Week 8: Literature review - RANS derivation and analysis
Aim: Literature review and RANS derivation analysis
Introduction: To understand the turbulent behaviour of the dynamic fluid is very complicated and intriguing in all classical physics. The basic theory behind the fluid flow is based on the Navier-stokes equations. In nature, most of the flows are turbulent and there as many factors which indicate the nature of the flow observed. A turbulent flow is a fluid motion in which random trajectories of pressure velocity and other parameters can be seen. There several reasons why the flow results turbulent in nature i,e due to the shear stress near the wall/boundary and high-velocity gradients. Some times it also possible to see the later mentioned nature in the laminar flow as well but that doesnt mean the flow is turbulent. In order to check whether a flow is laminar or turbulent, a Reynolds number is used which helps in predicting the flow pattern, which predicts the nature of the flow. At lower Reynolds number the flows tend to be laminar and at high Reynolds number, the flows tend to be turbulent.
Derivation Analysis:
For every flow calculation, three conservation of laws is considered. It starts with firstly conservation equation.
`∂ρ/∂t=∇.(ρu ⃗ )=0`..............................(1)
the second one is the conservation of momentum which is defined based on the following equation
`∂/∂t (ρu ⃗ )+∇.(ρu ⃗u ⃗ )= -∇p+∇.τ`..............................(2)
`In the above equation .τ represents the stress tensor.In this model conservation of energy is also taken into account.Influent this is defined as the following equation`
`∂/∂t (ρE)+∇.(ρu ⃗(ρE+p))=-∇.(∑_j▒〖h_j J_j 〗)+S_h`..........................(3)
The first term on the right-hand side of the equation represents heat production due to the conductivity. The last term Sh indicates the heat source due to the chemical reactions.
Flow is said to be turbulent if the Reynolds number which is defined below surpasses high value for a particular situation. Basically Reynolds number is the ratio between the inertial and viscous forces
`Re=ρuL/μ`...................(4)
`The idea behind RANS is to obtain the time-averaged solution for a turbulent flow problem.This could be achieved by separating the e scalar and vector variable into its means and fluctuating contribution. For velocity, it can be shown as follows T`
`u_i=(u_i ) ⃗+u_i^'` ..............................(5)
`here right hand side parameters are mean and fluctuating velocity components`
Scalar form can e written as follows :
`θ_i=(θ_i ) ⃗+θ_i^'`........................(6)
`ere theta represents scalar variable for pressure or energy `
Implementing equation 5 and 6 into the instantaneous compressible Navier stokes equation results in the following expression.
`∂ρ/∂t+∂/(∂x_i ) (ρu_i )=0`.......................(7)
`∂/∂t (ρu_i )+∂/(∂x_j ) (ρu_i u_j )=-∂p/(∂x_j ) [u((∂u_i)/(∂x_j )+(∂u_j)/(∂x_i )-2/3 ∂_ij (∂u_i)/(∂x_i ))]+∂/(∂x_i )(-ρ(u_i^' ) ̅(u_j^' ) ̅)`....................(8)
Equations 7 and 8 are called the RANS equations.Where`(u_i^' ) ̅(u_j^' ) ̅` is an extra unkown term called stress tensor , that represents the turbulence.Since we don't have an extra equations the model is not closed anymore.This is also known as the closure problem.In order to close equation (8) the reynolds stresses need to be modelled properly, which is done using the Reynolds submodels such as K-epsilon and RNG` model.Based on the type of the problem accordingly later mentioned submodels for chosen for further turbulence analysis.