2 Modelling
2.6 Integration Domain and Boundary Conditions
The correct choice of the integration domain and the boundary conditions is crucial for a successful simulation. In this chapter, the integration domains and boundary conditions for the simulation of the experiment and the steady case will be discussed.
2.6.1 Experiment
For the numerical investigation, the TH20.8 experiment is used as the experimental case. As shown in Fig. 3 (page 3), the geometry of the experiment is rotational symmetric. This is also true for the experimental boundary conditions. Therefore a two-dimensional integration domain is used to simulate the experiment (Fig. 15, left). The two-dimensional domain is realised with a 1° wedge with a thickness of 1 cell. The right-hand side of Fig. 15 shows the initial radial helium concentration at a given height. The used fluid is a variable composition mixture containing air as ideal gas and helium. The helium concentration is the passive scalar transported by the concentration equation and air is constraint. Buoyancy must be considered due to the variable density. The reference density for the buoyancy treatment is 0.179kg
m3 and the gravity constant is g=9.81m
s . The initialisation and reference pressure is 1.168 bar. The initialisation temperature is 24.3°C.
The first approach to generate the jet was to replace the inner part of the inner cylinder with an outlet boundary at the bottom of the inner cylinder and a velocity inlet boundary condition at the top of the nozzle. The helium is transferred from the outlet to the inlet, to prevent a loss of helium. A problem with this approach is that the influence of the increasing helium concentration on the generation of the jet cannot be covered. The fan, which is responsible for the jet generation, is applying a pressure gradient on the fluid. It acts as a momentum source.
If the density of the fluid decreases with an increasing concentration of helium, the volumetric flow rate will increase. This results in a greater jet velocity. Another minor problem is the shift of helium between outlet and inlet. Because the helium is bypassing the interior of the inner cylinder it is faster at the inlet than in reality. This can influence the result, because a higher helium concentration of the jet means a less sharp density gradient between jet and density layer, which results in a better mixing, which in turn leads to a higher helium concentration at the outlet. This effect is cumulative and can influence the result given the long transient.
An additional problem is associated with the nozzle as origin of the jet. As discussed in chapters 1.2.2 and 1.2.4, the mixing rate of a jet originating from a nozzle is larger compared to a long pipe as origin. The so called 'vena contracta' effect appears if the origin of a jet is a nozzle or orifice and must also be considered. Vena contracta is the constriction and acceleration of a jet depending of the opening angle of the nozzle (Alan Fox and Robert McDonald [79]).
Fig. 16 shows the behaviour of the jet close to the outlet of the nozzle dependent on the Fig. 15: Experimental Case: integration domain (left) and initial helium distribution (right)
9200 mm
3200 mm
Momentum Source
0 0.02 0.04 0.06 0.08
0 2 4 6 8 10
φ
z [m]
z
opening angle. Here, 90° is a long pipe and 0° an orifice. The velocity scale shows the maximum 10% of the stream-wise velocity. It can be seen, that the long pipe jet has already reached its maximum velocity at the outlet and shows no constriction. Decreasing the opening angle towards an orifice increases the constriction. Here, the jet is accelerating after leaving the nozzle or orifice. This effect cannot be captured if the measured velocity profile is the boundary condition at top of the nozzle, because the resulting jet can only decelerate after the boundary where the real jet will accelerate after leaving the nozzle.
To solve those problems, the interior of the inner cylinder is also part of the integration domain and there are no inlet or outlet boundaries. The fan is modelled as a momentum source. The volume of the fan in the 2D-wedge is 3.3⋅10−4m3 and the applied momentum is
10.43 kg
m2s2 , which is determined iteratively using velocity measurements of a pilot test of the inner cylinder. All walls are modelled smooth without slip. Symmetry boundary conditions are used for the symmetry axis and the sides.
2.6.2 Steady Case
For the steady case, two integration domains are used. A two-dimensional domain is used for the RANS-simulations and a three-dimensional domain for the large eddy simulation.
Fig. 17 shows the geometry of the steady case with the inlet and outlet boundaries on the left-hand side and the initial helium distribution on the right-left-hand side. The geometry of the steady state is a round cylinder with 1000 mm height and a radius of 500 mm. The air-inlet has a radius of 100 mm. The outlet is modelled as a nozzle to ensure a smooth outflow and to prevent non-physical behaviour, like recirculation across the outlet.
Fig. 16: Change of the stream-wise velocity of a jet depending on the origin
90° 79° 49° 0°
The boundary condition at the air inlet is a constant velocity of 2m
s with low turbulence intensity. The effects at the origin of the free jet in the experimental case, like the 'vena contracta' effect, can be neglected in the steady case. This is possible, because the reference investigation, the large eddy simulation, has the same boundary conditions as the simulations with RANS-models, and the influence of the origin of the jet on the mixing of the stable stratification is not the focus of this work.
The boundary condition at the helium inlet is a mass flow of helium. The mass of helium flowing into the domain is equal to the mass of helium leaving the domain through the outlet.
This way a steady state will be reached after a certain time. Using this approach to model the helium inlet provides a quantity to measure the quality of the mixing, since the helium mass flow increases with a better mixing.
Another approach to model the helium inlet would be a constant mass flow of helium, adjusted to the expected quality of the mixing. Using this approach gives an optically accessible way to measure the quality of the mixing, since the final density layer will be closer to the helium inlet, the better the mixing is. But there are several problems. The helium mass flow can not be determined a priori. So an initial simulation with the variable helium inlet has to be performed, to get an idea about the range of the helium mass flow. All simulations have to use the same helium mass flow to ensure comparability. This yields problems if the results of the used turbulence models deviates much. Finally, experience showed that the constant helium boundary needs significantly more time to converge than the
Fig. 17: Steady Case: integration domain (left) and initial helium distribution (right)
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
φ
z [m]
1000
500
Air-inlet
He-inlet
Outlet z
1000 mm
500 mm
variable boundary.
The initial helium distribution (Fig. 17, right) has a sharper density gradient and a larger helium concentration in the upper part compared to the experimental case. This leads to a more stable stratification and is necessary to keep the integration domain this small. If the stratification is less stable, the interaction area between jet and stratification consumes more space which would result in a larger integration domain and a longer simulation time. This would negate one important advantage of the steady state.
A summary of the initial and boundary conditions can be found in appendix A1.