RANS EQUATION

Reynolds Averaging

The Reynolds equation is derived in the Cartesian coordinate system. The flow is assumed to be incompressible & isentropic and is a function of x,y, z & t.

Turbulent flow is non-linear & difficult to solve.

In RANS we average with respect to the time parameter.

The velocity, pressure can be written as a function average & fluctuations

Generally, the following relations hold good

Averaging the velocity fluctuations

Reynolds Mass Continuity Equation

The mass conservation equation is

Applying RANS averaging in an incompressible, isotropic flow

Reynolds Momentum Equation

The Navier Stokes equation is

In an incompressible, isentropic scenario

Omitting the body force term & applying Reyonalds average in the above equation

The term accounts for the turbulent fluctuations.

From Eq. (2.2)

The total stress is given as

The Reynolds stress is denoted by the term in the Eq.(3.5).

Turbulent Viscosity

The Reyonalds sheer stress is defined as

The eddy viscosity or turbulent viscosity is

It has the units Pa.s.

The turbulent kinetic viscosity is

Physical Significance of Turbulent Viscosity

Let us consider the mixing of two fluids in 3 different scenarios

In case A there is a linear boundary, while in B&C there are curvilinear boundaries. The radius of curvature is small in case of B however, the radius of curvature is largest in the case of C.

Analyzing the rate at which the fluid mixes in all 3 scenarios.

Case A:-Due to the linear boundary, there is the low surface area the mixing is slowest. This especially seen in low-Reynolds number the laminar flow.

Case B:-Due to the curvature, the mixing rate is higher and the rate is even higher in the curved regions.

Case C:- Due to high curvature, the mixing rate is the highest. There is a development of swirls in the flow, ultimately leading to turbulence.

Molecular & Turbulent Viscosity

Molecular Viscosity

• The molecular viscosity is a fluid property and is given by

• It is the resistance to the movement of one layer of fluid over an adjacent layer of fluid.
• It is linear and is easy to measure.
• It is a fluid property and is independent of geometry.
• The flow is laminar & smooth.
• It is dominant in laminar flow regimes, where the Reynolds number is low.
• The velocity of flow is generally low & the molecular viscosity depends on the thickness of the fluid.

Turbulent Viscosity

From Eq. (4.1) turbulent viscosity is

• Turbulent viscosity is a function of fluctuations in velocity term and is influenced by randomness or disorderliness inflow.
• It is non-linear and difficult to estimate.
• It is dependent on flow domains such velocity, Reynolds number and is not a fluid property.
The flow is turbulent & is not smooth due to the presence of eddies & swirl.
• It is dominant in turbulent flow regimes, with high Reynolds number.
• The flow velocity is high and the turbulent viscosity depends on the velocity & flow conditions of the fluid.