×

### Let's log you in.

or

Don't have a StudySoup account? Create one here!

×

or

33

0

1

# 587 Class Note for M E 521 at PSU

Marketplace > Pennsylvania State University > 587 Class Note for M E 521 at PSU

No professor available

These notes were just uploaded, and will be ready to view shortly.

Either way, we'll remind you when they're ready :)

Get a free preview of these Notes, just enter your email below.

×
Unlock Preview

COURSE
PROF.
No professor available
TYPE
Class Notes
PAGES
1
WORDS
KARMA
25 ?

## Popular in Department

This 1 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 33 views.

×

## Reviews for 587 Class Note for M E 521 at PSU

×

×

### What is Karma?

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 02/06/15
The B nard Problem A Summary Author John M Cimbala Penn State University Latest revision 08 February 2008 1 Introduction and Problem Setup T AT cold Cons1der two 1nf1n1te statlonary parallel at plates separated by d1stance d The TU z dz lower plate is at temperature To and the upper plate is at temperature To AT d 1g 2L colder Initially the flow is at rest but this is an unstable situation since the warm 7 fluid on the bottom wants to rise We examine this problem using linear stability TU WW 2 7 quotm ana y51s 2 Summary of Linear Stability Analysis applied to this problem The inclass analysis follows Kundu Section 123 closely filling in some of the details I Step 0 Start with the Boussinesq equations NavierStokes equations for buoyant flows for total flow variables 7 0 1 quot3121 1 39de 7 of 0x 90 ary J ax 0x1 This represents 5 equations and 5 unknowns I Step 1 Generate the basic state equations 1b 2b and 3b Here we assume no flow so that U 0 P Pz irpog53i1 f1r nanchj fe W 3 u 7 7c Oxj 0x at 0x x 6x d hydrostatlc pressure although P does not vary w1th z exactly lmearly and T v To 7 F Z I Step 2 Add disturbances Q q and plug into 1 2 and 3 This generates total equations 1t 2t and 3t I Step 3 Subtract basic state equations from the total equations This generates disturbance equations 1d 2d and 3d I Step 4 Linearize the disturbance equations to generate the linearized disturbance equations 11 21 and 31 cu au 1 a a or airquot a 0 11 77 lg63aT39v 21 and 39 7l w 39 x1 7 2c 0 p0 0xx xiax Bt 6x390x equations and 5 unknowns now the disturbance variables are the unknowns since the basic state is known I Step 5 Solve the linearized disturbance equations ll 2 and 3 After some algebraic manipulation we can eliminate pressure from the equations and rewrite the energy and z m omentum equations as follows T39 r 39 PW JCVZT 3l vector notation and 3l This still represents 5 3V2wgav 2T4 WW 6 where ZEi FO Z atr39 quotH 39 39 H 052 ay This set of two equations and two unknowns w and T is uncoupled from the other linearized disturbance equations SemiNormalization of the equations Following Kundu s notation normalize only the independent variables x and t but do not rename the variables kind of confusing Also introduce the Prandtl number Pf E VK Equations 3l of rd2 at 1 7c and 6 become v wv2T39 7 and givzjv w go d VHT s The boundary conditions for these two equatlons w1th respect to z are w 0 T39 0 at z i where this 2 IS def1ned Z as the original 2 divided by d znew zodgimld Method of Normal Modes Assume disturbances that are periodic in x and y but not growing or decaying in x or y but may be periodic and ma be growing or decaying in t temporal instability Let the disturbances be of the form lcxla39 e1 1y and T39 Tzequot ll m where V and f are complex amplitudes and k and l are the x andy wviiz components respectively of wavenumber vector 1 For temporal stability analysis both k and lmust be real but 039 the complex growth rate can be complex otherwise spatial instability would also be possible Plug these disturbances into Eqs 7 and 8 to get 0397 D2 iKz f 39W 11 D2 7 K2 D2 7K2 7R a K2 12 Pr d rd 39 where DE Rayleigh number Ra amp K E k392 l2 and W E dz 16 K We are now down to 2 odes and 2 unknowns The boundary conditions for Equations 11 and 12 are W DW f 0 at z i w defined for convenience I Step 6 Examine stability Finally we solve 11 and 12 for the case of marginal stability to be done in class

×

×

### BOOM! Enjoy Your Free Notes!

×

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

Bentley McCaw University of Florida

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

Amaris Trozzo George Washington University

#### "I made \$350 in just two days after posting my first study guide."

Jim McGreen Ohio University

#### "Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

Parker Thompson 500 Startups

#### "It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

Become an Elite Notetaker and start selling your notes online!
×

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com