Fluid Kinematics

Kinematics of Fluid Flows

Kinematics:

The study of motion. Fluid kinematics:

The study of how fluids flow and how to describe fluid motion.

The relationship between pressure and flow velocity is important in many engineering applications. Blood pressure (flow of blood through veins and arteries) Weather forecasts (pressure readings of atmospheric flow patterns) Stirring a cup of coffee (pressure variations enhance mixing) Design of tall structures (pressure forces from the wind) Aircraft design (lift and drag) Flow systems (heating and air conditioning)

Summary/ Outline

Objectives: Mathematically describe the motion of a fluid Express the acceleration and vorticity of a fluid particle given the velocity components

Describe the deformation of a fluid particle Classify various fluid flows. Is a flow viscous, is it turbulent, is it incompressible, is it a uniform flow? Present several examples and numerous problems that demonstrate how fluid flows are described and how flows are classified

Descriptions of Fluid Motion Lagrangian description: Description of motion where individual particles are observed as a function of time

Descriptions of Fluid Motion cont: Eulerian description: Description of motion where the flow properties are functions of both space and time

Different Lines in Describing a Flow Field Pathline: The actual path traveled by an individual fluid particle over some time period.

Different Lines in Describing a Flow Field cont.

Streakline: The locus of fluid particles that have passed sequentially through a prescribed point in the flow.

Different Lines in Describing a Flow Field cont. Streamline: a line in the flow where the velocity vector is tangent to the streamline Streamtube: A tube whose walls are Streamlines

Acceleration of a fluid particle is found by considering a particular particle. Its velocity changes from V(t) at time t to V (t + dt). Angular Velocity: the average velocity of two perpendicular line segments of a fluid particle Vorticity: Twice the Angular Velocity

Classification of Fluid Flows Uniform Flow The velocity does not change along a fluid path.

Nonuniform Flow The velocity changes along a fluid path.

Classification of Fluid Flows cont. Steady Flow: The velocity at a given point on a fluid path does not change with time. Unsteady Flow: The velocity at a given point on a fluid path changes with time.

Classification of Fluid Flows cont. Laminar Flow: Well-ordered state of flow in which adjacent flow layers move smoothly with respect to each other. Turbulent Flow: Unsteady flow characterized by intense cross-stream mixing

Classification of Fluid Flows cont Incompressible Flow: The density of each Fluid particle remains constant. Compressible Flow: Density Variations influence the flow

Classification of Fluid Flows cont. Viscous Flow: The effects of viscosity are significant Inviscid Flow: Viscous effects do not significantly influence the flow.

Dimensional Flows One- Dimensional

Two- Dimensional

Three-Dimensional

Description of Fluid Flows •

LAGRANGIAN DESCRIPTION

•

EULERIAN DESCRIPTION

Lagrangian Description Pieces of the fluid are “tagged”. The fluid flow properties are determined by tracking the motion and properties of the particles as they move in time.

Joseph-Louis Lagrange

In the example shown, particles A and B have been identified. Position vectors and velocity vectors are shown at one instant of time for each of these marked particles. As the particles move in the flow field, their positions and velocities change with time, as seen in the animated diagram.

Eulerian Description

The Eulerian Description is one in which a control volume is defined, within which fluid flow properties of interest are expressed as fields. Leonhard Euler

In the Eulerian description of fluid flow, individual fluid particles are not identified. Instead, a control volume is defined, as shown in the diagram. Pressure, velocity, acceleration, and all other flow properties are described as fields within the control volume. In other words, each property is expressed as a function of space and time, as shown for the velocity field in the diagram.

In the Eulerian description of fluid flow, one is not concerned about the location or velocity of any particular particle, but rather about the velocity, acceleration, etc. of whatever particle happens to be at a particular location of interest at a particular time. Since fluid flow is a continuum phenomenon, at least down to the molecular level, the Eulerian description is usually preferred in fluid mechanics.

Note, however, that the physical laws such as Newton's laws and the laws of conservation of mass and energy apply directly to particles in a Lagrangian description. Hence, some translation or reformulation of these laws is required for use with an Eulerian description.



Pressure field - An example of a fluid flow variable expressed in Eulerian terms is the pressure. Rather than following the pressure of an individual particle, a pressure field is introduced, i.e. p = p(x,y,z,t).



Velocity field - An example of a fluid flow variable expressed in Eulerian terms is the velocity. Rather than following the velocity of an individual particle, a velocity field is introduced, i.e.



Acceleration field - An example of a fluid flow variable expressed in Eulerian terms is the acceleration. Rather than following the acceleration of an individual particle, an acceleration fieldis introduced, i.e.

EULARIAN VS LAGRANGIAN It is generally more common to use Eulerian approach to fluid flows. Measuring water temperature, or pressure at a point in a pipe. Lagrangian methods sometimes used in experiments. Throwing tracers into moving water bodies to determine currents (see movie Twister). X-ray opaque tracers in human blood. Bird migration example. Ornithologists with binoculars count migrating birds moving past a (Euler) or scientists place radio transmitters on the birds (Lagrange).



Either description method is valid in fluid mechanics, but the Eulerian description is usually preferred because there are simply too many particles to keep track of in a Lagrangian description

Methods of Visualizing Fluid Flows •

STREAMLINE

•

STREAKLINE

•

PATHLINE

Streamlines, streaklines and pathlines are used in the visualization of fluid flow. Streamlines mainly used in analytic work, streaklines and pathlines used in experimental work.



STREAMLINE

A streamline is a line everywhere tangent to the velocity vector at a given instant of time. (A streamline is an instantaneous pattern.) For example, consider simple shear flow between parallel plates. At some instant of time, a streamline can be drawn by connecting the velocity vector lines such that the streamline is everywhere parallel to the local velocity vector. In this example, streamlines are simply horizontal lines.



STREAKLINE

A streakline is the locus of particles which have earlier passed through a prescribed point in space. (A streakline is an integrated pattern.) For example, consider simple shear flow between parallel plates. A streakline is formed by injecting dye into the fluid at a fixed point in space. As time marches on, the streakline gets longer and longer, and represents an integrated history of the dye streak. In this example, streaklines are simply horizontal lines.



PATHLINE

A pathline is the actual path traversed by a given (marked) fluid particle. (A pathline is an integrated pattern.) For example, consider simple shear flow between parallel plates. A pathline is the actual path traversed by a given (marked) fluid particle. A pathline represents an integrated history of where the fluid particle has been. In this example, pathlines are simply horizontal lines.

Acceleration Acceleration The acceleration of a fluid particle is found by considering a particular particle shown in the figure. z V(t + dt) dV V(t) V(t)

V(t + dt)

y

𝑎= x

𝑑𝑉 𝑑𝑡

Velocity vector V 𝑉 = 𝑉 𝑥, 𝑦, 𝑧, 𝑡

𝑉 = 𝑢𝑖 + 𝑣𝑗 + wk

𝜕𝑉 𝜕𝑉 𝜕𝑉 𝜕𝑉 𝑑𝑉 = 𝑑𝑥 + 𝑑𝑦 + 𝑑𝑧 + 𝑑𝑡 𝜕𝑥 𝜕𝑦 𝜕𝑧 𝜕𝑡

𝑎=

𝜕𝑉 𝑑𝑥 𝜕𝑉 𝑑𝑦 𝜕𝑉 𝑑𝑧 𝜕𝑉 + + + 𝜕𝑥 𝑑𝑡 𝜕𝑦 𝑑𝑥 𝜕𝑧 𝑑𝑥 𝜕𝑡 𝑎=𝑢

𝑑𝑦 =𝑣 𝑑𝑡

𝑑𝑧 =𝑣 𝑑𝑡

𝜕𝑉 𝜕𝑉 𝜕𝑉 𝜕𝑉 + 𝑣 +𝑤 + 𝜕𝑥 𝜕𝑦 𝜕𝑧 𝜕𝑡

𝜕𝑢 𝜕𝑢 𝜕𝑢 𝜕𝑢 𝑎𝑥 = +𝑢 +𝑣 +𝑤 𝜕𝑡 𝜕𝑥 𝜕𝑦 𝜕𝑧 𝑎𝑧 =

𝑑𝑥 =𝑢 𝑑𝑡

𝜕𝑣 𝜕𝑣 𝜕𝑣 𝜕𝑣 𝑎𝑦 = +𝑢 +𝑣 +𝑤 𝜕𝑡 𝜕𝑥 𝜕𝑦 𝜕𝑧

𝜕𝑤 𝜕𝑤 𝜕𝑤 𝜕𝑤 +𝑢 +𝑣 +𝑤 𝜕𝑡 𝜕𝑥 𝜕𝑦 𝜕𝑧

𝑎=

𝐷𝑉 𝐷𝑡

𝐷 𝜕 𝜕 𝜕 𝜕 =𝑢 +𝑣 +𝑤 + 𝐷𝑡 𝜕𝑥 𝜕𝑦 𝜕𝑧 𝜕𝑡

Substantial Derivative or Material Derivative It is given a special name and special symbol (D/Dt instead of d/dt) because we followed a particular fluid particle, that is, we followed the substance (or material). It represents the relationship between a Lagrangian derivative in which a quantity depends on time t and Eulerian derivative in which a quantity depends on position (x, y, z, t). It can be used with other dependent variables; for example, DT/Dt would represent the rate of change of the temperature of a fluid as we followed the particle along.

𝑎=𝑢

𝜕𝑉 𝜕𝑉 𝜕𝑉 𝜕𝑉 + 𝑣 +𝑤 + 𝜕𝑥 𝜕𝑦 𝜕𝑧 𝜕𝑡

The time derivative term on the right side of the equations is called the local acceleration and the remaining terms on the right side form the convective acceleration. In a pipe, local acceleration results if, for example, a valve is being opened or closed; and convective acceleration occurs in the vicinity of a change in the pipe geometry, such as a pipe contraction or an elbow.

𝑑2𝑆 𝑑Ω 𝐴=𝑎+ + 2Ω ∗ 𝑉 + Ω ∗ Ω ∗ 𝑟 + ∗𝑟 𝑑𝑡 2 𝑑𝑡 Acceleration of a particle to a fixed frame

 If the acceleration of all fluid particles is given by A = a in a selected reference frame, it is inertial reference frame.  If A ≠ a, it is a noninertial reference frame.

z Ω

S

r

a V y

x Motion relative to noninertial reference frame

Angular Velocity and Vorticity

A fluid flow may be thought of as the motion of a collection of fluid particles. As a particle travels along it may rotate or deform.

Consider a moving fluid element which is initially rectangular. If the velocity varies significantly across the extent of the element, its corners will not move in unison, and the element will rotate and become distorted.

There are certain flows, or regions of a flow, in which the fluid particles do not rotate; such flows are of special importance, particularly in flows around objects, and are referred to as irrotational flows.

𝜕𝑢 𝑑𝑦 𝑢+ 𝜕𝑦 2 D 𝜕𝑣 𝑑𝑥 𝑣− 𝜕𝑥 2

𝑣+

v

A

u

B

𝜕𝑣 𝑑𝑥 𝜕𝑥 2 dy

C 𝜕𝑢 𝑑𝑦 𝑢− 𝜕𝑦 2

dx

Fluid particle occupying an infinitesimal parallelepiped at a particular instant.

Let us consider a small fluid particle that occupies an infinitesimal volume that has the xy-face as shown in the figure. The angular velocity Ω𝑧 about the z axis is the average of the angular velocity of line segment AB and line segment CD. 𝑢𝐷 − 𝑢𝐶 𝑣𝐵 − 𝑣𝐴 Ω𝐶𝐷 = − Ω𝐴𝐵 = 𝑑𝑦 𝑑𝑥 𝜕𝑢 𝑑𝑦 𝜕𝑢 𝑑𝑦 𝑢+ − 𝑢 − 𝜕𝑣 𝑑𝑥 𝜕𝑣 𝑑𝑥 𝜕𝑦 2 𝜕𝑦 2 𝑣+ − 𝑣 − Ω𝐶𝐷 = − 𝜕𝑥 2 𝜕𝑥 2 𝑑𝑥 Ω𝐴𝐵 = 𝑑𝑥 𝜕𝑢 𝜕𝑣 =− 𝜕𝑦 = 𝜕𝑥 Ω𝑧 = Considering the xz-face 1 𝜕𝑢 𝜕𝑤 Ω𝑦 = − 2 𝜕𝑧 𝜕𝑥

Ω𝑧 =

1 Ω + Ω𝐶𝐷 2 𝐴𝐵 1 𝜕𝑣 𝜕𝑢 − 2 𝜕𝑥 𝜕𝑦

Considering the yz-face 1 𝜕𝑤 𝜕𝑣 Ω𝑥 = − 2 𝜕𝑦 𝜕𝑧

A cork placed in a water flow in a wide channel (the x-y plane) would rotate with an angular velocity about the z-axis, given by the equation, Ω𝑧 =

1 𝜕𝑣 𝜕𝑢 − 2 𝜕𝑥 𝜕𝑦

Vorticity Vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along with the flow. It is common to define the vorticity ω to be twice the angular velocity. Its components are then, 𝜕𝑤 𝜕𝑣 𝜔𝑥 = − 𝜕𝑦 𝜕𝑧

𝜕𝑢 𝜕𝑤 𝜔𝑦 = − 𝜕𝑧 𝜕𝑥

𝜔𝑧 =

𝜕𝑣 𝜕𝑢 − 𝜕𝑥 𝜕𝑦

An irrotational flow possesses no vorticity; the cork mentioned above would not rotate in an irrotational flow.

The deformation of the particle of the figure is the rate of change of the angle that line segment AB makes with line segment CD. If AB is rotating with an angular velocity different from that of CD, the particle is deforming. The deformation is represented by the rate-ofstrain tensor; its component 𝜖𝑥𝑦 in the xy-plane is given by 1 𝜖𝑥𝑦 = (Ω𝐴𝐵 − Ω𝐶𝐷 ) 2 𝜖𝑥𝑦 =

1 𝜕𝑣 𝜕𝑢 ( + ) 2 𝜕𝑥 𝜕𝑦

For the xz-plane and the yz-plane we have 𝜖𝑥𝑧 =

1 𝜕𝑤 𝜕𝑢 ( + ) 2 𝜕𝑥 𝜕𝑦

𝜖𝑦𝑧 =

1 𝜕𝑤 𝜕𝑣 ( + ) 2 𝜕𝑦 𝜕𝑧

The fluid particle could also deform by being stretched or compressed in a particular direction. This normal rate of strain is measured by

𝑢𝐵 − 𝑢𝐴 𝜖𝑥𝑥 = 𝑑𝑥 𝜕𝑢 𝑑𝑥 𝜕𝑢 𝑑𝑥 𝑢+ − 𝑢 − 𝜕𝑥 2 𝜕𝑥 2 𝜖𝑥𝑥 = 𝑑𝑥

=

𝜕𝑢 𝜕𝑥

The symmetric rate-of-strain tensor can be displayed as

𝜖𝑖𝑗 =

𝜖𝑥𝑥 𝜖𝑥𝑦 𝜖𝑥𝑧

𝜖𝑥𝑦 𝜖𝑦𝑦 𝜖𝑦𝑧

𝜖𝑥𝑧 𝜖𝑦𝑧 𝜖𝑧𝑧

 A flow field is best characterized by its

velocity distribution, and thus a flow is said to be one-, two-, or threedimensional if the flow velocity varies in one, two, or three primary dimensions, respectively.

 A

typical fluid flow involves a threedimensional geometry, and the velocity may vary in all three dimensions, rendering the flow three dimensional [V (x, y, z) in rectangular or V (r, u, z) in cylindrical coordinates].

• The velocity profile develops fully and remains unchanged after some distance from the inlet (about 10 pipe diameters in turbulent flow, and less in laminar pipe flow, as in Fig. 1–24) and the flow in this region is said to be fully developed. The fully developed flow in a circular pipe is one-dimensional since the velocity varies in the radial r-direction but not in the angular u- or axial z-directions. That is, the velocity profile is the same at any axial zlocation, and it is symmetric about the axis of the pipe.

• Note that the dimensionality of the flow also depends on the choice of coordinate system and its orientation. The pipe flow discussed, for example, is one-dimensional in cylindrical coordinates, but two-dimensional in Cartesian coordinates—illustrating the importance of choosing the most appropriate coordinate system. • Also note that even in this simple flow, the velocity cannot be uniform across the cross section of the pipe because of the no-slip condition.

• However, the variation of velocity in certain directions

can be small relative to the variation in other directions and can be ignored with negligible error. In such cases, the flow can be modeled conveniently as being one- or two-dimensional, which is easier to analyze.

 Consider steady flow of a fluid entering from a large

tank into a circular pipe. The fluid velocity everywhere on the pipe surface is zero because of the no-slip condition, and the flow is two-dimensional in the entrance region of the pipe since the velocity changes in both the r- and z-directions, but not in the udirection.

• However, at a well-rounded entrance to the pipe, the velocity profile may be approximated as being nearly uniform across the pipe, since the velocity is nearly constant at all radii except very close to the pipe wall.

• A flow may be approximated as two-dimensional when the aspect ratio is large and the flow does not change appreciably along the longer dimension.



For example, the flow of air over a car antenna can be considered two-dimensional except near its ends since the antenna’s length is much greater than its diameter, and the airflow hitting the antenna is fairly uniform (Fig. 1–25).

 When two fluid layers move relative to each

other, a friction force develops between them and the slower layer tries to slow down the faster layer. This internal resistance to flow is quantified by the fluid property viscosity, which is a measure of internal stickiness of the fluid. Viscosity is caused by cohesive forces between the molecules in liquids and by molecular collisions in gases.  There is no fluid with zero viscosity, and thus all fluid flows involve viscous effects to some degree. Flows in which the frictional effects are significant are called viscous flows.

 In many flows of practical interest, there are regions

(typically regions not close to solid surfaces) where viscous forces are negligibly small compared to inertial or pressure forces.

 Neglecting the viscous terms in such inviscid flow regions

greatly simplifies the analysis without much loss in accuracy. The fluid sticks to the plate on both sides because of the no-slip condition, and the thin boundary layer in which the viscous effects are significant near the plate surface is the viscous flow region. The region of flow on both sides away from the plate and largely unaffected by the presence

• An inviscid flow is the flow of an ideal fluid that is assumed to have no viscocity. In fluid dynamics there are problems that are easily solved by using the simplifying assumption of an inviscid flow. • The flow of fluids with low values of viscocity agree closely with inviscid flow everywhere except close to the fluid boundary where the boundary layer plays a significant role

 The assumption of inviscid flow is generally

valid where viscous forces are small in comparison to the inertial forces. Such flow situations can be identified as flows with a Reynolds number much greater than one. The assumption that viscous forces are negligible can be used to simplify the Navier–Stokes solution to the Euler equations.

 While throughout much of a flow-field

the effect of viscosity may be very small, a number of factors make the assumption of negligible viscosity invalid in many cases.  Viscosity cannot be neglected near fluid boundaries because of the presence of aboundary layer, which enhances the effect of even a small amount of viscosity.

Turbulent and Laminar Flow Classification of Flow

Laminar A flow that is characterized to have a:  Uniform or consistent velocity in all points  A smooth flow  Particles do not cross each other  Re ≤ 2000 (2100)

Turbulent A flow that is characterized to have a: • Interrupted flow • Velocity at any point is not constant • Particles cross each other • Examples are: blood circulation inside arteries, oil transportation inside pipelines, lava movement, atmosphere, etc. • Re > 2000 (2100)

Osborne Reynold

Reynold’s Number  British engineer, physicist and educator best known for his work in hydraulics and hydrodynamics .

Where :

𝜐 - mean velocity (m/s) 𝐷- pipe diameter (m)  Also studied wave engineering and v- kinematic viscosity of the fluid (m2/s) tidal motions 𝜇- absolute or dynamic viscosity (Pa-s) ρ – density of the fluid (kg/m3)  Formulated the LUBRICATION (1886)

Sample Problem 1 (# 7-1)

Sample Problem 2 (# 7-2) 3 (# 7-3 Sample Problem

Water having kinematic viscosity v = 1.3 × 10−6 m2 /s flows in a 100-mm diameter pipe at a velocity of 4.5 m/s. Is the flow laminar or turbulent? •

for laminar condition Re ≤ 2000 𝜐𝐷 𝜐𝐷𝜌

Oil of specific gravity 0.80 flows in a 200 mm diameter •• At Recritical = =velocity in pipes, Re = v 𝜇 −2 •

𝜐𝐷

2000 Re = v

=

•

𝜐𝐷𝜌 = 𝜇 where: 4.5(0.1) −6

1.3 ×𝜐𝐷𝜌 10 𝜐𝐷 Re =𝜐𝐷 (𝑄𝐴=)𝐷 𝜇 = v =

pipe. Find the critical velocity. Use μ = 8.14 × 10

𝜐 = Q/A

Pa-s.

For laminar flow conditions, what size of pipe will deliver 6 liters per second of oil having kinematic viscosity of 6.1 × 10−6 m2 /s ?

Re = v346,153.85 > 2000 v 𝑣𝑐 0 2 1000 0 8 ( . )( )( . ) −2 8.14 × 10is the flow 0.006

2000 =

π 2𝐷 4 𝐷

2000 𝒗𝒄==6.11.0175 m/s turbulent × 10−6

D = 0.62618 m or 626.18, Say 626 mm Φ

Velocity Distribution in Pipes Laminar Flow - follows a parabolic curve with zero velocity at the walls. Where :

u=

𝛾ℎ𝐿𝑟 𝜐𝑐 4𝜇𝐿

𝜐𝑎𝑣𝑒 = 1 𝜐𝑐 2 u2= 𝜐𝑐 (1 𝑟 ) 𝑟u𝑜2= 𝜐 − 𝑐 𝑥

u - velocity at distance r 𝛾 - unit weight of the fluid ℎ𝐿 - head lost in the pipe 𝑟 - distance from the center 𝑟𝑜 - radius of the pipe 𝜇 - absolute viscosity 𝐿 - pipe length 𝜐𝑐 - centreline or max. velocity 𝜐 - average velocity

Velocity Distribution in Pipes Turbulent Flow - varies with Reynold’s number with zero velocity at the wall and increase more rapidly for a short distance 𝜏 from the walls. • u = 𝜐𝑐 – 5.75

𝑜

𝜌

𝑟𝑜 𝜏𝑜 𝑟𝑜− 𝑟 𝑓𝜐2 = 𝜌 8

log • • 𝜐𝑐

=

Where: 𝜏𝑜- maximum shearing stress

𝑣(1 + 1.33 𝑓 )

𝑓 – friction factor 𝜐 – mean velocity

• 𝑣 = 𝜐𝑐 − 3.75

𝜏𝑜 𝜌

To Find “ f“ Darcy-Weisbach Laminar Flow Formula

hf == • 𝑓

642 𝑓𝐿𝑣 2𝑔𝐷 𝑅 𝑒

=

Where:

64μ 𝑣𝐷ρ

For non-circular pipe:

•𝐷 =h4Rf = R=

32μ𝐿𝑣 ρ𝑔𝐷2

𝑐𝑟𝑜𝑠𝑠−𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑝𝑖𝑝𝑒,𝐴 𝑝𝑖𝑝𝑒 𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟,𝑃

 hf – minor head loss in the pipe  𝑓 - friction factor

 𝐿 - length of the pipe  𝑣 - mean or average velocity of flow  𝐷 - diameter of the pipe

 𝑔 - gravitational acceleration  R - hydraulic radius

Sample Problem 4 (# 718)

Oil of specific gravity 0.9 and dynamic viscosity μ = 0.04 Pa-s flows at the rate of 60 liters per second through 50 m of 120-mm-diameter pipe. If the head lost is 6 m, determine (a) the mean velocity of flow, (b) the type of flow, (c) the friction factor 𝑓, (d) the velocity at the center line of the pipe (a) Mean velocity 𝜐 = Q/A 0.060 = π 2 4

0.12

𝜐 = 5.305 m/s (b) Type of flow

Re = 𝜐𝐷𝜌 𝜇 0.9 = 5.305(0.12)(1000) 0.04 Re = 14,323.5 >

(c) Friction factor 𝑓𝐿𝑣2 hf = 2𝑔𝐷 6=

𝑓(50)(5.305)2 2(9.81)(0.12)

𝑓 = 0.01004 (d) Centerline velocity 𝜐c = 𝜐(1+1.33 𝑓) = 5.305(1+1.33 0.01004) 𝜐 c = 6.012 m/s

Sample Problem 5 (# 722) Glycerin (sp. gr. = 1.26 and μ = 1.49 Pa-s) flows through a rectangular conduit 300 mm by 450 mm at the rate of 160 liters per sec. (a) Is the flow laminar or turbulent? (b) Determine the head lost per kilometre length of pipe (a)

Non-circular conduits: D = 4R

(b) For laminar flow

Re =

𝜐𝐷𝜌 𝜇

𝑓 = 𝑅𝑒

64

𝜐 = Q/A = (

1.185 m/s D = 4R = 4(0.9)

0.160 )= 0.3(0.45)

4(0.3×0.45) = 2(0.45+0.3)

1.185(0.36)(1000)(1.26)

Re = 1.49 Re = 360.75 < 2000 (laminar)

64

= 360.75 𝑓 = 0.1774 𝑓𝐿𝑣2 2𝑔𝐷 0.1774(1000)(1.185)2 = 2(9.81)(0.36)

h𝑓=

h 𝑓 = 35.27 m

(ISOCHORIC FLOW)

 Fluid

motion with negligible changes in density.  No fluid is truly incompressible, since even liquids have their density increased through the application of sufficient pressure.  Density changes in a flow will be negligible if the Mach Number (Ma) of the flow is small.

MACH NUMBER (Ernst Mach) 𝑽 𝒄

Ma = < 0.3 where V = velocity of a fluid c = speed of sound of a fluid = 𝑘𝑅𝑇 

It tells how fast the object is moving, if it is moving equal to or faster than the speed of sound. Ma = 1 Ma < 1 Ma > 1 Ma >> 1

Sonic flow Subsonic flow Supersonic flow Hypersonic flow



Air at 68°F (20°C) has a speed of sound 780 mi/h (340 m/s). Thus inequality indicates that air is incompressible at velocities up to 228 mi/h (102 m/s).  It is nearly impossible to attain Ma=0.3 in liquid flow because of the very high pressures required. Thus liquid flow is incompressible.



A flow is said to be incompressible if the density of a fluid element does not change during its motion. It is a property of the flow and not of the fluid. The rate of change of density of a material fluid element is given by the material derivative

GOVERNING EQUATIONS NAVIER-STOKES EQUATIONS FOR INCOMPESSIBLE FLOW ARE:

INCOMPRESSIBLE FLOW IN 2D Continuity

→

X-Momentum → Y-Momentum →

SAMPLE PROBLEM The velocity distribution for a two-dimensional incompressible flow is given by u=

𝑥 − 2 2 𝑥 +𝑦

v=

𝑦 − 2 2 𝑥 +𝑦

Show that it satisfies continuity

The continuity equation for 2D Incompressible flow

𝜕𝑢 𝜕𝑥

+

𝜕𝑣 𝜕𝑦

=0

Then 2

𝜕𝑢 1 2𝑥 =− 2 + 𝜕𝑥 𝑥 + 𝑦 2 (𝑥 2 + 𝑦 2 )2

𝜕𝑣 1 2𝑦 2 =− 2 + 2 2 𝜕𝑦 𝑥 +𝑦 (𝑥 + 𝑦 2 )2 And their sum does equal to zero, satisfying the continuity.

COMPRESSIBLE FLOW

What is compressible flow?

 Compressible

flow (gas dynamics) is the branch of fluid mechanics that deals with flows having significant changes in fluid density.

 In

compressibility the speed of greater than 250 mph, the density of the air changes.

 When

a fluid flow is compressible, the fluid density varies with its pressure. Compressible flows are usually high speed flows with mach numbers greater than about 0.3. Examples includes, Aerodynamics applications such as flow over a wing or aircraft nacelle as well as industrial applications such as flow through a high-performance Valves.

What is mach?  Mach

is the ratio of the speed of a body to the speed of sound in the surrounding medium. It is often used with a numeral ( as Mach 1, Mach 2, and etc.) to indicate the speed of sound. And it is the contribution of Ernst Walfred Josef Wenzel Mach an Austrian physicist and philosopher in Physics.

Ernst Mach (18391916)

 The

Speed of Sound is called Mach 1

 Mach

is commonly used to represent an object’s speed, such as an aircraft or a missile, when it is travelling at the speed of sound or at multiples of it.

 Incompressible

flows do not have such a variation of density. The key differentiation between compressible and incompressible is the velocity of the flow. A fluid such as air that is moving slower than mach 0.3 is considered incompressible, even though it is a gas. A gas that is run through a compressor is not truly considered compressible unless its velocity exceeds mach 0.3.

Regimes of Flight  Mach

Number below 1 means the flow velocity is lower than the speed of sound and the sound is subsonic.

 Mach

Number 1 means the flow velocity is the speed of sound and the speed around that is transonic.

 Mach

Number above 1 means the flow velocity is higher than the speed of sound and the speed is supersonic.

 Mach

Number more than 3 is called

THE AIRCRAFT FASTER THAN Mach 1 (Speed of Sound)

Maximum Speed of Mach 1.6 (1,930Km/h)

Maximum Speed of Mach 2.35 ( 2,500 km/h)

Maximum Speed of Mach 2.5 ( 2,655km/h)

Maximum Speed of Mach 2.5 ( 2,655km/h)

Maximum Speed of Mach 2.83 ( 3000km/h)

Maximum Speed of Mach 3.02 ( 3,219km/h)

Maximum Speed of Mach 3.2 ( 3,370km/h)

Maximum Speed of Mach 3.2 (

Maximum Speed of Mach 3.2 ( 3,330km/h)

Maximum Speed of Mach 3.3 ( 3,540km/h)

Maximum Speed of Mach 6.72 ( 7,274km/h)

SOUND BARRIER

 The

Speed of Sound is given as,

Where k = 1.4 for air, R = 287J/Kg.K or gas constant, and T = reference temperature ( in absolute units)

The Mach Number is given as,

M = V/c where M is a mach number, V is a gas velocity, and “c” is the speed of sound.

Example #1.

Determine the speed of sound in Air at 120 degree celsius. C = 𝐾𝑅𝑇 = (1.4)(287)(393) C = 397.38 m/s