Force-amplified, single-sided diffused-interface immersed boundary kernel for correct local velocity gradient computation and accurate no-slip boundary enforcement

The current diffused-interface immersed boundary method (IBM) with a two-sided force distribution kernel cannot be used to correctly calculate the velocity gradients within the diffused solid-fluid interfaces. This is because the nonzero boundary force distributed to the fluid nodes modifies the momentum equation solved at these locations from the Navier-Stokes equations (NSEs). In this paper, this problem is analytically identified in simple plane channel flow. A single-sided force distribution kernel is used to restrict the boundary force in the solid region and restore NSEs in the fluid region for correct velocity gradient computation. In order to improve the no-slip boundary enforcement in IBM, an extremely simple force amplification technique is proposed. This technique requires no additional computation cost and can significantly reduce the necessary iterations to achieve accurate no-slip boundary enforcement. The single-sided kernel and the force amplification technique are examined in both laminar and turbulent flows. Compared to the standard IBM, the proposed methods not only produce correct velocity gradient results near a solid surface but also reduce numerical errors in the flow velocity and hydrodynamic force and torque results.

Figure 1

The physical plane channel flow (a) and its numerical realizations with different IBM algorithms: (b) standard IBM [3], (c) IBM with Lagrangian grid retraction [9], (d) IBM with single-sided force distribution strategy.

Figure 2

An ideal alignment of Lagrangian and Eulerian grid points around a solid boundary.

Figure 3

Comparisons of L2 errors of no-slip boundary enforcement on plane channel walls with and without the force amplification technique.

Figure 4

Profiles of velocity in a steady-state plane channel flow.

Figure 5

Profiles of velocity gradient in a steady-state plane channel flow.

Figure 6

Sketch of circular Taylor-Couette flow. The red frame indicates the edge of the computational domain. The two blue circles are the physical boundaries, which confine the fluid region in white color. The gray region is the virtual fluid region, which is physically irrelevant to the problem, but also filled with fluid with IBM.

Figure 7

Comparisons of L2 errors of no-slip boundary enforcement on (a) inner cylinder and (b) outer cylinder with and without the force amplification technique.

Figure 8

Comparison of velocity profiles with different DI-IBM algorithms in a steady-state Taylor-Couette flow.

Figure 9

Comparison of velocity gradient profiles with different DI-IBM algorithms in a steady-state Taylor-Couette flow.

Figure 10

The relative error of hydrodynamic torque on the inner and outer cylinders computed with different DI-IBM algorithms.

Figure 11

Orders of accuracy for velocity, velocity gradient, and hydrodynamic torque computation with different DI-IBM algorithms. The solid black line in the left plot shows the reference of the slope $-1.0$. The top and bottom black solid lines in the right plot are reference lines for the slope $-0.55$ and $-1.0$, respectively.

Figure 12

Sketch of a cylindrical particle settling under gravity in a vertical channel.

Figure 13

The L2 errors of no-slip enforcement with and without the force amplification technique: (a) $t=0.370\text{s}$, (b) $t=2.963\text{s}$.

Figure 14

Comparisons of the motion of a settling particle in a vertical channel: (a) particle trajectory, (b) particle angular velocity, (c) particle velocity in vertical direction, (d) particle velocity in horizontal direction.

Figure 15

Results of instantaneous hydrodynamic force in (a) the vertical direction and (b) the horizontal direction with different DI-IBM algorithms.

Figure 16

Results of instantaneous hydrodynamic force in (a) the vertical direction and (b) the horizontal direction with different ways of computing the inertia of virtual fluid inside the settling particle.

Figure 17

Decaying TKE (a) and dissipation rate (b) in a particle-laden decaying homogeneous isotropic turbulence.

Figure 18

Conditionally averaged TKE (a) and dissipation rate (b) as functions of distance from the particle surface.

Figure 19

Decaying particle kinetic energy.