Department of Mechanical and Biomedical Engineering
National University of Ireland, Galway
Galway, Ireland malachy.nestor@nuigalway.ie
Abstract — The Finite Volume Particle Method (FVPM) is a mesh-free method for fluid dynamics. The method is formulated from the integral form of the conservation equations, and particle interactions are described in terms of inter-particle fluxes. Boundaries can be implemented without the need for fictitious particles, and the method is conservative. The motion of the particles can be independent of the fluid velocity.
The advantages of Lagrangian particle motion are numerous, allowing free-surface and moving boundary problems to be simulated with relative ease. However, Lagrangian particle motion in FVPM has been problematic due to the development of highly non-uniform particle distributions, which affect the accuracy and robustness of the method.
In this article, a particle velocity correction is proposed which acts to maintain the uniformity of the moving particle cloud, regardless of the flow configuration. This is facilitated by FVPM due to the independence of particle and fluid velocities, in a similar manner to Arbitrary Lagrangian-Eulerian methods for mesh-based discretisations. The scheme is assessed for incompressible lid-cavity flow at Reynolds number 1000 (SPHERIC benchmark 3). In addition, the suitability of the method for moving boundary problems is demonstrated for incompressible flow over a moving square cylinder in a fixed rectangular walled enclosure (SPHERIC benchmark 6).
I. I NTRODUCTION
The Finite Volume Particle Method (FVPM) was introduced by Hietel et al. [5], and subsequently developed by Keck [6],
[7], Teleaga [15], and others. As in SPH, the fluid in FVPM is represented by particles. However, FVPM differs from SPH in that interactions between particles are defined in terms of fluxes. Particle-boundary interactions can also be described using fluxes, so boundary conditions can be enforced without the use of fictitious particles. The method is conservative in a similar manner to traditional finite volume methods. An interesting feature of the method is that the particle motion may be chosen arbitrarily, and may be independent of the fluid velocity.
Moving boundary problems in fluid dynamics represent a significant challenge for Eulerian mesh-based CFD methods.
Lagrangian motion of the computational nodes is a feature of mesh-free particle methods that offers the ability to handle moving boundary problems with relative ease. However, the use of Lagrangian particle motion means that the particle distribution is determined entirely by the velocity field. Highly distorted particle distributions can result in regions of high flow strain, which invariably leads to a degradation of the robustness and accuracy of any numerical discretisation. This has been demonstrated recently in SPH by Colagrossi et al.
[3] in the context of an incompressible moving boundary problem.
In this work, we argue that for incompressible flow in FVPM, this issue can be worse than the corresponding SPH case, and that some control needs to be exercised over the motion of the particles. We propose a particle motion correction, based on a previous formulation for one-dimensional problems [13], which maintains the uniformity of the particle distribution.
The resulting FVPM scheme is applied to incompressible flow problems with and without moving boundaries.
II. T HE F INITE V OLUME P ARTICLE M ETHOD
In FVPM, the fluid is represented by a set of N particles. A particle is defined by a normalised, compactly-supported test function ψ i
ψ i
( x , t ) =
W i
, (1)
P
N j =1
W j where W i
= W ( x − x i
( t ) , h ) is a compactly supported kernel function for particle i , centred at x i
, with a compact support radius 2 h . Each particle is associated with a volume
V i
, and a discrete value of any field variable φ i
:
V
φ i i
=
=
Z
ψ i
1
Ω
Z
V i Ω d x
φψ i d x ,
(2)
(3) where denotes the region coinciding with the particle support. The FVPM is derived from the integral form of the governing equations:
∂
∂t
Z
Ω
U d x +
Z
Ω
∇ · F d x = 0 .
U is the vector of conserved quantities
(4)
U =
µ ρ
ρ u
¶
, (5) where ρ is the fluid density, and u is the fluid velocity vector.
F is the flux vector
F =
µ ρ u T
ρ u ⊗ u + p I − τ
¶
, (6)
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ρ
0 a 2
0
γ
·µ ρ
ρ
0
¶
γ
− 1
¸
, (7) which is an approximation to the behaviour of incompressible fluids [10].
ρ
0 is a reference density, which is set equal to the initial fluid density, a
0 is a reference speed of sound, which is set to 10 times the expected maximum fluid velocity, and
γ = 7 is the usual choice for liquids.
The derivation of the FVPM is given by Hietel et al.
[5] and is not repeated here. The semi-discrete FVPM is written as d dt
( V i
U i
) = −
N
X
β ij
[ F ( U i
, U j
)] − β i b
F i b , j =1
(8) where β ij is a geometric coefficient which weights the interaction between a pair of particles:
β ij
=
Z
ψ i
∩ ψ j
( ψ i
∇ ψ j
− ψ j
∇ ψ i
) d x , (9) which is computed using numerical integration in the overlap region between each particle pair. The geometric coefficients are analogous to the interface areas between a pair of elements in a traditional finite volume discretisation.
F ( U i
, U j
) is a numerical approximation (e.g. an upwind or central approximation) to the relative flux F − U ˙
T between particles i and j , where ˙ is the particle motion velocity, which is independent of the fluid velocity u .
β metric coefficient, and F i b i b is the boundary geois the boundary flux. In FVPM, the implementation of boundary conditions is determined by these terms without the need for fictitious particles outside the fluid domain. In this article, the inviscid terms of (6) are evaluated using the AUSM+ scheme of Liou [9]. As discussed in Nestor and Quinlan [11], the viscous terms of (6) are computed using the consistency-corrected SPH approximation of Bonet and
Lok [1]. We also adopt the higher order discretisation of the inviscid terms presented in [11].
III. L AGRANGIAN PARTICLE MOTION IN FVPM
In SPH, the fluid density depends directly on the particle distribution. Typically, if SPH particles become clumped, a pressure gradient results which acts to resist non-physical particle clumping. In FVPM, the fluid density is determined by interparticle mass fluxes rather than (as in SPH) by particle position. Therefore, it is possible for FVPM particles to clump without any resulting density or pressure variation. Thus the particle clumping problem may be even more pronounced in
FVPM than in an equivalent SPH simulation.
As an example, Fig. 1 illustrates a hypothetical particle distribution that may arise during the course of a typical simulation. In SPH, this particle distribution would result in higher density values at particles A than at particles B. A density gradient, and therefore a pressure gradient, results between particles A and B, causing the particles to form into a more
(a) SPH
(b) FVPM
Fig. 1.
Illustration of how FVPM and SPH react to clumped particles in an incompressible fluid. For SPH, 1(a), a non-uniform particle distribution results in a non-zero pressure gradient due to the dependence of density on particle distribution. This causes the particles to approach a more uniform configuration for t > t
1
. For FVPM, 1(b), the fluid density is independent of the particle distribution, and therefore the particles may remain in the same position for t > t
1
.
uniform distribution. For a similar situation in FVPM, the fluid density is independent of the particle distribution, so particles
A don’t necessarily have a higher density than particles B.
Consequently, it is possible that the particles remain in the same non-uniform state. For pseudo-incompressible FVPM computations, we have observed empirically that uniform density distributions can occur even if the particles are clumped.
This argument highlights the fact that, although both methods offer Lagrangian particle motion, there is a mechanism present in SPH which helps to maintain the uniformity of the particle distribution for incompressible flows. The lack of such a mechanism in FVPM, and the resulting poor particle distribution, has been problematic in many FVPM computations using purely Lagrangian particle motion.
IV. P ARTICLE MOTION CORRECTION
In principle, FVPM allows the use of arbitrary particle velocities which are independent of the fluid velocity. For moving particle computations, non-Lagrangian particle motion was introduced to FVPM by Schick [13] in an effort to maintain adequate particle spacing for a one-dimensional problem with a discontinuous velocity field. In this work we propose and demonstrate a formulation for particle motion correction in two dimensions.
x i
In particle motion correction, the particle motion velocity is equal to the fluid velocity u i plus a correction velocity
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(a) Fully Lagrangian motion
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x
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(b) Corrected motion
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0
Fig. 2.
Close-up of particle positions and velocity vectors in the upper right corner of the lid-driven cavity at t ∗ = 3 .
3 and Re = 1000 with Lagrangian and corrected particle motion velocity (10).
-0.2
u ′ i
: x i
= u i
+ u
′ i
.
(10)
The correction velocity is given by u
′ i
= C r i
∆ t
R , where r i
=
1
M i
M i
X r ij j
(11)
(12) is the average particle spacing in the neighbourhood of i , ∆ t is the timestep, and C is a constant which is set to 1 / 1000 in this article. The term ¯ i
/ ∆ t represents the velocity required for a particle to move by the average particle spacing ¯ i in a single timestep ∆ t .
R is a dimensionless function of the inverse of the sum of particle spacings:
R =
M i
X j
1
µ r ij
¶
2 n ij
, r i
(13) where r ij and n ij are the distance and unit normal vector between particles i and j respectively.
M i of particle i .
denotes the neighbours
In FVPM, no fictitious particles are required in the implementation of boundary conditions, which leads to a ‘particle deficiency’ for (13) near boundaries. We remedy this problem by including an additional point in the summation of equation
(13) for particles that interact with the boundary. This point is selected as the midpoint of the intersection of the particle support radius with the boundary. This approach, though simple, has worked well for the test cases presented herein.
V. R ESULTS AND DISCUSSION
A. SPHERIC benchmark 3: Lid-driven cavity
The lid-driven cavity is a well-known test case for incompressible viscous flow. The test consists of a square cavity, with sides of length L and no-slip wall boundary conditions on all sides. The fluid is initially at rest. The horizontal boundary at y = L/ 2 is assigned a constant velocity u l which causes the fluid to circulate in the cavity and eventually reach a
-0.4
-0.4
-0.2
0 0.2
U/U l
FVPM: 50x50
FVPM: 75x75
FVPM: 100x100
SPH: 50x50
Ghia et al.
FVM Reference: 300x300 Mesh
0.4
0.6
0.8
1
Fig. 3.
Comparison of x component of velocity along the lid-driven cavity centreline x = 0 for Re = 1000 : FVPM with 50
×
50 , 75
×
75 and 100
×
100 particles, SPH with 50
×
50 particles, finite volume reference solution with
300
×
300 mesh, and high-resolution result of Ghia [4]. FVPM particles move with corrected motion (10).
steady state. The Reynolds number (based on the lid velocity u l and the cavity side L ) is 1000 . High-resolution numerical solutions to this flow are available in the literature. Results generated using the OpenFOAM 1.4.1 CFD package [12] with a 300 × 300 mesh, and the high-resolution numerical results of
Ghia [4], are used for the purposes of validation. Recent SPH results for 50 × 50 particles, first order consistency correction
[1], and the viscosity model of Cleary [2] are also included for comparison.
The FVPM results presented in this section are for 50 × 50 ,
75 × 75 or 100 × 100 particles. The particles are initially arranged in a non-uniform distribution by offsetting the particles from a cartesian arrangement by a randomly determined fraction of the smoothing length. The smoothing length is set to h = 0 .
7∆ x , where ∆ x denotes the corresponding spacing for a uniform cartesian distribution of particles.
Simulations with fully Lagrangian particle motion are hampered by the development of highly non-uniform particle distributions in the early stages of the transient regime. For
50 × 50 particles, Fig. 2(a) shows a close-up view of the particle positions and velocity vectors in the upper right corner of the cavity at non-dimensional time t ∗ = tu l
/L = 3 .
3 .
Even at this early stage, the particles have formed into a highly non-uniform distribution, and the simulation subsequently breaks down. Fig. 2(b) shows a similar situation, but with the corrected particle motion velocity of (10) rather than fully Lagrangian motion. The resulting particle distribution is relatively uniform and the computation proceeds until the flow
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0
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0.03
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(a) Velocity Magnitude, t = 5 s
8 9 10
0.02
0.01
0
0 10 20
FVPM: 50x50 Moving
FVPM: 50x50 Eulerian
SPH: 50x50
FVM Reference: 300x300 Mesh
30
Time (non-dimensional)
40 50
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1
0.5
0
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-1
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0
Umag [m/s]
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0.8
0.6
0.5
0.3
0.1
0
1 2 3 4 5 x [m]
6 7
(b) Velocity Magnitude, t = 8 s
8 9 10
Fig. 4.
Total kinetic energy values for lid-driven cavity flow at Re = 1000 :
Finite volume reference solution using 300
×
300 mesh, FVPM with 50
×
50
Eulerian and corrected particle motion, and SPH with 50
×
50 particles.
reaches steady state.
The steady state FVPM x -velocity profile along the cavity centreline at x = 0 is compared with the results of Ghia [4], the finite volume reference, and SPH with 50 × 50 particles in Fig. 3. Good agreement between the FVPM and reference velocity profiles is noted, and the FVPM solution converges toward the reference solution as the number of particles is increased. The FVPM solution for 50 × 50 particles is in closer agreement with the reference values than the corresponding
SPH solution.
As a further comparison, values for the non-dimensional total kinetic energy versus non-dimensional time are shown in Fig. 4. FVPM results are provided for 50 × 50 particles with both Eulerian and moving particles. All of the meshfree solutions underestimate the kinetic energy of the flow. However, the FVPM results are in closer agreement with the finite volume reference solution than the SPH results. There is only a slight difference in kinetic energy values between the Eulerian and moving particle FVPM cases. In this test, the particle motion correction brings the accuracy and robustness of the fully Eulerian method to a nearly fully Lagrangian method.
B. SPHERIC benchmark 6: Translating square cylinder
SPHERIC benchmark 6 [14] is undertaken as a test case for incompressible flow with moving boundaries. This test case consists of a translating square cylinder in a rectangular cavity.
The test case is considered for flow at Reynolds numbers 50 and 150, based on maximum cylinder velocity and length of side. The motion of the cylinder, prescribed in the benchmark, is characterised by acceleration from rest to a constant velocity of 1 .
0 m/s. The peak velocity is reached at approximately
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P [Pa]
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0
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1
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1
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0
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0
P [Pa]
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1
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0.3
0.1
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-0.2
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1
2
2
3
3
4
4
5 x [m]
(c) Pressure Distribution, t = 5 s
5 x [m]
6
6
7
7
(d) Pressure Distribution, t = 8 s
8
8
9
9
10
10
Fig. 5.
SPHERIC benchmark 6: Particle distribution coloured by velocity magnitude and pressure at t = 5 s and t = 8 s with Re = 50 and 20,000 particles.
t = 1 .
0 s. The simulation is terminated at t = 8 s when the cylinder becomes close to the end wall of the cavity. Contour plots of velocity magnitude and pressure at times t = 5 s and t = 8 s are provided in the benchmark specification, as well as histories of the pressure and viscous components of the drag coefficient for each Reynolds number. For the FVPM simulation, 20,000 particles are initially arranged in a uniform cartesian pattern and the smoothing length for all particles is set to h = 0 .
7∆ x . We adopt the corrected particle motion velocity of (10) throughout this section.
Figs. 5 and 6 show the FVPM solution for the velocity magnitude and pressure distribution at times t = 5 s and t = 8 s at Reynolds numbers 50 and 150 respectively. The
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Umag [m/s]
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1
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(a) Velocity Magnitude, t = 5 s
8 9 10
6
5
4
3
2
1
0
0
10
9
8
7
2 4
Time [s]
FVPM: Cd (pressure)
FVPM: Cd (viscous)
REF: Cd (pressure)
REF: Cd (viscous)
6
1.5
1
0.5
8
0
2 3 4 5 x [m]
6 7 8 9 10
(b) Velocity Magnitude, t = 8 s
(a) Re = 50
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1
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0
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-1
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0
P [Pa]
1.2
1
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0
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1
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0
P [Pa]
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1
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1
2 3 4 5 x [m]
(c) Pressure Distribution,
6 t = 5
7 s
8 9 10
10
9
8
7
6
5
4
3
2
1
0
0 2 4
Time [s]
FVPM: Cd (pressure)
FVPM: Cd (viscous)
REF: Cd (pressure)
REF: Cd (viscous)
6
1.5
1
0.5
8
0
2 3 4 5 x [m]
6 7 8 9 10
(b) Re = 150
(d) Pressure Distribution, t = 8 s
Fig. 6.
SPHERIC benchmark 6: Particle distribution coloured by velocity magnitude and pressure at t = 5 s and t = 8 s with Re = 150 and 20,000 particles.
Fig. 7.
SPHERIC benchmark 6: Drag coefficient history for Reynolds numbers 50 and 150: benchmark solution [14] and FVPM with 20,000 particles.
FVPM accurately reproduces the features of the benchmark level-set solution of Colagrossi [14] at both Reynolds numbers, including the regions of high velocity in the cylinder wake at t = 5 s, and near the upstream corners of the cylinder at t = 8 s.
The pressure distribution predicted by FVPM reproduces the overall trends in the benchmark solution, but is not in accurate agreement. We attribute the disparity to the use of the pseudoincompressible model in contrast with the fully incompressible fluid modelled in the benchmark. Early in the simulation, when the cylinder is accelerating to its peak velocity, we observe the propagation of a compression wave from the upstream face of the cylinder. This wave then reflects repeatedly between the end walls, resulting in pressure fluctuations throughout the domain. Although this results in disagreement with the reference solution, this behaviour is physically correct for a weakly compressible fluid. In light of the results presented by Lee et al.
[8] for this test case, we expect that a fully incompressible formulation of the method would lead to a more accurate prediction of the pressure field.
Fig. 7 shows a comparison between the reference and FVPM drag coefficients for both Re = 50 and Re = 150 . The results for the constant-velocity stage of the cylinder motion are characterised by fluctuations in the pressure component of the drag coefficient, which are due to the wave discussed above. The magnitude and duration of the drag peak in the acceleration stage are in reasonable agreement with the
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VI. C ONCLUSIONS
Lagrangian particle motion has been problematic in FVPM simulations due to the formation of unfavourable particle distributions. In this article we have presented arguments which suggest why this behaviour may be worse in FVPM than SPH for incompressible flow. We have proposed a particle motion correction procedure, based on a previously proposed method for one-dimensional problems, which maintains the uniformity of the particle distribution without affecting the fluid velocity.
This is facilitated by the independence of particle motion and fluid velocity in FVPM.
The scheme was assessed for lid-driven cavity flow at
Reynolds number 1000 (SPHERIC benchmark 3), and the centreline velocity profiles were found to be in good agreement with the reference solution. A comparison of the total kinetic energy values FVPM yielded closer agreement with the reference solution than SPH for this class of flow.
FVPM with the particle motion correction was evaluated for SPHERIC benchmark 6, a challenging moving boundary problem. The computed velocity field showed good agreement with the benchmark results. The pressure field agreed less accurately with the benchmark solution, due to the use of a pseudo-incompressible equation of state, in contrast with the incompressible fluid modelled in the benchmark model. The pressure fluctuations observed in the FVPM solution are due to waves generated by acceleration of the cylinder, and are physically plausible for a compressible fluid; they are not a form of numerical noise.
The FVPM is a relatively recent addition to the range of mesh-free methods available for CFD. The method offers the advantages of mesh-free discretisation coupled with many of the desirable properties of traditional finite volume discretisations, including conservation and convergence. The particle motion correction proposed in this paper allows FVPM to retain the advantages of a fully Lagrangian approach, while avoiding the poor particle distributions that often arise in
Lagrangian methods. Preliminary results suggest that this approach is a robust and accurate tool for the computation of moving boundary problems in fluid dynamics.
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