UFR 1-01 Best Practice Advice
Underlying Flow Regime 1-01 © copyright ERCOFTAC 2004
Best Practice Advice
Best Practice Advice for the UFR
The key physics to be captured in this UFR is flow expansion to ambient pressure via a complex shock structure.
The nature of the shock structure depends on the pressure ratio: For a pressure ratio, P0/P∞ , of between 2.08 to 3.85, a moderately under-expanded jet will be produced. Here the flow expands towards ambient pressure initially through a series of oblique shocks, then via a series of shock cells. For pressure ratios P0/P∞ in excess of 3.85, a highly under-expanded jet will be formed. In this case a normal shock is produced immediately downstream from the point of release, but followed again by a series of shock cells consisting of oblique and normal shocks. Once the jet's pressure has decayed to ambient, the flow relaxes to a classical free jet for all pressure ratios.
Numerical modelling issues
To ensure grid-independent resolution of the shock structure a mesh resolution of d/64 should be used (d being nozzle diameter) in the vicinity of the shocks. However with a stuctured Cartesian mesh a grid resolution of d/32 results in only a 2% difference in peak velocity compared to d/64, and should be adequate for most applications.
If possible use an adaptive grid to ensure efficient use of computational resource.
Discretisation schemes should be at least second-order accurate.
The flow beyond the first shock is insensitive to assumed inlet turbulence intensity and length scale. An assumed turbulence intensity of 5% of the jet velocity and a length scale of 0.05d can be used. The inlet profiles may be assumed to be uniform.
The jet axis can be simulated as a symmetry boundary condition. For further advice on boundary conditions see section 7.4
For the moderately under-expanded jet:
To resolve the full shock structure do not use a standard k-ε model, since it over-predicts mixing in the shock region and results in an over-rapid decay in axial velocity. Only the first few shock cells will be simulated if a standard k-ε model is used.
To resolve the full shock structure use a k-ε model with compressibility modifications due to Sarkar (1991), or a Reynolds stress model with the same modifications. The shock structure will then be captured significantly further downstream than with a standard k-ε model. In addition the axial turbulence intensity will be far better predicted.
If the sole object of simulations is the accurate prediction of the shock structure in an under-expanded jet, then use of a Sarker-modified k-ε model, but with the constant C2 set to 1.8 instead of 1.92, will result in the capture of almost all the shock structure and shock locations.
For the highly under-expanded jet:
To resolve the shock structure in the vicinity of the Mach disk and first shock cell, a standard k-ε model performs as well as a Sarkar-modified k-ε model, or a basic Reynolds stress model.
The two CFD studies reviewed here took differing approaches for setting both the type and location of the free atmospheric boundary conditions. Unfortunately the information which has been published is not sufficiently well documented to enable definitive advice to be extracted on this point.
Thus Cumber et al (1994, 1995) set the free boundary parallel to the jet axis to be at ambient pressure. The downstream boundary - if located well beyond the shock structure, i.e. sub-sonic out-flow, can also be treated the same way. The location of these free boundaries was tested to ensure they were not too close to the jet and thus affecting the computational result, but the actual location is not stated by Cumber et al. A sensitivity test to boundary location should therefore be undertaken if adopting this approach.
Cumber et al (1994, 1995) set the upstream boundary to be a wall, with an inlet for the jet. This clearly does not allow any entrainment from upstream. But in reality there is little entrainment in the shocked region of an under-expanded jet. Hence this approach seems to work. However it may not be an adequate approach if the region beyond the shock structure is also of interest and to be modelled, since this will entrain fluid from upstream. In this case the treatment of Bartosiewicz et al (2002) may be more appropriate.
Thus Bartosiewicz et al (2002) specify a co-flow at the upstream and outer boundary, and the downstream boundary is set either to ambient pressure for sub-sonic out-flow, or the flow extrapolated from the interior of the domain for super-sonic out-flow. However there is then some uncertainty in how to specify the co-flow velocity magnitude. Bartosiewicz et al (2002) do state the location of the free boundaries, but it is not clear whether these would always be appropriate - for instance whether they depend on the specified co-flow velocity magnitude. Hence it is again recommended that a sensitivity test to boundary location should be undertaken if adopting this alternative approach.
A flapping instability was observed in the highly under-expanded jet experiments. This means that it is difficult to draw firm conclusions on turbulence model performance. However all models which have been tested; a standard k-ε, a Sarkar-modified k-ε, and a Reynolds stress approach, appear to perform similarly.
Data and computational results are only available in the near-field (up to z/d 12) for the highly under-expanded jet. Hence this best practice advice on turbulence models is only valid in this region.
Recommendations for future work
A more extensive set of experiments for the highly under-expanded jet are needed. The existing data was gathered over three decades ago and consists only of mean velocity measurements and flow visualisation up to z/d 12. Data is needed further downstream. In addition, turbulence intensity should be measured to allow more in-depth assessment of turbulence models. The flapping instability observed in the existing data also needs investigation via new experiments to establish whether it is a characteristic solely of the particular experimental set-up, or a general feature of such flows.
© copyright ERCOFTAC 2004
Contributors: Chris Lea - Health & Safety Laboratory