Computational details

In this work, the k− ω SST model is chosen for turbulence closure, where ω is the specific dissipation rate. The effect of using this turbulence model is analysed in section 6.4, where the k− ω SST model is compared with k −  RNG model.

Standard values are used for the turbulence model coefficients, which are given in tables 6.1 and 6.2 for k− ω SST and k −  RNG models respectively.

Table 6.1: Coefficients of the k− ω SST model σ_k1^ω 1.176

σ_k2^ω 1.0 σ_ω1^ω 2.0 σ_ω2^ω 1.168

β1 0.075

β2 0.0828

β₁^∗ = β₁^∗ 0.09

κ 0.41

Table 6.2: Coefficients of the k−  RNG model

C_µ 0.085

σk= σ 0.719 σh = σm 0.9 C_1 1.42 C2 1.68 C3 0 or 1.42 C_4 -0.387

κ 0.4

E 9.0

η0 4.38

β 0.012

Since the combustion chamber is axisymmetric, only a one degree segment of the cylindrical domain is modelled to save computational costs. Furthermore, as spark ignition occurs at the centre of the chamber, only the top half of the cylinder is considered. Cyclic boundary conditions are imposed on the two geometrically identical faces and a symmetry boundary condition is applied at the mid-plane

where the cylinder is halved. Wall boundary conditions are applied for the top and side walls. The wall boundary is fixed at a temperature of 288 K since there was heat loss at the walls in the experiment.

The segment is one-cell thick and the grid spacing is uniform in both the axial and radial directions. The grid is refined until the solution did not show a significant change in the results and the spatial resolution in both axial and radial directions is about 0.18 mm. The size of the time-step is chosen to be 5 µs, which ensures the resolution of reaction, diffusion and convection time scales.

A certain number of cells are selected for spark ignition, which correspond to the spark gap and an ignition source term is included in the enthalpy equation. In this work the temperature is fixed at the burnt temperature, Tb, at these ignition cells during the spark duration. The ignition energy, Ei, can then be calculated using

Ei = ρuViCp(Tb− Tu) , (6.2) where Vi is the volume of ignition cells. Thus by changing the burnt temperature of the ignition cells and the volume of the cells, one could alter the ignition energy.

Different ignition energy values are tested since experimental ignition data was not available. Note that the spark duration is set to one time step size. In the progress variable based approach used in this work, in addition to supplying energy for spark ignition, hot products and cold reactants need to be specified for ignition. Here values of ˜c = 1 and fc⁰⁰² = 0.25 are prescribed for the ignition cells and ˜c = fc⁰⁰² = 0 is prescribed for the rest of the domain.

The initial temperature and pressure inside the combustion chamber are 325 K and 243 kPa respectively. Radial profiles of turbulence intensity, T I, and swirling velocity, V , obtained from the experimental measurements of Hamamoto et al.

(1988) are used to define the initial flow field. The integral length scale, Λ, is taken as 12.5 mm, which is 10% of the vessel diameter. The simulation results does not change significantly when this initial value of Λ is halved. Radial distribution of T I is used to calculate the turbulent kinetic energy and dissipation rate as

˜k = 3

2T I² and ε =˜ T I³

Λ . (6.3)

The governing equations are solved in STAR-CD using FVM. In this work, PISO method is chosen for the pressure-velocity coupling. The second-order MARS scheme (Asproulis, 1994) is used to discretise convective terms in the momen-tum and modelled turbulence equations. First-order upwind scheme is used to discretise the convective terms in enthalpy and ˜c and fc⁰⁰² equations. Accuracy of temporal discretisations lie between first- and second-order, in which the discreti-sation scheme is based on the fully-implicit Euler scheme and explicit deferred correctors.

6.4 Results and discussion

In this section, simulation results obtained using the strained and unstrained flamelet models are compared with the experimental data of Hamamoto et al.

(1988). Figure 6.6 shows these predictions, where the results from the EBU model available in STAR-CD is plotted as well. This figure shows that in the

2 4 6 8 10 12 14 16 18 20 22 24

0 5 10 15 20 25

p(bar)

time (ms) strainedexp

unstrained EBU

Figure 6.6: Pressure rise prediction using three different combustion models.

numerical simulations, pressure reaches a peak and then drops slightly. This drop is due to heat loss at the walls. This figure shows that the prediction from strained and unstrained flamelet models are reasonably good when compared

with the EBU model. In addition, these models are able to predict the temporal gradient of pressure, ∂p/∂t, reasonably well. The predicted pressure rise from the unstrained flamelet is 5% higher than the experimental value, while the EBU model overpredicts the pressure rise by 12%.

The time taken to generate the look-up table for the strained flamelet model is considerably longer than that for the unstrained flamelet model, since a number of strained laminar flames have to be calculated for each temperature and pressure condition considered. The plots in Figure 6.6 indicate that the unstrained flame is adequate for this problem and the improvements from the strained flamelet model are marginal. Therefore, only the results obtained using the unstrained flamelet model are discussed in the rest of this chapter.

It is important to note that the strained flamelet model gave considerably im-proved results when compared with the unstrained flamelet model for the flames simulated in Chapters 4 and 5. Even though the experimental flames simulated in Chapter 5 were conducted in a spherical bomb, they can be considered to be unconfined, since the experimental measurements were made before a consider-able pressure rise was observed and the diameter of the combustion vessel was more than three times larger than the one considered here. The experiments of Hamamoto et al. (1988) were performed in a closed vessel with pressure rise due to combustion. It is believed that confinement changes the turbulence stretch-ing effect on the flame, since it restricts the entrainment of air and changes the turbulence response of the flame.

Figure 6.7 shows the comparison of the mean scalar dissipation rate, ˜_c, and the mean reaction rate, ˙ω, across the flame brush using simulations from both strained and unstrained flamelet models. These quantities are taken across the centre line of the computational domain at a simulation time of 10 ms. These figures show that the difference in the quantities, ˜c and ˙ω, obtained from the strained and unstrained flamelet models are small, which could explain the small difference in the predictions from these two models for the confined flame simu-lated in this work.

0 50 100 150 200 250

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

˜ǫc(1/s)

˜ c unstrained

strained

(a)

0 100 200 300 400 500 600 700 800 900

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

˙ω(kg/m3s)

˜ c unstrained

strained

(b)

Figure 6.7: Comparison of flame quantities obtained using strained and un-strained flamelets. These quantities are taken along the centreline of the compu-tational domain at a simulation time of 10 ms; (a) mean scalar dissipation rate, and (b) mean reaction rate.