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Various authors have proposed relations for varying Reynolds number ranges. The drag on a sphere immersed in a fluid Table 1 At higher Reynolds numbers, semi-empirical equations are used to validate the method, Table | [30]. The Cartesian grid has 100 x 100 x 100 computational cells with a constant grid size. The grid size ranges from 0.2 m to 0.05 m, depending on the Reynolds number. For high Reynolds numbers, a small grid size is employed to resolve the small flow scales around the sphere. The time step and grid size in the simulations are chosen such that a constant CFL number of 0.1 is employed. The density of the fluid, p;, is 1.0 kg/m? and the viscosity, 1, is 0.1 N s/m*. The radius of the sphere is 0.5 m and the sphere is fixed in the center of the com- putational domain. In the inlet and on the side walls, the velocity of the fluid is set to the mean stream velocity U,, and in the outlet a Neumann boundary condition is employed. The fluid pressure is set to zero in the outlet and a Neumann pressure boundary condition is employed where velocity is specified. The same fluid boundary conditions are employed throughout this paper.

Table 1 Various authors have proposed relations for varying Reynolds number ranges. The drag on a sphere immersed in a fluid Table 1 At higher Reynolds numbers, semi-empirical equations are used to validate the method, Table | [30]. The Cartesian grid has 100 x 100 x 100 computational cells with a constant grid size. The grid size ranges from 0.2 m to 0.05 m, depending on the Reynolds number. For high Reynolds numbers, a small grid size is employed to resolve the small flow scales around the sphere. The time step and grid size in the simulations are chosen such that a constant CFL number of 0.1 is employed. The density of the fluid, p;, is 1.0 kg/m? and the viscosity, 1, is 0.1 N s/m*. The radius of the sphere is 0.5 m and the sphere is fixed in the center of the com- putational domain. In the inlet and on the side walls, the velocity of the fluid is set to the mean stream velocity U,, and in the outlet a Neumann boundary condition is employed. The fluid pressure is set to zero in the outlet and a Neumann pressure boundary condition is employed where velocity is specified. The same fluid boundary conditions are employed throughout this paper.

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Fig. 1. Definition of a control point, 7, which is the vertex in the triangulation of the IB which lies closest to the internal pressure point, 7). 7.x is defined as an external velocity point lying outside the IB in the flow domain, and 7, represents the internal immersed boundary (IIB) velocity points lying inside and close to the IB. Finally, a point lying inside the IB but not close to it is defined as an internal velocity point. 7. In the figure, the staggered velocity points in the x direction are shown. second-order accurate framework to an immersed boundary condition. The IBC constrains the velocity of the fluid to the local velocity of the IB, ip», exactly at the control point, 7,.. The IBC is employed at the internal immersed boundary (IIB) velocity points inside and close to the IB, 7, (see Fig. 1 for definition), such that a trilinear interpolation of the velocity field onto the control point, 7, gives the local velocity of the IB, dip, at every instant of time due to the implicit formulation of the IBC. As a result of the IBC, the velocity field is reversed over the IB and a fictitious velocity field is generated inside the IB. Due to the fictitious velocity field, a flux over the IB is generated, which is unphysical. Therefore, the fictitious velocity field is excluded in the continuity equation, resulting in no mass flux over the IB. Hence, the presence of the IB is accounted for both in the pressure correction equation and in the momentum equations. The fluid properties are extrapolated onto the triangle centers, which are then used to calculate the surface forces upon the one or more immersed bodies. In [32], a similar boundary condition has been developed but an other extrapolation scheme is employed and the fictitious velocity field inside the IB is included in the continuity equation, resulting in slower convergence rate and flux over the IB. Pek = GIANT: 215 eit ee ae Wee. Rees ee i i a ne ng See i ea
Fig. 1. Definition of a control point, 7, which is the vertex in the triangulation of the IB which lies closest to the internal pressure point, 7). 7.x is defined as an external velocity point lying outside the IB in the flow domain, and 7, represents the internal immersed boundary (IIB) velocity points lying inside and close to the IB. Finally, a point lying inside the IB but not close to it is defined as an internal velocity point. 7. In the figure, the staggered velocity points in the x direction are shown. second-order accurate framework to an immersed boundary condition. The IBC constrains the velocity of the fluid to the local velocity of the IB, ip», exactly at the control point, 7,.. The IBC is employed at the internal immersed boundary (IIB) velocity points inside and close to the IB, 7, (see Fig. 1 for definition), such that a trilinear interpolation of the velocity field onto the control point, 7, gives the local velocity of the IB, dip, at every instant of time due to the implicit formulation of the IBC. As a result of the IBC, the velocity field is reversed over the IB and a fictitious velocity field is generated inside the IB. Due to the fictitious velocity field, a flux over the IB is generated, which is unphysical. Therefore, the fictitious velocity field is excluded in the continuity equation, resulting in no mass flux over the IB. Hence, the presence of the IB is accounted for both in the pressure correction equation and in the momentum equations. The fluid properties are extrapolated onto the triangle centers, which are then used to calculate the surface forces upon the one or more immersed bodies. In [32], a similar boundary condition has been developed but an other extrapolation scheme is employed and the fictitious velocity field inside the IB is included in the continuity equation, resulting in slower convergence rate and flux over the IB. Pek = GIANT: 215 eit ee ae Wee. Rees ee i i a ne ng See i ea
Fig. 2. The markings of the staggered u velocities. The exterior velocity points, 7, are marked with a 0; the IIB velocity points, 7j,, are marked with a 1; and the internal velocity points, 7;, are marked with a 2. The two first steps are done once for a stationary method (i.e the IB does not move) as the points always are of the same type. Further more, it is only necessary to perform steps (a) and (b) for the initializing time step for a stationary IB method, because the coefficients are independent of the solution of the flow problem and
Fig. 2. The markings of the staggered u velocities. The exterior velocity points, 7, are marked with a 0; the IIB velocity points, 7j,, are marked with a 1; and the internal velocity points, 7;, are marked with a 2. The two first steps are done once for a stationary method (i.e the IB does not move) as the points always are of the same type. Further more, it is only necessary to perform steps (a) and (b) for the initializing time step for a stationary IB method, because the coefficients are independent of the solution of the flow problem and
Fig. 3. A triangle in the triangulation with the interior immersed boundary point, 7;,, the mirrored exterior normal point, 7., and the immersed boundary point, 7,, and their corresponding velocities i, ij, and uj». 7 is the normal and 7, is the centrum of the triangle. a aaa, ella © aaa aan aaa aaa aaa A ila ara velocity needs to be implicitly interpolated by trilinear interpolation. The IIB velocity can then be set to the reversed interpolated velocity plus the boundary velocity. The interior velocities are set to the boundary (IB) velocity. To ensure that there exists no mass flux over the IB, the same approach as in the first method is adopted. In previous methods [23] employing techniques similar to mirroring the flow on the IB, wiggles occur in the final solution. We propose that this is due to the presence of a unphysical mass flux and/or a unphysical boundary condition over the IB. To decrease the mass flux over the IB, Majumdar et al. [23] employ a Neu- mann boundary condition for pressure at the IB. This results in a zero pressure force over the IB, and thus a decreased mass flux. However, a mass flux still exists due to the flow and other forces. The drawback of employing a Neuman boundary condition for the pressure at the IB is a possible ill-conditioned coefficient structure at the IB; the diagonal coefficient can even be zero.
Fig. 3. A triangle in the triangulation with the interior immersed boundary point, 7;,, the mirrored exterior normal point, 7., and the immersed boundary point, 7,, and their corresponding velocities i, ij, and uj». 7 is the normal and 7, is the centrum of the triangle. a aaa, ella © aaa aan aaa aaa aaa A ila ara velocity needs to be implicitly interpolated by trilinear interpolation. The IIB velocity can then be set to the reversed interpolated velocity plus the boundary velocity. The interior velocities are set to the boundary (IB) velocity. To ensure that there exists no mass flux over the IB, the same approach as in the first method is adopted. In previous methods [23] employing techniques similar to mirroring the flow on the IB, wiggles occur in the final solution. We propose that this is due to the presence of a unphysical mass flux and/or a unphysical boundary condition over the IB. To decrease the mass flux over the IB, Majumdar et al. [23] employ a Neu- mann boundary condition for pressure at the IB. This results in a zero pressure force over the IB, and thus a decreased mass flux. However, a mass flux still exists due to the flow and other forces. The drawback of employing a Neuman boundary condition for the pressure at the IB is a possible ill-conditioned coefficient structure at the IB; the diagonal coefficient can even be zero.
ay Fig. 4. Visualization in two dimensions of the current discrete IIB velocity point, 7;,;, and the exterior normal velocity point, 7, with its surrounding staggered primed indexed velocity points corresponding to the interpolation cell.
ay Fig. 4. Visualization in two dimensions of the current discrete IIB velocity point, 7;,;, and the exterior normal velocity point, 7, with its surrounding staggered primed indexed velocity points corresponding to the interpolation cell.
Fig. 5. Triangle k in the triangulation of the immersed boundary with the exterior points, 7 and 7”. dS" is the area and 7i* is the normal of the triangle. and the other directions follow similarly. If a normal is zero in a certain direction, the resulting term gives no contribution to the total shear force.
Fig. 5. Triangle k in the triangulation of the immersed boundary with the exterior points, 7 and 7”. dS" is the area and 7i* is the normal of the triangle. and the other directions follow similarly. If a normal is zero in a certain direction, the resulting term gives no contribution to the total shear force.
Fig. 6. Triangle k in the triangulation of the immersed boundary with the projection of 7, onto the Cartesian coordinate axes (7,,7,,7:). 5. Results and discussion
Fig. 6. Triangle k in the triangulation of the immersed boundary with the projection of 7, onto the Cartesian coordinate axes (7,,7,,7:). 5. Results and discussion
Fig. 8. The simulated drag coefficient, Cy, for both methods displayed with the known relationships for sphere drag, Table 1. To further show that the viscous and the pressure parts of the surface force are correctly simulated, a grid refinement study of the skin friction, C;, and the pressure coefficient, C,, is performed. The dimensions of the
Fig. 8. The simulated drag coefficient, Cy, for both methods displayed with the known relationships for sphere drag, Table 1. To further show that the viscous and the pressure parts of the surface force are correctly simulated, a grid refinement study of the skin friction, C;, and the pressure coefficient, C,, is performed. The dimensions of the
Fig. 7. The total Stokes drag force, fy (—-), viscous force, fis: (------), and pressure force, fpress (-"-----"- ), are plotted against the simulated forces (VCIB “‘o” and MIB “ ”). From the figure it can be concluded that the simulations accurately reproduces the drag force for Re less than one. The drag force is slightly low due to the influence of the sliding wall.
Fig. 7. The total Stokes drag force, fy (—-), viscous force, fis: (------), and pressure force, fpress (-"-----"- ), are plotted against the simulated forces (VCIB “‘o” and MIB “ ”). From the figure it can be concluded that the simulations accurately reproduces the drag force for Re less than one. The drag force is slightly low due to the influence of the sliding wall.
Simulated drag coefficients as a function of the Reynolds number for the VCIB and MIB methods compared to a number of relation (Table 1) Table 2
Simulated drag coefficients as a function of the Reynolds number for the VCIB and MIB methods compared to a number of relation (Table 1) Table 2
Number of iterations required until convergence is obtained in the first time step if the fictitious velocity field is excluded or included in the pressure correction equation Table 3
Number of iterations required until convergence is obtained in the first time step if the fictitious velocity field is excluded or included in the pressure correction equation Table 3
Fig. 10. Top: The skin friction for different grid sizes. Bottom: Pressure coefficient for different grid sizes. (—-, -+--, ------—--, -+-, -x-, - 30, 40, 48, 60, 80, 100.
Fig. 10. Top: The skin friction for different grid sizes. Bottom: Pressure coefficient for different grid sizes. (—-, -+--, ------—--, -+-, -x-, - 30, 40, 48, 60, 80, 100.
The different computational domains and mesh sizes employed in the simulation of the flow around a single sphere Table 4
The different computational domains and mesh sizes employed in the simulation of the flow around a single sphere Table 4
The different time steps employed for different computational domains in the simulations Table 5
The different time steps employed for different computational domains in the simulations Table 5
Fig. 11. The 2-norm of the total error in the continuity equation is shown for each iteration step at the Reynolds number 10. ‘+’ represents the simulation when the velocity field inside the IB is included and ‘*’ where it is excluded (MIB method) in the pressure correction equation.
Fig. 11. The 2-norm of the total error in the continuity equation is shown for each iteration step at the Reynolds number 10. ‘+’ represents the simulation when the velocity field inside the IB is included and ‘*’ where it is excluded (MIB method) in the pressure correction equation.
Fig. 12. The simulated drag coefficient as a function of time for the four different Reynolds numbers and five different mesh spacings.
Fig. 12. The simulated drag coefficient as a function of time for the four different Reynolds numbers and five different mesh spacings.
Order of accuracy of the MIB method as a function of Reynolds numbers Table 6
Order of accuracy of the MIB method as a function of Reynolds numbers Table 6
Fig. 13. The relative error in the computed drag coefficient as a function of the mesh spacing, N/D. First-order accuracy (----) and second- order accuracy (------- ) are shown for reference.
Fig. 13. The relative error in the computed drag coefficient as a function of the mesh spacing, N/D. First-order accuracy (----) and second- order accuracy (------- ) are shown for reference.
Fig. 14. A simulation of a cluster of non-spherical particles of various sizes. The pressure distribution is visualized on the surface of the particles and the stream lines show the fluid flow.
Fig. 14. A simulation of a cluster of non-spherical particles of various sizes. The pressure distribution is visualized on the surface of the particles and the stream lines show the fluid flow.
Fig. A.1. A two-dimensional trilinear interpolation cell for the staggered velocities u onto the control point, 7; the dimension in the z direction follows analogy.
Fig. A.1. A two-dimensional trilinear interpolation cell for the staggered velocities u onto the control point, 7; the dimension in the z direction follows analogy.
Fig. B.1. A triangle of the immersed boundary, with the points, 7, the center, 7“, normal, 7i*, and area, ds", of triangle k. aloo Further, the center of the triangle is calculated from
Fig. B.1. A triangle of the immersed boundary, with the points, 7, the center, 7“, normal, 7i*, and area, ds", of triangle k. aloo Further, the center of the triangle is calculated from
Related topics:
EngineeringComputational PhysicsFinite Volume MethodsFluid structure interactionImmersed Boundary MethodsDrag ForceFinite Volume Method
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