Pipe expansion (with heat transfer)

Underlying Flow Regime 3-11 © copyright ERCOFTAC 2004

Best Practice Advice

Best Practice Advice for the UFR

Key Physics

The requirement to correctly predict heat transfer adds significantly to the demands on the modelling for this UFR as not only the flow and turbulence fields, but also temperature field must be accurately predicted, particularly in the near-wall region.

In common with predictions for the 2D backward-facing step (UFR3-15), calculations using the standard high Reynolds number k-ε model with wall functions will tend to predict the reattachment point as being too close to the expansion. The secondary recirculation bubble is also not predicted. With regard to the heat transfer, the peak Nusselt number is over-predicted by about 20%. Use of the k-ω model does not substantially improve the agreement with experiment for this UFR (Vieser et al. (2003)), and it also appears from this reference that the two-layer model results in an under-prediction of the peak Nusselt number by of the order of 30%.

Improvements in predictions are evident from use of a cubic low Reynolds number two equation turbulence model (Craft et al (1999)) provided the Yap length scale correction term is modified to remove the dependency on the wall distance. However, the Nusselt number dependence on Reynolds number may still not be adequate.

Use of the SST model improves the prediction of the peak Nusselt number, at the expense of a broader Nusselt number distribution around the peak

Numerical issues

Numerical dissipation should be minimised if a reasonable representation of this UFR is to be achieved.

The grid and grid resolution will be dependent on the type of wall modelling employed. If using a low Reynolds number formulation, then of the order of 120 grid nodes in the radial direction are required. With regard to the number of nodes required in the axial direction, Vieser et al (2003) used between 100 and 200 nodes downstream of the step, while Craft et al (1999) considered that approximately 85 nodes were sufficient to give mesh-independent results.

Computational Domain and Boundary Conditions

The computational domain should either be extended well upstream of the expansion, e.g. by at least 6 step heights, or fully developed pipe flow boundary conditions should be applied if the inlet boundary is placed closer to the expansion than this. The outlet boundary should be placed 30 to 40 step heights downstream of the step, and a pressure outlet boundary is sufficient at this location.

Physical Modelling

Although qualitative predictions of the main features of the flow and heat transfer can be made using the standard k-ε model, correct quantitative modelling of the flow requires alternative approaches and the BPA for the 2D backward-facing step (UFR3-15) should be consulted for further information on these. With regard to heat transfer, either a low Reynolds number two equation model or the SST model can improve predictions relative to the k-ε model. Other approaches documented for the impinging jet (UFR3-09) may also yield improvements in heat transfer predictions for the pipe expansion, but cannot be recommended here as results have not yet been documented for this UFR (see Section 7.6).

On the basis of the experimental evidence, it should also be noted that a check on the combined adequacy of the numerical scheme and the physical modelling is whether or not the secondary recirculation bubble in the corner between the expansion and the outer pipe is predicted. Furthermore, for varying inlet flow rates, the peak Nusselt number values should display a Re^0.69dependence on Reynolds number (Hutton and Szczepura (1987)).

Application Uncertainties

There should be no significant application uncertainties associated with the inlet boundary conditions provided that the inlet profiles closely approximate to those for fully developed pipe flow, and that the application is truly two dimensional (e.g. there are no swirl or buoyancy effects). Similarly, the geometry of the expansion should be well defined and, provided the modelled outlet boundary is placed sufficiently far downstream, there should be no significant application uncertainties associated with this either.

Although constant fluid physical properties will be adequate for most gases, the variations of fluid properties with temperature and/or pressure may have to be taken into account when modelling this UFR with water or some other liquids as the working fluid.

RECOMMENDATIONS FOR FURTHER WORK

It has been pointed out by Esch et al. (2003) that the number of CFD analyses aimed at heat transfer predictions is still low, with only about 1% of all industrial CFD simulations targeting the prediction of heat transfer to and from solid walls. There is consequently significant scope for further work in this field. For example, it is notable that the second moment closure scheme of Hanjalić and Jakirlić (1998) and the k-ε-υ² model of Durbin (1995) have yielded promising results for the 2D backward-facing step with isothermal flow, but to date no equivalent results for flows in a pipe expansion with heat transfer have been published.

Another candidate for further work is to apply the recently-developed two-component limit second moment closure model developed at UMIST and documented in the buoyancy-opposed wall jet Application Challenge (AC3-01) to this UFR. An alternative approach that is also currently being investigated is whether the introduction of second-moment closures for the Reynolds stresses and the generalized gradient diffusion hypothesis (GGDH) for the turbulent heat fluxes can result in further predictive improvements for this non-isothermal UFR.

Contributors: Jeremy Noyce - Magnox Electric