Week 8: Literature review RANS derivation and analysis

LITERATURE REVIEW RANS DERIVATION AND ANALYSIS

AIM: To write a literature review RANS derivation and analysis.

OBJECTIVE:

Apply Reynold's decomposition to the NS equations and come up with the expression for Reynold's stress.

• Explain your understanding of the terms Reynold's stress
• What is turbulent viscosity? How is it different from molecular viscosity?

THEORY AND SOLUTION:

Turbulence

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

Let's takes 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 flow, then after some disturbance, the region will be called as Transition 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 whereas in laminar it will be called the laminar boundary layer.

Turbulence Modeling Approaches

Converge offers two turbulence modeling approaches

Reynold-Averaged Navier Stokes (RANS)-Solves equation for an ensemble-averaged velocity field and the magnitude of the turbulent fluctuations.

Large Eddy Simulation (LES)-Solves equations for a filtered time-dependent velocity field that represents large scale turbulence motion

Reynolds-Averaged Navier - Stokes Equations (RANS):

Two alternative methods can be employed to transform the Navier-Stokes equations in such a way that the small-scale turbulent fluctuations do not have to be directly simulated: Reynolds averaging and filtering.
Both methods introduce additional terms in the governing equations that need to be modeled in order to achieve “closure”. (Closure implies that there are a sufficient number of equations for all the unknowns.)

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 in spite of turbulent fluctuations.

RANS equations represent transport equations for the mean flow quantities only, with all the scales of the turbulence being modeled. 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 variable 'Q' is resolved into a mean value of `barQ`plus a fluctuating value Q', where by definition,

`barQ = 1/T int_(t_0)^(t_0+T)Q dt`

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 `barQ`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.

`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)`

`ϕ(x,y,t) = bar ϕ(x,y)+ϕ'(x,y,t)`

Time averaged quantities:

`barQ(x,y) = 1/T int_(t_0)^(t_0+T)Q(x,y,t) dt`

mean values are not the function of 't' .

Moreover, the averaging of fluctuating terms lead to zero as shown below.

`1/Tint_0^Tu(x,y,t) dt= 1/Tint_0^Tbaru(x,y)dt+1/Tint_0^Tu'(x,y,t)dt`

and

`1/Tint_0^Tu(x,y,t) dt= baru(x,y)`

`1/Tint_0^Tbaru(x,y)dt = baru(x,y)`

implies

`int_0^Tu'(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 averaging apply for any two turbulent quantities 'f' and 'g'.

`bar(f') = 0`

`bar(barf)=barf`

`bar(fbarg)=barfbarg`

`bar(f'barg) = 0`

`bar(f+g) = barf+barg`

`bar(∂f)/(∂s) = (∂barf)/(∂s)`

`bar(fg)=barfbarg+bar(f'g')`

`bar(intfds) = intbarf ds`

Substituting Averged quantites in NS-equations:

Contnuity Equation:

`(∂u)/(∂x)+(∂v)/(∂y)=0`

substituting the averaged quantities

`(∂(baru+u'))/(∂x)+(∂(barv+v'))/(∂y)=0`

expanding the terms

`(∂baru)/(∂x)+(∂u')/(∂x)+(∂barv)/(∂y)+(∂v')/(∂y)=0`

averaging over time step 'T'

`1/Tint_0^T(∂baru)/(∂x)dt+1/Tint_0^T(∂u')/(∂x)dt+1/Tint_0^T(∂barv)/(∂y)dt+1/Tint_0^T(∂v')/(∂y)dt=0`

since the time integral of fluctuation component is zero and integral limit not a function of space.

`1/Tint_0^T(∂baru)/(∂x)dt+color(red)cancel(1/Tint_0^T(∂u')/(∂x)dt)+1/Tint_0^T(∂barv)/(∂y)dt+color(red)cancel(1/Tint_0^T(∂v')/(∂y)dt)=0`

The equation reduces to

`1/Tint_0^T(∂baru)/(∂x)dt+1/Tint_0^T(∂barv)/(∂y)dt=0`

since the `baru` and `barv` is the function of space only, the above equation can be re-written as:

`(∂baru)/(∂x)1/Tint_0^Tdt+(∂barv)/(∂y)1/Tint_0^Tdt=0`

and

`1/Tint_0^Tdt=(T-0)/T = 1`

implies, the continuity equation after the substitution of averaged quantities depends on only mean values of decomposed quantities as shown below.

`(∂baru)/(∂x)+(∂barv)/(∂y)=0`

x-Momentum Equation:

`ρ[(∂u)/(∂t)+u(∂u)/(∂x)+v(∂u)/(∂y)]=-(∂P)/(∂x)+ μ ((∂^2u)/(∂x^2)+(∂^2u)/(∂y^2))`

multiplying both sides by `1/ρ` and replacing `μ/ρ` with kinematic viscosity `ν`

`(∂u)/(∂t)+u(∂u)/(∂x)+v(∂u)/(∂y)=-1/rho(∂P)/(∂x)+ ν (∂^2u)/(∂x^2)+ ν (∂^2u)/(∂y^2)`

to convert the above equation in suitable form, we multiply x-component of velocity with continuity equation and add it to left hand side and its resultant is zero.

`(∂u)/(∂t)+u(∂u)/(∂x)+v(∂u)/(∂y)+u((∂u)/(∂x)+(∂v)/(∂y))=-1/rho(∂P)/(∂x)+ ν (∂^2u)/(∂x^2)+ ν (∂^2u)/(∂y^2)`

rearranging the terms in the above equation

`(∂u)/(∂t)+2u(∂u)/(∂x)+v(∂u)/(∂y)+u(∂v)/(∂y)=-1/rho(∂P)/(∂x)+ ν (∂^2u)/(∂x^2)+ ν (∂^2u)/(∂y^2)`

From Product rule,

`(∂(uv))/(∂y) =u(∂v)/(∂y)+v(∂u)/(∂y)`

and

`2u(∂u)/(∂x) = (∂u^2)/(∂x)`

substituting the terms of product rule in the reduced equation.

`(∂u)/(∂t)+(∂u^2)/(∂x)+(∂(uv))/(∂y)=-1/rho(∂P)/(∂x)+ ν (∂^2u)/(∂x^2)+ ν (∂^2u)/(∂y^2)`

substituting the averaged quantities

`(∂(baru+u'))/(∂t)+(∂(baru+u')^2)/(∂x)+(∂((baru+u')(barv+v')))/(∂y)=-1/rho(∂(barP+P'))/(∂x)+ ν (∂^2(baru+u'))/(∂x^2)+ ν (∂^2(baru+u'))/(∂y^2)`

expanding the terms

`(∂baru)/(∂t)+(∂u')/(∂t)+(∂baru^2)/(∂x)+(∂(u')^2)/(∂x)+(∂(2baru u'))/(∂x)+(∂(barubarv))/(∂y)+(∂(baru v'))/(∂y)+(∂(u'barv))/(∂y)+(∂(u'v'))/(∂y)=-1/rho(∂barP)/(∂x)-1/rho(∂P')/(∂x)+ ν (∂^2baru)/(∂x^2)+ ν (∂^2u')/(∂x^2)+ ν (∂^2baru)/(∂y^2)+ ν (∂^2u')/(∂y^2)`

averaging over time step 'T'

`1/Tint_0^T(∂baru)/(∂t)dt+1/Tint_0^T(∂u')/(∂t)dt+1/Tint_0^T(∂baru^2)/(∂x)dt+1/Tint_0^T(∂(u')^2)/(∂x)dt+1/Tint_0^T(∂(2baru u'))/(∂x)dt+1/Tint_0^T(∂(barubarv))/(∂y)dt+1/Tint_0^T(∂(baru v'))/(∂y)dt+1/Tint_0^T(∂(u'barv))/(∂y)dt+1/Tint_0^T(∂(u'v'))/(∂y)dt=-1/Tint_0^T1/rho(∂barP)/(∂x)dt-1/Tint_0^T1/rho(∂P')/(∂x)dt+ 1/Tint_0^Tν (∂^2baru)/(∂x^2)dt+ 1/Tint_0^Tν (∂^2u')/(∂x^2)dt+ 1/Tint_0^Tν (∂^2baru)/(∂y^2)dt+ 1/Tint_0^Tν (∂^2u')/(∂y^2)dt`

since the time integral of fluctuation component is zero and the rules mentioned above.

`1/Tint_0^T(∂baru)/(∂t)dt+color(red)cancel(1/Tint_0^T(∂u')/(∂t)dt)+1/Tint_0^T(∂baru^2)/(∂x)dt+1/Tint_0^T(∂(u')^2)/(∂x)dt+color(red)cancel(1/Tint_0^T(∂(2baru u'))/(∂x)dt)+1/Tint_0^T(∂(barubarv))/(∂y)dt+color(red)cancel(1/Tint_0^T(∂(baru v'))/(∂y)dt)+color(red)cancel(1/Tint_0^T(∂(u'barv))/(∂y)dt)+1/Tint_0^T(∂(u'v'))/(∂y)dt=-1/Tint_0^T1/rho(∂barP)/(∂x)dt-color(red)cancel(1/Tint_0^T1/rho(∂P')/(∂x)dt)+ 1/Tint_0^Tν (∂^2baru)/(∂x^2)dt+ color(red)cancel(1/Tint_0^Tν (∂^2u')/(∂x^2)dt)+ 1/Tint_0^Tν (∂^2baru)/(∂y^2)dt+ color(red)cancel(1/Tint_0^Tν (∂^2u')/(∂y^2)dt)`

The above equation reduces to

`1/Tint_0^T(∂baru)/(∂t)dt+1/Tint_0^T(∂baru^2)/(∂x)dt+1/Tint_0^T(∂(u')^2)/(∂x)dt+1/Tint_0^T(∂(barubarv))/(∂y)dt+1/Tint_0^T(∂(u'v'))/(∂y)dt=-1/Tint_0^T1/rho(∂barP)/(∂x)dt+ 1/Tint_0^Tν (∂^2baru)/(∂x^2)dt+ 1/Tint_0^Tν (∂^2baru)/(∂y^2)dt`

Upon integrating, the equation reduces to

`(∂baru)/(∂t)+(∂baru^2)/(∂x)+1/Tint_0^T(∂(u')^2)/(∂x)dt+(∂(barubarv))/(∂y)+1/Tint_0^T(∂(u'v'))/(∂y)dt=-1/rho(∂barP)/(∂x)+ ν (∂^2baru)/(∂x^2)+ ν (∂^2baru)/(∂y^2)`

Rearranging the terms

`(∂baru)/(∂t)+(∂baru^2)/(∂x)+(∂(barubarv))/(∂y)=-1/rho(∂barP)/(∂x)+ ν (∂^2baru)/(∂x^2)+ ν (∂^2baru)/(∂y^2)-color(red)(1/Tint_0^T(∂(u'v'))/(∂y)dt)-color(red)(1/Tint_0^T(∂(u')^2)/(∂x)dt)`

From the above equation it is clear that the momentum equation obtained with mean values of components is similar to momentum equation of actual transport properties except with the red terms.

If we consider that the `u'` value is 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(∂(u')^2)/(∂x)dt !=0`

Moreover, the above term can be approximated to zero in the boundary layer where the gradient along y is higher in comparison to the x-direction. Though we are not using this assumption in this case.

Similarly in the term, `1/Tint_0^T(∂(u'v'))/(∂y)dt` the fluctuations along the x-axis `u'` and the fluctuations along the y-axis `v'` are correlated. That is when a low momentum y-component is flowing in the positive y-direction then the high momentum (i.e., high velocity) x-component gets 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 negative value only. Hence, this term will also not give zero.

`ubrace(color(blue)((∂baru)/(∂t)))_("change w.r.t time")+ubrace(color(green)((∂baru^2)/(∂x)+(∂(barubarv))/(∂y)))_("convective terms")=ubrace(color(violet)(-1/rho(∂barP)/(∂x)))_("Pressure Gradient")+ ubrace(ν ((∂^2baru)/(∂x^2)+(∂^2baru)/(∂y^2)))_("Shear Stress")-ubrace(color(red)(1/Tint_0^T(∂(u'v'))/(∂y)dt)-color(red)(1/Tint_0^T(∂(u')^2)/(∂x)dt))_("Reynolds Stress")`

In the above equation, integral values are replaced by averages.

`(∂baru)/(∂t)+(∂baru^2)/(∂x)+(∂(barubarv))/(∂y)=-1/rho(∂barP)/(∂x)+ ν (∂^2baru)/(∂x^2)+ ν (∂^2baru)/(∂y^2)-(∂(bar(u'v')))/(∂y)-(∂(bar((u')^2)))/(∂x)`

The equation can be further rearranged as follows:

`(∂baru)/(∂t)+(∂baru^2)/(∂x)+(∂(barubarv))/(∂y)=-1/rho(∂barP)/(∂x)+ ν (∂^2baru)/(∂x^2)+ ν (∂^2baru)/(∂y^2)+1/ρ[(∂(-ρbar(u'v')))/(∂y)+(∂(-ρbar((u')^2)))/(∂x)]`

In the compact notation:

`"Stress term" = color(blue)(ubrace(μ∇^2u_i)_("shear stress gradient"))-color(red)(ubrace(ρ(∂bar(u_i^')bar(u_j^'))/(∂x_j))_("Reynolds Stress gradient")`

`= μ∂/(∂x_j)((∂u_i)/(∂x_j))-ρ∂/(∂ x_j)(bar(u_i^')bar(u_j^'))`

`= ∂/(∂x_j)[μ((∂u_i)/(∂x_j))-ρ(bar(u_i^')bar(u_j^'))]`

`"Total Shear Stress" = ubrace(μ((∂u_i)/(∂x_j)))_("Laminar Stress")-ubrace(ρ(bar(u_i^')bar(u_j^')))_("Turbulent Stress")`

In the 3D cartesian cordinate system the above equation will be as follows:

`(∂baru)/(∂t)+(∂baru^2)/(∂x)+(∂(barubarv))/(∂y)+(∂(barubarw))/(∂z)=-1/rho(∂barP)/(∂x)+ ν [(∂^2baru)/(∂x^2)+(∂^2baru)/(∂y^2)+(∂^2baru)/(∂z^2)]+1/ρ[(∂(-ρbar((u')^2)))/(∂x)+(∂(-ρbar(u'v')))/(∂y)+(∂(-ρbar(u'w')))/(∂z)]`

Hence the resulting Reynolds stress in the matrix form, along the x, y, and z axes is as follows:

`τ_(ij)^(Reynolds) = [[τ_(x x), τ_(xy), τ_(xz) ],[τ_(yx), τ_(yy), τ_(yz) ],[τ_(zx), τ_(zy), τ_(zz) ]]=[[-ρbar((u')^2),-ρbar((u'v')),-ρbar((u'w'))], [-ρbar((v'u')),-ρbar((v')^2),-ρbar((v'w'))], [-ρbar((w'u')),-ρbar((w'v')),-ρbar((w')^2)]]`

Reynold's 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 fluid momentum This stress offer significantly effects for the complex interactions in turbulent flows. This stress can be used to model the Reynolds Stress Transport Equation which will allow using the comparatively larger grid to flow analysis.

What is turbulent viscosity? How is it different 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 analogous in laminar flow to eddy viscosity in a turbulent flow. Eddy Viscosity (Turbulent Viscosity) is an imaginary concept, it does not exist as the molecular viscosity which is a well-defined transport coefficient as stated by the Kinetic Theory. The Turbulence 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 mean velocity, as happens in Newtonian laminar flows with the viscous stresses. However, the numerical methods stemming from this simplification (such as RANS) are low time consuming and can be used massively.