# UFR 3-11 Test Case

# Pipe expansion (with heat transfer)

Underlying Flow Regime 3-11 © copyright ERCOFTAC 2004

# Test Case

## Brief description of the study test case

The experimental configuration for the heat transfer studies by Baughn et al. (1984) is shown in the following sketch:

The geometry comprised a sudden pipe expansion with expansion ratios (d/D) ranging between 0.267 and 0.8. The experiments were run at Reynolds numbers (Re, based on the larger pipe diameter and bulk velocity) of between 5300 and 87000. Nusselt numbers (Nu) were varied between about 70 and 500. Velocity and turbulence measurements were not made in this series of tests as they are available in Szczepura (1985): the main focus in the work of Baughn et al. (1984) was on the Nusselt numbers downstream of the expansion.

It was found that, at least for higher values of Re, the shape of the axial variation in local Nusselt number was independent of Re and that peak values displayed a Reynolds number dependence of between Re^{0.67 }((Baughn et al (1984)) and Re^{0.69 }(Hutton and Szczepura (1987)).

In further experiments by Baughn et al. (1989), some of which are also reported by Yap (1987), heat transfer, mean velocity and temperature measurements were made with a expansion ratio of 0.4. Heat transfer measurements were made for downstream Reynolds numbers ranging from 4300 to 44500, and velocity and temperature measurements were made for Reynolds numbers from 4100 to 17400. The Nusselt numbers were similar to those used in the earlier work. In these experiments, the location of the peak Nusselt number was found to shift progressively upstream from 12 to 9 step heights from the expansion step as the Reynolds number was increased from 4300 to 44500.

## Test Case Experiments

The experiments of Baughn et al. (1984) used atmospheric air as the working fluid, with the upstream and downstream pipes being constructed from cast acrylic material to minimise axial heat conduction. The downstream pipe internal diameter, D, was fixed at 9.525 cm. Five different upstream pipe diameters ranging between 2.54 and 7.62 cm were used to obtain pipe expansion ratios (d/D) of 0.267, 0.4, 0.53, 0.667 and 0.8. The upstream pipe was at least 48d long to ensure an effectively fully developed velocity profile at inlet to the expansion. The downstream pipe had a heated length of 18D.

To the inner surface of the larger pipe was attached a thin transparent gold film, vacuum-deposited on a polyester sheet substrate. Passing alternating current through the gold film produced an essentially uniform heat flux; values were in the range between 80 and 720 W/m^{2}. These relatively low fluxes, combined with the low wall temperatures (typically 5 to 20°C above the bulk air temperature), ensured that buoyancy effects were negligible. To minimise external heat losses, the outside surface of the larger pipe was insulated.

Airflow rates were measured using either an orifice plate or a pitot-static probe, depending on the flow rate being used. Local wall temperatures were measured using 35 calibrated copper-constantan thermocouples located at different axial positions along the larger pipe. Most were placed axially along the top of the pipe, but some were distributed at other angular positions around the pipe to check the axisymmetry of the heat transfer to the flow.

The uncertainties in the experimental results were estimated at the 80% confidence level to be ±2.3% on Reynolds number, ±6.6% on peak Nusselt numbers and ±2% on the lowest Nusselt numbers. Circumferential variations in Nu around the pipe were shown to be less than 1%.

The subsequent series of experiments with a uniform wall temperature documented by Yap (1987) and Baughn et al. (1989) used many of the same components as the earlier tests. However, in this case the 9.525 cm inside diameter downstream pipe was made from aluminium, with electrical heating elements wrapped around the outside. This pipe had a wall thickness of 0.64 cm and a heated length of 16D, and was again externally insulated. Temperature measurements in these experiments were made using 18 chromel-alumel thermocouples, and the wall temperature was estimated to be constant to within ±0.2°C.

A sensor for measuring local time-average surface heat fluxes was installed in the downstream pipe. This comprised an electrically heated nickel/chromium ribbon, mounted flush with the inner surface of the pipe, which was thermally and electrically insulated from the pipe. Velocity and temperature measurements were made at the same axial location in the pipe as the heat flux sensor using traversing hot-wire or thermocouple probes. Measurements at different axial positions downstream of the expansion were made by moving the upstream pipe and expansion step relative to the downstream pipe. The measurements were made between 0.25H and 22H downstream of the step. The uncertainty on the Nusselt number was estimated at the 80% confidence level to be ±2.8%.

## CFD Methods

As noted in Section 2, this UFR received a considerable amount of attention in the 1980's as a test case for numerical and turbulence models. Probably as a consequence, it appears as one of the standard validation test cases in the documentation of most of the commercial CFD codes. The performance of a number of finite element and finite volume CFD codes on this UFR was reviewed at the time by Hutton and Szczepura (1987), both with regard to flow and turbulence modelling aspects, and also with reference to heat transfer predictions. The codes mostly used the standard k-ε turbulence model, but with different methods of treating the near-wall region. Their review highlighted the finding that standard wall functions cannot correctly predict the heat transfer distribution along the larger pipe, and that modifications are required to achieve satisfactory agreement with the experimental data.

Various CFD researchers have subsequently investigated the pipe expansion without heat transfer. Among them are Chang et al. (1995), who developed a low Reynolds number k-ε model that was reported to improve prediction of the location of the reattachment point. Rabbitt (1997) analysed the problem using standard, modified and non-linear k-ε models in a finite element code. This highlighted the limitations of using a turbulence model with only one length scale, as no significant improvements were evident in using the extended models. Hanjalić and Jakirlić (1998) applied two high Reynolds number second moment closure models (Launder-Reece-Rodi-Gibson (LRRG) and Speziale-Sarkar-Gatski (SSG)) and a new low Reynolds number model with wall-proximity modifications to plane and axisymmetric expansions. The low Reynolds number model was shown to improve predictions near the wall. Wagner and Friedrich (1999) used results from a DNS of a pipe expansion at a low Reynolds number of 360 (based on the inlet pipe diameter) to analyse the performance of various Reynolds stress models, including the low Reynolds number model of Hanjalić and Jakirlić (1998). This showed that the greatest discrepancies between the DNS data and the models arose in the pressure-strain terms for the velocity variances and in the dissipation rates.

Recent modelling of the pipe expansion with heat transfer has been carried out by Craft et al. (1999) using a modified non-linear eddy viscosity model, and is described in more detail below. Goldberg and Batten (2001) applied a linear k-ε-R model, where R is a pseudoeddy viscosity, to various flows including this UFR, together with a cubic k-ε turbulence model. For the pipe expansion, the k-ε-R model provided the better predictions of wall heat transfer. Modelling of this UFR using the shear stress transport (SST) model has been carried out by Vieser et al. (2002) and Esch et al. (2003), and again is described in more detail below.

The modelling of Craft et al. (1999) was based on a two-dimensional axisymmetric finite volume code, with pressure obtained using the SIMPLE algorithm and the HYBRID discretisation scheme. Turbulence was modelled using a cubic low Reynolds number two equation model with correction terms in the dissipation rate equation for the length scale of near-wall turbulence. In the work reported by Craft et al. (1999) the corrections were formulated to be independent of wall distance, to remove the need to introduce the wall distance in the correction proposed by Yap (1987).

The experiments documented by Yap (1987) and Baughn et al. (1989) were used as the basis for this modelling. The solution domain extended 6 step heights upstream of the expansion, and 29 step heights downstream. Uniform inlet conditions were prescribed. The computational mesh comprised 101 and 120 nodes in the axial and radial directions respectively. Grid sensitivity tests were carried out with grids of different sizes and degrees of non-uniformity, which showed that the flow and thermal predictions were free of numerical error. Computations were carried out at mean flow Reynolds numbers, based on the downstream pipe diameter and bulk velocity, of 17000 and 40000.

The analysis of Vieser et al. (2002) and Esch et al. (2003) was based around use of the commercial CFX software package and the SST approach to turbulence modelling developed by Menter (1994). This combines use of the k-ω model near the wall, which improves prediction in the near-wall region, with the k-ε model being used elsewhere. In addition, an automatic near-wall treatment for the ω-equation based models was introduced which could switch gradually from a wall function to a low Reynolds number formulation, depending on the grid density near the wall.

The experiments of Baughn et al. (1984) with a pipe expansion ratio of 0.4 were used as a basis for this modelling. The solution domain extended one step height upstream of the step, and 40 step heights downstream. The Reynolds number, based on the step height and the mean inflow velocity, was 20000. The inlet conditions were specified using profiles for fully developed pipe flow, derived from separate calculations. Grid dependency checks were performed by running the calculations on three successively refined meshes. Downstream of the expansion, the meshes contained between 101 and 401 nodes in the axial direction and between 121 and 481 nodes in the radial direction, leading to mean y^{+} values in the range 7.4 to 1.2.

© copyright ERCOFTAC 2004

Contributors: Jeremy Noyce - Magnox Electric