Numerical Parameters

to the integration point [80]. So with β=1 it is a second order upwind scheme. Because it is unbounded it may lead to non-physical oscillations in regions of rapid solution variation. The high resolution scheme is of second order where possible, with β=1. To prevent the non-physical oscillations it is decreasing the value of β where necessary.

The maximum number of iterations (coefficient loops) per time step is 20, the minimum number of iterations is two. The convergence criterion is a maximum residuum of 10^-3.

An important step for the post-processing of the experimental case is the definition of monitor points for the helium concentration. The position of those points is the position of the measuring points in the experiment, see Fig. 18. Using the monitor points, the mixing of helium can be analysed during the solver run. It would be possible to generate the mixing curves later with the CFX post-processor, but this requires transient result files in the frequency of the desired resolution of the curves. So it is highly recommended to use the monitor points, because here the values of every time step are available.

The numerical set-up for the two-dimensional steady case is similar to the set-up of the experimental case. It is also a transient calculation with the same time step of 0.01 seconds.

Once it reaches the steady state, small fluctuations of certain values, like Reynolds stresses can occur. Therefore a transient averaging of those values can be performed, since the average value is constant due to the steady state.

Fig. 18: Position of the monitor points in the experimental case 201

215

207

202 210

214 209

203

205 212

204 213

206 208

3.1.2 Steady Case – Large Eddy Simulation

The numerical set-up for the large eddy simulation is characterised by the aim to resolve the three-dimensional eddies in space and time. This leads to considerably higher computational effort compared to a RANS simulation, because a LES needs a much finer and three-dimensional grid and a smaller time step size.

For the transient set-up, the total time of the LES is 20 seconds calculated with a time step of a millisecond. The second order backward Euler scheme is used for the time discretisation.

After 10 seconds, a statistically steady state is reached. Transient averaging for post-processing and statistics is done over the last 10 seconds.

The spatial advection scheme is the central differences scheme.

As convergence criterion a maximum residuum of 10^-3 is used. The maximum number of iterations per time step is 10. Due to the large computational effort for a large eddy simulation, the number of iterations has to be more limited than for RANS simulations.

However, there have been no convergence issues with the smaller iteration limit.

For post-processing purposes, several preparations have to be done. To investigate the quality of the LES, a spectral analysis has to be performed (see chapter 2.5 and Fig. 14). To do so it is mandatory to define monitor points in the integration domain to monitor at least the velocity and the helium concentration. Fig. 19 shows the distribution of the monitor points for the LES. The horizontal position is at x=0 m and at x=0.3 m. The vertical position is at z=0.2 m, z=0.4 m, z=0.6m and z=0.8m. This distribution ensures time resolved data at various positions inside and beside the jet as well as below and above the density layer.

Fig. 19: Monitor Points of the Steady Case LES

X00 X03 Z02

Z04

Z06

Z08

It is necessary to have the data of every time step available to resolve the fluctuations with higher frequencies. It would be unreasonable to get this data through transient result files, due to the size of the files (500 MB) and the necessity to save every time step. The needed hard-drive space to have a transient result file every time step for the desired ten seconds statistical time would be approximately 5 TB (10 s * 1000 1/s * 500MB). Another issue would be the time to write the files to the hard-drive during the solver run and the time to load the files during post-processing.

Another mandatory preparation is the definition of a vector to access the turbulent scalar fluxes. They can only be accessed statistically by means of transient averaging since they are fluctuating values and the mean value has to be subtracted. This is an important reason to have a statistically steady case for the large eddy simulation. A transient simulation would require a spatial averaging to access the fluctuating scalar fluxes. This could be done circumferentially. The problem with this approach would be the low number of grid points to get information from near the symmetry axis and the jet.

To calculate the turbulent scalar fluxes, an expression for every spatial direction has to be defined as the product of velocity component and helium mass fraction.

• UX = Velocity u * He.Conservative Mass Fraction

VX and WX are defined similar to UX for their velocity component.

Next, an additional variable needs to be defined, here named UPHI.

• Variable Type: Specific

• Units: [ m / s ]

• Tensor Type: Vector

In the definition of the fluid models, the components of the vector must be defined as the respective expression. Finally, this vector, the velocity components and the helium mass fraction have to be assigned for transient averaging.

The turbulence mass fluxes can be post-processed by subtracting the product of the transient averaged velocity and helium mass fraction from the transient average of the product of both values. For example in x-direction:

• UPHI.Trnavg X - ( Velocity.Trnavg X * He.Conservative Mass Fraction.Trnavg) Or in a general formulation without using the CFX expression language:

_i=u_i− u_i⋅ (3-2)

This is true, because if the transient averaging interval is sufficiently large, u_i' '=0 and

' '=0 .

u_i− u_i⋅=





= ui⋅ui' '⋅ ui⋅' 'ui' '⋅' '− ui⋅

= u_i' '⋅' '= _i

(3-3)