Literature review RANS derivation and analysis

• Given Problem:-

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?

• Solution:-

NAVIER STOKES-

The movement of fluid in the physical domain is driven by various properties. For the purpose of bringing the behaviour of fluid flow to light and developing a mathematical model, those properties have to be defined precisely as to provide the transition between the physical and the numerical derivation. Velocity, pressure, temperature, density, and viscosity are the main properties that should be considered simultaneously when conducting a fluid flow examination. In accordance with the physical incidents such as combustion, multiphase flow, turbulent, mass transport, etc., those properties diversify enormously, which can be categorized into kinematic, transport, thermodynamic, and other properties.

It is relatively easy, for instance. to solve these equations for a flow between two parallel plates or for the flow in a circular pipe. For more complex geometries, however, the equations need to be solved. Depending on the flow regime of interest, it is often possible to simplify these equations. In other cases, additional equations may be required. In the field of fluid dynamics, the different flow regimes are categorized using a nondimensional number, such as the Reynolds number and the Mach number.

All the governing equations are based on the laws of conservation. The Navier-Stokes equations are applied a mathematical model to examine changes on those properties during dynamic and/or t hernial interactions. The equations are adjustable regarding the content of the problem and are expressed based on the principles of conservation of mass, momentum, and energy_

• Conservation of Mass: Continuity Equation
• Conservation of Momentum: Momentum Equation of Newton\'s Second Law
• Conservation of Energy: First Law of Thermodynamics or Energy Equation

The objective of this project is to review the available literature on turbulence models and to apply the Reynolds decomposition to Navier Stokes equation and derive the expression for Reynolds Stress. The turbulent viscosity term has been explained and the difference between turbulence viscosity and molecular viscosity is elucidated.

What is Turbulence?

Fluid flow can be split into three types:

1. Laminar Flow
2. Transitional Flow
3. Turbulent Flow

The type of flow can be determined using the Reynolds Number. A low Reynolds number corresponds to Laminar flow where the flow is in an orderly manner and parameters like pressure and velocity remain steady. A high Reynolds number corresponds to turbulent flow where the flow is chaotic and the parameters are fluctuating.

Turbulence Modelling:

The primary aim of turbulence modelling is to calculate Reynold\'s stress term and the turbulent viscosity. The effect of turbulence can be modelled by a method called Direct Numerical Simulation which involves solving the Navier-Stokes equation over the turbulent time step. But the turbulent time step is so small thus the computational cell size that will be required to solve the equation. Thus, a turbulence model is required to capture the effects of turbulence using a coarser grid.

Types of Turbulence Models:

There are three types of turbulence models:

1. Reynolds Averaged Navies Stokes Equation ANS)
2. Large Eddy Simulation (LES)
3. Direct Numerical Simulation (DNS)

Derivation of RANS equation-

Turbulent viscosity

Eddy viscosity or Turbulent viscosity is the proportionality factor describing the turbulent transfer of energy as a result of moving eddies, giving rise to tangential stresses. It is the fundamental idea of how we define viscosity in turbulent flow, i.e. internal fluid resistance_ turbulent viscosity expl sins what causes the internal friction.

Whereas Molecular viscosity is the same as 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 individual molecules not moving together in coherent groups. Molecular viscosity is analogous in laminar flow to turbulent viscosity in a turbulent flow.

Eddie Viscosity is an imaginary concept (also termed \'The Ansatz\' in turbulence)_ It does not exist as the molecular viscosity, which is a well-defined transport coefficient as stated by the Kinetic Th eery. The Eddie Viscosity hypothesis was posed for making the things simpler, in the sense that the turbulent Reynolds stresses which are nonlinear in velocity perturbations are simplified to be proportional to the gradients of the mean velocity, as happens in Newtonian laminar flows with the viscous stresses. The coefficient of proportionality is termed the Eddie Viscosity, which far from being a constant or fluid property, is a magnitude dependant on the flow field and its solution. The Eddie viscosity hypothesis is used in the numerical methods stemming from this simplification (such as FANS methods) are low-time consuming and can be used, sometimes massively, by computational fluid dynamicists to obtain approximate solutions of turbulent flows