An Adaptive-Grid Cartesian Cut-Cell Method

for Compressible Viscous Flows

D. Hartmann∗, M. Meinke†, and W. Schroder‡

Institute of Aerodynamics, RWTH Aachen University, Germany

In recent years, numerical methods based on non-boundary-conforming grids to solve the Euler and Navier-

Stokes equations have gained popularity since they offer the possibility to automatically generate the grid and

thus allow a very efficient simulation of flows with complex embedded boundaries [1, 2, 3, 4]. One of the

challenges of these methods is the representation of the embedded boundaries which do not coincide with grid

lines. A broad categorization of various existing methods for non-boundary-conforming grids is given by [2].

Numerical methods which rely on a discrete representation of the embedded boundaries without modifying

the continuous governing equations can be divided into finite-difference-based methods [3, 5] and finite-volume-

based methods also referred to as Cartesian grid or cut-cell methods [1, 4]. The inherent advantages of finite-

volume-based methods over the finite-difference counterpart are strict conservation of mass, momentum, and

energy even at the embedded boundaries and their amenability to adaptive mesh refinement [4, 6]. However, as

noted in [3, 5] general three-dimensional implementations of such cut-cell based Cartesian grid methods for the

Navier-Stokes equations have not been presented, yet.

The major obstacle towards three-dimensional implementations of cut-cell based Cartesian grid methods is

the complexity involved in formulating a finite-volume discretization on cells which are arbitrarily intersected

by the (embedded) boundaries of the computational domain. In this contribution a general formulation to impose

Dirichlet and von Neumann boundary conditions on the arbitrarily shaped cut cells at the (embedded) boundaries

of the computational domain is presented. Furthermore, general expressions to determine the viscous terms of

compressible flow at the embedded boundaries are derived. The application of boundary conditions is facilitated

by introducing ghost cells which can be freely positioned in space, which renders the presented method flexible

in terms of shape and size of the embedded boundaries. The cell centers of cut cells are shifted to the volumetric

center of the cell-fluid volume, such that the method resembles a grid-free approach near the boundaries. Small

cells, which are inherent in cut-cell Cartesian grid methods and may lead to numerical instability, are removed

by a cell-merging/cell-linking procedure. The cells are organized in a hierarchical octree data structure and by

solution-adaptive mesh refinement every cell can be refined or coarsened individually.

On the cut cells and at grid refinement interfaces the least-squares method is used to compute the cell center

gradients. To discretize the convective terms a second-order accurate modified AUSM scheme is used, while

the viscous terms are approximated by second-order accurate central differences. To integrate the compressible

Navier-Stokes equations in time a second-order 5-stage Runge-Kutta scheme is used. More details on the overall

numerical method which is used can be found in [4], where an earlier two-dimensional version of the code is

described. Some results for the flow past a sphere at various Reynolds numbers show that quality of the method.

More data and a more thorough discussion of the mathematics and the physics will be given at the colloquium.∗Corresponding author. Email address: d.hartmann@aia.rwth-aachen.de; tel.: +49 241 80 90396.†Email address: m.meinke@aia.rwth-aachen.de; tel.: +49 241 80 95328.‡Email address: office@aia.rwth-aachen.de; tel.: +49 241 80 95410.

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(a) (b)

Figure 1: Viscous compressible flow past a sphere computed with a Cartesian cut-cell method at M = 0.1 andReD = 200 and ReD = 300: (a) Computed streamlines at ReD = 200; (b) Visualization of the vortex structureat ReD = 300. Contours of the spanwise vorticity ωy computed in the center planes with respect to the sphereare projected as well as they are mapped onto an iso-surface of Q.

Table 1: Three-dimensional simulation of steady laminar flow past a sphere. Drag coefficient Cd and non-dimensional length of the recirculation region Lr/D for the flow at M = 0.1 and Reynolds numbers ReD = 25,ReD = 50, ReD = 100, ReD = 150, and ReD = 200.

Contribution ReD = 50 ReD = 100 ReD = 150 ReD = 200Cd Lr/D Cd Lr/D Cd Lr/D Cd Lr/D

Johnson and Patel [7] 1.57 0.41 1.08 0.88 0.90 1.20 0.78 1.46Marella et al. [8] 1.56 0.39 1.06 0.88 0.85 1.19 - -Present 1.568 0.400 1.085 0.872 0.882 1.202 0.768 1.443

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