Literature review RANS derivation and analysis

Objectives:

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

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

The Reynolds-averaged Navier-Stokes equations are time-averaged equations of motion for fluid flow. These equations can be used with approximations based on knowledge of the properties of flow turbulence to give appropriate time-averaged solutions to the Navier stokes equations.

Reynold's Decomposition: is a mathematical technique used to separate the expected value of a quantity from its fluctuations.

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

here, `bar u` denotes the expectation value of u (also called steady component, spatial or ensemble average) and `u' are the deviations from the expectation value (or fluctuations) For solving entire Navier stokes equation would require DNS approach which needs `10e-19` size cells grid. Due to these computational constraints, simplifications of the Navier Stokes equations are useful to parameterize turbulence that is smaller than the computational grid allowing larger computational domains.

Time averaged velocity for Steady flows `u_t(x) = (1/T)int_0^T u(x,t) dt`

Time averaged velocity for Unsteady flows `u_t(x, t_n) = (1/T)int_(t_n -1/2T)^(t_n +1/2T) u(x,t) dt`

For unsteady flow the T is the spontaneous time period which is shorter than entire time period, therefore the averaged velocity is time-dependent.

For simplicity, Reynolds decompositions to use for the derivation as follows

`u = U + u'`

`p = P + p'`

Averaging

• A simple averaging `U = barU` After averaging the averaged velocity would be constant with regard to the averaging operation
• `bar U = bar bar u = bar u = U` The averaging of the averaged velocity would be the averaged velocity itself
• The averaging of the fluctuating velocity is zero.
  • `u = U + u'`
  • `bar u = bar(U +u') = bar U + bar u' = U + bar u'`
  • Hence, `bar u' = 0`

The Averaging of the fluctuating velocity derivation with respect to time

`(del u)/(del t) = (del U)/(del t) + (del u')/(del t)`

`bar(del u)/(del t) = bar((del U)/(del t) + (del u')/(del t)) = bar(del U)/(del t) + bar(del u')/(del t)`

Since, `bar(del u)/(del t) approx bar(del U)/(del t)`

`cancel(bar(del u)/(del t)) = cancel(bar(del U)/(del t)) + bar(del u')/(del t) = 0`

Navier Stokes Equations for an Incompressible fluid flow:

`(del u_i)/(dt) + (del(u_i u_j))/(delx_j) = 1/rho f_i - 1/rho (del P)/(del x_i) + nu (del^2u_i)/(del x_j del x_j)`

where,

• `f_i =` the body force per volume
• `P =` The fluid Pressure
• `nu =` Kinematic viscosity
• `u_i u_j =` the tensor of the double velocity vector (`vec V vec V` )

Reynolds decompositions, we get

`del/(del t) (bar U_i + bar u'_i) + del/(del x_j) [( U_i + u_i') ( U_j+ u_j')] = 1/rho f_i - 1/rho del/(del x_i) (bar P + p') + nu del^2/(del x_j del x_j) (bar U_i + bar u'_i)`

Averaging the above equation

`ubrace(bar(del/(del t) (bar U_i + bar u'_i)))_(1) + ubrace(bar(del/(del x_j) [( U_i + u_i') ( U_j+ u_j')]))_(2) = ubrace(bar(1/rho f_i))_(3) - ubrace(bar(1/rho del/(del x_i) (bar P + p')))_(4) + ubrace(bar(nu del^2/(del x_j del x_j) (bar U_i + bar u'_i)))_(5)`

(1) `del /(del t) (bar(U_i + u_i')) = (delbarU_i)/(del t) + cancel((delbaru_i')/(del t) ) = (delbarU_i)/(del t) = (delU_i)/(del t)`

(2) `del / (del t) (bar( U_i U_j + u'_i u'_j + U_iu'_j + U_j u'_i)) = del / (del x_j) (bar( U_i U_j )) + del / (del x_j) (bar( u_i' u_j' )) + cancel(del / (del x_j) (bar( U_i u_j' ))) + cancel(del / (del x_j) (bar( U_j u_i')))`

`therefore del / (del x_j) (bar( U_i U_j )) + del / (del x_j) (bar( u_i' u_j' ))`

The averaging of a single fluctuating velocity is zero, but the averaging of the double fluctuating velocity is not zero.

(3) `1/rho barf_i = 1/rho f_i`

(4) `1/rho del/(del x_i) (bar (P + p')) = 1/rho bar(del P)/(del x_i) + cancel (1/rho bar(del p')/(del x_i)) = 1/rho bar(del P)/(del x_i)`

(5) `nu del^2/(del x_j del x_j) (bar (U_i + bar u'_i)) = nu del^2/(del x_j del x_j) bar U_i + cancel(nu del^2/(del x_j del x_j) bar u_i ') = nu del^2/(del x_j del x_j) bar U_i`

Putting (1) (2) (3) (4) & (5) together, we will come up with RANS

`(delU_i)/(del t) + del / (del x_j) (bar( U_i U_j )) + del / (del x_j) (bar( u_i' u_j' )) = 1/rho f_i - 1/rho bar(del P)/(del x_i) + nu del^2/(del x_j del x_j) bar U_i`

This equation can be written as,

`(delU_i)/(del t) + del / (del x_j) (bar( U_i U_j )) = 1/rho f_i - 1/rho bar(del P)/(del x_i) + del/(del x_j)(nu (del U_i)/(del x_j ) - bar (u'_i u'_j))`

This is Reynolds-averaged Navier-Stokes equation and Reynolds Stress tensor is `tau _(ij) = -bar(u'_iu'_j`

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 the 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.