RANS derivation and analysis

Content

1. Introduction
2. RANS Derivation
3. Findings

1.Introduction

SIGNIFICANCE OF TURBULENCE MODELLING:

Turbulence is a type of fluid flow that is unsteady, dissipative (in terms of energy), and diffusive (transport phenomenon) at high Reynolds numbers. Due to those divergences in turbulent flow, extremely small-scale fluctuations emerge in velocity, pressure, and temperature.

Turbulence modeling is the construction and use of a mathematical model to predict the effects of turbulence. Turbulent flows are commonplace in most real-life scenarios, including the flow of blood through the cardiovascular system, the airflow over an aircraft wing, the re-entry of space vehicles, besides others. In spite of decades of research, there is no analytical theory to predict the evolution of these turbulent flows. The equations governing turbulent flows can only be solved directly for simple cases of flow. For most real-life turbulent flows, CFD simulations use turbulent models to predict the evolution of turbulence. These turbulence models are simplified constitutive equations that predict the statistical evolution of turbulent flows.

Turbulent flow simulation methods

Direct Numerical Simulation

A direct numerical simulation (DNS) is a simulation in computational fluid dynamics in which the Navier-Stokes equations are numerically solved without any turbulence model. This means that the whole range of spatial and temporal scales of the turbulence must be resolved. All the spatial scales of the turbulence must be resolved in the computational mesh, from the smallest dissipative scales (Kolmogorov scales), up to the integral scale L, associated with the motions containing most of the kinetic energy.

he computational cost of DNS is very high, even at low Reynolds numbers. For the Reynolds numbers encountered in most industrial applications, the computational resources required by a DNS would exceed the capacity of the most powerful computer currently available. However, direct numerical simulation is a useful tool in fundamental research in turbulence. Using DNS it is possible to perform "numerical experiments", and extract from them information difficult or impossible to obtain in the laboratory, allowing a better understanding of the physics of turbulence. Also, direct numerical simulations are useful in the development of turbulence models for practical applications, such as sub-grid scale models for Large-eddy simulation (LES) and models for methods that solve the Reynolds-averaged Navier-Stokes equations (RANS). This is done by means of "a priori" tests, in which the input data for the model is taken from a DNS simulation, or by "a posteriori" tests, in which the results produced by the model are compared with those obtained by DNS. The biggest DNS in the world, up to this date, used 4096^{3} mesh points. It was carried out in the Japanese Earth Simulator supercomputer in 2002.

• DNS is possible but only for low Reynolds number flows (and simple geometries)
• The (time and space) details provided by DNS are NOT required for design purposes
• Time-averaged quantities are appropriate for engineering purposes
• Large scale resolution (not to the level of the smallest eddies) is enough for applications

RANS: (Reynolds Averaged Navier Stokes)

A mathematical model based on average values of variables for both steady-state and dynamic flows. The numerical simulation is driven by a turbulence model which is arbitrarily selected to find out the effect of turbulence fluctuation on the mean fluid flow.

RANS approach decomposes the instantaneous parameter into a Mean and an Averaged term and solves for ensemble-averaged velocity field and magnitude of turbulence fluctuations.

LES: (Large Eddy Simulation)

An average turbulence model between DNS and RANS in which filtered Navier-Stokes equations are used for large-scale eddies. An appropriate model is preferred to solve small-scale eddies. It solves for filtered tune dependant velocity field that represents large scale turbulent motion.

The principal idea behind LES is to reduce the computational cost by ignoring the smallest length scales, which are the most computationally expensive to resolve, via low-pass filtering of the Navier–Stokes equations. Such low-pass filtering, which can be viewed as a time- and spatial-averaging, effectively remove small-scale information from the numerical solution. This information is not irrelevant, however, and its effect on the flow field must be modeled a task which is an active area of research for problems in which small-scales can play an important role, such as near-wall flows, reacting flows, and multiphase flows.

2.RANS Derivation

Derivation of Reynolds Average Navier stokes

First, we convert Navier stokes equations into Reynolds Average Navier Stokes equations this is done when a governing equation is integrated with time greater than the turbulence time scale this process is called Averaging process. Then we apply Reynolds decomposition converts the set of obtained equations into a Reynolds Average Navier Stokes equation.

Let us go through the derivation

How Reynolds decomposition method is done here we are decomposing instantaneous velocity into mean and fluctuating components. These values are replaced in the Navier stokes equation.

"In order to prove the fluctuating term tends to zero"

We will start derivation by taking the continuity equation

Then Renolds Decomposition is applied where we are replacing instantaneous velocity with the mean and fluctuating velocity along with the X and Y component

Then we obtained the continuity equation with mean velocities.

Momentum Equation

Inertia Term=Pressure Term+Shear Stress

Now we will introduce the continuity equation in the left-hand side

As continuity equation is 0 it doesn't affect the equation

Applying Reynolds Decomposition for velocity and pressure terms

cancel out the fluctuating terms that tend to zero

Let us reduce the equation by assuming the below conditions

Along x-axis the velocity is too small, so we assume

Here v' is positive because u' is low-velocity particles at the solid boundary are transported upwards simply putting it u' became negative because low momentum fluid is moving into

higher momentum region because of that the even after time average the value doesn't tend to zero but it always remains negative for that reason.

After taking the above assumptions we get te below equation

Other Terms

3.Findings

Reynolds Stress

The mean forces (per unit area) imposed on the mean flow by turbulent fluctuations. They arise from the non-linear advection term when the Navier stokes equation is Reynolds decomposed and Reynolds averaged.

We consider a homogeneous fluid, whose density ρ is taken to be a constant. For such a fluid, the components τ'_ij of the Reynolds stress tensor are defined as:

Another often-used definition, for constant density, of the Reynolds stress components is:

It contains the terms of u' fluctuating velocity component in the X direction,v' fluctuating velocity component in the Y direction and Density

Turbulent Viscosity

The "eddy viscosity" is a flow property emerging from the increased momentum transport through the flow field due to turbulent velocity fluctuations simply we can put it as a coefficient used to achieve closure in the Reynolds equations for turbulent flow. The assumption is made that the Reynolds stresses are related to the velocity gradients of the flow by a viscosity analogous to the molecular viscosity, i.e. a turbulent or eddy viscosity.

How Turbulent Viscosity different from molecular viscosity

The utility of the analogy is strained by the fact that while the molecular viscosity is a property of the fluid, the eddy viscosity is a property of the flow. As such the specification of the eddy viscosity has more than a little of the air of the ad hoc about it since it is usually found via a trial-and-error procedure wherein it is varied until a numerically simulated flow reasonably replicates a known flow. The value thus obtained diagnostically is then used for prognostic simulations, a procedure that is questionable due to the above-mentioned fact of the eddy viscosity being a property of the flow rather than the fluid. That is, if the flow is remarkably different, then the eddy viscosity may also be remarkably different. The ability of eddy viscosity models to drain energy from large scales, thus simulating the dissipative nature of turbulence