Week 8: Literature review RANS derivation and analysis

AIM: to Derive from Reynolds Decomposition to the Navier Stokes Equation to come up with Reynold's stress.

INTRODUCTION:

What is RANS?

its Full form is Reynold's Average Navier Stokes.

this Equation Solves the Average Velocity Fields and the Magnitude of Turbulence Fluctutations.

THEORY:

Reynold's Number

It Quantifies the Turbulence Level in the system.

it is the Ratio of Intertia force to the Viscous Force.

Re=ρVD/μ

• R_e is the Reynolds number
• ρ is the density of the fluid
• V is the velocity of flow
• D is the pipe diameter
• μ is the viscosity of the fluid

Re<2000-----> Flow is Laminar

2000<=Re<=4000------> Flow is in a Transitional Phase

Re>4000--------> Flow is Turbulent

Reynold's Number is Dimensionless Unit

RANS DERIVATION

• U(x,y,z,t)= Ʉ(x,y,z)+U’(x,y,z,t)

Ʉ------->function of space

U’-------->Fluctuation’s space

• Ʉ(x,y)=[ ∫U(x,y,t)dt)/ t(int) ]

Note: in this integral

The Lower Limit =0;

The Upper Limits=t(int);

• Ṽ(x,y)=[ ∫ V(x,y,z)dt

• [ ∫ U’(x,y,z,t)dt ] / t(int) = 0;

• U(x,y,z,t)= Ʉ(x,y)+U’(x,y,z,t) ;

• {[∫ U(x,y,z,t)dt ] / t(int)]} =[ ∫ Ʉ(x,y) + ∫ U’(x,y,z,t) ]/ t(int)] ;

• Ʉ(x,y,z)= Ʉ(x,y,z)+ [ ∫ U’(x,y,z,t) ]/ t(int)];

• [ ∫ U’(x,y,z,t) ]/ t(int)] = 0;

=> ∂u/∂x + ∂v/∂y = 0;

Apply Decomposition Law

• ∂(Ʉ+U')/∂x + ∂(Ṽ+V')/∂y =0;
• [ ∫ {∂(Ʉ+U')/∂x}dt + {∂(Ṽ+V')/∂y}dt ]=0;

Now Expanding and Solving this Equation we will get.

also Cancel some of the Fluctuating Components

we will get Continuity Equation and ie

∂u/∂x + ∂v/∂y = 0;

MOMENTUM EQUATIONS

=> ∂U/∂t + U∂U/∂x + V∂V/∂y = -dp/dx + μ {∂^2(U)/∂(y^2)};

=> ∂U/∂t + U∂U/∂x + V∂V/∂y = [-dp/dx + μ {∂^2(U)/∂(y^2)}]/p;

=> ∂U/∂t + U∂U/∂x + V∂V/∂y = (-∂p/pdx) + γ{∂^2(U)/∂(y^2)};

we know that

∂(U.V)= U∂U/∂x + V∂V/∂y

Hence Sustitute this Red Marker Equation to Aqua Highlighter Equation

we get

=> ∂U/∂t + ∂(U.V)+2U∂U/∂x = (-∂p/∂x/p)+ γ{∂^2(U)/∂(y^2)};

Now more Simplify it, we get

=> ∂U/∂t + ∂(U.V)+ ∂(U^2)/∂y = (-∂p/∂x/p)+ γ{∂^2(U)/∂(y^2)};

Now Apply Reynold's Decomposition Equations

=> [∫ {∂(Ʉ+U')/∂t}dt + {∂(Ʉ+U')^2/∂x}dt + {∂[(Ʉ+U')(Ṽ+V')]/∂y}dt ] = [ ∫-{{∂(ṗ+p')/∂x}/p}dt + γ{∂^2(Ʉ+U')/∂(y^2)/ t(int) };

Our Average Fluctuating Quantity is going to zero hence our New Equation will become

∂U/∂t + ∂(Ʉ^2)/∂x + ∂(Ʉ.Ṽ) = (-∂ṗ/∂x)/p + γ{∂^2(Ʉ)/∂(y^2)}- [∫{∂(U'^2)/∂x}]dt - ∫ {∂(U'V')/∂y}dt]/t (int) ;

[∫{∂(U'^2)/∂x}]dt=0;

Hence the new Eqn will be

∂U/∂t + ∂(Ʉ^2)/∂x + ∂(Ʉ.Ṽ) = (-∂ṗ/∂x)/p + γ{∂^2(Ʉ)/∂(y^2)} - ∫ {∂(U'V')/∂y}dt]/t (int) ;

(-∂ṗ/∂x)/p--------> Pressure Terms

γ{∂^2(Ʉ)/∂(y^2)}--------> Viscous statics

∫ {∂(U'V')/∂y}dt]/t (int)------> Momentum Diffusivity due to Reynold's Stress.

CONCLUSION: Hence the Reynold's Stress has been Derived