Week 8: Literature review RANS derivation and analysis

Literature review RANS derivation and analysis

Objective:-

1. Apply reynold's decomposition to the NS equation and come up with the expression for reynold's stress.

2. Explain your understanding of the terms reynold's stress.

3. What is turbulent viscosity? How is it differ from molecular viscosity?

Theory:-

Navier-Stokes (NS) equation:-

The Navier-stokes equations mathematically express conservation of momentum and conservation of mass for Newtonian fluids. They are sometimes accompanied by an equation of state relating pressure, temperature and density. They arise from applying Isaac Newton's second law of fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term and a pressure term. The difference between them and the closely related Euler equations is that the Navier-Stokes equations take viscosity into account while the Eulers equation model only inviscid flow.

As a result, the Navier-Stokes equation are a parabolic equation and therefore have better analytic properties, at the expense of having less mathematical structure.

The Navier-Stokes equation are useful because they describe the physics of many phenomena of scientific and engineering interest. They may be used to model the weather, Oceans currents, water flow in a pipe and airflow around a wing. The Navier-Stokes equation, in their full and simplification forms, helps with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things. Coupled with Maxwell's equations, they can be used to model and study magnetohydrodynamics.

Reynolds-average Navier-stokes equations:-

The Reynolds-average Navier-stokes equations are the time-averaged equations of motion for fluid flow. The idea behind the equation is Reynolds decomposition, whereby an instantaneous quantity is decomposed into its time-averaged and fluctuating quantities, an idea first proposed by Osborne Reynolds.

The RANS equation is primarily used to describe turbulent flows. These equations can be used with approximations based on knowledge of the properties of flow turbulence to give an approximate time average solution to the Navier-Stokes equation.

Reynolds Decomposition:-

For the quantity `u` the decomposition would be

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

where `baru` denotes the expectation value of `u`called spatial, and `u^'` are the deviation from the expectation value or fluctuations. The fluctuation are defined as the expectation value subtracted from quantity `u` such that their time average equals zero.

The expected value, `baru` is often found from an ensemble average which is an average taken over multiple experiments under identical conditions. the expected value is also sometimes denoted u, but it is also often with the over-bar notation.

Direct numerical simulation, or resolving the Navier-stokes equation completely in `(x,y,z,t),` is only possible on small computational grids and small-time steps when Reynold numbers are low. Due to computational constraints, simplification of the Navier-Stokes equation is useful to parameterize turbulence that is smaller than the computational grid, allowing larger computational domains.

Reynolds decomposition allows the simplification of the Navier-Stokes equations by substituting in the sum of the steady components and perturbations to the velocity profile and taking the mean value. The resulting equation contains a nonlinear term known as the Reynolds stress which gives rise to turbulence.

Governing equation for the boundary layer:-

Continuity Equation:-

`(∂u)/(∂x) + (∂v)/(∂y) = 0`

Momentum equation:-

`rho ((∂u)/(∂t) + (u∂u)/(∂x) + (v∂u)/(∂y)) = (-dp)/(dx) + (mu∂^2u)/(∂y^2)`

Energy Equation:-

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

Q1. Apply Reynolds decomposition to the NS equation and come up with the expression for the reynold's stress.

Reynolds Decomposition:-

`ubrace (u(x,y,z,t))_("Instantaneous velocity")= ubrace (baru(x,y,z))_("Function of Spacial") + ubrace (u^'(x,y,z,t))_("Function of Spacial & time")`

• `baru = "Time average Quantity"`
• `u^' = "Fluctuating quantities"`

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

`w(x,y,z,t) = barw(x,y,z) + w^'(x,y,z,t)`

`baru(x,y) = 1/tint_0^t u(x,y,t)dt -> color(red) 1`

`barv(x,y) = 1/tint_0^tv(x,y,z)dt -> color(red) 2`

Integrate the instantaneous velocity of u and v with respect to time along x and y direction.

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

`1/tint_0^t u(x,y,z,t) dt = 1/tint_0^tbaru(x,y)dt +1/tint_0^tu^'(x,y,z,t)dt to color(red)3`

Apply equ 1 in 3,

`cancel(baru(x,y,z)) = cancel(baru(x,y,z)) + 1/tint_0^tu^'(x,y,z,t)dt`

`1/tint_0^tu^'(x,y,z,t) = 0`

The time integration on fluctuating components is resulting in zero.

Applying Reynold's Decomposition on Continuity Equation.

`text(continuity equation) to ((delu)/(delx)+ (delv)/(dely)) = 0`

`del/(delx)(baru+u^') + del/(dely)(barv+v^') = 0`

Apply time integration,

`1/tint_0^t(delbaru)/(delx)dt + cancel(1/tint_0^t(delu^')/(delx)dt) + 1/tint_0^t(delbarv)/(dely)dt + cancel(1/tint_0^t(delv^')/(dely)dt)`

fluctuating components is equal to zero.

`(delbaru)/(delx) 1/tint_0^t dt + (delbarv)/(dely) 1/tint_0^tdt = 0`

`(delbaru)/(delx) [cancel((t-0)/t)] + (delbarv)/(dely) [cancel((t-0)/t)] = 0`

`(delbaru)/(delx) +(delbarv)/(dely) = 0`

`therefore "The time average velocity components satisfy the continuity equation."`

Apply Reynold's Decomposition on Momentum Equation:-

`text(Momentum Equation) to rho((delu)/(delt) +(udelu)/(delx) +(vdelu)/(dely)) = (-d)/(dx) +mu(del^2u)/(dely^2)`

`rho` is moved from LHS to RHS

`(delu)/(delt) +(udelu)/(delx) +(vdelu)/(dely) = (1/rho)(-delp)/(delx) +(mu/rho) (del^2u)/(dely^2)`

`text(Kinematic viscosity) (nu) = mu/rho`

`(delu)/(delt) +(udelu)/(delx) + (vdelu)/(dely) = (1/rho)(-delp)/(delx) +nu(del^2u)/(dely^2)`

For further simplification, the term continuituity equation (zero term) is added on LHS.

`((delu)/(delt) + (udelu)/(delx) +(vdelu)/(dely) +u((delu)/(delx)+(delv)/(dely))) = (1/rho)(-delp)/(delx) + nu(del^2u)/(dely^2)`

`((delu)/(delt) +(2udelu)/(delx) +(vdelu)/(dely) +(udelv)/(dely)) = (1/rho) (-delp)/(delx) +nu(del^2u)/(dely^2)`

`text(The term) ((vdelu)/(dely) +(udelv)/(dely)) text(is converted into) ((del)/(dely)(uv))`

`(delu)/(delt) +(2udelu)/(delx) +(del)/(dely)(uv) = -(1/rho)(delp)/(delx) + nu (del^2u)/(dely^2)`

`(delu)/(delt) + (del)/(delx)(u^2) + (del)/(dely)(uv) = -(1/rho) (delp)/(delx) + nu (del^2u)/(dely^2)`

Apply time integration and Reynold's decomposation,

`1/t int_0^t(del)/(delt) (baru+u^')dt + (del)/(delx)(baru+u^')^2 dt + (del)/(dely)[(baru+u^')(barv+v^')]dt = 1/t int_0^t-(1/rho) [(del)/(delx) (barp + p^')]dt + nu(del^2)/(dely^2) (baru +u^') dt`

`1/t int_0^t (delbaru)/(delt) dt + cancel((delu^')/(delt)dt) + (delbaru^2)/(delx)dt + cancel(((delu^')^2)/(delx)dt) + cancel((del2baru u^')/(delx)dt) + (del)/(dely)(baru barv)dt + cancel((del)/(dely)(baruv^')dt) + cancel((del)/(dely)(u^'barv) dt) + cancel((del)/(dely)(u^'v^')dt) = 1/t int_0^t (-1)/(rho) (delbarp)/(delx) dt cancel((-1/rho)(delp^')/(delx)dt) + (nu2^2baru)/(dely^2)dt + cancel((nu2^2u^')/(dely^2)dt)`

`(delu)/(delt) +(del(baru^2))/(delx) +(del)/(dely)(barubarv) = -1/rho (delbarp)/(delx) + (nudel^2baru)/(dely^2) cancel(-1/t int_0^t (del)/(delx)(u^')dt) -1/t int_0^t(del)/(dely) (u^'v^')dt`

`(delbaru)/(delt) + (del(baru)^2)/(delx) + (del)/(dely)(barubarv) = -1/rho (delbarp)/(delx) + (nudel^2baru)/(dely^2) -1/tint_0^t (del)/(dely) (u^'v^') dt`

`ubrace((delbaru)/(delt) + (del(baru)^2)/(delx) + (del)/(dely)(barubarv))_("Inertial Stress") = ubrace(-1/rho (delbarp)/(delx))_("Pressure Terms") +1/rho (del)/(dely) [ ubrace((mu(delbaru)/(dely)))_("Viscous stress") + ubrace(rho(bar(u^'v^'))dt)_("Reynolds Stress")]`

Q2. Reynold's Stress.

• In the Navier-Stokes equation, the total stress tensor in a fluid is obtained from the time-averaging operation to account for turbulent fluctuations in fluid momentum.
• The Reynolds stress is quite complex, but it is generally accepted that they are the 'simplest' type of model. The Reynolds stress is by no means as well validated as the k-epsilon model, and because of the high cost of the computations, it is not so widely used in industrial flow calculation.
• The computational cost is very high and not as widely validated as the mixing length and k-epsilon model. The performs just as poorly as the k-epsilon model in some flows due to identical problems with the e-equation.
• It is also known as the second-order equation.

Q3. What is Turbulent Viscosity? How it is different from molecular viscosity?

• Turbulent viscosity (or) Eddy viscosity models are widely used for industrial, aeronautical, metrological and oceanographical applications. They are based on a constitutive equation, giving a relation between the Reynolds stress tensor and the mean velocity gradient tensor.
• The molecular viscosity is a property of the fluid, the eddy viscosity is a property of the flow.