FLUID KINEMATICS

By – Shabin George

What is Fluid Kinematics ? Branch of fluid mechanics which deals with response of fluids in

motion without considering forces and energies in them.

The study of kinematics is often referred to as the geometry of motion.

It is generally a continuous function in space and time.

Methods of describing Fluid Motion

Langrangian Method : It describes a defined mass (position, velocity, acceleration,

temperature , pressure, etc) as functions of time.

Eg. Track the location of a migrating bird.

Eulerian Method : It describes the flow field ( velocity, acceleration, pressure,

temperature, etc. ) as functions of position and time.

Eg. Counting the birds passing a particular location.

Acceleration Field

The acceleration of the particle is ascribable to two reasons. One reason is due to the fluid particle being convected from one

given location to another location in flow and the second location being of higher or lower velocity.

This is called as convective acceleration. It is also known as advective acceleration.

The second reason is due to the unsteadiness of the flow, that is due to the change in the local velocity of the fluid particle as a function of time. This is called as local acceleration.

Types of Flow Depending upon fluid properties

Ideal and Real flow.

Compressible and Incompressible flow.

Depending upon properties of flow

Laminar and turbulent flow.

Steady and unsteady flow.

Uniform and Non-uniform flow.

Rotational and Irrotational flow.

One, two and three dimensional flow.

Ideal and Real flow• Real fluid flows implies friction effects. Ideal fluid flow is

hypothetical.• It assumes no friction.

Velocity distribution of pipe flow

Compressible and Incompressible flows

Incompressible fluid flows assumes the fluid have constant density while in compressible fluid flow, density is variable and becomes function of temperature and pressure.

Laminar and turbulent flowThe flow in which adjacent layer do not cross to each other and move along the

well define path is called as laminar flow.

Laminar flow

Turbulent flow

Transition of flow from Laminar to turbulent

Reynold’s number is used todifferentiate between laminar andturbulent flows.

Steady and Unsteady flows Steady flow: It is the flow in which conditions of flow

remains constant w.r.t. time at a particular section but the condition may be different at different sections.

Flow conditions: velocity, pressure, density or cross-sectional area etc.

e.g., A constant discharge through a pipe.

Unsteady flow : It is the flow in which conditions of flow

changes w.r.t. time at a particular section. e.g. A variable discharge through a pipe.

Uniform and Non-uniform flow Uniform flow: The flow in which velocity at a given time

does not change with respect to space (length of direction of flow) is called as uniform flow.

e.g. Constant discharge though a constant diameter pipe.

Non – Uniform flow : The flow in which velocity at a given time

changes with respect to space (length of direction of flow) is called as non – uniform flow.

e.g., Constant discharge through variable diameter pipe.

Rotational & Irrotational Flow

Rotational Flows :-

The flow in which fuid particle while flowing along stream lines rotate about their own axis is called as rotational flow.

Irrotational Flows:-

The flow in which the fluid particle while flowing along stream lines do not rotate about their axis is called as irrotational flow.

One, Two and Three Dimensional Flows

Although in general all fluids flow three-dimensionally, with pressures and velocities and other flow properties varying in all directions, in many cases the greatest changes only occur in two directions or even only in one. In these cases changes in the other direction can be effectively ignored making analysis much more simple.

Flow is one dimensional : if the flow parameters (such as velocity, pressure, depth etc.) at a given instant in time only vary in the direction of flow and not across the cross-section.

One, Two and Three Dimensional Flows

Flow is two-dimensional : if it can be assumed that the flow parameters vary in the direction of flow and in one direction at right angles to this direction.

Flow is three-dimensional : if the flow parameters vary in all three direction of flow.

Visualization of flow Pattern• The flow velocity is the basic description of how a fluid moves in

time and space, but in order to visualize the flow pattern it is useful to define some other properties of the flow. These definitions correspond to various experimental methods of visualizing fluid flow. They are :

a. Streamlines b. Pathlinesc. Streak lines

Flow around cylindrical objectCAR surface pressure contoursand streamlines

Visualization of flow Pattern

Streamlines around a wing shaped body

Flow around a skiing athlete

Stream lineA Streamline is a curve that is everywhere tangent to the instantaneous local velocity vector.

It has the direction of the velocity vector at each point of flow across the streamline.

Character of Streamline : 1. Streamlines can not cross eachother. (otherwise, the cross pointwill have two tangential lines.) 2. Streamline can't be a folding line,but a smooth curve. 3. Streamline cluster density reflectsthe magnitude of velocity. (Densestreamlines mean large velocity;while sparse streamlines mean smallvelocity.

PATHLINE A Pathline is the actual path travelled by an individual fluid particle over some time period.

Same as the fluid particle's material position vector .

And the path of a particle same as Streamline for Steady Flow.

Streak line and Stream Tubes A streakline is the locus of

fluid particles that have passed sequentially through a prescribed point in the flow.

Easy to generate in experiments like dye in a water flow, or smoke in an airflow.

Streamtube : is an imaginary tube whose boundary consists of streamlines.The volume flow rate must be the same for all cross sections of the stream tube.

Circulation and Vorticity Circulation and vorticity are two primary measures of rotation in a

fluid. Circulation, which is a scalar integral quantity, is a macroscopic

measure of rotation for a finite area of the fluid. Vorticity however is a vector field that gives a microscopic measure of

the rotation at any point in the fluid.

VORTICITYCIRCULATION

The circulation, C, about a closed contour in a fluid is defined as the line integral evaluated along the contour of the component of the velocity vector that is locally tangent to the contour.

“Meaning” of Circulation : Circulation can be considered as the amount of force that pushes along a closed boundary or path. Circulation is the total “push” you get when going along a path, such as a circle.

Circulation

Vorticity Vorticity is the tendency for elements of the fluid to spin.

Vorticity can be related to the amount of circulation or rotation (or more strictly the angular rate of rotation) in a fluid.

Vertical Component of Vorticity : In large-scale dynamic meteorology, we are in concerned only

with the vertical components of absolute and relative vorticity.

The vertical component of vorticity is defined as the circulation about a closed contour in the horizontal plane divided by the area enclosed, in the limit where the area approaches zero.

Stream Function It is defined as the scalar function of space and time, such that its

partial derivative with respect to any direction gives the velocity component at right angles to that direction . It is denoted by psi and defined for two dimensional flow.

Properties of Stream Function :i. If stream function exists, it is a possible case of fluid flow which

may be rotational or irrotational.ii. If stream function satisfies the Laplace equation, it is a possible

case of an irrotational flow.

Laplace equation

Velocity Potential Function It is defined as a scalar function of space and time such that its

negative derivative with respect to any direction gives the fluid velocity in that direction. It is denoted by phi.

The negative sign indicates that the flow takes place in the direction in which phi decreases.

Flow Net A grid obtained by drawing a series of stream lines and

equipotential lines is known as a flow net. Flow net provides a simple graphical technique for studying two –

dimensional irrotational flows, when the mathematical calculation is difficult.

The stream lines and equipotential lines are mutually perpendicular to each other.

A flow net analysis assist in the design of an efficient boundary shapes. It is also used to calculate the flow at ground level.

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