Figure 1: Illustration of the wavy-vortex-ring pulsatile stenotic flow instability mode, at a Reynolds number close to onset. These show DNS results after the instability has reached final nonlinear saturation. Ur is the dimensionless pulse period.
Spectral Element Methods for Turbulent Flow Simulation; Large Eddy Simulation of Industrial Flows
Simulation of turbulent flow for engineering applications currently relies heavily on the use of turbulence models to predict the effect of all length scales of turbulence on the time-average flow. Turbulence models seek to be universal, but most of the turbulent momentum transport is produced by the largest coherent turbulent eddies of the flows which are heavily flow-dependent, so conventional models often fail to produce accurate predictions in engineering application. In direct numerical simulation, all dynamically relevant flow length scales are resolved and no turbulence model is required, so the accuracy of the results is comparable with those obtained by physical experiment. However, the cost of direct numerical simulation is prohibitive at Reynolds numbers appropriate to most engineering applications. Large eddy simulations are a compromise that employ a turbulence model to predict the influence of the finest length scales of turbulence on the large-scale flow structures, which are explicitly simulated.
To date most large eddy simulations have concentrated on simple geometries such as channels and pipes. One goal in this project is to extend the techniques to complex geometries typical of engineering applications, using a high-order finite element method to carry out the spatial discretisation, thus allowing geometries with arbitrary complexity to be modelled. The parallel code employed is well suited to distributed memory environments such as that provided by the APAC National Facility. Initial validation results are very good and we have progressed to simulate more challenging and representative flows such as those in cyclonic separators and turbulent combustors.
More recently, our attention has turned to direct simulations and stability analysis of biological flows, particularly cardiovascular flows. We will continue to use our existing parallel code and also employ a fully unstructured parallel spectral element code developed at Imperial College. Our goals here are (i) to understand fundamental fluid mechanics of transitional pulsatile flows and (ii) to publish turbulence and wall shear stress statistics for them. In this we will be working both with idealised (e.g. axisymmetric) and more complex (e.g. branching, reconstructed physiological shape) geometries. ; DNS and LES investigations of turbulent flows with complex geometries and/or rheologies. Project D86 is funded by CSIRO's contribution to APAC.
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Principal Investigator Hugh BlackburnManufacturing and Infrastructure Technology CSIRO and Bureau of Meteorology |
Project d77, d86 |
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Co-Investigators Shuhei FujimotoMechanical Engineering Hokkaido John Elston Mechanical Engineering Monash University Spencer Sherwin Department Aeronatics Imperial College London Nabil Noui-Mehidi CMIT CSIRO and Bureau of Meteorology Murray Rudman Manufacturing and Infrastructure Technology CSIRO and Bureau of Meteorology |
RFCD Codes 290501, 291803 |
Significant Achievements, Anticipated Outcomes and Future Work
We have in the past carried out quite a variety of different investigations of turbulent and transitional flows; follow this link for example publications. Here we will summarise the major new work of 2005. Our typical approach in this work has been to identify instability modes using linearised global stability analysis, which may subsequently serve as a starting point for direct numerical simulation (DNS).
With the adoption of the SGI Altix machine at APAC has come a substantial increase in capacity, allowing us to tackle a broader range of larger simulation projects. For example, here are outcomes of a test of parallel efficiency on the new machine with our standard production code. Principally owing to good data exchange bandwidth, speedup is substantially linear, and the code remains very efficient even up to a large number of nodes. On the previous machine, performance degradation was quite noticeable even at 40 nodes.
| NCPU | Walltime(minutes) | Time*NCPU | Efficiency | sec/step |
| 8 | 653 | 5224 | 1 | 7.8 |
| 16 | 309 | 4944 | 1.06 | 3.7 |
| 40 | 120 | 4800 | 1.09 | 1.4 |
| 80 | 62 | 4960 | 1.05 | 0.74 |
| 160 | 40 | 6400 | 0.82 | 0.48 |
Instability and transition in idealised stenotic flow
A straight tube with a smooth axisymmetric constriction is an idealised representation of a stenosed artery. We have examined the three-dimensional instabilities of steady flow, steady flow plus an oscillatory component, and an idealised vascular pulsatile flow in a tube with a 75% stenosis using both linear global stability analysis and DNS.
Following our report last year and publication of Sherwin & Blackburn (2005), we have now more fully characterised the pulse-period dependence of the two types of temporal Floquet modes of the vortex rings that are produced during successive pulses (period doubling via alternating vortex ring tilting, and azimuthal waves of individual rings). Using DNS at APAC, we have been able to demonstrate that for all the observed Floquet modes, turbulent transition ultimately progresses upstream to just a small number of tube diameters downstream of the stenosis, much closer than the downstream location of the greatest energy of the linear instability mode. We have also shown that the steady flow in a stenotic tube is highly susceptible to convective instability. This work has resulted in a new manuscript (Blackburn & Sherwin, under review). Since the onset of all the instabilities occurs at physiologically relevant values of parameters even in our highly idealised geometry, it is likely that they commonly lead to localised turbulent transition and associated regions of rapidly fluctuating wall shear stress in many places within the human arterial tree. Ur=1, Re=350; Widnall (wavy) mode
Figure 1: Illustration of the wavy-vortex-ring pulsatile stenotic flow instability mode, at a Reynolds number close to
onset. These show DNS results after the instability has reached final nonlinear saturation. Ur is the dimensionless
pulse period.
Symmetry breaking in flows with spatial and spatio-temporal symmetries
When a dynamical system is invariant under the action of a group of symmetries, there can be far-reaching consequences on its bifurcations; the symmetries of the system govern the types of possible bifurcations, as well as the symmetry properties of the bifurcating solutions themselves. So far our work has concentrated on flows with a Z2 spatio-temporal and an O(2) spatial symmetry, such as wakes of symmetric bodies, and flow in a rectangular cavity driven by periodic floor sliding. Apart from demonstrating the application of new ideas in symmetrical dynamical systems to the study of instability in fluid systems (Blackburn, Marques and Lopez 2005), a feature of particular interest is the predicted existence of modulated travelling waves, since these have never been a reported instability mode of such systems. Following the discovery that the presence of end wall constraint in experiments can prevent full development of travelling wave instabilities (Leung, Hirsa, Blackburn, Lopez & Marques 2005), we have concentrated efforts in computational studies of physically realisable geometries that will allow the modulated travelling waves to be observed. Figure 2 shows an example of such a flow (Blackburn & Lopez, in preparation).
Figure 2: A modulated travelling wave mode for a periodically driven cylindrical system.
When is a wavy wall rough?
All real pipes (more generally, all real solid walls) have some wall roughness; in many applications the degree of roughness can become quite significant. Even substantial amounts of wall roughness do not greatly affect the transition Reynolds number, but ultimately at high Reynolds numbers, the pipe friction factor becomes independent of Reynolds number, and is a function only of roughness height. While this behaviour is broadly understood to be produced by a change from viscosity-dominated to pressure-dominated drag with the onset of flow separation around roughness elements, the details of the mechanism are far from clear. With a recent theoretical linkage made to the phenomena of homogeneous isotropic turbulence (Gioia & Chakraborty, PRL 96(4), 2006), this has again become an important area of fundamental turbulence research.
While real roughness is always to some degree random, we are carrying out a systematic study of wall "roughness" by simulating turbulent flows in tubes with sinusoidal waviness (Blackburn, Ooi & Chong, in preparation), as illustrated in figure 3. Results to date suggest that the main elements of bulk behaviour are similar to that produced by sand roughness, as recorded in Nikuradse's classic experiments.
Figure 3: Turbulent flow in a tube with a wavy wall. Ret=314.
Instability of a stretched vortex flow
Stretched vortex flows are commonly found in aerospace and automotive flows (e.g. tip vortex flows), as well as a broader range of engineering applications in flows with swirl. Very often the flows are susceptible to vortex breakdown, instability and turbulent annihilation, which may be advantageous (if one wants the vortex to disappear, say in the case of wingtip vortices near runways) or disadvantageous (e.g. owing to fatigue loads produced on exposed structures downstream of the breakdown location). As a part of a larger study based on stability analysis developed by the PI, some DNS of these flows has been undertaken at APAC, as illustrated in figure 4.
Figure 4: Bi-helical instability of a Batchelor vortex in an adverse pressure gradient, illustrated using an isosurface of the magnitude of the intermediate eigenvalue of the velocity gradient tensor, coloured by axial velocity.
Computational Techniques Used
Most of our work to date is based around a parallel spectral element/Fourier Navier-Stokes solver written by the PI in the latter half of the 1990s. The spectral element method is a high-order finite element method that provides exponential spatial solution convergence for smooth problems. The method is parallelised across planes of data, and uses MPI for interprocess communication during pseudospectral formation of nonlinear terms in the Navier-Stokes equations. The code makes very heavy use of the BLAS, particularly matrix-matrix and matrix-vector multiplications, usually well-optimised in vendor-supplied libraries, and also the FFT, again often available as an optimised library. The code is used by APAC in benchmark testing. The code has been extended for use in steady and time-periodic global stability analysis, large eddy simulation, and direct simulation of turbulent non-Newtonian flows.
The vortex breakdown flow shown in figure 4 was computed using a fully unstructured (tetrahedral-based) parallel spectral element code written by colleagues at Imperial College London. Preliminary testing on APAC's SGI Altix shows a close to linear speedup when changing from 32 to 64 CPUs, although per-node memory limitations have so far circumvented more extensive testing. We hope to gain more experience with this code in future work.
Publications, Awards and External Funding
External Funding and Awards
Development and application of this code has received support from the Australian Academy of Science, and more recently the UK's Royal Academy of Engineering and the Engineering and Physical Sciences Research Council, which funded a UK visit by the PI in 2004. We currently have a new grant application before the EPSRC to fund another visit and development of new stability analysis techniques.
Publications