Beginning at the Beginning ... 5 Parabolic or Elliptic? - Or Somewhere In Between?

David Tatchell

Posted Jan 6, 2010

To complete my recollections of the time when I was working on my PhD at Imperial College during 1969-1972 …

First of all, a short theoretical interlude. As CFD theorists know, it is often useful to categorise classes of flow according to the form of the governing equations – as parabolic, hyperbolic, or elliptic. Leaving aside hyperbolic – supersonic – flows – I want to focus on the distinction between parabolic or elliptic flows as applied to low-speed subsonic flows.

The distinction derives from the form of the governing partial differential equations, but I find it easier to focus on the associated physical characteristics of the flow. Essentially, parabolic flows are flows which have a “one-way” nature – in one coordinate direction physical effects are only transmitted in the positive direction, not in the negative direction. The obvious example is transient flows – effects are transmitted in the forward direction in time – but not backwards in time. (As we all know, outside of Hollywood, future events cannot influence the past!)

So transient flows are parabolic in the time direction. What this means is that transient flows can be solved by progressing systematically from the start time to the end time. When the end time is reached, then we are done – there is no need to repeat the solution – there is no need to iterate in time.

Now – turning to steady flows – there are certain flows which have a one-way nature in a spatial direction (generally, unidirectional flows), which are also categorised as parabolic flows. All other steady-state (subsonic) flows are “elliptic”, meaning that effects can be transmitted in any direction in any coordinate direction. Sometimes the term “boundary-layer flows” is used as equivalent to spatially-parabolic flows, and elliptic flows are often referred to as “recirculating flows“.

The distinction is important because, all else being equal, spatially-parabolic flows are much easier and cheaper to solve than elliptic flows – because the solution can proceed incrementally in the flow direction without any need to iterate in that direction. Hence, when the Imperial College team was first embarking on 3D solutions in the late 1960s it started with 3D parabolic (boundary-layer) flows – and hence the formation of the 3D boundary-layer research group of which I was a member.

A few more words about exactly what makes a steady flow “parabolic”. The “one-way” requirement leads to three pre-conditions:

1. There must be a “main flow direction”, with no reverse flow in that direction.

2. The diffusive transport in the main flow direction must be small enough to be negligible.

3. For the momentum equation in the main flow direction, it must be possible (with negligible effect on the accuracy of the solution) to represent the pressure gradient term as a “mean pressure gradient” (or “gradient in mean pressure”, which boils down to the same thing) – rather than (as is rigorously correct) as a local gradient in local pressures.

These together mean that any possible “reverse-direction” effects in the main flow direction are eliminated, and the equations become parabolic. They are generally true for a classic “boundary layer” close to a solid surface, and also for many other “unidirectional flows“, such as the developing channel flows that I worked on in the early part of my PhD research.

As the final part of my project I was then asked to look at a more challenging situation – the mixing of a jet entering normal to a main flow, of the kind illustrated below.

Jet Entering Normal to a Main Flow

This class of flows was chosen for study because it occurs in a number of practical problems of (then) high interest – the one we most had in mind was the dilution region of a cylindrical or (even closer) annular gas turbine combustor.

Plus, it was a way of exploring the range of validity of our parabolic computational method. We were aware that, for cases where the jet velocity was as large as, or larger than, the main flow velocity, the flow would certainly be elliptic, with (among other elliptic features) some flow reversal just behind the jet entry. However, for lower jet velocities, there was the expectation/hope that a parabolic method would yield reasonable results, much more economically that a full elliptic approach. So it was a sort of “stress test” of the parabolic approach!

Specifically, the situation that I considered was a rectangular passage, with, part way down the passage, a square side jet entering normally at the centre of one of the long passage sides – as shown below.

The Configuration Computed and Measured (apologies for the non-SI units!)

As well as performing computations (using the SIMPLE-based solution method described earlier, with the “standard“ k-epsilon model), I also undertook a short experimental study of the configuration shown.

We looked at turbulent flow conditions, with jet velocity/main flow velocity ratios of 0.1 and 0.3 – in the expectation that these would be low enough for the parabolic approach to work.

The results were “interesting” to say the least! The computed flow fields were very different from the measurements, even for gross characteristics of the flow, such as the jet trajectory. While the measurements showed the jet essentially attaching itself to the near wall, the predictions showed the jet spreading out across the channel, and eventually reaching the far wall.

So – not a good result! Not a very positive note on which to complete my PhD work!

Of course, in research, a negative result is not necessarily a failure – so long as it contributes to a better understanding of the phenomena in question. In this case the conclusion was that, even for the modest jet/main flow velocity ratios we considered, the flow was in fact more “elliptic” than we had expected. Specifically, the measurements showed significant lateral variation in pressure across the passage in the region immediately downstream of the jet – in conflict with the third requirement cited earlier for the flow to be truly parabolic. It seemed likely that the effects of this wrong assumption in my parabolic computations had led to the observed errors in the solutions.

This led to the recognition that there are flows (such as this one) for which, while the first two requirements for parabolic flow are valid (or close enough), the third (relatively small lateral pressure variation) is not valid. Spalding identified these as an intermediate class of flows, between parabolic and elliptic, which he named initially “Semi-Elliptic” and then “Partially Parabolic”.

The point about these is that they can be solved more economically that “fully elliptic” flows, by a sort of composite method. The flow is treated as parabolic, except that the local (rather than mean) pressure gradient is used in the main-flow momentum equation. This means that the pressure has to be stored 3D – while other variables can be stored 2D. Solution is obtained by a marching integration sweep, as for a parabolic flow – but at the end of each complete sweep through the solution domain, the 3D pressure field is corrected (using a form of SIMPLE), and the solution sweep is then repeated using the new pressures. This is repeated until convergence.

There are two advantages over a “fully-elliptic” solution – convergence is obtained faster (because the iteration sweep exploits the essentially one-way character of the flow), and there is a much reduced storage requirement (because only pressure is solved 3D).

A method of this kind was developed by one of Spalding’s students, Pratap, soon after I completed my research, and was applied successfully to a range of practical situations, including my transverse jet case, and flows in curved pipes and passages. The method was then used, alongside SIMPLE-based parabolic and elliptic methods, at Imperial College and at CHAM, until the advent of the PHOENICS general-purpose CFD software in 1981.

This study completed my PhD project – which had encompassed the development of 3D boundary-layer computational methods, their successful application to a range of laminar and turbulent channel flows, and, finally, an exploration of the limitations of the parabolic, boundary-layer approach for certain classes of flows. Quite enough it seemed to me – and also, fortunately, to David Gosman, my PhD supervisor!

So, in the autumn of 1972 I joined CHAM, the commercial CFD operation then just beginning at Imperial College. More on this – in a sense “the birth of commercial CFD” – in future blogs …