Buoyancy-opposed wall jet

Application Challenge 3-01 © copyright ERCOFTAC 2004

Assessment of Computations

Isothermal Flows

The assessment first focuses on comparisons between the LES and experimental data for the case of ${\displaystyle V_{chan}/V{jet}}$= 0.077, in the form of contours of the vertical, (V) velocity components, shown in Figure 13. The comparison shows that there is close agreement between LES and the experiment as far as the width and the penetration length of the downward jet is concerned. The jet reaches a distance of about 0.54m along the right side of the channel (from the splitter plate at y=1.3m to about 0.76m). The validity of the RANS computations can thus be assessed through comparisons with the more detailed LES predictions as well as through comparisons with the available experimental data.

In Figure 14, the LES contours of the vertical velocity are compared with those of the RANS computations. The low-Re k–ε model returns a jet penetration length of about 0.74m, while use of the high-Re k–ε model with the standard wall-function approach results in a penetration length of about 0.98m. These first two comparisons would suggest that neither the effective viscosity approximation nor the assumptions involved in the standard wall function are appropriate for opposed wall-jet flows. The introduction of the analytical wall function results in good agreement between the high-Re and low-Re k–ε predicted V contours, indicating that this novel wall-function strategy is general enough to be used in the prediction of opposed wall-jet flows.

The remaining four sets of contours in Figure 14 were produced with second-moment closures. They indicate that while the introduction of the basic second-moment closure results in only modest predictive improvements in comparison with the effective-viscosity model, the more elaborate TCL version produces more substantial improvements. Indeed the combination of the TCL second-moment closure with the analytical wall function results in a jet penetration length that is very close to that of the LES computations. Comparisons between measured and computed profiles of the vertical mean velocity, shown in Figures 15, 16 and 17, also confirm these conclusions. There is generally close agreement between the measured profiles and those computed using either LES, or the TCL second-moment closure with the analytical wall function. Comparisons between computed and measured profiles of the cross-channel component of the mean velocity, shown in Figures 18 to 20, also lead to similar conclusions, though for this component the LES profiles are closer to those measured than those of the TCL model with the analytical wall functions. Finally, for this case, Figures 21 to 23 show comparisons between the computed and measured profiles of the turbulent kinetic energy. The measurements show that k reaches its maximum levels along the right hand side of the channel, in the region where the wall jet turns upwards. The EVM computations manage to capture both the overall trend and the actual levels. The second-moment closures return similar distributions, but where the jet turns upwards somewhat lower levels. The LES computations produce k distributions that are different to those measured, possibly because the simulation has not advanced a sufficiently large number of time steps to produce statistically independent values of the turbulent kinetic energy.

While the opposed wall jet flow is geometrically very simple, the resulting flow structure is complex enough for both the effective viscosity approximation and also the assumption of a logarithmic near-wall velocity variation to break down. In reviewing earlier numerical studies of wall jets developing in stagnant surroundings, Launder and Rodi (1983) showed that the effective-viscosity model produced a more rapid spreading rate than second-moment closures. This was attributed to the prediction of stronger mixing between the jet and the surroundings by the EVM, which in the case of the opposed wall jet, in contrast to the findings of the present study, should have resulted in the computation of a shorter penetration length. The question that therefore needs to be addressed is why EVM models in this case return a longer penetration length than second-moment closures. One major difference between the case of a self-similar wall jet in stagnant surroundings and a confined, opposed wall jet in a channel, is the presence of an adverse pressure gradient. A closer look at the vertical velocity profiles of Figures 15 to 17 shows that the TCL-AWF modelling combination above and just below the splitter plate ${\displaystyle {(y_{jet}{-}y<0.3m)}}$ predicts stronger downward flow on the outer side of the jet. Further downstream ${\displaystyle {(y_{jet}{-}y=0.4m)}}$ this model starts to predict a lower downward velocity along the right wall and also a lower upward velocity along the left wall. The implication is that the loss of the wall-jet’s downward momentum arises principally from entrainment of fluid into the outer layer of the wall jet (which reduces the level of the downward velocity) accompanied by a strongly rising pressure in the downward direction. We note from the vector plots of the velocity field, Figure 24, that the TCL closure does lead to a more rapid thickening of the wall jet than the k–ε scheme. Indeed, for the TCL predictions, notice that external fluid is drawn from well above the origin of the wall jet, for entrainment into the shear flow. The extra entrainment means that the TCL wall jet is less able to withstand the strong adverse pressure gradient, Figure 25(a). As shown by the shear stress profiles of Figure 26, the second-moment closures produce higher levels of the turbulent shear stress above the splitter plate than the EVM, most likely due to the effects of convective transport. This is consistent with flow entrainment shown in Figure 24. A further contributing factor to the loss of the wall jet’s momentum is the wall friction. We note in Figure 25 that, for the first 0.4m downstream from discharge, the TCL closure gives a friction factor that, on average, is 25% higher than the k–ε scheme. It can also be seen, in the same figure, that the introduction of the analytical wall function also increases the wall shear stress at the initial stages of the jet, which is again consistent with the shorter penetration length produced by this wall-function strategy.

For the case of ${\displaystyle V_{chan}/V_{jet}}$= 0.15, Figure 27 shows comparisons between the measured and the predicted contours of the vertical component of the mean velocity. As also seen in the earlier comparisons, the high-Re k–ε with the standard wall function produces a longer penetration length than the low-Re k–ε and with the introduction of the analytical wall function the high-Re k–ε predictions are brought to close agreement with those or the low-Re model. Replacing the EVM model with the TCL second-moment closure results in a further reduction of the predicted penetration length, but this reduction is not as substantial as in the case of ${\displaystyle V_{chan}/V_{jet}}$= 0.077. Moreover, the measured contours show that the penetration length is more than a factor of two shorter. The measured contours, however also show that the upward inlet velocity at the lower boundary is higher than the nominal inlet velocity, by about a factor of two. Whether this is the result of three-dimensional variations, or of experimental uncertainties in the determination of the inlet flow rates it is hard to say. Comparisons between the measured and predicted mean velocity profiles are presented in Figures 28 to 33. They also raise the same questions about the experimental conditions. The corresponding k-profile comparisons are presented in Figures 34 to 36. For the first 0.2m after the splitter plate all RANS models return the correct k levels and distribution. From 0.2m to 0.5m after the splitter plate, where the measured jet has turned upwards while the predicted jets continue to proceed downwards, the turbulence models over predict k levels.

Non-Isothermal Flows

So far, for the non-isothermal cases, computations have only been carried out using EVM models.

For the case of ${\displaystyle V_{chan}/V{jet}}$= 0.077, the vector plot comparisons of Figure 37, show that the three EVM models employed (low-Re k–ε, high-Re k–ε with standard wall-functions, and high-Re k–ε with analytical wall functions) return similar penetration lengths that are in agreement with that implied by the measurements. The spreading of the jet across the channel is, however, under-estimated. While the measurements show that once it separates from the wall, the jet turns gradually upwards, thus spreading across the channel, in the EVM predictions, the jet turns sharply upwards, making a tight 180° turn, and attaches itself on the other side of the splitter plate. In contrast to the available data, above the jet entry point, the fluid on the left half of the channel moves downwards. Comparisons of contour plots for temperature, turbulent kinetic energy and of the vertical velocity, shown in Figures 38 to 40, also present a similar picture. The measurements show that as soon as the jet turns high turbulence and temperature levels spread across the channel. The EVM predictions on the other hand show that initially the high temperature fluid is confined to the right side of the channel and only crosses to the left side further up above the jet entry location, as a result of the recirculating motion. High turbulence levels are also initially confined to the left side of the channel and as the second, upward directed wall jet develops along the splitter plate, high turbulence levels gradually spread to the left side of the channel. More detailed comparisons, that include profiles of the two mean velocity components and also the turbulent kinetic energy, are presented in Figures 41 to 43. They also confirm that the three EVM models produce similar predictions, especially the low-Re model and the high-Re model with the analytical wall function, which return the correct penetration length, but under-estimates the mixing of the jet. The latter trend is most obvious in the k profile comparisons.

The differences between the EVM predictions and the measured behaviour appear to originate from the failure of the EVM model to return the rapid spread of high turbulence levels across the channel as the wall jet turns upwards. This in turn inhibits the transfer of thermal energy across the channel, and leaves the cooler, heavier fluid on the left side causing the weak down flow present in the predictions. A possible cause of the failure of the EVM model to return the rapid spread of turbulence levels is the way that the contribution of buoyancy to the generation rate of turbulence is taken into account. In buoyant flows, there is an additional contribution to the generation rate of turbulence ${\displaystyle P{g}}$, the exact expression of which is

${\displaystyle P_{g}=-{\overline {\rho ^{\prime }u_{i}}}g_{i}=\rho {\overline {u_{i}\theta }}g_{i}/\Theta }$

Introduction of the effective diffusivity approximation for the turbulent heat fluxes leads to

${\displaystyle P_{g}=-{\frac {g}{\Theta }}{\frac {\mu _{t}}{\sigma _{\theta }}}{\frac {\partial \Theta }{\partial {y}}}}$

In this specific case where the y direction is the vertical direction

What the above expression shows is that as a consequence of the effective diffusivity approximation, the buoyant generation rate term is only active in regions with a temperature gradient in the vertical direction. Such a region occurs at the interface between the downward directed hot jet fluid and the upward-directed, cold channel fluid. Since at that interface the temperature gradient ${\displaystyle {\partial \Theta }/{\partial {y}}}$ is positive, ${\displaystyle P_{g}}$ would be negative, causing a reduction in turbulence levels. A more realistic way of representing the turbulent heat fluxes would be through the generalized gradient diffusion hypothesis, GGDH. This is routinely used with second-moment closures for the Reynolds stresses and, as shown by Ince and Launder (1989), can also be used with EVM closures for the flow field, though with a different value for the constant cθ. This approximation will result in

${\displaystyle P_{g}=-\rho {\frac {g_{i}}{\Theta }}c_{\theta }{\frac {k}{\varepsilon }}{\overline {u_{i}u_{j}}}{\frac {\partial \Theta }{\partial x_{j}}}}$

For the wall-opposed jet geometry the above expression becomes

${\displaystyle P_{g}=-\rho {\frac {g}{\Theta }}c_{\theta }{\frac {k}{\varepsilon }}\left({\overline {u\nu }}{\frac {\partial \Theta }{\partial x}}+{\overline {\nu ^{2}}}{\frac {\partial \Theta }{\partial y}}\right)}$

Consequently, use of GGDH will give rise to two contributions to the buoyant generation rate of turbulence, one due to the vertical temperature gradient, which is similar to that present when the effective diffusivity approximation is used, and one due to a horizontal temperature gradient. The latter will become active along the middle of the channel, after the jet has turned upwards, where ${\displaystyle {\partial \Theta }/{\partial x}}$ will be positive and the turbulent shear stress <uv> will be negative. This additional contribution would thus tend to increase turbulence levels, which means it is likely to improve agreement with the experimental data.

The corresponding comparisons for the higher velocity ratio, reveal a different behaviour, as far as the prediction of the penetration length is concerned. The vector and contour plots of Figures 44 to 47 show that the low-Re k–ε and also the high-Re k–ε with the analytical wall functions produce a penetration length, which is similar to what was measured. In these predictions, and the measurements, the jet starts turning upwards as soon as it is injected into the channel. These two models (the low-Re k–ε model in particular) also predict a spreading rate across the channel closer to that shown in the experimental data. The high-Re k–ε with the standard wall function, on the other hand, produces a penetration length that is twice as long and a jet that, as in the previous case of the lower velocity ratio, does not spread across the channel. Arguably, because with the standard wall function the jet is predicted to remain attached for longer, the predicted turbulence levels are higher. Along the left wall the EVM models again predict that the upward flow eventually reverses its direction, though at this higher velocity ratio this happens further up the channel. More detailed profile comparisons are shown in Figures 48 to 50. They confirm that there are now substantial differences between the predictions produced by the high-Re model with the standard wall function and those returned by the either the low-Re model or the high-Re model with the analytical wall function. It is interesting to note that the double peaks in the measured cross-channel velocity over the initial stages of the jet development, ${\displaystyle {(y_{jet}{-}y<0.06m)}}$ in Figure 49) are faithfully reproduced by the two EVM computations that do not make use of the standard wall-function. The resulting abrupt changes in the gradient of the measured vertical velocity over the same region (Figure 48) are also equally well predicted by the same two models. As far as the k profiles are concerned, the low-Re EVM model and the high-Re model with the analytical wall-function, in contrast to the high-Re model with the standard wall function, under-estimate the k levels within the jet. They do, however, predict that the high turbulence region extends further across the channel in comparison with the predictions obtained with the standard wall function, though not as far as what is indicated by the measurements.

One question that arises here, is why at the higher velocity ratio the predictions based on the standard wall function result in a longer penetration length than those of the other two near-wall treatments. The main difference as far as the flow dynamics are concerned is that at the higher velocity ratio the wall jet is subjected to a stronger adverse pressure gradient. Not surprisingly, the assumptions of the standard wall function (log-law, constant shear stress, fixed dimensionless thickness of the viscous sub-layer etc) become more restrictive as the adverse pressure gradient becomes stronger. It thus appears that modelling of near-wall turbulence is critical to the computation of both isothermal and non-isothermal opposed wall-jet flows and that the analytical wall function provides an effective and inexpensive alternative to the use of low-Reynolds-number models.

A further question that is currently being addressed is whether the introduction of second-moment closures for the Reynolds stresses and the GGDH for the turbulent heat fluxes can result in further predictive improvements for non-isothermal flows.

Further Work

Further numerical work is still in progress, focussing on non-isothermal flows. The aims are to provide an LES simulation of a non-isothermal case and also to extend use of the second-moment closures to the two cases already computed with effective-viscosity models. These computations will enable us to develop a better understanding of the non-isothermal cases and will also result in a more comprehensive evaluation of the capabilities of second-moment closures.

References

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ACKNOWLEDGEMENT

This work was funded under the HSE Generic Nuclear Safety Research programme and is published with the permission of the UK Nuclear Industry Management Committee (IMC). The authors gratefully acknowledge the financial assistance provided for this investigation. The Manchester University experiments were carried out under the terms of the research Contract entitled ‘CFD Quality and Trust – Generic Studies of Thermal Convection’. The UMIST computational studies were carried out under the terms of the research Contract entitled ‘CFD Quality and Trust – Model Evaluation, Refinement and Application Advice’.