Fully Developed Flow

Pressure is typically modeled with a jump condition at the interface that compensates for the streamwise pressure drop through the system.

When energy is present, a fully developed situation can be deemed to occur if the profiles of temperature and/or enthalpy become self-similar when using an appropriate scaling. In these cases, the self-similarity relations can be used to rescale the thermal field leaving the domain in order to reintroduce it at the inflow periodic boundary. This rescaling is done in Simcenter STAR-CCM+ by using the Fully Developed Flow Option. The treatment of the thermal field is more complex than pressure, since both convection and diffusion are taken into account. In addition, the limitations on which physical models are present in order for the fully developed condition to occur must also be considered.

In Which Situations Will Errors Be Incurred?

Errors occur with mesh issues, compressibility effects, property variations, and non-contiguous regions.

• Mesh Issues

  In addition to regular mesh refinement considerations, particularly in the wall-normal direction where large gradients occur, care must be taken to resolve the flow in the streamwise direction. Unlike simple fully developed duct flows without energy, which can often use as few as 3 - 5 cells in the streamwise direction, a larger number of cells is required when energy is considered. This requirement is true of the constant wall temperature heating condition, since the fluid heats up rapidly as the flow enters the domain. Grid refinement studies are recommended to confirm that the grid resolution is adequate.

• Compressibility Effects

  In truly compressible flows, viscous dissipation causes the fluid to heat up. With adiabatic walls, this heating effect causes the flow to accelerate. This scenario is called “fanno flow.” The fully developed energy treatment in Simcenter STAR-CCM+ is not designed for fanno flows or compressible duct flows with other heating situations. However, use of the ideal gas law is not precluded and if employed, the Mach number must be sufficiently low that these effects do not dominate and cause stability problems.

• Property Variations

  The existence of a similarity solution for energy in a fully developed duct flow is most easily demonstrated for constant property flows. However, the use of temperature-dependent fluid properties is not precluded, and small property variations do not have a large effect on the solution. However, caution must be exercised when using variable properties in case they affect the accuracy or stability of the solution.

• Non-Contiguous Regions

  Simcenter STAR-CCM+ permits regions to be non-contiguous, in general. However, the fully developed energy algorithm must search through all regions that are connected to the interface to evaluate contributions from associated cells and boundaries. For this reason, it is recommended that non-contiguous regions are not used. If in any doubt, use the Split Non-Contiguous function on the region.

What About Variable Heating Situations?

The similarity solutions that cause a fully developed energy situation in a duct are limited to:

• Constant wall heat flux
• Constant wall temperature

However, given the flexibility of the boundary conditions in Simcenter STAR-CCM+, no attempt is made to ensure that the values specified for wall heat flux or temperature are indeed constant. If a boundary condition is specified that deviates from the profile scaling that is used at the fully developed interface, an error is incurred. The significance of the error clearly depends on how incorrect the assumption is.

The specified wall heat flux situation is likely to be more forgiving than the specified wall temperature situation because the linear behavior of the energy equation permits superposition of solutions. Therefore a wall heat flux that varies periodically (such as a sinusoidal wave) most likely produces a well-behaved solution.

How Can One Verify That Fully Developed Energy Is Valid for a Specific Heating Situation?

The similarity solutions that cause a fully developed energy situation in a duct are limited to:

• Constant wall heat flux
• Constant wall temperature in constant property flows

Any deviations from these situations cast doubt on the validity of the approach. One way to check that the fully developed approach is valid is to create a simplified version of the domain and manually replicate it 20 or more times. Then, verify that the velocity and thermal fields are indeed periodic (using the appropriate scaling). This check requires a significant effort, but is necessary to remove residual doubt.

What Happens If Recirculation Occurs at the Fully Developed Interface?

Care has been taken to test the fully developed interface for an interface at which recirculating flow occurs. However, problems can arise if the portion of flow entering approaches the portion of the flow leaving. Therefore, if possible, it is preferable to place the interface at a location where recirculation does not occur.