Literature review on the Reynolds-averaged Navier Stokes derivation and analysis

Objective:

To apply Reynold\'s decomposition to the NS equations to get the RANS Equations

• Explain your understanding of the terms Reynold\'s stress
• What is turbulent viscosity? How is it different from molecular viscosity?

Theory for the turbulence:

The term \"turbulence” is used for a large variety of dynamical phenomena of fluids in motion whenever the details of the flow appear to be random and average properties are of primary interest. Just as wide-ranging are the theoretical methods that have been applied toward a better understanding of fluid turbulence.

• Turbulence is looked at as unsteady, irregular motion in which transport quantities (mass, momentum, scalar species) fluctuated in space and times. It is in contracts to laminar flows, which occur when a fluid flows in parallel layers, with no disruption between those layers.
• Turbulence is caused by excessive kinetic energy in the part of fluid flow, which overcomes the damping effect of the viscosity of the fluid. For this reason, turbulence is commonly realized in low viscosity fluids.

Properties associated with Turbulence :

1.Irregularity:

Irregular and random in nature, Requires statistical methods to resolve.

2.Diffusivity:

A measure of the capability of a substance or energy to be diffused or to allow something to pass by diffusion.Rapid mixing of heat, mass, momentum, etc. One of the most important properties

3.Large Re:

A large Reynolds number indicates that viscous forces are not important at large scales of the flow. With a strong predominance of inertial forces over viscous forces, the largest scales of fluid motion are undamped there is not enough viscosity to dissipate their motions. Turbulence always occurs at a high Re number. It often results from the instability of laminar flow as Re becomes too large.

4.3D vorticity fluctuation:

Rotational and 3D, characterized by a high level of fluctuating vorticity. Vorticity dynamics play a central role in the analysis.

5.Dissipation:

Turbulence is always dissipative, Viscous shear stresses perform deformation work at the expense of kinetic energy of the flow.

6.Continuum:

Governed by the equations of fluid mechanics.

7.Flows:

Turbulent flows are flows who characterized by properties of flows not fluids.

Methods of Analysis :

Let\'s take a look at instantaneous velocity measurements in turbulent flows

Mean and fluctuating turbulent velocity and pressure.

Fluctuations can be of small scale and high frequency. These are too expensive to simulate for practical engineering applications directly.

Additionally, instantaneous regulated equations can be combined or otherwise manipulated in order to eliminate small scales, which results in a changing set of equations which are less costly to solve computationally. Nevertheless, these modified equations have additional unknown variables and turbulence models in known quantities are needed to evaluate these variables.

Turbulence modeling:

Turbulence modeling is the construction and use of a mathematical model to predict the effects of turbulence. The equations governing turbulent flows can only be solved directly for simple cases of flow. For most real-life turbulent flows, CFD simulations use turbulent models to predict the evolution of turbulence.

Since we know the turbulence has a very low time-scale with N-S equations and we have a large number of estimated cells. we don\'t need a turbulence model. This process is quite impractical to attempt. That\'s why we need turbulence modeling in which even at total grid size and time steps we can capture the impact of turbulence.

The task of turbulence modeling:

We are trying to find approximate simplified solutions for the Navier stokes equations in the manner that either describes turbulence in terms of mean properties of limits the spatial/ temporal resolution requirements associated with the full model.

There are three modeling frameworks:

1. Direct Numerical Simulations(DNS).
2. Reynolds-Averaged Navier-stroke Equations(RANS)
3. Large Eddy Simulations(LES)

Reynolds Averaged Navier-Stroke Equations (RANS)

Two alternative methods can be employed to transform the Navier stokes equations in such a way that the small scale turbulent fluctuations do not have to be directly simulated. Reynolds averaging and filtering

Both methods introduce additional terms in the governing equations that need to be modeled in order to achieve \"Closure\".

(Closure implies that there are a sufficient number of equations for all the unknowns.)

The rationale of r Reynolds averaging is that we are not interested in the part of the flow solution that can be described as \"Turbulent fluctuations\" instead it is the mean (Velocity, pressure, lift, drag) that is of interest looking at turbulent flow it may best steady in the mean in spite of turbulent fluctuations.

RANS equations represent transport equations for the mean flow quantities only, with all the scales of the turbulence being modeled. The approach of permitting a solution for the mean flow variables greatly reduces the computational effort.

Reynold\'s Decomposition:

Instantaneous Velocity @ any point in Time and Space = Time Averaged Quantity + Fluctuating Component.

In 3-Dimensional Flows,

`u(x,y,z,t)=baru(x,y,z,t)+u\'(x,y,z,t) rightarrow 1`

`v(x,y,z,t) = barv(x,y,z,t)+v\'(x,y,z,t) rightarrow 2`

`w(x,y,z,t) = barw(x,y,z,t)+w\'(x,y,z,t) rightarrow 3`

Time-Averaged Quantities:

`baru(x,y) = 1/T*int_0^(T) u(x,y)dt rightarrow4`

Time averaging is appropriate when considering a stationary turbulence that is when the flow does not vary on the average in Time.

When Equation-4 is applied to equation-1 then,

`1/T*int_0^(T)u(x,y,z,t)dt=1/T*int_0^(T)baru(x,y,z,t)dt+1/T*int_0^(T)u\'(x,y,z,t)dt`

Instantaneous velocity

`1/T*int_0^(T)u(x,y,z,t)dt`

Time-averaged Quantity

`1/T*int_0^(T)baru(x,y,z,t)dt`

Fluctuating Quantity

`1/T*int_0^(T)u\'(x,y,z,t)dt`

when we integrate fluctuating terms for the longer time period this term would be zero.

Therefore, it is also assumed that the average of fluctuating quantities is zero.

`1/T*int_0^(T) u\'(x,y,z,t)dt =0rightarrow(a)`

Continuity Equations:

Instantaneous Velocities u,v,w at a point can be written as

`u =baru+u\'`

`v=barv+v\'`

`w=barw+w\'`

`P =barP+P\'`

Continuity Equations:

`(delu)/(delx)+(delv)/(dely)=0rightarrow5`

Substitute instantaneous velocities in Equations-5

`(del(baru+u\'))/(delx)+(del(barv+v\'))/(dely)=0`

Applying time-averaging for the above equations

`1/T*int_0^Tdel/(delx)(baru+u\')*dt+1/T*int_0^Tdel/(dely)(barv+v\')*dt=0`

`1/Tint_0^T(delbaru)/(delx)*dt+1/Tint_0^T(delu\')/(delx)*dt+1/Tint_0^T(delbarv)/(dely)*dt+1/Tint_0^T(delv\')/(dely)*dt =0`

According to equation-(a),time-averaging of the fluctuating component is zero.

`1/Tint_0^T(del(baru))/(delx)*dt+1/Tint_0^T(del(barv))/(dely)*dt = 0`

`(delbaru)/(delx)*1/Tint_0^T*dt+(delbarv)/(dely)*1/Tint_0^T*dt=0`

`(delbaru)/(delx)*1/T(T)_0^T+(delbarv)/(dely)*1/T(T)_0^T=0`

`(delbaru)/(delx)*1/cancel(T)(cancel(T)-0)+(delbarv)/(dely)*1/cancel(T)(cancel(T)-0)=0`

`(delbaru)/(delx)+(delbarv)/(dely)=0`

Therefore, the continuity equation satisfy the time-averaged flow and unsteady velocity fluctuations

`(delbaru)/(delx)=0 , (delbarv)/(dely)=0`

`(delu\')/(delx)=0 , (delv\')/(dely)=0`

Navier-Stokes equations:

X-Momentum Equations

`(delu)/(delt)+(del(u)(u))/(delx)+(deluv)/(dely)+(deluw)/(delz)=-(1/rho)*(delP)/(delx)+nu[(del^2u)/(delx^2)+(del^2u)/(dely^2)+(del^2u)/(delz^2)]rightarrow6`

Substitute instantaneous velocities in Equations-6, that is,

`u= baru+u\'`

`v=barv+v\'`

`w=barw+w\'`

`P=barP+P\'`

Then

`(delbaru)/(delt)+(delbarubaru)/(delx)+(delu\'u\')/(delx)+(delbarubarv)/(dely)+(delu\'v\')/(dely)+(delbarubarw)/(delz)+(delu\'w\')/(delz)=-(1/rho)(delbarP)/(delx)+nu[(del^2baru)/(delx^2)+(del^2baru)/(dely^2)+(del^2baru)/(delz^2)+cancel((del^2u\')/(delx^2))+cancel((del^2u\')/(dely^2))+cancel((del^2u\')/(delz^2))]`

`(delbaru)/(delt)+(delbarubaru)/(delx)+(delu\'u\')/(delx)+(delbarubarv)/(dely)+(delu\'v\')/(dely)+(delbarubarw)/(delz)+(delu\'w\')/(delz)=-(1/rho)(delbarP)/(delx)+nu[(del^2baru)/(delx^2)+(del^2baru)/(dely^2)+(del^2baru)/(delz^2)]`

`(delbaru)/(delt)+(delbarubaru)/(delx)+(delbarubarv)/(dely)+(delbarubarw)/(delz)=-(1/rho)(delbarP)/(delx)+nu[(del^2baru)/(delx^2)+(del^2baru)/(dely^2)+(del^2baru)/(delz^2)]-(delu\'u\')/(delx)-(delu\'v\')/(dely)-(delu\'w\')/(delz)rightarrow7`

Applying Reynolds stress tensor in Equation-7

`tau_(i,j) = -rho*[[bar(u\'^2),bar(u\'v\'),bar(u\'w\')],[bar(v\'u\'),bar(v\'^2),bar(v\'w\')],[bar(w\'u\'),bar(w\'v\'),bar(w\'^2)]]`

Therefore

Reynolds stress tensors

`tau_(x,x) = -rho[u\'u\']=>u\'u\'=-tau_(x,x)/rho`

`tau_(x,y) = -rho[u\'v\']=>u\'v\'=-tau_(x,y)/rho`

`tau_(x,z) = -rho[u\'w\']=>u\'w\'=-tau_(x,z)/rho`

Then Equation-7 becomes, the RANS equation

`(delbaru)/(delt)+(delbarubaru)/(delx)+(delbarubarv)/(dely)+(delbarubarw)/(delz)=-(1/rho)(delbarP)/(delx)+nu[(del^2baru)/(delx^2)+(del^2baru)/(dely^2)+(del^2baru)/(delz^2)]+(1/rho)[(del(tau_(x,x)))/(delx)+(del(tau_(x,y)))/(dely)+(del(tau_(x,z)))/(delz)]`

Finally we uptain the Reynolds-averaged Navier–Stokes (RANS) formulation.

Reynold\'s stress:

It results from turbulent transport of momentum. The momentum flux is equivalently stress as it causes deformation of fluid parcels, It is not because of the existence of real tangential force (as in viscous stress). It is the property of flow.

We consider a homogeneous fluid, whose density `rho` is taken to be constant.

For such a fluid, the components `tau_(ij)` of the Reynolds stress tensor is defined as

`tau_(i,j) = -rho*bar(u\')*bar(v\')`

where

`-rho*bar(u\')*bar(v\')`

It is Average advection of u\' and v\'

Advection of momentum or velocity fluctuations results in a spread of momentum by turbulence. This term describes the diffusive nature of turbulence.The negative divergence of the Reynolds stresses represents convection momentum transfer due to the random motion of macroscopic fluid packets(Eddies)

Reynolds stress tensor

To describe Reynold\'s Stress

`tau_(i,j) = -rho*[[bar(u\'^2),bar(u\'v\'),bar(u\'w\')],[bar(v\'u\'),bar(v\'^2),bar(v\'w\')],[bar(w\'u\'),bar(w\'v\'),bar(w\'^2)]]`

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 nonhomogeneous and it varies in space.

The relation between the

Turbulent stress:

`tau_(turb)`

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

Turbulent viscosity:

`mu_(turb)`

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

where

u\' and v\' are fluctuating components of velocity.

rho is the density of the fluid.

mu is the turbulent viscosity.

u bar is the mean velocity.

The 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 viscosity. The coefficient of molecular viscosity is the same value as dynamic viscosity. Molecular viscosity motions of individual molecules not moving together in coherent groups.

In general, the shearing stress of the turbulent flow may be written as in the following form.

`tau = tau_(lam)+tau_(turb)`

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

The first term of equations 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 fluctuations will be zero.