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Modelling Liquid Sloshing in Rotating
Tanks
Antoni Brentjes
July 3, 2015
Contents
1 Introduction 31.1 Method of approach . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Simulation of a gyroscope 52.1 Governing physics . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Gyroscope . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Numerical simulation of the gyroscope . . . . . . . . . . . . . . . 7
2.2.1 Advancing the rotation matrix . . . . . . . . . . . . . . . 82.2.2 Simulation algorithm . . . . . . . . . . . . . . . . . . . . . 9
2.3 Verification of the simulation . . . . . . . . . . . . . . . . . . . . 102.3.1 Small amplitude oscillation . . . . . . . . . . . . . . . . . 102.3.2 Steady precession . . . . . . . . . . . . . . . . . . . . . . . 122.3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Simulating flow in closed tanks 163.1 Available CFD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.1 THETA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1.2 FLUENT . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1.3 Other codes . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Rotating cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.1 THETA standard situation . . . . . . . . . . . . . . . . . 183.2.2 THETA Reynolds and mesh refinement . . . . . . . . . . 193.2.3 FLUENT standard situation . . . . . . . . . . . . . . . . 213.2.4 Annular mesh . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Multi-fluid spinup . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4.1 Causes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.4.2 Choice of solver . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Liquid & gyroscope comparison 284.1 No spin, 11.25 rad/s 6.25 mm excitation . . . . . . . . . . . . . . 28
4.1.1 Gyroscope setup . . . . . . . . . . . . . . . . . . . . . . . 294.2 80 RPM spin, 1.5 Hz 10 mm excitation . . . . . . . . . . . . . . . 304.3 80 RPM spin, 3 Hz 1 mm excitation . . . . . . . . . . . . . . . . 324.4 80 RPM spin, 0.5 Hz 25 mm excitation . . . . . . . . . . . . . . . 344.5 40 RPM spin, 1 Hz 25 mm amplitude excitation . . . . . . . . . 354.6 Gravity scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
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5 Discussion 395.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.1.1 Gyroscope parameters . . . . . . . . . . . . . . . . . . . . 395.2 Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.3 Gravity scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
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1. Introduction
The European Space Agency embarks on the development of the new Ariane 6launcher. The chosen concept will involve a cryogenic restartable upper stage,to allow launching multiple payloads into different orbits. Restarting a cryo-genic engine requires good control and settling of the fuel, as liquid propellantengines do not tolerate bubbles and interruptions of fuel/oxidizer flow. This ismade more difficult by the fact that the upper stage could be required to spinup for the release of either payload. The liquid fuel and oxidizer will have to becontrolled while going through spin, de-spin, pointing and delta-V manoeuvres.
Liquids in microgravity can be controlled by various means, ranging frompassive devices working with surface tension and membranes to active meansusing attitude and ullage thrusters. Each method has its advantages and dis-advantages, but one thing is common: any increase in launcher mass cuts intothe payload.
A linear model which accurately predicts the liquids’ behaviour would behighly valuable, allowing the guidance, navigation and control system to prop-erly account for tank contents. Reducing the complexity of a flow problem to asimple model is obviously not trivial, but there is a good starting point. Previ-ous research has shown that a pendulum can be used to approximate the bulkbehaviour of liquid sloshing in (quasi-)spherical tanks.
Pendulum models do however fail to properly predict liquid behaviour in arotating tank, which is one of the more critical cases in the proposed Ariane6 upper stage. Changing the model to a gyroscope by adding rotation andmoments of inertia is thus a logical next step. While or course not a linearsystem a gyroscope model would still be helpfull in designing a GNC system.Investigating the validity of such a model, and if viable, its requirements andlimitations was thus the main topic of interest for this internship.
1.1 Method of approach
The approach of this problem is based around an experiment which was beingbuilt during the time of the internship. This experiment consists of a transpar-ent tank, which can rotate around the vertical axis driven by a spin motor. Thistank is mounted on a frame which can translate along a single horizontal axis,driven by an oscillation motor. The tank can be partially filled with liquid,
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spun and oscillated while reaction forces are measured and the liquid filmed.The liquid can be replaced by a solid gyroscope suspended from the tank cen-tre. This experiment will be used to validate simulations, which can then beused to extrapolate beyond the earth-gravity case.
The first part of the theoretical investigation is the simulation of the be-haviour of a gyroscope. This simulation must be able to accurately predict thebehaviour of an arbitrary gyroscope, subject to various physical loads. Whenproven accurate this simulation will be used to validate the performance of theexperimental setup.
The second part is the simulation of liquid in closed tanks. Practically thisboils down to picking the right CFD software and settings. It turned out thiswas far less trivial than perhaps expected, and thus has a chapter devoted to it.
Last is comparing the behaviour of a gyro to that of a liquid in a rotatingspherical tank when excited in horizontal direction. Several cases are investi-gated and the results presented.
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2. Simulation of agyroscope
Two FORTRAN scripts simulating the behaviour of a gyroscope had been previ-ously written by my supervisor and another member of TEC-MPA (Dr. Schwaneand Dr. Wong). These simulation codes were based on semi-fixed and Euler-angle coordinate systems respectively. As they produced different results for afew specific cases the first step was a new analysis of the theory. It turned outthat the issues in part were caused by singularities in the coordinate transfor-mations. Due to my lack of knowledge of FORTRAN building onto one of theseexisting codes would have taken more effort than doing a rewrite into C++. Anew simulation was thus written, combining elements from both as well as otherconcepts to work around the found issues.
This chapter therefore covers the creation of a new gyroscope simulation codein three parts. First the relevant physics will be investigated. The second partwill go into the numerical methods chosen to simulate the system. Finally a fewspecific cases will be tested, solving them analytically and with the simulation.
2.1 Governing physics
The governing physics for a gyroscope, or in fact any spinning object, is con-servation of angular momentum. The equation for angular momentum in aninertial frame is given by:
d~L
dt= ~M with ~L = ¯̄I~ω (2.1)
Here ~ω denotes the angular velocity vector, ¯̄I the moment of inertia tensorand ~L the angular momentum vector. The usefullness of this form is limitedhowever, as ¯̄I depends on the the object’s orientation, and thus changes withtime for nonzero ~ω. Converting to a frame which rotates with the object allowsus to take ¯̄I as constant, at the cost of adding an extra term.(
d~L
dt
)in
=
(d~L
dt
)rot
+ ~ω × ~L = ¯̄I
(d~ω
dt
)rot
+ ~ω × ( ¯̄I~ω) (2.2)
Taking the x, y, z-axes parallel to the object’s principal axes of inertia sim-plifies the moment of inertia matrix to a diagonal matrix, i.e. Iij = 0∀ i 6= j.This simplifies equation (2.2) to the following:
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M1 = I1ω̇1 + (I3 − I2)ω2ω3
M2 = I2ω̇2 + (I1 − I3)ω3ω1
M3 = I3ω̇3 + (I2 − I1)ω1ω2
(2.3)
2.1.1 Gyroscope
A typical gyroscope consists of a disk or wheel mounted on a spin axis, whichmay be either free to gimbal, driven or fixed. For this model the gyro axis willbe suspended from one end, free to rotate in three axes. In the experimenta universal or ball joint will be used here, which will limit deflection, but thesimulation will not suffer this disadvantage.
Since this geometry is axially symmetric the conservation equations can besimplified even further. The 1 and 2 axes are taken perpendicular to the spinaxis, the 3 axis parallel, with the origin in the hinge location. Picking the hingepoint instead of the centre of mass as origin is convenient in this case, as itallows for the moments to be calculated using gravity and frame accelerationacting on the centre of mass. Hinge reaction forces could then be calculatedafterward from centre-of-mass acceleration if required.
Figure 2.1: Gyroscope with origin and axes
With this geometry the 1, 2, 3 axes are renamed x, y, z, and the z axis mo-ment of inertia to axial moment of inertia Iax. The x and y axis moments ofinertia are equal, and renamed radial moment of inertia Ira, calculated aroundthe hinge point. With these definitions equation (2.3) can be simplified furtherto:
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Mx = Iraω̇x + (Iax − Ira)ωyωz
My = Iraω̇y + (Ira − Iax)ωzωx
Mz = Iaxω̇z
(2.4)
As mentioned before the moments acting on the gyroscope can be deter-mined from the gravity and frame acceleration, which can be thought of asforces acting on the centre of mass. The gyro centre of mass is located in~l = (0, 0,−l)T , in the gyro’s frame. The force acting on it is ~F = mgyro(~g − ~a),with gyro mass mgyro, and the gravity and frame acceleration vectors taken inthe gyro’s frame. Additionally, to properly model a real system, frictional andviscous losses must be taken into account. This is done by including a damp-ing moment vector ~D. This damping vector represents a viscous resistance,~D = (ωxdra, ωydra, ωzdax)T , with dax and dra the axial and radial damping
coefficients. The combined moment acting on the gyro is then ~M = ~l × ~F + ~D.
Finally, the following system of differential equations can be written to de-scribe the gyro’s angular velocity:
dωxdt
=1
Ira[Mx + (Ira − Iax)ωyωz]
dωydt
=1
Ira[My + (Iax − Ira)ωzωx]
dωzdt
=Mz
Iax
(2.5)
2.2 Numerical simulation of the gyroscope
Equation (2.5) can technically be solved directly, but practically forms only halfthe problem. The other, admittedly less complicated half, consists of determin-ing the orientation and position of the gyro. The displacement of the hingepoint will be prescribed, which leaves a few options for the orientation. Amongthese are rotation matrices and Euler angles. Euler angles allow for an analyticderivation of gyroscopic precession and nutation, but are non-commutative andmay be non-uniquely defined. This makes them less than ideal for a numericalsimulation. A rotation matrix is less elegant, needing nine values to describewhat Euler angles can in three, but it is very convenient in the context of nu-merics.
The rotation matrix can be defined as:
¯̄R =
xg,x xg,y xg,zyg,x yg,y yg,zzg,x zg,y zg,z
(2.6)
Here ~xg, ~yg and ~zg are unit vectors forming the gyro-relative basis. The sub-scripts denote their components in the x, y, z directions of the inertial frame. Avector in the inertial frame is transformed into the gyro basis by multiplying it
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with ¯̄R, the reverse is done by multiplying by ¯̄RT .
The velocity of a point located in ~r rotating around an axis (through theorigin) is given by d/dt(~r) = ~ω × ~r. This can be used to complete the system ofdifferential equations describing the gyroscope’s motion:
d~xgdt
= ~ωin × ~xgd~ygdt
= ~ωin × ~ygd~zgdt
= ~ωin × ~zg
(2.7)
Note that here ~ωin denotes the gyro’s angular velocity taken in the inertialframe.
2.2.1 Advancing the rotation matrix
Several methods exist for the numerical integration of (systems of) ordinary dif-ferential equations, ranging from the simplest forward-Euler scheme to variousflavours of Runge-Kutta. Naively applying such a method to equations (2.5)and (2.7) is likely not to end well. Imagine simply stepping forward ~xg by afinite amount. The time-derivative of this vector is by definition perpendicularto the vector itself. Adding a finite perpendicular component will increase thevector length. This is a problem, as said vector forms a part of the orthonor-mal gyro basis. The requirement of orthonormality combined with the ninepartial differential equations for the basis vectors overdefines the system. Thefact that Euler rotations fully define orientation makes this clear: the rotationmatrix only contains three independent variables. An alternative solution mustbe found to overcome this.
As the orthonormality of the basis is a requirement it makes sense to discardthe ‘extra’ information and reconstruct the basis after stepping time forward.The z and x basis vectors are chosen to reconstruct the new basis from. ~zgprimarily, as it corresponds directly to the axis orientation, and ~xg to lock therotation around the axis. Here the assumption is made that the timestep timesthe angular velocity is much smaller than one. Thus, for angular step α it willhold that sinα ≈ tanα ≈ α and cosα ≈ 1. This implies that normalising oneof said basis vectors can be done without penalty. Furthermore it means thatfollowing a timestep the basis vectors will still be very close to orthogonal, andthus their cross product never zero.
Not using the stepped ~yg effectively discards three variables worth of super-fluous information. Then, the stepped z basis vector is taken, and normalized,discarding a further point. Following this the new ~yg is created from the cross-product of ~zg and the stepped ~xg, and normalised. This reduces the effectivelyretained information to three variables. Finally a new x basis vector is createdfrom the cross product of the new y and z vectors. In pseudocode this processgoes as shown below:
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~zg =~zg|~zg|
;
~yg =~zg × ~xg|~zg × ~xg|
;
~xg = ~yg × ~zg;
(2.8)
Note that this is essentially equivalent to projecting ~xg on the plane orthog-onal to (the normalised) ~zg, and taking ~yg orthonormal to those.
2.2.2 Simulation algorithm
A classic Runge-Kutta scheme was chosen to perform the time-integration inthis simulation, with the above correction to preserve frame orthonormality.This scheme is described by equation (2.9).
dy
dt= f(t, y)
yn+1 = yn +h
6(k1 + 2k2 + 2k3 + k4); tn+1 = tn + h
k1 = f(tn, yn)
k2 = f(tn +h
2, yn +
h
2k1)
k2 = f(tn +h
2, yn +
h
2k2)
k4 = f(tn + h, yn + hk3)
(2.9)
In this case y consists of the gyro-relative ~ω and the orthonormalized gyrobasis z and x vectors. The function f(y, t) can be described as follows:
1. The passed basis vectors are turned into a rotation matrix by the methodexplained in the previous subsection.
2. The force on the gyroscope, determined by it’s mass, gravity and a knownacceleration profile, is rotated into the gyro frame. From this force themoment acting on the gyro is calculated, and combined with the dampingmoments calculated from ~ω.
3. The current angular rate ~ω is rotated from the gyro frame to the inertialframe.
4. Equation (2.5) is solved to find the time-derivative of the angular velocity.This forms part of the function output.
5. Equations (2.7) are solved to find the time-derivatives of ~xg and ~zg. Thisforms the rest of the function output.
During every timestep this function is called four times, and the resultsstored as k1 through k4. At the end of each timestep the new state yn+1 iscalculated from the old state and these derivatives. The basis vectors of thenew state are orthonormalised, and the next timestep is started.
The simulation itself was written in C++, and is included digitally.
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2.3 Verification of the simulation
The simulation results needed verification before they could be used for any com-parison. To succesfully qualify a number of requirements had to be met. First,for each verification test the simulation results had to converge with timestepreduction. This mirrors the typical mesh-refinement studies in CFD, failure toconverge with reducing timesteps indicates either (yet) insufficient refinementor a systematic problem. Second, energy must be conserved in the system whendamping is disabled. Failure to do so in a timestep-converged solution wouldindicate a serious error. Finally, the simulation results should match the ana-lytical solution, for the test cases in which such is available.
2.3.1 Small amplitude oscillation
Simple oscillation on one axis, without spin, is governed by equation (2.10).
θ̈Ira = −mgl sin θ (2.10)
With θ the angle between the gyro axis and the downward vertical, l the gyroaxis length, m the gyro mass, g the gravity and Ira the known radial momentof inertia. From the Taylor series for a sine we know that sinα = α + O(α3).
Linearising equation (2.10) yields the solution θ = aeibt with b =√
mglIra
.
For this test the following parameters were used:
Parameter valuel 1mm 1 kgIax 1 kg.m2
Ira 2 kg.m2
g 9.81m/s2
dax, dra 0 kg.m2/s
Table 2.1: Parameters for small amplitude oscillation validation
The chosen initial condition is zero initial (angular) velocity, with the gyro-scope’s centre of mass displaced by 1mm in the x-direction. This correspondsto an angle with the downward vertical of 1 milliradian. The analytical solutionfor this case is then x = ∆x cos (tmglIra
), a cosine with an amplitude of 1mm and
a period of 2.84 s. This case is simulated using timesteps of 0.1, 0.01 and 10−3
seconds, for a physical time duration of 100 seconds.
Figure 2.2 shows the first ten seconds of the gyroscope’s motion togetherwith the analytical solution.
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0 1 2 3 4 5 6 7 8 9 10−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1x 10
−3
Time [s]
Dis
plac
emen
t [m
]
∆ t = 0.1s∆ t = 0.01s∆ t = 0.001sAnalytic solution
Figure 2.2: Small amplitude single axis oscillation of the gyroscope
Even at the coarsest timestep of 0.1 seconds the simulation results are visu-ally indistinguishable from the analytic solution. The small differences betweenthe simulation and analytic plots are due to the fact that all simulation resultsare sampled at 10Hz, and plotted as straight lines between data points.
Figure 2.3 shows the position error of the simulation results with respect tothe analytic solution, normalized by the oscillation amplitude.
0 10 20 30 40 50 60 70 80 90 100−5
−4
−3
−2
−1
0
1
2
3
4
5x 10
−3
Time [s]
Rel
ativ
e po
sitio
n er
ror
[−]
∆ t = 0.1s∆ t = 0.01s∆ t = 0.001s
(a) Position error overview
0 10 20 30 40 50 60 70 80 90 100−2
−1.5
−1
−0.5
0
0.5
1
1.5
2x 10
−5
Time [s]
Rel
ativ
e po
sitio
n er
ror
[−]
∆ t = 0.1s∆ t = 0.01s
(b) Short timestep error highlight
Figure 2.3: Normalised position error in simple oscillation test
Two things become apparent here. First is that, while visually indistinguish-able, the coarsest simulation diverges from the analytic solution by nearly half apercent in 35 oscillation periods. While this is not unreasonable the solution isstill far from timestep-convergence. The two finer simulations have a non-zeroerror identical within floating point precision of up to O(10−5). The magnitudeof this error is consistent with the expected error in the analytic solution due
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to the linearisation used to solve equation (2.10).
This leaves the final test, the conservation of energy. The normalised differ-ence in total system energy is plotted in figure 2.4.
0 10 20 30 40 50 60 70 80 90 100−18
−16
−14
−12
−10
−8
−6
−4
−2
0
2x 10
−4
Time [s]
Rel
ativ
e en
ergy
err
or [−
]
∆ t = 0.1s∆ t = 0.01s∆ t = 0.001s
Figure 2.4: Normalised energy difference in simple oscillation test
Again the coarse simulation fails the test, losing nearly two thousandth ofthe gyroscope’s energy over the first hundred seconds. Both finer simulationsdo not appear to consistently lose any energy. The graph shows some noise witha magnitude of 10−6, which is within the range of floating point error.
Alltogether it seems reasonable to conclude that both fine simulations cor-rectly model the behaviour of the gyroscope.The results do not change between0.01 and 0.001 s timesteps, total system energy is conserved within numericalaccuracy and the analytic solution is approached within linearisation limits.
2.3.2 Steady precession
The second test case is steady precession, during which axial spin and gravitydrive the gyroscope to trace circles around the vertical axis. An analytic solutionexists for this mode of motion, which is given as equation (2.11).
ωp = − mgl
Iaxωz+
(Ira − Iax)ω2p cos θ
Iaxωz(2.11)
with θ the angle of the gyro axis to the downward vertical and ωp the rateat which the gyro axis rotates around the vertical. The second part of the righthand side is typically neglected under the assumption |ωz| � |ωp|, which tendsto hold true for gyroscopes. For this validation however such an assumptionwill not do, and instead the case θ = π
2 will be used to the same effect.
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The same gyroscope properties are used as in the previous test-case, butshorter timesteps were used. The initial condition for this simulation is thegyroscope axis pointing in the x-direction. The gyroscope-basis y-axis pointsvertically up, the gyro z-axis in the negative x-direction. The initial rates ofrotation are ωz = −10 rad/s and ωy = ωp = 0.981 rad/s, with ωx = 0. Figure2.5 shows the x-coordinate of the gyroscope centre of mass as a function of time.
0 2 4 6 8 10 12 14 16 18 20−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Time [s]
X−
disp
lace
men
t [m
]
∆ t = 10e−3s
∆ t = 1e−3s
∆ t = 100e−6sAnalytic solution
Figure 2.5: Gyro x-coordinate in steady precession
As expected this figure traces a cosine, with each simulation result visuallyindistinguishable from the analytical solution. Figure 2.6 shows the absoluteerror between simulation and analytic solution.
0 10 20 30 40 50 60 70 80 90 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1x 10
−4
Time [s]
Err
or d
ista
nce
[m]
∆ t = 10e−3s
∆ t = 1e−3s
∆ t = 100e−6s
Figure 2.6: Absolute error in steady precession
As can be seen in this figure the absolute error increases quadratically forthe coarsest simulation, while in the finer two simulations the error does not
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increase above the noise level.
The energy conservation for this simulation is plotted in figure 2.7.
0 10 20 30 40 50 60 70 80 90 100−20
−15
−10
−5
0
5x 10
−7
Time [s]
Rel
ativ
e en
ergy
err
or [−
]
∆ t = 10e−3s
∆ t = 1e−3s
∆ t = 100e−6s
Figure 2.7: Normalised energy difference in steady precession
Again only the coarse simulation seems to develop a significant error, withonly small noise for the finer simulations. For this test case one final check canbe done: the gyro z coordinate, which should remain exactly zero in steadyprecession. This result is plotted in figure 2.8.
0 10 20 30 40 50 60 70 80 90 100
−10
−8
−6
−4
−2
0
2x 10
−11
Time [s]
Z−
disp
lace
men
t [m
]
∆ t = 1e−3s
∆ t = 100e−6s
∆ t = 10e−6s
Figure 2.8: Gyro z-coordinate in steady precession
Note that in this figure results are shown from simulations with yet smallertimesteps. This most precise check shows that even a supposedly accuratelytime-stepped simulation still has a non-zero drift.
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2.3.3 Conclusions
Both test case results indicate the simulation satisfies the verification require-ments. First, the simulations results converge on a final solution when time-stepsare reduced. Second, with sufficiently small time-steps energy is conserved wellover the used simulation intervals. Finally, the time-step converged solutiondoes in fact approach the analytic solution for the used test-cases.
One caveat remains however, suggested mainly by the results in figure 2.8.Every reduction in timestep seems to reduce the rate at which drift occurs, buttotal elimination does not happen in this test-case. For sufficiently long physicaltimes even the finest simulation might start drifting. Thus, for every performedsimulation a time-step refinement should be done, the difference between resultswill give an indication of accuracy.
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3. Simulating flow in closedtanks
The simulation of flow in closed systems is significantly more challenging thanin open systems. This is especially true for rotating systems, as will be the casehere. The root cause of this difficulty is that closed systems lack the stabilisa-tion of inflow and outflow. A slight numerical error or asymmetry in an opensystem will have an influence on the reaction forces and outflow momentumand energy. The constant inflow serves to flush the fluid affected by the errors,limiting their effects to what can be done in a single pass of the fluid.
In a closed system on the other hand there is no in- or outflow. All er-rors introduced will persist in the fluid. Especially in rotating systems this canpose a problem, for instance when an ‘inner’ layer of fluid is accelerated to befaster than the layer outside of it. Such an error would promote, for instance,Taylor-Couette instabilities, which can lead to the simulated flow taking on adramatically different topology than it should.
Writing a CFD code is far outside the scope of this internship. Thereforethis chapter will cover the testing and analysis of several existing CFD codes invarious configurations to try and find the best available option.
3.1 Available CFD
3.1.1 THETA
The THETA CFD code is an incompressible branch of TAU, currently in devel-opment by DLR. It was chosen because ESA has an interest in this code andongoing developments, and thus evaluation licenses were already available.
3.1.2 FLUENT
Fluent is a commercial CFD code owned by ANSYS. It has a graphical userinterface, built-in meshing (with multigrid capacity), setup and post-processingfunctionalities and everything else common to the ANSYS suite. On the backend it has a pressure and a density based solver, both supporting various dis-cretisation schemes, as well as physical models including volume-of-fluid. It waschosen due to being a commercial counterpart to TAU/THETA, as well as mypersonal familiarity with ANSYS products.
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3.1.3 Other codes
Other codes have been analysed in similar context prior to this internship, adetailed report of CFD-ACE specifically was available. These codes were nottried again, but where applicable comparisons will be made to these previoustests.
3.2 Rotating cylinder
The first test case is axial rotation of a cylindrical tank filled with a single fluid.This test case has been used previously, which would allow for good comparison.The expected result is the fluid settling into a solid rotation, with ~v = ~ω×~x anda parabolic pressure profile. Simple as the case may seem many codes fail toproduce the correct solution, with reasonable meshes and real liquid viscosities.This basic test case is described in the table below:
Tank height 2mTank diameter 1.414mRotation rate 1 rpmLiquid density 1000 kg/m3 (water)Liquid viscosity 8.94 ∗ 10−4kg/(m.s) (water)
Table 3.1: Basic cylindrical tank rotation parameters
These parameters and the used mesh are inherited from the CFD-ACE in-vestigation. The standard mesh is a butterfly grid with 92,000 elements, refinedtoward the outer wall and top/bottom surfaces. This mesh is pictured in figure3.1.
Figure 3.1: 92,000 element cylindrical butterfly grid
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3.2.1 THETA standard situation
The first test was simulating the basic case using THETA. A quadratic upwinddiscretisation scheme was used for momentum, implicit Euler for timestepping,which had previously proven to give the best results. Timestep size was set toprovide a CFL number of 0.15. The calculation was initialised with all liquidstationary.
Plotting the entire flowfield is not necessary to evaluate the results. Insteadthe x-direction velocity component is extracted along the y-axis. The expectedlinear profile is subtracted, and the result normalised by dividing by the outerwall velocity. This results in a dimensionless velocity error, which is plotted infigure 3.2.
Figure 3.2: Velocity error for THETA spinup test
Unsurprisingly the error profiles 100 and 1000 seconds after initialisationshow a large linear segment. This part corresponds to the part of the fluidwhich is still stationary, and therefore far from the expected steady state veloc-ity. After 10, 000 seconds of physical time the error is below 0.1 over the entiresection, indicating the fluid is mostly up to speed. Strangely the profile fails toconverge further, instead showing a fluctuating error profile.
This is somewhat of a problem, and thus as a second test the same case wasrun, this time initialised with all liquid in solid rotation. The results are shownin figure 3.3.
18
Figure 3.3: Velocity error for THETA initial rotation test
Again the error profile seems to randomly fluctuate by up to 0.08 error.
3.2.2 THETA Reynolds and mesh refinement
Two further tests were done on the cylindrical geometry using THETA. Thefirst involved increasing the rate of rotation to 10 rpm, and thus the Reynoldsnumber by a factor 10. As figure 3.4 shows fluctuations remain present, but nowtend toward positive error values for positive y. As x velocity should be pro-portional to −y due to rotation around z this indicates the fluid ‘underspeeds’.
The last test is done using the basic parameters, but using a finer mesh. Therefined mesh has 357,000 cells, but is otherwise similar to the standard mesh.The results in figure 3.5 show a maximum error reduction by roughly a factorfour, proportional to the mesh refinement. The error now tends toward over-speeding in the cylinder centre, and underspeeding near the walls. Fluctuationsremain however, no convergence is reached within 20, 000 s.
19
Figure 3.4: Velocity error for THETA fast spin test
Figure 3.5: Velocity error for THETA refined mesh test
20
3.2.3 FLUENT standard situation
Due to run-time constraints all FLUENT simulations were initialised with thefluid rotating with the tank. The default case was simulated in three configu-rations. The first two simulations used the pressure based solver, second orderupwind discretisation and second order implicit time-stepping. One of these sim-ulations used a relative (to the rotating frame of reference) velocity formulation,the second the absolute velocity formulation. The results of these simulationsare shown in figures 3.6 and 3.7.
The third simulation used the density based solver, with AUSM flux splittingand second order implicit time stepping. The results to this simulation areshown in figure 3.8.
Figure 3.6: Velocity error for FLUENT pressure based relative velocity formulation
Again the figures show clear non-zero errors arising in the solution. Un-like in the THETA simulations the FLUENT results seem to converge towardsymmetric error profiles. The two converged solutions have underspeeding er-ror profiles, some overspeeding is seen in the pressure based absolute velocitysimulation. The error magnitudes are similar to the standard mesh THETA so-lutions, with the density based solution being the best. The high underspeedingspikes at the walls are likely caused by the interpolation in Tecplot, using theFLUENT post-processor shows zero error at the wall.
3.2.4 Annular mesh
In the FLUENT simulations some features of the error profile coincide withthe mesh-block boundaries. This suggested the errors were possible caused oraffected by the element orientation or deformation. A butterfly mesh inevitablyhas a number of deformed elements, and element surfaces not normal or parallel
21
Figure 3.7: Velocity error for FLUENT pressure based absolute velocity formulation
Figure 3.8: Velocity error for FLUENT density based absolute velocity formulation
to the flow. Filling a cylinder with a perfectly regular mesh requires the innerelements to be prisms instead of hexes, and is likely to make these excessivelythin. To prevent this problem a new, annular geometry was made, which canbe turned into a regular mesh without these issues. To keep the case similarthe same height and outer diameter were used, and the mesh was made to have95,000 elements, similar to the cylindrical one. This mesh is shown in figure 3.9.
A set of simulations was performed with the same settings as used in the
22
Figure 3.9: 95,000 element regular annular grid
cylindrical tests. Results varied little between different solver settings, but thesame was not true between FLUENT (figure 3.11) and THETA (figure 3.12).
Switching to the regular grid reduces the maximum error by two ordersof magnitude for the FLUENT simulation. Some error remains, convergingon overspeeding near the inner and underspeeding near the outer wall. TheTHETA results are hardly different from the cylindrical test.
3.3 Multi-fluid spinup
Since the actual liquid sloshing problem obviously involves multiple fluids tobe simulated the VOF models must be tested as well. The choice was madeto do this using the same geometry as the physical test-setup will have: a 250mm diameter sphere. For this two spherical seven-block butterfly meshes weregenerated, having 100,000 and one million elements respectively. This sphere is
23
Figure 3.10: Velocity error for FLUENT annular grid test
Figure 3.11: Velocity error for THETA annular grid test
filled half with water and half with air, with initially zero velocity. The gravityis set to 9.81 ·10−3m/s2 and the tank is spun around the vertical axis at 10 rpm.These parameters were chosen so the effects of buoyancy, centrifugal forces andsurface tension were of similar magnitude.
Again the THETA simulation fails to converge on a steady solution, withfluctuations up to 10% of the outer wall velocity remaining present. This resultsin a very bumpy liquid surface (where V OF = 0.5), as seen in figure 3.12.
24
Figure 3.12: Velocity error for THETA annular grid test
FLUENT has several schemes for solving the volume fraction equation. Thetheory and user guide recommends using an explicit timestepping formulation,as this allows it to apply fluid interface reconstruction. These reconstructionmethods treat cells differently depending on the VOF variable being 0, 1, orin between. The two options tried were Geometric Reconstruction and theCICSAM scheme. The georec scheme ensures a sharp fluid boundary and isconsidered most accurate. The CICSAM scheme is considered the better optionfor treating flows with high ratios of viscosity between the fluids.
Both FLUENT solutions do seem to converge toward a steady solution,leaving a smooth liquid surface. Figure 3.13 shows the VOF variable on thecrossection, figure 3.14 the velocity error (as in the cylinder tests), both withthe liquid surface superimposed.
The differences between the results are clear. The CICSAM scheme doesdillute the fluid interface more than the georec scheme, but otherwise seems toperform fairy well on surface tension and interfluid shear. The georec schemeproduces a very crisp boundary, at the trade-off of very high error velocities,overspeeding by up to 30%. This significantly deforms the liquid surface, flat-tening it against the surface tension.
A THETA simulation was attempted on the million point grid, but aftermore than a week of computational time the physical time had only reached onetenth of the time to convergence. The results up to that point were similar tothe low-resolution THETA results in that significant velocity fluctuations werepresent, although lower in magnitude due to the finer grid. A fine FLUENT
25
(a) VOF with Geometric Reconstruction (b) VOF with CICSAM scheme
Figure 3.13: VOF in the FLUENT spinup simulations
(a) Error with Geometric Reconstruction (b) Error with CICSAM scheme
Figure 3.14: Velocity error in the FLUENT spinup simulations
calculation was not attempted due to these same performance constraints.
3.4 Conclusions
It seems reasonable to conclude that neither of the tested CFD codes is able toaccurately simulate such a closed rotating system with real liquid viscosities.
3.4.1 Causes
Several factors were found to influence the errors. Mesh refinement has beenshown to reduce the error, but doing so was impossible with available resourcesand should not have been necessary. The element shape and orientation withrespect to the flow does have a significant influence for FLUENT, but does nothelp improve THETA results. Making a rotationally symmetrical mesh for asphere is not possible however, and may not help as much when there is sloshing
26
in addition to rotation.
Increasing liquid viscosity by three or more orders of magnitude has beenshown to reduce the error dramatically in these and previously tested CFDcodes. These results are not shown here as the simulations required for thegyroscope comparison will need actual liquid viscosities. It does suggest theproblem may be in an imbalance in the momentum conservation equation andit’s discretisation. Further investigation into the problem and possible solutionswas deemed to be outside the scope for this internship. The results were insteadforwarded to the developers of THETA, but a solution could not be found intime.
3.4.2 Choice of solver
Despite the present issues it was decided to go ahead with the gyroscope com-parison study. It was expected that a qualitative gyro-fluid comparison waspossible despite the CFD inaccuracies. The converging error seen in FLUENTwas considered preferrable over THETA’s random fluctuations of the same mag-nitude. Therefore the FLUENT solver was chosen, with the CICSAM schemeas best option for liquid surface reconstruction.
27
4. Liquid & gyroscopecomparison
Having means for the simulation of both gyroscope and liquid the two systemscan now be compared. The testcases are meant to be reproduced in a physicalexperiment, and are designed with that in mind. Each of them will have earthgravity, start with the liquid in a steady state and then apply a harmonic exci-tation.
The mentioned experimental setup will consist of a spherical tank, mountedto allow it to be spun around the z-axis and translated along x. To match thecapabilities of the setup all simulations in this chapter will have the followingin common. The tank will be a 250 mm diameter sphere meshed as a sevensegment butterfly grid with 100,000 elements, as seen in previous chapter. Itwill be half filled with water and half with air, at room temperature sea levelconditions. In cases involving spin the initial condition will be the steady stateof the spinning tank. All cases will involve excitation, in the form of a cosinemotion of the tank along the x-axis.
For each of the gyroscope simulations the recommended timestep refinementstudy was done. All shown gyroscope simulation results are from simulationswith timesteps sufficiently small that increasing or decreasing them tenfold hasno influence on the result. The data from these refinement tests was omittedfrom the report for brevity.
4.1 No spin, 11.25 rad/s 6.25 mm excitation
The first test case is a simple driven oscillation, with the fluids in the tank ini-tially at rest. The tank will be excited harmonically with a 6.25mm amplitude(5% of the tank radius). The excitation frequency is set at 11.25 rad/s, whichis approximately the natural frequency of the system, according to Sumner [1].
This case is simulated for five seconds using FLUENT , an animation of theresults is included digitally. The liquid starts sloshing as a solid hemisphere,and quickly reaches a thirty degree slosh angle. At this point surface waves startforming, pushing the liquid up higher locally, but damping the centre-of-massoscillation. Figure 4.1 shows the liquid surface showing this local pushing. Nosplashing occurs, and after five seconds the sloshing seems reasonably stable in
28
amplitude, with some disturbances from running waves.
Figure 4.1: Sloshing in the driven oscillation case, t = 2.15 s
By integration the centre of mass of the tank can be found. This centre ofmass will be used to compare the gyroscope with the liquid.
4.1.1 Gyroscope setup
The gyroscope is parameterised by a length, mass, moments of inertia anddamping coefficients. A sensible initial guess for these parameters is to simplyfill in the values for a solid hemisphere with the density of water. Figure 4.2ashows a comparison between the CFD and the gyroscope simulation with naivelychosen parameters. The behaviour is clearly similar, but the amplitude andperiod are off. With some tuning a better match can be made, shown in figure4.2b. The input parameters are given in table 4.1.
Setting up the gyroscope naively causes it to oscillate faster than the liquiddoes, going out of phase. This is prevented by reducing the gyro axis length,which effectively reduces the system’s stiffness, slowing the natural frequency.For the purpose of the simulation this has the same effect as reducing the mass,or increasing the moments of inertia. The advantage of altering the axis lengthhowever is that contact forces (mass times acceleration) are preserved and thatthe resulting gyro will not necessarily be larger than the tank.
The improved match is still far from perfect. The low-damping gyro matchesthe CFD well during the first 2.5 seconds, after which the CFD results showreducing amplitude. The high-damping gyro matches reasonably with the CFD
29
Naive OptimizedAxis length 46.9 ∗ 10−3 m 37 ∗ 10−3 mGyro Mass 4.09 kg 4.09 kgAxial M.O.I 1) 12.8 ∗ 10−3 kg.m2 12.5 ∗ 10−3 kg.m2
Radial M.O.I. 12.8 ∗ 10−3 kg.m2 12.5 ∗ 10−3 kg.m2
Axial damping 1) 0 kg.m2/s 0 kg.m2/sRadial damping 0.01 kg.m2/s 0.01− 0.02 kg.m2/s
1) Not relevant in simple oscillation
Table 4.1: Gyroscope parameters for driven oscillation
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.03
−0.02
−0.01
0
0.01
0.02
0.03
Time [s]
Cen
tre−
of−
mas
s x−
coor
dina
te [m
]
CFDNaive
(a) CFD compared with naive gyro
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.025
−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
0.025
Time [s]
Cen
tre−
of−
mas
s x−
coor
dina
te [m
]
CFD0.01 damping0.02 damping
(b) CFD compared with improved gyro
Figure 4.2: CFD compared with gyro model in driven simple oscillation
results during the last second and a half, but does not reach the same peakamplitude at 2.5 seconds. This suggests significant non-linear damping effectsare present in the liquid which the solid gyroscope cannot match. The travellingsurface waves seen in the digitally included video are extra evidence of this.
4.2 80 RPM spin, 1.5 Hz 10 mm excitation
The experimental setup being built was specced to be able to reach 120 RPMspin and lateral excitation at 3 Hz with an amplitude up to 25 mm (20% ofthe radius). The first gyroscopic test case was chosen to be near the middle ofthat specification. Spin was set to 80 rpm instead of 60 to avoid a 2:3 periodmatchup between spin and excitation. Preliminary tests with the gyroscopesimulation resulted in relatively ‘nice’ motion - small excursions in both x- andy- directions with varying amplitudes.
The CFD results did not agree with the prediction. The liquid took onesecond to enter a clockwise (counter-rotating w.r.t spin) swirling mode, withsteadily increasing amplitude. After six-and-a-half seconds the swirl has grownsufficiently to start splashing, as seen in figure 4.3. An animation of this simu-lation is again included digitally.
A more quantitative analysis is done by determining the location of the cen-tre of mass of the liquid. Figure 4.4 shows a comparison between the liquid
30
Figure 4.3: Onset of splashing in first spun CFD
and the gyroscope simulation, set up with the parameters used to match theoscillation case. The gyroscope model shows harmonic oscillation in both x- andy-directions with a frequency of 1.5 Hz, with a slight beat (period ≈ 3 s). TheCFD results have much higher amplitude and a different phase and beat thanthe gyro model, even if the splashing part (t ≥ 6.5 s) is disregarded.
0 1 2 3 4 5 6 7 8 9 10−0.02
−0.01
0
0.01
0.02
0.03
0.04
Time [s]
Cen
tre−
of−
mas
s x−
coor
dina
te [m
]
CFDPrevious
(a) x-coordinate comparison
0 1 2 3 4 5 6 7 8 9 10−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
Time [s]
Cen
tre−
of−
mas
s y−
coor
dina
te [m
]
CFDPrevious
(b) y-coordinate comparison
Figure 4.4: CFD compared with previous gyro model in first spun case
With some fine-tuning a better matching set of parameters can be foundhere as well, given in table 4.2. The results of the gyroscope simulation usingthese parameters are shown in figure 4.5.
31
Gyro A Gyro BAxis length 40 ∗ 10−3 m 40 ∗ 10−3 mGyro Mass 4.09 kg 4.09 kgAxial M.O.I 11.2 ∗ 10−3 kg.m2 11.1 ∗ 10−3 kg.m2
Radial M.O.I. 11.8 ∗ 10−3 kg.m2 11.1 ∗ 10−3 kg.m2
Axial damping 0 kg.m2/s 0 kg.m2/sRadial damping 0.012 kg.m2/s 0.012 kg.m2/s
Table 4.2: Gyroscope parameters for the first spun case
0 1 2 3 4 5 6 7 8 9 10−0.02
−0.01
0
0.01
0.02
0.03
0.04
Time [s]
Cen
tre−
of−
mas
s x−
coor
dina
te [m
]
CFDGyro AGyro B
(a) x-coordinate comparison
0 1 2 3 4 5 6 7 8 9 10−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
Time [s]
Cen
tre−
of−
mas
s y−
coor
dina
te [m
]
CFDGyro AGyro B
(b) y-coordinate comparison
Figure 4.5: CFD compared with tuned gyro model in first spun case
Again the liquid’s behaviour can be approximated but not matched perfectly.The ‘A’ settings match the x-coordinate quite well for the first five seconds, buthave a too low amplitude in the y-direction. The ‘B’ settings mainly match they-coordinate for the first six seconds, but fare poorly for x. Overall, though,the general behaviour is matched decently during the non-splashing part of thesimulation, as is seen in the included animation. Still better matching setupsare likely to exist, but it is not worth pursuing these here, as no gyro will be ableto account for the splashing and surface waves occuring later in the simulation.
4.3 80 RPM spin, 3 Hz 1 mm excitation
The second spun test was performed at a higher excitation frequency with amuch lower amplitude, to prevent splashing and other behaviour likely to breakthe similarity to the gyroscope. A total of ten seconds of physical time were sim-ulated. The CFD-results indeed show low-amplitude behaviour without largedeformation of the liquid surface. This behaviour is best described as a coun-terclockwise (co-rotating) swirling mode, with a beat period of 2.2 seconds.Although no large waves or splashing occur the liquid surface does flatten not-icably, as is shown in figure 4.6.
Figure 4.7 shows the centre-of-mass comparisons for this simulation. In thisfigure the CFD, the ‘Gyro B’ from the previous case and a new ‘Gyro C’ con-figuration are compared. The gyroscope parameters are given in table 4.3.
32
(a) Initial (steady state) liquid crossection (b) t = 10 s liquid crossection
Figure 4.6: Flattening of the liquid surface in the second spun case
0 1 2 3 4 5 6 7 8 9 10−2
−1
0
1
2
3x 10
−3
Time [s]
Cen
tre−
of−
mas
s x−
coor
dina
te [m
]
CFDGyro BGyro C
(a) x-coordinate comparison
0 1 2 3 4 5 6 7 8 9 10−3
−2
−1
0
1
2
3x 10
−3
Time [s]
Cen
tre−
of−
mas
s y−
coor
dina
te [m
]
CFDGyro BGyro C
(b) y-coordinate comparison
Figure 4.7: CFD compared with tuned gyro model in second spun case
Again the ‘old’ gyro setup matches the CFD reasonably for one or two os-cillations, but it takes different settings to stay close for a longer period. The‘C’ setup matches the behaviour quite accurately during for one-and-a-half beatperiods, but after that the amplitude drifts away from the CFD simulation. A
33
Gyro B Gyro CAxis length 40 ∗ 10−3 m 40 ∗ 10−3 mGyro Mass 4.09 kg 4.09 kgAxial M.O.I 11.1 ∗ 10−3 kg.m2 10.9 ∗ 10−3 kg.m2
Radial M.O.I. 11.1 ∗ 10−3 kg.m2 12.0 ∗ 10−3 kg.m2
Axial damping 0 kg.m2/s 0 kg.m2/sRadial damping 0.012 kg.m2/s 0.0065 kg.m2/s
Table 4.3: Gyroscope parameters for the second spun case
perfect match for longer than this may not be possible without dynamicallychanging gyroscope parameters to match the liquid changing shape as men-tioned. The phase is accurately predicted by both ‘B’ and ‘C’ setups for theentire ten seconds, suggesting a form of stability, while the diverging amplitudeindicates chaos.
The x-coordinate again appears to be matched worse than the y-coordinate.This is due to the x-displacement caused by the excitation being larger than thesloshing displacement, causing apparent phase mismatches.
4.4 80 RPM spin, 0.5 Hz 25 mm excitation
As a counterpart to the previous case this case was designed to study the effectsof low frequency excitation. To keep the excitation energy levels similar theamplitude was boosted to the maximum available in the experimental setup.This still ends up as a slight reduction in energy content (by a factor 25/36).The liquid shows a combination of sloshing and gyroscopic motion, at very lowamplitudes. The centre of mass displacement for this simulation is shown infigure 4.8. The x-displacement is corrected for the tank excitation, which is anorder of magnitude larger than the sloshing effects and would otherwise domi-nate the graph.
0 1 2 3 4 5 6−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5x 10
−3
Time [s]
Cor
rect
ed c
entr
e−of
−m
ass
x−co
ordi
nate
[m]
CFD
Gyro B
Gyro C
(a) Corrected x-coordinate comparison
0 1 2 3 4 5 6−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5x 10
−3
Time [s]
Cen
tre−
of−
mas
s y−
coor
dina
te [m
]
CFD
Gyro B
Gyro C
(b) y-coordinate comparison
Figure 4.8: CFD compared with gyro models in third spun case
34
For this case only the initial slosh is reasonably predicted by the gyroscopemodel. No better matching gyroscope setups were found. This suggests that inthis case non-gyroscopic effects dominate the general behaviour of the system.
4.5 40 RPM spin, 1 Hz 25 mm amplitude exci-tation
This case was made to investigate the effects of slower spin on the gyroscopiceffect. Again a relatively low excitation frequency with high amplitude waschosen, although higher than the previous to ellicit stronger response. The ob-served behaviour of the liquid is a combination of strong, apparently gyroscopic,sloshing and low amplitude travelling surface waves. The sloshing has a strongbeat, with the liquid returning to an approximately steady state after threesloshes in a period just short of two seconds. The 2:3 synchronisation of spinand excitation is a likely cause of the observed beat.
0 1 2 3 4 5 6 7 8 9 10−0.01
−0.008
−0.006
−0.004
−0.002
0
0.002
0.004
0.006
0.008
0.01
Time [s]
Cor
rect
ed c
entr
e−of
−m
ass
x−co
ordi
nate
[m]
CFDGyro DGyro E
(a) Corrected x-coordinate comparison
0 1 2 3 4 5 6 7 8 9 10−8
−6
−4
−2
0
2
4
6
8x 10
−3
Time [s]
Cen
tre−
of−
mas
s y−
coor
dina
te [m
]
CFDGyro DGyro E
(b) y-coordinate comparison
Figure 4.9: CFD compared with tuned gyro model in fourth spun case
35
Figure 4.9 shows the centre-of-mass displacements for this case, with the x-displacement corrected for the excitation. The gyroscope model approximatesthe liquid behaviour accurately in this case, with little error thorughout the firstten seconds of physical time. In this case the need for different gyroscope pa-rameters is less surprising, as the lower spin rate causes a flatter liquid surface inthe steady state. The new gyroscope parameters are given in table 4.4. The ‘D’configuration provides a better match in y-displacement, the ‘E’ configurationperforms better for x.
Gyro D Gyro EAxis length 40 ∗ 10−3 m 40 ∗ 10−3 mGyro Mass 4.09 kg 4.09 kgAxial M.O.I 10.2 ∗ 10−3 kg.m2 10.35 ∗ 10−3 kg.m2
Radial M.O.I. 13.0 ∗ 10−3 kg.m2 13.02 ∗ 10−3 kg.m2
Axial damping 0 kg.m2/s 0 kg.m2/sRadial damping 0.002 kg.m2/s 0.002 kg.m2/s
Table 4.4: Gyroscope parameters for the fourth spun case
4.6 Gravity scaling
An attempt was made to investigate gravity scaling of the system. Simulatinglow gravity on earth can be done using neutral buoyancy tests. In these teststhe ‘air’ fraction of the tank would be filled with a liquid with a density slightlylower than that of water. Doing so also reduces inertial and centrifugal forces,requiring higher spin and excitation rates. As a test the default tank was sim-ulated, half filled with water and half with a water-like liquid with a density15/16 of water. The tank was spun at 80 RPM, making the test equivalent to1/16 gravity, 20 RPM. The resulting liquid surface and velocity error are shownin figure 4.10.
The corresponding low-gravity simulation was performed to check the buoy-ancy scaling results. This simulation has the tank half filled with water andhalf with air, rotating at 20 RPM. The gravity in this simulation is 1/16 g. Theresults are shown in figure 4.11.
36
Figure 4.10: Velocity error and liquid surface in neutral buoyancy test
Figure 4.11: Velocity error and liquid surface in 1/16 gravity test
The low gravity simulation shows a parabolic liquid surface, but has highvelocity error in the air-filled half. The buoyancy simulation shows somewhat
37
lower velocity error, again mainly in the tank upper half. The liquid surfacedoes not match the low gravity simulation nor theory. This deformation of theliquid surface is likely due to the pressure profile being affected by the velocityerror (and/or vice versa). While the air in the low-gravity simulation showseven higher velocity error its low density reduces the effect it has on the overallpressure profile, and thus the liquid surface shape.
Due to the large error already present in the steady solution it was decidednot to attempt sloshing simulations using buoyancy compensation. Such simu-lations would be guaranteed to be inaccurate, and thus deemed not worth doingwith currently available CFD.
38
5. Discussion
In this chapter the results, as well as several points encountered during the re-search will be briefly discussed.
5.1 Results
In every testcase the initial response of the system is predicted quite accurately,even with non-optimized gyroscope parameters. In that respect the idea of mod-elling liquid motion as a solid gyroscope can be considered valid.
Prediction of liquid motion over longer timeframes is not as succesful. Inthe first two cases waves and splashing occur, modes which the gyroscope modelobviously cannot match. The long term gyroscope prediction also fails for the0.5 Hz excitation case however. No clear conclusions can be drawn on the rea-son for this, but possible culprits are a too small excitation energy and doubtfulCFD accuracy.
The ability to predict instant response should be enough to allow a controlsystem to manage the liquid, provided the motions remain relatively calm. Sucha control system would need a means to determine the current state of the liq-uid, as predicting forward from the initial condition indefinitely is not possible.
5.1.1 Gyroscope parameters
For every simulation the gyroscope parameters giving the best prediction ofliquid motion have been different. This is to be expected between cases withdifferent spin, as the initial liquid shape is determined by that. It also appearshowever, that the portion of the liquid behaving in a gyro-like fashion dependson the mode of excitation. This means that predicting the liquid response to acase of which there is no prior knowledge by using the gyro model is limited toeven shorter timeframes.
5.2 Chaos
Figure 5.1 shows a trace of a gyroscope centre of mass. The gyroscope in thiscase is the same as the one used in the simulation validation (see table 2.1). It is
39
spun at 2 radians per second and excited horizontally at one radian per second,with a 1 metre amplitude. The initial position of the gyroscope is at z = −1, thex- and y-coordinates being 0. The trace shows the x- and y-position, relative tothe hinge point.
Despite the simplicity of the gyroscope system the resulting motion is quitecomplex. It is not unlikely that under some circumstances this motion is chaoticas well. If that were the case it would be impossible to predict the motion ofthe gyroscope indefinitely. It is certainly worth investigating whether and forwhich cases the gyroscope motion is chaotic. Those cases in which gyroscopemotion is chaotic would most likely yield chaotic liquid motion as well.
−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
X−displacement [m]
Y−
disp
lace
men
t [m
]
Figure 5.1: Trace of gyroscope centre of mass x and y positions
5.3 Gravity scaling
An attempt was made at investigating the gravity scaling of the problem usingneutral buoyancy. This failed due to errors in the numerics, but even withoutsuch issues this would not be trivial. Using buoyancy compensation reducesapparent gravity at the same rate as inertial ‘forces’. Therefore spin rates, aswell as excitation rates and amplitudes would need to be increased when scalingdown gravity, This in turn changes various similarity parameters such as theReynolds number.
Additionally, replacing the air, which can be reasonably considered masslesswith respect to the water, with a liquid of significant mass changes the inertialproperties of the system as well. Should this approach be used to perform grav-ity scaling all of this will have to be taken into consideration.
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Bibliography
[1] Experimentally determined pendulum analogy of liquid sloshing in sphericaland oblate-spheroidal tanks. Nasa Technical Notes, D-2737, 1965.
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