Week 8: Literature review - RANS derivation and analysis

1) Introduction

We know that Navier-stokes equations could be used in any fluid flow applications to effectively approximate the results, but when we encounter turbulent flows we need mesh elements with tiny dimensions, for instance in IC engines largest turbulent length scales equal to small values from 0.1mm to 1mm, in non-premixed combustion chamber mixing occures rapidly due to turbulence, however to capture the flow structure while mixing, our mesh element size should be 1e-6m (approximately 1micron). According to this, if we set mesh element size pretty small we probably will get fine results but the simulation will run for a really long time, which we do not want. Fortunately, instead of decreasing mesh element size we can use turbulent models (special mathematical models) to capture turbulent flow structures. one of these models is called \"RANS\" (Reynols averaged Navier stokes).

Derivation

Assumptions

• Incompressible flow
• Newtonian Fluid
• Flow dominant along X direction

For turbulent flow the velocity is divided into two components

1. Average flow - Average velocity over period of time t. `overline u(x,y,z)`
2. Fluctuating flow- Difference between actual and average velocity. `u'(x,y,z)`

Expression for Reynold's Stress :-

Continuity Equation :

`(partialu)/(partialx) + (partialv)/(partialy) = 0`

Momentum Equation :

`rho((partialu)/(partialt) +u (partialu)/(partialx) + v(partialu)/(partialy))=-(partialp)/(partialx) + mu(partial^2u)/(partialy^2)`

Energy Equation :

`rho_c (partial T)/(partial t) + u (partialT)/(partial x) + v (partial T)/(partialy) = k(partial T^2)/(partial x^2) + mu ((partial u)/(partial y))^2`

These all of the equation are valid inside the boundary layer. These final set of equation we can simulate boundary layer in any types of geometry but as we seen so far we always used a turbulence model so whenever we setup the shockflow problem or the conjugate heat transfer problem we always used a turbulence model

So why is that if we look at above 3 equations, if we solve them they are completly capable of giving you solution for Laminar and Turbulent flows and if we do that then people would say that we performing a direct numerical simulation, so DNS the technique where people solve the navier stocks equations and solve it and these are able to capturing everything including turbulence.

Split velocity component or anyother variables into a mean component and a fluctuating component . This is what called as the Reynolds decomposition.

`u(x,y,z,t) = overline u(x,y,z) + u'(x,y,z,t)`

`v(x,y,z,t) = overline v(x,y,z) + v'(x,y,z,t)`

`omega(x,y,z,t) = overline omega(x,y,z) + omega(x,y,z,t)`

Here,

`u` = Instantaneous velocity

`overline u`= Average velocity

`u'` = Velocity Fluctuation

Here `u , v``omega` is instantaneous velocity at any point in time and space. Here, i am taking a velocity and split into two component one component is just a function of space or we can say average time quantity and the other component is function of time and space or fluctuation .

`overline u (x,y) = 1/ t ( int_0^(t_"int") u(x,y,t)dt)`

`overline v (x,y) = 1/ t ( int_0^(t_"int") v(x,y,z)dt)`

Since `overline u` is the averaging operator

`overline u = 1/ t int_0^(t_"int") (overline u+ u')dt`

`overline u = 1/ t int_0^(t_"int") overline u dt + 1/ t int_0^(t_"int") u' dt`

so, `overline u` is not a function of time

`overline u = overlineu(1)+ 1/ t int_0^(t_"int") u'dt`

So, `1/ t int_0^(t_"int") u'dt = 0`

We also decompose the pressure

`p = overline p + p'`

Now we will substitute this decomposition into the Navier-Stokes equations and seek a new equation governing the dynamics of the time average flow field.

Conservation of Mass :

`(partialu)/(partialx) + (partialv)/(partialy) = 0`

Substitude `u = overline u + u'`

and `v = overline v + v'`

Apply decomposition

`partial/(partialx)(overline u + u' ) + partial/(partialy)(overline v + v' ) = 0`

Taking time average of whole equation

`1/(T) int_0^T partial/(partialx) [overlineu + u'] dt + 1/(T) int_0^T partial/(partialy) [overlinev + v'] dt = 0`

for futher simplification we got that,

`1/(T) int_0^T (partial) /(partial x) overline u dt +1/(T) int_0^T (partial ) /(partial x) u'dt + 1/(T) int_0^T (partial) /(partial y) overline v dt + 1/(T) int_0^T (partial) /(partial y) v'dt = 0`

Average fluctuations is equal to zero

So,

`1/(T) int_0^T (partial) /(partial x) overline u dt +cancel{1/(T)int_0^T (partial ) /(partial x) u' dt} + 1/(T) int_0^T (partial ) /(partial y) overline v dt + cancel{1/T) int_0^T (partial ) /(partial y) v' dt = 0`

`1/(T) int_0^T (partial) /(partial x) overline u dt + 1/(T) int_0^T (partial ) /(partial y) overline v dt = 0`

By solving the integration

`(partial overlineu) /(partial x) [ [T-0] / (T) ]+ (partial overline v) /(partial y) [ [T-0] / (T) ]= 0`

`(partial overline u)/(partialx) + (partial overline v)/(partialy) = 0`

So by the above equation we said that the average field flow obeys continuity equation.

Conservation of Momentum :

`rho((partialu)/(partialt) +u (partialu)/(partialx) + v(partialu)/(partialy))=-(partialp)/(partialx) + mu(partial^2u)/(partialy^2)`

`(partialu)/(partialt) +u (partialu)/(partialx) + v(partialu)/(partialy)= -1/rho(partialp)/(partialx) + mu/rho(partial^2u)/(partialy^2)`

Adding continuity equation inside the equation

`u ( (partialu)/(partialx) + (partialv)/(partialy) ) = 0`

`(partialu)/(partialt) +u (partialu)/(partialx) + v(partial u)/(partial y) + u ((partial u)/ (partial x) + (partial v)/ (partial x))= -(partial p)/(rho partial x) + nu (partial^2u)/(partialy^2)`

`(partialu)/(partialt) + 2u (partialu)/(partialx) + v(partial u)/(partial y) + u (partial v)/ (partial x)= -(partial p)/(rho partial x) + nu (partial^2u)/(partialy^2)`

Product rule

`partial/ (partial y)(u v)= u (partialu)/(partialx) + v(partial u)/(partial y)`

Put into the equation

`(partialu)/(partialt) + 2u (partialu)/(partialx) + partial/ (partial y)(u v) = -1/rho(partial p)/( partial x) + nu (partial^2u)/(partialy^2`

`(partialu)/(partialt) + (partialu)/(partialx) (u^2)+ partial/ (partial y)(u v) = -1/rho(partial p)/(partial x) + nu (partial^2u)/(partialy^2`

Substitude `u = overline u + u'`

and `v = overline v + v'`

`partial /(partial t) (overline u + u') + (partial u)/(partial x) (overline u + u')^2+ partial/ (partial y)[ (overline u + u') + (overline v + v') ] = -1/rho (partial )/ (partial x) (overline p + p')+ nu (partial^2)/(partial y^2)(overline u + u')`

`(partial overline u) /(partial t) + (partialu')/(partialt) + (partial overline u^2)/(partial x)+ (partial u'^2)/(partial t)+ (partial (2 overline u .u'))/(partial x ) +(partial(overline u overline v))/(partial y)+ (partial (overline u v'))/(partial y)+(partial (u' overline v))/(partial y) + (partial (u'v'))/(partial y) = -1/rho(partial overline p)/(partial x) -1/rho (partial p')/(partial x) + nu (partial^2overline u )/(partial y^2) + nu(partial^2u')/(partial y^2)`

Eliminating the velocity fluctuation term from the above equation, we get

`(partial overline u) /(partial t) + cancel((partial u')/(partialt)) + (partial overline u^2)/(partial x)+ (partial u'^2)/(partial t)+ cancel((partial (2 overline u u'))/(partial x )) + (partial(overline u overline v))/(partial y) +cancel((partial (overline u v'))/(partial y))+ cancel((partial (u' overline v))/(partial y)) + (partial (u'v'))/(partial y) = =1/rho(partial overline p)/(partial x)cancel -1/rho(partial p')/(rho partial x) + nu (partial^2overline u )/(partial y^2)+ cancel(nu(partial^2u')/(partial y^2)`

`(partial overline u) /(partial t) + (partial overline u^2)/(partial x)+ (partial u'^2)/(partial t)+ (partial(overline u overline v))/(partial y) + (partial (u'v'))/(partial y) = -1/rho (partial overline p)/( partial x) + nu (partial^2overline u )/(partial y^2dt`

Arranging the equation

`(partial overline u) /(partial t) + (partial overline u^2)/(partial x)+ (partial(overline u overline v))/(partial y) = -1/rho (partial overline p)/( partial x) + nu (partial^2overline u )/(partial y^2) - (partial u'^2)/(partial x) - (partial (u'v'))/(partial y)`

Neglecting `u'^2`

`(partial overline u) /(partial t) + (partial overline u^2)/(partial x)+ (partial(overline u overline v))/(partial y) = -1/rho (partial overline p)/( partial x) + nu (partial^2overline u )/(partial y^2) - (partial (u'v'))/(partial y)`

so.. `nu = m/rho` --->

Rearranging the above term

`(partial overline u) /(partial t) + (partial overline u^2)/(partial x)+ (partial(overline u overline v))/(partial y) = -1/rho (partial overline p)/( partial x) + 1/rho (partial)/(partial y) (mu (partial overline u)/(partial y)- rho u'v')`

`ubrace{((partial overline u) /(partial t) + (partial overline u^2)/(partial x)+ (partialoverline u overline v)/(partial y))}_"Inertial term" = ubrace (-1/rho (partial overline p)/( partial x))_("Pressure term") + 1/rho (partial)/(partial y) [ubrace(mu (partial overline u)/(partial y))_("Molecular viscosity") - ubrace (rho u'v')_("Turbulent viscosity")]`

Unsteady Reynolds Average N-S equation :

`rho ((partial overline u) /(partial t) + (partial overline u^2)/(partial x)+ (partial(overline u overline v))/(partial y)) = -(partial overline p)/( partial x) +(partial)/(partial y) (mu (partial overline u)/(partial y)- rho u'v')`

In the above equation `rho , u', v'`represents as added stress on the fluid.

Reynolds Shear Stress

Reynolds stress is a quantity which depends upon the density of the fluid and the fluctuating
component of the velocity. It arises due to turbulence in the flow.

It does not depend on the viscosity of the fluid.

Solving Transport equation for courser grid can be possible if we are able to determine these
stresses equation depending different Reynolds number and fluid properties can be formulated by
experimentation as the remaining part of transport equation is basic navier stokes equation.
RANS Turbulence models are developed by experimentally calculating these stresses for various
conditions.

Molecular Viscosity

Molecular viscosity is the property of fluid which is the measure of the unit deformation rate of the
fluid because of the shear force that arises between layers of fluid due to the cohesive force of the
molecules.

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.

Turbulence viscosity

Turbulence viscosity is measure of the unit transverse resistance that is created by fluctuating
component of the velocity across the flow.

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.