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Monday, 17 April 2017

Derivation for continuity equation in integral form

Derivation for continuity equation in integral form

Integral form of continuity equation

In Aerodynamics there are three fundamental equations

1. Continuity equation
2. Momentum equation
3. Energy equation

We will study all the equations one by one. 

Let's talk about Continuity equation once.

1. Continuity equation 

Continuity equation is simply conservation of mass of the flowing fluid. Consider fluid flowing through the pipe. It is really not possible that fluid entering from one end of pipe vanishes while coming out of other end of the pipe(Except if its magical fluid, just kidding). This is a same thing which continuity equation tells us. That mass of flowing fluid is conserved.

Let 
'm' be the Mass of the fluid
'V' be the Volume of the fluid
'ρ' be the Density of the fluid
Click here to know more about fluid properties

As we know Density is equal to ration of mass and volume

hence 

ρ = m/V           (1)

So mass becomes,

m = ρ x V           (2)

Volume can be written as Area times thickness

i.e V = A x t         (3)

Where,

'A' is Cross section are of pipe
't' is thickness of fluid column in pipe

So Mass becomes,

m = ρ x A x t           (4)     (Replacing V by A x t)          

To find of mass flow rate, differentiation above equation with respect to time

'' be the mass flow rate

Hence

 = ρ x A x v         (5) (Differentiation of t with respect to time gives velocity of the fluid)           

Considering mass flow rate we got is for small section of fluid

So to find mass flow rate for entire fluid

We will write mass flow rate from equation (5) in integral form

   
From equation (2) we know that


m = ρ x V

 So taking Elemental volume '∀' instead of 'V'

 m = ρ x ∀             (6)

 To find mass flow rate integrating equation (6) with respect to time, we get



Equating both the mass flow rate equations we get,


using gauss divergence theorem,





Since the volume∀  does not change with time, the sequence of differentiation and integration in the first term of can be interchanged. Therefore

This is integral form of continuity equation

We can also write it as,


This is Continuity equation!


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Properties of fluids

Properties of fluids

Fluid Properties


Before going into complex aerodynamics it is mandatory to know about basic fluid property. Let's take a look at fluid property one by one.

Pressure:


Pressure is simply force applied on the body per unit area.
. 

Fluid property - Pressure
P= F/A
where A tends to zero
The unit of Pressure is N/m^2.
It is a point property and the pressure acting on a fluid varies from point to point.

Fluid property - Pressure




Density:


Fluid property- Density
In very simple terms, consider density as a property of fluid to measure how thick fluid is. Density is given by mass of an element in space to the volume of space occupied by it . So over here density is the ratio of the mass of the fluid element being  considered, ‘m’ to the volume of  fluid over which the mass is being considered, ‘v’.
Density = m/v
The unit of Density is kg/m^3. The density of air at sea level is about 1.2 kg/m3












Volume:



Fluid property - Volume

Lets talk about volume now. You have definitely come across this term many times before in your life. To define volume, let us consider a jar of water having a volume of 20 liters. This means that the water jar can maximum contain 20 liters of water. This signifies that volume is the measurement of the amount of a fluid that is being contained in a system. Volume is 3D quantity. If you multiply area of any shape by its thickness you get volume. If jar is in rectangular shape, it can be calculated as the product of length, breadth and height(thickness) of fluid. In other way volume is taken as the ratio between the mass (m) of the fluid being considered,  to the density(ρ) of the fluid being considered.

V = m/ρ

The unit for Volume is m^3


Temperature:

Fluid property - Temperature

In the field of aerodynamics, temperature is considered only for high speed flight. Because in high speed flight, production of heat due to friction is considerable. We will talk in depth about this in our forthcoming posts. Now coming back to temperature in relation to subsonic flow. Let us begin with a simple example, what happens to a young kid when you give him a little too much chocolate to have? He will get hyper and find a way to expend all his energy until he gets tired, yes? (Just wanted to tell about excitation) Similarly, when the molecules of a fluid are excited by an external stimulus, they start to move rapidly, colliding with one and other. Due to this rapid collision, heat is generated and the temperature of the flow increases. 
Fluid property - Temperature

High energy means high temperature!

Velocity:

                             
Fluid property - Velocity
Consider a fluid flowing with some velocity ‘u’. Now what do we mean when we say ‘flowing with some velocity’? It basically means that the fluid is being displaced from one point to another or the rate of change of displacement. When we talk about the velocity of a solid, say a ball, we can safely say that the entire ball is being displaced with the same velocity. But in the case of fluids, it is not the same, velocity is a point property and it varies from point to point. This is mainly due to the physical property of a fluid i.e. Viscosity. 


Viscosity is simply resisting force to the motion of the fluid. It is analogous to friction force in solid.

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Saturday, 18 March 2017

What is Control Volume and Control Surface?

 What is Control Volume and Control Surface?

CV= Control Volume, CS= Control Surface - Wikimedia
Control Volume is simply mathematical representation of 3D space.

In fluid mechanics and thermodynamics, the control volume is a volume fixed in space or moving with constant velocity (no acceleration) through which the fluid (it could be gas or liquid) flows.

The surface enclosing the control volume is referred to as the control surface.  Control surface is boundary of Control Volume.

Mathematically Control Volume is denoted by ∂V.
In the figure above, CV is Control Volume.

Mathematically Control Surface is denoted by ∂S.
In the figure above, CS is Control Surface.

Relation between Control Volume and Control Surface: -


Relation between Control Volume and Control Surface can be given by Gauss Divergence Theorem








For more information about Control Volume and Control Surface Visit

What is the magnus effect?

What is the Magnus effect?

 

Understanding Magnus Effect 

 

Magnus Effect - Wikimedia

Magnus effect is one of the important effect to study in Aerodynamics. Magnus effect is an effect on on the rotating object placed in airflow due to its rotation. Let's discuss about Magnus effect.

Discovery: -

 

The Magnus effect is named after Heinrich Gustav Magnus, the German physicist who discovered it.

Heinrich Gustav Magnus - Wikimedia

Definition: -

 

Encyclopedia Britannica defines Magnus effect as follows

Magnus effect, generation of a sidewise force on a spinning cylindrical or spherical solid immersed in a fluid (liquid or gas) when there is relative motion between the spinning body and the fluid.

Oxford Dictionary defines Magnus effect as

The force exerted on a rapidly spinning cylinder or sphere moving through air or another fluid in a direction at an angle to the axis of spin. This force is responsible for the swerving of balls when hit or thrown with spin. 

 Explanation: -

 

Consider streamline airflow over simple stationary cylinder

Airflow over stationary cylinder - Wikimedia
Airflow will simply go around the cylinder as shown in figure above.

Now, Check out the airflow over rotating cylinder

Airflow over rotating Cylinder - Wikimedia

(Consider left side of image is front of the cylinder and right side of the image is back of the cylinder)
When Airflow hits rotating cylinder, velocity of the air will increase in the direction of the rotation of the cylinder due to friction between surface of the cylinder and the air. In the figure above, cylinder is rotating in the clockwise direction. So due to friction, cylinder will throw the air backwards. This increases the velocity of the air at the backside of the cylinder. As we know according to Bernoulli's principle higher the velocity, lower the pressure. Hence, Increased velocity behind the cylinder will reduce the pressure behind it(cylinder). And due to pressure difference between front and back of the cylinder, cylinder will move backwards (Towards the region with less pressure). This effect is known as Magnus effect.

Application of the Magnus Effect: -

 

Flettner aircraft: - A flettner aircraft is a type of aircraft which uses a Flettner rotor to provide lift. The rotor consists of a spinning cylinder with circular end plates and it spins about a spanwise horizontal axis. Magnus effect creates lift.

Flettner Aircraft Application of Magnus effect - Wikimedia

Tuesday, 14 March 2017

Streamlines, Streaklines, Pathlines and Timelines

Different types of Field lines : - Streamlines, Streaklines, Pathlines and Timelines

Streamlines - Wikimedia

 

Field Lines

Field lines are simply graphical representation of various types of flows. Field lines can be divided into following types
  • Streamlines
  • Streaklines
  • Pathlines
  • Timelines
 Lets discuss about Stramlines, Streaklines, Pathlines and Timelines one by one

Streamlines

Streamlines are graphical representation of fluid flow in such a way that tangent to any point on curve gives direction of velocity at that point.


Streaklines

Assume that the fluid is flowing through the fixed source. And fluid particles A, B, and C started to flow through source at same time. So at any instant of time, the line formed by joining instantaneous location of fluid particles A, B and C is known as Streakline. Line in black is streakline in the figure.
Streakline - Aircraft Explained


Pathlines

Pathline is shown in the figure below. Pathlines are formed by tracing path of an individual fluid particle. So, pathlines are simply path taken by individual fluid particle in fluid flow.
This is like recording path of the fluid particle for certain period of time. In the figure below Point A, B, C, D, and E are positions of fluid particle at different interval of time.
Pathline - Aircraft Explained

Timelines

 Timeline is formed by joining adjacent fluid particles in fluid flow at any instant of time

Saturday, 11 March 2017

Reynolds number

Reynolds number

Laminar, Turbulent, Transition flow Reynolds number - Wikimedia
Reynolds number is a dimensionless quantity used to determine the type of flow. From the value of Reynolds number one can tell whether flow is laminar or turbulent or flow is in transition.

Laminar flow

The flow in which fluid flows smoothly such that fluid layers are parallel to each other or no to streamlines intersect each other, such type of flow is known as laminar flow

Turbulent flow

The flow in which fluid flows in zig-zag manner and fluctuate irregularly in such a way that its velocity changes irregularly, such type of flow is known as turbulent flow

Laminar and Turbulent flow (Reynolds number) - Wikimedia

To know more about different types of fluid flow check out this post

Osborne Reynolds studied various conditions for types of flow such as Turbulent flow, Laminar flow and flow in transition. Osborne Reynolds described dependency of flow on Reynolds number.


Osborne Reynolds - Wikimedia


Reynolds numbers is defined as ratio of inertial forces to viscous forces.

Numerically Reynolds number can be given by,

Reynolds number

    Re is Reynolds number
    ρ is the density of the fluid (kg/m3)
    u is the velocity of the fluid with respect to the object (m/s)
    L is a characteristic linear dimension (m)
    μ is the dynamic viscosity of the fluid (Pa.s or N.s/m2 or kg/(m.s))
    ν is the kinematic viscosity of the fluid (m2/s).

For flow of fluid through pipe through fixed diameter D,
When value of Reynolds number is approximately less than 2000, then flow is Laminar.
When value of Reynolds number is approximately in between 2000 to 4000, then flow is in Transition.
When value of Reynolds number is approximately more than 4000, then flow is Turbulent.

Behavior of fluid flow for various values of Reynolds number is shown

Behavior of flow for various values of Reynolds number - Wikimedia

Different types of flow

Types of flow
Types of fluid flow - Pixabay

According to nature of flows and their dependency on other factors such as density, velocity gradient, etc. flows are divided into various types. Let's discuss today about various types of flows.

Incompressible Flow

Incompressible flow is type of flow in which density of fluid remains constant. It means fluid is incompressible. Practically incompressible flow is not possible. But in aerodynamics when velocity of air is less than 0.3 mach (370.44 Kmph) flow is considered to be as incompressible flow.

Compressible Flow

Compressible flow is flow in which density of fluid changes with respect to distance. For example, consider fluid flowing over body. So the fluid will have higher density at a place where fluid collides with body.

Steady flow

If fluid parameters such as velocity, acceleration, etc does not change with respect to time, Such type of flow is know as Steady flow.

Unsteady Flow

If fluid parameters such as velocity, acceleration, etc changes with respect to to time, such a  type of flow is known as Unsteady flow.

Uniform flow

If fluid parameters such as velocity, acceleration, etc does not change with respect to space, such type of flow is known as Uniform flow. 

Non-Uniform flow

If fluid parameters such as velocity, acceleration, etc changes with respect to space, such type of flow is known as Non-Uniform flow

Rotational flow

when fluid particles while flowing rotates about their own axis, such a flow is known as Rotational flow.

Irrotational flow

When fluid particles while flowing does not rotate about their own axis, such a type of flow is known as Irrotational flow.

Viscous flow

When viscosity of fluid is considered in fluid flow, such type of flow is known as Viscous flow. Viscosity is a resisting force to flowing fluid.

Inviscid flow

When viscosity of fluid is not considered in fluid flow, such type of flow is known as Inviscid flow.

Streamline flow

The flow in which velocity is constant or varies with regular manner, such type of flow is known as Streamline flow.

Laminar flow

The flow in which fluid flows smoothly such that fluid layers are parallel to each other or no to streamlines intersect each other, such type of flow is known as laminar flow.

Laminar and turbulent type of flow - Wikimedia

Turbulent flow

The flow in which fluid flows in zig-zag manner and fluctuate irregularly in such a way that its velocity changes irregularly, such type of flow is known as turbulent flow.

The flow is Turbulent flow or Laminar flow this can be determined from the value of Reynolds number. To know more about Reynolds number Checkout this post.

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