In fluid dynamics, a turbulent regime refers to irregular flows in which eddies, swirls, and flow instabilities occur. It is governed by high momentum convection and low momentum diffusion. It is in contrast to the laminar regime, which occurs when a fluid flows in parallel layers with no disruption between the layers. The turbulence regime is extremely frequent in natural phenomena and human applications; some examples are the rise of a cigarette’s smoke, waterfalls, blood flow in arteries, and most of the terrestrial atmospheric recirculation. In terms of human applications, the turbulent regime occurs in the aerodynamics of vehicles, but also in many industrial applications such as heat exchangers, quenching processes or continuous casting of steel.

Reynolds Number

The real onset of scientific studies on turbulence can be found in the work of Osborne Reynolds in the second half of the 19\(^{th}\) century. Reynolds showed the transition between a laminar and a turbulent regime through a set of experimental investigations. He also suggested that this transition was directly linked to the ratio between inertial and viscous forces. This ratio was computed by George Gabriel Stokes in 1851 and has been named “Reynolds number” in honor of Osborne Reynolds who popularized it. This dimensionless number is defined as:

$$ Re = \frac{\rho ud}{\mu}=\frac{ud}{\nu} \tag{1}$$

where:

\(\rho\) is the density of the fluid

\(u\) is the macroscopic velocity of the flow

\(d\) is the characteristic length (or hydraulic diameter)

\(\mu\) is the dynamic viscosity of the fluid

\(\nu\) is the kinematic viscosity of the fluid.

Turbulent flows occur when \(Re\) exceeds a certain threshold (dependent on the application’s topology and flow physics) called “Critical Reynolds number”.

Turbulence Structure

In 1920, Lewis Fry Richardson summarized his works about the structure of turbulence for meteorological applications through a celebrated rhyme published in Weather Prediction by Numerical Process\(^1\):

“Big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls and so on to viscosity”\(^1\)

This principle was motivated by energetical considerations; big eddies are highly inertial and tend to be unstable. Their motion feeds smaller eddies thanks to a local transfer of kinetic energy. These smaller eddies undergo the same process, giving rise to even smaller eddies that inherit the energy of their parent eddy, and so on. This transfer of energy is usually called “energy cascade” and it is mainly inertial, thus almost no energy dissipation occurs until reaching a sufficiently small length scale such that the viscosity of the fluid can effectively dissipate the kinetic energy. This latter scale of the turbulence exhibits a local laminar regime and is characterized by a low value of \(Re\). This process has been depicted in Figure 1. Richardson’s studies highlight an important feature of turbulent flows: they are energy demanding. A turbulent flow will dissipate energy and decay to a laminar flow unless it is fed by an external source of energy.

Governing Equations

Turbulence scale

The complexity of turbulence and its aleatory behavior led scientists to use statistical models to describe turbulence flows. In 1941, Kolmogorov enhanced Richardson’s theory\(^2\). Kolmogorov postulated that for high enough Reynolds number, the small scale eddies are isotropic, while large eddies may be anisotropic (or anyway, dependent on the specific domain’s topology). This assumption is very important because it means that the statistical analysis of small eddies is independent of any specific geometry and thus it is universal for all turbulent flows. Under this hypothesis, Kolmogorov statistically described the main features of the smallest turbulence scale (known as “Kolmogorov microscales”) as follows:

\(\nu\) ,\(\tau\) and \(u\) are the characteristic length, period (or “turnover time”) and velocity of the smallest eddies respectively

RANS

It is normally believed (but not proven) that Navier-Stokes (NS) equations can model any kind of flow, turbulent flows included. The problem is that for very high values of \(Re\), the resolution of NS equation is very challenging and not stable. Thus a small perturbation in the parameter like the initial or boundary conditions may lead to a completely different solution. This problem is partially overcome by the use of the Reynolds-Averaged Navier-Stokes Equations (RANS)\(^3\). Let’s consider the Navier-Stokes equation for an incompressible Newtonian fluid:

where \(u\) is the velocity, \(p\) is the pressure of the fluid and the material parameters are considered uniform. The principle is to consider the flow as the sum of mean flow and turbulent/unsteady flow. The steady mean velocity can be computed as the Favre average of the global velocity:

where \(U\) is the mean velocity and \(u’\) is the turbulent flow velocity (or turbulent scales). \(T\) is the averaging time-scale, which must be small enough to have a good approximation of the problem, but also sufficiently higher than the turbulence time-scale, i.e. the Kolmogorov’s turnover time. By substituting the averaged quantities in the Navier-Stokes equation, we obtain the RANS equations:

where \(\left( u_{i}’ u_{j}’\right) _j\) is usually called “Reynolds stress” and represents the effect of the small-scale turbulence on the average flow. The RANS equations have no unique solution because they are not in a close form, the unknowns being more than the equations. Thus, additional equations are needed to close the problem. The most common strategy used in CFD is to relate the Reynolds stress to the shear rate by the Boussinesq relationship:

where \(\mu_t\) is the turbulent viscosity, which is usually computed through turbulence models\(^4\).

Applications

Turbulent flows are present in many natural phenomena (e.g. river flows, atmospheric streams, natural convection) and human applications (e.g. wind flow in a city, aerodynamic analyses, continuous casting, quenching process, cooling/heating systems). In this section, two small applications done using SimScale have been reported. The first (Figure 2) is an analysis of the flow around a cylinder. The case can be applied to the flow of a river around a bridge column, but it also has great academic value since it is a widely used benchmark for validation. Furthermore, it is often used to show the different regimes (linked to the presence, size, and frequency of eddies) at different problem configurations.

The second case is the aerodynamic analysis of a motorbike with the rider (Figure 3). The high velocities and the low viscosity of air encourage the development of a turbulent regime, even if it is avoided as much as possible in order to reduce the drag linked to the detachment of eddies behind the vehicle.

References

Richardson, L. F. Weather Prediction by Numerical Process (Cambridge Univ. Press, 1922)

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