Dr WCDK Fernando 1
LEARNING OUTCOMES • Identify the importance of ideal fluid flow analysis • Discuss various ways to visualize flow fields • Explain fundamental kinematic properties of fluid motion and deformation • Discuss the concepts of vorticity, rotationality & irrotationality • Describe simple ideal flows • Describe and sketch combined flow patterns WCDKF-KDU
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WHAT IS AN IDEAL FLUID?
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Ideal fluid
Real fluid
Imaginary
Real or practical
Incompressible
Compressible
Non-viscous
Viscous WCDKF-KDU
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INTRODUCTION
• Fluid Kinematics deals with the motion of fluids without considering the forces and moments which create the 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.
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FLUID FLOW
• Lagrangian Description
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FLUID FLOW • Eulerian Description
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FLUID FLOW • Eulerian Description – Pressure field p = p(x,y,z,t) – Velocity field – Acceleration field
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VELOCITY
• In the rectangular coordinate system, Directions
Velocity components
X
u
dx/dt
y
v
dy/dt
z
w
dz/dt
V ui vj wk
V u v w 2
2
r xi yj zk WCDKF-KDU
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1
2
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Ex 1 • The velocity components expressed in m/s in a fluid flow are known to be u = (6xy2+t), v = (3yz+t2+5), w = (2+3ty) where x, y, z are given in metres and time t in seconds. Set up an expression for the velocity vector at point P (4, 1, 2) m at T = 3 S. Also determine the magnitude of velocity for this flow field at the given location and time. WCDKF-KDU
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ROTATIONAL & IRROTATIONAL FLOWS
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Kinematic Description • In fluid mechanics, an element may undergo four fundamental types of motion. a)Translation b)Rotation c)Linear strain d)Shear strain • Because fluids are in constant motion, motion and deformation is best described in terms of rates.
y
TRANSLATION
+ x
D
A
dy
B
dx
C
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y
TRANSLATION
+ x
A’
D’
B’
C’
D
A
dy vdt B
C
dx udt
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y
ROTATION
+ x
D
A
dy
B
dx
C
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y
ROTATION • Angular rotation of element about z-axis is defined as the average counterclockwise rotation of the two sides BC and BA D A
+ x
A’
db
dy
B
dx
C
B’
D’
da
C’ WCDKF-KDU
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ROTATION
y
+
1 d z da db 2
x
u dydt y
A’
db
B’
D’
da
v dxdt x C’
u dydt u 1 y db tan dt dy y v dxdt v 1 x dt da tan dx x
d z 1 v u dt 2 x y
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y
EXTENSIONAL STRAIN (DILATATION) +
x D
A
dy
B
dx
C
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EXTENSIONAL STRAIN (DILATATION) •
Extensional strain in x-direction is defined as the fractional increase in length of the A’ D’ horizontal side of the element y +
x
A
D
dy
B’
B
dx
C
u dx dxdt x
u dx dxdt dx u x Extensional strain rate in x-direction dt dt WCDKF-KDU xx dx x
C’
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y
SHEAR STRAIN
+ x
D
A
dy
B
dx
C
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SHEAR STRAIN • Defined as the average decrease of the angle between two lines which y are initially perpendicular in the unstrained state (AB and BC) + x
D
A
db
dy
da B
dx
C
Shear-strain increment Shear-strain rate WCDKF-KDU
1 da db 2 1 da db xy 2 dt dt 21
DISTORTION OF A MOVING FLUID ELEMENT
v dxdt x
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DISTORTION
• Average angular displacement 1 v dt u dt 2 x y 1 v u • Mean rate of rotation 2 x y
• The quantity v u is known as the Vorticity x y (Ω ). • ω=½Ω v u • For irrotational flow, ω = 0 0 WCDKF-KDU
x v u x y
y
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A STEADY IRROTATIONAL FLOW IS CLASSIFIED AS POTENTIAL FLOW.
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CIRCULATION • Circulation is the line integral of tangential velocity around a closed contour in the flow field. A measure of the rotation within a finite element of a fluid
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CIRCULATION Circulation is considered positive in an anticlockwise direction.
V d l V cos a d l WCDKF-KDU
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Calculate the circulation within a small fluid element with area xy
v u udx vdy u x v x y u y x v y x y WCDKF-KDU 27
v u udx vdy u x v x y u y x v y x y
v u xy x y v u lim relative vorticity xy x y
xy 0
A Circulation per unit area equals the vorticity in flow. WCDKF-KDU
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Ex 2 • Determine the circulation Τ around a rectangle defined by x=1, y=1, x=5 and y=4 for the velocity field u = 2x + 3y and v = -2y.
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FLOW VISUALIZATION • Flow visualization is the visual examination of flow-field features. • Important for both physical experiments and numerical (CFD) solutions. • Numerous methods – Streamlines and streamtubes – Pathlines – Streaklines – Timelines – Refractive techniques – Surface flow techniques WCDKF-KDU
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STREAMLINES • A line in the fluid whose tangent is parallel to at a given instant t. • Steady flow : the streamlines are fixed in space for all time. • Unsteady flow : the streamlines are changing from instant to instant. WCDKF-KDU
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STREAMLINES • A Streamline is a curve that is everywhere tangent to the instantaneous local velocity vector. dx dy dz u ,v ,w dt dt dt
• Equation of a general streamline dx dy dz u v w
STREAMLINES • For 2-D flow,
dx dy u v dy v dx u
• Streamlines do not cross, otherwise the fluid particle will have two velocities at the point of intersection. • The flow is only along the streamline and not cross it.
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STREAM-TUBE • is the surface formed instantaneously by all the streamlines that pass through a given closed curve in the fluid.
Since no fluid can penetrate the streamlines, the flow passing through each of the sections would be same. WCDKF-KDU
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PATHLINE • A line traced by an individual fluid particle • For a steady flow the pathlines are identical with the streamlines. A Pathline is the actual path traveled by an individual fluid particle over a time period.
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STREAKLINE • A streakline 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. • For steady flow : The streamline, the pathline, and the streakline are the same.
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STREAKLINES • A Streakline is the locus of fluid particles that have passed sequentially through a prescribed point in the flow. • Easy to generate in experiments: dye in a water flow, or smoke in an airflow.
COMPARISON • For steady flow, streamlines, pathlines, and streaklines are identical. • For unsteady flow, they can be very different. – Streamlines are an instantaneous picture of the flow field – Pathlines and Streaklines are flow patterns that have a time history associated with them. – Streakline: instantaneous snapshot of a timeintegrated flow pattern. – Pathline: time-exposed flow path of an individual particle. WCDKF-KDU
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Ex 3 • Determine the equation of streamline for a two dimensional flow field for which the velocity components are given by i. u = a and v = a where a is a non-zero constant. The streamline passes through the point (1, 3). ii. u = y/b2 and v = x/a2. The streamline passes through the point (a, 0).
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Ex 4 • In a steady fluid flow, the velocity components are u = 2kx, v = 2ky, w = -4kz. Find the equation of streamline passing through the point (1, 0, 1).
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VELOCITY POTENTIAL FUNCTION • Imagine that a function φ exist such that its derivative in any direction gives the velocity in that direction
u
x v y
• The function φ is called the velocity potential function and lines of constant potential function are termed equipotential lines. 42 WCDKF-KDU
VELOCITY POTENTIAL FUNCTION • Since φ is a function of x and y alone,
d dx dy u.dx v.dy x y
• For an equipotential line (φ = constant), dφ = 0 u.dx v.dy 0 v.dy u.dx dy u dx v WCDKF-KDU
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VELOCITY POTENTIAL FUNCTION v u x y x y y x 0 xy yx Hence the velocity potential function, φ exists when the flow is irrotational. 2
2
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Differential Equation of Continuity • The fluid is continuous both in space & time. • For an incompressible fluid, the density ρ would be constant. • For 3-D incompressible flow u v w 0 x y z • For 2-D incompressible flow
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• When φ exists, 0 x x y y 2 2 x y 2
2
Φ satisfies the Laplace Equation WCDKF-KDU
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Ex 4 • Which of the following velocity fields pertain to the motion of steady, two-dimensional flow of an incompressible fluid
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STREAM FUNCTION • Mathematically, the stream function for a flow in the x – y is defined as a function of x and y such that the velocity components are given by,
u y v x where ψ is the value of stream function. WCDKF-KDU
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STREAM FUNCTION Considering the continuity of flow u v 0 x y 0 x y y x 2 2 0 xy yx 2 2 xy yx WCDKF-KDU
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STREAM FUNCTION Show that ψ satisfies the Laplace Equation for irrotational flow
v u 0 x y
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Ex 5 A fluid flows along a flat surface parallel to the xdirection. The velocity u varies linearly with y, the distance from the flat surface and u=Ay a) Find the stream function of the flow b) Determine whether the flow is irrotational
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Ex 6 If φ=3xy, find x and y components of velocity at (1, 3) and (3, 3). Determine the discharge passing between streamlines.
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PROPERTIES OF φ AND ψ Property Continuity equation Irrotationality condition
ψ Automatically satisfied Satisfied if ………..
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φ Satisfied if ………………… Automatically satisfied
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PROPERTIES OF φ AND ψ • Streamlines and equipotential lines are orthogonal to each other.
The gradient of the equipotential line = -u/v the gradient of a stream line = v/u WCDKF-KDU
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FLOW THROUGH A BEND
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