Immersed boundary method for the problem of interaction between elliptic body and viscous fluid flow

02001


Introduction
Nowadays, the simulation of the viscous fluid flow with particle of complex shape is an important and difficult applied problem.Here, the complexity of the shape means that particle has an elliptical or star-like shape.In this type of the problem, the geometry of the fluid flow area changes with time.Therefore, it is necessary to trace the particle boundaries and to change the flow area respectively.The correct calculation of the particle movement and rotation requires a high accuracy in calculation of the forces and torques acting on it by the fluid.
To overcome these problems, the immersed boundary method (IBM) is currently gaining popularity [1].The main idea of the IBM is that the fluid flow is modelled on the uniform Eulerian grid covering the whole fluid area, and the presence of immersed particles is described by the moving Lagrangian mesh.In general, their nodes do not coincide and some interpolation procedures are used.The interaction between particles and fluid flow described by a force introduced into momentum equations.The distribution of this force near the immersed boundary is calculated in such a way to fulfill the no-slip condition on the immersed surface.
We use the Navier-Stokes equations for a viscous incompressible fluid, which are solved using a modification of the SIMPLE method.In this modification the fields of velocity, pressure and forces of interaction between particles and fluid are simultaneously calculated in iterations.The movement and rotation of particle are described by  Corresponding author: d.kuranakov@g.nsu.ru the system of the Newton-Euler equations for absolutely rigid body.We concentrate at 2D simulation case.

Model equations
The flow is governed by the momentum and continuity equations where is the fluid velocity, is the fluid stress tensor, is the fluid density, and is the gravity acceleration.The force is the mentioned above immersed boundary force.
For a Newtonian incompressible fluid, the stress tensor is , where is the unit tensor, is the viscous stress tensor, is the fluid dynamic viscosity.The position and motion of a rigid body are characterized by the position of its mass center, the vector of the body rotation relative to its initial orientation, the velocity and the angular velocity .The coordinate and velocity of an arbitrary point of the rigid particle can be decomposed into translational and rotational parts p p , , = + = +  X X r U U ω r (3) where is the radius-vector from the body center to the arbitrary point of the particle Vectors and are oriented in the perpendicular direction to the flow plane.
The evolution of the movement velocity and rotational velocity are described by the Newton-Euler equations Here, the property of the potential gravity field is used.In (6) is the density of the particle, is the volume of the particle with the surface , is the mass moment of inertia, is the outward normal vector to the surface .The vector is the repulsive force acting from the solid wall of the vessel if the particle comes close to one of the boundaries.The magnitude of is computed as follows Here is the distance from particle boundary to the vessel wall, is the critical distance, is the repulsion parameter.Both and are the model parameters introduced to correctly describe particle-wall collisions.The value of is usually chosen sufficiently small, no more than the size of one-two cells of Eulerian grid.The value of is chosen to prevent particle-wall overlapping.
Expressing from (1), one can rewrite (6) as follows ( ) Eqs. ( 1), ( 5) and ( 8) with necessary boundary conditions form the mathematical model.The appropriate numerical method to solve simultaneously all model equations is developed.This numerical method is realized as a C++ code and further is named numerical model.
The developed numerical model is verified on the well-known benchmark problem of unsteady flow past a cylinder at a moderately high Reynolds number.A moderately high Reynolds number means that the Karman vortex street after cylinder is formed, but there are no turbulence vortices in the flow.Good agreement between our calculations and the results of other authors [2][3][4][5][6] is obtained.The results guarantee the accuracy in calculating the interaction force between the fluid and the fixed immersed body.
On the other hand, this benchmark cannot allow us to check the case with the movement of the particle in the fluid.Thus, we consider a problem of sedimentation of a particle in the fluid.

Results
The considered problem is the sedimentation of single particle in a rectangular vessel (tank).The size of the vessel 2 12  length units.All computations were done in dimensionless variables and further "units" will be omitted.The initial position of the particle lays on the midline of the vessel near its top wall.At the initial time the fluid is at rest.The first case is settling of a circular particle of radius 0.125 r = (Fig. 2, a).The particle density is p 1.25

 =
, and the fluid one is f 1  = .Thus, the particle is slightly heavier than the fluid.The fluid viscosity is 0.01  = .Other two cases are with the same elliptical particle, but they differ by horizontal (Fig. 2, b) and vertical (Fig. 2, c) initial orientation of the particle.The elliptical particle has eccentricity 0.9 e = , compression ra- Fig. 2 shows the trajectories of particles, their position at different time moments, as well as the distribution of the vorticity field corresponding the same time moment.While the round particle settles, a Karman vortex street is formed (Fig. 2, a).With the considered flow parameters, the maximum particle sedimentation velocity was max 9 v  , which approximately corresponds to the Reynolds number Re 225  .The formation of the Karman vortex street leads to oscillations of the transversal force acting on the particle.This leads to very low oscillations of the particle trajectory.
For the elliptical particle, the flow pattern changes significantly.Initially horizontal particle settles slower due to the significantly increased drag force.The increased rate of the transversal force oscillations causes the oscillations of the particle trajectory and orientation.However, the orientation of the particle is kept almost horizontal.
The most interesting and complicated case is the sedimentation of the elliptical particle oriented vertically at the initial time.This orientation is not stable and the particle starts to rotate.Rotation causes the particle to move towards right wall of the vessel.The small changes in the initial position can cause the collision with the left wall of the vessel.So this starting orientation is not stable.After the collision, the particle returns to the vessel midline, gains a horizontal orientation, and oscillates in a quasistationary regime approximately in the same way as in the previous case with an initially horizontal particle orientation.The total sedimentation time is bigger than in the previous case.The first stage with wall collision takes additional time.From the mechanical point of view, we explain this behavior using the followings simulations.We consider a rectangular channel of the same size as the vessel in previous computations.The bottom wall now is the inflow boundary, the upper one is the outflow boundary.Without particle the simple Poiseuille fluid flow runs through the channel.The maximal value of the inflow velocity is set max 1 v = .The kinematic viscosity of the fluid is chosen smaller 0.001  = .This is done to consider flows with almost the same particle Reynolds number Re 250  as in the sedimentation case.A particle is placed at the midline of the vessel at height 3 h = .It is fixed and cannot rotate.We consider several cases differed by the turning angles: 0  =  (Fig. 2, b), 90  =  (Fig. 2, c), 45  =  (Fig. 2, d).For comparison, we place the same simulation result for the circular particle (Fig. 3, a).Fig. 3 shows the streamlines and the vorticity field distributions.Karman vortex street is formed downstream every particle.It can be seen that the shape and orientation of the particle significantly affect the magnitudes of the vorticity field and the distortion of the fluid streamlines.If the elliptic particle is oriented along the streamlines, it is easier to flow around it, and the vorticity field is less contrast (Fig. 3, b).If the particle is oriented across the streamlines, it is more difficult to flow around it and the flow is much more disturbed (Fig. 3, c The influence of the particle shape and orientation on the values of drag D f and lift L f forces and torque T has been studied, and is presented in Fig. 4, and the average values and amplitudes of these parameters are collected in Tab. 1.In all symmetrical cases the average value of torque is zero.The amplitudes of oscillation grow with the drag force grow from 0.000145 to 0.011 .In asymmetrical cases the torque also oscillates, but around the average non-zero value.In all asymmetrical cases the value of torque is positive, so the flow rotates the particle counterclockwise to the orientation across the direction of the flow.This phenomenon is used by skydivers to orient themselves across the direction of falling, for example.
Therefore, the vertical orientation, which has the minimal flow resistance, is unstable.The particle in any other orientations try to reorient to the horizontal ones, which has the maximal flow resistance.

Conclusion
We have developed the numerical model of non-stationary fluid flow with particle of complex shape.Using the developed model, we studied the sedimentation of the elliptical particle in the rectangular vessel filled by fluid and showed the influence of shape and its initial orientation on the process.The Reynolds number in our computations was moderately high.The Karman vortex street is formed after the settling particle.This causes oscillations of the drag and lift force which amplitudes significantly depend on the shape and orientation of the particle.The elliptical particle tends to rotate to the equilibrium position in such a way that its semi-major axis is orthogonal to the streamlines of the flow.This is caused by reorienting torque acting from the fluid.The sedimentation of the elliptical particle is slower than the round one because it rotates to the equilibrium position where the drag force is the highest.The sedimentation of the elliptical particle from the stable orientation tends to the quasi-stationary regime with stable oscillations of the trajectory and particle orientation.Starting settling particle from the unstable orientation causes complicated scenario of sedimentation in the beginning of the process, but finally it tends to the quasistationary regime the same as in the stable case.
and elliptical particles have nearly the same volume. α

2 E3SFig. 2 .
Web of Conferences 459, 02001 (2023) https://doi.org/10.1051/e3sconf/202345902001XXXIX Siberian Thermophysical Seminar a b c Trajectory of particles sedimentation, its position at different time moments, and the amplitude of the vorticity field at certain time momemt shown by shades of gray (the darker color means the bigger value of vorticity): circular particle (a), elliptical particle with horizontal (b) and vertical (c) initial orientation.

Fig. 3 .
).If the particle is rotated to 45  =  then flow asymmetry arises: the streamlines on the right are more perturbed than on the left.Karman vortex street shown by vorticity field and streamlines around the fixed particle in the channel: circular particle (a), elliptical particle turned to the angle 0 ) relative to the flow direction.

Fig. 4 .
Fig. 4. Drag force D f , lift force L f and torque T forces versus time for various types of particles: circular particle (circle), elliptical particle turned on angle 0  =  (square),

Table 1 .
Drag D f , lift L f forces and torque T values.