Week 5 - Turbulence modelling challenge
Part 1: Explain different turbulence models & why DNS isn’t feasible according to Kolmogorov's scale.
There are two main models are available to model turbulence which are as follows:
- First Order Models:
First-order models assume an analogy between laminar & turbulent flows. They assume that the average turbulent flow field is similar to the laminar flow field which results in lower accuracy.
It has three subtypes of models based on the number of equations the model is solving.
- Zero Equation Models
- One Equation Models (Spalart Allmaras model)
- Two Equations Models (K-epsilon model & k-omega model)
- Second-Order Models:
These models are solving the transport equations additionally to derive Reynolds stress terms in 3D.
As these models are solving additional equations to work in 3D & also PDEs are of the 2nd order, it has the highest accuracy but are computationally very expensive. And because of a large number of extra equations, these models are sensitive to boundary conditions as well. These models are commonly used in industries.
This type also has subtypes.
- Algebraic Stress Model
- Reynolds Stress Model
The main difference between the first-order & second-order models is that the first-order model uses the PDEs of 1st order to derive Reynolds stress terms while the second-order model are using the PDEs of 2nd order because of the accuracy of the second-order models is more than the first-order models but computationally more expensive.
Detail explanation of each model of First Order Models is as follows:
Zero Equation Model:
- No PDE is used to solve the turbulent viscosity.
- Fails to account for transient behaviour of turbulent viscosity which leads to lower accuracy results.
- Directly solves turbulent viscosity using a Mixing length model.
- Mixing length model assumes eddy viscosity terms to be dimensionally similar to two scales: Kolmogorov length scales & Kolmogorov velocity scales
One Equation Model:
- This model uses one set of equations to solve the turbulent kinetic energy in order to obtain turbulent viscosity.
- The derived turbulent viscosity is substituted into the RANS model to solve the turbulence.
- Spalart Allmaras Model (S-A model) is one eqation model. This S-A model is specifically designed for internal & external flows when the flow is attached, but not good when flows with strong adverse gradients to detaching flows.
Two Equation models:
These models solve two PDEs to determine the following quantity based on the model used.
- Turbulent Kinetic Energy & Turbulent Dissipation Rate (K-epsilon Model)
- Turbulent Kinetic Energy & Specific Dissipation Rate (K-omega Model)
K - epsilon Model:
This model has three different variants:
(a) Standard k-epsilon model:
- This is the most widely used model in the industries.
- It uses wall functions to calculate epsilon near the wall.
- It is a very known model to have very good convergence & memory efficient characteristics.
- It is a very good model for attached & turbulent flows and also for solving external flow around complex bodies.
- This model performs very poorly for separated flows.
(b) Realizable K-epsilon Model:
- Realizable & RNG types of K-epsilon models modify the existing epsilon equations to determine Reynolds Stress terms.
- It solves some of the constraints on Reynolds stresses.
- It shows consistent results with the physics of turbulent flows.
(c) Renormalized Group (RNG) K-epsilon Model:
- This model uses the renormalized group theory to improve the epsilon equations.
- The effect of swirl in turbulent flow is introduced using this model.
- It uses empirical relations to find a modified version of Prandtl Number to determine turbulent viscosity.
k-omega Model:
- It is a very popular model in the industry.
- This model does not contain terms that are defined at the wall. Hence, wall functions are not required at the wall to capture turbulence flows.
- It has two types of variants: one is the Standard K-omega model & the other is the SST K-omega model.
(a) Standard K-omega model:
- It is widely adopted in aerospace & turbomachinery components.
(b) SST (Shear Stress Turbulence) K-omega model:
- It is a go-to model in all the other industries than aerospace & turbomachinery.
- SST model uses a blending function to transition from the standard k-omega model to a higher Reynolds model away from the wall to better approximate turbulent flows.
Part 2: The objective is to determine which way the ball is going to move. The velocity contour graph & vector plot will be presented.
Given Inputs in problem:
Air velocity: 60 m/s
Procedure:
The methodology of the simulation is as follows:
1) Import the geometry in Ansys Design Modeler.
2) Mesh generation. Element size = 7mm
3) Fluent set-up.
General Setting:
Viscous Model :
Assigning Material :
Initial Boundary Conditions :
Initialization :
4) Post-processing in CFD post. Make the following plots:
A) Contour Plot.
B) Streamline Plot.
C) Vector Plot.
Solution:
Max velocity:
Contour Plot:
Streamline Plot:
Vector Plot:
Observation :
- By inspecting the velocity vectors we can determine the direction of the ball and say it's going down.
- The dimples on the golf ball help to reduce the drag by reducing the wake region (recirculation).
- It also helps the flow to be attached and tries to separate the flow away from the golf ball.