Week 8: Literature review - 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 equation of boundary layers:

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.ln 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:

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 i s turbulent viscosity? How i s i t different from molecular viscosity?
What i s Turbulence?
In fluid dynamics, a turbulent regime refers to i rregular flows i n which eddies, swirls, and
flow i nstabilities occur. I t i s i n contrast to the l aminar regime, which occurs when a fluid
flows i n parallel l ayers, with no disruption between the l ayers. The turbulence regime i s
extremely frequent i n human applications.In terms of human applications, turbulent
regime occurs i n the aerodynamics of vehicles, but also i n many i ndustrial applications
such as heat exchangers etc.
Why do we need turbulence model?
To understand about turbulence model,lets take an example of an I C engine i n order to
captured the turbulence effect we need a computational cell around 1e-5m.Its i mpossible
to run a I C 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 i dea behind the equations i s
Reynolds decomposition, whereby an i nstantaneous quantity i s decomposed i nto i ts
time-averaged and fluctuating quantities, an i dea 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 i n a turbulent flow field i s shown i n above i mage. The velocity at
any i nstant can be divided i nto a mean component and i nstantaneous component.
The variation of velocity i n a turbulent flow field i s shown above. The velocity at any
instant can be divided i nto a mean component and i nstantaneous component.