Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering PDF Download

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Reynolds decomposition of turbulent flow :

  • The Experiment: In 1883, O. Reynolds conducted experiments with pipe flow by feeding into the stream a thin thread of liquid dye. For low Reynolds numbers, the dye traced a straight line and did not disperse. With increasing velocity, the dye thread got mixed in all directions and the flowing fluid appeared to be uniformly colored in the downstream flow.

The Inference: It was conjectured that on the main motion in the direction of the pipe axis, there existed a superimposed motion all along the main motion at right angles to it. The superimposed motion causes exchange of momentum in transverse direction and the velocity distribution over the cross-section is more uniform than in laminar flow. This description of turbulent flow which consists of superimposed streaming and fluctuating (eddying) motion is well known as Reynolds decomposition of turbulent flow.

  • Here, we shall discuss different descriptions of mean motion. Generally, for Eulerian velocity u , the following two methods of averaging could be obtained.

    (i) Time average for a stationary turbulence:

Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering

(ii) Space average for a homogeneous turbulence: 

Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering

For a stationary and homogeneous turbulence, it is assumed that the two averages lead to the same result:  Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering  and the assumption is known as the ergodic hypothesis.

  • In our analysis, average of any quantity will be evaluated as a time average . Take a finite time interval t1. This interval must be larger than the time scale of turbulence. Needless to say that it must be small compared with the period t2 of any slow variation (such as periodicity of the mean flow) in the flow field that we do not consider to be chaotic or turbulent .

Thus, for a parallel flow, it can be written that the axial velocity component is
 

Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering                               (32.1)                                      

 

As such, the time mean component  Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering (y)determines whether the turbulent motion is steady or not. The symbol Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering  signifies any of the space variables.

  • While the motion described by Fig.32.6(a) is for a turbulent flow with steady mean velocity the Fig.32.6(b) shows an example of turbulent flow with unsteady mean velocity. The time period of the high frequency fluctuating component is t1 whereas the time period for the unsteady mean motion is t2 and for obvious reason t2>>t1. Even if the bulk motion is parallel, the fluctuation ' being random varies in all directions. 
  • The continuity equation, gives us

Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering

Invoking Eq.(32.1) in the above expression, we get

Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering

Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering

 

Since  Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering  , Eq.(32.2) depicts that and components of velocity exist even for the parallel flow if the flow is turbulent. We have-

Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering                                 (32.3)

 

Contd. from Previous slide

  • However, the fluctuating components do not bring about the bulk displacement of a fluid element. The instantaneous displacement is U'dt , and that is not responsible for the bulk motion. We can conclude from the above

Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering  

Due to the interaction of fluctuating components, macroscopic momentum transport takes place. Therefore, interaction effect between two fluctuating components over a long period is non-zero and this can be expressed as

Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering

Taking time average of these two integrals and write

Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering                                       (32.4a)

 

    

 

and  Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering                      (32.4b)                    

 

  • Now, we can make a general statement with any two fluctuating parameters, say, with ' and g' as

    Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering                                        (32.5a)

       Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering                                   (32.5b)

The time averages of the spatial gradients of the fluctuating components also follow the same laws, and they can be written as

 

Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering                                        (32.6)                 

 

  • The intensity of turbulence or degree of turbulence in a flow is described by the relative magnitude of the root mean square value of the fluctuating components with respect to the time averaged main velocity. The mathematical expression is given by

          Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering                             (32.7a)

    The degree of turbulence in a wind tunnel can be brought down by introducing screens of fine mesh at the bell mouth entry. In general, at a certain distance from the screens, the turbulence in a wind tunnel becomes isotropic, i.e. the mean oscillation in the three components are equal,

                 Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering    

In this case, it is sufficient to consider the oscillation u' in the direction of flow and to put

             Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering                                                         (32.7b)

This simpler definition of turbulence intensity is often used in practice even in cases when turbulence is not isotropic.

Following Reynolds decomposition, it is suggested to separate the motion into a mean motion and a fluctuating or eddying motion. Denoting the time average of the u component of velocity by Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering and fluctuating component as Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering, we can write down the following

 

Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering

By definition, the time averages of all quantities describing fluctuations are equal to zero.

Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering                                                     (32.8)

 

The fluctuations u', v' , and w' influence the mean motion Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering in such a way that the mean motion exhibits an apparent increase in the resistance to deformation. In other words, the effect of fluctuations is an apparent increase in viscosity or macroscopic momentum diffusivity .

  • Rules of mean time - averages

If and g are two dependent variables and if s denotes anyone of the independent variables x, y

Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering

The document Introduction to Turbulent Flow - 2 | Fluid Mechanics for Mechanical Engineering is a part of the Mechanical Engineering Course Fluid Mechanics for Mechanical Engineering.
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FAQs on Introduction to Turbulent Flow - 2 - Fluid Mechanics for Mechanical Engineering

1. What is turbulent flow?
Ans. Turbulent flow is a type of fluid flow characterized by chaotic, irregular motion of the fluid particles. In this type of flow, the fluid moves in a random manner, with eddies and vortices forming within the flow. Turbulent flow is often observed at high velocities or in situations where there are disturbances in the flow, such as obstacles or changes in direction.
2. How is turbulent flow different from laminar flow?
Ans. Turbulent flow and laminar flow are two types of fluid flows that differ in the behavior of fluid particles. In laminar flow, the fluid particles move in a smooth, orderly manner, with layers of fluid sliding past each other. On the other hand, turbulent flow is characterized by chaotic, irregular motion of the fluid particles, with mixing and eddies forming within the flow. The transition from laminar to turbulent flow is typically influenced by factors such as velocity, viscosity, and the presence of disturbances in the flow.
3. What are the factors that can cause turbulent flow?
Ans. Several factors can cause turbulent flow. These include high velocities, changes in flow direction, presence of obstacles or rough surfaces, and disturbances in the flow such as sudden expansions or contractions. In addition, factors such as fluid viscosity and density play a role in determining whether the flow remains laminar or becomes turbulent. For example, fluids with higher viscosity are more likely to exhibit laminar flow, while fluids with lower viscosity are more prone to turbulent flow.
4. How is turbulent flow important in civil engineering?
Ans. Turbulent flow has significant implications in civil engineering, particularly in areas such as fluid dynamics, hydraulic design, and transport of pollutants in water bodies. Understanding turbulent flow is crucial for designing efficient and safe hydraulic structures, such as dams, channels, and pipelines. It also plays a role in predicting the dispersion and mixing of pollutants in rivers and estuaries. By studying and analyzing turbulent flow patterns, civil engineers can make informed decisions in designing and managing various hydraulic systems.
5. What are the challenges in studying turbulent flow?
Ans. Studying turbulent flow is a complex task due to its chaotic and unpredictable nature. The behavior of turbulent flow is highly dependent on various factors, including the geometry of the flow, boundary conditions, and fluid properties. Additionally, turbulent flows often involve a wide range of length scales, from large eddies to small-scale fluctuations, making it challenging to capture all the details and accurately model the flow. As a result, researchers and engineers utilize experimental techniques, computational fluid dynamics, and mathematical models to gain insights into turbulent flow behavior.
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