Week 8: Literature review - RANS derivation and analysis
Derivation and analysis of Reynolds-averaged Navier-Stokes equations
I. Aim
The aim of the current project is to:
i. Apply Reynolds decompostion and Reynolds-averaging to the Navier-Stokes equations and derive the expression for Reynolds stress.
ii. Explain Reynolds stress.
iii. Explain turbulent viscosity.
II. Theoretical background
The Navier-Stokes equations given below govern the constant-density flow (incompressible flow) of a Newtonian fluid (fluid which obeys Newton's law of viscosity) in the absence of body forces:
Continuity equation: `(delu)/(delx)+(delv)/(dely)+(delw)/(delz)=0`
X-Momentum equation: `rho((delu)/(delt)+u(delu)/(delx)+v(delu)/(dely)+w(delu)/(delz))=-(delp)/(delx)+mu((del^2u)/(delx^2)+(del^2u)/(dely^2)+(del^2u)/(delz^2))`
Y-Momentum equation: `rho((delv)/(delt)+u(delv)/(delx)+v(delv)/(dely)+w(delv)/(delz))=-(delp)/(dely)+mu((del^2v)/(delx^2)+(del^2v)/(dely^2)+(del^2v)/(delz^2))`
Z-Momentum equation: `rho((delw)/(delt)+u(delw)/(delx)+v(delw)/(dely)+w(delw)/(delz))=-(delp)/(delz)+mu((del^2w)/(delx^2)+(del^2w)/(dely^2)+(del^2w)/(delz^2))`
In tensor notation, the above equations can be represented as below:
Continuity equation: `(delu_i)/(delx_i)=0`
Momentum equation: `rho((delu_i)/(delt)+u_j(delu_i)/(delx_j))=-(delp)/(delx_i)+mu(del^2u_i)/(delx_j^2)`
The above equations can be used to describe the behavior of both the laminar flow and turbulent flow. In other words, if the above equations are numerically solved using a sufficiently small time-step and grid size, the turbulence in the flow can be captured accurately. The type of simulation which accomplishes this task is termed as Direct Numerical Simulation (DNS). However such simulations are quite expensive computationally. This is because the turbulence in the flow spans over a wide range of length- and time-scales; and in order to capture the smallest scale of turbulence effectively, quite small time-step and grid size are required. Performing such simulations for engineering systems, such as flow over an aircraft, flow through a turbine, flow inside an internal combustion engine etc., is quite impractical. Hence an alternate approach is adopted for capturing turbulence in engineering systems. In this approach, the mean flow behavior is described by solving the governing equations without the knowledge of random flow behavior. These governing equations (for the mean flow) are known as the Reynolds-averaged Navier-Stokes (RANS) equations, and are derived from the Navier-Stokes equations using the following two operations:
1. Reynolds decomposition: It is the decomposition of turbulent flow variables into their mean and fluctuating flow components.
Hence if `f(x,y,z,t)` represents the turbulent flow variable, `bar(f)(x,y,z)` represents the mean flow component and `f'(x,y,z,t)` represents the fluctuating flow component, then:
`f(x,y,z,t)=bar(f)(x,y,z)+f'(x,y,z,t)`
2. Reynolds-averaging: It is the act of performing time-average (time-integration) of turbulent flow variables over a sufficiently long time interval, such that the resultant quantity describes the mean flow behavior.
Hence if `f(x,y,z,t)` represents the turbulent flow variable, then Reynolds-averaging yields:
`bar(f)(x,y,z)=1/Tint_0^Tf(x,y,z,t)dt`
Henceforth the operation of Reynolds-averaging will be denoted by an overbar.
Some properties of Reynolds-averaging which are useful in the derivation of RANS equations are as follows:
i. `bar(f+g)=bar(f)+bar(g)`
ii. `bar(f')=0`
iii. `bar(bar(f))=bar(f)`
iv. `bar(bar(f)g')=bar(f)bar(g')=0`
v. `bar(bar(f)^2)=bar(bar(f)bar(f))=bar(f)bar(f)=bar(f)^2`
III. Derivation of RANS equations
The Navier-Stokes equations for the incompressible flow of a Newtonian fluid in the absence of body forces are given below in the tensor notation:
Continuity: `(delu_i)/(delx_i)=0`
Momentum: `rho((delu_i)/(delt)+u_j(delu_i)/(delx_j))=-(delp)/(delx_i)+mu(del^2u_i)/(delx_j^2)`
Let us first consider the continuity equation.
`(delu_i)/(delx_i)=0`
Performing Reynolds decomposition for `u_i` we get:
`(del(bar(u_i)+u_i'))/(delx_i)=0`
Performing Reynolds-averaging on the above equation we get:
`bar((del(bar(u_i)+u_i'))/(delx_i))=0`
`(del(bar(bar(u_i)+u_i')))/(delx_i)=0`
Using property i. we get:
`(del(bar(bar(u_i))+bar(u_i')))/(delx_i)=0`
Using properties ii. and iii. we get:
`(delbar(u_i))/(delx_i)=0`
Hence the mean flow is governed by the same continuity equation without the addition of removal of any terms.
Let us consider the momentum equation:
`rho((delu_i)/(delt)+u_j(delu_i)/(delx_j))=-(delp)/(delx_i)+mu(del^2u_i)/(delx_j^2)`
`(delu_i)/(delt)+u_j(delu_i)/(delx_j)=-1/rho(delp)/(delx_i)+nu(del^2u_i)/(delx_j^2)`
Upon multiplying the continuity equation with `u_i` and adding it to the LHS of the above equation we get:
`(delu_i)/(delt)+u_j(delu_i)/(delx_j)+u_i(delu_j)/(delx_j)=-1/rho(delp)/(delx_i)+nu(del^2u_i)/(delx_j^2)`
`(delu_i)/(delt)+(del(u_iu_j))/(delx_j)=-1/rho(delp)/(delx_i)+nu(del^2u_i)/(delx_j^2)`
Performing Reynolds decomposition for `u_i` and `p` we get:
`(del(bar(u_i)+u_i'))/(delt)+(del((bar(u_j)+u_j')(bar(u_i)+u_i')))/(delx_j)=-1/rho(del(bar(p)+p'))/(delx_i)+nu(del^2(bar(u_i)+u_i'))/(delx_j^2)`
`(del(bar(u_i)+u_i'))/(delt)+(del(bar(u_i) \ bar(u_j)+bar(u_i)u_j'+u_i'bar(u_j)+u_i'u_j'))/(delx_j)=-1/rho(del(bar(p)+p'))/(delx_i)+nu(del^2(bar(u_i)+u_i'))/(delx_j^2)`
Performing Reynolds-averaging on the above equation we get:
`bar((del(bar(u_i)+u_i'))/(delt)+(del(bar(u_i) \ bar(u_j)+bar(u_i)u_j'+u_i'bar(u_j)+u_i'u_j'))/(delx_j))=bar(-1/rho(del(bar(p)+p'))/(delx_i)+nu(del^2(bar(u_i)+u_i'))/(delx_j^2))`
Using property i. we get:
`bar((del(bar(u_i)+u_i'))/(delt))+bar((del(bar(u_i) \ bar(u_j)+bar(u_i)u_j'+u_i'bar(u_j)+u_i'u_j'))/(delx_j))=bar(-1/rho(del(bar(p)+p'))/(delx_i))+bar(nu(del^2(bar(u_i)+u_i'))/(delx_j^2))`
`(delbar((bar(u_i)+u_i')))/(delt)+(delbar((bar(u_i) \ bar(u_j)+bar(u_i)u_j'+u_i'bar(u_j)+u_i'u_j')))/(delx_j)=-1/rho(del(bar(bar(p)+p')))/(delx_i)+nu(del^2bar((bar(u_i)+u_i')))/(delx_j^2)`
Using property i. again we get:
`(del(bar(bar(u_i))+bar(u_i')))/(delt)+(del(bar(bar(u_i) \ bar(u_j))+bar(bar(u_i)u_j')+bar(u_i'bar(u_j))+bar(u_i'u_j')))/(delx_j)=-1/rho(del(bar(bar(p))+bar(p')))/(delx_i)+nu(del^2(bar(bar(u_i))+bar(u_i')))/(delx_j^2)`
Using properties ii. and iii. we get:
`(delbar(u_i))/(delt)+(del(bar(u_i) \ bar(u_j)+bar(u_i'u_j')))/(delx_j)=-1/rho(delbar(p))/(delx_i)+nu(del^2bar(u_i))/(delx_j^2)`
Rearranging the above equation we get:
`(delbar(u_i))/(delt)+(del(bar(u_i) \ bar(u_j)))/(delx_j)=-1/rho(delbar(p))/(delx_i)+nu(del^2bar(u_i))/(delx_j^2)-(del(bar(u_i'u_j')))/(delx_j)`
`(delbar(u_i))/(delt)+bar(u_j)(delbar(u_i))/(delx_j)+bar(u_i)(delbar(u_j))/(delx_j)=-1/rho(delbar(p))/(delx_i)+nu(del^2bar(u_i))/(delx_j^2)-(del(bar(u_i'u_j')))/(delx_j)`
The third term on the LHS is nothing but `bar(u_i)` multiplied by the Reynolds-averaged continuity equation. Hence:
`(delbar(u_i))/(delt)+bar(u_j)(delbar(u_i))/(delx_j)=-1/rho(delbar(p))/(delx_i)+nu(del^2bar(u_i))/(delx_j^2)-(del(bar(u_i'u_j')))/(delx_j)`
`rho((delbar(u_i))/(delt)+bar(u_j)(delbar(u_i))/(delx_j))=-(delbar(p))/(delx_i)+mu(del^2bar(u_i))/(delx_j^2)-(del(rhobar(u_i'u_j')))/(delx_j)`
Hence the momentum equation for mean flow is same as the momentum equation for turbulent flow, except the last term. This last term is a derivative of the Reynolds stress tensor which is given by:
`tau_(ij,Reynolds)=-rhobar(u_i'u_j')`
Reynolds stresses manifest as a consequence of performing time-averaging of the momentum equation for turbulent flow. They originate out of the term for turbulent convection. Hence their origin is not physical, like viscous stresses, but mathematical. In this sense, they are not actually stresses but are termed so because they cause momentum diffusion, similar to the viscous stresses. It can be noted that the RANS equations are a set of four equations (`i=1,2,3`) with 10 unknowns (`p; bar(u_i), i=1,2,3; and bar(u_i'u_j'), i=1,2,3, j<=i`). Hence for closure of the turbulence problem, the extra six components of Reynolds stresses (containing unknown fluctuating flow variables) need to be modeled. Hence turbulence modeling boils down to modeling the Reynolds stresses appearing in the RANS equations.
The molecular viscosity (`mu`) diffuses momentum at the molecular level. This effect appears as viscous stress in the governning equations. Since the diffusive nature of Reynolds stresses is similar to viscous stresses, they are often modeled using a turbulent viscosity as follows:
`rhobar(u_i'u_j')=mu_T((delbar(u_i))/(delx_j)+(delbar(u_j))/(delx_i))`
However it should be noted that unlike molecular viscosity, the turbulent viscosity is not a physical property of the material and is only used for mathematical modeling.