28 333.64 506.43ρ∞ [kg/m3] 1.413 1.537 1.633 9.750T∞ [K] 246.2 233.9 227.5 158.5

Figure 6. Detailed view of the mesh for the free flightconfiguration.

2.5. Simulation procedure

Several steady-state computations have been carried outto find the trim angle αtrim of each configuration wherethe pitching moment coefficient Cm is equal to zero. Asolution close to the trim position (θinit) is used as the ini-tial restart solution for the unsteady coupled CFD/RBDsimulation. The pitch damping sum Cmq+Cmα can thancomputed from the time-history of the pitch oscillationamplitude θ(t) [5],

Cmq + Cmα =2u∞

12ρ∞u

2∞ArefL2

ref

f(θ0) (1)

using logarithmic decrements

f(θ0) =2 Iyy

t0,i+1 − t0,iln(θ0,i+1

θ0,i

). (2)

θ0,i describes the i-th local extremum of the pitch oscil-lation amplitude and (t0,i+1− t0,i) the time between twosucceeding extremal values. The pitching moment slopeCmα is computed from the fitted steady-state results. An-gular frequencies ω expected close to the trim point resultfrom

ω =

√CmαI

12ρ∞u2

∞ArefLref . (3)

3. RESULTS

3.1. Grid convergence study

As already mentioned a grid convergence study has beenperformed for the free flight case. On several grids withincreasing spatial resolution steady-state computationsfor various angles of attack have been executed for run19. Figure 7 shows the pitching moment coefficient ver-sus the angle of attack for the six grids. There is no

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

55 56 57 58 59 60 61 62 63

Cm

α [ °]

0.65x106 grid points1.03x106 grid points2.07x106 grid points3.96x106 grid points5.37x106 grid points8.95x106 grid points

Figure 7. Pitching moment coefficient for run 19(M = 0.95) versus angle of attack computed on differentgrids.

-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0.000

0.001

0.002

0.003

0 50000 100000 150000 200000 250000 300000

Cm

time steps

CFD/RBD coupling with stingCFD/RBD coupling w/o sting

0.001

0.002

170000 185000 6⋅10-4

8⋅10-4

160000 195000

Figure 8. Pitching moment coefficient for run 19(M = 0.95) versus number of time steps.

significant difference between the three finest grids. Forthe further study the grid with 5.37 million points havebeen used. The grid including the support sting has asimilar resolution. Figure 8 shows the evolution of thepitching moment coefficient during the numerical itera-tion process. At the end of each physical time step theflow solution is converged. The numerical convergencefor the wind tunnel case is much worse compared to theconvergence of the free flight simulation. The number ofinner iteration for the dual time stepping scheme is nearlythree times higher. It is supposed that this is caused by thehigher complexity of the model geometry and the appli-cation of the Chimera technique.

3.2. Steady-state results

For both configurations and all flow conditions steady-state simulations have been carried out for different an-gles of attack. The results for the pitching moment forrun 18, 19, 23 and 10 are shown in Fig. 9–12. The valuesof the pitching moment are fitted by a second order poly-

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

57 58 59 60 61 62 63 64 65 66

Cm

α [ °]

Fitted steady-state CFD with stingFitted steady-state CFD w/o sting

Figure 9. Steady-state pitching moment coefficient versusangle of attack for run 18 (M = 0.80).

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

55 56 57 58 59 60 61 62

Cm

α [ °]

Fitted steady-state CFD with stingFitted steady-state CFD w/o sting

Figure 10. Steady-state pitching moment coefficient ver-sus angle of attack for run 19 (M = 0.95).

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

58 59 60 61 62 63 64

Cm

α [ °]

Fitted steady-state CFD with stingFitted steady-state CFD w/o sting

Figure 11. Steady-state pitching moment coefficient ver-sus angle of attack for run 23 (M = 1.10).

nomial. Trim angles, pitching moment slopes, expectedangular frequencies and the inital pitch angles for the cou-pled simulations are summarized in Tab. 2. In the con-sidered range of angle of attack the sting has the largest

Table 2. Steady-state results for free flight (FF) and windtunnel (WT) configuration.

run 18 run19 run 23 run 10

M∞ 0.80 0.95 1.10 2.01αtrim [ ◦] FF 59.7 56.4 69.5 41.9

WT 63.5 60.7 – 42.0Cmα [1/rad] FF -0.2786 -0.2843 -0.3026 -0.0838

WT -0.3144 -0.2273 – -0.0854ω [1/s] FF 68.4 83.3 101.47 54.9

WT 72.6 74.5 – 55.5θinit [ ◦] FF 60.0 56.5 60.5 43.0/50.0

WT 63.5 60.5 62.0 43.0

-0.005

-0.004

-0.003

-0.002

-0.001

0.000

0.001

0.002

0.003

0.004

40 41 42 43 44 45

Cm

α [ °]

Fitted steady-state CFD with stingFitted steady-state CFD w/o sting

Figure 12. Steady-state pitching moment coefficient ver-sus angle of attack for run 10 (M = 2.01).

effect for the subsonic cases run 18, 19 and 23. For run18 and 19 the sting induces an increase of the trim angleof about 4◦. For run 23 the presence of the support stingleads to indifferent static behavior of the vehicle. There isno significant influence of the wind tunnel support stingin supersonic flow.

3.3. Unsteady coupled results

To ensure to have a sufficient number of sampling pointfor the simulation of the unsteady pitch oscillation a phys-ical time step of 2 ms has been chosen. Figure 13 to 16show in comparison the pitch oscillation history for thefree flight and wind tunnel configuration for each flowcondition. In all cases the vehicle oscillates around thetrim position. In addition to the physical time step of 2 msa simulation with a time step size of 1 ms has been car-ried out for run 19 (Fig. 13). Also for large oscillationamplitudes the difference is small.

For the transonic cases run 18 and 19 the pitch oscillationamplitude escalates with time. Figure 13 and 14 showthat the escalation rate is much higher for the free flightconfiguration compared to the wind tunnel configurationwith sting. The support sting has a damping effect on thepitch oscillation. Nevertheless the dynamic behavior forboth configurations in this flow condition is dynamicallyunstable.

For an increasing Mach number a change of the unstabledynamic behavior without sting to a slightly stable typeof pitch motion with sting is observed for run 23 in Fig.15. For the supersonic case run 10 the pitch oscillationamplitude decreases in time. Both configurations are dy-namically stable. In order get data for large oscillationamplitudes for run 10 a second simulation with an initialpitch angle of 50◦ has been performed. The results arealso shown in Fig.16.

Due to the fact that the initial starting solution for thesimulation of run 18 was very close to the trim angle

45

50

55

60

65

70

75

80

0.0 0.2 0.4 0.6 0.8 1.0 1.2

θ [ °

]

t [s]

CFD/RBD coupling with stingTrim angle with sting

CFD/RBD coupling w/o stingTrim angle w/o sting

63.0

64.5

0.64 0.68

Figure 13. Time-history of pitch oscillation amplitude forrun 18 (M = 0.80).

45

50

55

60

65

70

0.0 0.2 0.4 0.6 0.8 1.0 1.2

θ [ °

]

t [s]

CFD/RBD coupling with stingTrim angle with sting

CFD/RBD coupling w/o sting (∆ t = 2ms)CFD/RBD coupling w/o sting (∆ t = 1ms)

Trim angle w/o sting

60

65

0.95 0.98

Figure 14. Time-history of pitch oscillation amplitude forrun 19 (M = 0.95). The small picture shows the differencebetween the two simulated physical time step of 2 ms and1 ms

54

56

58

60

62

64

66

68

0.0 0.2 0.4 0.6 0.8 1.0 1.2

θ [ °

]

t [s]

CFD/RBD coupling with stingCFD/RBD coupling w/o sting

Trim angle w/o sting

Figure 15. Time-history of pitch oscillation amplitude forrun 23 (M = 1.10).

32

34

36

38

40

42

44

46

48

50

52

0.0 0.2 0.4 0.6 0.8 1.0 1.2

θ [ °

]

t [s]

CFD/RBD coupling with stingTrim angle with sting

CFD/RBD coupling w/o sting (θinit = 43°)CFD/RBD coupling w/o sting (θinit = 50°)

Trim angle w/o sting

Figure 16. Time-history of pitch oscillation amplitude forrun 10 (M = 2.01).

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

40 50 60 70 80 90

Cm

q+C

mα.

θ0 [ °]

CFD/RBD coupling with stingCFD/RBD coupling w/o sting

0.8

1.2

63.2 63.8

Figure 17. Pitch damping sum versus pitch oscillationamplitude for run 18.

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

45 50 55 60 65 70 75

Cm

q+C

mα.

θ0 [ °]

CFD/RBD coupling with stingCFD/RBD coupling w/o sting

Figure 18. Pitch damping sum versus pitch oscillationamplitude for run 19 (M = 0.95).

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

50 55 60 65 70

Cm

q+C

mα.

θ0 [ °]

CFD/RBD coupling with stingCFD/RBD coupling w/o sting

-0.14

0.06

60.5 62.5

Figure 19. Pitch damping sum versus pitch oscillationamplitude for run 23 (M = 1.10).

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

32 34 36 38 40 42 44 46 48 50

Cm

q+C

mα.

θ0 [ °]

CFD/RBD coupling with stingCFD/RBD coupling w/o sting (θinit = 43°)CFD/RBD coupling w/o sting (θinit = 50°)

Figure 20. Pitch damping sum versus pitch oscillationamplitude for run 10 (M = 2.01).

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

Cm

q+C

mα.

M

dynamically stabledynamically unstable

Oscillation amplitude <1° with stingOscillation amplitude ≈2° with stingOscillation amplitude ≈1° w/o stingOscillation amplitude ≈3° w/o sting

Figure 21. Averaged pitch damping sum versus Machnumber for diiferent pitch oscillation amplitudes.

Table 3. Steady-state results for free flight (FF) and windtunnel (WT) configuration.

run 18 run19 run 23 run 10

M∞ 0.80 0.95 1.10 2.01Cmq + Cmα FF (1◦) 1.964 1.742 1.287 -0.561

[1/rad] WT (1◦) 1.006 1.018 -0.042 -0.372FF (3◦) 1.707 1.431 1.084 -0.561

WT (2◦) – 0.837 – –

the reached maximum amplitude is very small. For allfreestream conditions the sting has a small influence onthe frequency of the oscillation. Figure 17 to 20 showthe pitch damping sum computed from the time-historiedata using Eq. (2) and (3). The analysis has been donefor local maxima and minima separately. For the tran-sonic freesteam the pitch damping sum decreases rapidlywith increasing oscillation amplitudes this is mainly ob-served for the free flight cases where higher amplitudeshave been achieved. For run 18, 19 and 23 the pitchdamping sum is reduced by a factor of two for an oscil-lation amplitude of ±10◦ compared to oscillations withvery small amplitudes.

For practical reasons the pitch damping sum is averagedin a region around the trim angle. The averaging has beendone for two ranges of amplitude, one for small oscilla-tions up to 1◦ and one for larger oscillations up to 3◦, ifdata available. The averaged values are listed in Tab. 3and plotted versus Mach number in Fig. 21. The plotshows that the suport sting has a stabilizing effect on thepitch oscillation for all transonic cases. Especially for run23 the pitch damping sum changes the sign from positiveto negative what means that the behavior changes fromunstable to slightly stable. It also shows a higher damp-ing for larger oscillation amplitudes. For the supersoniccase the dependancy of the pitch damping sum from theoscillation amplitude is less severe. It seems that the sup-port sting decreases the damping slightly.

4. CONCLUSION

Steady-state as well as fully coupled CFD/RBD simula-tions have been carried out to investigate the static anddynamic stability of a re-entry vehicle in transonic andlow supersonic flow at high angles of attack. The studyfocusses on the influence of the support sting, holding thevehicle in the wind tunnel, to the pitch damping deriva-tives. For all investigated transonic cases the results showthat the vehicle is dynamically unstable. The presence ofthe sting increases the trim angle and decreases the esca-lation rate of the pitch oscillation. The results also showa strong dependancy of the pitch damping derivative onthe pitch oscillation amplitude. The pitch damping sumdecreases with larger pitch oscillation amplitudes. In su-personic flow the influence of the sting and the the oscil-

lation amplitude is small.

ACKNOWLEDGMENTS

These activities have been carried out in the framework ofthe Technology Research Program of the European SpaceAgency. The author would like to thanks ESA for thesupport and for the funding of this project.

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