Week 8: Literature review - RANS derivation and analysis

RANS (Reylond avergaed navier stokes equation) :

Introduction:

It is a modification of navier stokes equation so that it can be used to model turbulence . If we use the DNS i.e. the direct numerical system for integration , then a lot of time will be taken to complete the simulation because the turbulent time scale between which the velocity fluctuates is very less. Hence such low time step will lead to a very slow simulation .Hence , a averaged time interval is used which is actually relevant from output point of view and is large enough to complete the simulation is less time with effects of turbulence. this is why we use turbulence modelling. RANS is an approach in turbulence modelling. Findinf the reylond stress and specifically the time interval is the biggest important factor in turbulence modelling.

Explanation:

Governing equation inside a boundary layer:

Continuity equation: ∂u/∂x+ ∂v/∂y =0

Momentum equation: ρ(∂u/∂t+u∂u/∂x+v∂u/∂y)=-∂p/∂x+μ(∂^2u/∂y^2)

Reylond's decomposition:

Dividing into a fluctuating and a time averaged component.

u_av=time-averaged velocity component

u'=fluctuating component of velocity

u(x,y,z,t)=u_av(x,y,z)+u'(x,y,z,t)

v(x,y,z,t)=v_av(x,y,z)+v'(x,yx,z,t)

w(x,y,z,t)=w_av(x,y,z)+w'(x,y,z,t)

time interval= t

u_av=(1/t)*( ∫u(x,y,z,t). dt)

Applying these in navier stokes equation:

In continuity equation:

∂(u_av+u')/∂x+∂(v_av+v')/∂y=0

Integrate entire equation wrt time &do time averaging:

(1/t)*(∫(∂(u_av+u')/∂x + ∂(v_av+v')/∂y)*dt)=0

(1/t)*(∫(∂u_av/∂x)*dt+(1/t)*(∫∂v_av/∂y)*dt=0 because

(1/t)*(∫(∂u'/∂x)*dt)=0 and (1/t)*(∫(∂v'/∂y)*dt)=0

integration is done from o to t

so, ∂(u_av)/∂x+∂(v_av)/∂y=0....equation 1

this means time averaged values satisfy continuity equation because equation 1 resembles to the structure of original continuity equation.

Momentum equation:

(1/t)*(((∫∂(u_av+u')/∂t +∫(∂((u_av+u')^2)/∂x)*dt +∫(∂(u_av+u')(v_av+v')/∂y)))*dt = (-1/t)*(∫1/ρ*(∂(p_av+p')/∂x)*dt) + (1/t)*∫(1/ρ*μ*∂^2(u_av+u')/∂y^2)*dt

∂u_av/∂t+ ∂u_av^2/∂x + (1/t)*∫( ∂(u'^2)/∂x*dt)+(1/t)*(∫(∂(2*∂(u_av*u')/∂x)*dt)+(1/t)*[(∫∂((u_av)*(v_av))/∂y*dt)+ ∫∂(u_av*v')/dy *dt + ∫∂(v_av*u')/∂y*dt +∫∂(u'*v')/∂y*dt]=

1/ρ*(∂(p_av)/∂x)+ (1/t)*∫(1/ρ*μ*∂^2(u_av)/∂y^2)*dt.......equation2

now, some fluctuating terms got cancelled because there integration over time and averaging is zero.

In , the above equation 2, the term

(1/t)*∫( ∂(u'^2)/∂x*dt) is neglected because the variation , the gradients in x direction are very less as compared to y.

(1/t)*(∫(∂(2*∂(u_av*u')/∂x)*dt) term is zero because the fluctuating term's integration over time is going to be zero.

∫∂((u_av)*(v_av))/∂y*dt) term is irrespective of time so, ∂(u_av)*(v_av)/∂y

∫∂(u_av*v')/dy *dt and ∫∂(v_av*u')/∂y*dt are also zero because of time integration of fluctuating terms.

∫∂(u'*v')/∂y*dt is negative term and not zero because the fluctuation in y direction influences the fluctuation in x direction inversely.So, the final equation is

∂u_av/∂t+ ∂u_av^2/∂x+∂(u_av*v_av)/∂y=-1/ρ(∂p/∂x)+ (μ/ρ)*(∂^2(u_av)/∂y^2)-(1/t)*(∫∂(u'v')/∂y*dt)

i.e. ∂u_av/∂t+ ∂u_av^2/∂x+∂(u_av*v_av)/∂y=-1/ρ(∂p/∂x)+(1/ρ)*(∂/∂y( μ*(∂(u_av)/∂y)-(ρ/t)*(∂(u'v')/∂y*dt)) is the final reylond's avergaed navier strokes equation.

This term μ*(∂(u_av)/∂y) is the viscous stress, this is the momentum diffusion because of molecular viscosity and

(ρ/t)*(∫∂(u'v')/∂y*dt) is the reylond stress used in turbulence modelling, it shows the momentum diffusion because of the turbulence. Reylond stress is the most important value to be found in the turbulence modelling. With high or low value of reylond number we find location of our first computational cell in the flow.

Molecular viscosity:dominating in the viscous-sub laminar layer

It is momentum diffusion due to molecular motion . Molecular viscosity can be understood as the friction between molecules. The more the viscosity, the more the energy required for the fluid to flow.

Turbulent viscosity:dominating in the turbulent region in boundary layer

It is the momentum diffusion because of turbulent mixing of particles. In turbulent flow, fluid moves in random fashion unlike laminar flow and eddies develop which make the fluid more viscous or difficult to flow . Hence, this develops turbulent viscosity.

The ratio of molecular and turbulent viscosity is basically used to find the type of flow i.e. laminar or turbulent. Inside the viscous laminar region the visous stresses dominates and inside the turbulent boundary layer , the reylond stress dominates.