Particle Shifting

In this second step a pressure to correct the velocity of the particles is calculated.

The final velocity is

þu^t+1_a =þua,∗+þaa,∗∗∆t (4.61)

with the acceleration of the corrector step indicated by∗∗ and the position is þx^t+1_a =þu^t_a+þu^t+1_a ∆t+ 0.5·þaa,∗∗·(∆t)². (4.62) In the DIDF approach we need to solve the Pressure Poisson equation two times per time step. Therefore the numerical effort increases drastically because the solution step of the LES takes approx. 70 % of the hole computation time. Furthermore the approach is not applicable to free surface flows [Hu15]. On the other hand the DIDF approach improves the conservation of volume because both particle position and velocity are considered as criterion for incompressibility.

We use this approach in some simulations where volume conservation is of major importance especially in the presence of stagnation points. In all other cases we allow small errors in the conservation of volume to reduce computational effort.

In the following we call this approach DIDF to indicate the density invariant and divergence free nature.

Nevertheless particles still remain on their trajectory and split up at stagnation points. Therefore another approach to regularize the particles is necessary.

4.5 Numerical stability 81

total shifting vectorþra,s is

þra,s =Cα þRa (4.63)

with the arbitrary constant C = 0.04, the shifting magnitude α = |þumax|dt, the magnitude of the maximum particle velocity|þumax|, the time stepdt, and the shifting vector

Rþa=

N b

X

b

¯ r_a²

r²_abþrˆab. (4.64)

þrˆab, rab and r¯a are the unit distance vector between particle a and b, the distance between both particles and the average particle spacing in the neighborhood defined as

¯ ra = 1

N b

N b

X

b

rab. (4.65)

In this approach we assume that all particles represent the same volume. Therefore a geometrically uniform distribution is the best particle distribution.

Some properties of the fluid are connected to the particles and their position. If we shift the particles we need to correct the hydrodynamic variables Ψ to be consistent and not influence the balance equations. According to Xu et al. a Taylor series of the hydrodynamic variable

Ψa^′ =Ψa+δþraa^′ ·(∇Ψ)_a+· · · (4.66) is used to approximate the property of a particle at the new positiona^′ after shifting.

We use a linear interpolation and neglect quadratic and higher terms. Hydrodynamic variables used in this thesis are the velocity, concentration and temperature. The density is not a hydrodynamic variable in this context because we calculate the density by summation of masses in the neighborhood instead of calculating the continuity equation directly. Therefore we are able to calculate the density at any point.

The particle shifting technique is independent of the time integration and does not

affect the balance equations because we shift the particles only at the very end of a time step and correct the particle properties accordingly.

Originally the shifting technique by Xu et al. assumes that every particle is an identical interpolation point in the domain. In a multi-phase system this is not valid any more because of the presence of an interface. In SPH the interface between two phases is defined by the type (color) of the particles. Therefore the interface is implicitly moved when particles change their position. If we shift the particles then we implicitly shift the interface between phases as well. To overcome this shortcoming we modify Eq. 4.63 near an interface between two phases and apply shifting only in the tangential direction to the interface. This can be done by only consider the tangential part of Eq. 4.63. A similar extension was proposed by Lind.

et al. [Lin12]. They modified the shifting technique for free surface flows based on a Fickian approach.

In the last part of this section we compare the 3 previously introduced approaches for numerical stability and combinations of them. One common test case in literature is a two-dimensional Taylor-Green vortex. In this test case all boundaries are periodic and initially a divergence-free velocity profile

ux =sin(2πx)cos(2πy) (4.67)

uy =−cos(2πx)sin(2πy) (4.68)

is imposed. The vortex decays due to viscous dissipation of the fluid. Details about this test case are found in [Lin12;Xu09]. We consider a case with Reynolds number Re= 1000.

The magnitude of velocity and the particle distribution are shown in Fig. 4.5at time t = 0.24s for different combinations of the previous approaches. Fig. 4.5a shows the results without any numerical stability. We see that particles are aligned on their streamline and large void spaces are formed between the streamlines. The other 3 cases in the top line (Fig. 4.5b-d) are cases where the corrected SPH and DIDF approach without particle shifting is used. Except for some small deviations we get the same particle distribution as in Fig. 4.5a. We note that the velocity distribution

4.5 Numerical stability 83

(a)Without numeri-cal corrections.

n

(b) With corrected SPH approach.

n

(c) With DIDF approach.

n

(d) With DIDF approach and cor-rected SPH ap-proach.

(e) With Particle Shifting approach.

n

nnnn nnnn nnnn nnnn nnnn nnnn

(f) With corrected SPH approach and Particle Shifting approach.nnnn nnnn nnnn

(g) With DIDF ap-proach and Particle Shifting approach.

nnnn nnnn nnnn nnnn nnnn

(h) With corrected SPH approach, DIDF approach and Particle Shifting approach.

Figure 4.5: Taylor-Green vortex simulation with different numerical stability ap-proaches. Particle distribution shown at time t= 0.24s. The Reynolds number is Re= 1000. Color indicates the magnitude of the velocity of the particles. Red indicates

|u|= 1m/s and blue|u|= 0m/s. Colors are distributed by an unsaturated rainbow color scheme.

is good but the simulation tend to diverge. If we apply the particle shifting approach alone or in addition with any of the other stability approaches (Fig. 4.5e-h) the particle distribution as well as the velocity distribution is very smooth. As shown by Xu et al. the accuracy is largely improved.

In this thesis we use the corrected SPH and particle shifting approach where particle movement or surface tension is involved. The DIDF approach is only used in simulations where volume conservation is crucial since this is the main advantage of the approach.