Natural and mixed convection boundary layers on vertical heated walls (B)

Underlying Flow Regime 3-07 © copyright ERCOFTAC 2004

Test Case

Brief description of the study test case

The test case consists of either upward or downward flow of air in a heated tube. The inside diameter of the tube is 76.12mm. The unheated length before the heated test section is either 3.805 or 3.926m long, depending on whether the air flow is downwards or upwards respectively. The heated test section has a length of 8.08m.

The parameters that define the flow regime are the inlet Reynolds number, Grashof number (based on wall heat flux), buoyancy parameter, wall temperature and ambient temperature. The principal derived quantity, by which the success, or failure, of CFD calculations is judged, is Nusselt number. This is derived from the heat flux, wall temperature and bulk air temperature, defined as

The numerical results are mainly presented in terms of Nusselt number development along with the experimental data for each test condition.

Test Case Experiments

The basic test section is the same for all cases. The heated length is 8.08m long and consists of two stainless steel tube sections carefully welded together. The main upper part is 6.571m long with inside diameter 76.12mm and wall thickness 1.90mm. The lower part is 1.51m long and has the same inside diameter as the upper part but a slightly different wall thickness of about 1.83mm.

The heated length is resistance heated using AC electrical power from the mains supplied by a variable voltage system. 58 chromel-alumel thermocouples were resistance welded on the outside tube wall of the heated section. In total there are 43 measuring positions along the test section, 38 positions with one thermocouple at each position and another five positions with four thermocouples at each position to check the circumferential uniformity of temperature around the tube.

In the experiments, the thermal boundary condition was, strictly speaking, that of a slightly non-uniform prescribed heat flux pattern as a result of the temperature dependence of the resistivity of the stainless steel tube. However, a uniform wall heat flux was attempted by carefully selecting a tube of very uniform wall thickness. Careful ultra-sonic measurements of tube wall thickness were made to check its uniformity.

Accurate hot film calibration was a necessary pre-cursor to flow rate measurement. A resulting explicit equation provided the required correction for each case.

Appendix 5 of Li (1994) provides estimates of the accuracy of the wall temperature, the ambient air temperature (the air temperature at each section is derived from the known heat flux distribution), the local heat flux and the local Nusselt number. The random errors in temperatures, local heat flux and local Nusselt number are 0.45°C, 1%, and generally less than 5% respectively.

CFD Methods

Li

Li (1994) describes calculations of the experiments using the Launder-Sharma two equation k-ε turbulence model (Launder and Sharma, 1974). Temperature dependent property functions for viscosity, thermal conductivity and density are incorporated.

In his calculations, the differential equations were discretised following the finite volume/finite difference scheme of Leschziner (1982) under which the differential equations are integrated over a control volume prior to the application of the finite difference approximation. The discretisation procedure converts the set of differential equations into a set of algebraic equations which are then solved using the Gaussian elimination method. Solution of the parabolic equation set is obtained by starting with an assumed distribution of the dependent variables at the first station and marching in the streamwise direction.

Li (1994) does not describe the numerical mesh he uses. However, as is well known, it is necessary to solve the equations down to y⁺ values below about two everywhere on the walls, otherwise the Launder-Sharma model gives poor predictions.

Gerasimov

“Standard” wall functions usually give unreliable prediction of heat transfer, in conjunction with any turbulence model. UMIST (Craft et al, 2002) have recently developed an analytic wall function that includes a more physical representation of the near-wall region compared with the standard approach. The reader is referred to Craft et al (2002) or to the Application Challenge documentation provided by British Energy and Magnox Electric for more details. Broadly, the analytic wall function incorporates the effects of,

(a) convection parallel and normal to the wall,

(b) fluid property variation across the wall layer, including a parabolic variation of molecular viscosity across the viscous sub-layer,

(c) the inclusion of any pressure gradient and buoyancy force,

(d) viscous sub-layer thickening and thinning,

(e) a Prandtl number which may not be close to unity (e.g. water at room conditions) and its significant variation with temperature.

Gerasimov (2002) has undertaken calculations of the test case using the analytic wall function approach. He used the TEAM code (Huang and Leschziner, 1983), which uses TDMA to resolve a set of discretised equations. Initially, he repeated some of Li (1994)’s cases, using the Launder-Sharma low Reynolds number turbulence model, for both buoyancy-opposed and buoyancy-aided flows and results were identical. The predictions using the analytic wall function are compared with the experimental results and results from solving the Launder-Sharma k-ε turbulence model up to the wall. Note that the analytic wall function is used in conjunction with the solution of the low-Reynolds number equations outside the wall layer.

Entry conditions correspond to isothermal fully developed flow. The outlet boundary was positioned well downstream of the heated section to ensure that the results are not affected by the outlet boundary condition. Thermal boundary conditions of uniform wall heat flux are imposed.

Regarding grids for buoyancy-affected air flows in pipes:

WF calculations grid: 12×190 nodes

LRN calculations grid: 12 sections of 45×61 nodes, superimposed by 6 nodes, which is equivalent to 45×660 nodes. The whole domain was split into 12 sections to make it converge faster.

Regarding grid independence:

WF calculations: number of nodes in axial direction; 300, 250, 200, 175 and 150 were tried. There was no difference between the results of the 300 to 175 node cases. 190 was chosen for reassurance. In the radial direction, it is normal to have 10 nodes between the axis and the near-wall cell to capture all of the outer gradients within the mean flow. Most of the variation for the number of nodes in the radial direction was related to the change of the near-wall cell size. This variation is shown in the Figures. For higher Reynolds numbers (50000-100000) the number of nodes in the radial direction, within the main flow, was increased to 15. Results were pretty much the same as with 10-11 nodes.

LRN Calculations: At the highest Reynolds number, the first node needed to be placed within y+=1. This is why it was necessary to have 45 nodes in the radial direction. In the axial direction, 61 nodes were needed for each section to obtain grid-independent results and a reasonable convergence rate (to avoid extremely elongated cells).

The calculations were repeated with the domain fully resolved (without sectioning) as well, but convergence was very slow. The sectioning was suitable only forbuoyancy-aided flows, as velocity vectors are always aligned with the axis of the domain.

The buoyancy-opposed flows, however, required calculations throughout the whole domain due to the deviation of velocity vectors from the axial direction or even due to possible reversal of the flow.

Contributors: Mike Rabbitt - British Energy