Fundamental 2-D Equations Derivations-libre

Derivations of Some 2-D Fundamental Equations of Fluid Mechanics

By, TEY WAH YEN

Equations involved :

• Continuity Equations Cartesian Coordinate

Cylindrical Coordinate

• Momentum Equations Cartesian Coordinate


• Navier-Stokes Equations Cartesian Coordinate


• Stream Vorticity Functions Cartesian Coordinate


Continuity Equations

(Cartesian Coordinate)

兼な = �懸捲穴捲兼に = 岫�懸捲 + 絞絞捲 �懸捲穴検岻穴捲兼ぬ = �懸検穴検兼ね = �懸検 + 絞絞検 �懸検穴捲穴検兼結 = ��痛穴捲穴検


(Cartesian Coordinate) 兼な − 兼に + 兼ぬ − 兼ね = 兼結 �懸捲穴捲 − 岫�懸捲 + 絞絞捲 �懸捲穴検岻穴捲 + �懸検穴検 − 岫�懸検 + 絞絞検 �懸検穴捲岻穴検 = − 項�項建穴捲穴検絞絞捲 �懸捲穴捲穴検 +

絞絞検 �懸検穴捲穴検 =− 項�項建穴捲穴検 � 絞絞捲懸捲穴捲穴検 + 懸捲絞絞捲 � 穴捲穴検 + 懸検絞絞検 � 穴捲穴検 + � 絞絞検懸検穴捲穴検 =− ��痛穴捲穴検絞絞捲懸捲穴捲穴検 + 絞絞検懸検穴捲穴検 =− ��痛穴捲穴検

for incompressible flow: �士姉�姉 + �士姿�姿 = 宋


(Cylindrical Coordinate)


(Cylindrical Coordinate) 兼堅,件券 = �懸堅堅穴� 兼堅,剣憲建 = 岫�懸堅 + 絞絞堅 �懸堅穴堅岻岫堅 + 穴堅岻穴� 兼�,件券 = �懸�穴堅兼�,剣憲建 = 岫�懸� + 絞絞� �懸� 穴�岻穴堅兼結 = 項�項建堅穴�穴堅


(Cylindrical Coordinate) 兼�,沈津 − 兼�,墜通痛 + 兼�,沈津 − 兼�,墜通痛 = 兼勅 �懸�堅穴� − 岫�懸� + �� 懸� 穴堅岻岫堅 + 穴堅岻穴� + �懸�穴堅 −岫�懸� + �� 懸� 穴�岻穴堅 = ��痛堅穴�穴堅 �� 堅�懸� 穴堅穴� + �� 懸� 穴堅穴� = − ��痛堅穴堅穴� 怠� 岫 �� 堅�懸� 穴堅穴� + �� 懸� 穴堅穴�岻 = − ��痛穴堅穴�

For incompressible flow: 層� 岫��士�� 岻 + 層� 岫�士�� 岻 = 宋

Momentum Equations

(Cartesian Coordinate)

Momentum Equations

(Cartesian Coordinate) X-direction momentum equation �捲 = �捲捲 + 絞�捲捲絞捲穴捲に − �捲捲 − 絞�捲捲絞捲穴捲に穴検+ �検捲 + 絞�検捲絞検穴検に − �検捲 − 絞�検捲絞検穴検に穴捲兼�捲 = 絞�捲捲絞捲穴捲穴検 + 絞�検捲絞検穴捲穴検 ��捲穴捲穴検 = 絞�捲捲絞捲穴捲穴検 + 絞�検捲絞検穴捲穴検 ��四�嗣 = ��姉姉�姉 + �滋姿姉�姿

Momentum Equations

(Cartesian Coordinate) Y-direction momentum equation �検 = �検検 + 絞�検検絞検穴検に − �検検 + 絞�検検絞検穴検に穴捲+ �捲検 + 絞�捲検絞捲穴捲に − �捲検 + 絞�捲検絞捲穴捲に穴検兼�検 = 絞�検検絞検穴捲穴検 + 絞�捲検絞捲穴捲穴検 ��検穴捲穴検 = 絞�検検絞検穴捲穴検 + 絞�捲検絞捲穴捲穴検 ��士�嗣 = ��姿姿�姿 + �滋姉姿�姉

Momentum Equations

(Cylindrical Coordinate) For r-momentum �堅 = �堅堅 − �堅堅 − 絞�堅堅絞堅穴堅堅穴� + ��堅穴�− �� + 絞絞堅 �� 穴堅堅 + 穴堅穴� 兼�堅 = 絞�堅堅絞堅穴�穴堅 + 絞岫堅��堅岻絞� 穴�穴堅 ��検堅穴�穴堅 = 絞�堅堅絞堅穴�穴堅 + 絞岫堅��岻絞検穴�穴堅 ��士��嗣 = �� + �岫�滋��岻��

Momentum Equations

(Cylindrical Coordinate) For �-momentum �� = �� − �� − 絞��絞� 穴堅堅 + 穴堅穴� + ��堅穴�− �� + 絞絞堅 �� 穴� 穴堅兼�� = 絞岫堅��岻絞� 穴�穴堅 + 絞��絞堅穴堅穴� ��堅穴�穴堅 = 絞岫堅��岻絞� 穴�穴堅 + 絞��絞堅穴�穴堅 ��士��嗣 = �岫��岻�� + �岫滋��岻��

Navier-Stokes Equations

(Cartesian Coordinate) �1 = � − 絞�絞捲 (穴捲2

) �2 = � +絞�絞捲 (

穴捲2

) �喧堅結嫌嫌憲堅結訣堅�穴件結券建 = �1 − �2

m�捲 = �1 − �2= (�1 − �2) 穴検穴権 � �憲�建穴捲穴検穴権 = = -(絞�絞捲)穴捲穴検穴権 � �憲�建 = -

絞�絞捲

Consider an infinitesimal

fluid particle with

variation of pressure in x

and y direction to derive

the Euler’s Formula…


(Cartesian Coordinate) �1 = � − 絞�絞検穴検2

�2 = � +絞�絞検穴検

2 �喧堅結嫌嫌憲堅結訣堅�穴件結券建 = �1 − �2 − �訣

m�検 = �1 − �2 − �訣= (�1 − �2) 穴検穴権 - 兼訣 � �懸�建穴捲穴検穴権 = = -(絞�絞検)穴捲穴検穴権 - 兼訣 � �懸�建 = - 絞�絞検 - �訣


(Cartesian Coordinate) Constitutive relationship for Viscous Flow:

�沈珍 = 2航綱沈珍, in which 航 = 穴検券�兼件潔懸件嫌潔剣嫌件建検剣血建月結血健憲件穴綱掴掴 = 鳥通鳥掴綱掴槻 = 綱槻掴 = 怠態岫鳥通鳥槻 + 鳥塚鳥掴岻綱槻槻 = 鳥塚鳥槻

For fluid acceleration, recall that: ��建 = 穴�穴建 + 憲穴�穴捲 + 懸穴�穴検 + 拳穴�穴権


(Cartesian Coordinate) 鳥�猫猫鳥掴 + 鳥�熱猫鳥槻 = 鳥鳥掴に航鳥通鳥掴 + 鳥鳥槻に航岫怠態岫鳥通鳥槻 + 鳥塚鳥掴岻 �岫�通�痛岻= に航鳥鉄通鳥掴鉄 + 鳥鳥掴岫航鳥通鳥槻 + 鳥塚鳥掴岻 �岫鳥通鳥痛 + 憲鳥通鳥掴 + 懸鳥通鳥槻岻= に航鳥鉄通鳥掴鉄 + 航鳥鉄通鳥槻鉄 + 鳥鉄塚鳥掴鳥槻 �岫穴憲穴建 + 憲穴憲穴捲 + 懸穴憲穴検岻=に航穴態憲穴捲態 + 航穴態憲穴検態 + 穴態懸穴捲穴検 �岫穴憲穴建 + 憲穴憲穴捲 + 懸穴憲穴検岻=航岫穴態憲穴捲態 + 穴態憲穴検態岻 + 航穴穴捲岫穴憲穴捲 + 穴懸穴検岻


(Cartesian Coordinate) 鳥�熱熱鳥槻 + 鳥�猫熱鳥掴 = 鳥鳥槻に航鳥塚鳥槻 + 鳥鳥掴に航岫怠態岫鳥通鳥槻 + 鳥塚鳥掴岻 �岫�塚�痛岻= に航鳥鉄塚鳥槻鉄 + 鳥鳥掴岫航鳥通鳥槻 + 鳥塚鳥掴岻 �岫鳥塚鳥痛 + 憲鳥塚鳥掴 + 懸鳥塚鳥槻岻= に航鳥鉄塚鳥槻鉄 + 航鳥鉄通鳥掴鳥槻 + 鳥鉄塚鳥掴鉄 �岫穴懸穴建 + 憲穴懸穴捲 + 懸穴懸穴検岻=に航穴態懸穴検態 + 航穴態憲穴捲穴検 + 穴態懸穴捲態 �岫穴懸穴建 + 憲穴懸穴捲 + 懸穴懸穴検岻=航岫穴態懸穴捲態 + 穴態懸穴検態岻 + 航穴穴検岫穴憲穴捲 + 穴懸穴検岻


(Cartesian Coordinate) Recall from continuity equation in which 鳥通鳥掴 + 鳥塚鳥槻 = 0, and then

for incompressible flow: �岫四�四�姉 + 士�四�姿岻=侍岫�匝四�姉匝 + �匝四�姿匝岻 �岫四�士�姉 + 士�士�姿岻=侍岫�匝士�姉匝 + �匝士�姿匝岻


(Cartesian Coordinate) Combining with the Euler’s Equation, the complete に-D

Cartesian Navier-Stokes Equations are: 憲穴憲穴捲 + 懸穴憲穴検= − な�穴�穴捲 + 荒岫穴態憲穴捲態 + 穴態憲穴検態岻憲穴懸穴捲 + 懸穴懸穴検= − な�穴�穴検 − 訣 + 荒岫穴態懸穴捲態 + 穴態懸穴検態岻

or in other words: ��嗣 = − 層�∆� − �� + 治�匝�


(Cylindrical Coordinate) Consider an infinitesimal

fluid particle with

variation of pressure in r

and � direction to derive

the Euler’s Formula…

�1 = � − 絞�絞堅 (穴堅2

) �2 = � +絞�絞堅 (

穴堅2

) �喧堅結嫌嫌憲堅結訣堅�穴件結券建 = �1 − �2 − �訣

m�堅 = �1 − �2= (�1 − �2) 堅穴� + m訣堅 � �懸堅�建堅穴堅穴� = = -(絞�絞堅 )堅穴堅穴� + m訣堅 � �懸堅�建 = - 絞�絞堅 + �訣堅


(Cylindrical Coordinate) �1 = � − 絞�絞� 穴�2

�2 = � +絞�絞� 穴�

2 �喧堅結嫌嫌憲堅結訣堅�穴件結券建 = �1 − �2 − �訣

m�検 = �1 − �2 − �訣= (�1 − �2) 穴堅 - 兼訣� � �懸��建堅穴堅穴� = = -(絞�絞�)穴堅穴� - 兼訣� � �懸��建 = -

1堅絞�絞検 – �訣�


(Cylindrical Coordinate) Constitutive relationship for Viscous Flow:

�沈珍 = 2航綱沈珍 , in which 航 = 穴検券�兼件潔懸件嫌潔剣嫌件建検剣血建月結血健憲件穴綱�� = 鳥塚�鳥� 綱�� = 綱�� = 怠態岫堅鳥鳥� 岫塚�� 岻 + 怠� 鳥塚�鳥� 岻綱�� = 怠� 鳥塚�鳥� + 塚��


(Cylindrical Coordinate) な堅穴岫堅��岻穴堅 + 穴��穴� = な堅に航穴穴堅堅穴懸�穴堅 + 穴穴� 岫航堅穴穴堅懸�堅 + な堅穴懸�穴� 岻穴懸�穴建 + 懸� 穴懸�穴堅 + 懸�堅穴懸�穴� − 懸�態に= 航に堅穴穴堅堅穴懸�穴堅 + 航な堅穴穴� 岫穴懸�穴堅 + 堅懸� 穴岫な堅岻穴堅 + な堅穴懸�穴� 岻穴懸�穴建 + 懸� 穴懸�穴堅 + 懸�堅穴懸�穴� − 懸�態に= 航に堅穴穴堅堅穴懸�穴堅 + 航な堅岫穴態懸�穴堅穴� − 堅穴穴� 懸�堅態 + な堅穴態懸�穴�態岻


(Cylindrical Coordinate) 穴懸�穴建 + 懸� 穴懸�穴堅 + 懸�堅穴懸�穴� − 懸�態に= 航に堅穴穴堅堅穴懸�穴堅 + 航な堅穴態懸�穴堅穴� + に堅態穴懸�穴� + な堅態穴態懸�穴�態穴懸�穴建 + 懸� 穴懸�穴堅 + 懸�堅穴懸�穴� − 懸�態に= 航な堅穴穴堅堅穴懸�穴堅 + 航な堅穴穴堅岫穴懸�穴� + 堅穴懸�穴堅岻+ に航堅態穴懸�穴� + 航堅態穴態懸�穴�態


(Cylindrical Coordinate)

な堅穴岫��岻穴� + 穴岫堅�堅�岻穴堅 = な堅に航穴穴� な堅穴懸�穴� + 懸堅堅 + 穴穴堅岫航堅に穴穴堅懸�堅 + 穴懸堅穴� 岻穴懸�穴建 + 懸堅穴懸�穴堅 + 懸�堅穴懸�穴� + 懸堅懸�に= 航堅に堅岫穴穴� 穴懸�穴� 岻 + に堅穴懸堅穴� + 航穴穴堅岫堅穴懸�穴堅 + 堅に懸� 穴岫な堅岻穴堅 + な堅穴懸堅穴� 岻穴懸�穴建 + 懸堅穴懸�穴堅 + 懸�堅穴懸�穴� + 懸堅懸�に= 航堅に堅岫穴穴� 穴懸�穴� 岻 + に堅穴懸堅穴� + 航穴穴堅岫堅穴懸�穴堅 − 懸� + な堅穴懸堅穴� 岻


(Cylindrical Coordinate) 穴懸�穴建 + 懸堅穴懸�穴堅 + 懸�堅穴懸�穴� + 懸堅懸�に= 航堅に堅岫穴穴� 穴懸�穴� 岻 + に堅穴懸堅穴� + 堅穴に懸�穴堅に − 穴懸�穴堅 + な堅穴に懸堅穴�穴堅穴懸�穴建 + 懸堅穴懸�穴堅 + 懸�堅穴懸�穴� + 懸堅懸�に= 航堅な堅穴穴� 穴懸�穴� + 堅穴懸堅穴堅 + な堅穴に懸�穴�に + に堅穴懸堅穴�+ 堅穴に懸�穴堅に − 穴懸�穴堅


(Cylindrical Coordinate) Noted that from continuity equation: 鳥塚�鳥� + 堅鳥塚�鳥� = 0,

and then, 堅鳥鉄塚�鳥�鉄 − 鳥塚�鳥� =r 鳥鳥� 鳥塚�鳥� + 塚�� = 堅鳥鳥� 怠� 鳥岫�塚�岻鳥�

By substituting both the equation here to the

unsolved cylindrical Navier-Stokes equations,


(Cylindrical Coordinate) The cylindrical coordinate Navier-Stokes Equations evolved into: 穴懸�穴建 + 懸� 穴懸�穴堅 + 懸�堅穴懸�穴� − 懸�態に= 航穴穴堅な堅穴岫堅懸�岻穴堅 + に堅態穴懸�穴� + な堅態穴態懸�穴�態穴懸�穴建 + 懸� 穴懸�穴堅 + 懸�堅穴懸�穴� + 懸�懸�に= 航穴穴堅な堅穴岫堅懸�岻穴堅 + な堅態穴態懸�穴�態 + に堅態穴懸�穴�


(Cylindrical Coordinate) Combining the previous slide’s equations with the Euler’s equations, then the complete incompressible Navier-Stokes Equations are: 士� �士�� + 士�� 士�� − 士�匝匝= -

層�� + �� + 児 �� 層� �岫�士�岻�� + 匝�匝 �士�� + 層�匝 �匝士��匝

士� �士�� + 士�� 士�� + 士�士�匝= - 層� 層��姿 +��+ 児 �� 層� �岫�士�岻�� + 層�匝 �匝士��匝 + 匝�匝 �士��

Stream Vorticity Functions

(Cartesian Coordinates) Vorticity Function: 降 = 穴懸穴捲 − 穴憲穴検

And recall the stream functions: u = 鳥�鳥槻 and v = − 鳥�鳥掴

By substituting both the stream function into vorticity

function then the vorticity equation will become: −磁 = �匝痔�姉匝 + �匝痔�姿匝


(Cartesian Coordinates) And from x-component Navier-Stokes Equation: 憲穴憲穴捲 + 懸穴憲穴検= − な� 穴�穴捲 + 荒岫穴態憲穴捲態 + 穴態憲穴検態岻

By taking the first derivative with respect to y, and

equation evolves: 憲鳥鉄通鳥掴鳥槻 + 懸鳥鉄通鳥槻鉄=− 怠� 鳥�鳥掴鳥槻 + 荒鳥典通鳥掴鉄鳥槻 + 鳥典通鳥槻典 ------(1)


(Cartesian Coordinates) Similarly, from y-component Navier-Stokes Equation: 憲穴懸穴捲 + 懸穴懸穴検= − な� 穴�穴検 − 訣 + 荒岫穴態懸穴捲態 + 穴態懸穴検態岻

Assume there’s no gravit� force, b� taking the first derivative with respect to x, and equation evolves: 憲鳥鉄塚鳥掴鉄 + 懸鳥鉄塚鳥掴鳥槻=− 怠� 鳥�鳥掴鳥槻 + 荒岫鳥典塚鳥掴典 + 鳥典塚鳥槻典岻 -------(2)


(Cartesian Coordinates) (2)-(1): 憲鳥鉄塚鳥掴鉄 − 憲鳥鉄通鳥掴鳥槻 + 懸鳥鉄塚鳥掴鳥槻 − 懸鳥鉄通鳥槻鉄 =荒岫鳥典塚鳥掴典 + 鳥典塚鳥槻典 − 鳥典通鳥掴鉄鳥槻 + 鳥典通鳥槻典 ) 憲穴穴捲岫穴懸穴捲 − 穴憲穴検岻 + 懸穴穴検岫穴懸穴捲 − 懸穴憲穴検岻=荒岫穴穴捲穴態懸穴捲態 − 穴態憲穴検態 + 穴穴捲穴態懸穴捲態 − 穴態憲穴検態岻

Recall that 降 = 鳥塚鳥掴 − 鳥通鳥槻, then 四�磁�姉 + 士�磁�姿 =児岫�匝磁�姉匝 + �匝磁�姿匝岻

For compressible flow: �磁�嗣 + 四�磁�姉 + 士�磁�姿 =児岫�匝磁�姉匝 + �匝磁�姿匝岻


(Cylindrical Coordinates) Vorticity Function: 降 = な堅穴岫堅懸�岻穴堅 − な堅穴懸�穴�

And recall the stream functions: 懸� = 怠� 鳥�鳥� 懸� = − 鳥�鳥�

By substituting both the stream function into vorticity function then the vorticity equation will become:


(Cylindrical Coordinates)

降 = 怠� 鳥岫�塚�岻鳥� − 怠� 鳥塚�鳥� 降 = 塚�� + 鳥塚�鳥� − 怠� 鳥塚�鳥� 降 = 怠� 岫− 鳥�鳥�岻 + 鳥鳥� 岫− 鳥�鳥�岻 − 怠� 鳥鳥� 岫怠� 鳥�鳥�)

-磁 = 層� �痔�� + �匝痔��匝 + 層�匝 �匝痔��匝


(Cylindrical Coordinates) And from r-component Navier-Stokes Equation: 穴懸�穴建 + 懸� 穴懸�穴堅 + 懸�堅穴懸�穴� − 懸�態に= − な� 絞�絞堅 + 訣� + 荒穴穴堅な堅穴岫堅懸�岻穴堅 + に堅態穴懸�穴� + な堅態穴態懸�穴�態

Taking the first derivation with respect to �: 鳥鉄塚�鳥�鳥痛 + 懸� 鳥鉄塚�鳥�鳥� + 塚�� 鳥鉄塚�鳥�鉄 =− 怠� 鳥�鳥�鳥� + 荒鳥鳥� 怠� 鳥鉄塚�鳥�鉄 + 鳥塚�鳥� + 怠�鉄鳥鉄塚�鳥�鉄 ----(3)


(Cylindrical Coordinates) And from �-component Navier-Stokes Equation: 穴懸�穴建 + 懸� 穴懸�穴堅 + 懸�堅穴懸�穴� + 懸�懸�に= −

な堅な� 絞�絞検 +訣�+ 荒穴穴堅な堅穴岫堅懸�岻穴堅 + な堅態穴態懸�穴�態 + に堅態穴懸�穴�

Taking the first derivation with respect to r: 鳥鉄塚�鳥�鳥痛 + 懸� 鳥鉄塚�鳥�鳥� + 塚�� 鳥鉄塚�鳥�鉄 =− 怠� 怠� 鳥�鳥�鳥� + 荒鳥鳥� 怠� 鳥鉄塚�鳥�鉄 + 鳥塚�鳥� + 怠�鉄鳥典塚�鳥�典 ----(4)


(Cylindrical Coordinates) Upon dividing equation(3) by r, then (4) – (3)/r, by some

rearrangement: 穴穴建穴懸�穴堅 − な堅穴懸�穴� + 懸� 穴穴堅穴懸�穴堅 − な堅穴懸�穴� + 懸�堅穴穴� 穴懸�穴堅 − な堅穴懸�穴�= 荒穴態穴堅態岫穴懸�穴堅 − な堅穴懸�穴� 岻+な堅穴穴堅岫穴懸�穴堅 − な堅穴懸�穴� 岻+ な堅態穴態穴�態岫穴懸�穴堅− な堅穴懸�穴� 岻

Then the stream vorticity function in cylindrical coordinate is: �磁�嗣 + 士� �磁�� + 士�� 磁�� = 児 �匝磁��匝 +層� �磁�� + 層�匝 �匝磁��匝

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