Fluid Mechanics

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Fluid Mechanics I

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Review

• Fluid definition

• Fluid as a continuum

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Review

• Fluid properties:

– Density

– Viscosity ???

• Fluid flow parameters: لایس نادیم

Velocity تعرس Pressure راشف???

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Review

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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

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Kinematic Concepts of Flow Field

• A more common method of describing fluid flow is the Eulerian description 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)

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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

Ԧ𝑎_{𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒} = 𝑑𝑉_{𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒}

𝑑𝑡 = 𝑑𝑉

𝑑𝑡 = 𝑑𝑉(𝑥_{𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒} 𝑡 , 𝑦_{𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒} 𝑡 , 𝑧_{𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒} 𝑡 , 𝑡) 𝑑𝑡

= 𝜕𝑉

𝜕𝑡 𝑑𝑡

𝑑𝑡 + 𝜕𝑉

𝜕𝑥_{𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒}

𝑑𝑥_{𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒}

𝑑𝑡 + 𝜕𝑉

𝜕𝑦_{𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒}

𝑑𝑦_{𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒}

𝑑𝑡 + 𝜕𝑉

𝜕𝑧_{𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒}

𝑑𝑧_{𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒} 𝑑𝑡

Ԧ𝑎_{𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒}(𝑥, 𝑦, 𝑧, 𝑡) = 𝑑𝑉

𝑑𝑡 = 𝜕𝑉

𝜕𝑡 + 𝑢 𝜕𝑉

𝜕𝑥 + 𝑣 𝜕𝑉

𝜕𝑦 + 𝑤 𝜕𝑉

𝜕𝑧

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Kinematic Concepts of Flow Field

• Steady state

𝒂_𝒙 = 𝝏𝒖

𝝏𝒕 + 𝒖𝝏𝒖

𝝏𝒙 + 𝒗𝝏𝒖

𝝏𝒚 + 𝒘𝝏𝒖

𝝏𝒛 𝒂_𝒚 = 𝝏𝒗

𝝏𝒕 + 𝒖𝝏𝒗

𝝏𝒙 + 𝒗𝝏𝒗

𝝏𝒚 + 𝒘𝝏𝒗

𝝏𝒛 𝒂_𝒛 = 𝝏𝒘

𝝏𝒕 + 𝒖𝝏𝒘

𝝏𝒙 + 𝒗𝝏𝒘

𝝏𝒚 + 𝒘𝝏𝒘

𝝏𝒛 Local acceleration

(steady flow=0)

Convective (advective) acceleration

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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

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Flow Visualization

Spinning baseball

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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.

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Flow Visualization

• In 2D:

𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝒇𝒐𝒓 𝒂 𝒔𝒕𝒓𝒆𝒂𝒎𝒍𝒊𝒏𝒆: 𝒅𝒙

𝒖 = 𝒅𝒚

𝒗 = 𝒅𝒛 𝒘

𝒔𝒕𝒓𝒆𝒂𝒎𝒍𝒊𝒏𝒆 𝒊𝒏 𝒕𝒉𝒆 𝒙𝒚 − 𝒑𝒍𝒂𝒏𝒆: 𝒅𝒚

𝒅𝒙 𝒂𝒍𝒐𝒏𝒈 𝒂 𝒔𝒕𝒓𝒆𝒂𝒎𝒍𝒊𝒏𝒆

= 𝒗 𝒖

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Kinematic Concepts of Flow Field

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Kinematic Concepts of Flow Field

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Kinematic Concepts of Flow Field

Velocity vectors for the velocity field.

The stagnation point is indicated by circle.

Solid black curves represent the approximate shapes of some streamlines, based on the calculated velocity vectors.

The shaded region represents a portion of the flow field that can approximate flow into an inlet.

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Flow Visualization

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Flow Visualization

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Flow Visualization

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Flow Visualization

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

Pathlines

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Flow Visualization

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

𝒅𝒙

𝒅𝒕 = 𝒖; 𝒅𝒚

𝒅𝒕 = 𝒗; 𝒅𝒛

𝒅𝒕 = 𝒘

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Flow Visualization

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

Streaklines

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Fluid Deformation

A B

C

𝒖_𝑩 𝒖_𝑨

A A’ B B’

𝒗_𝑪

𝒗_𝑨

A C A’

C’

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Fluid Deformation

A B

C

𝒗_𝑩 𝒗_𝑨

A B

𝒖_𝑪

𝒖_𝑨

A C

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Fluid Deformation

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

Translation: لاقتنا

Shear strain: یشرب شنرک Rotation: نارود

Linear strain: یطخ شنرک

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Fluid Deformation

هعلاطم • لایس

یتح زا

هعلاطم دماج

هدیچیپ رت

تسا هب

نیا لیلد هک

بلغا ره

4 تلاح تکرح

فیرعت

هدش رد نآ هدید یم

دوش .

رییغت ای و لایس تکرح هک تسا رتهب دنتسه تکرح لاح رد میاد لایس یاهناملا هکییاجنآ زا • نآ لکش

تارییغت خرن بلاغ رد ار (rate of motion/deformation)

مینک نایب .

هنآ تسا رتهب درک نایب و تفرگ هزادنا یتحار هب لایس رد ار اهرتماراپ نیا ناوتب هکنیا یارب • رب ار ا

میروآ تسدب لایس تعرس تاقتشم و لایس تعرس بسح .

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

shear strain rate (rate of shear strain)

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Fluid Deformation

• Rate of translation vector in Cartesian coordinates:

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

نارود خرن • :

رد مه رب دومع علض ود نارود خرن نیگنایم ناملا

لایس

𝑽 = 𝒖 Ԧ𝒊 + 𝒗 Ԧ𝒋 + 𝒘 𝒌

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Fluid Deformation

A B

C

𝒗_𝑩 𝒗_𝑨

A B

𝜕𝑣

𝜕𝑥 = 𝑣_𝐵 − 𝑣_𝐴

𝑥_𝐵 − 𝑥_𝐴 > 0

𝒖_𝑪

𝒖_𝑨

A C

𝜕𝑢

𝜕𝑦 = 𝑢_𝐶 − 𝑢_𝐴

𝑦_𝐶 − 𝑦_𝐴 < 0

𝑟𝑎𝑡𝑒 𝑜𝑓 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛: 1 2(𝜕𝑣

𝜕𝑥 − 𝜕𝑢

𝜕𝑦)

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Fluid Deformation

Ԧሶ𝛼 = 𝑑 𝑑𝑡

𝛼_𝑎 + 𝛼_𝑏

2 = 1

2(𝜕𝑣

𝜕𝑥 − 𝜕𝑢

𝜕𝑦)

Ԧሶ𝛼

= 1 2

𝜕𝑤

𝜕𝑦 − 𝜕𝑣

𝜕𝑧 Ԧ𝑖 + 1

2

𝜕𝑢

𝜕𝑧 − 𝜕𝑤

𝜕𝑥 Ԧ𝑗 + 1

2

𝜕𝑣

𝜕𝑥 − 𝜕𝑢

𝜕𝑦 𝑘

Rate of rotation in Cartesian coordinates:

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Fluid Deformation

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

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

خرن • شرب : نیگنایم خرن

کیدزن ندش

ود علض دومع

رب مه رد ناملا لایس

𝜀_𝑥𝑥 = 𝜕𝑢

𝜕𝑥 ; 𝜀_𝑦𝑦 = 𝜕𝑣

𝜕𝑦; 𝜀_𝑧𝑧 = 𝜕𝑤

𝜕𝑧 Linear Strain Rate in Cartesian coordinates:

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Fluid Deformation

A B

C

𝒗_𝑩 𝒗_𝑨

A B

𝜕𝑣

𝜕𝑥 = 𝑣_𝐵 − 𝑣_𝐴

𝑥_𝐵 − 𝑥_𝐴 > 0

𝒖_𝑪

𝒖_𝑨

A C

𝜕𝑢

𝜕𝑦 = 𝑢_𝐶 − 𝑢_𝐴

𝑦_𝐶 − 𝑦_𝐴 < 0

𝑟𝑎𝑡𝑒 𝑜𝑓 𝑠ℎ𝑒𝑎𝑟: 1

2(𝜕𝑣

𝜕𝑥 + 𝜕𝑢

𝜕𝑦)

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Fluid Deformation

𝜀_𝑥𝑦 = 1 2

𝜕𝑢

𝜕𝑦 + 𝜕𝑣

𝜕𝑥 𝜀_𝑧𝑥 = 1

2

𝜕𝑤

𝜕𝑥 + 𝜕𝑢

𝜕𝑧 𝜀_𝑦𝑧 = 1

2

𝜕𝑣

𝜕𝑧 + 𝜕𝑤

𝜕𝑦

Shear Strain Rate in Cartesian coordinates:

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Fluid Deformation

Linear Strain+Shear Strain: Deformation

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Fluid Deformation

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Fluid Deformation

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Fluid Deformation

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Fluid Deformation

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Fluid Deformation

• Rate of rotation vector in Cartesian coordinates:

Vorticity and Rotationality

Ԧሶ𝛼 = 1 2

𝜕𝑤

𝜕𝑦 − 𝜕𝑣

𝜕𝑧 Ԧ𝑖 + 1 2

𝜕𝑢

𝜕𝑧 − 𝜕𝑤

𝜕𝑥 Ԧ𝑗 + 1 2

𝜕𝑣

𝜕𝑥 − 𝜕𝑢

𝜕𝑦 𝑘 𝑽𝒐𝒓𝒕𝒊𝒄𝒊𝒕𝒚 𝒗𝒆𝒄𝒕𝒐𝒓: 𝝎 = 𝜵 × 𝑽 = 𝒄𝒖𝒓𝒍(𝑽)

𝝎

= 𝝏𝒘

𝝏𝒚 − 𝝏𝒗

𝝏𝒛 Ԧ𝒊 + 𝝏𝒖

𝝏𝒛 − 𝝏𝒘

𝝏𝒙 Ԧ𝒋 + 𝝏𝒗

𝝏𝒙 − 𝝏𝒖

𝝏𝒚 𝒌

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Fluid Deformation

• Two-dimensional flow in Cartesian coordinates:

• Two-dimensional flow in cylindrical coordinates:

𝝎 = 𝝏𝒗

𝝏𝒙 − 𝝏𝒖

𝝏𝒚 𝒌

𝝎 = 𝟏 𝒓

𝝏(𝒓𝒖_𝜽)

𝝏𝒓 − 𝝏𝒖_𝒓

𝝏𝜽 𝒌

𝝎 = 𝟏 𝒓

𝝏𝒗_𝒛

𝝏𝜽 − 𝝏𝒗_𝜽

𝝏𝒛 𝒊_𝒓 + 𝝏𝒗_𝒓

𝝏𝒛 − 𝝏𝒗_𝒛

𝝏𝒓 𝒊_𝜽 + 𝟏 𝒓

𝝏(𝒓𝒗_𝜽)

𝝏𝒓 − 𝟏 𝒓

𝝏𝒗_𝒓

𝝏𝜽 𝒊_𝒛

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Fluid Deformation

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

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

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Fluid Deformation

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

flow with heat transfer.

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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:

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Fluid Deformation

𝐹𝑙𝑜𝑤 𝐴 − 𝑆𝑜𝑙𝑖𝑑 𝐵𝑜𝑑𝑦 𝑅𝑜𝑡𝑎𝑡𝑖𝑜𝑛: 𝑢_𝑟 = 0 𝑎𝑛𝑑 𝑢_𝜃 = 𝜔𝑟

𝐹𝑙𝑜𝑤 𝐵 − 𝐿𝑖𝑛𝑒 𝑉𝑜𝑟𝑡𝑒𝑥: 𝑢_𝑟 = 0 𝑎𝑛𝑑 𝑢_𝜃 = 𝑘 𝑟

𝐹𝑙𝑜𝑤 𝐴 − 𝑆𝑜𝑙𝑖𝑑 𝐵𝑜𝑑𝑦 𝑅𝑜𝑡𝑎𝑡𝑖𝑜𝑛: 𝝎 = 𝟏 𝒓

𝝏 𝝎𝒓^𝟐

𝝏𝒓 − 𝟎 𝒌 = 2ω𝑘

𝐹𝑙𝑜𝑤 𝐵 − 𝐿𝑖𝑛𝑒 𝑉𝑜𝑟𝑡𝑒𝑥: 𝝎 = 𝟏 𝒓

𝝏 𝒌

𝝏𝒓 − 𝟎 𝒌 = 0

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Fluid Deformation

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

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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

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Fluid Deformation

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