×

### Let's log you in.

or

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

×

### Create a StudySoup account

#### Be part of our community, it's free to join!

or

##### By creating an account you agree to StudySoup's terms and conditions and privacy policy

Already have a StudySoup account? Login here

## Foundations of Fluid Mechanics I

by: Chester Goldner III

17

0

0

# Foundations of Fluid Mechanics I M E 521

Marketplace > Pennsylvania State University > Mechanical Engineering > M E 521 > Foundations of Fluid Mechanics I
Chester Goldner III
Penn State
GPA 3.79

Staff

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

Purchase these notes here, or revisit this page.

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

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

×
Unlock Preview

### Preview these materials now for free

Why put in your email? Get access to more of this material and other relevant free materials for your school

COURSE
PROF.
Staff
TYPE
Class Notes
PAGES
0
WORDS
KARMA
25 ?

## Popular in Mechanical Engineering

This 0 page Class Notes was uploaded by Chester Goldner III on Sunday November 1, 2015. The Class Notes belongs to M E 521 at Pennsylvania State University taught by Staff in Fall. Since its upload, it has received 17 views. For similar materials see /class/233073/m-e-521-pennsylvania-state-university in Mechanical Engineering at Pennsylvania State University.

×

## Reviews for Foundations of Fluid Mechanics I

×

×

### 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: 11/01/15
The B nard Problem A Summary Author John M Cimbala Penn State University Latest revision 08 February 2008 1 Introduction and Problem Setup Consider two infinite stationary parallel at plates separated by distance d The T Anmld 2 012 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 uid on the bottom wants to rise We examine this problem using linear stability TU WW 2 quot12 analysis 2 Summary of Linear Stability Analysis applied to this problem The inclass analysis follows Kundu Section 123 closely filling in some of the details Step 0 Start with the Boussinesq equations NavierStokes equations for buoyant flows for total flow variables q 012 z 3931 0f 01 u 7 7 517aT7T 1 a 0 Mg 0 i This represents 5 equations and 5 unknowns Step 1 Generate the basic state equations 1b 2b and 3b Here we assume no flow so that U 0 P Pz d 0 airquot a f u 7c OxJOxJ at Bx vaxjaxj 3 hydrostatic pressure although P does not vary with z exactly linearly and Z Step 2 Add disturbances q Q q and plug into 1 2 and 3 This generates total equations 1t 2t and 3t Step 3 Subtract basic state equations from the total equations This generates disturbance equations 1d 2d and 3d Step 4 Linearize the disturbance equations to generate the linearized disturbance equations 11 21 and 31 au 1 0 0 0T7 BZTquot 11 7 pg53ocT39V 39 21 and 7l w 2c at p0 0xx x at 0x 0x equations and f unknowns now the disturbance vari bles are the unknowns since the basic state is known 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 0 2 u 3l This still represents 5 V2141 gaVHZT39 VVAW 6 where E 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 I Z L 2 and 6 become if iw39V2T39 7 and V2 V2 VHZT39 8 at K Pr 0t V The boundary conditions for these two equatlons w1th respect to z are w 0 T39 0 at z i where this 2 1s 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 may be rowin or decaying in t temporal instability Let the disturbances be of the form and T39 TZ8 md m where V and f are complex amplitudes and k and l are the x andy 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 quot a 12 PI d rd4 Fdz d Rayleigh number Ra amp K k2 l2 and W E W defined for convenience Z K17 K We are now down to 2 odes and 2 unknowns into Eqs 7 and 8 to get 77032 71212 W 11 7D2 712 D2 7K2W7RaK2 where D The boundary conditions for Equations 11 and 12 are W DW T 0 at z i Step 6 Examine stability Finally we solve 11 and 12 for the case of marginal stability to be done in class B nard and Taylor Cells 1 B nard cells buoyancy driven convection cells Author John M Cimbala Penn State University Latest revision 15 February 2008 139 Buoyancydriven convection rolls Differential interferograms show side views of convective instability of silicone oil in a rectangular box of relative dimensions 1041 heath from below At the top is the classical Ray leigh Bonard situation uniform heating produces rolls parallel to the shorter side In the middle photograph the temperature cliflcrcnce and hence the amplitude of motion increase from right to left At the bottom the box is rotating about a vertical axis Canal 6 Kirchurt 1979 Oertel 19822 From Van Dyke M An Album afFluidMa an Stanford CA The Parabolic Press 1982 p 82 l 140 Circular buoyancydriven convection cells Sili cone oil containing aluminum powder is covered by a uniformly cooled glass plate which eliminates surface tensinn effects The circular boundary induces circular rolls In the left photograph the copper bottom is uni formly heated at 29 times the critical Rayleigh number giving regular rolls At the tight the bottom is hotter at the rim than at the center This induces an overall circula tion whichI superimposed on regular circular rolls pro duces alternately larger and smaller rolls Koschmietler 1974 1966 2 Taylor cells in the narrow gap between two concentric cylinders inner cylinder rotating 127 39 39 39 39 Taylor vnr re L39 quot pu dcl 39 39 a xe 39 39 one of relative radius 0727 The top and bottom plates are xed The rotation speed is 91 times that at which Taylor predicts the onset of the regularly spaced toroidal vottices seen here The ow is radially inward on the heavier dark hoti zontal rings and outward on the ner ones The motion was started impulsively giving narrower votticcs than would result mm a smooth start Burklialter E Kascllmledm 1974 a we 5 WM mm 125 Laminar Taylor vortices in a narrow gap A larger inner cylinder in the apparatus D the right gives a radius ratio of 0396 Again only the inner cylinder rotates The upper photograph shows the center section of axisymmertic vortices at 116 times the critical speed In the lower at 85 times the critical speed the ow is doubly riodic with SD waves around the circumference drifting with the rotation Koschmiedm 1979 From Van Dyke MAn Album afFluidJlIotion Stanford CA The Parabolic Press 1982 76 Elementary Planar Irrotational Flows in Complex Variables Author John M Cimbala Penn State University atest revision 17 October 2007 Note Consider steady incompressible irrotational Newtonian uid ow in which gravity is neglected The ow is assumed to be twodimensional in the x y or r Bplane Summary of the Equations Elementary Planar Irrotational Flows f rm stream in the x direct39 Uniform stream in an arbitrar direction C Line vortex at the origin

×

×

×

### You're already Subscribed!

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!"

Jennifer McGill UCSF Med School

#### "Selling my MCAT study guides and notes has been a great source of side revenue while I'm in school. Some months I'm making over \$500! Plus, it makes me happy knowing that I'm helping future med students with their MCAT."

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!"

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

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.