Aim:- Apply reynold's decomposition to the NS equations and evaluate the expression for reynold's stress, also explain the term reynolds stress.

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

Introduction

Turbulence is an irregular motion of the air resulting from eddies and vertical currents. It may be as insignificant as a few annoying bumps or severe enough to momentarily throw an airplane out of control or to cause structural damage. Turbulence is associated with fronts, wind shear, thunderstorms, etc.

Characteristics of Turbulent Flow

• Turbulent flow tends to occur at higher velocities, low viscosity and at higher characteristic linear dimensions. If the Reynolds number is greater than Re > 3500, the flow is turbulent.
• Irregularity: The flow is characterized by the irregular movement of particles of the fluid. The movement of fluid particles is chaotic. For this reason, turbulent flow is normally treated statistically rather than deterministically.
• Diffusivity: In turbulent flow, a fairly flat velocity distribution exists across the section of pipe, with the result that the entire fluid flows at a given single value and drops rapidly extremely close to the walls. The characteristic which is responsible for the enhanced mixing and increased rates of mass, momentum and energy transports in a flow is called “diffusivity”.
• Rotationality: Turbulent flow is characterized by a strong three-dimensional vortex generation mechanism. This mechanism is known as vortex stretching.
• Dissipation: A dissipative process is a process in which the kinetic energy of turbulent flow is transformed into internal energy by viscous shear stress.

Turbulent Flow Velociy Profile

Turbulence Modeling

As we know the time scale is associated with turbulence is very less if we have navier stocks equation with very small time step and we have large number of computational cells then we dont need turbulent model . This process is quite impractile to attempt. That why we need to use turbulent modelling also we can capture the effect of turbulence even at coarser grid size and larger time step.

Three modeling frameworks :

1. Direct Numerical simulation(DNS)

2. Reynolds-Averaged navier-Stockes equations(RANS)

3. Large Eddy Simulation (LES)

Reynolds-averaged Navier-stokes equations

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 flow. 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-stokes equation.

Reynolds Decomposition

Instantaneous Velocity at any point in time and space = Time averaged Quantity + Fluctuating Component

In 3D flow,

`u(x,y,z,t)=overline u(x,y,z)+u prime(x,y,z,t)------1)`

`v(x,y,z,t)=overline v(x,y,z)+v prime(x,y,z,t)------2)`

`z(x,y,z,t)=overline z(x,y,z)+z prime(x,y,z,t)-------3)`

Time- Averaged Quantities:

`overline u(x,y) =1/(t_(i n t))*int_0^(t_(i n t)) u(x,y,z) dt`

The average quantity does not change with time

From equation--1)

`1/t_(i n t)*int_0^(t_( i n t)) u(x,y,z,t) dt= 1/t_(i n t)*int_0^(t_( i n t)) u(x,y,z) dt+ 1/t_(i n t)*int_0^(t_( i n t)) u prime(x,y,z,t) dt`

where `overline u` is not function of time

hence when we integrate fluctuating component its always we zero.

`1/t_(i n t)*int_0^(t_( i n t)) u prime (x,y,z,t) dt=0`

Hence we integrate fluctuating terms for longer time period this term would we zero.

Continuity Equation

`(du)/dx+(dv)/dy=0`

`u= overline u+uprime`

`v= overline v+vprime`

`d/dx(overline u+uprime)+d/dy(overline v+vprime)=0`

Taking time average both side

`1/t_(i n t)*int_0^(t _(i n t))d/dx[(overline u+uprime)dt]+ 1/t_(i n t)*int_0^(t _(i n t))d/dy[(overline v+vprime)dt]=0`

`1/t_(i n t)*int_0^(t _(i n t))d/dxoverline (u) dt+ 1/t_(i n t)*int_0^(t _(i n t))d/dxu prime dt+1/t_(i n t)*int_0^(t _(i n t))d/dyoverline (v) dt+1/t_(i n t)*int_0^(t _(i n t))d/dyv prime d=0`

Since time integral of fluctuating component is zero and integral limit is not a function of x.

`1/t_(i n t)*int_0^(t _(i n t))d/dxu prime dt=0`

`1/t_(i n t)*int_0^(t _(i n t))d/dyv prime d=0`

The above equation becomes

`1/t_(i n t)*int_0^(t _(i n t))d/dxoverline (u) dt+1/t_(i n t)*int_0^(t _(i n t))d/dyoverline (v) dt=0`

`1/t_(i n t)*d/dxoverline (u)int_0^(t _(i n t)) dt+1/t_(i n t)*d/dyoverline (v)int_0^(t _(i n t)) dt=0`

`d/dx(overline u)+d/dx (overline v)=0`

That means time average component satisfies continuity equation

Conservation of momentum

Conservation form of N-S equation is

`rho.((du)/dt+u(du)/dx+v(du)/dx)=-(dp)/dx+mu.(d^2u)/dy^2`

where,`nu=mu/rho`

Dividing `rho` both side

`((du)/dt+u(du)/dx+v(du)/dx)=-1/rho(dp)/dx+mu/rho.(d^2u)/dy^2`

`((du)/dt+u(du)/dx+v(du)/dx)+u((du)/dx+(dv)/dx)=-1/rho(dp)/dx+mu/rho.(d^2u)/dy^2`

`((du)/dt+u(du)/dx+v(du)/dx)+u((du)/dx+(dv)/dx)=-1/rho(dp)/dx+nu(d^2u)/dy^2`

We know that

`u((du)/dx+(dv)/dx)=0`

`(du)/dt+2u(du)/dx+v(du)/dy+u.(dv)/dy=-1/rho(dp)/dx+nu(d^2u)/dy^2`

`(du)/dt+2u(du)/dx+v(du)/dy+u.(dv)/dy=-1/rho(dp)/dx+nu(d^2u)/dy^2`

`(du)/dt+(du^2)/dx+d/dy(u*v)=-1/rho(dp)/dx+nu(d^2u)/dy^2`

Put, `u=overlineu+ u' & v= overlinev +v'`

Applying Reynolds Decomposition

`1/t_(i n t) int_0^(t_(i n t))[d/dt(overline u+u')+d/dx(overlineu+u')^2+d/dy(overlineu+u')*(overlinev+v')].dt=1/t_(i n t) int_0^(t_(i n t))[-1/rhod/dx(overlinep+p')+nu*d^2/dy^2(overlineu+u')]*dt`

Since time averaged of fluctuation term is zero, equation becomes

`1/t_(i n t)int_0^(t_( i n t))[d/dt(overlineu).dt+d/dtu'dt+d/dx overlineu^2dt+d/dxu'^2dt+2d/dx(overlineu*u')*dt+d/dy(overlineu*overlinev)dt+d/dy(overlineu*v')dt+d/dy(u'*overlinev)dt+d/dy(u'*v')dt]`

`=1/t_(i n t)int_0^(t_(i n t))[-1/rhod/dx(overlinep)dt-1/rhod/dxp'.dt+nud^2/dy^2(overlineu).dt+nud^2/dy^2(u').dt]`

Time average fluctuating component is zero, after cleaning the fluctuating component above equation becomes

`1/t_(i n t)int_0^(t_( i n t))[d/dt(overlineu).dt+d/dx overlineu^2dt+d/dxu'^2dt+d/dy(overlineu*overlinev)dt+d/dy(u'*v')dt] =1/t_(i n t)int_0^(t_(i n t))[-1/rhod/dx(overlinep)dt+nud^2/dy^2(overlineu).dt]`

`1/t_(i n t)int_0^(t_( i n t))[d/dt(overlineu).dt+d/dx overlineu^2dt+d/dy(overlineu*overlinev)dt] =1/t_(i n t)int_0^(t_(i n t))[-1/rhod/dx(overlinep)dt+nud^2/dy^2(overlineu).dt]-1/t_(i n t)int_0^(t_( i n t))d/dxu'^2dt-1/t_(i n t)int_0^(t_( i n t))d/dy(u'*v')dt`

Assume changes along X-direction is negligible compare to changes along Y- direction inside boundary layer hence

`1/t_(i n t)int_0^(t_(i n t))d/dxu'^2.dt=0`

Equation becomes

`1/t_(i n t)int_0^(t_( i n t))[d/dt(overlineu).dt+d/dx overlineu^2dt+d/dy(overlineu*overlinev)dt] =1/t_(i n t)int_0^(t_(i n t))[-1/rhod/dx(overlinep)dt+nud^2/dy^2(overlineu).dt]-1/t_(i n t)int_0^(t_( i n t))d/dy(u'*v')dt`

RHS of the above equation is

`RHS=-1/rhod/dx(overlinep)dt+nu(d^2overlineu)/dy^2.dt-1/t_(i n t)int_0^(t_( i n t))d/dy(u'*v')dt`

`RHS=-1/rhod/dx(overlinep)dt+1/rhod/dy[mu(doverlineu)/dy-rho/t_(i n t)int_0^(t_( i n t))(u'*v')dt`

`1/rho (doverlinep)/dx-----`pressure term

`mu.(doverlineu)/dy---------`Molecular viscocity

`rho/t_(i n t)int_0^(t_(i n t))(u'*v').dt-------`Momentum diffusivity.

Reynolds stress

The Reynolds stress is the component of the total stress tensor in a fluid obtained from the averaging operation over the navier stockes-equation to account for turbulent fluctuations in fluid momentum

We consider a incompressible fluid whose density`rho`is constant.

We consider a homogeneous fluid, whose density `rho` is taken to be a constant. For such a fluid, the components τ'_ij of the Reynolds stress tensor are defined as:

`tau_(i ,j=)rho.overline(u'.v')`

where `rho.overline(u'.v')` is average advection of u' & v'

Advection of momentum of velocity fluctuation results in a spread of momentum by turbulence.

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

Turbulent viscosity

The coefficient that relates the average shear stress and rate of velocity gradient in the turbulent flow is called Turbulent viscosity. The viscosity in turbulence is non homogeneous and it varies in space.

The relation between turbulent stress`(tau_(turb))` and turbulent viscosity(`mu_t`) is

`tau_(turb)=-rho*overline(u'*v')`

`tau_(turb)=mu_t*(du)/dy`

Where u' and v' are fluctuating component of velocity, `rho`is density of fluid ,`overline u` is mean velocityin the direction of flow and `mu`is the turbulent viscosity.

Turbulent viscosity is a turbulent transfer of momentum by eddies giving rise to internal fluid friction which is analogous to the action of molecular viscosity in laminar flow but takes place in much larger space.

Molecular viscosity is the same as the viscosity. The coefficient of molecular viscosity is the same value as dynamic viscosity. Molecular viscosity is the transport of mass motion momentum solely by the random motions of individuals molecules not moving together in coherent group.

The shearing stress of the turbulent flow may be written as in the following form:

`tau=tau_(laminar)+tau_(turb u l e n t)`

`tau= mu.(du)/dy+(-rho*overline(u'*v'))`

The first term of equation is the result of the viscous effect and the second term is the

result of the turbulence effect. If the flow is laminar u' and v' velocity flucations will be zero.