• # Differences between turbulent flow and laminar flow

Posted by on April 2, 2023 at 8:17 AM

A turbulent flow differs from a laminar flow because inside it there are vortical structures of different size and speed that make the flow not predictable in time even if the motion remains deterministic. That is, the motion is governed by the laws of deterministic chaos: if we were able to know “exactly” all the velocity field at a given time and we were able to solve the Navier-Stokes equations we could get all the future motion fields. But if we know the field with a very small inaccuracy, this after a certain time would make the solution found completely different from the real one.

For example, in the case of motion in a cylindrical duct, in case of turbulent regime the fluid moves in a disordered way, but with an average speed of advance almost constant on the section. In the case of laminar motion instead the trajectories are straight and the velocity profile is parabolic or Poiseuille. The Reynolds number for which the transition from laminar to turbulent regime occurs in this case is Re = 2300. However, this value is strictly dependent on the amplitude of disturbances present in the flow before the transition to the turbulent regime. Therefore, it is theoretically possible to obtain laminar flows for higher values of the Reynolds number.

Turbulent motions possess peculiar characteristics such as:

• Irregularity. Turbulent flows are always highly irregular. For this reason, turbulence problems are usually treated statistically rather than deterministically. A turbulent flow is chaotic, but not all chaotic flows can be called turbulent.
• Diffusivity. The large availability of kinetic energy in turbulent flows tends to accelerate the homogenization (mixing) of fluid mixtures. The characteristic responsible for increasing mixing and increasing the rate of transport of mass, momentum, and energy in a flow is called diffusivity. Turbulent diffusion is usually parameterized by a turbulent diffusion coefficient. This turbulent diffusion coefficient is defined in a phenomenological sense, by analogy with molecular diffusivities, but has no real physical meaning, being dependent on the particular characteristics of the flow, and not a true property of the fluid itself. Furthermore, the concept of turbulent diffusivity assumes a constitutive relationship between turbulent flow and the gradient of an average variable similar to the relationship between flow and gradient that exists for molecular transport. At best, this assumption is only an approximation. However, turbulent diffusivity turns out to be the simplest approach for the quantitative analysis of turbulent flows and many models have been postulated to calculate it. For example, in large bodies of water such as oceans this coefficient can be found using Richardson’s four-thirds power law and is governed by random walk principles. In rivers and large ocean currents, the diffusion coefficient is given by variations of Elder’s formula.
• Rotationality. Turbulent flows possess non-zero vorticity and are characterized by an important mechanism of three-dimensional vortex generation known as vortex stretching. In fluid dynamics, these are essentially vortices subject to “stretching” associated with an increase in the component of vorticity in the stretching direction due to conservation of angular momentum. Furthermore, vortex stretching is the fundamental mechanism upon which the turbulent energy cascade relies to establish and maintain an identifiable structure function. In general, the stretching mechanism involves thinning of the vortices in the direction perpendicular to the stretching direction due to volume conservation of the fluid elements. As a result, the radial length scale of the vortices decreases and the larger structures in the flow break down into smaller structures. The process continues until the small-scale structures are small enough to transform their kinetic energy into heat due to the molecular viscosity of the fluid. According to this definition, turbulent flow in the strict sense is therefore always rotational and three-dimensional. However, this definition is not shared by all scholars in the field, and it is not uncommon to come across the term two-dimensional turbulence, which is rotational and chaotic, but lacks the vortex stretching mechanism.
• Dissipation. In order to maintain a turbulent flow, a persistent source of energy is required because turbulence dissipates it rapidly, converting kinetic energy into internal energy by viscous shear stresses. In the turbulent regime, the formation of vortices having widely varying length scales occurs. Most of the kinetic energy of turbulent motion is contained in large-scale structures. Energy “precipitates” from these large-scale structures to smaller-scale structures by a mechanism due to purely inertial effects and essentially invisible. This process continues, creating smaller and smaller structures thus corresponding to a hierarchy of vortices. At the end of this process, structures are formed that are small enough to be affected by the effects due to molecular diffusion, and then viscous energy dissipation takes place. The scale at which this occurs is the Kolmogorov scale. This process is referred to as an energy cascade, for which the turbulent flow can be modeled as a superposition of a spectrum corresponding to the vortices and fluctuations of the velocity field i on an average flow. Vortices can be in a sense defined as coherent structures of the velocity, vorticity, and pressure fields. Turbulent flows can be viewed as consisting of an entire hierarchy of vortices over a wide range of length scales, and this hierarchy can be described quantitatively by the energy spectrum, which measures the kinetic energy of the velocity fluctuations for each length scale (each corresponding to a wave number). The scales in the energy cascade are generally unpredictable and highly non-symmetrical.
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