ME 6123 Mechanics of Inviscid Incompressible Fluidteacher.buet.ac.bd/toufiquehasan/ME 6123-Lecture...

ME 6123

Mechanics of Inviscid Incompressible Fluid

- Dr. A.B.M. Toufique Hasan

Assistant Professor

ME 6123: Mechanics of Inviscid Incompressible Fluid 11

Assistant Professor

Department of Mechanical Engineering, BUET

Lecture # 1

May 19, 2013

Course Outline:1. Kinematics of fluid medium2. The fundamental hydrodynamic equations for ideal fluid3. The simplest cases of motion of an ideal fluid4. Vortex motion of an ideal fluid5. The plane motion of a body in an ideal fluid6. The three dimensional motion of a body in a ideal fluid


Reference Books:1. Advanced Fluid Mechanics - W.P. Graebel2. Introduction to Fluid Mechanics - Fox and McDonald's3. Inviscid Incompressible Flow - J.S. Marshall4. Incompressible Flow - Panton5. Fluid Mechanics - F.M. White

For high Re flow about streamlined bodies viscous effects are

confined to boundary layer and wake region. The analysis of such

high Reynolds number (Re) flows past streamlined bodies can

greatly be simplified by the fact that viscous effects are generally

important only within thin regions immediately adjacent to solid

boundaries.

Outside these viscous boundary layers the flow can well

approximated as inviscid (ideal flow).


approximated as inviscid (ideal flow).

Kinematics

Kinematics of fluid medium is the branch of fluid mechanics thatdeals with quantities involving space and time only. It treats variablessuch as displacement, velocity, acceleration, deformation, androtation of fluid elements without referring to the forces responsiblefor such a motion.

Figure shows a typical finite fluid element, within which it is consideredto have an infinitesimal particle of mass dm and initial volume dxdydz,at time t, and as it (and the infinitesimal particle) may appear after atime interval dt. The finite element has moved and changed its shapeand orientation. Note that while the finite element has quite severedistortion, the infinitesimal particle has changes in shape limited to


distortion, the infinitesimal particle has changes in shape limited tostretching/shrinking and rotation of the element’s sides.

As the element moves in a flow field, several things may happen to it.Certainly the element translates; it undergoes a linear displacement.The element may also rotate. In addition, the element may deform.The deformation may be subdivided into two parts- linear and angular

deformation.

Linear deformation involves a deformation in which planes of the elementwere originally perpendicular remain perpendicular. Angulardeformation involves a distortion of the element in which planes thatwere originally perpendicular do not remain perpendicular.


were originally perpendicular do not remain perpendicular.

Fig. Representation of Fluid Motion

Fluid Translation: Acceleration of a fluid Particle

A general description of particleacceleration can be obtained byconsidering a particle moving ina velocity field.

Consider the particle at time t, is atthe position x, y, z and has avelocity corresponding to thevelocity at that point in space attime t is u(x,y,z,t).


time t is u(x,y,z,t).

At t+∆t, the particle has moved to a new position, with coordinates x+∆x,y+ ∆y, z+ ∆z, and has a velocity given by u(x+∆x ,y+ ∆y,z+∆z,t+ ∆t). Thenthe expression for the acceleration vector becomes-

The above equation provides a means for calculating the threecomponents of acceleration (ax , ay , az) from the three components ofvelocity (u,v,w). For example, the x- component of acceleration ax,which involves only the x- component of velocity u, is given by

To evaluate the limit, use a Taylor series expansion to write u(x +∆x,y+∆y, z +∆z, t +∆t) in terms of the nearby value u(x, y, z, t). For acontinuous function u(x,y,z,t) of several variables, the Taylor series

….. (1)


continuous function u(x,y,z,t) of several variables, the Taylor seriesexpansion is

+ HOT

+ HOT

….. (2)

Considering equations (1) and (2), the following relations can be obtained:

Similarly considering other two components of velocity, the followingrelations can be obtained:


convective

acceleration

temporal

acceleration

total

acceleration

It is shown that a component of acceleration is given by the sum of onetemporal and three spatial derivative terms involving the correspondingvelocity component. This suggests that fluid acceleration is present

in virtually all flows of engineering interest.


Steady/Unsteady flow through a nozzle. Convective acceleration of a fluid

particle exists.

It is customary in fluid mechanics to use the symbol D( )/Dt to representthe combination of derivatives:

or

zk

yj

xi

∂

∂+

∂

∂+

∂

∂=

=∇

operatorntdel/gradie;


is commonly called the substantial derivative. It often is called the total

derivative, material derivative or particle derivative.

zyx ∂∂∂

Using this concept, the three-components of acceleration are

or in a compact form as a vector equation:

xu

tDt

D

∂

∂+

∂

∂=

uuu

wkvjui ++=u

yv

uu

tDt

D

∂

∂+

∂

∂+

∂

∂=

uuuu

Considering the velocity field in Cartesian coordinate system as

For 1-D unsteady flow:

For 2-D unsteady flow:

The acceleration (total) of a fluid particle is:


zw

yv

xu

tDt

D

∂

∂+

∂

∂+

∂

∂+

∂

∂=

uuuuuFor 3-D unsteady flow:

zw

yv

xu

Dt

D

∂

∂+

∂

∂+

∂

∂=

uuuuFor 3-D steady flow:

For this case, the acceleration is not necessarily

zero. Thus a fluid particle may undergo a

convective acceleration due to its motion, even in a

steady velocity field.

In cylindrical coordinates, the three components of acceleration (a r , aθ ,az) are given in terms of the three components of velocity (vr, vθ, vz) as


zz

rrr

zr

∂

∂+

∂

∂+

∂

∂=

=∇

ˆ1ˆˆ

) ,,(operatorntdel/gradie;

θθ

θ

A flow field is given by

Find the velocity and acceleration at point (2,1,3). Is the flow steady/unsteady?

Problems:

jxixV23 22 −=

r


Find the velocity and acceleration at point (2,1,3). Is the flow steady/unsteady?

In vector calculus, divergence is a vector operator that measures the magnitude

of a vector field's source or sink at a given point, in terms of a signed scalar. It is a

local measure of its "outgoingness”.

Divergence of u u⋅∇≡

In vector calculus, the curl is a vector operator that describes the infinitesimal

rotation of a 3-dimensional vector field. At every point in the field, the curl of that

field is represented by a vector. The attributes of this vector (length and direction)


field is represented by a vector. The attributes of this vector (length and direction)

characterize the rotation at that point.

Curl of u u×∇≡

In mathematics the Laplace operator or Laplacian is a differential operator given

by the divergence of the gradient of a function. It is usually denoted by the

symbols ∇·∇, ∇2 or ∆.

2

2

2

2

2

2

zyx ∂

∂+

∂

∂+

∂

∂≡∆

ME 6123 Mechanics of Inviscid Incompressible Fluidteacher.buet.ac.bd/toufiquehasan/ME 6123-Lecture...

Documents

Transcript of ME 6123 Mechanics of Inviscid Incompressible Fluidteacher.buet.ac.bd/toufiquehasan/ME 6123-Lecture...