4.4 Numerical Schemes
5.1.1 Computational Configurations and Numerical Setups
The setup of our computations is based on the following conception of the spray combustion process: after injection from a nozzle into a hot turbulent flow, the fuel jet breaks up into many ligaments and droplets. A subsequent, so-called secondary break-up of the ligaments and large droplets yields a relatively fine, dispersed spray.
Large turbulent eddies transport such droplet clouds. This kind of droplet cloud can be simplified as a droplet array for our fully resolved DNS investigation. Here, only regular droplet arrays are investigated. They consist of single component kerosene (C12H23) droplets. The droplets are uniformly arranged in 3 droplet layers perpen-dicular to the flow direction in the computational domain. The distance between the inlet face of the computational domain and the first layer of droplets and between
the third layer droplet and the outlet face of the computational domain are 10dand 16d, respectively. This is similar to the validation studies in [1, 125]. There, it was recommended to set the inflow stream at least ten droplet radii upstream of the droplets. One plane across one layer of droplets along the flow direction of a 12-droplet array is visualized in Fig. 5.1 where dis the droplet diameter, rc represents the inter-droplet distance,xis the axial distance from the droplet upstream andris the radial distance from the droplet diameter on a plane perpendicular to the flow direction. Inter-droplet space 1 is defined as the space between the first and second layer while inter-droplet space 2 is located in between the second and third layer.
Outflow
r
cx
cyclic d
Inflow p
∞U'/U
T
cyclic
∞ ∞
Inter-droplet space 1
Inter-droplet space 2
x
r
cr r
Figure 5.1: Schematic diagram of the computational domain containing a regular droplet array
The array is infinite in both cross-stream directions which is realized by periodic boundary conditions. The mean relative flow is perpendicular to the droplet array.
The parameters U∞ and U0 are the inflow mean velocity and the root-mean-square (RMS) turbulent fluctuating velocity. For the cases investigated here, the inflow stream is humid air with N2, O2 and H2O (76.4 : 23.4 : 0.2 on a mass basis) at a pressure of p∞ = 15bar and a temperature of T∞ = 1000K. These conditions are similar to the ambient conditions in [150, 151, 152]. The very high resolution needed for the boundary layers around the droplets requires computational savings elsewhere: therefore, a 4-step chemical mechanism with seven species (C12H23, O2, CO, H2, CO2, H2O, N2) as presented in [61, 62] by Jones and Lindstedt is used to approximate the combustion of kerosene. This mechanism was also adopted in the earlier fully resolved DNS study by Zoby et al. [158] and other studies on LES of turbulent spray combustion [33, 63]. A comparison of the flamelet computations of this 7-species mechanism and Luche’s 134-species mechanism [80] in Fig. 5.2 shows that the peak temperature can differ by up to 100K and disparities on the rich side can be as large as 250K. However, the reaction mechanism will not affect the
5.1. DEFINITION OF THE DOMAIN SIZE AND MESH RESOLUTION 44
500 1000 1500 2000 2500 3000
0 0.2 0.4 0.6 0.8 1
T(K)
f Nf,0=10
Luche 4-step
500 1000 1500 2000 2500 3000
0 0.2 0.4 0.6 0.8 1
T(K)
f Nf,0=100
Luche 4-step
Figure 5.2: Comparisons of the flamelet solutions for the 7-species [62] and Luche’s 134-species [80] mechanisms for two different dissipation rates,Nf,0 = 10 andNf,0 = 100.
validation of the scaling laws: the difference in heat release equally affects the DNS results and the scaling relationships as changes of the DNS changes the parameters of Nf,KSZ (Eq. (2.42)) that are input to the scaling laws. The thermodynamic and transport properties of the gas mixture are calculated by polynomials and semi-empirical equations [50]. The initial droplet temperature is set to Ts = 300K.
Under stationary conditions, the surface temperature of the droplets reaches the wet-bulb temperature [111]. This wet-bulb temperature is computed by energy and mass conservation across the liquid-vapour interface. Using the boundary conditions for the liquid as introduce in Sec. 4.1, the interface temperature stays about 20K below the boiling point temperature and is nearly uniform throughout the droplet.
Therefore, thermodynamic and transport properties for the liquid can be assumed constant [112, 143, 151]. The average properties of kerosene (C12H23) are similar to the properties of dodecane (C12H26) [49] and the corresponding expressions are taken from [50].
Periodic boundary conditions in the two cross-stream directions mimic a quasi-infinite droplet arrangement. The mean convective flow is perpendicular to the droplet array. A homogeneous isotropic turbulent velocity field is superposed to the mean flow at the inflow of the computational domain. The perturbations are calculated from a modified von-Karman spectrum and details of the algorithm can be found in Sec. 4.3. The initial integral length scale is restricted to half the domain size and limits the range of the turbulence spectrum that can be realized in the DNS. The domain size (larger scales) will be changed to investigate its effect on inter-droplet mixing.
Figure5.3(a) is a schematic illustration of the mesh module for a single droplet
block. A droplet array can be constructed by a succession of these modules. The mesh adjacent to the droplet on the gas side is spherical and extended by 7d from the droplet center. Then it is gradually transformed into a Cartesian configuration.
The mesh on the liquid side is extruded from the surface mesh. The smallest cells with the size of ∆xmin are located at the droplet surface and the largest cells with the size of ∆xmax are located at the inlet and outlet faces of the domain. The time-step is controlled to ensure the maximum Courant number to be smaller than 0.15.
A PISO algorithm with an additional non-orthogonal correction is used to solve the pressure-velocity coupling and a second-order least squares scheme is employed to discretize the pressure gradient. This ensures the stability of the non-orthogonal corrections in the PISO algorithm. These options avoid potential flow instabilities originating from the mesh non-orthogonality at the conjunction between the spher-ical and Cartesian meshes. Details of the numerspher-ical schemes for the discretization of governing equations can be found in Sec. 4.4. Since the DNS is required to re-solve the smallest turbulent scales and the boundary layer at the droplet surface, the computational cost is extremely high, especially for high Reynolds number con-ditions. Therefore, the characteristic cell size should be determined for the single droplet block before the droplet arrays are constructed. The balance between the accuracy of solution and the computational cost can be assessed for the DNS of droplet array combustion. The mesh independence study is performed for the single droplet domain as shown in Fig. 5.3(b). The domain is 30d and 20d (with the
d x
min7d x
max(a)
20d
10d 20d
cyclic
cyclic
(b)
Figure 5.3: Three dimensional grid and schematic diagram of one single droplet domain
droplet diameter d= 100µm) in the flow and cross-stream directions, respectively.
The droplet is positioned at 1/3 of the length of the domain. Three different mesh resolutions are tested for one of our setups with the highest droplet Reynolds num-ber (inflowRed,∞ = 17.2 which is calculated based on the droplet diameter (d) and
5.1. DEFINITION OF THE DOMAIN SIZE AND MESH RESOLUTION 46 Table 5.1: Characteristic quantities of three different mesh resolutions
Mesh ∆xmin/d ∆xmax/η # cells
1 1/10 1.5 55K
2 1/20 0.75 420K
3 1/40 0.375 3400K
the inflow mean velocity (Ud,∞)) and the smallest Kolmogorov length scale (inflow η/d = 0.42). The characteristic quantities of the three meshes are listed in Tab. 5.1.
Mesh 2 is the standard mesh. Meshes 1 and 3 are coarsened and refined by a factor of 2, respectively.