**Introduction**

Nearly all macroscopic flows in engineering practice have a turbulent nature. Flows through turbines and engines, power generation by water, and petroleum production from the reservoirs all resemble turbulent regimes.

Turbulence involves fluctuations that are not easy to predict, at least fully. However, some general conclusions about turbulence can be drawn. In spite of the common association of turbulence with randomness, this is not completely true. Turbulence is not straightforward, but some basic features that are characteristic for all turbulent flows are:

(1) Fluctuations: Turbulent flows contain fluctuations. Turbulent fluctuations appear to be irregular, chaotic, and unpredictable.

(2) Nonlinearity: Turbulence occurs when the relevant nonlinearity parameter, e.g., the Reynolds number *Re*, exceeds a critical value. Small perturbations can grow spontaneously when critical value is reached. These perturbations develop complicated flow structures. Eventually flow reaches a nonrepeating unpredictable state.

(3) Vorticity: Turbulence is characterized by fluctuating vorticity. Identifiable structures in a turbulent flow, are called eddies. Turbulence involves a range of eddy sizes and the size range increases with higher Reynolds number.

(4) Dissipation: In turbulent flow, fluctuation energy and vorticity are transferred from large scale structures to smaller ones via nonlinear interactions, until energy is converted into heat, i.e., dissipated. The action of the motion and the viscous interaction between the eddies of smallest size cause the dissipation. Turbulence therefore requires a continuous supply of energy in order to balance this energy loss. Otherwise, it will fade out.

(5) Diffusivity: Macroscopic mixing and agitation characterize the turbulent flow. These processes stimulate the rapid rate of mixing and diffusion of species, momentum and heat. Equivalent laminar flows lack such fluctuations.

In short, turbulence can be summarized as a dissipative flow state characterized by nonlinear fluctuating three-dimensional vorticity.

**Energy in turbulent flow**

Turbulence rapidly dissipates kinetic energy. The energy loss in the turbulent flow is due to fluctuations of the velocity field. Mean kinetic energy is converted into turbulent kinetic energy, i.e., mean flow velocity is redistributed into the velocity of eddies of various sizes which are more chaotic. In turn, turbulent kinetic energy converts to heat via viscous dissipation.

The average turbulent kinetic energy cascades down the eddy scale size. The energy received by the largest eddies from the mean flow must be balanced by the kinetic energy dissipation rate, ε. Second-tier eddies, which are smaller than the largest, extract energy from the largest eddies by the same mechanism that the largest eddies extract energy from the mean flow. Since the scales of the large eddies are much larger than the sizes of small eddies, they do not interact directly with the large eddies or the mean field. Therefore, turbulence in the range of smallest size eddies is nearly isotropic. The energy cascade process continues until the Reynolds number of the eddies becomes of order of unity. Then viscous effects are important and viscous dissipation of energy into heat takes place.

Kolmogorov (1941) hypothesized that the statistics of small scales are isotropic and depend on two parameters only, the kinematic viscosity and the average rate of kinetic energy dissipation per unit mass of fluid. He derived that the size of the smallest eddies, before they dissipate into heat, can be determined from the dimensional analysis:

(1)

Where

is kinematic viscosity, m^{2}/s

is energy dissipation rate per unit mass, m^{2}/s^{3}.

His second hypothesis was that, at scales much smaller than the largest eddies and much larger than , there must exist an inertial subrange of turbulent eddy sizes for which plays no role; in this range the statistics depend only on a single parameter .

**Turbulence in single-phase flow**

The discrimination between laminar and turbulent regime plays a decisive role for the friction pressure drop or the head loss. The Reynolds number determines the flow regime.

(2)

Where

- fluid density, kg/m^{3}

– fluid velocity, m/s

– pipe diameter, m

- dynamic viscosity, kg/m·s

We discriminate between following regimes:

**Re**__<__2000 : Laminar flow- 2000 <
**Re**__<__4000 : Transition between laminar and turbulent flow - 4000 <
**Re**: Turbulent flow.

In single-phase laminar flow one may show that the frictional pressure gradient at constant flow velocity and constant pipe diameter is given by

(3)

is usually replaced with ** f_{D}**, which stands for friction factor (Darcy friction factor).

Friction becomes larger in the turbulent flow. This is due to the velocity profile becomes more uniform (although fluctuating), causing a larger velocity fall-off towards the pipe wall and thus a larger shear. It may seem as if the viscosity apparently increases. This implies that a new friction factor is required. There are many empirical equations to determine the turbulent friction factor. One of the most well-known equations is Colebrook & White friction factor which takes into account pipe roughness:

(4)

This equation is implicit in ** f_{D} **and has to be solved numerically. The representative form of this equation takes shape of a Moody Diagram.

**Turbulence in multiphase flow**

Multiphase flow is the dynamics of several phases (gas, liquid and solids) flowing in a common stream. Multiphase flows are in most cases turbulent, and turbulence is still not a completely solved problem in fluid mechanics. In single-phase flow, we discriminate between laminar and turbulent flow. In multiphase flow, we discriminate in addition between flow regimes that are characteristic for the time and space distribution of different phases.

In horizontal flow, we discriminate between the following flow regimes:

- Stratified flow
- Slug flow
- Dispersed bubble flow
- Annular flow.

**Fig. 2** shows examples of two-phase gas-liquid flow. At low velocities, the gas and liquid are separated as in stratified flow. At high velocities, gas and liquid become mixed. Slug flow is an example of a flow regime in between, representing both separation and mixing. Slug flow is consequently referred to as an intermittent flow regime.

The same comments that apply to horizontal flow are valid in vertical flow. The big difference is that in vertical flow it is not possible to obtain stratified flow.

Regarding low shear applications, the liquid-liquid two-phase turbulent flow presents the most interest. One of the common areas to find such a flow regime is in the petroleum production. Crude oil is rarely flowing alone from the reservoir. On average, for each produced barrel of oil there are two-three barrels of co-produced water. Water can originate from the reservoir water layer or it can be injected water used for reservoir pressure support. Reservoir fluids experience high level of turbulence via the flow acceleration during production. This is mainly due to rapid pressure drop and the flow through the narrowing radii.

For a mixture of oil and water flowing at high velocity it is most common to form a dispersed bubble type flow. As mentioned earlier, turbulent flow consists of eddies of different size range. These eddies have direct impact on the bubbles or droplets that are formed in the fluid stream. Eddies that have larger size than droplets, transport these droplets through the flow field. Eddies, which are smaller or equal to the size of the droplets, cause droplet deformation and break-up. It can be viewed as eddies collide with droplets and break them if they have sufficient energy to overcome the droplets internal forces. Most of the droplet break-up happens in the inertial subrange of the eddy sizes. According to the hypothesis of Kolmogorov, only energy dissipation rate, , governs the properties of eddies of this size range.

At the same time, turbulent flow induces droplet-droplet interaction, which is important for the coalescence mechanism. When two droplets collide, this may lead to coalescence, resulting in a bigger droplet size.

The coalescence and the break-up of droplets determine the droplet-size distribution in an oil-water mixture. If the energy dissipation rate is constant, shear forces acting on the fluid mixture balance the mechanism of coalescence and break-up, creating more less stable range of droplet sizes existing in the flow. In general, the higher the energy dissipation rate and shear forces acting on the fluid mixture, the smaller the average droplet size of the dispersed phase.

**Additional reading**

Kundu, P.K., Cohen, I.M., Dowling, D.R., 2012. Fluid Mechanics, 5th Ed. Academic Press.

Kolmogorov, A.N. 1941. Dissipation of energy in locally isotropic turbulence. Compt. Rend. Acad. Sci. USSSR, Vol. 32, No. 1

Brennen, C.E., 2005.Fundamentals of Multiphase Flows. Cambridge University Press.