General comments on performance of LES It is the main task of turbulence modelling to develop computational pro-

3.8.7 General comments on performance of LES It is the main task of turbulence modelling to develop computational pro-

cedures of sufficient accuracy and generality for engineers to predict the Reynolds stresses and the scalar transport terms. The inherent unsteady nature of LES suggests that the computational requirements should be much larger than those of classical turbulence models. This is indeed the case when LES is compared with two-equation models such as the k– ε and k–ω models. However, RSMs require the solution of seven additional PDEs, and Ferziger (1977) noted that LES may only need about twice the computer resources compared with RSM for the same calculation. With such modest differences in computational requirements the focus switches to the achiev- able solution accuracy and the ability of the LES to resolve certain time- dependent features ‘for free’. Post-processing of LES results yields informa- tion relating to the mean flow and statistics of the resolved fluctuations. The latter are unique to LES, and Moin as well as Meinke and Krause (both in Peyret and Krause, 2000) gave examples of flows where persistent large-scale vortices have a substantial influence on flow development, e.g. vortex shed- ding behind bluff bodies, flows in diffusing passages, flows in pipe bends and tumble swirl in engine combustion chambers. The ability to obtain fluctuat- ing pressure fields from LES output has also led to aeroacoustic applications for the prediction of noise from jets and other high-speed flows.

As an illustration of the most advanced LES capability we show results from Moin (2002) for a gas turbine. Figure 3.17 shows a detail of the com- bustor geometry and the computational grid, which is unstructured to model all the details in this very complex geometry. Figure 3.18 shows contours of instantaneous velocity magnitude on a mid-section plane and on four further perpendicular cross-sections as indicated on the diagram. The physics of the turbulent flow is also highly complex, involving combustion, swirl, dilution jets etc. Flow instabilities have serious consequences for combustion, and the information generated by LES calculations is uniquely applicable to the development of this technology.

Figure 3.17 LES computations on Pratt & Whitney gas turbine – detail of combustor geometry and computational grid Source: Moin (2002)

CHAPTER 3 TURBULENCE AND ITS MODELLING

Figure 3.18 LES computations on Pratt & Whitney gas turbine – instantaneous contours of velocity magnitude on sectional planes Source: Moin (2002)

LES has been around since the 1960s, but sufficiently powerful comput- ing resources to consider application to industrially relevant problems have only recently become available. Inclusion of LES in commercial CFD is even more recent, so the range of validation experience is limited. Most code ven- dors usually state that care must be taken with the interpretation of results generated with their LES models. Furthermore, it should be noted that the methodology for the treatment of non-commutativity effects in non-uniform and unstructured grids is comparatively recent, as are treatments for com- pressible flow and turbulent scalar fluctuations. This research does not yet appear to have been incorporated in finite volume/LES codes. Geurts and Leonard (2005) give a survey of the main issues that need to be addressed to control error sources and generate robust LES methodology for application to industrially relevant complex flows. It is likely that the pace of develop- ments will increase as computing resources become more powerful and as the CFD user community becomes more aware of the advantages of the LES approach to turbulence modelling.

3.9 Direct numerical simulation

The instantaneous continuity and Navier–Stokes equations (3.23) and (3.24a–c) for an incompressible turbulent flow form a closed set of four equa-

tions with four unknowns u, v, w and p. Direct numerical simulation

(DNS) of turbulent flow takes this set of equations as a starting point and develops a transient solution on a sufficiently fine spatial mesh with sufficiently small time steps to resolve even the smallest turbulent eddies and the fastest fluctuations.

Reynolds (in Lumley, 1989) and Moin and Mahesh (1998) listed the potential benefits of DNSs:

• Precise details of turbulence parameters, their transport and budgets at any point in the flow can be calculated with DNS. These are useful for the development and validation of new turbulence models. Refereed databases giving free access to DNS results have started to emerge (e.g. ERCOFTAC, http://ercoftac.mech.surrey.ac.uk/dns/homepage.html; Turbulence and Heat Transfer Lab of the University of Tokyo, http://www.thtlab.t.u-tokyo.ac.jp; the University of Manchester, http://cfd.me.umist.ac.uk/ercoftac).

3.9 DIRECT NUMERICAL SIMULATION 111

• Instantaneous results can be generated that are not measurable with instrumentation, and instantaneous turbulence structures can be visualised and probed. For example, pressure–strain correlation terms in RSM turbulence models cannot be measured, but accurate values can be computed from DNSs.

• Advanced experimental techniques can be tested and evaluated in DNS

flow fields. Reynolds (in Lumley, 1989) noted that DNS has been used to calibrate hot-wire anemometry probes in near-wall turbulence.

• Fundamental turbulence research on virtual flow fields that cannot occur in reality, e.g. by including or excluding individual aspects of flow physics. Moin and Mahesh (1998) listed some examples: shear-free boundary layers developing on walls at rest with respect to the free stream, effect of initial conditions on the development of self-similar turbulent wakes, the study of the fundamentals of reacting flows (strain rates of flamelets and distortion of mixing surfaces).

On the downside we note that direct solution of the flow equations is very difficult because of the wide range of length and time scales caused by the appearance of eddies in a turbulent flow. In section 3.1 we considered order- of-magnitude estimates of the range of scales present in a turbulent flow and found that the ratio of smallest to largest length scales varied in proportion to Re 3/4 . To resolve the smallest and largest turbulence length scales a direct

simulation of a turbulent flow with a modest Reynolds number of 10 4 would require of the order of 10 3 points in each co-ordinate direction. Thus, since turbulent flows are inherently three-dimensional, we would need computing meshes with 10 9 grid points (N ≅ Re 9/4 ) to describe processes at all length scales. Furthermore, the ratio of smallest to largest time scales varies as Re 1/2 , so at Re = 10 4 we would need to run a simulation for at least 100 time steps. In practice, a larger number of time steps would be needed to ensure the passage of several of the largest eddies in order to obtain meaningful time- average flow results and turbulence statistics.

Speziale (1991) estimated that the direct simulation of a turbulent pipe flow at a Reynolds number of 500 000 requires a computer which is 10 mil- lion times faster than a (then) current generation Cray supercomputer. Moin and Kim (1997) estimated computing times of 100 hours to 300 years for tur-

bulent flows at Reynolds numbers in the range 10 4 to 10 6 based on high- performance computer speeds of 150 Mflops available at that time. This confirmed that it started to become possible to compute interesting turbulent flows with DNS based on the unsteady Navier–Stokes equations. Advanced supercomputers at present (2006) have processor speeds of the order 1–10 Tflops. If the performance scaling can be maintained across such a wide range of speeds, this would reduce computing times to minutes or hours. We briefly review progress in this rapidly growing area of turbulence research.