# On the Link between Diesel Spray Asymmetry and Off-Axis Needle Displacement
## Materials and Methods
### Primary Break-Up  
#### 2.4.1 Core Injection Approach under Nozzle Flow Local Information  
The primary break-up rate in the spray domain considers two independent mechanisms and is determined by the liquid turbulence in the nozzle orifice and by the aerodynamic conditions in the spray region. According to this approach, the turbulent fluctuations in the liquid jet create the initial perturbations on the surface. These grow under the action of aerodynamic pressure forces until they detach as atomized droplets. The coherent liquid core region at the nozzle exit, where primary break-up occurs, is calculated from a mass balance at the volume elements of the liquid core. The liquid core itself is modeled through a blob-injection scheme, in which the blob diameter is constant and the blob number varies according to the mass loss due to primary break-up. The determination of mass loss from this region is based on the rate-approach \(\mathrm{d}R/\mathrm{d}t\), where \(\mathrm{d}R/\mathrm{d}t\) is viewed as the artificial-radius change of the assumed blob injection. The rate \(\mathrm{d}R/\mathrm{d}t\) is an artificial dimension for the mass loss because the number of blobs decreases while the blob radius stays constant. In the current scheme, such a concept is applied locally, by means of the separate multi-phase nozzle flow simulation (Figure 5, left). For each \"\(j\)\" face of the nozzle orifice, the rate \(\mathrm{d}R_{\mathrm{j}}/\mathrm{d}t\) is calculated from the two-phase nozzle flow simulation; the atomization length scale \(L_{\mathrm{A,j}}\) is equal to the turbulent length scale \(L_{\mathrm{Tj}}\) as reported in Equation (8). The turbulent length-scale \(L_{\mathrm{Tj}}\) and the atomization time scale \(\tau_{\mathrm{A,j}}\) are determined locally for each j face. The expression for \(L_{\mathrm{Tj}}\) is given by Equation (9), whereas the expression for the local atomization time scale \(\tau_{\mathrm{A,j}}\) is given by Equation (10):  
\[\frac{\mathrm{d}R_{\mathrm{j}}}{\mathrm{d}t}=\frac{L_{\mathrm{A,j}}}{\tau_{ \mathrm{A,j}}}, \tag{8}\]  
\[L_{\mathrm{Tj}}=C_{\mathrm{2}}C_{\mathrm{u}}\frac{k_{\mathrm{j}}^{1.5}}{ \varepsilon_{\mathrm{j}}}, \tag{9}\]  
\[\tau_{\mathrm{A,j}}=C_{\mathrm{1}}\tau_{\mathrm{Tj}}+C_{\mathrm{3}}\tau_{ \mathrm{W,j}}. \tag{10}\]  
The local turbulent time scale \(\tau_{\mathrm{Tj}}\) is calculated from Equation (11) and the local aerodynamic time and length scale, \(\tau_{\mathrm{W,j}}\) and \(L_{\mathrm{W,j}}\), are calculated according to Equation (12):  
\[\tau_{\mathrm{Tj}}=C_{\mathrm{u}}\frac{k_{\mathrm{j}}}{\varepsilon_{\mathrm{j }}}, \tag{11}\]  
\[\tau_{\mathrm{Wj}}=\frac{L_{\mathrm{W,j}}}{\sqrt{\frac{\rho_{\mathrm{1}}\rho_{ \mathrm{N}}|V_{\mathrm{N}}-V_{\mathrm{j}}|^{2}}{\left(\rho_{\mathrm{N}}+\rho_{ \mathrm{1}}\right)^{2}}-\frac{\sigma}{\left(\rho_{\mathrm{N}}+\rho_{\mathrm{1 }}\right)L_{\mathrm{W,j}}}}}\text{ and }L_{\mathrm{W,j}}=2L_{\mathrm{Tj}}. \tag{12}\]  
The local diameter of the product droplets resulting from this model is proportional to the turbulent length scale, Equation (13):  
\[D_{\mathrm{d,j}}=2L_{\mathrm{Tj}}. \tag{13}\]  
The adopted model gives different break-up rates \(\mathrm{d}R_{\mathrm{j}}/\mathrm{d}t\) and different product droplet diameter for each face of the nozzle orifice. Each face of the orifice determines the break-up rate on an idealized blob injection surface; therefore, the local rate of change \(\mathrm{d}R_{\mathrm{j}}/\mathrm{d}t\) is mapped onto the corresponding blob surface and represents a certain fraction of the mass loss (Figure 5, right). The target phase of the liquid droplet phases is determined by the local product droplet diameter \(D_{\mathrm{d,j}}\). These droplet diameters are sorted into five different predetermined droplet size classes [39].