introduction of Turbulent Flow
Introduction: low descriptions such as Poiseuille's law are valid only for conditions of laminar flow. At some critical velocity, the flow will become turbulent with the formation of eddies and chaotic motion which do not contribute to the volume flow rate. This turbulence increases the resistance dramatically so that large increases in pressure will be required to further increase the volume flow rate. Experimental studies have characterized the critical velocity for a long straight tube in the form
Turbulent Flow Modeling: The behavior of fluid flow is described by well-established partial differential equations – the Navier-Stokes equations - which are, essentially, particular forms of Newton’s laws of motion, supplemented by an equation describing the conservation of mass. Except for very simple conditions, these equations need to be solved numerically with the aid of computers (often super-computers). To this end, the predefined flow domain is covered by a numerical mesh, which defines nodes at mesh cross-sections and finite volumes or finite elements which are patches of area of volume cells around nodes or between consecutive mesh lines. The differential flow-governing equations are then approximated, using numerical discretization schemes, as sets of algebraic equations, each pertaining to a node, finite volume or finite element. The collection of coupled algebraic equation are then solved, by linear-algebra methods, on a computer to yield discrete values of velocity and pressure at mesh nodes. Additional transport equations may be solved, alongside the equations of motion and mass conservation, to calculate the temperature, species concentration, chemical reaction and multi-phase interactions (solid particles, bubbles, liquid droplets, etc.).
The collection of theoretical, numerical and computational techniques that facilitate this process is called Computational Fluid Dynamics.
Turbulence: In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic and stochastic  property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time. Nobel Laureate Richard Feynman described turbulence as "the most important unsolved problem of classical physics."
Flow in which the kinetic energy dies out due to the action of fluid molecular viscosity is called laminar flow. While there is no theorem relating the non-dimensional Reynolds number (Re) to turbulence, flows at Reynolds numbers larger than 5000 are typically (but not necessarily) turbulent, while those at low Reynolds numbers usually remain laminar. In Poiseuille flow, for example, turbulence can first be sustained if the Reynolds number is larger than a critical value of about 2040. Moreover, the turbulence is generally interspersed with laminar flow until a larger Reynolds number of about 3000. In turbulent flow, unsteady vortices appear on many scales and interact with each other. Drag due to boundary layer skin friction increases. The structure and location of boundary layer separation often changes, sometimes resulting in a reduction of overall drag. Although laminar-turbulent transition is not governed by Reynolds number, the same transition occurs if the size of the object is gradually increased, or the viscosity of the fluid is decreased, or if the density of the fluid is increased.
Features: Turbulence is highly characterized by the following features:
- Irregularity: Turbulent flows are always highly irregular. This is why turbulence problems are always treated statistically rather than deterministically. Turbulent flow is always chaotic but not all chaotic flows are turbulent.
- Diffusivity:The readily available supply of energy in turbulent flows tends to accelerate the homogenization (mixing) of fluid mixtures. The characteristic which is responsible for theenhanced mixing and increased rates of mass, momentum and energy transports in a flow is called "diffusivity".
- Dissipation: To sustain turbulent flow, a constant source of energy supply is required because turbulence dissipates rapidly as the kinetic energy is converted into internal energy by viscous shear stress.
- Energy cascade: Turbulent flow can be realized as a superposition of a spectrum of velocity fluctuations and eddies upon a mean flow. The eddies are loosely defined as coherent patterns of velocity, vorticity and pressure. Turbulent flows may be viewed as made of an entire hierarchy of eddies over a wide range of length scales and the hierarchy can be described by the energy spectrum that measures the energy in velocity fluctuations for each wave number. The scales in the energy cascade are generally uncontrollable and highly non-symmetric. Nevertheless, based on these length scales these eddies can be divided into three categories.
- Integral length scales: Largest scales in the energy spectrum. These eddies obtain energy from the mean flow and also from each other. Thus these are the energy production eddies which contain the most of the energy. They have the large velocity fluctuation and are low in frequency. Integral scales are highly anisotropic and are defined in terms of the normalized two-point velocity correlations. The maximum length of these scales is constrained by the characteristic length of the apparatus. For example, the largest integral length scale of pipe flow is equal to the pipe diameter. In the case of atmospheric turbulence, this length can reach up to the order of several hundred kilometers.