Week 8: Literature review - RANS derivation and analysis

Objective:

To find the expressions for Reynolds stress by applying Reynolds decomposition to the Navier-Stokes equations.

Introduction

Turbulence Modeling is the construction and use of a mathematical model to predict the effects of turbulence. Turbulence 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. It is relatively easy, for instance, to solve these equations for a flow between parallel plates or for the flow in a circular pipe or annular flow pipe. For more complex geometries, however, the equations need to be resolved depending upon the flow regime of interest, it is often possible to simplify these equations. In other cases, additional equations and empirical correlations may be required. In the field of fluid dynamics, the different flow regimes are categorized using a non-dimensional number such as the Reynolds number and the Mach number.

As the turbulent time step is very small, 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 fluctuating part. Averaging the equations gives the Reynolds Averaged Navier Stokes equations which governs the mean flow. The non linearity of Navier Stokes equations means the velocity fluctuations appear in the RANS equations which is actually the  non linear term along with the term which is the convective acceleration. This term is known as the Reynolds stress.

All the governing equations are based on the 3 laws of conservation . The Navier- Stokes equations are applied to a turbulence model to evaluate the changes in the fluid parameters during dynamic or thermal interactions. The equations are expressed based on the principles of conservation of mass, momentum, and energy.

Reynolds-averaged Navier Stokes Equations (RANS) : Although statistical theory and numerical simulation (DNS) are viable options, most of the research on turbulent flow analysis in the past century has used the concept of time averaging. Applying the time averaging to the basic equations of motion yields the

Reynolds equations, which involve both mean and fluctuating quantities. One then attempts to model the fluctuation terms by relating them to mean properties or their gradients. This approach may now be yielding diminishing returns.

 Lumley (1989) gives a stimulating discussion of how time averaging might outlive its usefulness. The Reynolds equations are far from obsolete, however, and form the basis of most engineering analyses of turbulent flow.

Need of RANS:

Most of the engineering simulations need to be computationally effective (less computational effort). To capture the minute effects of turbulence DNS cannot be applied effectively (high computation effort required) and it requires the mesh size to be of order (1e-9 or less) to be in compliance with CFL number and small-time steps, it is going to be computationally inviable. Hence, to be computationally viable, engineers depend upon the time-averaged solutions, where the turbulence of engineering needs is captured effectively. Here the time step used is large compared to the relevant period of fluctuations of components. Hence a coarse grid than the one used for DNS can be used which will reduce the computational effort.

 Reynold’s Decomposition: Following the idea of Reynolds (1895), we assume that the fluid is in a random unsteady turbulent state and work with the time averaged or mean equations of motion. Any variable 'Q' is resolved into a mean value of plus a fluctuating value Q', where by definition

Instantaneous quantity at any point in space and time = Time-averaged Quantity + Fluctuating component. where T is large compared to the relevant period of fluctuations. The mean value itself may vary slowly with time as shown in the image below.

Let us consider only incompressible turbulent flow with constant transport properties but with possible significant fluctuations in velocity, pressure, and temperature.

Substituting averaged quantities in Navier Stokes Equation:

Which implies that the continuity equation after the substitution of averaged quantities depends on only mean values of decomposed quantities

From the above equation it is clear that the momentum equation obtained with mean values of components is similar to momentum equation of actual transport properties except with the red terms. If we consider the  value is fluctuating between -1 and 1 then by averaging it the value comes equal to 0 but if we consider the same case for then the square of -1 and 1 will be between 0 and 1 which when integrated will not give zero.

Hence,

Blue term~ changes w.r.t time

Green term~Convective term

Red term~ Reynold’s Stress term

Black term~Shear stress

Yellow Term~ Pressure Gradient

Reynolds stress is a component of stress tensor in fluid which is obtained by averaging operation over the Navier Stokes equation used to account for turbulent fluctuation in the fluid.

Turbulent Viscosity is a function of fluctuations in velocity and is influenced by the randomness of flow. It is non-linear and difficult to estimate. It depends upon flow regimes and is not a property. It is dominant in turbulent flow regimes with high Reynolds numbers.

Molecular Viscosity is a resistance to the movement of one layer of fluid over the adjacent layer. It is linear and easy to measure. It is the fluid property and independent of geometry.

Essentially, the ratio of turbulent to molecular viscosity gives an indication of how strong the Reynolds stresses are, as compared to molecular stresses. So it sounds reasonable to use this ratio as a measure of turbulence. Typically, an eddy viscosity ratio of more than 100 to 1000 indicates a turbulent flow. Although, the evolution of turbulence models has revolved around the definition of eddy viscosity, and hence for different models may have different values of it.

The red term in the last reduced equation denotes turbulent flux, blue term in the RHS denotes molecular flux, the green and the yellow term denotes viscous and turbulent stress dissipation respectively.

Conclusion:

The above RANS modeled NS-equations contain many new unknowns involving time-averaged quantities of fluctuating velocities and temperature. Therefore solutions cannot be attempted without additional empirical  equations and without knowledge of turbulence modeling . The above forms the basis for the modeling of any available time averaging turbulence models viz., (k-epsilon , k-omega etc.,) Hence by applying Reynold's decomposition to the NS equations, wecan derive the expression for RSM.

Below here is the attached word file drive link where the report was written in first hand

https://drive.google.com/file/d/1w4J5vc7uUgUWJ7LcJlzxdbk-O4Z1tOdQ/view?usp=sharing