To integrate the equations in time, a temporal differencing of the equations is performed. For a sequence of discrete times t n ( n =0, 1, 2,...), the time interval for the next step is given as Δ t n = t n+ 1 - t n is the time step, and the integer n is referred to as the time "cycle" number. Throughout this documentation, we use a superscript n on a variable to represent "the value at time t n ". Additionally, when we use the abbreviated " Δ t " without a superscript, we mean Δ t n . Using this notation, the difference approximation to the derivative δ Q/ δ t is the first-order expression: (Q n+ 1 - Q n )/ Δ t.
During time integration, Ansys Forte employs three stages of solution for each time step. The time stepping employs the operator-splitting method to separate the chemistry and spray source terms and the flow transport. The flow transport solution is based on the Arbitrary-Lagrangian-Eulerian (ALE) method [4] . The hydrodynamic time step is adaptively controlled to maximize the solution efficiency, accuracy and stability. The three stages considered are:
Stage 1: This stage solves the chemistry and spray source terms (Equation 2–1) in the species and energy transport equations ()Equation 2–5. The calculation is based on the Lagrangian coordinate in which cells move with the fluid and spray droplets are followed through collision, oscillation, and break-up processes, along with mass and energy terms due to the chemistry, gas and spray interactions.
Stage 2: The second stage is a coupled, implicit solution of the acoustic-mode terms (which include the pressure gradient in the momentum equation (Equation 2–3)) and velocity dilation terms in the mass (Equation 2–2) and energy equations), the spray momentum source term, and the terms due to diffusion of species mass, momentum, and energy. This stage of calculation is also based on the Lagrangian coordinate. As a result, cells move with the fluid, and convection across cell boundary is not considered. The calculation also includes the remaining source terms in the turbulence equations (Equation 2–20, Equation 2–21, Equation 2–25).
Stage 3: The third stage is a rezone stage, in which the flow field is frozen and then remapped onto a revised computational mesh after wall motion is accounted for. The convective transport is calculated by moving the Lagrangian mesh result from stage 2 to the revised mesh relative to the fluid. The convection terms in the governing equaitons are calculated in this stage, as flow across each cell boundary is considered.