Week 8: Literature review - RANS derivation and analysis

Need of Turbulence Modelling

Accurate solution of turbulent phenomenon requires much finer grid size. The trivial process would require huge computational resources. Thus, in order to arrive approximate results, time averaging convept is used to solve the turbulent phenomenon. RANS or Reynold averages Navier Stokes model is one of those approaches.

Derivation of expression of Reynold's Stress

The Navier Stokes Equation in 2 Dimention is-

The Continuity Equation

`{delu}/{delx} + {delv}/{dely} = 0`

The X-Momentum Equation

`rho({delu}/{delt} + u{delu}/{delx} +v{delv}/{dely}) = -{delp}/{delx} + mu{del^2u}/{dely^2}`

The Energy Equation

`rhoc({delT}/{delt} + u{delT}/{delx} + v{delT}/{dely}) = -k{del^2T}/{delx^2} + mu{del^2u}/{dely^2}`

Assuming `phi` to be a flow parameter, `bar{phi}` is the mean component of the flow variable and `phi'` is the fluctuating component.

Now, the flow parameters velocity can be expressed in terms of mean and fluctuating components.

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

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

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

Substituting the above expressions in continuity equation:

`{del}/{delx}(bar(u) + u') + {del}/{dely}(bar(v) + v') = 0`

time averaging,

`{1}/{t}int({del}/{delx}(bar(u) +u') + {del}/{dely}(bar(v)+ v')) = 0`

approximating the fluctuating component to zero,

`{1}/{t}int{delbar(u)}/{delx}dt + cancel({1}/{t}int{delu'}/{delx}dt)^0 + {1}/{t}int{delbar(v)}/{dely}dt + cancel({1}/{t}int{delv'}/{dely}dt)^0 = 0`

`{1}/{t}{delbar(u)}/{delx}intdt + {1}/{t}{delbar(v)}/{dely}intdt = 0`

`{1}/{t}{delbar(u)}/{delx}[t]_0^t + {1}/{t}{delbar(v)}/{dely}[t]_0^t = 0`

`{delbar(u)}/{delx} + {delbar(v)}/{dely} = 0`

Momentum Equation:-

`rho({delu}/{delt} + u{delu}/{delx} +v{delv}/{dely}) = -{delp}/{delx} + mu{del^2u}/{dely^2}`

Adding the product of x component of velocity to the continuity equation to the left side of momentum equation

`{delu}/{delt} + u{delu}/{delx} +v{delv}/{dely} + u({delu}/{delx} + {delv}/{dely})= -{1}/{rho}{delp}/{delx} + {mu}/{rho}{del^2u}/{dely^2}`

Adding the like terms,

`{delu}/{delt} + 2u{delu}/{delx} +v{delv}/{dely} + u{delv}/{dely}= -{1}/{rho}{delp}/{delx} + {mu}/{rho}{del^2u}/{dely^2}`

Applying Product rule (vdu/dx+udv/dx = duv/dx)

`=> {delu}/{delt} + {delu^2}/{delx} + {deluv}/{dely}= -{1}/{rho}{delp}/{delx} + {mu}/{rho}{del^2u}/{dely^2}`

Time Averaging the momentum equation,

Substituting the expression of velocity described earlier in the momentum equation.

`{1}/{t}int_0^t({del(bar(u)+u')}/{delt}) + {1}/{t}int_0^t({del}/{delx}(bar(u) + u')^2 )+ {1}/{t}int_0^t({del}/{dely}((bar(u) + u')(bar(v) + v')) )= {1}/{t}int_0^t(-{1}/{rho}{del(bar(p) + p')}/{delx} + nu{del^2(bar(u)+u')}/{dely^2})dt`

Where, `nu` is the Kinematic viscosity

`{1}/{t}int_0^t({delbar(u)}/{delt}dt + cancel({delu'}/{delt})^0dt + {delbar(u)^2}/{dx}dt + {delu'^2}/{dx}dt + cancel(2{del(u')(bar(u))}/{dy}dt)^0 +{delbar(u)bar(v)}/{dely}dt + cancel({del(bar(u)v')}/{dely}dt)^0 + cancel({delbar(v)u'}/{dely}dt)^0 + {del(u'v')}/{dely}dt) `= `{1}/{t}int_0^t(-{1}/{rho}{delbar(p)}/{delx}dt - cancel({1}/{rho}{delp'}/{delx})^0 + nu{del^2bar(u)}/{dely^2}dt + cancel(nu{del^2u'}/{dely^2}dt)^0)`

Rearranging the above equations:

`{delbar(u)}/{delt} + {delbar(u)^2}/{delx} + {del(bar(u)bar(v))}/{dely} + {1}/{t}int_0^t{delu'^2}/{delx}dt + {1}/{t}int_0^t{delu'v'}/{dely}dt= -{1}/{rho}{delbar(p)}/{delx} + nu{del^2bar(u)}/{dely^2}`

Since the thickness of boundarylayers are very less, so the gradient of velocity compoenent in the x direction is very small, thus

`{1}/{t}int_0^t{delu'^2}/{delx}dt` can be assumed zero

Now, the renyolds average NS equation can be written as,

`{delbar(u)}/{delt} + {delbar(u)^2}/{delx} + {del(bar(u)bar(v))}/{dely} = -{1}/{rho}{delbar(p)}/{delx} +{1}/{rho}{del}/{dely}(mu{delbar(u)}/{dely} - {rho}/{t}int_0^t(u'v')dt)`

The reynolds stress is,

`-{rho}/{t}int_0^t(u')(v')dt`

`=> -rhobar(u')bar(v')`

The stress can be expressed as combination of its turbulent and laminar form

`tau = tau_(" laminar") + tau_(" turbulent")`

What is turbulent viscosity? How is it different from molecular viscosity?

The coefficient the relates the average shear stress and rate of velocity gradient in the turbulent flow is called turbulent viscosity. The viscosity in the turbulent flow is non homogeneous and it varies in space.

Turbulent viscosity is not the same as molecular viscosity, as eddy viscosity is a function of flow not of fluid. The turbulent viscosity is the measure of a fluid's resistance to deformation at a given rate and hence is a property of the fluid.

The viscous shear stress is caused by the velocity of the fluid as a result of intermolecular forces and hence its a property of fluid not the flow.

`tau_(" turbulent") = -rhobar(u')bar(v')`

`tau_(" turbulent") = mu_t{delbar(u)}/{dely}`

`mu_t` is the Turbuelnt Viscosity