Literature review - RANS derivation and analysis

        Literature Review- Reynolds Averaged Navir Stokes(RANS) Derivation and Analysis

Aim:  To derive the Reynolds Averaged Navir Stokes(RANS) Equations.

Objective: To find the expressions for reynolds stress by applying Reynolds decomposition to the Navier-Stokes equations. Also understanding the difference between the turbulent viscosity and molecular velocity.

Literature Review:

The fluid flow is bascially divided into three types i.e. Laminar flow, Transient flow, and Turbulence flow. Generally the flow can be determined using Reynolds Number. A low Reynolds Number corresponds to laminar flow where the flow is in asequential manner and parameters such as pressure and velocity remains constant. A high Reynolds Number corresponds to turbulent  flow where the flow is chaotic and the paarameters are fluctuating.

Turbulence Modeling is the construction and use of a mathematical model to predict the effects of turbulence. Turbunce modelling is used to calculate the Reynolds stress and turbulent Viscosity. The effect of turbulence can be modelled using a method called Direct Numerical Simulations (DNS),where the Navier Stokes equation is solved over a turbulent time step. As the turbulent time step isvery small,  the computational cell size is required to solve. Hence, the turbulence model is used to capture the effect of turbulence in a coarse grid size.

The Navier Stokes equations govern the velocity and pressure of a fluid flow. Ina turbulent flow, each of these quantities may be decomposed into a mean part and a luctuating part. Averaging the equations gives the Reynolds Averaged Navier Stokes equations wich govern the mean flow. The non linearity of Navier Stokes equations means the velocity fluctuations appear in the RANS equations in the non linear term from the convective acceleration. This term is known as the Reynolds stress.

The Navier Stokes equation govern the motion of fluids and can be seen as Newton's second law of motion for fluids and can be seen as Newtons second law of motion for fluids. The equation consists of solving the contunity equation , momentum equation and energy equation of fluid together.This is important in the fluid flow modelling.

Governing Equations of boundary layer :

The Reynolds Average Navier Stokes equations are the time averaged equations of the motion for fluid flow. Here the quantity is decomposed into its time averaged and fluctuating quantities. Generally RANS equations are used to describe turbulent flows. These equations can be used with approximations based on the properties of the flow turbulence to give approximate time-averaged solutions to the Navier -Stokes equations. Turbulence Modeling is the construction and use of a mathematical model to predict the effects of turbulence. The Navier Stokes equations govern the velocity and pressure of a fluid flow. Ina turbulent flow, each of these quantities may be decomposed into a mean part and a fluctuating part. Averaging the equations gives the Reynolds Averaged Navier Stokes equations wich govern the mean flow. The non linearity of Navier Stokes equations means the velocity fluctuations appear in the RANS equations in the non linear term from the convective acceleration. This term is known as the Reynolds stress.In  fluid dynacims,the reynolds stress is the component of the the total stress tensor in a fluid obtained from the averaging operation over the Navier-Strokes  equation to account for turbulent fluctuations in fluid momentum. If we find frequency of the fluctuation and inverse that we can calculate Turbulent Time-scale. The time scale is small and occur over a very small distance. Using Turbulence Model we can capture the effect of turbulence but using courser grid and larger time-step. (i.e.) Taking actual governing equation s, integrate it over a time much larger than the turbulent time-scale. This is called as the Averaging Process. Now on applying Reynolds Decomposition, converts the original set of equations into the form of Reynolds Average Navier Stokes equation. Then use these equations along with turbulence model to simulate the turbulent flows. Here, we get unknown terms that should be molded.

RANS decomposes flow variables into mean and fluctuating terms where the mean component and the fluctuating component. Where, Mean component is the function space  and  Fluctuating component is the function of space and time.         

Reynolds Decomposition:

Time-Averaged Quantities:

As we are integrating, we get average quantities and these average quantities doesn't change with time. 

Hence , 

Here, when we integrate fluctuating terms for longer time period then the fluctuation tend to zero.

Now apply the Integral rule to the Contunity equation:-

Contunity Equation:   

Therefore, the Reynold's decomposition satisfies the contunity equation condition.

Momentum Equation:

 Reynolds Stress:

Reynold's stress is the component of the total stress tensor in a fluid obtained from the averaging operation over the Navier-Stokes equation to account for turbulent fluctuations in fluid momentum. This stress offers a significant effects of the complex interactions in the turbulent flows.In turbulent flow the Reynolds stresses are usually large compared to viscous stresses.  Reynolds stress provides the averaged effect of turbulent convection, which is highly diffusive.Reynolds stress tensor in the RANS equations representsa combination of mixing due to turbulent fluctuation and smoothing by averaging. The normal stresses are always non zero because they contain squared velocity fluctuations. The shear stresses would be zero if fluctuations wer statistically independent.

Turbulent Viscosity:

The turbulent viscosity hypothesis assumes that the Reynolds stresses can be related to the mean velocity gradients and turbulent  viscosity by gradient diffusion hypothesis, in a manner analogous to the relationship between the stress and strain tensors in laminar Newtonian flow. The turbulent viscosity is not homogeneous i.e. it varies in space. It is however assumed to be isentropic. It is same in all directions.

   Ina turbulent fluid, linear interface between different fluids breaks apart to form sim-scale structures which are called as eddies. As these grow and diminish the size, they effectively alter the surface area of the interface between the fluids with various properties, thus by altering the net transfer of momentum and scalar properties through the interfaces.

Molecular Viscosity:

 It isa resistance to the movement of one layerof fluid over the adjacent layer. It is linear and easy to measure.It is always independent of geometry. Molecular viscosity is the transport of mass motion momentum by random motions of individual molecules not moving together in coherent groups. Moleculer viscosity is analogous in laminar flow to eddy viscosity in turbulent flow. 

The turbulent viscosity is an imaginary concept but the molecular viscosity is a dynamic viscosity that is present in real fluids. Molecular viscosity depends on the properties of fluid but the turbulent viscosity depends on the fluid flow. Turbulent viscosity is dominant in regions with high Reynolds Number  whereas molecular velocity is dominant in the regions with low Reynolds number.Molecular Viscosity is linear and hence easy to measure.

Conclusion:

Hence by applying Reynold's decomposition to the NS equations, we have derived for the expression for Reynold's Stress. The theory behind the Reynold's stress is also studied.