In this work, the lung is considered as a conjugate fluid-porous domain, where the fluid region is a truncated airway tree and the porous region is the remainder of the lung. In the fluid region, the flow is governed by the continuity and Navier-Stokes equations, while in the porous region it is governed by their volume-averaged counterparts. At the interface between the fluid and porous regions, the equations are coupled by appropriate interface conditions which ensure a balance of fluxes across interfaces, as well as a balance of viscous and pressure forces. The basic numerical approach taken in this work follows that presented in Chapter 2, which has been shown to be an effective method of coupling fluid and porous
CHAPTER 5. CONJUGATE FLUID-POROUS FLOWS WITH MOVING
BOUNDARIES: APPLICATION TO THE HUMAN LUNG 147
regions where general unstructured grids are required to discretize the domain of interest. Since breathing is driven by the motion of the boundary of the lung, the computational mesh must be considered to be moving for the cases under consideration in this work. Thus, modifications to the original method proposed in Chapter 2 must be made in order to formulate the method in an arbitrary Lagrangian-Eulerian (ALE) framework. Writing the governing mass and momentum equations for the fluid region in integral form, taking into account the mesh motion, results in [32, 33]
Z ∂Ωu·ndS=0 (5.1) and ∂ ∂t Z ΩρfudV+ Z ∂Ωρfu(u−us)·ndS=− Z ∂ΩpndS+ Z ∂Ωµf∇u·ndS, (5.2)
whereΩdenotes an arbitrary control volume in space bounded by the control surface∂Ω. Differential elements of the control volume and control surface are denoted dV and dS, respectively, and the unit-normal vector to the surface∂Ωis denoted n. The field variables
u and p denote the velocity vector and pressure, respectively, while ρf and µf are the
density and dynamic viscosity of the fluid. Finally, usis the velocity of the control surface,
which is used to account for mesh motion. Note that the mesh motion does not change the continuity equation since the fluid is considered to be incompressible [32].
Similarly, the integral forms of the mass and momentum equations presented in the previous chapter are, for a moving control volume, given as
Z ∂Ωhui ·ndS=− Z Ω ∂ε ∂tdV (5.3)
and ∂ ∂t Z ΩρfhuidV+ Z ∂Ω ρf ε hui(hui −us)·ndS=− Z ∂Ωεhpi fndS+Z ∂Ωµf∇hui ·ndS − Z Ω εµf K huidV, (5.4)
wherehuidenotes the extrinsically volume-averaged velocity vector andhpif denotes the intrinsically averaged pressure, according to the standard definitions of volume-averaged quantities [34, 35]. In this case, the permeability tensor, K, is assumed to be diagonal such that the scalar permeability K may be used. Note that one might consider using a tensor permeability to bias the flow in the direction of ducts, however, for the purposes of this work it is assumed that the orientation of the ducts is random such that the flow has no preferred direction within the parenchyma. Also, as discussed in the previous chapter, the Forchheimer drag term is neglected since the Reynolds number in the lung parenchyma is very low. Furthermore, for simplicity of the governing equations, the porosity has been assumed to vary temporally, but not spatially, although the addition of a spatially varying porosity is relatively straightforward to implement numerically after expansion of the dif- ferential form of the convection term by the product rule. The variation in porosity will be further discussed in Sec. 5.3.4 which deals with the estimation of physical parameters of the lung parenchyma. Note that the porous-fluid interface conditions are the same as those for a fixed mesh, discussed in Chapter 2, and thus are not described here.
One final consideration when calculating flows on moving meshes is that one must ensure that the surface velocity, us, is selected in such a way that volume is conserved in
order to avoid artificial mass sources in the domain. This concept is expressed through the ‘geometric conservation law’ (GCL) [32, 33, 36–38], which is given as
∂ ∂t Z ΩdV− Z ∂Ωus·ndS=0. (5.5)
CHAPTER 5. CONJUGATE FLUID-POROUS FLOWS WITH MOVING
BOUNDARIES: APPLICATION TO THE HUMAN LUNG 149
Discretization of Eqs. 5.1 to 5.4 is carried out in the same way as for a stationary mesh using a spatially second-order finite-volume method, as in Chapter 2, with the appropriate modifications to the transient and convection terms to account for the changing cell volume and control surface velocity [32, 33]. For the moving grid calculations presented herein, a first-order backward Euler method was used for the transient terms, which are discretized for a general scalar,φ, as
∂ ∂t Z ΩφdV = φPVP−φPoVPo ∆t , (5.6)
where φ =ρfu in the fluid region and φ =ρfhui in the porous region, φPo indicates the
value ofφ at the previous timestep at the centroid of the control volume P, and VPand Vpo
are the volumes of control volume P at the current and previous timesteps, respectively. Additionally, the mass flux through a discrete control surface, which was calculated as
˙
mip=ρfhˆui ·nipAipfor fixed meshes, is now computed as
˙
mip=ρfhˆui ·nipAip−ρfus,ip·nipAip, (5.7)
to account for the motion of the control surface through which the mass flux is calculated, where us,ip is the surface velocity at the integration point ip and nip is the unit normal
vector to the discrete control surface at ip with area Aip.
Geometric conservation is enforced by computing the surface velocity in such a way that the discretized form of Eq. 5.5 is satisfied to machine precision, i.e.
VP−VPo ∆t = Nip
∑
ip=1 us,ip·nipAip= Nip∑
ip=1 δVip ∆t , (5.8)where Nipis the number of discrete control surfaces surrounding P andδVipis the volume
to the current timestep. Thus, selecting
us,ip·nipAip=
δVip
∆t (5.9)
for all control surfaces guarantees the satisfaction of the GCL since the sum of all swept volumes equals the change in volume represented on the left side of Eq. 5.8. Note that the GCL has been shown using a first-order backward Euler discretization of the transient terms, however, this may be extended to higher order temporal schemes if desired.
In addition to discretization of the governing equations, a procedure for updating the locations of the mesh nodes, based on prescribed boundary motions, is required. Several strategies have been proposed that fall into the class of ‘spring-analogy’ methods which considers the mesh edges as tension springs [38, 39] or torsion springs [40, 41] in order to solve for the nodal displacements. Spring analogy methods, however, will eventually fail given large enough boundary displacements [42]. To accommodate larger mesh motions, algorithms have been proposed which use a linear elasticity analogy [42, 43], which are exceptionally robust but also computationally expensive. For the purposes of this work, the motion of the mesh nodes is determined by numerically solving Laplace’s equation with a variable diffusion coefficient and Dirichlet conditions on all domain boundaries [44], which is reasonably robust for large deformations, is relatively straightforward to implement, and is not overly expensive computationally. The Laplace equation, given by
∇·(Γ∇v) =0, (5.10)
whereΓ is the mesh stiffness and v is the mesh displacement, is discretized using a cell- centred finite-volume method, similar to that used to solve all other transport equations. Since the resulting displacements are stored at the cell centres, they must be interpolated to the nodal locations in order to reposition the nodes. Although using a nodal formulation may be a more natural choice for this problem, no particular difficulties were encountered
CHAPTER 5. CONJUGATE FLUID-POROUS FLOWS WITH MOVING
BOUNDARIES: APPLICATION TO THE HUMAN LUNG 151
Figure 5.1: An illustration of a control volume, P, with the relevant geometric parameters for
evaluating the normal derivative noted.
as a result of the cell-centred formulation employed in this work. The mesh stiffness, Γ, is taken to be inversely proportional to the cell volume such that larger cells absorb more of the motion and smaller cells move more like a rigid body. SinceΓis different for each control volume, an approach similar to that for diffusion across a fluid-porous interface is adopted. Discretization of Eq. 5.10 using the finite-volume method, for a control volume P results in
Nip
∑
ip=1
ΓP∇v|ip·nipAip=0, (5.11)
where again ip refers to the integration points surrounding the volume P. The gradient term is given as ∇v|ip,P·nip= vnb−vP (DP,ip·nip)−ΓΓnbP(Dnb,ip·nip) + (Dnb,ip−(Dnb,ip·nip)nip) (DP,ip·nip)−ΓΓnbP(Dnb,ip·nip) ·∇v|nb − (DP,ip−(DP,ip·nip)nip) (DP,ip·nip)−ΓΓnbP(Dnb,ip·nip)· ∇v|P, (5.12) where the first term on the right side of Eq. 5.12 can be treated implicitly, while the re- maining terms are explicit, and the relevant geometric parameters are shown in Fig. 5.1 for the volume P. Note the definitions of the displacements are DP,ip =xip−xP and
centre of P, and cell centre of nb, respectively.