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 turbulent viscosity and molecular velocity.

Theory:

Reynolds Number: The Reynolds number is the ratio of inertia force to viscous force within a fluid that is being subjected to relative internal movement due to different fluid velocities, which is known as a boundary layer in the case of a bounding surface such as the interior of a pipe.

Reynolds Decomposition: Reynold decomposition is a mathematical technique to spilt the flow variables into main components and fluctuating components.

Types of fluid flow:

The fluid flow is basically divided into three types:

• Laminar flow
• Transient flow
• Turbulence flow

The flow can be determined using Reynolds Number. A low Reynolds Number corresponds to laminar flow where the flow is in a sequential manner and parameters such as pressure and velocity remain constant. A high Reynolds Number corresponds to turbulent flow where the flow is chaotic and the parameters are fluctuating.

Turbulence modeling: Turbulence modeling is used to calculate Reynolds stress term and Turbulent viscosity. The effect of turbulence can be modeled using a method called Direct Numerical Simulation(DNS) which involves solving the Navier-Stokes equation over a 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 size.

Three different turbulence models used are:

• Reynold Averaged Navier Stokes equation (RANS)
• Large eddy simulation(LES0
• Direct Numerical Simulation(DNS)

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

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 meaningful part and a fluctuating part. Averaging the equations gives the Reynolds Averaged Navier Stokes equations which govern the mean flow. The nonlinearity 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 dynamics, the Reynolds stress is the component of 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 the frequency of the fluctuation and inverse we can calculate the Turbulent Time-scale. The time scale is small and occurs over a very small distance. Using Turbulence Model we can capture the effect of turbulence but using a coarser grid and a larger time-step. (i.e.) Taking actual governing equations, integrate them over a time much larger than the turbulent time scale. This is called 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 a 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 the Mean component is the function of 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 don't change with time. 

Hence, 

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

Now apply the Integral rule to the Continuity equation:

Continuity 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 significant effects on the complex interactions in 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 represents a combination of mixing due to turbulent fluctuation and smoothing by averaging. The normal stresses are always nonzero because they contain squared velocity fluctuations. The shear stresses would be zero if fluctuations were 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 the 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 the same in all directions.

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

Molecular Viscosity:

It is resistant to the movement of one layer of 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. Molecular viscosity is analogous in laminar flow to eddy viscosity in a turbulent flow. 

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

Conclusion:

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