Literature review - RANS derivation and analysis.

Aim

To apply Reynold's decomposition to the NS equations to get the RANS equations and expression for Reynold’s stress.
1. Understanding of the term Reynold's stress.
2. What is turbulent viscosity? How is it different from molecular viscosity?

Objective

The objective of this review is to apply the Reynold's decomposition process to Navier Stokes (NS) equations and derive an expression for Reynolds's stress and also to understand the difference between the turbulent viscosity and molecular viscosity.

Introduction

Types of fluid flow

Fluid flow can be split into three types-

• Laminar flow
• Transient flow
• Turbulence flow

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

Theory of turbulence

Turbulence is an unsteady, irregular motion in which transport quantities (mass, momentum, scalar species) fluctuates in space and time. It is in contrast to laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between those layers.

Turbulence is caused by excessive kinetic energy in parts of a fluid flow, which overcomes the damping effect of the fluid's viscosity. For this reason, turbulence is commonly realized in low viscosity fluids.

Turbulence analysis

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

Mean and fluctuating velocity and pressure.

Fluctuations can be of small scale and high frequency. They are computationally too expensive to simulate directly in practical engineering applications.
Instead, the instantaneous governing equations can be time-averaged, ensemble-averaged, or otherwise manipulated to remove the small scales, resulting in a modified set of equations that are computationally less expensive to solve. However, the modified equations contain additional unknown variables, and turbulence models are needed to determine these variables in terms of known quantities.

Turbulence modelling

Turbulence modelling is used to calculate Reynold’s stress and turbulent viscosity. The effect of turbulence can be modelled using a method called as Direct Numerical Simulation (DNS) which involves in solving Navier-Stokes equation over turbulent time step. But the turbulent time step is very small thus computational cell size is required to solve. Therefore, turbulence model is used to capture the effect of turbulence in coarse gird size.

Three different turbulence models used are: -

• Reynold’s Averaged Navier-Stokes equation (RANS)
• Large eddy simulation (LES)
• Direct Numerical Simulation (DNS)

Navier–Stokes Equations

The Navier-Stokes equations govern the motion of fluids and can be seen as Newton's second law of motion for fluids. The equations consist of solving the continuity equation, momentum equation and energy equation of a fluid together. They are of prime importance in fluid flow modelling.

• Continuity Equation:

• Momentum Equation:

• Energy Equation:

Reynold’s Averaged Navier-Stokes 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, those are Reynold’s averaging and filtering. Both methods introduce additional terms in the governing equations that need to be modelled in order to achieve “closure”. (Closure implies that there are a sufficient number of equations for all the unknowns.)

The rationale for Reynold’s averaging is that we are not interested in the part of 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 modelled. The approach of permitting a solution for the mean flow variables greatly reduces the computational effort.

1. To get RANS equations and Reynold’s stress expression by applying Reynold’s decomposition to the NS equations.

Reynold’s Decomposition

The Reynold’s averaged Navier-Stokes equations are time-averaged equations of motion for fluid flow. The idea behind the equation is Reynold’s decomposition whereby an instantaneous quantity is decomposed into its time-averaged and fluctuating quantities. For example, for a quantity in the decomposition would be,

where, the momentary velocity component is u(x,y,z,t), the time-averaged value is and the fluctuating velocity is u'(x,y,z,t).

To apply the Reynold’s decomposition to the Navier-stokes equation, we need to substitute the Reynold’s decomposition method by introducing time-averaged values and fluctuating values into the basic Navier-stokes equation.

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 the turbulent viscosity. The viscosity in the turbulent flow is non-homogenous, and it varies in space.

The turbulent transfer of momentum by eddies giving rise to an internal fluid friction, in a manner analogous to the action of molecular viscosity in laminar flow, but taking place on a much larger scale. It is greater for flows with more turbulence.

In a turbulent fluid, a linear interface between two different fluids breaks apart to form small-scale structures, also known as eddies. As these eddies grow and diminish in size (by energy cascades), they effectively alter the surface area of the interface between the fluids with different properties, thus altering the net transfer of momentum and scalar properties through the interfaces.

A computer performs a fluid dynamics simulation by numerical discretizing the Navier-Stokes equation over a domain that is split into grid boxes, as shown below.

Computer simulation of the surface of ocean, containing eddies.

The size of the grid boxes (also called resolution) determines the smallest scale of the eddies that the machine can resolve. In other words, if the grid size of the domain is 1 km, then the machine won’t be able to capture the eddies smaller than 1 km. Which means that the net effective momentum transfers due to small eddies occurring within that box, is unknown to the machine. In order to estimate the net effective mixing at each of these grid boxes, we specify an effective viscosity and multiply it with the shear (or velocity gradient) of the fluid surrounding the grid box. This effective viscosity is called turbulent viscosity.

The product of this turbulent viscosity and the shear gives us the net diffusive flux of fluid across the grid box. For momentum, that would be a diffusive momentum flux, and for temperature/density it would be a scalar flux.

• Molecular viscosity

It is a fluid property and is given by,

Molecular viscosity is the same as viscosity. The coefficient of molecular viscosity is the same value as dynamic viscosity (µ). 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 eddy viscosity in turbulent flow. It can be construed like some sort of internal friction between two layers of the fluid developed while flowing.

• Difference

1. The turbulent viscosity is an imaginary concept but molecular viscosity is a dynamic viscosity that will be present in all real fluids.
2. The molecular viscosity depends only on the properties of the fluid but the turbulent viscosity depends on the properties of fluid flow like velocity, pressure, Reynold’s number etc.
3. The turbulent viscosity will only present in turbulent flow regions whereas molecular viscosity is present in all types of flows.
4. In molecular viscosity the velocity of the fluid flow is low and it depends on the fluid viscosity but in turbulent viscosity the fluid flow velocity is high and its dependent on the fluid flow conditions.
5. Turbulent viscosity is dominant in regions with high Reynold’s number (i.e., turbulent regions) whereas molecular viscosity is dominant in the regions with low Reynold’s number (i.e., laminar flow regions).
6. Turbulent viscosity is non-linear and difficult to estimate whereas molecular viscosity is linear and easy to measure.

Conclusion

Hence using Reynold’s decomposition on Navier-Stokes equation the RANS equations were obtained along with the expressions of Reynold’s stress, molecular viscosity, pressure and inertial terms. Also, the concept of Reynold’s stress was studied and understood. The concept of turbulent viscosity and molecular viscosity is understood and difference between the two is studied as well.