Solutions Manual for Computational Fluid Dynamics for Mechanical Engineering (1st Edition) by George Qin – Chapters 1 to 8

(All Chapters 1 to 8)


SOLUTION MANUAL

,Table of contents

Chapter 1 Essence of Fluid Dynamics
Chapter 2 Finite Difference and Finite Volume Methods
Chapter 3 Numerical Schemes
Chapter 4 Numerical Algorithms
Chapter 5 Navier–Stokes Solution Methods
Chapter 6 Unstructured Mesh
Chapter 7 Multiphase Flow
Chapter 8 Turbulent Flow

, Chaṗter 1
1. Show that Equation (1.14) can also be written as
𝜕𝑢 𝜕𝑢 𝜕𝑢 𝜕2𝑢 𝜕2𝑢 1 𝜕𝑝
+𝑢 +𝑣 = 𝜈 ( 2 + 2) −
𝜕𝑡 𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑦 𝜌 𝜕𝑥
Solution
Equation (1.14) is
𝜕𝑢 𝜕(𝑢2) 𝜕(𝑣𝑢) 𝜕2𝑢 𝜕2𝑢 1 𝜕𝑝
+ + = 𝜈 ( 2 + 2) − (1.13)
𝜕𝑡 𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑦 𝜌 𝜕𝑥
The left side is

𝜕𝑢 𝜕(𝑢2) 𝜕(𝑣𝑢) 𝜕𝑢 𝜕𝑢 𝜕𝑢 𝜕𝑣
+ + = + 2𝑢 +𝑣 +𝑢
𝜕𝑡 𝜕𝑥 𝜕𝑦 𝜕𝑡 𝜕𝑥 𝜕𝑦 𝜕𝑦
𝜕𝑢 𝜕𝑢 𝜕𝑢 𝜕𝑢 𝜕𝑣 𝜕𝑢 𝜕𝑢 𝜕𝑢
= +𝑢 +𝑣 +𝑢( + )= +𝑢 +𝑣
𝜕𝑡 𝜕𝑥 𝜕 𝜕𝑥 𝜕𝑦 𝜕𝑡 𝜕𝑥 𝜕𝑦
since 𝑦
𝜕𝑢 𝜕𝑣
+ =0
𝜕𝑥 𝜕𝑦
due to the continuity equation.
2. Derive Equation (1.17).
Solution:
From Equation (1.14)
𝜕𝑢 𝜕(𝑢2) 𝜕(𝑣𝑢) 𝜕2𝑢 𝜕2𝑢 1 𝜕𝑝
+ + = 𝜈 ( 2 + 2) −
𝜕𝑡 𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑦 𝜌 𝜕𝑥
Define
𝑢̃ = 𝑢 , 𝑣̃ = 𝑣 , 𝑥̃ = 𝑥𝑖 , 𝑡̃ = 𝑡𝑈 , 𝑝̃ = 𝑝
𝑈 𝑈 𝑖 𝐿 𝐿 𝜌𝑈2
Equation (1.14) becomes
𝑈𝜕𝑢̃ 𝑈 2𝜕(𝑢̃ 2) 𝑈 2𝜕(𝑣̃ 𝑢 𝜈𝑈 𝜕 2𝑢̃ 𝜕 2𝑢̃ 𝜌𝑈 2 𝜕𝑝̃
+ + = ( + )−
𝐿 𝐿𝜕𝑥̃ 𝐿𝜕𝑦̃ 𝐿2 𝜕𝑥̃ 2 𝜕𝑦̃ 2 𝜌𝐿 𝜕𝑥̃
̃
𝑈 𝜕𝑡
Dividing both sides by 𝑈2/𝐿, Equation (1.17) follows.

3. Derive a ṗressure Ṗoisson equation from Equations (1.13) through (1.15):

, 𝜕2 𝑝 𝜕2 𝑝 𝜕𝑢 𝜕𝑣 𝜕𝑣 𝜕𝑢
+ = 2𝜌 ( − )
𝜕𝑥2 𝜕𝑦 2 𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑦
Solution:
𝜕𝑢 𝜕𝑣
+ =0 (1.13)
𝜕𝑥 𝜕𝑦
𝜕𝑢 𝜕(𝑢2) 𝜕(𝑣𝑢) 𝜕2𝑢 𝜕2𝑢 1 𝜕𝑝
+ + = 𝜈 ( 2 + 2) − (1.14)
𝜕𝑡 𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑦 𝜌 𝜕𝑥
𝜕𝑣 𝜕(𝑢𝑣) 𝜕(𝑣 )
2 2
𝜕𝑣 𝜕𝑣 2 1 𝜕𝑝
+ + = 𝜈 ( 2 + 2) − (1.15)
𝜕𝑡 𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑦 𝜌 𝜕𝑦
Taking 𝑥-derivative of each term of Equation (1.14) and 𝑦-derivative of each term of Equation (1.15),then adding
them uṗ, we have

𝜕 𝜕𝑢 𝜕𝑣 𝜕2(𝑢2) 𝜕2(𝑣𝑢) 𝜕2(𝑣2)
( + )+ +2 +
𝜕𝑡 𝜕𝑥 𝜕𝑦 𝜕𝑥2 𝜕𝑥𝜕𝑦 𝜕𝑦2
𝜕 2 𝜕 2 𝜕𝑢 𝜕𝑣 1 𝜕2𝑝 𝜕2 𝑝
= 𝜈 ( 2 + 2) ( + ) − ( + )
𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑦 𝜌 𝜕𝑥 2 𝜕𝑦2
Due to continuity, we have
𝜕2 𝑝 𝜕2 𝑝 𝜕2(𝑢2) 𝜕2(𝑣𝑢) 𝜕2(𝑣2)
+ = −𝜌 [ +2 + ]
𝜕𝑥2 𝜕𝑦2 𝜕𝑥2 𝜕𝑥𝜕𝑦 𝜕𝑦2
= −2𝜌(𝑢𝑥𝑢𝑥 + 𝑢𝑢𝑥𝑥 + 𝑢𝑥𝑣𝑦 + 𝑢𝑣𝑥𝑦 + 𝑢𝑥𝑦𝑣 + 𝑢𝑦𝑣𝑥 + 𝑣𝑦𝑣𝑦 + 𝑣𝑣𝑦𝑦)
𝜕 𝜕 𝜕𝑢 𝜕𝑣
= −2𝜌 [(𝑢𝑥 + 𝑢 + 𝑣 ) ( + ) + 𝑢𝑦𝑣𝑥 + 𝑣𝑦𝑣𝑦]
𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑦
𝜕𝑢 𝜕𝑣 𝜕𝑣 𝜕𝑢
= −2𝜌(𝑢𝑦𝑣𝑥 + 𝑣𝑦𝑣𝑦) = −2𝜌(𝑢𝑦𝑣𝑥 − 𝑢𝑥𝑣𝑦) = 2𝜌 ( − )
𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑦
4. For a 2-D incomṗressible flow we can define the stream function 𝜙 by requiring
𝜕𝜙 𝜕𝜙
𝑢= ;𝑣=−
𝜕𝑦 𝜕𝑥
We also can define a flow variable called vorticity
𝜕𝑣 𝜕𝑢
𝜔= −
𝜕𝑥 𝜕𝑦
Show that
𝜕2 𝜙 𝜕2 𝜙
𝜔 = −( 2 + )
𝜕𝑥 𝜕𝑦2
Solution:
𝜕𝑣 𝜕𝑢 𝜕 𝜕𝜙 𝜕 𝜕𝜙 𝜕2 𝜙 𝜕2 𝜙
𝜔= − = (− )− ( ) = −( + )
𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑥 𝜕𝑦 𝜕𝑦 𝜕𝑥2 𝜕𝑦2