Week 5 - Turbulence modelling challenge

The ratio of inertia force to viscous force is called as Reynolds number.

1. Laminar flow : 

• Occur at low Reynolds number 
• Viscous forces are dominant
• Characterized by smooth, constant fluid motion.

    2. Turbulent flow:   

• Occurs at a high Reynolds number
• Inertia forces are dominant.
• Produce chaotic eddies, vortices, and other instabilities      

    3. Transition flow :      

• Energy and momentum transfer are strongly influenced by either inertial or nonlinear effects that are unique to the simulation. 

According to the order of PDEs, the Turbulent models are classified into  two types

1. First-order model 
2. Second-order model

1. First-order model:   Assumes an analogy between laminar and turbulent flows, Average turbulent flow field is similar to the corresponding laminar flow. Uses first-order PDEs to solve for turbulent viscosity. 

         First-order models:

• Zero equation model 
• One equation model (Spalart Allmaras model)
• Two equation models: 1. k-ε">εε model,  2. k-ω">ωω model
• Zero equation model:  The zero equation model does not employ PDEs to account for turbulent viscosity
• One equation model (Spalart Allmaras model):

 Turbulent Kinetic energy is determined from

        

The turbulent viscosity is determined by,

The turbulent viscosity is substituted in RANS

 The Spalart Allmaras model is designed especially for internal flow and external aerodynamic applications.

Especially when the flow is attached.Not good for flow with the strong adverse gradient 

• Two equation model: Two PDEs are used to solve for 

       1. Turbulent Kinetic energy and Turbulent dissipation rate

       2. Turbulent Kinetic energy and Specific dissipation rate

       Most commonly used method in industries.

• k-ε">−ε-ε Model: The K-epsilon model is good for turbulent flows with no flow separation.

         1. Standard k ε">εεmodel:  Most widely used model for industrial applications. Uses wall functions for calculating ε">εε near walls. Known to have very good convergence and memory efficiency. Good for attached and turbulent flow.Good in solving external flow around complex bodies. Performs poorly for separated flow.

        2. Realizable k -ε">εεmodel: Modify the ε">εε equation. Solve some of the constraints on Reynolds stresses. Shows consistent results with the physics of the turbulent flow.  

        3. Renormalized Group ( RNG) k-ε">εε model: Improved ε">εε equation. The effect of swirl is introduced. It uses empirical relations to find a modified version of the Prandtl Number.

• k -ω">ωω model: The k-omega model is most widely utilized in industry. This model does not contain terms undefined at the wall Wall function is not required at the wall. 

      1. Standard k-ω">ωω model: Most widely adopted in aerospace and turbomachinery components.

      2. SST k-ω">ωωmodel: Shear stress transfer k-omega model. It is a better variant of the traditional k-ω">ωω model. Uses a blending function to transition from the standard model to a higher Reynolds model away from the wall.

• Second order Model: Solves transport equations for Reynolds stress term. For 3D additional equations needs to be solved. Most accurate result.computationally expensive. This model is highly sensitive to Boundary conditions. 

      1. Algebraic stress model

      2. Reynolds stress model

According to Kolmogorov's scale.DNS isn’t feasible.

Ans: Solve NS equations at very small scales 

  According to the Kolmogorov scale,

• Grid size must be smaller than this.
• In 3D Each dimension of the element must be smaller than the length determined by the Kolmogorov scale.

          

    Number of elements = VΔV">V/ΔV

      where, V=Structural overall volume

                 ΔV">ΔV= Volume of smallest elements 

 Number of elements ∝">∝ Re^9/4

• This leads to a very large number of elements in 3D 
• So DNS is still performed in 2D  
• Using this method for industrial application is next to impossible.

Material: Air 

Velocity: 60 m/sec 

Viscous Model:

Velocity Contour:

Pressure Contour:

The velocity vectors the direction of the ball as it's going down and the dimples on the golf ball help to reduce the drag by reducing the wake region. It also helps the flow to be attached and tries to separate the flow away from the golf ball