Literature review on RANS derivation

Objective:

• To apply reynold\'s decomposition to the NS equations and come up with the expression for reynold\'s stress
• Explain your understanding of the terms reynold\'s stress
• What is turbulent viscosity? How is it different from molecular viscosity?

What is Turbulence?

In fluid dynamics, a turbulent regime refers to irregular flows in which eddies, swirls, and flow instabilities occur. It is in contrast to the laminar regime, which occurs when a fluid flows in parallel layers, with no disruption between the layers. The turbulence regime is extremely frequent in human applications.In terms of human applications, turbulent regime occurs in the aerodynamics of vehicles, but also in many industrial applications such as heat exchangers etc.

Why do we need turbulence model?

To understand about turbulence model,lets take an example of an IC engine in order to captured the turbulence effect we need a computational cell around 1e-5m.Its impossible to run a IC engine Simulation which having 1e-5 mesh size.Here Turbulence model comes into picture,turbulence model will captured the turbulence effect with the coarses grid.In order to obtain turbulence model there two approche which discribed below

Turbulence Modelling Approche

• Reynols-Averaged Navier Strokes (RANS)
• Large Eddy Simualtion (LES)

In our case we will discuss about the Reynols-Averaged Navier Strokes (RANS)

RANS: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, an idea first proposed by Osborne Reynolds The RANS equations are primarily used to describe turbulent flows. These equations can be used with approximations based on knowledge of the properties of flow turbulence to give approximate time-averaged solutions to the Navier Strokes Equation.

The variation of velocity in a turbulent flow field is shown in above image. The velocity at any instant can be divided into a mean component and instantaneous component.

The variation of velocity in a turbulent flow field is shown above. The velocity at any instant can be divided into a mean component and instantaneous component.

`u underset(i)()(x,y,z,t)=u underset(m)()(x,y,z)+u underset(f)()(x,y,z,t)`

`v underset(i)()(x,y,z,t)=v underset(m)()(x,y,z)+v underset(f)()(x,y,z,t)`

`w underset(i)()(x,y,z,t)=w underset(m)()(x,y,z)+w underset(f)()(x,y,z,t)`

This process is called Reynold’s Decomposition.
The subscript m indicates mean component and f indicates fluctuating component.

The mean component can be given as:

`u underset(m)()(x,y,z)=frac{1}{t underset(Int)()}int_0^(t underset(Int)())u(x,y,z,t)dt`

It is the time average of the instantaneous velocity over the turbulence time step.
The fluctuating component is:

`frac{1}{t underset(Int)()}int_0^(t underset(Int)())u underset(f)()(x,y,z,t)dt=0`

Now,we will apply Reynolds decomposition to Navier strokes equation which as follows

Continuity Equation

`frac{delu}{delx}+frac{delv}{dely}`

Applying Reynolds decomposition to continuity equation

`frac{del(u underset(m)()+u underset(f)())}{delx}+frac{del(v underset(m)()+v underset(f)())}{dely}=0`

Integrating over the time interval

`frac{1}{t underset(Int)()}[int_0^(t underset(Int)() )frac{del(u underset(m)()+u underset(f)())dt}{delx}+int_0^(t underset(Int)() )frac{del(v underset(m)()+v underset(f)())dt}{dely}]=0`

Splitting the integrals

`frac{1}{t underset(Int)()}int_0^(t underset(Int)() )frac{del(u underset(m)())dt}{delx}+frac{1}{t underset(Int)()}int_0^(t underset(Int)() )frac{del(u underset(f)())dt}{delx}+frac{1}{t underset(Int)()}int_0^(t underset(Int)() )frac{del(v underset(m)())dt}{delx}+frac{1}{t underset(Int)()}int_0^(t underset(Int)() )frac{del(v underset(f)())dt}{dely}=0`

Since,the integral of fluctuating component is zero.Hence we can write above equation as

`frac{1}{t underset(Int)()}int_0^(t underset(Int)() )frac{del(u underset(m)())dt}{delx}+frac{1}{t underset(Int)()}int_0^(t underset(Int)() )frac{del(v underset(m)())dt}{delx}=0`

Which is nothing but

`frac{del(u underset(m)())}{delx}frac{1}{t underset(Int)()}int_0^(t underset(Int)()}dt+frac{del(v underset(m)())}{delx}frac{1}{t underset(Int)()}int_0^(t underset(Int)()}dt=0`

Thus the continuity equation can be simplified as,

`frac{del(u underset(m)())}{delx}+frac{del(v underset(m)())}{dely}=0`

Momentum Equation

`rho(frac{delu}{delt}+ufrac{delu}{delx}+vfrac{delu}{dely})=-frac(delp}{delx}+mufrac{del^2u}{delx^2}`

Taking density term into other side we get,

`frac{delu}{delt}+ufrac{delu}{delx}+vfrac{delu}{dely}=-frac{1}{rho}frac(delp}{delx}+nu frac{del^2u}{delx^2}`

`frac{delu}{delt}+ufrac{delu}{delx}+vfrac{delu}{dely}+u(frac{delu}{delx}+frac{delv}{dely})=-frac{1}{rho}frac(delp}{delx}+nu frac{del^2u}{delx^2}`

`frac{delu}{delt}+2ufrac{delu}{delx}+vfrac{delu}{dely}+ufrac{delv}{dely}=-frac{1}{rho}frac(delp}{delx}+nu frac{del^2u}{delx^2}`

As we know,

`frac{del(uv)}{dely}=ufrac{delv}{dely}+vfrac{delu}{dely}`

`frac{delu}{delt}+2ufrac{delu}{delx}+frac{del(uv)}{dely}=-frac{1}{rho}frac(delp}{delx}+nu frac{del^2u}{delx^2}`

Simplifying above equation we get,

`frac{delu}{delt}+frac{delu^2}{delx}+frac{del(uv)}{dely}=-frac{1}{rho}frac(delp}{delx}+nu frac{del^2u}{delx^2}`

Now,applying Reynolds decomposition into above equation and integrating it over time interval

`frac{1}{t underset(Int)())int_0^(tunderset(Int)()) frac{del(u underset(m)()+u underset(f)())dt }{delt}+frac{1}{t underset(Int)())int_0^(tunderset(Int)()) frac{del(u underset(m)()+u underset(f)())^2}{delx}dt+frac{1}{t underset(Int)())int_0^(tunderset(Int)()) frac{del(u underset(m)()+u underset(f)())(v underset(m)()+v underset(f)())}{dely}dt=-frac{1}{t underset(Int)())int_0^(tunderset(Int)()) frac{1}{rho}frac(del(p underset(m)+p underset(f)())}{delx}dt+nu frac{1}{t underset(Int)())int_0^(tunderset(Int)()) frac{del^2(u underset(m)()+u underset(f)())}{delx^2}dt`

Now by neglecting the integrals of fluctuating terms and by time integrating the rest of the terms we get the equation

`frac{delu underset(m)()}{delt}+frac{delu^2 underset(m)()}{delx}+frac{del(u underset(m)()v underset(m)())}{dely}=-frac{del p underset(m)()}{rho delx}+nu frac{del^2 u underset(m)()}(dely^2}-frac{1}{t underset(Int)()}int_0^(t underset(Int)())(frac{delu^2 underset(f)()}{delx}+frac{del(u underset(f)()v underset(f)())}{dely}dt)`

Inside the boundary layer the change in velocity across the x-axis, i.e. the velocity gradient, is not as large as the yaxis and hence the square of x-velocity can be taken as zero. Using this assumption and by simplifying the above equation we get

`frac{delu underset(m)()}{delt}+frac{delu^2 underset(m)()}{delx}+frac{del(u underset(m)()v underset(m)())}{dely}=-frac{del p underset(m)()}{rho delx}+frac{1}{rho}frac{del}{dely}(mufrac{delu underset(m)()}{dely}+frac{rho}{t underset(Int)()}int_0^(t underset(Int)())(u underset(f)() v underset(f)())dt)`

Momentum diffusivity consists of two components which is,

1.Molecular viscosity i.e.`mufrac{delu underset(m)()}{dely` is also called as viscous stress.

2.Momentum diffusivity due to turbulence i.e `frac{rho}{t underset(Int)()}int_0^(t underset(Int)())(u underset(f)() v underset(f)())dt` is called as Reynolds stress.

Reynolds stress is the component of total stress tensor obtained after doing averging operation of Navier strokes equation to account for turbulent fluctuation in fluid.And to solve this reynolds stress term their are two turbulence model are available which are k-e & K-w

What is turbulent viscosity?How is it different from molecular viscosity?

Turbulent viscosity is also called as Eddy viscosity which occur in turbulence.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.Eddy viscosity is a function of the flow, not of the fluid.

Molecular viscosity is same as viscosity.The coefficient of molecular viscosity is same as dynamic viscosity.Molecular viscosity is transport of mass motion momentum solely by the random motions of individual molecules not moving togther in coherent groups.Molecular viscosity is analogous in laminar flow to eddy viscosity in turbulent flow.

The molecular viscosity is microscopic property of fluid.And the eddy viscosity is the flow property emerging from the increased momentum transport through the flow field due to turbulent velocity fluctuations.