• Turbulent motion is an irregular motion of fluid particles in a flow field. However, for homogeneous and isotropic turbulence, the flow field can be described by time-mean motions and fluctuating components. This is called Reynolds decomposition of turbulent flow.

• In a three dimensional flow field, the velocity components and the pressure can be expressed in terms of the time-averages and the corresponding fluctuations. Substitution of these dependent variables in the Navier-Stokes equations for incompressible flow and subsequent time averaging yield the governing equations for the turbulent flow. The mean velocity components of turbulent flow satisfy the same Navier-Stokes equations for laminar flow. However, for the turbulent flow, the laminar stresses are increased by additional stresses arising out  of the fluctuating velocity components. These additional stresses are known as apparent stresses of turbulent flow or Reynolds stresses.In analogy with the laminar shear stresses, the turbulent shear stresses can be expressed in terms of mean velocity gradients and a mixing coefficient known as eddy viscosity. The eddy viscosity (ν_t) can be expressed as , where l is known as Prandtl's mixing length.

• For fully developed turbulent duct flows at high Reynolds numbers, the velocity profile is given by

where is the time mean velocity at any and is the friction velocity given by . The constants A₁ and D₁ are determined experimentally. For the smooth pipes, A₁ and D₁ are 2.5 and 5.5 respectively. Corresponding friction factor, f is given by