1. Transport Equations for Turbulent Quantities
P M V Subbarao
Professor
Mechanical Engineering Department
I I T Delhi
Conservation Equations for Symptoms ....

2. Steady Turbulent flow

3. One-Equation Model by Prandtl
A one-equation model is an enhanced version of the algebraic models.
This model utilizes one turbulent transport equation originally developed by Prandtl.
Based on purely dimensional arguments, Prandtl proposed a relationship between the dissipation and the kinetic energy that reads
where the turbulence length scale lt is set proportional to the mixing length, lm, the boundary layer thickness  or a wake or a jet width.
The velocity scale is set proportional to the turbulent kinetic energy as suggested independently.
Thus, the expression for the turbulent viscosity becomes:
with the constant C to be determined from the experiment.

4. A Segment of Reconstructed Turbulent Flame in SI Engines

5. Large Scales: Parents Vortices

6. Creation of Large Eddies an I.C. Engines
There are two types of structural turbulence that are recognizable in an engine; tumbling and swirl.
Both are created during the intake stroke.
Tumble is defined as the in-cylinder flow that is rotating around an axis perpendicular with the cylinder axis.
Swirl is defined as the charge that rotates concentrically about the axis of the cylinder.

7. Instantaneous Energy Cascade in Turbulent Boundary Layer.
A state of universal equilibrium is reached when the rate of energy received from larger eddies is nearly equal to the rate of energy of when the smallest eddies dissipate into heat.

8. Transport equation for turbulent kinetic Energy
x-momentum equation for incompressible steady turbulent flow:
Reynolds averaged x-momentum equation for incompressible steady turbulent flow:
subtract the second equation from the second equation to get
Multiply above equation with u and take Reynolds averaging

9. Similarly:
Define turbulent kinetic energy as:

10. Turbulent Kinetic Energy Conservation Equation
The Cartesian index notation is:
Boundary conditions:

11. Taylor Hypothesis
Taylor proposed an hypothesis that the energy transport contribution of small size eddies that are carried by a large scale eddy Taylor proposed an hypothesis:
The energy transport contribution of small size eddies that are carried by a large scale eddy, compared with the one produced by a larger eddy, is negligibly small.
In such a situation, the transport of a turbulence field past a fixed point is due to the larger energy containing eddies.
It states that “in certain circumstances, turbulence can be considered as “frozen” as it passes by a sensor”.

12. Spectral Representation of Turbulent Flows

13. Kolmogorov Hypotheses
Kolmogorov established his universal equilibrium theory based on two similarity hypotheses for turbulent flows.
The first hypothesis states that for a high Reynolds number turbulent flow, the small-scale turbulent motions are isotropic and independent of the detailed structure of large scale eddies.
Furthermore, there is a range of high wavenumbers where the turbulence is statistically in equilibrium and uniquely determined by the energy dissipation and the kinematic viscosity .
With this hypothesis and in conjunction with dimensional reasoning, Kolmogorov arrived at length (), time () and the velocity (v) scales. Considering the Kolmogorov’s length and velocity scales, the corresponding Kolmogorov’s equilibrium Reynolds number is

14. Structure of Equilibrium Turbulence

15. Structure of Equilibrium Turbulence

16. One and Two Equation Turbulence Models
The derivation is again based on the Boussinesq approximation
The mixing velocity is determined by the turbulent turbulent kinetic energy
The length scale is determined from another transport equation ex.

17. Second equation

18. Dissipation of turbulent kinetic energy
The equation is derived by the following operation on the Navier-Stokes equation
The resulting equation have the following form

19. The k-ε model Eddy viscosity
Transport equation for turbulent kinetic energy
Transport equation for dissipation of turbulent kinetic energy
Constants for the model

20. Dealing with Infected flows
The RANS equations are derived by an averaging or filtering process from the Navier-Stokes equations.
The ’averaging’ process results in more unknown that equations, the turbulent closure problem
Additional equations are derived by performing operation on the Navier-Stokes equations
Non of the model are complete, all model needs some kind of modeling.
Special care may be need when integrating the model all the way to the wall, low-Reynolds number models and wall damping terms.