A novel interpolation-free sharp-interface immersed boundary method

Abstract

This paper describes a novel 2^nd order direct forcing immersed boundary method designed for simulation of 2D and 3D incompressible flow problems with complex immersed boundaries. In this formulation, each cell cut by the immersed boundary (IB) is reshaped to conform to the shape of the IB. IBs are modeled as a series of 2D planes in 3D space that connect seamlessly at the edges of the cut cells, in a way that mimics conformal grid. IBs are represented in a continuous and consistent fashion from one cell to another, thus eliminating spatial pressure oscillations originating from inconsistent description of the IB as well as the traditional stair-step problem, leading to a more accurate resolution of the boundary layer. Boundary conditions are enforced at the exact location of the IB devoid of interpolation, which guarantees sound simulations even on grids with high aspect ratio, and enables simulations of flow packed with multiple IBs in close proximity. Boundary conditions for each phase across the IB are enforced independently, yielding a unique capability to solve flows with zero-thickness IBs. Simulations of a large number of 2D and 3D test cases confirm the prowess of the devised immersed boundary method in solving flows over multiple loosely/closely-packed IBs; stationary, moving and highly morphing IBs; as well as IBs with zero-thickness. Results show that predictions from the proposed scheme agree remarkably well with theoretical models and experimental data for both integral variables and local flow fields and they are often with less than 1% deviation from solutions obtained by conformal grid of similar resolution.

1. Introduction

Recent innovative engineering designs such as folding wings in unmanned aerial-aquatic vehicles (e.g. Wu et al. (2019)), biomimetic shape optimization of high-speed train (Kim & Lee (2015)), whale inspired serrated-edge wind turbine (Lee et al. (2019); Mayer et al. (2019)) among others, are accompanied by a significant increase in structural complexity. Desire to simulate fluid interaction with such designs has led to an increased interest in immersed boundary methods since they can accurately represent complex geometries with moving and/or morphing boundaries on a simple Cartesian grid.

Immersed boundary (IB) method was first developed by Peskin (1972) to simulate cardiac mechanics and associated blood flow. Since Peskin (Peskin (1972)) introduced this method, numerous modifications and refinements have been proposed and a number of variants of this approach now exist. Following Mittal & Iaccarino (2005), IB methods can be classified into two categories: continuous forcing methods (CFM) and discrete forcing methods (DFM). In CFMs, solid boundaries is modeled as massless Lagrangian nodes on Eulerian mesh (Peskin (1972, 1977); Yang et al. (2016); Bao et al. (2017); Patel & Natarajan (2018); Yang et al. (2019); Amiri et al. (2020); Tschisgale & Fröhlich (2020)). Velocity on a Lagrangian node is imposed by interpolating the velocities at surrounding Eulerian grids through discrete delta function (DDF). CFMs are attractive because they are formulated independently of underlying spatial discretization. However, the relationship between stress and deformation on Langrangian nodes is formulated by constitutive laws (e.g. Hookes law) making them challenging to employ in cases with rigid boundaries since constitutive relations are not well-posed in the rigid limit. Additionally, CFMs smears IBs a few cells into the fluid domain making them less desirable for simulation at high Reynolds number.

In DFM, the discretized momentum equation is solved without any modification due to the presence of IBs. The solution for obtained intermediate velocity is then used to determine the forcing term. Depending on how the IB force is implemented to impose no-slip boundary condition, DFMs are further categorized into two categories: indirect forcing methods and direct forcing methods. Indirect forcing methods extract the forcing term from a numerical solution for which prior estimate can be determined. IB force is then redistributed over a few cells in the neighborhood of IB in the same manner as CFM but DDF is replaced with a smooth distribution function. This family of DFM was introduced by Fadlun et al. (2000) for laminar flow and later extended by Verzicco et al. (2000, 2002) to cover turbulence flow. Recent work in this category include the works of Allen & Zerroukat (2019); Abdol Azis et al. (2019); Mao et al. (2020). The major advantages of these methods are the absence of user-defined parameters in the forcing and the elimination of associated stability constraints encountered when CFM are employed in simulations with rigid boundaries. However, forcing extends into the fluid region similar to CFMs due to the use of a distribution function.

Direct forcing methods model IB as a ”sharp interface” even in cases where the IB is not aligned to the grid lines. This is accomplished by modifying the computational stencil near the IB to directly impose no-slip BC on the IB. Direct forcing IB methods include ghost-cell methods (Tseng & Ferziger (2003); Mittal et al. (2008); Chaudhuri et al. (2011); Lodato et al. (2016); Zhang et al. (2019)) and cut-cell methods (Udaykumar et al. (1996, 2001); Gao et al. (2007); Schneiders et al. (2013); Patel & Lakdawala (2018)).

Ghost-cell methods enforce no-slip BC by use of a ghost-cell (a cell whose center is in the solid domain but has at least one of its neighbors in the fluid domain). Conservative properties in the ghost cells are reconstructed by an implicit interpolation scheme (Pan et al. (2018)). Although the IB is modeled as a sharp interface, accuracy of the IBs’ representation depends on the accuracy of the reconstruction and hence the number of cells used in interpolation. In Mittal et al. (2008) for instance, 8 cells are used in the neighborhood of a ghost cell for reconstruction purposes. This makes it quite challenging to implement and parallelize on existing codes especially ones employing domain decomposition since all cells required for reconstruction may not be available in one processor domain. Existing solvers (e.g. OpenFOAM) might utilize small computational molecule in which information is shared only across cell faces. In such cases, implementation of interpolation algorithm is computationally expensive because data required for interpolation may not be readily available and might require inter-processor communication which will in return impact parallel efficiency negatively (Liu & Stevens (2020)). Additionally, accuracy of interpolation deteriorates with the increase in the grid’s aspect ratio. Picot & Glockner (2018) observed that ghost-cell methods are most accurate on a grid with the aspect ratio of about one. Another major drawback of ghost-cell method is that a given ghost cell is not unique to one boundary cell and also determining the ghost-cell for a particular boundary cell is not straight forward (Mittal et al. (2008)). As a result, IB is not captured in a continuous fashion from one boundary cell to another (Maitri et al. (2018)) and implementing boundary conditions that require unique treatment for each boundary cell is very challenging e.g. simulation of flow over a bluff body with constant heat flux. Moreover, spatial discontinuities lead to spatial pressure oscillation as reported by Horng et al. (2018), who observed source-sink pressure distribution in cells cut by IB. Seo & Mittal (2011) reported that such oscillation could originate from inconsistent description of the fluxes between momentum equation and pressure equation, leading to classical stair-step problem.

Cut-cell method was introduced by Clarke et al. (1986) for inviscid flow and extended by Udaykumar et al. (1996) to cover viscous flow. In this method, the boundary cell is reshaped to conform to the shape of the IB (Schneiders et al. (2016)). This way, the need for a complex interpolation technique is eliminated while retaining a sharp interface. This method accurately represents IB and eliminates the stair-step problem (Vreman (2020)). However, it is complex to implement (Vreman (2020); Seshadri & De (2019)) and is currently limited to 2D geometries or 3D extruded geometries. Cut-cell immersed boundary methods also suffer severe time-step restriction.

In addition to aforementioned limitations for ghost-cell and cut-cell IB methods, they also suffer severe numerical oscillations when employed to simulate flow with moving boundaries (Uhlmann (2005); Seo & Mittal (2011); Lee et al. (2011); Luo et al. (2012); Liu & Hu (2014)) due to lack of history of conservative variables in a freshly vacated cell as boundary cell changes from solid to fluid domain. Various methods have been proposed to curb spurious oscillations. In Muldoon & Acharya (2008), an optimization scheme was proposed to minimize the deviation from the interpolation stencil to represent the immersed boundary and at the same time ensure a divergence-free solution. A field-extension technique was proposed by Yang et al. (2006) that extends the solution of governing equations to solid cells near the interface by extrapolation from nearby fluid cells. Mittal et al. (2008) employed the interpolation algorithm used to predict values of conservative properties in ghost-cells to provide values of conservative variables in freshly vacated cells. A dynamic weighted interpolation schemes was proposed by Liu & Hu (2014) to smoothen the pressure oscillations. Since most of the techniques proposed rely on interpolation and therefore inherit the aforementioned interpolation-based problems, a standard state of the art technique for addressing ”fresh cell” problem is still elusive.

Computational efficiency is another challenges facing IB methods especially in simulations of internal flow where the number of cells in the solid domain is in the same order as the number of cells in the fluid domain (Kumar et al. (2016)). Attempt to improve computational efficiency by eliminating the solid part of the domain from computation matrix was made in Zhu et al. (2019) where a graphical partitioned method was devised to determine which cells should be solved. However, this method requires storage of flags (active switches) that distinguishes active cells. It also requires a run-time decision to determine the status of the cell in question.

In the light of these challenges, the current study proposes a novel interpolation-free direct forcing IB method, hybrid immersed boundary (HIB), that borrows from ghost-cell and cut-cell methods without inheriting their weakness. In the next section the governing equations and the details of proposed technique are presented.

2. Numerical procedure and mathematical equations

2.1. Governing equation and discretization

The non-dimensional continuity and momentum equations governing the incompressible fluid flow are expressed by Equations 1 and 2.

$$\nabla \cdot \mathbf{U}=0$$ (1)

$$\frac{\partial \mathbf{U}}{\partial t}+\nabla \cdot (\mathbf{U}U)=-\nabla p+\frac{1}{Re}{\nabla }^2\mathbf{U}+\mathbf{f}$$ (2)

where U and p are non-dimensional velocity and pressure, respectively. f is the dimensionless virtual force term.

Equation 2 can be written in form of a general transport equation for a quantity ϕ where ϕ = U represents the momentum equation.

$$\frac{\partial \varphi }{\partial t}+\nabla \cdot (\mathbf{U}\varphi )-\nabla \cdot \left({\mathrm{\Gamma }}_{\varphi }\nabla \varphi \right)=S_{\varphi }(\varphi )+{\mathbf{f}}_{\varphi }$$ (3)

where Γ_ϕ is the diffusion coefficient and S_ϕ(ϕ) is a source term. f_ϕ is a source term associated with the impact of immersed boundary on transport equation.

Finite volume formulation with collocated grid is employed to discretize Equation 3 in a computational stencil shown in Figure 1.

$$a_P{\varphi }_P+a_E{\varphi }_E+a_W{\varphi }_W+a_N{\varphi }_N+a_S{\varphi }_S=a_P^0{\varphi }_P^n+S_{\varphi }{\forall }_P+{\int }_{\forall }{\mathbf{f}}_{\varphi }d\forall$$ (4)

where $a_I=\frac{f_i}{2}-D_i$, I = E, W, N, S, i = e, w, n, s, $a_P^0=\frac{{\forall }_P}{\mathrm{\Delta }t}$, $a_P=a_P^0+\sum \left(f_i-a_I\right)$ and f_i = U_i · A_i is the volume flux leaving the control volume through face i; U_i and A_i are the velocity and area vectors of face i, respectively. $D_i=\frac{{\mathrm{\Gamma }}_{\varphi }\left|{\mathbf{A}}_i\right|}{d_i}$ and d_i is the center to center distance between the two cells sharing face i. ${\varphi }_P^n$ is the value of ϕ in cell P from previous time step.

Figure 1:

Sketch of immersed boundary (IB) and its representative virtual faces (VF) separating two phases in a computational domain: Black cell:- phase 1; Green cells:- phase 2; Blue cells:- Immersed boundary cells (IBC)

Implicit Euler scheme is employed for discretization of temporal term while central differencing is employed for diffusion term. For convective term, linear-upwind scheme is employed in the field while central differencing is used in cells cut by IBs.

2.2. Immersed boundary treatment

Consider IB separating two phases (”Physically distinct, mechanically separable portion that is separated from other phases by an IB”) present in the boundary cell P in Figure 2. The cells cut by IB are reshaped to conform to the shape of the IB leading to modification of coefficient matrix and source vector.

Figure 2:

Illustration of stencils for evaluation of cell face fluxes.

Reshaping cell P changes the flux of phase 1 leaving cell P through face i to $f_i^1={\mathbf{U}}_i\cdot \left(1-{\eta }_i\right){\mathbf{A}}_i$ where η_i is the area fraction that is covered by a phase other than the phase occupying the boundary cell. Similarly, diffusion of phase 1 across face i is modified as $D_i^1=\frac{{\mathrm{\Gamma }}_{\varphi }\left(1-{\eta }_i\right)\left|{\mathbf{A}}_i\right|}{d_i}$. This results in a modified matrix coefficient $a_I^1=\frac{f_i^1}{2}-D_i^1=\left(1-{\eta }_i\right)a_I$. The main assumption is that the center of the face does not change due to reshaping of the boundary cell.

Reshaping cell P also alters its volume occupied by phase 1 to $V_P^1=V_P-\sum V_i^{-}+\sum V_i^{+}$ where $V_i^{-}$ is the volume of the boundary cell P that is occupied by phase 2 and is donated to its neighbor I over face I as shown in Figure 2 while $V_i^{+}$ is the volume of neighboring cell I that is occupied by phase 1 and is donated to boundary cell P over face i (detailed in section 2.6.4). Thus, the coefficient $a_P^0=\frac{V_P}{\mathrm{\Delta }t}$ is replaced by $a_P^{01}=\frac{V_P^1}{\mathrm{\Delta }t}$. The procedure on how to determine volume of the cell in new configuration as well as face fractions η_i is detailed in section 2.6.3. Similar to cell faces, the center of boundary cell is assumed not to change due to reshaping of the cell.

Addition of cell volumes from neighboring cells link the boundary cell to other cells that originally did not share a face. In Figure 2, cell P is connected to cell SW by accepting $V_s^{+}$ from cell S. Since the matrix entry linking P to SW does not exist, convective and diffusive fluxes associated with face sw are explicitly solved and moved to the RHS of Equation 4 in the procedure referred to as ”flux correction” which is detailed in section 2.6.5.

Finally, the boundary conditions on the IB which is governed by Equation 5 is discretized which results in modification of diagonal coefficient $a_P^{IB}$ and source term $S_{\varphi }^{IB}$ for the boundary cell P.

$${\alpha \frac{d\varphi }{dn}|}_{IB}+\beta {\varphi }_{IB}=\gamma$$ (5)

where α, β and γ are constants depending on the desired boundary condition while n is the normal distance between the cell center to the IB.

The diagonal entry is therefore determined by $a_P^1=a_P^{01}+\sum \left(f_i^1-a_I^1\right)+a_P^{IB}$ which is used to replace a_P in Equation 4. $S_{\varphi }^{IB}$ is added to the RHS of Equation 4 and the resulting system shown in Equation 6 is solved for ϕ.

$$a_P^1{\varphi }_P+a_E^1{\varphi }_E+a_W^1{\varphi }_W+a_N^1{\varphi }_N+a_S^1{\varphi }_S=a_P^{01}{\varphi }_P^n+S_{\varphi }{V}_P^1+S_{\varphi }^{IB}$$ (6)

2.3. Momentum predictor

Following discretization procedure described in section 2.1, the transport of momentum yields Equation 7.

$$a_P^1{\mathbf{U}}_P^{*}+a_E^1{\mathbf{U}}_E^{*}+a_W^1{\mathbf{U}}_W^{*}+a_N^1{\mathbf{U}}_N^{*}+a_S^1{\mathbf{U}}_S^{*}=a_P^{01}{\mathbf{U}}_P^n+S_{\mathbf{U}}{V}_P^1+S_{\mathbf{U}}^{IB}$$ (7)

where U^∗ is the guessed velocity.

Equation 7 can be written as Equation 8 by grouping off-diagonal terms together.

$$a_p{\mathbf{U}}_p^{*}=\mathbf{H}\left(\mathbf{U}*\right)-\nabla p^n$$ (8)

$$\mathbf{H}\left(\mathbf{U}\ast \right)=a_p^{01}{\mathbf{U}}_p^n+\sum _{nb}{a}_{nb}{\mathbf{U}}_{nb}^{*}+S_{\mathbf{U}}{V}_P^1+S_{\mathbf{U}}^{IB}$$ (9)

where a_p and a_nb are the diagonal and off-diagonal matrix coefficients, respectively, $a_{nb}{\mathbf{U}}_{nb}^{*}$ is the contribution of neighboring cell nb.

Equation 8 is used to predict guessed velocity U^∗ using pressure from previous timestep pⁿ

2.4. Mass conservation

In immersed boundary formulation, derivation of pressure Poisson equation from mass conservation (Equation 1) can lead to inconsistent description of the geometry. In Figure 2 for instance, discretization of Equation 1 using central differencing method yields Equation 10.

$$f_e+f_w+f_s+f_n=\sum _{nb}{f}_i=0$$ (10)

Equation 10 enforces mass conservation ignoring the interaction between the IB and the boundary cell leading to stair-step problem detailed by Seo & Mittal (2011). In the current numerical scheme, Equation 10 is modified to reflect the action of IB

$$\sum _{nb}{f}_i^1+f_{IB}^P+\sum _{nb}{f}_{IB}^I+\sum _{nb}{f}_i^I=0$$ (11)

where $f_{IB}^P$ is the volume flux leaving through the IB due to motion of IB and/or porous IB, $f_{IB}^I$ is the volume flux leaving through the IB of cell portion accepted from neighboring cell $I\left(V_I^{+}\right)$ and $f_i^I$ is the volume flux leaving through face i of cell portion accepted from neighboring cell I (see Figure 2).

Seo & Mittal (2011) suggested similar correction for pressure equation to curb mass conservation errors and subsequently pressure oscillations but their modification was only limited to pressure equation while boundary equation on velocity field were imposed by classical ghost-cell method described in Mittal et al. (2008). The current scheme on the other hand restructures the grid near the immersed boundary completely for both velocity and pressure field.

The flux $f_{IB}^P$ consists of two components, i.e. $f_{IB}^P=f_{IB}^1+f_{IB}^2$. The first component originates form the porous nature of boundary i.e. blowing/sucution velocity, V_W, at the IB. This component is modeled as $f_{IB}^1={\mathbf{V}}_W\cdot {\mathbf{A}}_{IB}$ where A_IB is the area vector on IB in the cut cell. The second omponent originates from pressure boundary condition at a moving IB shown in Equation 12.

$${\frac{dp}{d\zeta }|}_{IB}=-\overrightarrow{\zeta }\cdot \frac{D{\mathbf{U}}_B}{Dt}$$ (12)

where U_B is the velocity of the IB and $\overrightarrow{\zeta }$ is a unit vector in direction normal to the IB. Descritization of Equation 12 yields the second component of $f_{IB}^P$ i.e. $f_{IB}^2$. This component $\left(f_{IB}^2\right)$ ensures that the IB moves smoothly through the cell during the duration of a time-step.

2.5. Pressure corrector

From Equation 8, the volume flux leaving the boundary cell through face i is given by Equation 13

$$f_i^1={\mathbf{U}}_i\cdot \left(1-{\eta }_i\right){\mathbf{A}}_i=\left(1-{\eta }_i\right){\mathbf{A}}_i\cdot \left({\left[\frac{H\left(\mathbf{U}*\right)}{a_p}\right]}_i-{[\nabla p]}_i\right)$$ (13)

At the IB faces, flux is calculated using Equation 13 since the velocity and pressure gradients are known from the boundary conditions. Using Equations 11 and 13 yields 14

$$\sum _{nb}\left(1-{\eta }_i\right){\mathbf{A}}_i\cdot \left[{\left(\frac{1}{a_p}\right)}_i{\left(\nabla p*\right)}_i\right]=\sum _{nb}\left(1-{\eta }_i\right){\mathbf{A}}_i\cdot {\left(\frac{\mathbf{H}\left(\mathbf{U}*\right)}{a_p}\right)}_i$$ (14)

Equation 14 is solved iteratively until the desired convergence is achieved. In the current scheme, two iterations are sufficient for most cases. The resulting pressure after two iterations is taken as the new pressure field (Equation 15) while the new velocity field is calculated from Equation 16.

$$p^{n+1}=p**$$ (15)

$${\mathbf{U}}_p^{n+1}={\mathbf{U}}_p^{*}-\frac{1}{a_p}\nabla p**$$ (16)

where the subscript, ∗, denotes the number of iteration.

Equation 8, 14 and 15 are solved in an in-house solver; CFDFOAM (Heydari & Sadat-Hosseini (2020)) which is build as an extension of OpenFOAM.

2.6. Immersed boundary procedure

2.6.1. Domain marking

The current IB treatment begins by marking the computational domain into sub-domains corresponding to the number of different phases present in computational domain as shown in Figure 1. A cell is considered to be in one phase if all of its eight vertices are in that particular phase. A cell that has any two of its vertices in different phases is considered as an immersed boundary cell (IBC). This definition is slightly different from the classical ghost-cell method in which a cell is considered to be in one phase if its center is in that particular phase. In ghost-cell methods, at least one phase must be solid since virtual force acts in a ghost-cell; a cell in the solid domain with at least one neighbor in the fluid domain.

For flow over stationary IBs, domain marking is done once for the entire simulation and therefore no appreciable overhead due to immersed boundary treatment. For simulations with moving and/or morphing IBs, domain marking needs to be done after every time-step. However, entire computational domain is examined only in the initial time-step while in the subsequent time-steps, only cells in the vicinity of IB are examined as the IB can not move by more than one cell per time-step due to CFL condition. Since the number cells cut by IBs is much less than the number of cells in the computational domain, the computational overhead associated with domain marking is also negligible for moving/morphing IBs.

2.6.2. Fictitious cells formation

A layer of fictitious cells around the IB are formed for the purpose of implementing desired boundary conditions. The idea of employing fictitious cells to enforce boundary conditions on primary variables (pressure and velocity) was proposed by Hirt et al. (1975) using a novel MAC (Maker and Cell).

The proposed numerical scheme seeks to restructure the computational stencil around the IB so that the faces of IBC coincide with the IB. Faces separating IBCs belonging to different phases are masked and replaced by virtual faces (VFs) (see Figure 1) which are created as 2D planes coinciding with the IB. The center of IBC in question in each phase, say P for phase 1, is mirrored to a point P′ across the VF. A fictitious cell (FC) is constructed around P′ in which the virtual force acts to impose the desired boundary conditions onto the computational domain as shown in Figure 3. Therefore, the center of FC, P′, is repositioned to ensure its image point coincides with the center of the IBC, P, hence ousts the need for interpolation. Similarly, the center of cell E in phase 2 is mirrored to image point E′ where boundary conditions for phase 2 are imposed.

Figure 3:

Formation of fictitious cells (FCs). P is the center of IBC (immersed boundary cell) which is mirrored across virtual face, VF, to form an image point P′; d_P is the normal distance between P and V F; d_E is the normal distance between E and V F

Since every FC is unique to one phase and one IBC as opposed to classical ghost-cell method, governing equations are solved separately in each phase with heterogeneous boundary conditions imposed at the VF for each of the two phases separated by the VF. This makes HIB capable of modeling fluid-fluid phases (e.g. flow around a zero-thickness IBs) and fluid-structure phases where each phase has a unique set of governing equations (e.g. conjugate heat transfer problem). In fact, HIB’s prowess in solving flow with heat and mass transfer is explored in Kingora & Sadat-Hosseini (2021, 2022). The uniqueness of FC is handy in problems that require unique boundary conditions at different locations of IB e.g. cases with specified heat flux.

Velocity at the image point (e.g P′) is represented by Equation 17.

$${\mathbf{U}}_{P^{\prime }}=2{\mathbf{U}}_P-{\mathbf{U}}_{IB}$$ (17)

where ${\mathbf{U}}_{P^{\prime }}$ is velocity at the image point, U_P is the velocity in the IBC and U_IB is the velocity of IB. Velocity at the IB, U_IB, is calculated from the imposed boundary conditions governed Equation 5.

2.6.3. Immersed boundary cell splitting

Cell splitting begins by modeling IB as piecewise segments of 2D planes in 3D space (see Figure 4). By defining an intersection point as a point which IB intersects with an edge of IBC in question (points N, U and V in Figure 4), the IB-plane (plane NUV in Figure 4) is represented by Equation 18.

$$Ax+By+Cz=D$$ (18)

where A, B, C and D are constants.

Figure 4:

Schematic of IBC and positively defined cut plane NUV. Letters represent position vectors from the origin

In 3D, the procedure for determining the volume of an irregular shaped IBC, ∀_BC, and area of the virtual face, A_VF, is not trivial as there are 108 different ways in which the IB can intersect the IBC yielding 108 distinct configurations. They classified into two categories: configurations in which all constant A, B and C in Equation 18 are non-zero and configurations in which at least one of the constants is zero.

If any of the constants A, B or C equal to zero and the other two are not equal to zero, 36 different cell configurations are possible which can be treated into three basic cell shapes shown in Figure 5. Other configurations are treated by rotating these basic shapes referred to as 2D-extruded geometries in the literature. Determination of the area of virtual faces and the resulting volume of IBC for these cases is straight forward and requires no special attention.

Figure 5:

Sketch of three basic types 2D-extruded cut-cell

If any two of the constant A, B or C are equal to zero and the remaining one is not equal to zero, 3 basic configurations are possible in which the IB is parallel to either x, y or z axis. Treatment of IBC with these configurations is also trivial. Finally in the case of a null plane where IB only touches the one vertex of IBC resulting to a full or empty IBC, the treatment is also trivial making a total of 40 configurations in which determination of IBC volume and area of VF is straight forward in what is termed as the limiting cases since at least one of the constant A, B and C is zero.

If all three constants A, B and C are non-zero, volume and face areas are calculated as detailed below for a positively defined IB plane i.e. A ≥ 0, B ≥ 0, C ≥ 0 or A ≤ 0, B ≤ 0, C ≤ 0.

Volume of IBC.

Consider Figure 4. Let point O be the origin such that the position vector of point P,Q and R is $\overrightarrow{P}=OP$, $\overrightarrow{Q}=OQ$ and $\overrightarrow{R}=OR$, respectively. Since the volume of IBC shown in Figure 4 is a combination of tetrahedrons, the volume of IBC, V_IBC is determined by Equation 19

$$\begin{gathered} {\forall }_{IBC}=\frac{\overrightarrow{P}\cdot (\overrightarrow{Q}\times \overrightarrow{R})}{6}-\mathrm{\Phi }(\overrightarrow{S},\overrightarrow{R},\overrightarrow{F},\overrightarrow{I})-\mathrm{\Phi }(\overrightarrow{W},\overrightarrow{P},\overrightarrow{E},\overrightarrow{H})- \\ \mathrm{\Phi }(\overrightarrow{L},\overrightarrow{Q},\overrightarrow{G},\overrightarrow{J})+\mathrm{\Phi }(\overrightarrow{M},\overrightarrow{H},\overrightarrow{G},\overrightarrow{N})+\mathrm{\Phi }(\overrightarrow{T},\overrightarrow{E},\overrightarrow{F},\overrightarrow{U})+\mathrm{\Phi }(\overrightarrow{K},\overrightarrow{J},\overrightarrow{I},\overrightarrow{V}) \end{gathered}$$ (19)

where

$$\mathrm{\Phi }(\overrightarrow{s},\overrightarrow{u},\overrightarrow{v},\overrightarrow{w})=\{\begin{array}{ll}\frac{(\overrightarrow{u}-\overrightarrow{s})\cdot [(\overrightarrow{v}-\overrightarrow{s})\times (\overrightarrow{w}-\overrightarrow{s})]}{6}\hfill & \overrightarrow{u}\ge \overrightarrow{s}\hfill \\ 0\hfill & \overrightarrow{u}<\overrightarrow{s}\hfill \end{array}$$

and letters represent the position vector of their respective points in Figure 4. Equation 19 determines the volume of different cell shapes without knowledge of cell centers or any information of the nature of surfaces enclosing it which is different from what was proposed by Seo & Mittal (2011) where the centers and areas of planes enclosing the control volume needs to be determined

Area of virtual face.

Since the virtual face NUV in Figure 4 is a combination of triangles, area of the virtual face is given by Equation 20

$$\begin{gathered} A_{IB}=\frac{(\overrightarrow{P}-\overrightarrow{Q})\times (\overrightarrow{R}-\overrightarrow{Q})}{2}-\mathrm{\Psi }(\overrightarrow{S},\overrightarrow{R},\overrightarrow{F},\overrightarrow{I})-\mathrm{\Psi }(\overrightarrow{W},\overrightarrow{P},\overrightarrow{E},\overrightarrow{H})- \\ \mathrm{\Psi }(\overrightarrow{L},\overrightarrow{Q},\overrightarrow{G},\overrightarrow{J})+\mathrm{\Psi }(\overrightarrow{M},\overrightarrow{H},\overrightarrow{G},\overrightarrow{N})+\mathrm{\Psi }(\overrightarrow{T},\overrightarrow{E},\overrightarrow{F},\overrightarrow{U})+\mathrm{\Psi }(\overrightarrow{K},\overrightarrow{J},\overrightarrow{I},\overrightarrow{V}) \end{gathered}$$ (20)

where

$$\mathrm{\Psi }(\overrightarrow{s},\overrightarrow{u},\overrightarrow{v},\overrightarrow{w})=\{\begin{array}{ll}\frac{(\overrightarrow{v}-\overrightarrow{u})\times (\overrightarrow{w}-\overrightarrow{u})}{2}\hfill & \overrightarrow{u}\ge \overrightarrow{s}\hfill \\ 0\hfill & \overrightarrow{u}<\overrightarrow{s}\hfill \end{array}$$

and letters represent the position vector of their respective points in Figure 4

Equation 19 and 20 determine the volume of any IBC and the area of any VF, respectively, for 17 different configurations in which the cut plane is positively defined. For a negatively defined cut plane (51 configurations), the IBC is rotated to make cut plane positively before employing Equation 19 and 20 to determine the volume of IBC and area of the VF, respectively: For a cut plane with A ≤ 0, B ≥ 0, C ≥ 0 or A ≥ 0, B ≤ 0, C ≤ 0, the plane becomes positively defined by rotating IBC 90° about y-axis. Similar technique is applicable in the cases where A ≥ 0, B ≤ 0, C ≥ 0 or A ≤ 0, B ≥ 0, C ≤ 0 in which IBC is rotated by 180° and cases where A ≥ 0, B ≥ 0, C ≤ 0 or A ≤ 0, B ≤ 0, C ≥ 0 in which IBC is rotated by −90° hence covering all 51 distinct 3D configurations with a negatively defined cut plane.

2.6.4. Rejected phases treatment

The result of cell-splitting process is a remnant cell whose phase does not constitute the primary phase and is hence occupied by the secondary phase (see Figure 6a). Some authors treat this cell as a new cell with a new discretization procedure (Udaykumar et al. (1996)). However, this attracts a severe time-step restriction since the cell in question might be too small, and for the CFL (Courant–Friedrichs–Lewy) condition to be satisfied, a very small time-step needs to be used. In HIB, original cells (e.g. P) donates (Donor Cell, DC) remnant cell chunks occupied by the secondary phase to their neighbor (e.g. Cell N or E). In return, a neighbor of IBC can accept (Acceptor Cell, AC) the rejected secondary phase (RP in Figure 6a) if the primary phase of such neighbor is similar to the rejected secondary phase of the IBC in question i.e., N and E are viable ACs for IBC, P. The influence of immersed boundary on AC is captured by reshaping the AC in question to reflect new shape configuration. Assuming cell E is determined as optimal AC for the IBC, P (see Figure 6a), the acceptance process involves masking original face, e, separating DC (IBC P) from AC (cell E), and introducing a VF along the IB to replace masked face (MF), e, as shown in Figure 6a. Considering the new configuration, AC becomes an IBC in its own right and is treated as such.

Figure 6:

Illustration of a novel delinking-Linking process designed to minimize spatial pressure oscillation

In the event that a DC has more than one viable AC, the AC whose shared face has maximum area occupied by the rejected secondary phase is chosen to accept the rejected secondary phase and other viable ACs are simply delinked from the DC in question. Considering immersed boundary cell P in Figure 6a, convective and diffusive fluxes across faces n is set to zero (delinking). Among its two viable ACs (Cells N and E), the cell whose shared face has a larger area occupied by the rejected secondary phase is chosen to form a trapezium as shown in cell E.

2.6.5. Flux correction

Delinking process discussed in the previous section involves masking face n. Consequently, the new face area of face ne does not really separate cells E and NE. A more accurate treatment is to split new face area A_ne into original area separating E and NE, say ${\mathbf{A}}_{ne}^0$, and a portion ${\mathbf{A}}_{ne}^{\prime }={\mathbf{A}}_{ne}-{\mathbf{A}}_{ne}^0$, separating E and N which were originally not neighbors (i.e. did not share any face). In HIB, a link is formed between E and N by explicit discretization of diffusion and convection term in E and N corresponding to area ${\mathbf{A}}_{ne}^{\prime }$, while the face ne retains its original area ${\mathbf{A}}_{ne}^0$. For face n where the action of virtual force resulted in delinking cell N from P(DC), F_pn corrects the flux from N to E (AC) (see Figure 6b), which were originally not linked but now share a portion of face n. This ensures phase consistency and seamless representation of IB, thus eradicating source/sink pressure oscillation originating from discontinuous representation of IB from one IBC to another.

The aforementioned method is applicable to situations where an IBC is a viable AC to another IBC. For instance N in Figure 6b clearly donates its volume occupied by phase 1 to NW while NW donates its volume occupied by phase 2 to N. HIB is designed as a commutative procedure such that piecewise addition and subtraction of areas and volumes accurately represent the new configuration of IBCs. Each IBC is therefore treated independently and is treated to one VF for every RP. For the case of cells N and NW in Figure 6b, two virtual faces are formed each corresponding to RP of each IBC, and consequently four FCs.

Lastly, F_IB shown in Figure 6b is the flux associated with movement of IB within IBC. It ensures a continuous motion of IB within the IBC during the duration of a time step. F_IB also represent flux from one domain to another in cases where a thin porous IB allows inter-domain exchange of fluids.

2.6.6. Solid phase treatment

The procedure described so far is generic and can be used to impose heterogeneous boundary conditions on any IB even with zero thickness separating two fluid phases. If one of the phases present in the computational domain is solid, the current method implements a special procedure geared towards stability, accuracy and efficiency.

In the solid domain, velocity U_S is prescribed beforehand or calculated from hydrodynamic forces on the solid. Either way, U_S is a known quantity and is not determined from the governing equation of fluid flow. Pressure inside the solid domain p_S is also invariant. Therefore, the governing equations reduce to:

$$a_P{\mathbf{U}}_P=a_P{\mathbf{U}}_S$$ (21)

$$\nabla p_S^{*}=0$$ (22)

where the coefficient a_P is retained for stability reasons.

Equation 21 and 22 logically eliminate the solid part on the domain from the velocity and pressure matrix, respectively. This improves the efficiency of simulations since only fluid cells are solved. The gain in efficiency is more pronounced in simulations of internal flows where the number of cells in the solid domain is in the same order as the number of cells in the fluid domain. HIB inherently eliminates solid part without storage of extra flags or run-time decisions as opposed to the graphical partitioned method proposed by Zhu et al. (2019).

Equation 21 and 22 also eliminate spurious flow inside the IB which makes computation more stable and converge faster. Spatial pressure oscillations that result in source/sink patterns in the vicinity of IBs are eliminated leading to a more accurate solution in the boundary layer.

2.6.7. Phase correction

In simulation of flows over stationary boundaries, each IBC is always occupied by the same primary phase hence the time history of conservative properties in the IBC is well documented as long as CFL condition is satisfied. A problem emerges in the case of moving and/or morphing boundary where IBC changes its primary phase in consecutive time-steps hence time histories of pressure and velocity in that IBC represent the local histories of previous primary phase. If not carefully treated, these values lead to a wrong solution and cause severe pressure oscillations which lead to numerical instability or even total lack of convergence. IBC that changes its primary phase between consecutive time steps is termed as ”Freshly vacated cell” (FVC).

In HIB, velocity in FVC is treated from consideration of mass flux across faces of FVC. Consider Figure 7 in which a cell, P, changes its primary phase over one time step. At time t = n, P is in phase 1 and cell E is its AC as shown in Figure 7a. After solving the governing equations and obtaining divergent free velocity (say time t = n^∗), same configuration shown in Figure 7a persist only the IB has moved such that P falls in phase 2. At this point, velocity in the AC is divergent free as shown in Equation 23.

$$f_{pn}^{n*}+f_{ne}^{n*}+f_{ee}^{n*}+f_{se}^{n*}+f_{IB}^{n*}=0$$ (23)

where F is the flux. The superscript and the subscript represents the time step and the face, respectively.

Figure 7:

Sketch of freshly vacated: F_xx is the flux across face xx

f_IB is the flux associated to movement of IB while f_pn is flux associated with flux correction procedure described in section 2.6.5. These terms do not change in the course of a time step whereas all other terms are determined from the new velocity field.

Since the AC at time t = n^∗ extends to cover the center of the DC, phase correction is designed to split the AC into two parts i.e. one part associated to AC, E, and the other associated with DC, P, shown in Figure 7b. The part belonging to DC, P, will act as the initial value of velocity field at time t = n + 1, in which P will be in phase 2.

Consider cell E (AC) in Figure 7b. Since mass is conserved, flux across its faces can be described by Equation 24.

$$f_e^{n+1}+f_{ne}^{n+1}+f_{ee}^{n+1}+f_{se}^{n+1}=0$$ (24)

Similarly for cell P (DC)

$$f_e^{n+1}+f_n^{n+1}+f_{sp}^{n+1}+f_{IB}^{n+1}=0$$ (25)

Let $f_n^{n+1}=f_{np}^{n*}$ such that flux between E and NE is not affected by phase correction which would affect global mass conservation. Given that $f_{sp}^{n+1}=0$ (S is in different phase with P at time t = n^∗), Equation 25 is solved to yield flux across MF, $f_e^{n+1}$ as shown in Equation 26. The beauty of this equation is that mass is not only conserved in the AC, E, but also in the FVC, P, which ensures global and local mass conservation. Equation 25 is valid regardless of the change of configuration in the neighboring IBCs for instance in this case it is valid whether cell S changes its phase or not.

$$f_e^{n+1}=-f_{pn}^{n*}-f_{IB}=-f_n^{n+1}-f_{IB}$$ (26)

Equation 26 is the backbone of phase correction. In other words, flux across masked face, f_MF is equal to the summation of fluxes entering the AC due to flux correction and sumation of fluxes across IBs, f_IB.

Having determined fluxes across all faces of a FVC, P, velocity is determined from Equation 27

$${\mathbf{U}}_P^n=\left(1+\frac{d_P}{d_{AC}}\right)\frac{F_{MF}}{{\mathbf{A}}_{MF}}-\frac{d_P}{d_{AC}}{\mathbf{U}}_{AC}$$ (27)

where F_MF is flux across the masked face, U_AC is velocity in the AC, A_MF is the area of MF, d_P is the normal distance from P to MF and d_AC is the normal distance from the center of AC to MF. Having determined the velocity in FVC, pressure is determined explicitly from the semi-discretized momentum equation (Equation 14) and boundary conditions.

3. Results and discussion

3.1. Order of accuracy studies

To determine the order of convergence of the current numerical infrastructure, various simulations are performed on three systematically refined grid sizes: coarse grid, medium grid and fine grid. The instantaneous values of absolute velocity near the immersed boundary for each grid size is evaluated using Equation 28

$$M_x=\sum _i^{N\times N}\left(\rho \times \mathrm{\Delta }{\forall }_i\times {\left|U_x\right|}_i\right)$$ (28)

where M_x is the integral of absolute x-component of velocity, |U_x|_i is the absolute value of x-component of velocity in the i^th cell’s, ρ = 1 Kg/m³ is the density of working fluid, Δ∀_i = Δx_iΔy_iΔz is the volume of the i^th cell. M_y is calculated in analogous fashion to M_x. The solution obtained with the fine grid is taken as the exact solution and errors are evaluated using Equation 29.

$${ϵ}_2=M_1-M_2,{ϵ}_3=M_1-M_3$$ (29)

where ϵ₂ and ϵ₃ are the errors in medium and coarse grids, respectively, M₁, M₂ and M₃ are the integral of absolute velocity for the fine, medium and coarse grids, receptively.

The order of accuracy n is evaluated from Equation 30

$$n=-\frac{\ln \left({ϵ}_2\right)-\ln \left({ϵ}_3\right)}{\ln \left(r_G\right)}$$ (30)

where r_G is the grid refinement ratio.

3.1.1. Cylinder/Sphere rotating in a cavity

A cylinder rotating in a squire cavity is considered as the first case for order of accuracy study. A sketch of domain and boundary condition is presented in Figure 8. Rectangular computational domain stretching from (−D,−D) to (D, D) is employed with uniform grid of size Δx = Δy = 0.01D. A cylinder of diameter D = 1 is placed at the origin. Initially, the cylinder and surrounding fluid (ν = 0.05m²/s) is at rest. The cylinder is then impulsively rotated with angular velocity ω = 1. A small time step (Δt = 0.0001) is used in this simulation.

Figure 8:

Sketch of domain and boundary conditions of a cylinder rotating about its own axis in a square cavity

Figure 9 shows the instantaneous values of x-component of integral of absolute velocity, M_x, for the fine, medium and coarse grid. Plot of y-component of integral of absolute velocity, M_y, is exactly similar to the Figure 9 hence it is not shown to avoid duplication.

Figure 9:

Plot of Mx with time for a cylinder rotating in a cavity

Values of M_x and M_y for various grid sizes at steady state are tabulated in Table 1. n ≈ 2 is obtained for both velocity component which infers that the current scheme represents 2D curved surfaces with second order accuracy.

Table 1:

Grid convergence study for a cylinder rotating in a cavity as shown in Figure 8

	Fine grid (320 × 320)	Medium grid(160 × 160)	Coarse grid (80 × 80)	rG	ϵ 2	ϵ 3	n
Mx	0.143	0.136	0.117	2	7.06 × 10−3	25.89 × 10−3	1.87
My	0.143	0.136	0.117	2	7.06 × 10−3	25.89 × 10−3	1.87

Similar analysis of a sphere rotating in a cavity is shown in Table 2. n ≈ 2 is obtained for all velocity component which confirms that the current scheme represents 3D curved surfaces with second order accuracy.

Table 2:

Grid convergence study for a sphere rotating in a cavity

	FG (160 × 160 × 160)	MG(113 × 113 × 113)	CG (80 × 80 × 80)	rG	ϵ 2	ϵ 3	n
Mx	9.02 × 10−2	8.05 × 10−2	7.14 × 10−2	$\sqrt{2}$	9.76 × 10−3	18.84 × 10−3	1.88
My	9.02 × 10−2	8.05 × 10−2	7.14 × 10−2	$\sqrt{2}$	9.76 × 10−3	18.84 × 10−3	1.88
Mz	2.66 × 10−2	2.06 × 10−2	1.81 × 10−2	$\sqrt{2}$	6.07 × 10−3	11.38 × 10−3	1.81

3.1.2. Oscillating sphere in a cavity

A sphere of diameter D = 1m is placed in a cavity of dimension 2D × 2D × 2D. At time t = 0, the fluid in the cavity is at rest and the sphere’s center coincide with the center of the cavity. At time t ≥ 0, the sphere’s motion is prescribed by Equation 31.

$$x(t)=A\sin (2\pi ft)$$ (31)

where A = 0.25D is the amplitude of oscillation, f = 1Hz is the frequency of oscillation, x(t) is the instantaneous x location of the sphere and t is time in seconds.

Since the flow is unsteady, the order of accuracy is determined at every time-step using the three solutions as shown in Figure 10. The change in order of accuracy of n_x between 1.8–2.2 comes from the fact that the motion of the sphere is periodic and when U = 0, errors on different grids cannot be easily analyzed. All velocity components exhibit 2^nd order of accuracy as observed from Figure 10. This proves that the current numerical scheme is capable of representing complex moving curved surfaces with 2^nd order of accuracy.

Figure 10:

Order of accuracy for sphere oscillating in a cavity

3.2. Pressure oscillation study

The purpose of this study is to demonstrate the significance of flux correction procedure discussed in section 2.6.5 in reducing pressure oscillation. A cylinder rotating in a cavity described in section 3.1.1 is employed in this study.

Figure 11a shows a plot of pressure contours after 1s for a case without flux correction with grid density of $\frac{50}{D}$ and time-step of 10⁻⁴. Severe pressure oscillations (spikes of maximum amplitude of 10 Pa) are observed (see Figure 11c). Horng et al. (2018) observed similar source-sink oscillations in the cells cut by IB and observed that the source and sink are almost equal in strength hence they do not affect mass conservation. Seo & Mittal (2011) suggested that these osculations originate because of stair-step problem in pressure equation which leads to errors in mass conservation. In either case, the repercussion of such oscillations on the stability of numerical solution as well as the accuracy of resolving boundary layer is unclear, especially at high Re. Since HIB is intended for applications at moderate and high Re, a novel flux correction procedure described in section 2.6.5 is devised to fix this problem.

Figure 11:

Pressure oscillation study due to stair-step problem and inconsistent description of immersed boundary

Figure 11b shows a plot of pressure contours after 1s for a case with flux correction. Minimal pressure oscillations (spikes of maximum amplitude of 0.8 Pa) are observed compared to the case where flux correction is not employed and this is because IB is captured in a continuous fashion. The observed small oscillations are due to explicit discretization of the flux correction term described in section 2.6. A seamless pressure would result if the procedure is implemented implicitly but this would require enlarging the computational molecule which is not in the spirit of current immersed boundary method. Even with explicit treatment of flux correction tern, pressure spikes are reduced at least ten folds as shown in Figure 11c.

Figure 11d shows profile of C_P along the surface of the cylinder for various grid densities (grid points per diameter). It is evidence that as grid becomes more dense, the magnitude of pressure spikes reduces. Plot of maximum pressure spike vs grid density in logarithmic scale is shown in Figure 11e. The graph is almost linear with gradient of one. Plot of maximum pressure spikes against log of time-steps for mesh of density $\frac{71}{D}$ is shown in Figure 11f. For time-steps ranging from 10⁻⁴ – 10⁻², which suffices for most practical purposes, the magnitude of maximum pressure spikes are fairly independent of time-step size. Below time-step of 10⁻⁴, magnitude of pressure spikes increases with reduction of time-step size.

Mass conservation error is evaluated by summing the flux across a control volume of dimensions 1.5D × 1.5D around the IB. Mass conservation error of $\mathcal{O}$10⁻⁷ is recorded for both cases with flux correction and without. This insinuates that mass conservation is governed by the tolerance of solving pressure matrix which was set to 10⁻⁶ in this study. Horng et al. (2018) reported that sinks and sources in the vicinity of IB due to pressure oscillations are of equal strength therefore do not affect mass conservation which is collaborated by the present control volume analysis.

3.3. Validation of HIB with stationary immersed boundaries

3.3.1. Steady flow over an isolated stationary flat plate

The plate is modeled as an IB of infinitesimal thickness with height D. A rectangular computational domain stretching from (−10D,−15D) to (70D,15D) is used with the plate’s center located at the origin. Neumann boundary conditions are applied on all sides of the computational domain. A non-uniform Cartesian grid is employed with minimum cell size of Δx = Δy = 0.01D in the vicinity of the IB.

Dirichlet boundary conditions imposed for velocity field at the inlet. Reynolds number based on plate’s height is set to Re = 20 to ensure flow remains steady. The critical Re beyond which separation occurs is 0.4 and flow remains steady until Re ≈ 25 (Taneda (1968)).

Figure 12 shows a schematic of twin vortex formed behind the flat plate. A separated region of flow appears to the rear of the plate consisting of two counter-rotating vortices. Steady state velocity profile at the center line is shown in Figure 13a. The length between the plate and the rear stagnation point, L_W, is taken as the length of recirculation bubble.

Figure 12:

Steady closed wake geometry of flow over a normal flat plate

Figure 13:

Flow over a stationary flat plate at Re = 20

Figure 13b shows time evolution of L_W. Initially, L_W increases rapidly almost in a linear fashion before it asymptotically approaches a constant value. Evolution of position of vortex core (a,b) is shown in Figure 13c. a increases rapidly with time in the initial stage to a maximum value and then reduces asymptotically to a constant value. b on the other hand behave more like L_W which increases asymptotically to a fixed value without a local maximum.

At steady state, the coefficient of drag C_D predicted by the current scheme is 2.06 which is only 1.44% lower than the theoretical value (C_D = 2.09) calculated by Dennis et al. (1993) which empathizes the accuracy of the developed scheme in solving flow around IBs with zero-thickness.

3.3.2. Steady flow over an isolated circular cylinder

Simulation of steady flow over a stationary circular cylinder of diameter D, in unbounded uniform flow is performed on a rectangular computational domain extending from (−10D,−10D) to (20D,10D). The center of cylinder located at the origin. Such a large computational domain is employed to minimize the effect of outer boundary on wake development. A non-uniform Cartesian grid is employed with minimum cell size of Δx = Δy = 0.01D in the vicinity of the cylinder. Neumann boundary conditions are applied on all sides of the computational domain analogous to Figure 29.

Figure 29:

Schematic of cylinders arrangement and boundary conditions

The geometry of closed-wake (Figure 14) at Re = 20, Re = 30 and Re = 40 based on the cylinder diameter is examined. Initially, fluid is at rest. Flow is then allowed to enter the computational domain and wake evolution is studied at various Re.

Figure 14:

Visualisatiion of a steady state closed wake at Re = 40

Figure 14 shows the geometry of a closed wake at low Re. Laminar separation occurs at an angle θ_s from the rear stagnation point. The wake consists of 2D symmetric vortices which grow with time to reach steady state condition. Coordinates of vortex core $(a,\frac{b}{2})$ shown in Figure 14 evolve with time towards a steady value.

Evolution of vortex core $(a,\frac{b}{2})$ as well as separation angle θ_s with dimensionless time $\tau =\frac{tU}{D}$ is shown in Table 3. θ_s, a and b increase with time in a linear fashion initially before asymptotically approaching a steady state value analogous to L_w shown in Figure 15a. At Re = 40, all geometric parameters (θ_s, a,b) predicted in the current study are within 3% deviation from the experimental data, which emphasizes on the accuracy of the current scheme.

Table 3:

Evolution of closed-wake geometrical parameters at low Reynolds number

Re			τ = 4	τ = 5	τ = 6	τ = 8	τ = 10	τ = 12
40	θs	Coutanceau & Bouard (1977)	51°	52.1°	52.7°	53.1°	53.4°	53.4°
		Present study	52.2°	53.2°	52.8°	53.5°	54.1°	54.1°
40	a	Coutanceau & Bouard (1977)	0.62	0.69	0.73	0.75	0.75	0.75
		Present study	0.63	0.71	0.73	0.73	0.73	0.73
40	b	Coutanceau & Bouard (1977)	0.55	0.56	0.58	0.59	0.59	0.59
		Present study	0.54	0.55	0.56	0.57	0.58	0.58
20	θs	Coutanceau & Bouard (1977)	43.1°	44.1°	44.7°	44.9°	44.9°	44.4°
		Present study	43.3°	43.4°	43.6°	43.8°	43.9°	44.1°
20	a	Coutanceau & Bouard (1977)	0.33	0.33	0.33	0.33	0.33	0.33
		a Present study	0.36	0.37	0.37	0.36	0.36	0.36
20	b	Coutanceau & Bouard (1977)	0.44	0.45	0.46	0.46	0.46	0.46
		Present study	0.42	0.42	0.42	0.42	0.42	0.43

Figure 15:

Steady flow over a cylinder

At Re = 20, deviation is within 10% for (a,b) while θ_s is predicted within 1% error. This is because at low Re, the wake parameters are much smaller and evolve rather quickly making them difficult to measure in experiment. From experiment of Coutanceau & Bouard (1977), the authors suggested that Re = 20 may fall in different flow regime as it exhibits a substantial deviation from theoretical values. In fact, the authors reported that the wake does not grow steadily but has a maximum value which is larger than fully established wake. In general, predictions obtained in this study agrees well with experimental values even for transient part of flow.

Figure 15a compares time evolution of wake length, L_w, predicted in present study with experimental data published by Coutanceau & Bouard (1977). L_w is taken as the distance along the centerline where reverse flow prevails. L_w initially grows linearly with time but later approaches a constant value asymptotically. Wake evolution predicted by the current study is within 2% deviation form experimental results for Re = 20 and Re = 40. For Re = 30 the predicted wake evolution approaches asymptotic value slightly sooner than experimental data but approaches the same steady state value.

Figure 15b shows a plot of pressure coefficient C_p around the cylinder. In the current study, pressure at the surface is obtained by applying governing equation explicitly in the boundary cell in fluid domain with zero gradient boundary condition at the IB. The values of C_p predicted in the current study are in good agreement with conformal grid solution (Dennis & Chang (1970)) as well as experimental results published by Grove et al. (1964). The predicted total drag force induced by pressure and friction forces is 1.57 which is exactly identical to the experimentally obtained value by Tritton (1959) as shown in Table 4. The contributions of friction (C_Df ) and pressure forces (C_Dp) to the total drag are 35% and 65%, respectively.

Table 4:

Forces and geometric parameter of a steady state closed wake at Re = 40

Author	LW	a		θs	CD
Present study	2.29	0.75	0.59	53.4°	1.57
Coutanceau & Bouard (1977)(Exp.)	-	0.73	0.58	54.1°	-
Tritton (1959) (Exp.)	-	-	-	-	1.57
Riahi et al. (2018) (Num)	2.35	0.7	0.6	53.7°	1.58
Gautier et al. (2013)(Num)	2.24	0.71	0.59	53.6°	1.49
Chiu et al. (2010) (Num)	2.27	0.73	0.6	53.6°	1.52
Taira & Colonius (2007) (Num)	2.33	0.73	0.6	53.7°	1.54
Brehm et al. (2015) (Num)	2.26	0.72	0.58	52.9°	1.51
Linnick & Fasel (2005)(immersed interface)	2.28	0.72	0.6	53.6°	1.54
Wang & Zhang (2011)(Num)	2.36	0.72	0.6	53.8°	1.54
Dennis & Chang (1970)(Num)	2.35	-	-	53.8°	1.52

Table 4 tabulates the drag coefficients and geometric parameters of fully established wake at Re = 40. C_D as predicted by the current numerical scheme against other studies: immersed boundary method on compressible flow OpenFOAM solver (Riahi et al. (2018)); finite difference method (Dennis & Chang (1970)); immersed interface method (Wang & Zhang (2011)); pseudo-spectral method (Gautier et al. (2013)); differentially interpolated immersed boundary method of Chiu et al. (2010); projection-based immersed boundary method (Taira & Colonius (2007)) among others. Values predicted by the current method agrees well with values predicted from other studies.

3.3.3. Unsteady flow over an isolated circular cylinder

Figure 16 shows vortex shedding behind the cylinder at Re = 100. A quasi-steady wake is observed with alternating positive and negative vorticity.

Figure 16:

Visualisatiion of a vortex street behind the cylinder at Re = 100

Table 5 compares values of C_D, C_L and Strauhal number (St) obtained in the current study with the ones reported in literature. Values obtained by the current method are within range of the values obtained by other numerical methods as shown in Table 5. Value of St predicted by the current scheme (0.167) is exact compared to the experiment of Williamson (1989).

Table 5:

Drag and lift coefficients (C_D and C_L) of a quasi-steady wake at Re = 100

Author	CD	max CL	St
Present study	1.385 ± 0.010	0.344	0.167
Williamson (1989) (Exp)	-	-	0.166
Berger & Wille (1972) (Exp)	-	-	0.16 – 0.7
Liu et al. (1998) (Conformal)	1.35 ± 0.012	0.339	0.165
Linnick & Fasel (2005)(immersed interface)	1.34	0.34	0.165
Tseng & Ferziger (2003)(immersed boundary)	1.42	0.29	0.166
Li et al. (2016)(immersed boundary)	1.36	0.33	0.165
Xu (2008)(immersed interface)	1.42 ± 0.010	0.353	0.172
Uhlmann (2005)(Immersed boundary)	1.45	0.34	0.169

3.3.4. Flow in a 3D convergent divergent nozzle

To test the integrity of implemented numerical infrastructure in 3D. Flow in a 3D convergent-divergent nozzle shown in Figure 17 is simulated and results are compared with conformal grid solution. U = (1, 0, 0) and Δp = 0 are prescribed at the inlet while ΔU = 0 and p = 0 are prescribed at the outlet. No-slip boundary conditions for velocity and zero-gradient pressure boundary condition are prescribed on all curved surfaces.

Figure 17:

Sketch 3D nozzle geometry

A uniform Cartesian grid is employed for this purpose with grid spacing Δx = Δy = Δz = 0.01D where D is the diameter of the neck region of the nozzle. The nozzle is composed of two cylindrical end sections of diameter 2D and length D, converging and diverging cones of length D, and the neck section which is a cylinder diameter D and length of 2D. Reynolds number based on the inlet velocity and diameter of the neck region is set to $Re=\frac{UD}{\nu }=100$.

Figures 18a, 18b and 18c show the profiles of pressure, longitudinal velocity and radial velocity predicted at various x locations, respectively. As expected, these profiles are completely symmetrical owing to the geometry of the nozzle. Pressure and velocity profiles are in remarkable agreement with conformal grid prediction (CG) with less than 1% deviation.

Figure 18:

Plot of pressure and velocity profile at several locations in the nozzle compared with conformal grid solution (CG)

3.3.5. Flow over an isolated sphere

Simulations of flow over an isolated sphere at various Re are carried out in a computational domain extending from (−10D,−10D,−10D) to (20D,10D,10D). Nonuniform Cartesian grid was employed with uniform grid spacing Δx = Δy = Δz = 0.01D in the vicinity of the sphere.

For all Re values considered in this study, flow is steady, axisymmetric and topologically similar to the plot of streamlines, as presented in Figure 19. Flow separates from the surface of the sphere and rejoins at a stagnation point at distant L_W from the sphere forming a closed wake and a toroidal vortex center at distance a from the sphere. This forms a stable axisymmetric vortex ring with a core of radius b as shown in Figure 19. The wake geometry predicted by the current study is compared with experimental results published by Taneda (1956) as well as results of Johnson & Patel (1999) who performed numerical simulations as well as experimental studies.

Figure 19:

Wake visualization of a steady axisymmetric flow

Figure 20a shows a plot of recirculation length, L_W vs. Re as predicted by the present study versus experiment results. As Re increases, L_W is observed to increase which is in agreement with experiment of Taneda (1956). Since Taneda (1956) performed a series of experiments on a sting-mounted sphere over a range of Re that do not exactly match the current setting, comparison is only qualitative. Johnson & Patel (1999) had experimental and numerical setting exactly similar to the present numerical study and quantitatively speaking, values of L_W predicted in this study are within 2% deviation from the ones reported in Johnson & Patel (1999) (see Table 6). Similar observations are made for location of vortex core,a, and separation angle, θ_S as shown in Figure 20b and 20c, respectively.

Figure 20:

Steady axisymmetric flow over a sphere

Table 6:

Closed-wake geometrical parameters for steady state flow over a sphere

Author	Re	50	100	150	200
Present study	θs	38.8°	52.9°	60.1°	64.5°
Johnson & Patel (1999)		40.0°	53.1°	59°	63°
Mittal (1999)		39.8°	53.2°
Present study	LW	0.4	0.87	1.21	1.45
Johnson & Patel (1999)		0.4	0.88	1.21	1.45
Mittal et al. (2008)			0.84	1.17
Mittal (1999)		0.44	0.87
Marella et al. (2005)		0.39	0.88
Luo et al. (2012)			0.91	1.23
Present study	a	0.64	0.76	0.83	0.88
Johnson & Patel (1999)		0.63	0.75	0.83	0.88
Luo et al. (2012)			0.76	0.83
Present study	b	0.42	0.58	0.66	0.72
Johnson & Patel (1999)		0.42	0.58	0.64	0.73
Present study	CD	1.57	1.12	0.92	0.81
Johnson & Patel (1999)		1.57	1.1	0.91	0.81
Marella et al. (2005)		1.56	1.06	0.85

Table 6 shows comparison between the wake geometry and drag predicted by the present study with previously published results. Values of L_W, a,b and θ_S are within acceptable range with values in literature. Closed wake geometrical parameters predicted by HIB are within 2% deviation from the experimental and numerical values published by Johnson & Patel (1999) which emphasizes HIB’s prowess in solving 3D flows. Drag predicted by the current study is also in excellent agreement with data reported by Johnson & Patel (1999).

3.4. Validation of HIB with moving immersed boundaries

3.4.1. Stokes problem

Flow induced by shear motion of a flat plate in an infinite fluid domain is simulated and results are validated against analytical solutions. The working fluid is of density ρ = 1Kg/m³ and viscosity ν = 0.01m²/s. Initially, both fluid and the plate are at rest.

A rectangular computational domain extending from(−10,−10) to (10,10) is used with the plate placed at the x−axis. Neumann boundary conditions are applied on all sides of the computational domain. At the plate, no-slip and zero-gradient boundary conditions are imposed for velocity and pressure fields, respectively. A non-uniform Cartesian grid is employed with minimum cell size of $\frac{\mathrm{\Delta }x}{8}=\mathrm{\Delta }y=\frac{1}{160}\text{m}$ in the vicinity of the plate, i.e. the grid aspect ratio is 8 near the IB to examine performance of the current numerical scheme in grid with high aspect ratio.

Stokes 1^st problem.

The plate is suddenly moved with uniform velocity U₀ = 1m/s for time t > 0s. Profile of velocity at time t = 3s, 6s and 30s is compared with the analytical solution and the two values are in excellent agreement as shown in Figure 21a

Figure 21:

Stokes 1^st and 2^nd problem

Stokes 2^nd problem.

Plate motion for time t > 0s is described by Equation 32. Velocity profile at various phase angles predicted by the current numerical scheme compares well with analytical solution as shown in Figure 21b.

$$u(0,t)=U_0\cos (\omega t)$$ (32)

The results show that the current numerical scheme is capable of accurately modeling IB on grid with high aspect ratio.

3.4.2. Flow induced by an impulsively started flat plate

Flow induced by an impulsive flat plate in a quiescent fluid is computed in order to evaluate the accuracy of the current study in enforcing boundary conditions at the exact location of the interface for a moving IB. Results predicted in this study are compared with the land mark experimental results of Taneda & Honji (1971) where an impulsively started thin flat plate in a water tank is photographed using a flash synchro-socket of a camera and recorded on the oscillograph paper.

The plate is modeled as an IB of infinitesimal thickness with height D. A rectangular computational domain stretching from (−10D,−15D) to (70D,15D) is used with the plate’s center located at the origin. Neumann boundary conditions are applied on all sides of the computational domain. A non-uniform Cartesian grid is employed with minimum cell size of Δx = Δy = 0.01D in the vicinity of the IB.

Initially, the plate is placed with its center at the origin. The plate is then suddenly moved at a constant velocity, U, in the normal direction. Reynolds number based on the height of the plate, D, is set to $(Re=\frac{UD}{\nu }=40)$, similar to the experimental setting of Taneda & Honji (1971).

Figure 22a shows a plot of vorticity contours at dimensionless time $\tau =\frac{Ut}{D}=1$ after the plate is impulsively started. Flow separates at each edge to form a vortex pair which elongates in the flow direction with time. The twin vortices glow in a symmetrical fashion for τ < 8 after which flow becomes unstable and the twin vortices become asymmetric. Figure 22b shows a plot of streamlines at the initial stages of development with the coordinate frame of reference attached to the plate.

Figure 22:

Flow visualization fo an impulsively started normal flat plate with coordinate frame of reference moving with the plate

Time evolution of position of vortex core (a,b) and recirculation length, L_W, are tabulated in Table 7. Initially, vortex core is exactly halfway the recirculation length but continues to move backward towards the wake stagnation point, analogous to experimental observation reported by Taneda & Honji (1971). Time evolution of L_W predicted by the current scheme is compared with experimental values of Taneda & Honji (1971) and numerical simulation of Koumoutsakos & Shiels (1996) in Figure 23a. All three studies are in excellent agreement.

Table 7:

Evolution of closed-wake geometrical parameters for an impulsively started flat plate

Re		τ = 1	τ = 2	τ = 3	τ = 4	τ = 5	τ = 6	τ = 7
40	LW	0.91	1.48	1.89	2.27	2.59	2.89	3.18
	a	0.46	0.74	0.98	1.12	1.29	1.35	1.51
	b	0.77	0.78	0.83	0.86	0.87	0.93	0.94
100	LW	0.92	1.44	1.83	2.17	2.45	2.71	2.94
	a	0.48	0.74	0.89	1.01	1.14	1.26	1.37
	b	0.81	0.79	0.85	0.9	0.94	0.99	1.02
500	LW	0.93	1.38	1.69	1.91	2.11	2.26	2.39
	a	0.48	0.71	0.80	0.86	0.91	0.98	1.08
	b	0.87	0.82	0.88	0.95	0.98	1.05	1.06
1000	LW	0.93	1.37	1.66	1.89	2.06	2.18	2.29
	a	0.49	0.71	0.78	0.84	0.88	0.98	1.07
	b	0.88	0.83	0.86	0.94	0.99	1.04	1.02

Figure 23:

Flow induced by impulsively started normal flat plate at Re = 40 with coordinate frame of reference moving with the plate

Figure 23b shows plots of velocity profile at the centerline at different time instances with the coordinate frame attached to the moving plate. The point of inflection always coincides with the position of the plate despite the plate location not coinciding with grid faces. Maximum deviation of inflection point from actual position of IB is $\mathcal{O}$10⁻⁴Δx. This shows that boundary conditions are imposed at the exact position of the IB.

3.4.3. Steady flow due to a rotating sphere

Steady flow induced by a sphere of diameter D rotating with a constant angular velocity Ω about z axis in unbounded domain is simulated to test the capability of the current numerical scheme in representing 3D moving curved surfaces. Velocity profile along the equator is compared to the theoretical solution derived using vorticity stream function formulation in spherical coordinate (Dennis et al. (1980)).

A cubical computational domain extending from (−12D,−12D,−12D) to (12D,12D,12D) is employed with non-uniform grid distribution. The smallest grid size is Δx = Δy = Δz = 0.01D. Zero-gradient boundary conditions are employed on all sides of the domain for both velocity and pressure fields. Reynolds number based on the sphere’s diameter $Re=\frac{\mathrm{\Omega }{D}^2}{4\nu }$ is set to Re = 50 and Re = 100.

Figure 24 shows velocity profile along the equator at steady state. At the surface, radial velocity U_r = 0 since there is no penetration. As the distance from the sphere increases along the equator, U_r increases due to inflow of fluid at the poles which in turn induces outflow along the equator. U_r reaches a maximum value and then decays gradually thereafter. This observation is consistent with Dennis et al. (1980) and Gilmanov & Sotiropoulos (2005). Velocity profile predicted by the current numerical scheme is in excellent agreement with theoretical result.

Figure 24:

Radial velocity profile along the equator for Re = 50 and Re = 100

3.4.4. Flow around oscillating circular cylinder in quiescent fluid

The proficiency of HIB in solving flows with moving IBs with curved surfaces is assessed by solving flow induced by an oscillating circular cylinder of diameter D in quiescent fluid. The predicted velocity profiles are compared with experimental data reported by Dutsch et al. (1998). Their data has been used to validate various numerical schemes (Yang & Balaras (2006); Wang & Zhang (2011); Liu & Hu (2014); Yuan et al. (2015); Li et al. (2016); Xin et al. (2018); Chi et al. (2020)).

A rectangular computational domain extending (−12D,−12D) to (12D,12D) is employed with non-uniform grid distribution. The smallest grid size is Δx = Δy = 0.01D. Zero gradient boundary conditions is employed on all sides of the computational domain for both velocity and pressure. Reynolds number is set to $Re=\frac{U_{max}D}{\nu }=100$ while Keulegan-Carpenter number is set to $KC=\frac{U_{max}}{fD}=5$, where f is the frequency of oscillation, and U_max is the maximum velocity of the cylinder.

Initially, the cylinder is placed with its center(X_C, Y_C) at the origin. The cylinder is then moved impulsively in simple harmonic motion according to the Equation 33. The set up is left to run until periodic vortex shedding is established before data is recorded.

$$X_C=\frac{KC}{2\pi }\sin (\frac{2\pi }{KC}t)$$ (33)

Figure 25 and 26 show plots of instantaneous contours of vorticity and pressure, respectively, at various phase angles over one cycle. As the cylinder moves in positive x direction, the lower and upper boundary layers develop which separate at the same position on the cylinder (see Figure 25a). This produces two standing counter-rotating vortices of the same magnitude but opposite sign. When the cylinder reaches maximum displacement (see Figure 25b), vortex production diminishes as the cylinder starts to move in negative x direction. A new boundary layer is formed as the cylinder proceeds in negative x direction both at the upper and lower side which separate at the same position but in a reverse fashion to when the cylinder is moving in positive x direction(see Figure 25c). Backward motion of the cylinder also causes splitting of vortex pair produced by the forward motion, which results in vortex shedding. This sequence is repeated in every cycle producing a stable, symmetric and periodic vortex shedding and two stagnation points at the front and back of the cylinder. This pattern is consistent with the corresponding experimental results reported in Dutsch et al. (1998).

Figure 25:

Contours of Vorticity at different phase angles

Figure 26:

Contours of pressure at different phase angles

Figure 27 shows the time-history of pressure drag (C_Dp) over four periods of oscillation which agrees excellently with numerical data published by Dutsch et al. (1998). Immersed boundary methods exhibit fluctuations in the temporal history of pressure drag when employed to solve flow with moving immersed boundaries due to freshly-vacated cells problem (Liao et al. (2010); Seo & Mittal (2011); Kumar et al. (2016)). A smooth pressure history is predicted by the current scheme which demonstrates the capability of our novel ”Phase correction” procedure in addressing the problem of freshly vacated cells. Control volume analysis of mass conservation is evaluated by summing the flux across a control volume of dimensions 2D × 2D around the IB. Decimal mass conservation error of $\mathcal{O}$10⁻⁷ is recorded which proves that addition of virtual volume flux, $f_{IB}^p$, in Equation 11 due to the motion of IB does not affect mass conservation.

Figure 27:

Time history of pressure drag

Figure 28 shows a plot of instantaneous velocity profiles in the y direction at various x positions for various phase angles. Both components of velocity predicted by the current numerical scheme are in good agreement with experimental data reported by Dutsch et al. (1998). This shows the current methods capability in solving flow with moving and/or morphing IBs.

Figure 28:

Profiles of velocity at several X position at different phase angles

3.5. Validation of HIB with multiple immersed boundaries

3.5.1. Flow over three side-by-side cylinders

Flow over three identical side-by-side cylinders at Re = 90 was performed experimentally by Sooraj et al. (2019) using PIV (Particle Image Velocimetry) which provide a unique opportunity to validate simulation with multiple IBs.

Figure 29 shows boundary conditions and cylinders’ arrangement in the computational domain. A non-uniform Cartesian grid is employed with the smallest grid size of Δx = Δy = 0.01D in the vicinity of the cylinders. Center to center distance between consecutive cylinders, S, is set to S = 4D.

Plots of vorticity and time-averaged velocity are presented in Figures 30a and 30b, respectively. Profile of average velocity at x = 4D is presented in Figure 30c. Three distinct wake regions are observed corresponding to the three cylinders. Prediction of velocity profile from the current numerical scheme is in excellent match to the experimental data provided by Sooraj et al. (2019).

Figure 30:

Unsteady flow over three interacting cylinders with S = 4D cylinder at Re = 90

3.5.2. Flow over a circular colony of cylinders

To demonstrate the robustness of the current scheme in solving flow with multiple closely-packed IBs, simulation of uniform unbounded flow over a circular colony of cylinders arranged in concentric ring is presented. Such flow has been simulated by Nicolle & Eames (2011); Chang & Constantinescu (2015); Chang et al. (2018).

A rectangular computational domain extending from (−210D,−210D) to (620D,210D) is used with the colony’s center located at the origin. Similar boundary conditions with the ones shown in Figure 29 are employed. The colony’s diameter and Reynolds number are set to D_G = 21D and $Re=\frac{U{D}_G}{\nu }=2100$, respectively, where D is the diameter of each member cylinder. A total of ten scenarios are simulated (C₇,C₂₀,C₃₉,C₆₄,C₉₅,C₁₃₃,C₁₇₇,C₂₂₇,C₂₈₄,C_S,) with the subscript corresponding to the number of cylinders within the colony apart from C_S which correspond to a solid body. Solid fraction $\varphi =N_C{\left(\frac{D}{D_G}\right)}^2$ varies from ϕ = 0.0159 to ϕ = 0.644, where N_C is the number of cylinders in the colony. Figure 31 shows a plot of quasi-steady vorticity contours for various colonies.

Figure 31:

Flow over a circular colony of cylinders at Re = 2100 based on the diameter of the array, ϕ is the solid fraction, C_N is uniformly distributed N cylinders in the array

Contours forC₇ shows a weakly interacting group of cylinders. Flow features resemble those of an isolated cylinder with each body having a well-defined vortex street. Colony C₂₀ depicts more interaction between wakes of various cylinders within the colony with cylinders sitting in the wake of others depict a strongly coupled vortex field.

Colony C₃₉ shows a strong interaction between wakes of various cylinders in the colony to form a well-defined free shear layer downstream of the colony. As the number of cylinders within the colony is increased beyond N_C > 64, the detached shear layer formed downstream of the colony become unstable and degenerated into coherent lumps of counter-rotating vortices. Beyond N_C > 133, wake bleeding significantly stops to the point that counter-rotating vortices attach to the colony resembling a solid body.

Cases C₁₇₇, C₂₂₇ and C₂₈₄ show the capability of the present method to resolve flow over multiple IBs in close proximity. The minimum distance between any two cylinders for case with 284 is only 0.1D which is mathematically the maximum number of cylinders that can be distributed in a colony of D_G = 21D without cylinders coming into contact. The solid fraction for this cases is ϕ = 0.644. Previous authors (Nicolle & Eames (2011); Chang et al. (2018); Chang & Constantinescu (2015)) reported results up to ϕ = 0.3 corresponding to C₁₃₃ which is below half the solid fraction achieved in this study.

Figures 32a and 32b show variation of average drag (C_D) and lift (C_L) coefficients, respectively, with solid fraction, ϕ. Variation of Strauhal number, $St=\frac{fU}{D_G}$ with ϕ is shown in Figure 32c. C_D increases with ϕ until ϕ ≈ 0.5 where it has a peak value of 1.86 and then drops with further increase in ϕ. C_L increases with ϕ for low values of ϕ since wakes from individual cylinders are in phase and act independently. As wakes interact more vigorously, (0.08 < ϕ < 0.15), wakes from various cylinders are out of phase and lift force cancel out leading to a region of zero lift as shown in Figure 32b. Further increase in ϕ results in an increase in C_L as all cylinders in the colony act as a single solid. St in this region (ϕ > 0.2) shows that the colony acts as a single solid body as there is no noticeable change in St in this region. The prediction of drag and lift coefficients as well as Strauhal number show a close agreement with available data in literature.

Figure 32:

Hydrodynamic forces acting on a colony of cylinders at Re = 2100 for various solid fractions.

3.6. Flow over morphing immersed boundaries

3.6.1. Cardiovascular Flows

Flow in a simplified left ventricle model is simulated to investigate the capability of the proposed numerical scheme in solving fluid-structure interaction problems involving highly morphing and moving 3D structures. The left-ventricle’s cavity is modeled as semi-prolate-spheroid while the aorta and the left atrium are modeled as straight tubes shown in Figure 33a. The geometry is non-dimensionlized with major axis of the prolate-spheroid (2H) which is taken as one unit at the beginning of filling process (diastole) in a cardiac cycle. At the beginning of the diastolic phase, the height of the left ventricle is H while its diameter at the equatorial plane is D_v = 0.5H. The aorta and mitral orifice are of diameters D_a = 0.1H and D_m = 0.3H, respectively. The aortic and mitral valves are modeled as semi-permeable membranes at the equatorial plane, that only allow flow out of the left ventricle and into the left ventricle, respectively. The centerline of the aorta intersects the equatorial plane at coordinates (−0.075,0,0) while the center of the mitral orifice is located at (0.15,0,0). Four IBs are modeled in this case: Aorta (tube), left atrium (tube), equatorial plane and prolate-spheroid wall. Half a prolate spheroid has been employed previously to study complex cardiovascular flow in the left ventricle with tremendous success notably by Baccani et al. (2002); Domenichini et al. (2005, 2007); Domenichini (2008); Zheng et al. (2012). In this study, the dimension and orientation of the mitral orifice and the aorta remain unchanged throughout the cardiac cycle. This is different from Baccani et al. (2002); Domenichini et al. (2005) who adjusted the diameter of mitral orifice to maintain a ratio $\frac{D_m}{D_v}=0.8$ throughout the cardiac cycle.

Figure 33:

Simplified left ventricle model

At time t > 0, volume of the left ventricle cavity expands according to volume shown in Figure 33b which is obtained by integration of flow rate given in Zheng et al. (2012). Expansion occurs in such a manner that the ratio $\frac{H}{D_v}=2$ is maintained. The aortic valve remains closed in diastolic phase (0 ≤ t/T ≤ 0.675) while the mitral valve remains closed in systolic phase (0.64 ≤ t/T ≤ 1). The rate of flow passing the aortic valve and the mitral valve is studied over one period, T.

Figure 33c shows the time history of volumetric flow rate through the aortic and mitral valves with dimensionless time $\tau =\frac{t}{T}$ over one cardiac cycle. The net flow rate is observed to be about zero ($\mathcal{O}$10⁻⁴) which is expected as the net change in volume of the left ventricle’s cavity over one cardiac cycle is zero.

Figure 34 shows the instantaneous vorticity at various times. At 0 ≤ τ ≤ 0.2, a vortex ring is formed at the mitral orifice. At 0.2 ≤ τ ≤ 0.3, a secondary vortex ring is formed which grows as the primary vortex continues to be convected deeper into the left ventricle. At 0.3 ≤ τ ≤ 0.6, the primary vortex ring continuously convects towards the apex and finally reconnects with the secondary vortex. This observations are consistent with the flow structure observed from simulation of Domenichini et al. (2007); Zheng et al. (2012).

Figure 34:

Instantaneous vorticity at various time. $\tau =\frac{t}{T}$

4. Conclusions

A novel 2^nd order interpolation-free immersed boundary method for incompressible flow has been described. The technique, hybrid immersed boundary (HIB), entails reshaping of the cells cut by the IB to conform to the shape of IB in a collocated grid finite volume formulation. Boundary conditions are enforced at the exact location of the IB devoid of interpolation by employing specially designed fictitious cells hence the accuracy of IB’s representation is not influenced by grid aspect ratio. Fictitious cells are unique for each boundary cell which ensures that the IB is represented in continuous and consistent fashion from one cell to another. The technique is designed for a small computational molecule and is therefore easy to implement and parallelize.

Pressure oscillations due to freshly vacated cells is addressed through a novel phase correction procedure that provide the history of primitive variable in such cells. Source-sink oscillations that appear in cells cut by immersed boundary due to stair-step problem are addressed by a novel flux correction procedure. The proposed technique yields reduction in pressure oscillations at least ten folds.

The proficiency of HIB in solving internal and external flows including cases with highly moving/morphing IBs and cases with IBs of infinitesimal thickness on uniform grids as well as non-uniform grids with high aspect ratio has been demonstrated through a large number of 2D and 3D test cases. Values of integral variables as well as local flow fields predicted by the current numerical model agree remarkably well with theoretical models and experimental data. In addition, HIB results match conformal grid solutions with similar grid density in both efficiency and accuracy with maximum deviation of less 1% for both pressure and velocity fields. Moreover, the potential of HIB in simulating flows with multiple IBs is demonstrated by modeling flow over a colony of cylinders arranged in concentric rings. HIB is able to solve closely-packed colony of 284 cylinders with solid fraction of 0.644, which is mathematically the maximum number of cylinders that could be distributed in such colony without the cylinders coming into contact.