100% satisfaction guarantee Immediately available after payment Both online and in PDF No strings attached
logo-home
Previously searched by you
Previously searched by you
logo
Solutions Manual for Computational Fluid Dynamics for Mechanical Engineering, 1st Edition by George Qin (All Chapters) A+ $12.99   Add to cart
Quickly navigate to
Preview
  2.  
  3. Computational Fluid Dynamics For Mechanical Engine
  4.  
  5. Computational Fluid Dynamics for Mechanical Engine

Exam (elaborations)

Solutions Manual for Computational Fluid Dynamics for Mechanical Engineering, 1st Edition by George Qin (All Chapters) A+

 6 views  0 purchase
  • Course
  • Computational Fluid Dynamics for Mechanical Engine
  • Institution
  • Computational Fluid Dynamics For Mechanical Engine
  • Book
  • Computational Fluid Dynamics for Mechanical Engineering

Solutions Manual for Computational Fluid Dynamics for Mechanical Engineering, 1st Edition by George Qin (All Chapters) A+ ..

Preview 4 out of 116  pages

Document information

  • October 10, 2024
  • 116
  • 2024/2025
  • Exam (elaborations)
  • Questions & answers

Subjects

  • solution
  • manual for
  • computational
  • fluid
  • dynamics
  • 1st edition
  • by george qin
  • 2024
  • for mechanical engineering
  • computational fluid dynamics for mechanical engi

Connected book

book image

Book Title:

Author(s):

  • Edition:
  • ISBN:
  • Edition:

Written for

  • Computational Fluid Dynamics for Mechanical Engine
All documents for this subject (1)

Seller

avatar-seller
ASolution

Reviews received

Content preview

Created By: A Solution


Solutions Manual for Computational Fluid Dynamics for

Mechanical Engineering, 1st Edition by George Qin (All

Chapters) A+

Chapter 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
+ + =
𝜕𝑥 𝜕𝑡
𝜕𝑡 𝜕𝑦 + 2𝑢 𝜕𝑥 + 𝑣 𝜕𝑦 + 𝑢 𝜕𝑦

𝜕𝑢 𝜕𝑢 𝜕𝑢 𝜕𝑢 𝜕𝑣 𝜕𝑢 𝜕𝑢 𝜕𝑢
= +𝑢 +𝑣 +𝑢( + ) = +𝑢 +𝑣
𝜕𝑡 𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑦 𝜕𝑡 𝜕𝑥 𝜕𝑦
since


𝜕𝑢 𝜕𝑣
+ =0
𝜕𝑦
due to the continuity equation. 𝜕𝑥


2. Derive Equation (1.17).
Solution:
From Equation (1.14)

𝜕𝑢 𝜕(𝑢2) 𝜕(𝑣𝑢) 𝜕2𝑢 𝜕2𝑢 1 𝜕𝑝
+ + = 𝜈( + )−
𝜕𝑡 𝜕𝑥 𝜕𝑦 𝜕𝑥2 𝜕𝑦 2 𝜌 𝜕𝑥
Define
𝑥𝑖 𝑡𝑈 𝑝
𝑢= 𝑢, 𝑣 = ,𝑣𝑥 = ,𝑡 = ,𝑝 =
𝑈 𝑈 𝑖 𝐿 𝐿
Equation (1.14) becomes

1

,Created By: A Solution


𝜌𝑈2

𝑈𝜕 𝑢 𝑈 2 𝜕(𝑢2 ) 𝑈2𝜕(𝑣𝑢 ) 𝜈𝑈 𝜕 2 𝑢 𝜕 2 𝑢 𝜌𝑈2 𝜕𝑝
+ + = ( + )−
𝐿 𝐿𝜕𝑥 𝐿𝜕𝑦 𝐿2 𝜕𝑥2 𝜕𝑦2
𝑈 𝜕𝑡 𝜌𝐿 𝜕𝑥

Dividing both sides by 𝑈2/𝐿, Equation (1.17) follows.

3. Derive a pressure Poisson equation from Equations (1.13) through (1.15):




2

,Created By: A Solution
𝜕2𝑝 𝜕2𝑝 𝜕𝑢 𝜕𝑣 𝜕𝑣 𝜕𝑢
+ = 2𝜌 ( − )
𝜕𝑥 2 𝜕𝑦 2 𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑦
Solution:
𝜕𝑢 𝜕𝑣
+
𝜕𝑦
=0 (1.13)
𝜕𝑥
𝜕𝑢 𝜕(𝑢2) 𝜕(𝑣𝑢)
𝜕2𝑢 𝜕2𝑢 1 𝜕𝑝
+ +
= 𝜈 ( 2 + 2) − (1.14)
𝜕𝑡 𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑦 𝜌 𝜕𝑥
𝜕𝑣 𝜕(𝑢𝑣) 𝜕(𝑣 ) 2 𝜕2𝑣 𝜕2𝑣 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 up, we have

𝜕 𝜕𝑢 𝜕𝑣 𝜕2(𝑢2) 𝜕2(𝑣𝑢) 𝜕2(𝑣2)
( + )+ +2 +
𝜕𝑥 𝜕𝑦 𝜕𝑥2 𝜕𝑦2
𝜕𝑡 𝜕𝑥𝜕𝑦

𝜕2 𝜕2 𝜕𝑢 𝜕𝑣 1 𝜕2𝑝 𝜕2𝑝
= 𝜈( 2+ 2) ( + ) − ( 2 + 2)
𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑦 𝜌 𝜕𝑥 𝜕𝑦
Due to continuity, we have

𝜕2𝑝 𝜕2𝑝 𝜕2(𝑢2) 𝜕2(𝑣𝑢) 𝜕2(𝑣2)
] +
𝜕𝑥 2 + 𝜕𝑦2
= −𝜌 [
𝜕𝑥2 𝜕𝑥𝜕𝑦
+2 𝜕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 incompressible 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

𝜕2𝜙 𝜕2𝜙
𝜔 = − ( 2 + 2)
𝜕𝑥 𝜕𝑦
Solution:
𝜕𝑣 𝜕𝑢 𝜕 𝜕𝜙 𝜕 𝜕𝜙 𝜕2𝜙 𝜕2𝜙
𝜔= − = (− )− ( ) = −( + )
𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑥 𝜕𝑦 𝜕𝑦 𝜕𝑥2 𝜕𝑦2



3

, Created By: A Solution




Chapter 2
𝑑𝜙
1. Develop a second-order accurate finite difference approximation for ( )
on a non-uniform
𝑑𝑥 𝑖
mesh using information (𝜙 and 𝑥 values) from mesh points 𝑥𝑖−1, 𝑥𝑖 and 𝑥𝑖+1. Suppose 𝛿𝑥𝑖 =
𝑥𝑖+1 − 𝑥𝑖 = 𝛼𝛿𝑥𝑖−1 = 𝛼(𝑥𝑖 − 𝑥𝑖−1).

Solution:

Assume close to the 𝑖𝑡ℎ point, 𝜙(𝑥) = 𝜙𝑖 + 𝑏(𝑥 − 𝑥𝑖) + 𝑐(𝑥 − 𝑥𝑖)2 + 𝑑(𝑥 − 𝑥𝑖)3 …

Then 𝑑𝜙 = 𝑏 + 2𝑐(𝑥 − 𝑥 ) + ⋯ and 𝑑𝜙 = 𝑏.
( )
𝑑𝑥 𝑑𝑥 𝑖


Now 𝜙𝑖+1 = 𝜙(𝑥𝑖+1) = 𝜙𝑖 + 𝑏(𝑥𝑖+1 − 𝑥𝑖) + 𝑐(𝑥𝑖+1 − 𝑥𝑖)2 + ⋯ = 𝜙𝑖 + 𝑏Δ𝑥𝑖 + 𝑐Δ𝑥2 + 𝑑Δ𝑥3 …
𝑖 𝑖
And 𝜙𝑖−1 = 𝜙(𝑥𝑖−1) = 𝜙𝑖 + 𝑏(𝑥𝑖−1 − 𝑥𝑖) + 𝑐(𝑥𝑖−1 − 𝑥𝑖 )2 + ⋯ = 𝜙𝑖 − 𝑏Δ𝑥𝑖−1 + 𝑐Δ𝑥2 − 𝑑Δ𝑥3 …
𝑖−1 𝑖−1
So Δ𝑥2 𝜙𝑖+1 − Δ𝑥2𝜙𝑖−1 = (Δ𝑥2 − Δ𝑥2)𝜙𝑖 + 𝑏Δ𝑥𝑖Δ𝑥𝑖−1(Δ𝑥𝑖 + Δ𝑥𝑖−1) + 𝑑Δ𝑥2Δ𝑥2 (Δ𝑥𝑖 +
𝑖−1 𝑖 𝑖−1 𝑖 𝑖 𝑖−1
Δ𝑥𝑖−1) + ⋯
Δ𝑥2 𝜙𝑖+1−Δ𝑥2𝜙𝑖−1−(Δ𝑥2 −Δ𝑥2)𝜙𝑖
And 𝑏 = 𝑖−1 𝑖 𝑖−1 𝑖 − 𝑑Δ𝑥𝑖Δ𝑥𝑖−1 + ⋯
Δ𝑥𝑖Δ𝑥𝑖−1(Δ𝑥𝑖+Δ𝑥𝑖−1)
𝑑𝜙
A 2nd order finite difference for ( )
is therefore
𝑑𝑥 𝑖

𝑑𝜙 Δ𝑥2 𝜙𝑖+1 − Δ𝑥2𝜙𝑖−1 − (Δ𝑥2 − Δ𝑥2)𝜙𝑖 𝜙 + (α2 − 1)𝜙𝑖 − α2𝜙𝑖−1
( ) =𝑏≈ = 𝑖+1
𝑑𝑥 𝑖 Δ𝑥𝑖Δ𝑥𝑖−1(Δ𝑥𝑖 + Δ𝑥𝑖−1) α(α + 1)Δ𝑥𝑖−1




4

The benefits of buying summaries with Stuvia:

Guaranteed quality through customer reviews

Guaranteed quality through customer reviews

Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.

Quick and easy check-out

Quick and easy check-out

You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.

Focus on what matters

Focus on what matters

Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!

Frequently asked questions

What do I get when I buy this document?

You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.

Satisfaction guarantee: how does it work?

Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.

Who am I buying these notes from?

Stuvia is a marketplace, so you are not buying this document from us, but from seller ASolution. Stuvia facilitates payment to the seller.

Will I be stuck with a subscription?

No, you only buy these notes for $12.99. You're not tied to anything after your purchase.

Can Stuvia be trusted?

4.6 stars on Google & Trustpilot (+1000 reviews)

75632 documents were sold in the last 30 days

Founded in 2010, the go-to place to buy study notes for 14 years now

Start selling
$12.99
  • (0)
  Add to cart