### Introduction to Turbulence and Turbulence Modeling

Rate this item
15 May

Understanding the turbulent behaviour of fluids is one of the most fascinating, forbidding & critical problems in all of classical physics. Turbulence is omnipresent, as most of the fluid flows are turbulent in nature right from the microscopic level at interior of biological cells to the macroscopic scales of the geophysical and astrophysical phenomena including planetary interiors, oceans and atmospheres that represent the dominant physics of turbulent fluid flows.

Turbulence has been the topic of interest for many great researchers from the 19th and 20th centuries, however still remains ambiguous as we yet do not understand completely as to how or why it occurs, nor we can predict their behavior with any degree of reliability. For many researchers it thus becomes a topic of both inherent intellectual challenge and by the practical utility of thorough understanding its nature. In this blog series on Turbulence Modeling, here we shall try to have an overview of turbulence phenomena before getting into the details of various existing turbulence models used for CFD simulations in the blogs to follow.

##### Fluid Flows

The smoke coming out from a source like a lighted cigarette has its first few units of length as smooth and then it starts fluctuating randomly in all direction as it continues to rise, as shown in below image. The first case is said to be laminar flow and the latter is known as turbulent flow. Most flows encountered in engineering practice are turbulent, in nature and thus it is important to understand how exactly turbulence affects the flow or flow parameters. The turbulent flows are however complex mechanisms dominated by fluctuations and chaotic property changes. Despite tremendous amount of work done in this area by many researches, turbulent flows are still not fully understood and thus at times one has to rely on experiments and the empirical or semi empirical correlations developed for various conditions.

##### Reynolds Pipe Experiment

Experimentally the formal study of turbulent flow began with the famous Reynolds Pipe Experiment conducted by the great Anglo-Irish innovator Osborne Reynolds (1842-1912). A brief description of the experiment is explained as follows. The apparatus contained a large pipe through which water is allowed to flow and through a small glass tube a die is injected. Initially when the die is injected at low velocities, the die streak forms a straight and smooth line as shown in figure (below), termed as laminar flow regime. When the velocity was increased slightly the flow is still laminar but somewhat wavy, this regime is termed as transitional flow regime and at high velocities it is found that there was fluctuations and mixing, this regime is termed as turbulent flow regime. Reynolds also found that, the changes in the nature of the flow not only depended on velocity but also on viscosity. Thus based on all these experimental evidence Reynolds derived a dimensionless number to distinguish the different regimes, that is known as the Reynolds number.

$Re = \frac{\rho V D}{\mu} = \frac{Inertial Force}{Viscous Force} = \frac{Promotes Turbulent flow}{Promotes Laminar flow}$

Typically for flow through pipe if:

• $Re = \mbox{Re \leq {2300} \Rightarrow flow is laminar}$
• $Re = \mbox{2300 \leq Re\leq {4000} \Rightarrow flow is in transition}$
• $Re = \mbox{Re \textgreater 4000 \Rightarrow flow is turbulent}$

Unlike laminar flow, turbulent flow cannot be defined with a set of guidelines, but instead can be expressed in terms of certain characteristics. Basically a turbulent flow is identified by three characteristics:

• 3 dimensional
• Fluctuating
• Chaotic

##### Turbulent flows

Turbulent ﬂow is always three-dimensional, unsteady, viscous phenomenon that occurs at high Reynolds number. However when the equations are time averaged, the ﬂow can be treated as two-dimensional if the geometry is two-dimensional. Turbulent disturbances can be thought of as a series of three-dimensional eddies of different sizes that are in constant interaction with each other. There is no exact defination of a turbulent eddy, but one can suppose that it exists in a certain region in space for a certain turbulent time and that it is subsequently destroyed by the cascade process or by dissipation and has a characteristic velocity and length, called as velocity scale and length scale. The region covered by a large eddy may well enclose also smaller eddies. The largest eddies are of the order of the ﬂow geometry i.e. boundary layer thickness, jet width, etc and at the other end of the spectra we have the smallest eddies which are dissipated by viscous forces (stresses) into thermal energy resulting in a temperature increase.

In turbulent ﬂow the diffusivity increases and also increases the exchange of momentum for example in boundary layers, and reduces or delays thereby separation at bluff bodies such as cylinders, airfoils and cars. The increased diffusivity also increases the resistance (wall friction) and heat transfer in internal ﬂows such as in channels and pipes. When we say the turbulent ﬂows are dissipative, we mean that the kinetic energy in the small (dissipative) eddies are transformed into thermal energy. The small eddies receive the kinetic energy from slightly larger eddies, that receive their energy from even larger eddies and so on. The largest eddies extract their energy from the mean ﬂow and this process of transferring energy from the largest turbulent scales (eddies) to the smallest is called the energy cascade process.

##### Turbulent Flow in Pipes

Let us consider an example to get insights of the nature of turbulent flows in pipes. Consider the fluid flow coming out of a tap to be turbulent and assume that the flow rate is constant and thus the velocity of the flow at the outlet is expected to be constant with respect to time. When a pitot tube is inserted in the flow to measure the velocity, the result obtained is represented in the form of a velocity vs time graph as shown:

The zig-zag line in the above graph shows the velocity fluctuation which is a property of turbulent flow i.e., unsteadiness. If we consider averaging i.e. by taking a small interval $\Delta {t}$ and taking average of the values in between, the values obtained are steady in nature as shown by the dotted line. The averaging concept thus makes the problem much simpler. Though strictly speaking no turbulent flows are steady in nature, but if averaged values of flow variables are in steady state then the turbulent flow can be said to be steady. From above graph it can be observed that, the instantaneous or actual values of the velocity fluctuate about an average value, which suggests that the velocity can be expressed as the sum of an average value $\bar{u}$ and a fluctuating component $u'$

$\bar{u} = \bar{u} + u'$

This is also the case for other properties such as the velocity component $\v$ in the $\y$

$\bar{v} = \bar{v} + v'$ , $\bar{T} = \bar{T} + T'$ ; and the averaged value can be defined over a time interval $\Delta {t}$ as:

$\bar{u}=\frac{1}{\Delta{t}} {\int_t^{t=\Delta{t}} {u dt}}$

Now the first things that comes in mind is to determine the shear stress similar to the manner in laminar flow using the expression.

$\tau = \mu \frac{du'}{dy}$

But the experimental studies show that this is not the case, and the shear stress is much larger due to the turbulent fluctuations. More detailed analysis shows that the actual shear stress would be as given below:

$\tau = \mu \frac{du'}{dy} - \rho u' v'$

where; $\u'$ and $\v'$ are fluctuating components in $\x$ and $\y$ directions, thus from above expression it is convenient to think of the turbulent shear stress as consisting of two parts: the laminar component, which accounts for the friction between layers in the flow direction (expressed as, $\tau = \mu \frac{du'}{dy}$) and the turbulent component, which accounts for the friction between the fluctuating fluid particles and the fluid body. Then the total shear stress in turbulent flow can be expressed as

$\tau_{total} = \tau_{lam} + \tau_{turb}$

The term $\rho u' v'$ is called Reynolds stresses or turbulent stresses. The Reynolds stress is a complicated term and its determination in terms of the known quantities is considered as one of the most complicated problems in fluid mechanics. Many semi-empirical formulations have been developed that model the Reynolds stress in terms of average velocity gradients, the models are called turbulence models.

##### Turbulence Models

The origin of the time-averaged Navier-Stokes equations dates back to the late nineteenth century when Reynolds (1895) published results from his research on turbulence. The earliest attempts of developing a mathematical description of the turbulent stresses, which is the core of the closure problem, were performed by Boussinesq (1877) with the introduction of the eddy viscosity concept. Prandtl (1925) later introduced the concept of the mixing-length model, which prescribed an algebraic relation for the turbulent stresses. To develop a more realistic mathematical model of the turbulent stresses, Prandtl (1945) introduced the first one-equation model by proposing that the eddy viscosity depends on the turbulent kinetic energy $k$, solving a differential equation to approximate the exact equation for $k$. This one equation model improved the turbulence predictions by taking into account the effects of flow history. Kolmogorov (1942) introduced the first complete turbulence model, by modeling the turbulent kinetic energy $k$, and introducing a second parameter $omega$ that he referred to as the rate of dissipation of energy per unit volume and time, which is termed as two equation model. Rotta (1951) pioneered the use of the Boussinesq approximation in turbulence models to solve for the Reynolds stresses. This approach is called a second-order or second-moment closure.

A turbulence model is a procedure to close the system of mean flow equations. For most engineering applications it is unnecessary to resolve the details of the turbulent fluctuations. We only need to know how turbulence affected the mean flow. In particular we need expressions for the Reynolds stresses. For a turbulence model to be useful it should fulfill the following criteria:

• Must have wide applicability,
• Be accurate,
• Simple,
• economical to run.

The scope of turbulence models available today is extensive. However despite over a century of research in this field, there are no successful universal turbulence models available. Turbulence modelling can be divided in to the primary following fields:

• Reynolds Averaged Navier-Stokes (RANS) modelling, where turbulence is modelled using the Reynolds Averaged Navier-Stokes (RANS) equations. The RANS equations are derived by averaging the Navier-Stokes and continuity equations.
• Large Eddy Simulation (LES), where the large energy containing eddies are computed directly and the effect of small scale eddies are modelled using subgrid-scale models.
• Direct Numerical Simulation (DNS), where the full scale Navier-Stokes equations are solved on the grids fine enough, and time steps small enough to capture the full length of turbulent scales.

A schematic description of the above is shown below:

(Turbulence models and their applications)

Few of the popular turbulence models are listed below:

• Algebraic models
1. Mixing Length Model
2. Baldwin Lomax Model
• Half Equation Model
1. Johnson-King Model
• One equation model
1. Mixing Length One Equation Model
2. Spalart Allmaras Model
• Two Equation Models
1. Standard $k-\epsilon$ Model
2. Standard $k-\omega$ Model
3. BSL Model
4. SST Model
5. LRN $k-\epsilon$
6. $k-\tau$ Model
• Explicit Algebraic Stress Models
• Second Order Closure Models
• Detached Eddy Simulation
• Large Eddy Simulation
• Direct Numerical Simulation

Of the above listed models we shall further discuss a few important models that are used for CFD simulations in the upcoming blogs.

For a more clear understanding I would recommend one to go through the below video by the National Committee for Fluid Mechanics on basics of fluid mechanics to have some experimental insights and illustrations.

##### References

 Turbulence Modeling for CFD By David C. Wilcox A First Course in Turbulence By Henk Tennekes and John L. Lumley Turbulent Flows By Stephen B. Pope Computational Fluid Mechanics and Heat Transfer By Richard H. Pletcher, John C. Tannehill and Dale Anderson

##### Dr. Ganesh Visavale

Dr. Ganesh is a leading researcher in computational engineering, sustainable energy and its application in process industry. He has done immence contribution in LearnCAx. Before he joined post-doctoral fellowship at IIT Delhi, he was the general manager at LearnCAx. He was instrumental in conceptualization, development and implementation of online education from CCTech for CAx professional. Ganesh has a number of publications both in international journal and conference proceedings. Before CCTech, he held the position of Associate Scientist in solar thermal division at Sardar Patel Renewable Energy Research Institute, Anand.

Ganesh holds Ph.D and M.Chem. Engineering from Department of Chemical Engineering, Institute of Chemical Technology, Mumbai (formerly UDCT Mumbai).

Introduction to Turbulence and Turbulence Modeling - 4.8 out of 5 based on 24 votes

### Latest from Ganesh Visavale

 Career Path to CFD Engineer Views : 12624 Reviewing Governing Equations of Fluid Dynamics Views : 9041

### Related items

 Tutorial: CFD Simulation of Unsteady Flow Past Square Cylinder Author : Trushna Dhote Views : 10933 Tutorial: CFD Simulation of Backward Facing Step Author : Trushna Dhote Views : 9946 Y Plus / Y+ Calculator for First Cell Wall Distance for Turbulent Flows Author : Sanket Dange Views : 20161 How to Learn CFD ? – The Beginner’s Guide Author : Vijay Mali Views : 38674 Democratization of CFD Education, Truly!! Author : Vijay Mali Views : 3033

Refresh