Week 8: Literature review - RANS derivation and analysis
AIM:
To write a literature review for RANS derivation and analysis
OBJECTIVE:
Apply Reynolds decomposition to the Navier Strokes equations and to come up with the expression for Reynolds stress.
And to explain the terms in Reynold's stress.
To explain turbulent viscosity and explain how is it different from molecular viscosity.
DESCRIPTION:
TURBULENCE:
Turbulence is the state of chaotic and unstable fluid flow just opposite to liminar flow. Y+ is the non-dimensional distance of the first cell from the wall.
Lets take an example of a flat plate under zero pressure gradient and front left side the uniform velocity is coming to the plate. Firstly it will be laminar flor then after some time the disturbance occurs, the region will be called as Transient region. When the disturbance is growing too much the flow will change in Turbulent flow. The boundary layer in the turbulent region will be called a Turbulent boundary layer where as in laminar it will be called as laminar boundary layer.
TURBULENCE MODEL APPROACHES:
Converge uses two boundary modelling approaches
RANS-REYNOLD AVERAGED NAVIER STROKES: Solves the equation for an ensemble-averaged velocity field and the magnitude of the turbulent fluctuations.
LES-LARGE EDDY SIMULATION: Solves the equations for a filtered time-dependent velocity field that represents large scale turbulence motion.
REYNOLDS AVERAGED NAVIER STROKES EQUATIONS (RANS)
Two alternative methods can be employed to transform the Navier strokes equations in such a way that the small scale turbulent fluctuations do not have to be directly simulated using 'Reynolds averaging and filtering'.
The rationale for Reynolds averaging is that we are not interested in the part of flow solution that can be described as "Turbulent fluctuations" instead, it is the mean (velocity, pressure, lift, drag) that is of interest. Looking at turbulent flow, it may best steady in the mean inspite of turbulent fluctuations.
RANS equations represent transport equations for the mesn flow quantities only, with all the scales of the turbulence being modelled. The approach of permitting a solution for the mean flow variables greatly reduces the computational effort.
REYNOLDS 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 varibale 'Q' is resolved into a mean value of `barQ` + fluctuating value of 'Q', where by defination
`barQ = 1/ T int_(t0) ^(T+ t) Q d t`
Instanataneous 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 `barQ` itself may vary slowly with time as shown in the image below.
Let us consider only incompressible turbulent flow iwth constant transport properties but with possible significant fluctuations in velocity, pressure and temperature.
`u(x,y,t) = baru (x,y) + u' (x,y,t)`
`v(x,y,t) = barv (x,y) + v' (x,y,t)`
`P(x,y,t) = barP (x,y) + P' (x,y,t)`
`T(x,y,t) = barT (x,y) + T' (x,y,t)`
`phi(x,y,t) = barphi (x,y) + phi' (x,y,t)`
TIME AVERAGED QUANTITIES:
`barQ ( x , y ) = 1 /Tint_(t0)^(T+ t) Q ( x , y , t ) d t`
Mean values are not the function of 't'
Moreover the averaging of fluctuating terms lead to zero as shown beloe
`1/Tint_0^T u(x,y,t)dt = 1/Tint_0^T baru(x,y)dt+1/Tint_0^T u'(x,y,t)dt`
and
`1/Tint_0^T u(x,y,t)dt=baru(x,y)`
`1/Tint_0^T baru(x,y)dt=baru(x,y)`
implies
`int_0^T u'(x,y,t)dt=0`
Before substituting these functions in the NS equations we can verify from basic integral relation that the following rules of avergaing apply for any two turbulent quantities 'T' and 'g'.
`barf'=0`
`barf=barf`
`bar(fg)=barf barg`
`bar(f'g)=0`
`bar(f+g)=barf+barg`
`bar((delf)/(delg))=bar((delf)/(delg))`
`bar(fg)=barfbarg + barf' barg'`
`barintfds=intbarf ds`
Substitutibg averaged quantities in NS equations
Continuity equation is `(delu)/(delx)+(delv)/(dely)=0`
substituting the averaged quantities `(del(baru+u'))/(delx)+(del(barv+v'))/(dely)=0`
expanding the terms as `(delbaru)/(delx)+(delu')/(delx)+(delbarv)/(dely)+(delv')/(dely)=0`
Averaging over time step 't'
`1/Tint_0^T (delbaru)/(delx)dt+1/Tint_0^T (delu')/(delx)dt+1/Tint_0^T (delbarv)/(delx)dt+1/Tint_0^T (delv')/(delx)dt=0`
Since the time integral of fluctuation component is zero and integral limit not a function of space
`1/Tint_0^T (delbaru)/(delx)dt+cancelcolor(red)(1/Tint_0^T (delu')/(delx)dt)+1/Tint_0^T (delbarv)/(delx)dt+cancelcolor(red)(1/Tint_0^T (delv')/(delx)dt)=0`
The equation reduces to `1/Tint_0^T (delbaru)/(delx)dt+1/Tint_0^T (delbarv)/(delx)dt=0`
Since the `baru and barv` is the function of space only the above equation can be written as
`(delbaru)/(delx)1/Tint_0^Tdt+(delbarv)/(delx)1/Tint_0^T dt=0`
and `1/Tint_0^T dt = (T-0)/T = 1`
This implies the continuity equation after the substitution of averaged quantities depends on only means values of decomposed quantities as shown below
`(delbaru)/(delx)+(delbary)/(dely)=0`
X-MOMENTUM EQUATION:
`rho[(delu)/(delt)+u(delu)/(delx)+v(delu)/(dely)]=-(delP)/(delx)+mu((del^2u)/(delx^2)+(del^2u)/(dely^2))`
Multiplying both sides by `1/rho` and replacing by `mu/rho` with kinematic viscosity `v`
`(delu)/(delt)+u(delu)/(delx)+v(delu)/(dely)=-(delP)/(delx)+v(del^2u)/(delx^2)+v(del^2u)/(dely^2)`
To convert the above equations in suitable form we multiply x-component of velocity with continuity equation and add it to left hand side and its resultant is zero
`(delu)/(delt)+u(delu)/(delx)+v(delu)/(dely)+u((delu)/(delx)+(delv)/(dely))=-(delP)/(delx)+v(del^2u)/(delx^2)+v(del^2u)/(dely^2)`
Rearranging the terms in the above equation
`(delu)/(delt)+2u(delu)/(delx)+v(delu)/(dely)+u(delv)/(dely)=-(delP)/(delx)+v(del^2u)/(delx^2)+v(del^2u)/(dely^2)`
From the product rule
`(del(uv))/(dely)=u(delv)/(dely)+v(delu)/(dely)` and `2u(delu)/(delx)=(delu^2)/(delx)`
Substituting the terms of product rule in the reduced equation
`(delu)/(delt)+(delu^2)/(delx)+(deluv)/(dely)=-(delP)/(delx)+v(del^2u)/(delx^2)+v(del^2u)/(dely^2)`
Substituting the averaged quations
`(del(baru+u'))/(delt)+(del(barv+v')^2)/(delx)+(del((baru+u')(baru+u')))/(dely)=-1/P(del(barP+P'))/(delx)+v(del^2(baru+u'))/(delx^2)+v(del^2(baru+u'))/(dely^2)`
Expanding the terms
`(delbaru)/(delt)+(delu')/(delt)+(delbaru^2)/(delx)+(delu'^2)/(delx)+(del2baru u')/(delx)+(delbaru barv)/(dely)+(delbaru v')/(dely)+(delu' barv)/(dely)+(delu' v')/(dely)=-1/rho(delbarP)/(delx)-1/rho(delP')/(delx)+v(del^2baru)/(delx^2)+v(del^2u')/(delx^2)+v(del^2baru)/(dely^2)+v(del^2u')/(dely^2)`
Averaging over time step T
`1/Tint_0^T(delbaru)/(delt)dt+1/Tint_0^T(delu')/(delt)dt+1/Tint_0^T(delbaru^2)/(delx)dt+1/Tint_0^T(delu'^2)/(delx)dt+1/Tint_0^T(del2baru u')/(delx)dt+1/Tint_0^T(delbaru barv)/(dely)dt+1/Tint_0^T(delbaru v')/(dely)dt+1/Tint_0^T(delu' barv)/(dely)dt+1/Tint_0^T(delu' v')/(dely)dt=-1/Tint_0^T1/rho(delbarP)/(delx)dt-1/Tint_0^T1/rho(delP')/(delx)dt+1/Tint_0^Tv(del^2baru)/(delx^2)dt+1/Tint_0^Tv(del^2u')/(delx^2)dt+1/Tint_0^Tv(del^2baru)/(dely^2)dt+1/Tint_0^Tv(del^2u')/(dely^2)dt`
Since the time integral of fluctuation component is zero and the rules mentioned above
`1/Tint_0^T(delbaru)/(delt)dt+cancelcolor(red)(1/Tint_0^T(delu')/(delt)dt)+1/Tint_0^T(delbaru^2)/(delx)dt+1/Tint_0^T(delu'^2)/(delx)dt+cancelcolor(red)(1/Tint_0^T(del2baru u')/(delx)dt)+1/Tint_0^T(delbaru barv)/(dely)dt+cancelcolor(red)(1/Tint_0^T(delbaru v')/(dely)dt)+cancelcolor(red)(1/Tint_0^T(delu' barv)/(dely)dt)+1/Tint_0^T(delu' v')/(dely)dt=-1/Tint_0^T1/rho(delbarP)/(delx)dt-cancelcolor(red)(1/Tint_0^T1/rho(delP')/(delx)dt)+1/Tint_0^Tv(del^2baru)/(delx^2)dt+cancelcolor(red)(1/Tint_0^Tv(del^2u')/(delx^2)dt)+1/Tint_0^Tv(del^2baru)/(dely^2)dt+cancelcolor(red)(1/Tint_0^Tv(del^2u')/(dely^2)dt)`
The above equation reduces to
`1/Tint_0^T(delbaru)/(delt)dt+1/Tint_0^T(delbaru^2)/(delx)dt+1/Tint_0^T(delu'^2)/(delx)dt+1/Tint_0^T(delbaru barv)/(dely)dt+1/Tint_0^T(delu' v')/(dely)dt=-1/Tint_0^T1/rho(delbarP)/(delx)dt+1/Tint_0^Tv(del^2baru)/(delx^2)dt+1/Tint_0^Tv(del^2baru)/(dely^2)dt`
Upon integrating the equation reduces to
`(delbaru)/(delt)+(delbaru^2)/(delx)+1/Tint_0^T(delu'^2)/(delx)dt+(delbaru barv)/(dely)+1/Tint_0^T(delu' v')/(dely)dt=-1/rho(delbarP)/(delx)+v(del^2baru)/(delx^2)+v(del^2baru)/(dely^2)`
Rearranging the terms
`(delbaru)/(delt)+(delbaru^2)/(delx)+(delbaru barv)/(dely)=-1/rho(delbarP)/(delx)+v(del^2baru)/(delx^2)+v(del^2baru)/(dely^2)-color(red)(1/Tint_0^T(delu'^2)/(delx)dt)-color(red)(1/Tint_0^T(delu' v')/(dely)dt)`
From the above equations it is clear that the momentum equation obtained with mean vales of components is similar to momentum equation of actual transport properties except with the red terms
If we consider that the u' value if fluctuating between -1 and 1 then by averaging it the value of u' comes equal to 0 but if we consider the same case for `u'^2` then the square of -1 and 1 will be between 0 and 1 which when integrated will not give zero. Hence
`1/Tint_0^T(delu'^2)/(delx)dt !=0`
Moreover the above term can be approximated to zero in the boundary layer where the gradient along y is higher in comparision to the x-direction. Though we are not using this assumption in this case
Similarly in the term `1/Tint_0^T(delu' v')/(dely)dt` the fluctuations along the x-axis u' and the fluctuations along the y-axis v' are correlated. That is when a low moementum y-component is flowing in the positive y-direction then the high momentum (i.e high velocity) x-component ges slowed down which is caused by v'. So, the positive increase in the v' is going to cause a negative result in the u' and the product of two will come out to be negatve value only. Hence, this term will also not give zero.
`color(blue)((delbaru)/(delt))+color(green)((delbaru^2)/(delx)+(delbaru barv)/(dely))=color(orange)(-1/rho(delbarP)/(delx))+v((del^2baru)/(delx^2)+(del^2baru)/(dely^2))-color(red)(1/Tint_0^T(delu'^2)/(delx)dt)-color(red)(1/Tint_0^T(delu' v')/(dely)dt)`
The above terms are related to as follows
`color(blue)((delbaru)/(delt))`= change w.r.t time
`color(green)((delbaru^2)/(delx)+(delbaru barv)/(dely))` = convective terms
`color(orange)(-1/rho(delbarP)/(delx))` = pressure gradient
`v((del^2baru)/(delx^2)+(del^2baru)/(dely^2))` = shear stress
`color(red)(1/Tint_0^T(delu'^2)/(delx)dt)-color(red)(1/Tint_0^T(delu' v')/(dely)dt)` = Reynold's stress
In the above equation integral values are replaced by averages
`(delbaru)/(delt)+(delbaru^2)/(delx)+(delbaru barv)/(dely)=-1/rho(delbarP)/(delx)+v(del^2baru)/(delx^2)+v(del^2baru)/(dely^2)-(del(bar(u'v')))/(dely)-(del(bar(u')^2))/(delx)`
The equation can be further rearranged as follows
`(delbaru)/(delt)+(delbaru^2)/(delx)+(delbaru barv)/(dely)=-1/rho(delbarP)/(delx)+v(del^2baru)/(delx^2)+v(del^2baru)/(dely^2)+1/rho[(del(bar(-rho u'v')))/(dely)+(del(-barrho(u')^2))/(delx)]`
In the compact notation
Stress term = `color(blue)(mugrad^2u underset(i)`-`color(red)((rho(bar(del ui' uj')))/(delxj))`
Where `color(blue)(mugrad^2u underset(i)` = Shear stress gradient
`color(red)((rho(bar(del ui' uj')))/(delxj))` = Reynolds stress gradient
=`mu(del)/(delxj)(delui)/(delxj)-rho(del)/(delxj)(bar(ui' uj'))`
`=(del)/(delxj)[mu(delui)/(delxj)-rho(bar(ui' uj'))]`
Total Shear stress = `mu(delui)/(delxj)-rho(bar(ui' uj'))`
Where `mu(delui)/(delxj)` = Laminar stress
`rho(bar(ui' uj'))` = Turbulent stress
In the 3D cartesian co-ordinate system the above equation will be as follows
`(delbaru)/(delt)+(delbaru^2)/(delx)+(delbaru barv)/(dely)=-1/rho(delbarP)/(delx)+v[(del^2baru)/(delx^2)+(del^2baru)/(dely^2)+(del^2baru)/(delz^2)]+1/rho[(del(bar(-rho u'v')))/(dely)+(del(-barrho(u')^2))/(delx)+(del(bar(-rho u'w')))/(delz)]`
Hence the resulting Reynolds stress in the matrix form along x,y and z axes is as follows
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-strokes equations to account for turbulent fluctuations in fluid momentum. This stress can be used to model the Reynolds stress transport equation which will allow using the comparatively larger grid to flow analysis.
TURBULENT VISCOSITY AND HOW IT DIFFERS FROM MOLECULAR VISCOSITY:
The Turbulent transfer of momentum by eddies creates internal fluid friction. It is the fundamental idea of how we define viscosity in turbulent flow i.e internal fluid resistance. Eddy viscosity explains what causes internal friction.
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 analagous in laminar flow to eddy viscosity in a turbulent flow. Eddy viscosity (turbulent viscosity) is an imaginary concept, it doesn not exist as the molecular viscosity which is a well-defined transport coefficient as stated by the kinetic theory. The turbulent viscosity hypothesis was posed for making things simpler, in the sense that the turbulent Reynolds stresses are simplified to be proportional to the gradients of the mesn velocity, as heppens in Newtonian laminar flows with the viscous stresses. However, the numerical methods from this simplification (such as RANS) are low time consuming and can be used widely.