Week 8: Literature review - RANS derivation and analysis

Objective

To apply Reynolds decomposition to the Navier Stokes equation and to determine the expression for Reynolds stress.

 

Navier Stokes Equations

The movement of fluid in the physical domain is driven by various properties. For the purpose of bringing the behaviour of fluid flow to light and developing a mathematical model, those properties have to be defined precisely as to provide the transition between the physical and the numerical derivation. Velocity, pressure, temperature, density and viscosity are the main properties that should be considered simultaneously when conducting the fluid flow examination. In accordance with the physical incidents such as combustion, multiphase flow, turbulence, mass transport etc those properties diversify enormously. They can be categorized into kinematic, transport, thermodynamic and other properties.

It is relatively easy to solve these equations for a flow between parallel plates or for the flow in a circular 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 regimes 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.

All the governing equations are based on the laws of conservation. The Navier Stokes equations are applied to a mathematical model to examine changes in those properties during dynamic and thermal interactions. These equations are adjustable regarding the content of the problem and are expressed based on the principles of conservation of mass, momemtum and energy.

Note - All equations are in a 2D cartesian coordinate system

 

Govering equations for incompressible flow

Continuity equation - Conservation of mass

$\frac{`(\partial u)/(\partial x)+(\partial v)/(\partial y)=0`}{}$

 

X - momemtum equation - Conservation of momentum - Newton's second law

$\frac{1}{T}\underset{0}{{\int }^T}[\frac{\partial [\overline{u}+u\text{'}]}{\partial t}+2\frac{\partial {[\overline{u}+u\text{'}]}^2}{\partial x}+\frac{\partial [\overline{u}+u\text{'}][\overline{v}+v\text{'}]}{\partial y}]=\frac{1}{T}\underset{0}{{\int }^T}[-\frac{1}{\rho }\frac{\partial P}{\partial x}+\gamma \frac{{\partial }^{2}u}{\partial y^2}]$

 

Energy/Transport equation - Conservation of energy (First law of thermodynamics)

$`\rho C\_P[(\partial \varphi )/(\partial t)+u(\partial \varphi )/(\partial x)+v(\partial \varphi )/(\partial y)]=K[\; (\partial ^2\varphi )/(\partial x^2)+(\partial ^2\varphi )/(\partial y^2)]+\mu \; ((\partial u)/(\partial y))^2`{}^{}$

 

 

Turbulence

Turbulence is the state of chaotic and unstable fluid flow. Y plus (Y+) is the non dimensional distance of the first cell from the wall.

Let's take the example of a flat plate under zero pressure gradient. Assuming that a uniform velocity comes on to the plate, first the flow will be laminar, then there will be a transition region between laminar and turbulent flow, after which the flow is completely turbulent.

The boundary layer in the turbulent region is called a turbulent boundary layer whereas in the laminar region it is called a laminar boundary layer.

The governing equations are valid in the boundary layer.

 

Reynolds number Re is the ratio of the inertial force to the viscous force

$\frac{1}{T}\underset{0}{{\int }^T}[\frac{\partial \overline{u}}{\partial t}dt+\frac{\partial u\text{'}}{\partial t}dt+\frac{\partial \overline{u}}{\partial x}dt+\frac{\partial u\text{'}}{\partial x}dt+2\frac{\partial \overline{u}u{\text{'}}^2}{\partial x}dt+\frac{\partial \overline{u}\overline{v}}{\partial y}dt+\frac{\partial \overline{u}v\text{'}}{\partial y}dt+\frac{\partial u\text{'}v\text{'}}{\partial y}dt+\frac{\partial u\text{'}\overline{v}}{\partial y}dt]=\frac{1}{T}\underset{0}{{\int }^T}[-\frac{1}{\rho }\frac{\partial \overline{P}}{\partial x}dt-\frac{1}{\rho }\frac{\partial P\text{'}}{\partial x}dt+\gamma \frac{{\partial }^{2}u\text{'}}{\partial y^2}dt+\gamma \frac{{\partial }^2\overline{u}}{\partial y^2}dt]$

 

External flow - Re < 5e5 - Laminar flow, Re > 5e5 - Turbulent flow

Internal flow - Re < 2300 - Laminar flow, Re > 4000 - Turbulent flow

 

Viscous sub layer - Viscous stress term dominates

Buffer layer - Transitional region

Log law region - Reynolds shear stress term dominates

 

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 Stokes equations to account for turbulent fluctuations in the fluid momentum.

$\tau \text{'}\underset{uv}{}=-\rho .u\text{'}.v\text{'}$

 

 

It is for this reason, that turbulence modeling is necessary for capturing the flow the boundary layer.

 

Turbulence Modeling Approaches

Converge offers two turbulence modeling approaches

Reynolds Averaged Navier Stokes (RANS)

Large Eddy Simulations (LES)

 

We shall focus here only on the RANS approach.

 

Reynolds Averaged Navier Stokes (RANS)

Although statistical theory and numerical simulations (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 fluctuation quantities. One then attempts to model the fluctuation terms by relating them to mean quantities or their gradients. This approach may now be yielding diminishing returns. The Reynolds equations however, are far from obsolete and form the basis of most engineering analyses of turbulent flow.

Need of RANS

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

 

Reynolds Decomposition

We assume that the fluid is in a random unsteady turbulent state and work with time averaged or mean equations of motion. Any variable Q is resolved into a mean value of $\overline{Q}$

plus a fluctuating value Q', where by definition

$\stackrel{}{}$

 

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 $\overline{Q}$

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.

$\mathrm{`u(x,\; y,z,\; t)=bar\; u(x,y,z)+\; u\text{'}(x,y,\; z,t)`}$

 

$\mathrm{`v(x,\; y,z,\; t)=bar\; v(x,y,z)+\; v\text{'}(x,y,\; z,t)`}$

 

$\mathrm{`w(x,\; y,z,\; t)=bar\; w(x,y,z)+\; w\text{'}(x,y,\; z,t)`}$

 

 

Here

u = instantaneous velocity

$\stackrel{}{}$= Average velocity

u' = Velocity fluctuation

 

The mean time average velocity is a function of space, while the fluctuating velocity is a function of both space and time.

 

On integration

$\stackrel{}{}$

 

$\stackrel{}{}$

 

 

Integration of the fluctuating velocity term gives zero. This can be illustrated as follows

$\frac{\mathrm{`frac\{1\}\{t\; underset("int")()\}\; underset("0")(int^(T)\; )u\text{'}(x,y,z,t)dt=0`}}{}$

 

Using the Reynolds decomposition principle

$\stackrel{}{}$

 

$\mathrm{`cancel(bar\; u)=cancel(baru)+\; frac\{1\}\{T\}\; underset("0")(int^(T)\; )\; baru\; dt`}$

 

 

We get

$\frac{\mathrm{`frac\{1\}\{T\}\; underset("0")(int^(T)\; )\; baru\; dt=0`}}{}$

 

 

Applying Reynolds decomposition on Continuity equation

$\frac{`frac\{delu\}\{delx\}+\; frac\{delv\}\{dely\}=0\; -`}{}$Continuity equation

Applying Reynolds decomposition

$\frac{`frac\{del(baru+\; u\text{'})\}\{delx\}+\; frac\{del(barv+v\text{'})\}\{dely\}=0`}{}$

 

 

On integration with respect to time and doing time averaging

$\frac{\mathrm{`frac\{1\}\{T\}int(frac\{del(baru+\; u\text{'})\}\{delx\}+\; frac\{del(barv+v\text{'})\}\{dely\})=0`}}{}$

 

 

On splitting, we get

$\frac{\mathrm{`frac\{1\}\{T\}intfrac\{delbaru\}\{delx\}dt+frac\{1\}\{T\}intfrac\{delu\text{'}\}\{delx\}dt+frac\{1\}\{T\}intfrac\{delbarv\}\{dely\}dt\; +frac\{1\}\{T\}intfrac\{delv\text{'}\}\{dely\}dt=0`}}{}$

 

 

Since the integral of fluctuating terms is zero

$\mathrm{`frac\{1\}\{T\}intfrac\{delbaru\}\{delx\}dt+cancel(frac\{1\}\{T\}intfrac\{delu\text{'}\}\{delx\}dt)+frac\{1\}\{T\}intfrac\{delbarv\}\{dely\}dt\; +cancel(frac\{1\}\{T\}intfrac\{delv\text{'}\}\{dely\}dt)=0`}$

 

$\frac{`frac\{delbaru\}\{delx\}frac\{1\}\{T\}intdt\; +frac\{delbarv\}\{dely\}frac\{1\}\{T\}intdt\; =0`}{}$

 

 

Applying limits and integrating

$\frac{`frac\{delbaru\}\{delx\}frac\{1\}\{T\}underset(0)int^Tdt\; +frac\{delbarv\}\{dely\}frac\{1\}\{T\}underset(0)int^Tdt\; =0`}{}$

 

$\frac{`frac\{delbaru\}\{delx\}frac\{[T-0]\}\{T\}+frac\{delbarv\}\{dely\}frac\{[T-0]\}\{T\}=0`}{}$

 

 

We finally get

$\frac{`frac\{delbaru\}\{delx\}+frac\{delbarv\}\{dely\}=0`}{}$

 

Therefore, the continuity equation is satisfied.

 

Applying Reynolds decomposition on Momentum equation

$\mathrm{`rho[frac\{delu\}\{delt\}\; +frac\{udelu\}\{delx\}+frac\{vdelu\}\{dely\}]=-frac\{delp\}\{delx\}+mufrac\{del^2u\}\{dely^2\}`}\frac{{}^{}}{}$

 

$`[frac\{delu\}\{delt\}\; +frac\{udelu\}\{delx\}+frac\{vdelu\}\{dely\}]=frac\{1\}\{rho\}[-frac\{delp\}\{delx\}+mufrac\{del^2u\}\{dely^2\}]`$

 

 

Replacing `frac{mu}{rho}= gamma` by kinematic viscosity

Adding a extra term on LHS

$`[frac\{delu\}\{delt\}\; +frac\{udelu\}\{delx\}+frac\{vdelu\}\{dely\}]+u[frac\{delu\}\{delx\}+frac\{delv\}\{dely\}]=-frac\{1\}\{rho\}frac\{delp\}\{delx\}+gammafrac\{del^2u\}\{dely^2\}`\frac{{}^{}}{}$

 

 

On simplifying

$`[frac\{delu\}\{delt\}\; +2frac\{udelu\}\{delx\}+frac\{vdelu\}\{dely\}+frac\{udelv\}\{dely\}]=-frac\{1\}\{rho\}frac\{delP\}\{delx\}+gammafrac\{del^2u\}\{dely^2\}`\frac{{}^{}}{}$

 

 

Using product rule by substituting the cross product

$`[frac\{delu\}\{delt\}\; +2frac\{udelu\}\{delx\}+frac\{deluv\}\{dely\}]=-frac\{1\}\{rho\}frac\{delP\}\{delx\}+gammafrac\{del^2u\}\{dely^2\}`\frac{{}^{}}{}$

 

$`[frac\{delu\}\{delt\}\; +2frac\{delu^2\}\{delx\}+frac\{deluv\}\{dely\}]=-frac\{1\}\{rho\}frac\{delP\}\{delx\}+gammafrac\{del^2u\}\{dely^2\}`\frac{{}^{}}{}$

 

 

Applying Reynolds decomposition

 

 `frac{1}{T}underset{0}{int^T}[frac{del[baru+u']}{delt} +2frac{del[baru+u']^2}{delx}+frac{del[baru+u'][barv+v']}{dely}]=frac{1}{T}underset{0}{int^T}[-frac{1}{rho}frac{delP}{delx}+gammafrac{del^2u}{dely^2}]`

 

`frac{1}{T}underset{0}{int^T}[frac{delbaru}{delt} dt+frac{delu'}{delt}dt+frac{delbaru}{delx}dt+frac{delu'}{delx}dt+2frac{delbaru u'^ 2}{delx}dt+frac{delbarubarv}{dely}dt+frac{delbaruv'}{dely}dt+frac{delu'v'}{dely}dt+frac{delu'barv}{dely}dt]=frac{1}{T}underset{0}{int^T}[-frac{1}{rho}frac{delbarP}{delx}dt-frac{1}{rho}frac{delP'}{delx}dt+gammafrac{del^2u'}{dely^2}dt+gammafrac{del^2baru}{dely^2}dt]`

 

`frac{1}{T}underset{0}{int^T}[frac{delbaru}{delt} dt+cancel(frac{delu'}{delt}dt)+frac{delbaru}{delx}dt+frac{delu'}{delx}dt+cancel(2frac{delbaru u'^ 2}{delx}dt)+frac{delbarubarv}{dely}dt+cancel(frac{delbaruv'}{dely}dt)+frac{delu'v'}{dely}dt+cancel(frac{delu'barv}{dely}dt)]=frac{1}{T}underset{0}{int^T}[-frac{1}{rho}frac{delbarP}{delx}dt-cancel(frac{1}{rho}frac{delP'}{delx}dt)+cancel(gammafrac{del^2u'}{dely^2}dt)+gammafrac{del^2baru}{dely^2}dt]`

 

`[frac{delbaru}{delt} +frac{delbaru^2}{delx}+frac{delbarubarv}{dely}]=-frac{1}{rho}frac{delbarP}{delx}+gammafrac{del^2baru}{dely^2}-frac{1}{T}underset{0}{int^T}frac{del(u')^2}{delx}dt-frac{1}{T}underset{0}{int^T}frac{del(u'v')}{dely}dt`

 

`[frac{delbaru}{delt} +frac{delbaru^2}{delx}+frac{delbarubarv}{dely}]=-frac{1}{rho}frac{delbarP}{delx}+gammafrac{del^2baru}{dely^2}-frac{1}{T}underset{0}{int^T}frac{del(u'v')}{dely}dt`

 

`[frac{delbaru}{delt} +frac{delbaru^2}{delx}+frac{delbarubarv}{dely}]=-frac{1}{rho}frac{delbarP}{delx}+frac{1}{rho}frac{del}{dely}(mufrac{delbaru}{dely}-rho.u'.v')`

 

 

The last of the diffusion terms represents the momentum diffusivity due to turbulence (Reynolds stress). This is the only unknown term, and therefore turbulence modeling is needed to capture it.

 

What is turbulent viscosity? How is it different from molecular viscosity

The turbulent viscosity is also known as eddy viscosity. In the study of turbulence in fluids, a common stategy is to ignore the small scale vortices (eddies) in the motion and to calculate a large scale motion with an effective viscosity, called the "eddy viscosity", which characterizes the transport and dissipation of energy in the smaller scale flow. In contrast to the viscosity of the fluid itself, which must be positive by the second law of thermodynamics, the eddy viscosity can be negative.

 

Molecular viscosity is the same as viscosity. The coefficient of molecular viscosity is the same 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 eddy viscosity in turbulent flow.

 

The dynamic or molecular viscosity is a macroscopic fluid property, and the effy viscosity or turbulent viscosity is a flow property emerging from the increased momentum transport through the flow field due to turbulent velocity fluctuations.