# Turbulent times

According to the legendary physicist Richard Feynman, turbulence is the last great unsolved problem of classical physics. Even so, computers allow us to study it in great detail, but you must apply such tools with great care. Paul Schreier tries to bring some order to the world of software intended for these studies of chaos

Scientific Computing World: June/July 2010

When you study aerodynamics or indeed any fluid flow, in most cases you’re actually studying turbulence. Laminar (‘smooth’) flow is not terribly interesting in the real world, whereas turbulent flow is very common in nature and occurs nearly everywhere: in rivers, in the oceans, around our cars and airplanes, even in stars and galaxies.

We’re all familiar with the difference between laminar and turbulent flow in our daily lives. When water flows slowly out of a faucet, the stream is smooth and regular, because the water molecules move at more or less the same speed in the same direction. This is laminar flow. When you increase the flow’s speed, the stream becomes rough and irregular; the water molecules tend to move in different directions – this flow is turbulent. Similarly, when you drive your car faster, turbulence builds up behind it and drag develops. How easily a fluid becomes turbulent depends to a large extent on its viscosity; the more viscous a fluid, the less likely it is to become turbulent.

Water and air, which have a low viscosity, can become turbulent relative easily; in contrast, oil and molten glass or metal tend not to become turbulent.

### A key parameter

Another way of specifying this behaviour is with the Reynold’s number, which is a measure of the way in which a moving fluid encounters an obstacle. It is proportional to the fluid’s density and speed as well as the size of the obstacle; it is inversely proportional to the fluid’s viscosity. A small Reynolds number defines a flow in which the fluid moves slowly or has a large viscosity to keep it organised. In such a situation, the fluid can move around an obstacle smoothly with laminar flow. On the other hand, a large Reynolds number indicates a flow in which the fluid can’t get around an obstacle without breaking up into turbulent swirls and eddies. You can describe such turbulent flow as dominated by the fluid’s inertia – the tendency of each portion of fluid to follow a path determined by its own momentum.

If a flow appears to be random, how can you study it? Not very easily. George Papanicolaou, mathematics professor at Stanford University [1], explains: ‘Simply put, turbulence is very hard. Every hard problem in classical physics finds itself embedded in turbulence. It is nonlinear, chaotic, stochastic. And there is no separation of scales – you must deal with a very large number of scales of irregularities. It’s just a mess. In most other physics problems, you can get control by reducing them to simpler problems that you can understand. You can separate scales, for instance, and determine that certain scales are not important. You can limit the phenomena. Or perhaps the inhomogeneity, the chaotic behaviour, is not there all the time, so you can somehow approach it. In turbulence all these things happen at once, and you don’t know how to separate them out.’

### How CFD handles turbulence

Despite this statement, any CFD (computational flow dynamics) software must include some way to handle turbulent flow. Getting vendors to discuss their offerings, however, isn’t always easy. One supplier commented that all vendors provide basically the same capabilities, and the differences are in the software’s usability and technical support. ‘There is some truth to this statement,’ comments Professor Michael Leschziner, who heads the Turbulent Flow Modelling and Simulation Group at Imperial College London. He adds that everyone is solving the same basic equations, and numerical modelling has nothing to do with the core physics, but instead works with the underlying partial differential equations. There are differences, however, in the numerical machinery in how various software approaches the solution method and in the solvers they use.

Leschziner adds that there are roughly 120 different turbulence models in existence to handle various conditions. Many of these are incorporated in different packages, many of which contain similar sets of models. Users tend to compare Package A with Package B and ask, ‘why don’t you have Turbulence Model X?’ This motivates uniformity in the market. Thus, many packages have common basic models that are well known to all, but there are many more models that are not included in these packages. These are the ones that are more difficult to incorporate into the software, make the software unwieldy and are more difficult to solve; the result is that a software house doesn’t want to include them and get a reputation for unstable models that don’t converge.

Another point Leschziner makes is that you can’t make definitive statements about which model is the best one to use. The software reveals trends, so you can try different models to see their sensitivity and how they vary with each other. In every situation, models are influential, but not decisive. He adds, however, that users often simply choose one and just cross their fingers. It’s quite difficult to make a rational decision as to which one to use. He believes that to understand turbulence to this degree requires a decade of experience.

Along these lines, Leschziner points to one of the best resources he is aware of, one of the few documents targeting people who are users but not turbulence experts. It comes from ERCOFTAC (the European Research Community on Flow Turbulence and Combustion), and it consists of two Best Practice Guidelines, one for single-phase flows and another for dispersed multi-phase flows. (Both are available at www.ercoftac.org).

### Several major classes

Even so, it’s instructive to recognise that simulation and modelling approaches can be broken down into several major classes (Figure 1). Much of this breakdown is based on the fact that turbulence involves a wide range of turbulence eddy sizes. With DNS (direct numerical solution), the software is expected to solve the actual Navier-Stokes flow equations and resolve all scales and frequencies. Because of the massive computations required, this approach is possible only for simple cases; otherwise, simulation times could run into the months or years, even with today’s hardware. It’s just not practical for real industrial applications.

Fig. 1: Major classes of turbulence simulations and models (courtesy of Ansys)

To cut down on computation times, the other approaches vary in which turbulence eddy sizes are actually calculated directly and which are modelled with time averages of turbulent structures that produce mean values. At the other end of the class chart is RANS, Reynolds Averaged Navier-Stokes simulations, where all the turbulence eddies are examined in a time-averaged fashion. In between are a wide range of models that solve larger eddies directly but handle the remaining smaller ones in a time-averaged fashion. There are, of course, a multitude of approaches for deciding how to split up the work (Figure 2), and these options help account for the many turbulence models found in various software packages and even more in the literature.

Even so, there are many possible selections of turbulence models in today’s CFD software, which can be daunting. To help users, says André Braune, customer support team leader for power generation and turbomachinery, Ansys has released a handful of stable, validated and broadly applicable turbulence models in its CFD software. When guiding customers to select among them, Ansys puts the models in the following hierarchy, going from simplest (and often less accurate) to most complex (and most accurate):

• Zero-equation models, for instance when constant eddy viscosity is applied throughout the entire domain;
• 1-equation models, e.g. for external flows of wings;
• 2-equation models, belonging to the RANS/URANS category and still today’s industry standard. Ansys’ recommendation here is the SST (Shear Stress Transport) model, a combination of k-epsilon and k-omega turbulence models with a shear stress limiter and automatic wall treatment. Braune says this is the workhorse of many steady-state and unsteady-state simulations; it is efficient and accurate for a wide range of applications;
• Reynolds stress models, which are used less often, because they require significantly more computational efforts than 2-equation models and are less stable in terms of solver convergence. However they have advantages in some areas such as strong streamline curvature and vorticities;
• Scale Adaptive Simulation (SAS), an enhancement and extension of the SST model. It can provide a LES-like behaviour in detached flow regions, but it falls back to the URANS (SST) solution for areas where the requirements for mesh or time discretisation to resolve the eddies are not fulfilled;
• Detached Eddy Simulation (DES), a hybrid approach combining LES and RANS models, to improve prediction capabilities of turbulence models in regions with high separation potentials; and
• Large Eddy Simulation (LES).

### Auto-selection option

On the specifics of turbulence modelling for aerodynamics in Star-CCM+, CD-adapco offers a wide selection of models ranging from the near ubiquitous k-epsilon model, through k-omega SST (widely used in vehicle aerodynamics) and on to the more sophisticated (and hence computationally expensive) Reynolds stress models (RSM), large eddy and detached eddy simulation (DES and LES) types. Also available are models for quite specific subsets of the aerodynamics market such as Spalart-Allmaras, which is commonly used in the aerospace industry. For each of these there are, of course, a range of options with different ‘flavours’ of each model type, which can be chosen to capture specific features within the flow field.

Fig 2: Turbulence models vary in the degree to which they can resolve small eddies, but the tradeoff is computational efforts. Here three methods (SAS, DES, URANS SS T) for the same computational mesh and boundary conditions result in different flow fields (copyright Robert Bosch GmbH)

Another option that can be critical in computational aerodynamics is the ability to model the transition from laminar flow to turbulence, and like many other packages Star-CCM+ can identify and model the onset of transition.

While Star-CCM+ provides many options, which can be somewhat daunting to inexperienced users, the software can auto-select recommend models to help guide the setup. If the user is more experienced and has need of a specific model, this is also available. The basic premise is that it is as easy to do the very complex as it is to do the very simple (and everything in between).

Even so, there is a crucial need for strong customer support says Joel Davison, Star-CCM+ solution champion. He suggests that users look for a company that provides a comprehensive support organisation with every user allocated a dedicated support engineer.

### Enhanced Turbulence Modelling

Aerodynamic testing is a classic flow-simulation application, and besides the ability to simulate airflow over the entire range of engineering-relevant Reynolds numbers – including subsonic, transitional, supersonic and even hypersonic flows – Mentor Graphic’s Concurrent FloEFD offers a range of additional physical models for characterisation of a design’s aerodynamic behaviour. This includes a special automatic laminar/turbulent modelling process, an innovative model created to simulate near-wall physics in a very efficient way.

In addition, the firm’s FloEFD product supports a dual-wall function modified k-epsilon model that addresses the key weaknesses of standard k-epsilon models such as low Reynolds number flows and high curvature flows.

The firm also provides what it terms Enhanced Turbulence Modelling (ETM), which consists of modifications of the k-epsilon model and modifications of the wall function approach to specifying the wall boundary conditions for the Navier-Stokes equations.

In addition to turbulence, when simulating fluid flows it is also necessary to simulate fluid boundary layers of these flows over solid bodies or walls, which is a substantial drawback also due to high velocity and temperature gradients across these layers. To solve the Navier-Stokes equations with conventional turbulence models without resolving the fluid boundary layer by the computation mesh, a wall functions approach can be used. The fluid wall friction and heat flux from the fluid to the wall serve as the wall boundary conditions for the Navier-Stokes equations. Naturally, the main flow’s properties are used as the boundary layer’s external boundary conditions.

FloEFD employs Van Driest’s universal profiles to describe turbulent boundary layers along with two approaches (2-scale wall functions) to fit the boundary layer calculation to the main flow properties depending on whether the fluid mass centres of the near-wall mesh cells are located inside or outside the boundary layer. These two approaches allow the software to overcome the restriction on the mesh density near the walls and use Cartesian meshes.

#### References

1. ScienceWatch, July/Aug 1995, Stanford University’s George Papanicolauo Seeks Order in Turbulence