Implicit Advection in FLOW-3D
This article highlights developments to be released in an upcoming version of FLOW-3D.
A new implicit advection technique has been incorporated into FLOW-3D for release in Version 9.2. This note describes its advantages and points out certain limitations related to the accuracy of implicit methods. A more complete description is available in a Flow Science technical note.
Two options are provided for modeling advection implicitly controlled by a new input parameter, impadv.
When using FLOW-3D to simulate transient problems, especially those involving sharp free surfaces and/or fluid-fluid interfaces the appropriate implicit advection option is impadv=1. In this case, the program will limit time-step size by those fluid velocities at a free surface where the velocity is normal to the surface and the fluid fraction at that location has changed by more than 5% in the preceding cycle. Otherwise, the surface velocities will not impose a limit on time-step size.
When the user wants to accelerate the approach to steady-state conditions, the second option, impadv=2, which has no advection velocity limit, can be used. However, users should take caution when selecting impadv=2, because the presence of free surfaces or sharp fluid-fluid interfaces in their problems could introduce poor results. If there are no sharp interfaces, then the choice of impadv=1 or 2 makes no difference.
It should always be kept in mind that implicit techniques invariably introduce some amount of smoothing or damping of the computed results beyond what is expected in explicit simulations. To retain as much accuracy as possible, therefore, the new implicit method is only used at locations where instabilities could develop; everywhere else the usual explicit methods are used.
The two examples below illustrate both the advantages and limitations of the implicit advection technique.
Figure 1: Steady flow in 2-D tank with
small inlet/outlet. The implicit case is
4.3 times faster than the explicit case.
Figure 2a: NADCA part (left) at end
of filling. Color indicates temperature.
Figure 2b. Close-up of gate region
controlling the time-step size.
A Simple Test
A simple two-dimensional test nicely illustrates the implicit advection option. A rectangular tank 30 cm wide and 20 cm high is outfitted with two internal baffles, one from the bottom and the other from the top. A small (1 cm) inlet port is defined at the middle of the left side of the tank and a corresponding outlet is located on the right side. The inlet/outlet flow speeds are fixed at 30 cm/s and the tank is initially filled with water at rest (i.e., there is no free surface. The purpose of the simulation is to determine steady flow conditions.
An explicit computation using a grid of 60 by 40 elements, (see Fig.1), required 310 seconds of CPU time to reach near steady conditions at t=30 seconds. The maximum stable time step size, δt=0.0033 seconds, is controlled by the horizontal flow at the edge of the outlet. With the implicit option impadv=1, the CPU time was 72 seconds (4.3 times faster than the explicit case) and the time-step size was always at least an order of magnitude larger than the explicit value.
The converged steady-state results are in very close agreement. More iterations were required to get the flow started (including about 17 cases of SOR pressure iteration failure), which reduce the time-step size to the 0.03 s range (an order of magnitude larger than the explicit value). Once convergence is achieved and steady conditions are approached, the time-step size increases to 0.1 s by the end of the computation.
A More Complex Case – High-Pressure Die Casting
A filling simulation of an alternator housing through the high-pressure die casting process was simulated. In this process, generally there is a thin gate region where the largest flow velocities are recorded. This flow region is relatively steady during the majority of the die filling process. Such a situation should be ideal for the implicit advection technique to show a significant speedup without loss of accuracy.
In this example, the gate region does not completely fill (see Fig. 2b) until the very end of the simulation. Thus, the time-step size for accurate free surfaces with implicit advection must take into consideration the free surface velocities at the gate, which means that a large speedup might not be possible. In fact, because the problem includes some restrictions on the free surface velocities that must be treated explicitly—namely that they are normal to a surface and that the local fluid fraction must be changing by more than 5% in a cycle—the actual limitation is not so bad. The free surface located at the gate arises from a flow separation that is relatively stationary, so the time-step size does not have to be limited much to maintain accuracy. In fact, the implicit computation uses a time-step size that is roughly 5 to 10 times larger than the explicit time-step size, and the overall CPU time is over 40% less that needed for the explicit computation. Not a huge increase, but a factor of nearly 2x in speed is certainly worthwhile.