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Week 8: Literature review - RANS derivation and analysis
Reynolds-averaged Navier Stokes Equation The Reynolds-averaged Navier–Stokes equations or RANS equations are a time-averaged equations of motion for fluid flow. The idea behind the equations is Reynolds decomposition, whereby an instantaneous quantity is decomposed into its time-averaged and fluctuating quantities,…
Sudharsan Vijayan
updated on 31 May 2020
Reynolds-averaged Navier Stokes Equation
The Reynolds-averaged Navier–Stokes equations or RANS equations are a time-averaged equations of motion for fluid flow. The idea behind the equations 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 equations are primarily used to describe turbulent flows. These equations can be used with approximations based on knowledge of the properties of flow turbulence to give approximate time-averaged solutions to the Navier–Stokes equations.
Reynolds Decomposition
u(x,y,z,t)=ˉu(x,y,z)+u′(x,y,z,t)
v(x,y,z,t)=ˉv(x,y,z)+v′(x,y,z,t)
w(x,y,z,t)=ˉw(x,y,z)+w′(x,y,z,t)
Contiunity Equation
The Contiunity Equation is as follows
∂u∂x+∂v∂y=0
∂∂x(ˉu+u′)+∂∂y(ˉv+v′)=0
1tint∫∂∂x(ˉu+u′)dt+∂∂y(ˉv+v′)dt=0
1tint∫∂ˉu∂xdt+1tint∫∂u′∂xdt+1tint∫∂ˉv∂ydt+1tint∫∂v′∂ydt=0
we can assume that 1tint∫∂u′∂xdt&1tint∫∂v′∂ydt=0
Therefore we have
1tint∫∂ˉu∂xdt+1tint∫∂ˉv∂ydt=0
Applying the time integral
1tint∫tint0∂ˉu∂xdt+1tint∫tint0∂ˉv∂ydt=0
Hence we get the final equation
∂ˉu∂x+∂ˉv∂y=0
Momentum Equation
The momentum equation is given as follows
ρ(∂u∂t+u∂u∂x+v∂u∂y)=-∂p∂x+μ∂2u∂y2
Taking rho to the otherside we get
∂u∂t+u∂u∂x+v∂u∂y=1ρ-∂p∂x+μρ∂2u∂y2
we know that mu/rho is gamma
∂u∂t+u∂u∂x+v∂u∂y=1ρ-∂p∂x+γ∂2u∂y2
Multiplying the Xcomponent of velocity to LHS of the equation
∂u∂t+u∂u∂x+v∂u∂y+u(∂u∂x+∂v∂y)=1ρ-∂p∂x+γ∂2u∂y2
On simplifing
∂u∂t+2u∂u∂x+v∂u∂y+u(∂v∂y)=1ρ-∂p∂x+γ∂2u∂y2
By product rule ∂∂y(uv)=u∂v∂y+v∂u∂y
∂u∂t+2u∂u∂x+∂∂y(uv)=1ρ-∂p∂x+γ∂2u∂y2
∂u∂t+∂∂x(u2)+∂∂y(uv)=1ρ-∂p∂x+γ∂2u∂y2
Applying Time Intergration
1tint∫tinto∂∂t(ˉu+u′)dt+∂∂x(ˉu+u′)2dt+∂∂y(ˉu+u′)(ˉv+v′)dt=1tint∫tinto-1ρ∂∂x(ˉp+p′)dt+γ∂2∂y2(ˉu+u′)dt
1tint∫tinto∂ˉu∂tdt+∂u′∂tdt+∂ˉu2∂xdt+∂u′2∂xdt+(∂∂x(2ˉuu′))dt+(∂∂y(ˉuˉv))dt+(∂∂y(ˉuv′))dt+(∂∂y(u′ˉv))dt+(∂∂y(u′v′))dt=1tint∫tinto(-1ρ∂p∂xdt-1ρ∂p′∂xdt+γ∂2ˉu∂y2dt+γ∂2u′∂y2dt
from the above equation some terms are zero
∂u′∂tdt&(∂∂x(2ˉuu′))dt&(∂∂y(ˉuv′))dt&(∂∂y(u′ˉv))dt&-1ρ∂p′∂xdt&γ∂2u′∂y2dt
Therfore we get the following equation
∂ˉu∂t+∂ˉu2∂x+∂∂y(ˉuˉv)=-1ρ∂p′∂x+γ∂2u′∂y2-1tint∫tinto(∂∂x)ˉu2dt-1tint∫tinto(∂∂y)ˉuˉvdt
On simplifying we
∂ˉu∂t+∂ˉu2∂x+∂∂y(ˉuˉv)=-1ρ∂p′∂x+γ∂2u′∂y2-1tint∫tinto(∂∂y)ˉuˉvdt
On simplifying the RHS we get
`(delbaru)/(delt)+(delbaru^2)/(delx)+del/(dely)(baru barv)= -1/rho (delp')/(delx)1/rho del/(dely)(mu(del u')/(dely) + rho/(ti nt) int_o^(ti nt) barubarv dt)`
Reynolds Stress
It is the total stress tensor in fluid obtained from averaging the navier stokes equation in the account for turbulent fluctuation in a momentum of the fluid.
Turbulent Viscosity
In a turbulent fluid, a linear interface between two different fluids breaks apart to form small-scale structures, also known as eddies and this viscosity is called turbulent viscosity.A coefficient that relates the average shear stress within a turbulent flow of water or air to the vertical gradient of velocity.
Molecular 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 turbulent flow.
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