FLUID KINEMATICS

1
FLUID KINEMATICS

• BY
• GP CAPT NC CHATTOPADHYAY

2
Fluid Kinematics

• Velocity Field
• Continuity Equation

3
Fluid Kinematics

• What is fluid kinematics?
• Fluid kinematics is the study on fluid motion in
  space and time without considering the force
  which causes the fluid motion.
• According to the continuum hypothesis the local
  velocity of fluid is the velocity of an
  infinitesimally small fluid particle/element at a
  given instant t. It is generally a continuous
  function in space and time.

4
Velocity Field

• Eulerian Flow Description
• Lagrangian Flow Description
• Streamline
• Pathline
• Streakline

5
In the Eulerian Method

• The flow quantities, like p,u,T,? are described
  as a function of space and time without referring
  to any individual identity of the fluid particle
  (ALL PARTICLES ARE CONSIDERED)

6
Streamline

• A line in the fluid whose tangent at a point is
  parallel to the instantaneous velocity vector
  at a given instant t. So, the tangent indicates
  the velocity at that point.
• The family of streamlines at time t are solutions
  of dx/ u dy/v dz/w (so, the equations of
  stream lines are..)
• Where u,v,w are functions of x,y,z,t and u,v,w
  are velocity components in the respective
  direction

7
TYPES OF FLOW

• Steady flow the streamlines are fixed in space
  for all time. d(k)/dt0 (NO CHANGE W.R.T. TIME)
• Unsteady flow the streamlines are changing from
  instant to instant. d(k)/dt0

8
Flow Dimensionality

• Most of the real flows are
• 3-dimensional and unsteady u (x,y,z,t)
• For many situations simplifications can be made
• 2-dimensional unsteady and steady flow
• u (x,y,t)
• 1-dimensional unsteady and steady flow
• u (x,t)

9
In the Lagrangian Method

• The flow quantities are described for each
  individually identifiable fluid particle moving
  through flow field of interest. The position of
  the individual fluid particle is a function of
  time. (STUDY OF EACH PARTICLE IS CUMBERSOME)

10
Pathline

• A line traced by an individual fluid particle
  r(t)
• For a steady flow the path lines are identical
  with the streamlines.

11
Streakline

• A streak line consists of all fluid particles in
  a flow that have previously passed through a
  common point. Such a line can be produced by
  continuously injecting marked fluid (smoke in
  air, or dye in water) at a given location.
  (locus of particles at a particular station)
• For steady flow The streamline, the path line,
  and the streak line are the same.

12
Stream-tube and Continuity Equation

• Stream-tube
• Continuity Equation of a Steady Flow

13
Stream-tube

• Is the surface formed instantaneously by all the
  streamlines that pass through a given closed
  curve in the fluid.

14
DEFINITIONS

• LAMINAR FLOW IS LAMINAR IFPARTICLES MOVE IN
  DEFINED LAYERS IN DEFINED PATH,(NO CROSSING OF
  LAYERS, flow on ac skin)
• TURBULENT PARTICLES MOVE IN A ZIG- ZAG WAY
  (PARTICLES CROSS EACH OTHER/ LAYERS, high speed
  flow in pipe)

15
DEFINITIONS

• ROTATIONAL PARTICLES ROTATE ABOUT OWN AXIS
  (FLOW NEAR SOLID BOUNDARY, ROTATING TANK)
• IRROTATIONAL PARTICLES MAINTAIN SAME
  ORIENTATION (FLOW ON LOW SPEED AEROFOIL)

16
STREAM FUNCTION VELOCITY POTENTIAL FUNCTION

• STREAM FUNCTION (?) A FUNCTION IN THE 2 D
  FLOW FIELD WHOSE DERIVATIVES REPRESENT VELOCITIES
  ALONG RESPECTIVE AXES. DIFFERENCE BETWEEN TWO
  NEIGHBOURING STREAM FUNCTIONS INDICATE VOLUMETRIC
  FLOW i.e. ?1 ?2 VOL THROUGH THE STREAM
  LINES.
• ALSO, u ? ?/? y and v - ? ?/?
  x,
• VELOCITY POTENTIAL FUNCTION (F) A SCALER
  FUNCTION WHOSE NEGATIVE DERIVATIVES REPRESENT
  RESPECTIVE VELOCITIES. IT INDICATES IRROTATIONAL
  OR P0TENTIAL FLOW. MATHEMATICALLY,
• u - ? F /? x and v - ? F /?y, w -
  ? F/?z

17
DEFINITIONS

• Continuity
• Matter cannot be created or destroyed - (it is
  simply changed in to a different form of matter).
  This principle is know as the conservation of
  mass and we use it in the analysis of flowing
  fluids.
• The principle is applied to fixed volumes, known
  as control volumes For steady flow -
  Mass entering per unit time Mass leaving per
  unit time

18
CONTINUITY
A liquid is flowing from left to right and the
pipe is narrowing in the same direction. By the
continuity principle, the mass flow rate must be
the same at each section - the mass going into
the pipe is equal to the mass going out of the
pipe. So we can write
19
Continuity Equation of a Steady Flow

• For a steady flow the stream-tube formed by a
  closed curved fixed in space is also fixed in
  space, and no fluid can penetrate through the
  stream-tube surface, like a duct wall.

20
Considering a stream-tube of cylindrical cross
sections with velocities
perpendicular to the cross
sections and densities
at the respective cross sections
and assuming the velocities
and densities are constant across the whole cross
section , a fluid mass closed
between cross section 1 and 2 at an instant t
will be moved after a time interval dt by
to the cross section 1 and
2 respectively. Because the closed mass
between 1 and 2 must be the same between 1 and
2, and the mass between 1 and 2 for a steady
flow can not change from t and tdt, the mass
between 1 and 1 moved in dt, i.e
must be the same as the mass between 2 and 2
moved in the same time dt i.e

21
(No Transcript)
22
CONTINUITY OF FLOW
23
DISCHARGE

• Discharge and mean velocity
• If we know the size of a pipe, and we know the
  discharge, we can deduce the mean velocity AS
  Um Q/A
• If the area of cross section of the pipe at point
  X is A, and the mean velocity here is Um. During
  a time t, a cylinder of fluid will pass point X
  with a volume Q. The volume per unit time (the
  discharge) will thus be

Discharge in a pipe
24
DEFINITIONS

• Mass flow rate MASS/ TIME

• Volume flow rate (Discharge)
• Simply called flow rate The symbol
  normally used for discharge is Q. The
  discharge is the volume of fluid flowing per
  unit time. Multiplying this by the density of
  the fluid gives us the mass flow rate.
  Consequently, if the density of the fluid for
  example is 850 and time is 1 sec for 0.857 cubic m

then
25
Derivation of the Continuity Equation

• Lets start with a small, fixed volume of fluid
  somewhere in the middle of a flow stream. This
  elemental volume has sides of lengths Dx, Dy and
  Dz (see Figure 1).
•  

The rate of mass entering a face is the product
of the density, the fluid velocity and the face
area. For example, on the side facing the reader,
the density (r) is multiplied by the velocity in
the x direction (u) and the area of the face Dy
Dz. Thus, the mass flux entering the volume
through this face is
26
CONTINUITY EQUATION

• The mass leaving the volume on the opposite side
  of the volume is again the product of density,
  velocity and area, but the density and velocity
  may have changed as the fluid passed through the
  volume. We will express these changes as small
  quantities (since our volume is small enough),
  i.e., ? ? ? and u ? u. The mass flux leaving
  that face is thus
• Performing the same analysis on the mass entering
  the volume through the other faces of the volume
  gives us
• Similarly, the mass fluxes leaving the volume on
  the opposite faces are

27
CONTINUITY EQUATION

• All of these added together must equal the mass
  of fluid accumulating in the volume,
• Putting all of these together, we have

28
CONTINUITY EQUATION

• Multiplying out the quantities in parentheses
  results in the cancellation of some terms and the
  appearance of higher-order terms such as ??, ?u
  ?x ?y ?z . Since the quantities preceded by ?
  are very small, products of these quantities will
  be extremely small, depending on the number of ?
  terms included in the product. The terms with
  four of these will be much smaller than the terms
  with only three ? terms. Thus, all higher order
  terms are neglected. This leaves
• which, when divided by and rearranged,
  yields

29
CONTINUITY EQUATION
The application of basic calculus (taking the
limit as ?t tends to 0) allows us to write this
equation as
The Continuity Equation may be simplified
for some common flow situations as follows. If
the fluid may be treated as incompressible (as is
the case with water or in low velocity air
flows), the density will be constant. The
Continuity Equation then becomes
30
CONTINUITY

• Another example of the use of the continuity
  principle is to determine the velocities in pipes
  coming from a junction.

Total mass flow into the junction Total mass
flow out of the junction r1Q1 r2Q2 r3Q3 When
the flow is incompressible (e.g. if it is water)
r1 r2 r
31
PROBLEM

• If pipe 1 diameter 50mm, mean velocity 2m/s,
  pipe 2 diameter 40mm takes 30 of total discharge
  and pipe 3 diameter 60mm. What are the values of
  discharge and mean velocity in each pipe?

32
Practice numericals

• As discussed