Lagrangian and Eulerian approaches
There are two ways to describe fluid motion. One is called Lagrangian, where one follows all
fluid particles and describes the variations around each fluid particle along its trajectory. The
other is Eulerian, where the variations are described at all fixed stations/spaces as a function of
time. In the second, different particles pass the same station at different times. Lagrangian is a
system approach and Eulerian is control volume approach.
Types of Flows
Steady and unsteady flow: A flow is defined as steady when its fluid characteristics like
velocity, density, and pressure at a point do not change with time. Mathematically the time
derivative of the properties must be zero. A flow is defined unsteady, when the fluid
characteristics velocity, pressure and density at a point changes with respect to time.
Uniform and Non-Uniform Flow: Uniform flow is the type of fluid flow in which the velocity
of the flow at any given time does not change with respect to space [Along the length of
direction of flow].
A non-uniform flow is a type of fluid flow in which the velocity of the flow at any given time
changes with respect to space.
Compressible and Incompressible Flows: A compressible flow is a type of flow in which the
density of the fluid changes from one point to another point. This means the density is not
constant. Incompressible flow is that type of flow in which the density of the fluid is constant
from one point to another. Liquids are generally incompressible and gases are compressible.
Rotational and Irrotational Flows: A type of flow in which the fluid particles rotate about
their own axis while flowing along the streamlines is called a rotational flow. If the fluid
particles while flowing along the streamline do not rotate about their own axis, then the flow
is called irrotational flow.
One, Two and Three Dimensional Flows: One-dimensional fluid flow is a fluid flow in
which, the flow parameter such as velocity is expressed as a function of time and one space
coordinates. That is,
u = f(x,t), v=0; w=0;
In this type, the velocity along y and z directions i.e. v and w are considered negligible.
Two-dimensional flow is that type of flow in which the velocity is a function of time and two
rectangular space co-ordinates. The velocity of flow along the third direction is considered
negligible. That is,
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u = f(x,y,t); v = g(x,y,t); w = 0;
Three-dimensional flow is the type of flow in which the velocity is a function of time and three
mutually perpendicular rectangular space coordinates (x, y, and z). That is,
u = f(x,y,z,t); v = g(x,y,z,t); w = h(x,y,z,t)
Laminar and Turbulent Flow: Laminar flow is defined as a type of flow in which the fluid
particles move along a well-defined streamline or paths, such that all the streamlines are
straight and parallel to each other. In a laminar flow, fluid particles move in laminas. The layers
in laminar flow glide smoothly over the adjacent layer. Turbulent flow is a type of flow in
which the fluid particles move in a zig-zag manner. The movement in zig-zag manner results
in high turbulence and eddies are formed. This results in high energy loss. Laminar and
Turbulent flow in a pipe flow is characterized based on the Reynolds number.
Streamline, Streak line, and Path line:
Streamline, path line, and streak line are used to describe a flow and visualize it.
A streamline is one that drawn is tangential to the velocity vector at every point in the flow at
a given instant.
The equation of streamline is
where u,v, and w are the velocity components in x, y, and z directions respectively.
By definition, different streamlines at the same instant in a flow do not intersect, because a
fluid particle cannot have two different velocities at the same point. A bundle of streamlines is
called stream tube.
Streaklines are the locus of points of all the fluid particles that have passed continuously
through a particular spatial point in the past. Dye is steadily injected into the fluid at a fixed
point extends along a streakline. Eg; Reynolds experiment.
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Pathline is the line traced by a given particle. (Lagrangian concept)
In a steady flow, the streamline, pathline, and streakline all coincide. In an unsteady flow, they
can be different. Streamlines are easily generated mathematically while pathlines and
streaklines are obtained through experiments. Streamlines are an instantaneous feature while
streaklines and pathlines have time history.
Velocity and Acceleration in Fluid:
Velocity and Acceleration are vector functions that are used to explain the kinematics of fluid
flow. Both velocity and acceleration have their respective components in three directions, i.e
x, y, and z. Both the components are dependent on the space-co-ordinates (x,y, and z)and time
‘t’).
V = (u,v,w)= ui+vj+wk
Where u, v, and w are functions of x,y,z and t.
The magnitude of velocity is √𝑢2 + 𝑣 2 + 𝑤 2 . The acceleration is,
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In compact form it can be written as
Where,
, is gradient operator.
The first term called local or temporal acceleration results from velocity changes with respect
to time at a given point. Local acceleration results when the flow is unsteady. For steady flow
it will be zero.
The second term is called convective acceleration because it is associated with spatial gradients
of velocity in the flow field. Convective acceleration results when the flow is non-uniform, that
is, if the velocity changes along a streamline. The convective acceleration terms are nonlinear
which causes mathematical difficulties in flow analysis; also, even in a steady flow, the
convective acceleration can be large if spatial gradients of velocity are large.
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Circulation and Vorticity:
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 is a vector field, which
gives a microscopic measure of the rotation at any point in the fluid. Vorticity is defined as the
local rotation or spin of the fluid element about an axis through the element.
angular velocity of the fluid element is given by,
The rotation vector is given as
(half the curl of rotation)
The curl of the velocity field is the vector quantity that is equal to twice of the angular velocity.
That vector quantity is the vorticity that gives us the measure of the local rotation of the fluid
element. Which is given by
If we consider a closed path c around the fluid flow then circulation is defined as the line
integral of the tangential velocity around c.
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, is the circulation.
By applying the Stoke’s theorem, circulation can be related to the vorticity as;
This shows that the circulation actually is the surface integral of the all the vorticities of an area
bounded by the curve c. In other words, circulation is the flux of vorticity. And conversely,
vorticity at a point can be considered as the circulation per unit area.
Vorticity is a vector quantity and gives the measure of local rotation while circulation is a scalar
quantity and it gives the measure of global rotation. Circulation can be actually thought of as
the ‘push’ that can be felt while moving along a closed path or boundary. For example, take
water in a bucket or tank and then stir it so that the flow becomes like that of a whirlpool. Now,
if you keep any object in that flow, the object will experience the push and that is circulation.
If we gather all the circulation in an area and calculate it around a single point, it is called
Vorticity.
Stream Function and Potential Function:
Stream function is a scalar function of space and time whose derivative with respect to any
direction would give the velocity component at right angles to that direction.
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Substituting in continuity (2D) equation,
we get,
, as it satisfies the continuity equation, the existence of a stream function
proves a possible case of fluid flow. This flow can be either rotational or irrotational.
If the rotational component is given by the formula, wz, then we get the Laplace relation for
Stream Function
A stream function of a fluid satisfying a Laplace equation is supposed to have an irrotational
flow. The difference between the stream function values at any two points gives the volumetric
flow rate (or volumetric flux) through a line connecting the two points.
Velocity potential function is a scalar function of space and time, whose negative derivative,
with respect to any direction, gives the velocity component in that direction. If, Phi is
Here, u, v, and w are the velocity components of the fluid flow along x, y, and z directions.
Substitute in continuity equation,
That is, the velocity potential function satisfies the Laplace equation, then it corresponds to
some case of fluid flow.
The line along which the velocity potential function is constant is called an equipotential line.
The slope of equipotential line is given by dy/dx = -u/v.
Streamlines are defined as the lines along which the stream function is constant. Hence the
slope at any point on the streamline is given by dy/dx=v/u;
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The product of slope of an equipotential line and streamline is obtained as -1. This means, both
the lines are orthogonal to each other. Hence, knowing the value of the stream function, the
velocity potential value is determined.
The velocity potential function and stream function are related as,
Comparing both the equations,
Flow nets:
A flow net is a grid obtained by drawing a series of equipotential lines and streamlines. Flow
net is very much useful in analysing the two-dimensional, irrotational flow problems. Usually,
the flow net is a square mesh. However, in regions, where the boundaries converge or diverge
or bend, the flow net does not contain squares.
The
flow
nets
can
be
constructed
only
in
the
following
situations
(i) The flow should be steady. This is so because the streamline pattern for unsteady flow does
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not
remain
constant,
it
changes
from
instant
to
instant.
(ii) The flow should be irrotational, which is possible only when the flowing fluid is an ideal
fluid
(having
no
viscosity)
or
it
has
negligible
viscosity.
(iii) The flow should not be governed by the force of gravity, because under the action of
gravity, the shape of the free surface changes constantly, and hence no fixed flow net pattern
can be obtained.
Applications of Flow net :………………………………………………………………......
The flow net helps in depicting and analyzing the behaviour of irrotational flow. Many flow
phenomena which cannot be analyzed easily by mathematical means can be analyzed by
drawing flow nets. The following are some of the important uses of flow net analysis (i) For given boundaries of flow, the velocity and pressure distribution can be determined, if
the velocity distribution and pressure at any reference section are known.
(ii) Loss of flow due to seepage in earth dams and unlined canals can be evaluated.
(iii) Uplift pressures on the underside (bottom) of the dam can be worked out.
(iv) Outlets can be designed for their streamlining.
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