Turbulent flows are characterised by three-dimensional motion of the fluid on a wide range of scales both in time and space. Mathematically, this means that a very small distance between discretized points is required which results in the computations becoming longer and more expensive. This is why a number of approximations have been introduced to represent the turbulence and effectively make the simulation more feasible.
The different approaches that have been developed to compute turbulent flows can be broadly categorised into the following three groups: 1) Direct Numerical Simulations, ‘DNS’, 2) Large Eddy Simulations, ‘LES’, and 3) Reynolds-Averaged Navier-Stokes, ‘RANS’. In addition to these technique, hybrid methods combining for example LES and RANS, etc also exist, the discussion of which are beyond the scope of this course.
A) Direct Numerical Simulation (DNS)
In Direct Numerical Simulation (DNS), all the fluid scales of motion are resolved by solving the Navier-Stokes and continuity equations. Since all the fluid scales need to be taken into account, DNS requires a very fine grid spacing. DNS provides highly detailed flow data, but requires supercomputing machines. Since the range of turbulent eddy sizes grows rapidly as the Reynolds number increases, computational requirements soon become prohibitive. The second-order accuracy of most finite volume schemes is often not sufficient, and other methods are frequently employed. These generally restrict one to using simple geometries. Whilst useful for fundamental flow studies, DNS is not therefore a tool for routine engineering calculations and is not an option for complex industrial flow problems.
B) Large Eddy Simulation (LES)
Large Eddy Simulation (LES) is another technique for computing turbulent flows in which large scales are resolved, while the small scales are modelled/approximated. LES emerged over 40 years ago, initially for meteorological applications. The idea behind LES technique is that large scales in the flow are anisotropic (i.e. directional) and thus not universal and need to be resolved, while small scales can be approximated since they are isotropic (i.e. non-directional), dissipative and more universal. The large scales can be separated from the small scales through applying a filter to the velocity field and decomposing it into filtered (resolved) and modelled (sub-grid) components. The size of this filter is determined by the resolution of the grid. Although LES has proved to be accurate on a range of industrial and non-industrial applications whilst requiring less computational resources than DNS, the cost of LES still exceeds the cost of a RANS simulation (introduced below) by at least few orders of magnitude. Therefore it has been predicted that LES calculations for complex geometries, especially at high Reynolds-number may not be feasible for several decades to come.
C) Reynolds-Averaged Navier-Stokes (RANS)
In most engineering situations it is the average velocity, pressure, etc. that are of interest, and the fine details of all the turbulent eddies are not required. Therefore, if instantaneous flow parameters are not required, a more practical alternative to DNS and LES is the Reynolds-Averaged Navier-Stokes (RANS) technique.
As originally proposed by Reynolds, the velocity and pressure fields can be split into a mean (or average) and a fluctuating part:
In many cases the flow field may be steady on average, in which case the decomposition of variables can be easily done by defining
The decomposed velocities, pressure, etc. can be substituted into the Navier-Stokes equations. This forms the basis of the RANS approach.
Averaging the resulting expressions leads to a set of equations governing the behaviour of the mean flow field. Details of this derivation are not given here, but the result is a set of equations rather similar to the original Navier-Stokes system:
Continuity
Momentum
However, the Reynolds stresses are not known (and cannot be calculated, since we are not now resolving the turbulent eddies). Turbulence models must therefore be introduced to approximate the Reynolds stresses (will be discussed in the next post).
RANS is currently the most economical and flexible, and thus the most commonly-adopted, approach for predicting turbulent flows and is widely used in industry for design and analysis of various flow problems, even though its accuracy is strongly dependent on the choice of turbulence model. Since it is not possible to have a universal turbulence model, it is very important for CFD users to know which turbulence models are the more accurate and thus reliable for use in various flow problems. This need has resulted in extensive CFD verification and validation assessments, and still remains a topic of research and debate across the international community.