
For timedependent flows, the discretized form of the generic transport equations is of the following form:
where  
=  conservative form of transient derivative of transported variable  
=  density  
=  velocity vector (= in 2D)  
=  surface area vector  
=  diffusion coefficient for  
=  gradient of (= in 2D)  
=  source of per unit volume 
The temporal discretization of the transient derivative in the Equation 25.425 is described in Section 25.3.2, including firstorder and secondorder schemes in time. The pressurebased solver in FLUENT uses an implicit discretization of the transport equation (Equation 25.425). As a standard default approach, all convective, diffusive, and source terms are evaluated from the fields for time level n+1.
In the pressurebased solver, the overall timediscretization error is determined by both the choice of temporal discretization (e.g., firstorder, secondorder) and the manner in which the solutions are advanced to the next time step (timeadvancement scheme). Temporal discretization introduces the corresponding truncation error; , , for firstorder and secondorder, respectively. The segregated solution process by which the equations are solved one by one introduces splitting error. There are two approaches to the timeadvancement scheme depending on how you want to control the splitting error.
Iterative TimeAdvancement Scheme
In the iterative scheme, all the equations are solved iteratively, for a given timestep, until the convergence criteria are met. Thus, advancing the solutions by one timestep normally requires a number of outer iterations as shown in Figure 25.1.1 and Figure 25.4.1. With this iterative scheme, nonlinearity of the individual equations and interequation couplings are fully accounted for, eliminating the splitting error. The iterative scheme is the default in FLUENT.
The Frozen Flux Formulation
The standard fullyimplicit discretization of the convective part of Equation 25.426 produces nonlinear terms in the resulting equations. In addition, solving these equations generally requires numerous iterations per time step. As an alternative, FLUENT provides an optional way to discretize the convective part of Equation 25.425 using the mass flux at the cell faces from the previous time level n.
The solution still has the same order of accuracy but the nonlinear character of the discretized transport equation is essentially reduced and the convergence within each time step is improved.
To use this feature, turn on the Frozen Flux Formulation option in the Solver panel.

This option is only available for singlephase transient problems that use the segregated iterative solver and do not use a moving/deforming mesh model.

NonIterative TimeAdvancement Scheme
The iterative timeadvancement scheme requires a considerable amount of computational effort due to a large number of outer iterations performed for each timestep. The idea underlying the noniterative timeadvancement (NITA) scheme is that, in order to preserve overall time accuracy, you do not really need to reduce the splitting error to zero, but only have to make it the same order as the truncation error. The NITA scheme, as seen in Figure 25.4.2, thus does not need the outer iterations, performing only a single outer iteration per timestep, which significantly speeds up transient simulations. However, the NITA scheme still allows for an inner iteration to solve the individual set of equations.
FLUENT offers two versions of NITA schemes; the noniterative fractional step method (FSM) ([ 13], [ 91], [ 156], and [ 157]) and the noniterative PISO method [ 154] . Both NITA schemes are available for firstorder and secondorder time integration.

In general, the NITA solver is not recommended for highly viscous fluid flow.
