1. General

[GLOSS]Turbulence[/GLOSS] is the most complicated kind of fluid motion, making its precise definition difficult. A fluid motion is described as turbulent if it is rotational, intermittent, highly disordered, diffusive and dissipative.

For further details see the associated Combustion File,

CF110.

Turbulence should be accounted for in numerical modelling techniques (CFD) to allow for a proper representation of phenomena occurring in industrial combustion systems.

Different numerical methods can be applied to describe turbulent flows, such as: Reynolds averaged Navier-Stokes equations (e.g., the k-ε model), large eddy simulation (LES) or direct numerical simulation (DNS).

The first method is popular and has been widely used for industrial purpose. LES turbulence model is increasing its capability to be applied for industrial research, while DNS is the most advanced model for R&D purposes and has provided a lot of information on vortex dynamics, coherent structures and transition, compressible or rotating flows, but encounters serious problems on computer technological limitations.

This CF describes the main characteristics of the DNS turbulence model.

2. The DNS turbulence model

The most exact approach to model turbulence is to solve the Navier-Stokes equations, without averaging or approximation other than necessary [GLOSS]numerical discretisation[/GLOSS]. This approach, called DNS (Direct Numerical Simulation), is also the simplest approach from the conceptual point of view and attempts to resolve all the dynamically important scale of motions contained in a high Reynolds Number flow. The output can be viewed as equivalent to a short-duration laboratory experiment.

In a DNS, the computational domain must be at least as large as the largest turbulent eddy. This is the integral scale L of the turbulence, i.e. the distance over which the fluctuating component of the velocity remains correlated.

But a valid simulation must be also performed on the smallest scales, where the kinetic energy dissipation occurs. So, the size of an elemental cell must not be larger than a [GLOSS]viscosity[/GLOSS] determined scale, called the Kolmogorov scale η. The DNS model has to include nonlinear convective terms, because vortex interaction is a nonlinear mechanism. It must as well be time-dependent and three-dimensional.

The spectral method is one of the available numerical procedures to integrate non-linear partial differential equations. For example, a DNS code could use the [GLOSS]Fourier spectral method[/GLOSS] in all three spatial directions and a third order [GLOSS]Runge-Kutta method[/GLOSS] in time.

For homogeneous turbulence, the simplest type of turbulence, a uniform grid can be used, with a number of grid points in each direction (N) equal at least to L/η. This ratio is proportional to
R_eL^3/4, the Reynolds number based on the magnitude of the velocity fluctuations and the integral scale. Therefore, an approximate evaluation will show that the number of grid points required for a DNS in all three directions
(N³) is proportional to R_eL^9/4; which is extremely large for natural phenomena. For example, as shown in Table 1, the direct simulation of flows with
R_eL ~ 10⁶ (not unusual in engineering science) would yield
N³ of the order of 10¹³.

Table 1 – Numerical values of N against various values of R_eL

The results of a DNS contain very detailed and useful information about the flow. However, DNS is too expensive and complicated to be implemented in engineering applications. DNS can be used to produce data in place of experimental work and trigger statistical analysis or create a “numerical flow visualisation”.

Some examples of application of the DNS approach:

• Understanding the mechanisms of turbulence production, energy transfer, and dissipation in turbulence flow;
• Simulation of the production of aerodynamic noise;
• Understanding the effects of compressibility on turbulence;
• Understanding the interaction of combustion and turbulence;
• Controlling and reducing drag on a solid surface.

One of the important features of DNS is the possibility to compute statistical quantities that are useful in assessing models, allowing comparison with experiments and visualisation of the same flow.

This is rarely possible in laboratory experiments. Furthermore, it is possible to control the external variables and thus, test control methods that are difficult (or impossible) to implement in a laboratory.

It is therefore the technique applied to provide an accurate insight into the physics of various flows when investigating turbulence effects (even when compared to experimental investigations).

In the longer term, the study of turbulence in combusting flows with the DNS approach may increase engineering capabilities to indicate possible technical improvements in burner design and system control.

Sources

Joel H. Ferziger, Milovan Perić, “Computational Methods for Fluid Dynamics”, Springer, 1997.