**3.2 Numerical setup**

190 Advanced Fluid Dynamics

Fig. 3. The sketch of the test rig and the analogy of the mass-spring-damper system and the

Fig. 4. Coupled Helmholtz-resonators and oscillating masses connected with springs and

combustion chamber as Helmholtz resonator

damping elements

In order to compute the resonance characteristics of the system a series of LESs at discrete forcing frequencies had to be completed.

In the case of the single resonator these were taken for a basic configuration corresponding to the experiments, for variation of the geometry of the resonator neck and for variation of the fluid temperature. The compressible flow in the chamber of the basic geometry was simulated for five different frequencies in the vicinity of the resonance frequency. A detailed description of the investigated cases is omitted here as it is not in the focus of this chapter and it can be found in (Magagnato et al., 2005).

In the case of the coupled resonators one configuration was investigated. Ten LESs were calculated at different excitation frequencies, because the domain of interest is a broader frequency range than in the case of the single resonator. A detailed description of this investigation can be found in (Pritz et al., 2009).

### **3.2.1 Numerical method**

The main goal of the numerical investigation was to predict the damping coefficient of the system which is an important input for the reduced physical model. In order to provide an insight into the flow mechanics inside the system LES were carried out. LES is an approach to simulate turbulent flows based on resolving the unsteady large-scale motion of the fluid while the impact of the small-scale turbulence on the large scales is accounted for by a subgrid scale model. By the prediction of flows in complex geometries, where large, anisotropic vortex structures dominate, the statistical turbulence models often fail. The LES approach is for such flows more reliable and more attractive as it allows more insight into the vortex dynamics. In recent years the rapid increase of computer power has made LES accessible to a broader scientific community. This is reflected in an abundance of papers on the method and its applications.

The solution of the fully compressible Navier-Stokes equations was essential to capture the physical response of the pulsation amplification, which is mainly the compressibility of the gas volume in the chamber. Viscous effects play a crucial role in the oscillating boundary layer in the neck of the Helmholtz resonator and, hereby, in the damping of the pulsation. The pulsation and the high shear in the resonator neck produce highly anisotropic swirled flow. Therefore it is improbable that a URANS can render such flow reliably.

The LESs of this system were carried out with the in-house developed parallel flow solver called SPARC (Structured PArallel Research Code) (Magagnato, 1998). The code is based on three-dimensional block structured finite volume method and parallelized with the message passing interface (MPI).

In the case of the combustors the fully compressible Navier-Stokes equations are solved. The spatial discretization is a second-order accurate central difference formulation. The temporal integration is carried out with a second-order accurate implicit dual-time stepping scheme (Zou & Xu, 2000). For the inner iterations the 5-stage Runge-Kutta scheme was used. The time step was Δ*t*=2·10-5 *s* and Δ*t*=2·10-6 *s* for the single resonator and for the coupled resonators, respectively. This was a compromise in order to resolve the turbulent scales and compute the pulsation cycles within the permitted time. The Smagorinsky-Lilly model was chosen as subgrid-scale model (Lilly, 1967). Later investigations with MILES approach and dynamic Smagorinsky model show no significant difference in the results. This proofs that the mesh was sufficiently fine in the regions which are responsible for the damping of the pulsation, thus the modelling of the SGS structures has a minor influence there.

Stability Investigation of Combustion Chambers with LES 193

At the inlet a partially pulsated mass flow rate was prescribed. The rate of pulsation was set

The definition of the outflow conditions at the end of the exhaust pipe is particularly difficult. The resolved eddies can produce a local backflow in this cross section occasionally. In particular, by excitation frequencies in the proximity of the resonant frequency there is a temporal backflow through the whole cross section, which has been observed by the

Fig. 6. Third finest mesh extracted to the symmetry plane (distortions were caused by the

The change of the direction of the flow changes the mathematical character of the set of equations. For compressible subsonic flow four boundary values must be given at the inlet and one must be extrapolated from the flow area. At the outlet one must give one boundary value and extrapolate four others. Since these values are a function of the space and time, their determination from the measurement is impossible. Further the reflection of the waves must be avoided also at the outlet. For these reasons the outflow boundary was set not at the end of the exhaust gas pipe, but in the far field. In order to damp the waves in direction to

At the solid surfaces the no-slip boundary condition and an adiabatic wall were imposed. For the first grid point *y*+<1 was obtained, the turbulence effect of the wall was modelled with the van Driest type damping function. The geometry of the computational domain and the boundary conditions are shown in Fig. 5. The entire computational domain contains

The definition of the computational domain and the boundary conditions in the case of the coupled resonators were very similar. The geometry of the configuration chosen for the numerical investigation of the coupled resonators is illustrated in Fig. 7. The observation windows (for operations with flame) and the inserted baffle plates increased the complexity of the geometry and hence the generation of the mesh significantly. There were baffle plates placed in the burner plenum and in the combustion chamber to avoid the jet of the nozzle and of the resonator neck to flow directly through the system, furthermore to achieve a

about 4.3·106 grid points in 111 blocks. A coarsened mesh is shown in Fig. 6.

to 25%.

experimental investigations as well.

extraction in Tecplot)

the outlet boundary mesh stretching was used.

The Full Multigrid (FMG) method is used with four grid levels to achieve faster the statistically stationary state. The FMG method implies grid sequencing and a convergence acceleration technique. The number of cells on a grid level is eight time less then on the next finer grid level.
