Downward flow in a heated annulus

Application Challenge 3-11 © copyright ERCOFTAC 2004

Best Practice Advice for the AC

Key Fluid Physics

In these buoyancy-opposed mixed convection flows, the buoyancy has the effect of increasing the heat-transfer relative to that which would be found in forced convection at the same Reynolds number. Heat is transferred from the heated core to the downward flowing water, and the velocity magnitude in the near-wall region is thus reduced as the buoyancy becomes stronger. At sufficiently high levels of buoyancy flow reversal can occur adjacent to the wall.

The UFR associated with this AC is “Mixed Convection Boundary Layers on Vertical Heated Walls”.

Application Uncertainties

Measurement Errors

The accuracy of the temperature data depends on the suitability of the sampling time and the accuracy of the thermocouples employed. A bounding estimate of the error in the temperature data is 0.5oC. This translates into an error of less than 2% on Nusselt number.

There are no major sources of uncertainty, and a CFD model is straightforward to assemble.

Computational Domain and Boundary Conditions

The problem can be treated as axi-symmetric.

COMPUTATIONAL BOUNDARY CONDITIONS

Fully developed pipe inlet conditions were applied, although tests with uniform inlet conditions returned indistinguishable results – because the 23 diameters over which the flow can develop before the heated section of pipe ensured that fully developed flow conditions were achieved by the beginning of the heated section. Zero streamwise gradients were applied at the outlet plane.

Regarding the thermal boundary conditions, adiabatic conditions were applied on the outer wall, together with the initial and end sections of the inner wall, whilst a constant heat flux was applied to the middle section of the inner wall, to match the experimental conditions.

Computational Domain

The above figure shows the computational domain.

Discretisation and Grid Resolution

Use a higher order scheme (second order or above).

For the low-Reynolds-number model a grid of 252 (streamwise) by 60 (radial) nodes was employed, which was sufficient (at these low bulk Reynolds numbers) to ensure that the value of y+ at the near-wall node was less than unity, and to produce grid-independent results. With the wall function approaches, the number of radial grids could be reduced to 12 whilst still obtaining grid-independent results. Because of the low bulk Reynolds numbers of these flows, a fairly large near-wall cell of around 10% of the annular gap width was used. This ensured that, when the algebraic wall function was employed, the first grid node lay in the fully turbulent flow region. However, with the standard wall function, even these large near-wall cells did, in some cases, result in the near-wall node still lying within the viscosity-affected region. It is worth noting at this point that the analytical wall function is, in fact, designed to be fairly insensitive to the size of the near-wall cell (and can even be applied when the near-wall node does not lie in the fully turbulent region). Predictions obtained with the standard wall function, on the other hand, are known to show a dependence on the size of the near-wall cell.

Physical Modelling

• Use the low-Reynolds number k-ε turbulence model of Launder and Sharma (1974), including property variation with temperature. It is also likely that higher order turbulence models are acceptable, but only in conjunction with an analytic wall function. However this has not been checked.

• For water flows it is important to include the variation of molecular properties with temperature.

• Where buoyant effects are significant the computational studies have highlighted the need for an accurate modeling of the near-wall sublayer. The usual form of wall functions, based on a universal logarithmic wall law, do not account for these buoyancy effects and hence fail to predict the enhancement of heat transfer as the buoyancy parameter is increased.

• The choice of wall model is the most important consideration to obtain accurate predictions. The low-Reynolds number model equations can be solved up to the wall. However, this is an expensive approach because it requires resolution of the flow and temperature down to the wall. The alternative is to use an analytical wall function that captures the important physical phenomena, in conjunction with the solution of the low-Reynolds number model equations outside the wall layer. 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.

The alternative approaches suggested above will give good results for most flows. However, if there is weak buoyancy influence and the properties vary significantly near the wall then the results would under-predict the experimental values by ~20%.