Project

LITERATURE REVIEW ON RANS DERIVATION ANALYSIS

LITERATURE REVIEW ON RANS DERIVATION & ANALYSIS

Introduction:

The Reynolds-averaged Navier–Stokes equations (or RANS equations) are time-averaged equations of motion for fluid flow. The idea behind the equations is reynolds decomposition, whereby an instantaneous quantity is decomposed into its time-averaged and fluctuating quantities
The RANS equations are primarily used to describe turbulent flow.

Objective:

In this project we have to apply reynold\'s decomposition to the NS equations and come up with the expression for reynold\'s stress. And also explain the reynold\'s stress, turbulent viscosity, molecular viscosity.

Navier–Stokes Equations:

Reynolds Decomposition:

The basic tool required for the derivation of the RANS equations from the instantaneous Navier–Stokes equations is the Reynolds decomposition. Reynolds decomposition refers to separation of the flow variable into the mean and the fluctuating component

RANS Derivation:

Momentum Equation:

Applying the integral rule to the navier-strokes momentum equation,

Multiplying both sides by density (1/ρ),

Replacing the (μ/ρ) with the kinematic viscosity (γ),

Here we are adding the x-component of velocity multiplied by the continuity equation,

Combining both the x-components equations,

For further simplification, We are applying the product rule,

Applying the above value in the combined equation and simplifying it,

Now we are integrating the entire equation w.r.t. time (t) & also applying the reynold\'s decomposition,

Splitting the entire equation into multiple integrals,

Upon splitting we can eliminate the time average of the fluctuating components as its equal to zero,

Finally, since integration in time removes the time dependence of the resultant terms, the time derivative must be eliminated,

On further simplification we get the RANS equation as follows,

The below expression is the inertial term,

And its the pressure term,

And its the molecular viscosity or the viscous stress,

Finally its the momentum diffusivity due to turbulence and also known as the Reynold\'s stress,

Reynold\'s Stress:

In fluid dynamics, the Reynolds stress is the component of the total stress tensor in a fluid obtained from the averaging operation over the Navier–Stokes equations to account for turbulent fluctuations in fluid momentum.

Turbulent Viscosity:

The turbulent transfer of momentum by eddies giving rise to an internal fluid friction, in a manner analogous to the action of molecular viscosity in laminar flow, but taking place on a much larger scale. It is greater for flows with more turbulence.

How did it differ from Molecular Viscosity:

Molecular viscosity is the transport of mass motion momentum solely by the random motions of individual molecules not moving together in coherent groups.

The molecular viscosity is depending upon the fluid property but in the turbulent viscosity it is depending on the fluid flow like velocity, pressure, reynolds number etc.,
In molecular viscosity the velocity of the fluid flow is low and it depends on the fluid viscosity.
But in turbulent viscosity the fluid flow velocity is high and its dependent on the fluid flow conditions.

Ship work like this
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