Review

• Fluid definition

• Fluid as

a continuum

Review

• Fluid properties:– Density

– Viscosity ???

• Fluid flow parameters: سیالمیدان

Velocity سرعت Pressure فشار???

Review

Kinematic Concepts of Flow Field

• Kinematics: how the fluid flows?

• Lagrangian description: we follow a mass of fixed identity.

Lagrangian & Eulerian Descriptions

Difficult!

From a microscopic point of view, a fluid is composed of billions of

molecules that are continuously banging into one another

Kinematic Concepts of Flow Field

• A more common method of describing fluid flow is the Euleriandescription of fluid motion.

𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑓𝑖𝑒𝑙𝑑: 𝑃 = 𝑃(𝑥, 𝑦, 𝑧, 𝑡)

𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑓𝑖𝑒𝑙𝑑: 𝑉 = 𝑉(𝑥, 𝑦, 𝑧, 𝑡)

𝐴𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑓𝑖𝑒𝑙𝑑: Ԧ𝑎 = Ԧ𝑎(𝑥, 𝑦, 𝑧, 𝑡)

Kinematic Concepts of Flow Field

• Collectively, these (and other) field variables define the flow field:

𝑉 𝑥, 𝑦, 𝑧, 𝑡 = 𝐢 𝑢 𝑥, 𝑦, 𝑧, 𝑡 + 𝐣 𝑣 𝑥, 𝑦, 𝑧, 𝑡 + 𝐤 𝑤 𝑥, 𝑦, 𝑧, 𝑡

Ԧ𝑎𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 =𝑑𝑉𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒

𝑑𝑡=𝑑𝑉

𝑑𝑡=𝑑𝑉(𝑥𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑡 , 𝑦𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑡 , 𝑧𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑡 , 𝑡)

𝑑𝑡

=𝜕𝑉

𝜕𝑡

𝑑𝑡

𝑑𝑡+

𝜕𝑉

𝜕𝑥𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒

𝑑𝑥𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒

𝑑𝑡+

𝜕𝑉

𝜕𝑦𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒

𝑑𝑦𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒

𝑑𝑡+

𝜕𝑉

𝜕𝑧𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒

𝑑𝑧𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒

𝑑𝑡

Ԧ𝑎𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒(𝑥, 𝑦, 𝑧, 𝑡) =𝑑𝑉

𝑑𝑡=𝜕𝑉

𝜕𝑡+ 𝑢

𝜕𝑉

𝜕𝑥+ 𝑣

𝜕𝑉

𝜕𝑦+ 𝑤

𝜕𝑉

𝜕𝑧

Kinematic Concepts of Flow Field

• Steady state

𝒂𝒙 =𝝏𝒖

𝝏𝒕+ 𝒖

𝝏𝒖

𝝏𝒙+ 𝒗

𝝏𝒖

𝝏𝒚+𝒘

𝝏𝒖

𝝏𝒛

𝒂𝒚 =𝝏𝒗

𝝏𝒕+ 𝒖

𝝏𝒗

𝝏𝒙+ 𝒗

𝝏𝒗

𝝏𝒚+ 𝒘

𝝏𝒗

𝝏𝒛

𝒂𝒛 =𝝏𝒘

𝝏𝒕+ 𝒖

𝝏𝒘

𝝏𝒙+ 𝒗

𝝏𝒘

𝝏𝒚+ 𝒘

𝝏𝒘

𝝏𝒛

Local acceleration

(steady flow=0)

Convective (advective) acceleration

Review

• We then discuss various ways to visualize flow fields:

streamlines, streaklines, pathlines

• and we describe three ways to plot flow data:

Profile plots, vector plots, and contour plots

Flow Visualization

Flow Visualization

Spinning baseball

Flow Visualization

• Consider an infinitesimal arc length: 𝒅𝒓 = 𝒅𝒙 Ԧ𝒊 + 𝒅𝒚 Ԧ𝒋 + 𝒅𝒛 𝒌

along a streamline;

𝒅𝒓 must be parallel to the local velocity vector: 𝑽 = 𝒖 Ԧ𝒊 + 𝒗 Ԧ𝒋 + 𝒘 𝒌 by definition of the streamline.

Flow Visualization

• In 2D:

𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝒇𝒐𝒓 𝒂 𝒔𝒕𝒓𝒆𝒂𝒎𝒍𝒊𝒏𝒆:𝒅𝒙

𝒖=𝒅𝒚

𝒗=𝒅𝒛

𝒘

𝒔𝒕𝒓𝒆𝒂𝒎𝒍𝒊𝒏𝒆 𝒊𝒏 𝒕𝒉𝒆 𝒙𝒚 − 𝒑𝒍𝒂𝒏𝒆:𝒅𝒚

𝒅𝒙𝒂𝒍𝒐𝒏𝒈 𝒂 𝒔𝒕𝒓𝒆𝒂𝒎𝒍𝒊𝒏𝒆

=𝒗

𝒖

Kinematic Concepts of Flow Field

Kinematic Concepts of Flow Field

Kinematic Concepts of Flow Field

Velocity vectors for the velocityfield.

The stagnation point is indicated bycircle.

Solid black curves represent theapproximate shapes of somestreamlines, based on the calculatedvelocity vectors.

The shaded region represents aportion of the flow field that canapproximate flow into an inlet.

Flow Visualization

Flow Visualization

Flow Visualization

Flow Visualization

• A pathline is the actual path traveled by an individual fluid particle over some time period.

Pathlines

Flow Visualization

• Pathlines can also be calculated numerically for a known velocity field:

𝒅𝒙

𝒅𝒕= 𝒖;

𝒅𝒚

𝒅𝒕= 𝒗;

𝒅𝒛

𝒅𝒕= 𝒘

Flow Visualization

• A streakline is the path of fluid particles that have passedsequentially through a prescribed point in the flow.

Streaklines

Fluid Deformation

A BC

𝒖𝑩𝒖𝑨

A BA’ B’

𝒗𝑪

𝒗𝑨

A

C

A’

C’

Fluid Deformation

A BC

𝒗𝑩𝒗𝑨

A B

𝒖𝑪

𝒖𝑨A

C

Fluid Deformation

• In fluid mechanics, as in solid mechanics, an element mayundergo four fundamental types of motion or deformation, asillustrated in two dimensions (a) translation, (b) rotation, (c) linearstrain (d) shear strain.

Translation: انتقال

Shear strain: برشیکرنشRotation: دوران

Linear strain: کرنش خطی

Fluid Deformation

حالت حرکت تعریف 4مطالعه سیال حتی از مطالعه جامد پیچیده تر است به این دلیل که اغلب هر •.شده در آن دیده می شود

شکل آن از آنجاییکه المانهای سیال دایم در حال حرکت هستند بهتر است که حرکت سیال و یا تغییر•.بیان کنیم(rate of motion/deformation)را در غالب نرخ تغییرات

ا را بر برای اینکه بتوان این پارامترها را در سیال به راحتی اندازه گرفت و بیان کرد بهتر است آنه•.حسب سرعت سیال و مشتقات سرعت سیال بدست آوریم

velocity (rate of translation), angular velocity (rate of rotation),

linear strain rate (rate of linear strain),

shear strain rate (rate of shear strain)

Fluid Deformation

• Rate of translation vector in Cartesian coordinates:

• Rate of rotation (angular velocity) at a point is defined as theaverage rotation rate of two initially perpendicular lines that

intersect at that point.

سیال المانمیانگین نرخ دوران دو ضلع عمود بر هم در : نرخ دوران•

𝑽 = 𝒖 Ԧ𝒊 + 𝒗 Ԧ𝒋 + 𝒘 𝒌

Fluid Deformation

A BC

𝒗𝑩𝒗𝑨

A B

𝜕𝑣

𝜕𝑥=𝑣𝐵 − 𝑣𝐴𝑥𝐵 − 𝑥𝐴

> 0

𝒖𝑪

𝒖𝑨A

C

𝜕𝑢

𝜕𝑦=𝑢𝐶 − 𝑢𝐴𝑦𝐶 − 𝑦𝐴

< 0

𝑟𝑎𝑡𝑒 𝑜𝑓 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛:1

2(𝜕𝑣

𝜕𝑥−𝜕𝑢

𝜕𝑦)

Fluid Deformation

Ԧሶ𝛼 =𝑑

𝑑𝑡

𝛼𝑎 + 𝛼𝑏2

=1

2(𝜕𝑣

𝜕𝑥−𝜕𝑢

𝜕𝑦)

Ԧሶ𝛼

=1

2

𝜕𝑤

𝜕𝑦−𝜕𝑣

𝜕𝑧Ԧ𝑖

+1

2

𝜕𝑢

𝜕𝑧−𝜕𝑤

𝜕𝑥Ԧ𝑗

+1

2

𝜕𝑣

𝜕𝑥−𝜕𝑢

𝜕𝑦𝑘

Rate of rotation in Cartesian coordinates:

Fluid Deformation

• Linear strain rate is defined as the rate of increase in length perunit length.

• Shear strain rate at a point is defined as half of the rate of decreaseof the angle between two initially perpendicular lines thatintersect at the point.

سیالالماندرهمبرعمودضلعدوشدننزدیکنرخمیانگین:برشنرخ•

𝜀𝑥𝑥 =𝜕𝑢

𝜕𝑥; 𝜀𝑦𝑦 =

𝜕𝑣

𝜕𝑦; 𝜀𝑧𝑧 =

𝜕𝑤

𝜕𝑧

Linear Strain Rate in Cartesian coordinates:

Fluid Deformation

A BC

𝒗𝑩𝒗𝑨

A B

𝜕𝑣

𝜕𝑥=𝑣𝐵 − 𝑣𝐴𝑥𝐵 − 𝑥𝐴

> 0

𝒖𝑪

𝒖𝑨A

C

𝜕𝑢

𝜕𝑦=𝑢𝐶 − 𝑢𝐴𝑦𝐶 − 𝑦𝐴

< 0

𝑟𝑎𝑡𝑒 𝑜𝑓 𝑠ℎ𝑒𝑎𝑟:1

2(𝜕𝑣

𝜕𝑥+𝜕𝑢

𝜕𝑦)

Fluid Deformation

𝜀𝑥𝑦 =1

2

𝜕𝑢

𝜕𝑦+𝜕𝑣

𝜕𝑥

𝜀𝑧𝑥 =1

2

𝜕𝑤

𝜕𝑥+𝜕𝑢

𝜕𝑧

𝜀𝑦𝑧 =1

2

𝜕𝑣

𝜕𝑧+𝜕𝑤

𝜕𝑦

Shear Strain Rate in Cartesian coordinates:

Fluid Deformation

Linear Strain+Shear Strain: Deformation

Fluid Deformation

Fluid Deformation

Fluid Deformation

Fluid Deformation

Fluid Deformation

• Rate of rotation vector in Cartesian coordinates:

Vorticity and Rotationality

Ԧሶ𝛼 =1

2

𝜕𝑤

𝜕𝑦−𝜕𝑣

𝜕𝑧Ԧ𝑖 +

1

2

𝜕𝑢

𝜕𝑧−𝜕𝑤

𝜕𝑥Ԧ𝑗 +

1

2

𝜕𝑣

𝜕𝑥−𝜕𝑢

𝜕𝑦𝑘

𝑽𝒐𝒓𝒕𝒊𝒄𝒊𝒕𝒚 𝒗𝒆𝒄𝒕𝒐𝒓: 𝝎 = 𝜵 × 𝑽 = 𝒄𝒖𝒓𝒍(𝑽)

𝝎

=𝝏𝒘

𝝏𝒚−𝝏𝒗

𝝏𝒛Ԧ𝒊

+𝝏𝒖

𝝏𝒛−𝝏𝒘

𝝏𝒙Ԧ𝒋

+𝝏𝒗

𝝏𝒙−𝝏𝒖

𝝏𝒚𝒌

Fluid Deformation

• Two-dimensional flow in Cartesian coordinates:

• Two-dimensional flow in cylindrical coordinates:

𝝎 =𝝏𝒗

𝝏𝒙−𝝏𝒖

𝝏𝒚𝒌

𝝎 =𝟏

𝒓

𝝏(𝒓𝒖𝜽)

𝝏𝒓−𝝏𝒖𝒓𝝏𝜽

𝒌

𝝎 =𝟏

𝒓

𝝏𝒗𝒛𝝏𝜽

−𝝏𝒗𝜽𝝏𝒛

𝒊𝒓 +𝝏𝒗𝒓𝝏𝒛

−𝝏𝒗𝒛𝝏𝒓

𝒊𝜽 +𝟏

𝒓

𝝏(𝒓𝒗𝜽)

𝝏𝒓−𝟏

𝒓

𝝏𝒗𝒓𝝏𝜽

𝒊𝒛

Fluid Deformation

• If the vorticity at a point in a flow field is nonzero, the fluidparticle that happens to occupy that point in space is rotating; theflow in that region is called rotational.

• Likewise, if the vorticity in a region of the flow is zero (ornegligibly small), fluid particles there are not rotating; the flow inthat region is called irrotational.

Fluid Deformation

Rotation of fluid elements is associated with wakes, boundary layers, flow through turbomachinery (fans, turbines, compressors, etc.), and

flow with heat transfer.

Fluid Deformation

• Not all flows with circular streamlines are rotational. To illustrate this point, we consider two incompressible, steady, two-dimensional flows, both of which have circular streamlines in the rθ-plane:

Fluid Deformation

𝐹𝑙𝑜𝑤 𝐴 − 𝑆𝑜𝑙𝑖𝑑 𝐵𝑜𝑑𝑦 𝑅𝑜𝑡𝑎𝑡𝑖𝑜𝑛: 𝑢𝑟 = 0 𝑎𝑛𝑑 𝑢𝜃 = 𝜔𝑟

𝐹𝑙𝑜𝑤 𝐵 − 𝐿𝑖𝑛𝑒 𝑉𝑜𝑟𝑡𝑒𝑥: 𝑢𝑟 = 0 𝑎𝑛𝑑 𝑢𝜃 =𝑘

𝑟

𝐹𝑙𝑜𝑤 𝐴 − 𝑆𝑜𝑙𝑖𝑑 𝐵𝑜𝑑𝑦 𝑅𝑜𝑡𝑎𝑡𝑖𝑜𝑛: 𝝎 =𝟏

𝒓

𝝏 𝝎𝒓𝟐

𝝏𝒓− 𝟎 𝒌 = 2ω𝑘

𝐹𝑙𝑜𝑤 𝐵 − 𝐿𝑖𝑛𝑒 𝑉𝑜𝑟𝑡𝑒𝑥: 𝝎 =𝟏

𝒓

𝝏 𝒌

𝝏𝒓− 𝟎 𝒌 = 0

Fluid Deformation

Flow A is rotational. Physically, this means that individual fluid particles rotate as they revolve around the origin

Fluid Deformation

By contrast, the vorticity of the line vortex is identically zero everywhere (except right at the origin, which is a mathematical singularity). Flow B is irrotational. Physically, fluid particles do

not rotate as they revolve in circles about the origin

Fluid Deformation

Fluid Deformation