Week 8: Literature review - RANS derivation and analysis

AIM

To determine the following term

NS Equation with expression output for Reynold's Stress

understanding of the terms Reynolds stress

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

Turbulent

In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and fluid velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between those layers

REYNOLDS AVERAGED NAVIER-STOKES EQUATIONS
the hand of a time-averaging of the NS equations and the continuity equation for incompressible
fluids, the basic equations for the averaged turbulent flow will be derived in the following. The flow
field can then be described only with help of the mean values.

The momentary velocity component is u, the time-averaged value is named`bar(u)` and the fluctuating
velocity has the letter u′ . With help of this definition, the decomposition can mathematically be written

u = `bar(u)` + u′ , v =`bar(v)` + v′ , w = `bar(w)` + w′ , p = `bar(p)`+ p′

From Governing equation

Continuity

`(∂u)/(∂x )+(∂V)/(∂y)=0`

Momentum

`ρ(∂u⁄∂t+u(∂u)/(∂x) +v(∂u)/(∂y)=−(dp)/(dx)+μ(∂^2 u)/(∂y^2 )`

Energy

`ρ_C (∂T⁄∂t+u∂T⁄∂x+v (∂T)/(∂y))=k(∂^2u)/(∂x^2 )+μ(∂u⁄∂y)^2`

Reynolds Decomposes

`u(x,y,z,t)=bar(u)(x,y,z)+u^′ (x,y,z,t)`

`v(x,y,z,t)=bar(v)(x,y,z)+v^′ (x,y,z,t)`

`w(x,y,z,t)=bar(w)(x,y,z)+w^′ (x,y,z,t)`

Time integral

`bar(U)(x,y)=1/t_(INT) ∫_0^(t_(INT))u(x,y,z)dt`

`bar(v)(x,y)=1/t_(INT) ∫v(x,y,z)dt`

`1/t_(INT) ∫_0^(t_(INT))u^′ (x,y,z,t)dt=0`

`u(x,y,z,t)=bar(u) (x,y)+u′(x,y,z,t)`

`1/t_(INT) ∫_0^(t_(INT))u(x,y,z,t)dt= 1/t_(INT) ∫_0^(t_(INT))u (x,y) +1/t_(INT) ∫_0^(t_int)u′(x,y,z,t)`

`baru(x,y,z)=baru(x,y,z)+1/t_(INT) ∫_0^(t_(INT))u′(x,y,z,t)`

`1/t_(INT) ∫_0^(t_(INT))u′(x,y,z,t)=0`

Apply Reynolds Decomposition

Continuity equation

`(∂u)/(∂x) +(∂V)/(∂y)=0`

`(∂(baru+u^′ ))/(∂x) +(∂(¯v+v^′ ))/(∂y)=0`

`1/t_(INT) ∫(∂(baru+u^′ ))/(∂x) dt +(∂(barv+v^′ ))/(∂y) dt=0`

`1/t_(INT) ∫(∂baru)/(∂x)dt+1/t_(INT) ∫(∂u^′)/(∂x)dt+1/t_(INT) ∫(∂barv)/(∂y)dt+1/t_(INT) ∫(∂v^′)/(∂y)dt=0`

`((∂baru)/(∂x)) 1/t_(INT) ∫dt+(∂barv)/dy 1/t_(INT) dt=0`

`((∂baru)/(∂x) 1/t_(INT) ∫_0^(t_(INT))dt+((∂barv)/(∂x)) 1/t_(INT) ∫_0^(t_int)dt=0`

`((∂baru)/(∂x)) ((T_(INT)−0))/t_(INT) +(∂barv)/(∂x) ((T_(INT)−0))/t_(INT)`

Finally came up with

`(∂baru)/(∂x) +(∂barv)/(∂y)=0`

Momentum Equation

`ρ(∂u⁄∂t)+(u∂u)/(∂x) +(v∂u)/(∂y)=−(dp)/(dx)+μ(∂^2 u)/(∂y^2 )`

`(∂u⁄∂t)+(u∂u)/(∂x) +(v∂u)/(∂y)=1/ρ−(dp)/(dx)+μ/ρ (∂^2 u)/(∂y^2 )`

`∂u⁄∂t+(u∂u)/(∂x) +(v∂u)/(∂y)=1/ρ−ϑ(dp)/(dx)+(∂^2 u)/(∂y^2 )`

`(∂u⁄∂t+(u∂u)/(∂x)+(v∂u)/(∂y)+u((∂u)/(∂x)+(v∂v)/dy))=1/ρ−(dp)/(dx)+ϑ(∂^2 u)/(∂y^2 )`

`(∂u⁄∂t+2∗(u∂u)/(∂x)+(v∂u)/(∂y)+(u ∂v)/(∂x)=−1/ρ (dp)/dx+ϑ(∂^2 u)/(∂y^2 )`

Apply Product Rule

`∂/(∂y) (uv)=u (∂v)/(∂y)+v(∂u)/(∂y)`

`(∂u)/(∂t)+2u (∂u)/(∂x)+∂/(∂y) (uv)=−1/ρ (∂p)/(∂x)+(ϑ ∂^2 u)/(∂y^2 )`

`(∂u)/(∂t)+(∂u^2)/(∂x)+∂/(∂y) (uv)=−1/ρ (∂p)/(∂x)+(ϑ∂^2 u)/(∂y^2 )`

`1/t_(INT) ∫_0^(t_(INT))∂/(∂t) (baru+u^′ )dt+ ∂/(∂x) (baru+u^′ )dt+∂/(∂y) [(baru+u^′ )(barv+v^′ )]dt =−1/t_(INT) ∫_0^(t_(INT))−1/ρ [2(barp+p^′ )]/(dx)+ϑ∂^2/(∂y^2 ) (baru+u^′ )`

Simplified

`(∂baru)/(∂t)+(∂bar(u^2 ))/(∂x)+∂/(∂y) (barubarv)dt=−1/ρ (∂barp)/(∂x)+ϑ (∂^2 baru)/(∂y^2) dt−1/t_(INT) ∫_0^(t_(INT))∂/(∂x) (u^′ )^2 dt−1/t_(INT) ∫_0^(t_(INT))∂/(∂y) (u^′ v^′ )dt`

`(∂baru)/(∂t)+(∂bar(u^2 ))/(∂x)+∂/(∂y) (baru barv)dt=−1/ρ (∂barp)/(∂x)+ϑ (∂^2 baru)/(∂y^2) dt−1/t_(INT) ∫_0^(t_(INT))∂/(∂y) (u^′ v^′ )dt`

`(∂baru)/(∂t)+(∂bar(u^2 ))/(∂x)+∂/(∂y) (baru barv)=−1/ρ (∂barp)/(∂x)+1/ρ ∂/(∂y)[(μ∂baru)/(∂y)+1/t_(INT) ∫_0^(t_(INT))(u^′ v^′ )dt]`

above menstion as

Inertial term=pressure term +molecular viscosity+ momentum diffusivity because of turbulent

(viscous stress) (Reynolds Stress)

Reynolds stress

the Reynolds stress is the component of the total stress tensor in a fluid obtained from the averaging operation over the Navier-stroke equation to account for turbulent fluctuations in the fluid moments.

They arise from the nonlinear advection term when the Navier Strock equation is Reynolds decomposed and Reynolds averaged. The general form of the Reynolds stress tensor is

`tau_(ij)=-rhobaru_i baru_j`

`baru_i baru_j` is velocity correlation

`x_(iorj)` the momentum flux in direction

`[[bar(u'u'),bar(u'v'),bar(u'w']],[bar(v'u'),bar(v'v'),bar(v'w']],[bar(w'u'),bar(w'v'),bar(w'w']]]`

second order tensor

Reynolds stress equation model (RSM), also referred to as second-moment closures are the most complete classical turbulent model. In these models, the eddy-viscosity hypothesis is avoided and the individual components of the Reynolds stress tensor are directly computed

turbulent viscosity & 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 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 eddy viscosity in a turbulent flow

Conclusion

RANS is a closure loop problem for turbulent for computer expense compared to Large Eddy Simulation

According to Log law, the selection of Turbulence modeling varies bases on that RANS model varies

Due to the mystery of turbulence, there is various models is calculated according to boundary layer and Reynolds number

Reference

https://www.sciencedirect.com/topics/earth-and-planetary-sciences/eddy-viscosity

https://www.sciencedirect.com/topics/engineering/eddy-viscosity

https://glossary.ametsoc.org/wiki/Reynolds_stresses