Summary of above lecture along with applications of turbulence are described below.
Why Turbulence ?
There is no universally accepted answer for reason behind turbulence. Many scientific searches to find out reason behind turbulence of flow have ended up in vain. Take a look at a famous witticism made by Heisenberg regarding this.
But engineers and scientists have developed a good understanding on nature of turbulence and way to quantify effect of it. So here we will learn ‘How Turbulence’ instead of ‘Why Turbulence’.
How to distinguish a Turbulent flow ?
All turbulent flows have got following 3 characteristics
- 3 dimensional
- Chaotic – With eddies and vortices
So if a fluid flow under consideration has got all 3 above characteristics it is turbulent in nature, otherwise flow is laminar.
A Daily Life Experience to Predict Turbulence
To understand nature of turbulence we will consider a daily life experience, a tap water problem. Consider following 3 cases, where in each case flow rate of water increases. It is clear that as flow rate increases turbulence of flow also increases. So finding number one turbulence increases with increase in flow velocity.
Fig.1 Increase in turbulence of flow as flow rate of water is increased
If you replace water in tap by a fluid which is more viscous in nature(oil), you will find that flow is not turbulent even at high flow rate. So finding number two turbulence decreases with increase in fluid viscosity.
Fig.2 Decrease in turbulence of flow as flow as viscosity of fluid is increased
From above findings it can be summarized that turbulence increases with increase in flow velocity and decrease in fluid viscosity. Flow velocity increases with increase inertial force on the fluid and if fluid viscosity is high viscous force in fluid will also be high. So it can be summarized that turbulence increases with increase in inertial force and decrease in viscous force.
Concept of Reynolds number
Ratio of inertial force to viscous force is know as Reynolds number .
It is clear that when Reynolds number increases turbulence increases. So Reynolds number is the criterion which decides whether a flow is laminar or turbulent. For this pipe problem Reynolds number can be represented as
Where D is diameter of pipe. So you can define a Critical Reynolds number for a particular problem above which flow is turbulent and below which flow is laminar
More analysis – Concept of Averaging
Consider a turbulent tap water case with constant flow rate input. If you measure velocity at tap outlet for this case you will find that velocity is highly unsteady as shown in figure below.
Fig.3 Fluctuating velocity field at outlet of a turbulent flow problem
This is one big characteristic of turbulent flow, strictly speaking all flow variables in a turbulent flow are unsteady in nature.
But if you do a mathematical operation called averaging in this case on flow velocity, the result becomes steady in nature. So you could say a turbulent flow is in steady state if averaged flow variable is in steady state.
Fig.4 Result of averaging operation in constant flow input flow problem
Averaging is defined as follows
Where time interval used for integration should be carefully chosen. It should be small enough to take care of any unsteadiness in flow, at the same time it should be big enough to take care of any fluctuation in the flow.
An engineer always speak about averaged quantities when he comes across a turbulent flow. Because averaged quantities are pretty enough for his purpose. Knowledge of actual fluctuating value of a turbulent flow might be useful in scientific world, but for an engineer it is of no use most of the time. Figure below shows averaging operation in a turbulent-unsteady flow.
It is clear from above figure that actual velocity can have 2 components, one average component and another fluctuating component. Similarly one can define averaging for any other flow variable say pressure,temperature,other components of velocity etc.
Shear stress in a Turbulent Flow & Turbulence Modeling
Let us consider a turbulent pipe flow case, if you want to determine shear stress near pipe wall, first thing you have to obtain is averaged velocity profile near wall as shown in figure below.
Fig.6 Average velocity profile and inter layer mixing in a turbulent flow
Assuming this is 2 dimensional flow case one can express shear stress parallel to flow direction as
Thus shear stress has got 2 components. First component which is similar to shear stress in a laminar case is known as laminar shear stress. Second component arises due to mixing of different fluid layers in a turbulent flow as shown in figure above. This is known as turbulent shear stress or Reynolds stress. So shear stress in a turbulent flow can be represented as
One can note Reynolds stress is in terms of fluctuating parts of velocity components, which are unknown to the user. Determination of Reynolds stress in terms of known quantities (averaged quantities)is considered to be one of the toughest problem in fluid mechanics. And this is known as Turbulence Modeling.
Applications Utilizing Effect of Turbulence
Most of the time turbulence has positive effect on engineering devices. It increases convective heat transfer, it increases mixing and reduces drag around a body.
Heat Transfer Enhancement
Convective heat transfer coefficient increases drastically when the flow becomes turbulent, due to effective mixing of different fluid layers in the flow. This behaviour is shown in following figure.So it is a common practice among designers to covert laminar flows into turbulent by introducing suitable vortex generators in the flow.
Fig.:Increase in heat transfer coefficient due to turbulence
Coefficient of drag around a body reduces by a huge amount when flow changes from laminar to turbulent.This phenomenon is shown in following figure.This is the reason why golf ball has got lot of dimples on it.This irregularities on surface of the ball will help in transforming laminar flow into turbulent and reduces drag, with low drag ball can travel more distance.
Fig.8 Change in drag coefficient over a sphere when flow changes from laminar to turbulent