Literature review RANS derivation and analysis
What is Navier stokes equations? and why are we dealing with them?
The movement of fluid in the physical domain is driven by various properties. For the purpose of bringing the behavior of fluid flow to light and developing a mathematical model, those properties have to be defined precisely as to provide transition between the physical and the numerical domain. 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.
All the governing equations are based on the laws of conservation. The Navier-Stokes equations are applied mathematical model to examine changes on those properties during dynamic and/or thermal 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
![\"Conservation](\"https://www.simscale.com/docs/_images/1-conservation-equations.jpg\")
Figure 1: Conservation Equation.
Turbulence flow
Turbulent flow is a type of fluid (gas or liquid) flow in which the fluid undergoes irregular fluctuations, or mixing, in contrast to laminar flow, in which the fluid moves in smooth paths or layers. In turbulent flow the speed of the fluid at a point is continuously undergoing changes in both magnitude and direction. we can determine whether a flow is turbulent or laminar by the Reynolds number.
There are three ways to describe the turbulent flow using mathematical models. they are.
- Direct numerical simulation.
- Large eddy simulation
- Reynolds average Navier-Stokes equations(RANS)
Direct numerical simulation is proved to be the most accurate of all the models to predict the behaviour of the turbulent flow however its downside is taking large amount of the computational power and time. Large eddy simulation has been in the development stage. Some research orgainzations are working on it and the results we got from this model are very promising.
Reynolds average Navier-Stokes equations(RANS)
In order to describe the behaviour of the turbulent flow Osborne Reynolds has came up with the concept of time averaging and decomposing the equations of the motion of the fliud flow. This method of decomposition is called Reynolds decomposition. Upon time averaging and decomposing we get 6 additional unknows along with the 4 variable unknows. These 6 unknows appear due to time averaging of momentum equations and because of this unknows we approach closure problem of turbulence. In order to get rid of these unknows there are certain models which can be used to model turbulence.
Reynolds decomposition
It is a process of separating the flow variables like velocity into mean component(time averaged) and a fluctuating component.
The momentary velocity components is u , the time-averaged value is named `baru` and the fluctuating velocity has the letter u\' . With help of this definition the decomposition can mathematically be written as:
Eg:- velocity w.r.t space and time = mean velocity w.r.t space and flucuating w.r.t space and time
Analogously for the density and the temperature will be considered constant.
The time-averaged values of the fluctuating values are defined to be zero:
This averaging method takes the mean values at a fix place in space and averaged over a time span that is large enough for the mean values to be independent of it.
Time average `baru` of the variable u is defined by
From the above equation we have derived some quations that can be used as the rules for the decomposition of the NS equations. Before writing the rules lets consider two variables A and B. `barA` and `barB` are mean variables. \'a\' and \'b\' be fluctuating components of the variables.
using above rules we will Split each instantaneous quantity into its averaged and fluctuating components,
continuity equation:-
upon splitting and writing down the whole equation we get.
and time averaging the above equation gives.
Understanding rules before transformation and reduction.
The averaged derivatives of the fluctuations are also zero according to these rules, so that the timeaveraged continuity equation is
Now the NS equations will be time-averaged. The averaging will be done for the x-component. Beforehand a small transformation of the advection term from
X-momentum equation:-
On expanding and splitting the equation we get
Applying the rules and reducing the equation we get:
Further small transformations, for example a repeated application of the product rule and the continuity equation to the advection term, lead to a form of the time-averaged NS equations for all three directions as:
In the tensor form:
In the above equation we know all the terms and we can derive the expresions for them. However for the last term of the above equation we have no idea how to deal with it. so to derive an expression for it we will focus on last two terms instead of only one to get an understanding and derive an equation
The expression in the brackets above corresponds to the total shear stress:
If we compare to the Navier-Stokes equations, it is evident that besides the viscous part an additional term has been added to the total shear stress. This term results from the time-average and is generally the dominant part of the total shear stress. Since the term only appears due to the Reynolds average, it is called Reynolds stress or apparent turbulent shear stress. This lead to the closure of the equation system, an approximation for the Reynolds stresses has to be done, which sets in relation the apparent shear stresses with the velocity field of the average flow.
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 explains what causes the internal friction.
where as 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 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 Theory. 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 RANS methods) are low-time consuming and can be used, sometimes massively, by computational fluid dynamicists to obtain approximate solutions of turbulent flows