Cimbala Introducing CFD in Undergrad Fluids

Introducing CFD in Introducing CFD in Undergraduate Fluid MechanicsUndergraduate Fluid MechanicsJohn Cimbala, Mechanical Engr., Penn State Univ.

ISTEC Meeting, Cornell UniversityJuly 25-26, 2008, Ithaca, NY

with collaboration from:Shane Moeykens, Strategic Partnerships Manager, ANSYS.Ajay Parihar, FlowLab Support Engineer, ANSYS. Sujith Sukumaran, FlowLab Support Engineer, ANSYSSatyanarayana Kondle, FlowLab Support Engineer, ANSYS.

IntroductionIntroductionz It has become important in recent years to

introduce the fundamentals of CFD in intro-level undergraduate fluid mechanics classes due to the changing requirements of the job market for graduating engineers

z At a minimum, it is desirable to teach the fundamental steps required to obtain a useful CFD solution

zMany instructors want to include CFD in their undergrad fluids course, but dont know how and/or think they cant afford the class time

Many of them will use CFD in their jobs, whether they know anything

about CFD or not!

z Undergraduate fluid mechanics textbook, Fluid Mechanics: Fundamentals and Applications, by Y. A. engel and J. M. Cimbala, McGraw-Hill, 2006

z Chapter 15: Introduction to CFD

Our first attempt to Our first attempt to introduce CFD to introduce CFD to

undergradsundergrads

zThe CFD chapter introduces: grids boundary

conditions residuals etc.

engelengel--CimbalaCimbala textbooktextbook

zThe CFD chapter introduces: grids boundary

conditions residuals etc.

engelengel--CimbalaCimbala textbooktextbook

zThe CFD chapter introduces: grids boundary

conditions residuals etc.

engelengel--CimbalaCimbala textbooktextbook

zThe CFDchapterintroduces: grids boundary

conditions residuals etc. just the basics, not anything about

numerical algorithms, stability, etc. how to use CFD as a tool.

engelengel--CimbalaCimbala textbooktextbook

z The engel-Cimbala book includes FlowLabas a textbook companion, where CFD exercises are employed to convey important concepts to the student

z 46 FlowLab end-of-chapter problems are included in Ed. 1, Chapter 15 (Intro to CFD)

z FlowLab exercises jointly developed by John Cimbala and Fluent Inc. (now part of ANSYS).

z FlowLab & these FlowLab templates are free to students who use the engel-Cimbala book

Intro to CFD using Intro to CFD using FlowLabFlowLab

What is What is FlowLabFlowLab??zA virtual (CFD) fluids laboratoryz Simple to use with a very fast learning curvez Runs pre-defined exercises (templates)z Setup, solution, and post-processing are all

performed in the same interfacez Students vary only one or two parameters in

each template (to look at trends, compare boundary conditions, grid resolution, etc.)

zEach homework problem, along with its corresponding FlowLab template, has been carefully designed with two major learning objectives in mind: Enhance the students understanding of a

specific fluid mechanics concept Introduce the student to a specific

capability and/or limitation of CFD through hands-on practice

FlowLabFlowLab TemplatesTemplates

Original Templates for Ed. 1Original Templates for Ed. 1z FlowLab HW problems only in CFD chapterzMost templates are too complex to compare

with analytical calculations (e.g., flow over cylinders, flow through diffusers, etc.)

z Emphasis mostly on CFD grid resolution, extent of computational domain, BCs, etc.

z In the first edition, the primary emphasis of the FlowLab templates was as a CFD learning tool, with only a secondary emphasis on learning fluid mechanics

New Templates for Ed. 2New Templates for Ed. 2zNew FlowLab templates in almost all chapters

goal is to introduce students to CFD early onzMost new templates compare CFD calculations

with analytical calculationsz The primary emphasis is learning fluid

mechanics, with a secondary emphasis on CFDzNew templates are intentionally more simplezHomework problems show a progression in

difficulty and level of sophistication, often based on the same base problem or theme

Examples: New homework & templates, Ed. 2Examples: New homework & templates, Ed. 2z End-of-chapter homework problem, Chap. 2

2-89 A rotating viscometer consists of two concentric cylinders an inner cylinder of radius Ri rotating at angular velocity (rotation rate) i , and a stationary outer cylinder of inside radius Ro . In the tiny gap between the two cylinders is the fluid of viscosity . The length of the cylinders (into the page in the sketch) is L. L is large such that end effects are negligible (we can treat this as a two- dimensional problem). Torque (T) is required to rotate the inner cylinder at constant speed.

(a) Showing all your work and algebra, generate an approximate expression for T as a function of the other variables. (b) Explain why your solution is only an approximation. In particular, do you expect the velocity profile in the gap to remain linear as the gap becomes larger and larger (i.e., if the outer radius Ro were to increase, all else staying the same)?

Fluid: ,

i

Rotating inner cylinder Stationary outer cylinder

Ro Ri This is a standard analytical problem as

found in most undergraduate fluids books.

They are able to obtain an analytical (approximate) solution for a small gap.

z Solution (from solutions manual)2-89 (a) We assume a linear velocity profile between the two walls as sketched the inner wall is moving at speed V = i Ri and the outer wall is stationary. The thickness of the gap is h, and we let y be the distance from the outer wall into the fluid (towards the inner wall). Thus,

where

Since shear stress has dimensions of force/area, the clockwise (mathematically negative) tangential force acting along the surface of the inner cylinder by the fluid is

But the torque is the tangential force times the moment arm Ri . Also, we are asked for the torque required to turn the inner cylinder. This applied torque is counterclockwise (mathematically positive). Thus,

and y du Vu Vh dy h

= = =- and o i i ih R R V R= =

2 2i ii io i

RVF A R L R Lh R R

= = =

3 32 2T i i i iio i

L R L RFRR R h = = =

V

Outer cylinder

h

Inner cylinder

y u

Analytical solution for a

small gap

2-90

z Another end-of-chapter homework problem, Chap. 2

Consider the rotating viscometer of the previous problem. We make an approximation that the gap (distance between the inner and outer cylinders) is very small. Consider an experiment in which the inner cylinder radius is Ri = 0.0600 m, the outer cylinder radius is Ro = 0.0602 m, the fluid viscosity is 0.799 kg/ms, and the length L of the viscometer is 1.00 m. Everything is held constant in the experiment except that the rotation rate of the inner cylinder varies. (a) Calculate the torque in Nm for several rotation rates in the range -700 to 700 rpm. Discuss the relationship between T and i (is the relationship linear, quadratic, etc.?). (b) Run FlowLab with the template Concentric_inner. Set the rotation rate to the same values as in Part (a), and calculate the torque on the inner cylinder for all cases. Compare to the approximate values of Part (a), and calculate a percentage error for each case, assuming that the CFD results are exact. Discuss. In particular, how good is the small-gap approximation? Note: Be careful with the sign (+ or -) of the torque.

This is one of their first exposures to CFD through FlowLab

They calculate torque as a function of rpm

z Solution (from solutions manual)2-90 (a) Note that we must convert the rotation rate from rpm to radians per second so that the units are proper. When i is -700 rpm, we get

For h = 0.0002 m, the torque is calculated (using the equation derived in the previous problem). Note, however, that since we are calculating the torque of the fluid acting on the cylinder, the sign is opposite to that of the previous problem,

where we have rounded to three significant digits. We repeat for various other values of rotation rate, and summarize the results in the table below.(b) The FlowLab template was run with the same values of i . The results are compared with the manual calculations in the table. The agreement between manual and CFD calculations is excellent for all rotation rates. The relationship between torque and rotation rate is linear, as predicted by theory.

rot 2 rad 1 min rad700 73.304min rot 60 s si

= =

( )( ) ( )

3 3

3

2

2 2T

rad2 1.00 m 0.799 kg/m s 73.304 0.0600 mNs

0.0002 m kg m/s 397.445 N m 397. N m

i i i i

o i

L R L RR R h

= = =

=

We run various rpm cases, both manually and with CFD

z Solution (from solutions manual - continued)

Discussion Since the gap here is very small compared to the radii of the cylinders, the linear velocity profile approximation is actually quite good, yielding excellent agreement between theory and CFD. However, if the gap were much larger, the agreement would not be so good.

Agreement between analytical and CFD results is excellent

Analytical FlowLab

2-91

z Another end-of-chapter homework problem, Chap. 2

Consider the rotating viscometer of the previous problem. We make an approximation that the gap (distance between the inner and outer cylinders) is very small. Consider an experiment in which the inner cylinder radius is Ri = 0.0600 m, rotating at a constant angular rotation rate of 300 rpm. The fluid viscosity is 0.799 kg/ms, and the length L of the viscometer is 1.00 m. Everything is held constant in the experiment except that different diameter outer cylinders are used. The gap distance between inner and outer cylinders is h = Ro Ri . (a) Calculate the torque in Nm for the following gaps: 0.0002, 0.0015, 0.0075, 0.02, and 0.04 m. (b) Run FlowLab with the template Concentric_gap. Set the gap to the same values as in Part (a), and calculate the torque on the inner cylinder for all cases. Compare to the approximate values of Part (a), and calculate a percentage error for each case, assuming that the CFD results are exact. Discuss. In particular, how good is the small-gap approximation? Note: Use absolute value of torque to avoid sign inconsistencies.

This is the next problem in this series

This time we vary gap size at a fixed rpm

z Solution (from solutions manual)2-91 (a) First we convert the rotation rate from rpm to radians per second so that the units are proper,

When h = 0.0002 m, the torque is calculated (using the equation derived in the previous problem). Note, however, that since we are calculating the torque of the fluid acting on the cylinder, the sign is opposite to that of the previous problem,

where we have rounded to three significant digits. We repeat for various other values of gap distance h, and summarize the results in the table below.

(b) The FlowLab template was run with the same values of h. The results are compared with the manual calculations in the table and plot shown below. Note: We use absolute value of torque for comparison without worrying about the sign.

rot 2 rad 1 min rad300 31.416min rot 60 s si

= =

( )( ) ( )

3 3

3

2

2 2T

rad2 1.00 m 0.799 kg/m s 31.416 0.0600 mNs

0.0002 m kg m/s 170.336 N m 170. N m

i i i i

o i

L R L RR R h

= = =

=

We run various gap size cases, both manually and with CFD

z Solution (from solutions manual - continued)

The agreement between analysis and CFD is

great for small gap sizes

But the agreement is not so good for the larger gap sizes

z Solution (from solutions manual - continued)The agreement between manual and CFD calculations is excellent for very small gaps (the percentage error is less than half a percent for the smallest gap). However, as the gap thickness increases, the agreement gets worse. By the time the gap is 0.04 m, the agreement is worse than 50%. Why such disagreement? Remember that we are assuming that the gap is very small and are approximating the velocity profile in the gap as linear. Apparently, the linear approximation breaks down as the gap gets larger.

Discussion We used a log scale for torque so that the differences between manual calculations and CFD could be more clearly seen.

Students realize that their simple small-gap approximation breaks down as the gap gets larger. At this point in their study

of fluid mechanics, they do not know how to calculate this flow exactly for arbitrary gap size and rpm that is not

learned until Chapter 9, the differential equations chapter.

Examples: New homework & templates, Ed. 2Examples: New homework & templates, Ed. 2z End-of-chapter homework problem, Chap. 9

Fluid: ,

i

Rotating inner cylinder Stationary outer cylinder

Ro Ri

9-92 An incompressible Newtonian liquid is confined between two concentric circular cylinders of infinite length a solid inner cylinder of radius Ri and a hollow, stationary outer cylinder of radius Ro . (see figure; the z axis is out of the page.) The inner cylinder rotates at angular velocity i . The flow is steady, laminar, and two- dimensional in the r- plane. The flow is also rotationally symmetric, meaning that nothing is a function of coordinate (u

and P are functions of radius r only). The flow is also circular, meaning that velocity component ur = 0 everywhere. Generate an exact expression for velocity component u

as a function of radius r and the other parameters in the problem. You may ignore gravity. Hint: The result of Problem 9-91 is useful.

Now we advance to Chapter 9 problems

This is a follow-up problem to those of Chapter 2 just discussed

First, an analytical solution

z Solution (from solutions manual)9-92 The solution is fairly long and not repeated in its entirety here. The Navier- Stokes equations are solved analytically for this simple geometry, and the boundary conditions are applied. Here are the last few lines of the solution:

The solution:

Apply one boundary condition:

Apply another boundary condition:

Solve for the constants of integration:

The final equation is

This is a closed-form analytical solution. In the next problem, we compare with CFD.

21 2

Cru Cr

= +2

21 2 10 or 2 2

o o

o

R RCC C CR

= + = 2

21 1 12 2 2

i i oi i

i i

R R RCR C C CR R

= + = 2 2 2

1 22 2 2 2

2 i i o i i

o i o i

R R RC CR R R R

= = 2 2

2 2i i o

o i

R Ru r

rR R =

Analytical solution for any

size gap

9-93

z Another end-of-chapter homework problem, Chap. 9

Glycerin ( = 1259.9 kg/m3, and = 0.799 kg/ms) flows between two concentric cylinders as in the previous problem. The inner radius is 0.060 m, and the inner cylinder rotates at 300 rpm. The outer cylinder is stationary. Recall from Chapter 2 that when the gap between the cylinders is small, the tangential velocity of the fluid in the gap is nearly linear. When the gap is large, however, we expect the linear approximation to fail. Run FlowLab with the template Concentric_gap. Run two cases: (a) a small gap of 0.001 m and (b) a large gap of 0.060 m. For each case, plot and save the velocity profile data. Compare to the analytical prediction for both cases. Is there good agreement? How good is the linear approximation? Discuss.

This is the next problem in this series

Now we use FlowLab (actually the same template as in Chapter 2) to compare CFD-generated velocity profiles to those generated

analytically. We do this for two gaps, a small gap and a large gap.

9-93 (a) Small gap (gap = 0.001 m): We apply the equation from the previous problem to calculate the tangential velocity as a function of radius,

and we plot the velocity profile, u

as a function of r, in the plot below. We run FlowLab for the same geometry and conditions, and plot the velocity profile on the same plot for comparison. The agreement is excellent (less than 0.02% error at any radius). This is not surprising since the flow is laminar, steady, etc. CFD does a very good job in this kind of situation. The small errors are due to lack of complete convergence and a mesh that could be a little finer. The profile is nearly linear as expected since the gap is small.

z Solution (from solutions manual)

2 2

2 2i i o

o i

R Ru r

rR R =

Students compare the analytical (exact) solutions to those obtained by FlowLab for both cases, small gap and large gap.

z Solution (from solutions manual - continued)

The small gap results show excellent agreement as expected, and the velocity profile is nearly linear since the gap is so small.

z Solution (from solutions manual - continued)(b) Large gap (gap = 0.06 m): We repeat for the larger gap case. The plot is shown below. Again the agreement is excellent, with errors less than 0.1% for all radii, but the profile is not linear the linear approximation breaks down when the gap can no longer be considered small.

This time, the large gap results show excellent agreement as

well since we have not made a small-gap approximation; but

the profile is not linear.

z Solution (from solutions manual - continued)Discussion Problems such as this in which a known analytical solution exists are great for testing CFD codes. The fluid properties did not enter into the calculations viscosity affects only the transient solution, not the final flow field.

Note how this one simple problem yields several homework problems even across chapters.

Students get a feel for using CFD and compare the results with analytical analysis.

They see where their simplified analysis works well and where it breaks down (e.g., small gap approximation breaks down when the gap is too large).

These types of analytical/FlowLab problems have been added to nearly all the chapters in Ed. 2.

Here is the mesh that FlowLab generates for the same geometry as in the exact analysis.

FlowLab Details for this problem

Residual plot (iteration takes only a couple minutes)

They look at velocity magnitude contours

They plot velocity magnitude vs. radial position.

They save these data points to an Excel file.

Live Live FlowLabFlowLab DemonstrationDemonstrationWe will demonstrate the templates called

Concentric_gap

and Submerged_plate_angle

[These templates will have corresponding end- of-chapter homework problems in Ed. 2 of the

engel-Cimbala undergraduate fluids textbook.]

If time, also show some other templates live.

SummarySummaryz It is possible to introduce the fundamentals of CFD

into an undergraduate fluids course(I do it in only one class period, plus homework)zFlowLab software enables students to experience

CFD without getting bogged down in the detailszEach FlowLab exercise has two objectives:

Enhance understanding of fluid mechanics Teach the capabilities and limitations of CFD

zMost of the new templates in Ed. 2 of the fluids textbook by engel and Cimbala compare analytical solutions to those obtained with CFD for enhanced learning and good exposure to CFD

Homework is the key can introduce students to CFD

without taking much class time

How to Integrate CFD into an How to Integrate CFD into an Undergraduate Fluids CourseUndergraduate Fluids CoursezDevelop the continuity and Navier-Stokes

equations for fluid flow, as usualzShow how to solve simple problems

analytically (solve N-S equations): Couette flow between plates Fully developed pipe flow Etc.zThen, introduce CFD as a way to do the

same thing, but with a computer.

This is what is normally done in an introductory fluid mechanics course.

This is what is new added to the course.

How to Integrate CFD into an How to Integrate CFD into an Undergraduate Fluids CourseUndergraduate Fluids CoursezThe CFD lecture takes only about one

class period, where we briefly explain: Computational domain and types of grids Boundary conditions and initial guesses The concept of residuals and iteration Post-processing (contour plots, etc.)

zIn-class live demonstration of FlowLabzAssign homework requiring FlowLabThe homework is where students get hands-on CFD practice

Sample lecture notes fromSample lecture notes fromFall 2005, Penn StateFall 2005, Penn State

(the lecture where CFD was (the lecture where CFD was presented for the first time)presented for the first time)

These notes are directly from my lecture notes, given using a tablet PC, and posted on the Internet for students to download

StillStill--Slide BackSlide Back--Up toUp to Live Demonstration of Live Demonstration of

FlowLabFlowLab templatetemplate Diffuser_angleDiffuser_angle

Example: Flow through a conical Example: Flow through a conical diffuserdiffuser

z Fluid Mechanics Learning Objective: Compare pressure recovery in conical diffusers of half-angle 5 to 90

z CFD Objective: Observe streamline patterns and flow separation as diffuser half-angle increases; compute pressure recovery for all cases

Flow through a conical diffuser Flow through a conical diffuser

V

x

D1

L1

D2

L2

V Axis x

Pin Pout

Wall Wall

Geometry and dimensions

Computational domain, assuming axisymmetric flow

User Interface for User Interface for FlowLabFlowLab

Graphical display window

Main working window

Overview window

Result table

Display options

Operation options

Flow through a conical diffuserFlow through a conical diffuser

Diffuser section x

Hybrid mesh for the 5o half-angle conical diffuser

Flow through a conical diffuser (continued)Flow through a conical diffuser (continued)

X-Y plot of residuals for the conical diffuser case, = 5o

Flow through a conical diffuserFlow through a conical diffuser

x (a) = 5o

x(b) = 7.5o

x (c) = 10o

x(d) = 12.5o

x (e) = 15o

x(f) = 17.5o

x (g) = 20o

x(h) = 25o

x(i) = 30o

x(j) = 45o

x(k) = 60o

x(l) = 90o

Streamlines through conical diffusers of various half-angles

Flow through a conical diffuserFlow through a conical diffuser P 5 -49.1371

7.5 -47.7787 10 -44.9927

12.5 -42.4013 15 -39.6981

17.5 -37.6431 20 -36.0981 25 -32.7173 30 -29.9919

32.5 -23.2118 35 -21.6434

37.5 -21.0490 45 -19.6571 60 -18.7252 75 -18.1364 90 -18.3018

Pressure difference from inlet to outlet of a conical diffuser as a function of diffuser half-angle

Flow through a conical diffuserFlow through a conical diffuser

-50.0

-40.0

-30.0

-20.0

-10.0

0 50 100 (degrees)

P (Pa)

Pressure difference from inlet to outlet

of a conical diffuser as a function of

diffuser half-angle

Flow through a conical diffuserFlow through a conical diffuser

(a) = 5o

(b) = 30o

(c) = 45o Pressure contours through a conical diffuser of three different half-angles. Colors range from dark blue at

-60 Pa to bright red at 0 Pa gage pressure.

Flow through a conical diffuserFlow through a conical diffuser

(a) = 5o

(b) = 30o

(c) = 45o Contours of turbulent kinetic energy through a conical

diffuser of three different half-angles. Colors range from dark blue at 0 m2/s2 to bright red at 3.5 m2/s2

StillStill--Slide BackSlide Back--Up toUp to Live Demonstration of Live Demonstration of

FlowLabFlowLab templatetemplate Block_meshBlock_mesh

Example: Flow over a rectangular blockExample: Flow over a rectangular block

z Fluid Mechanics Learning Objective: Compare drag coefficient with empirical results

z CFD Objective: Learn to refine a mesh until grid independence is achieved

Introducing CFD in Undergraduate Fluid MechanicsIntroductionOur first attempt to introduce CFD to undergradsengel-Cimbalatextbookengel-Cimbalatextbookengel-Cimbalatextbookengel-CimbalatextbookIntro to CFD using FlowLabWhat is FlowLab?FlowLab TemplatesOriginal Templates for Ed. 1New Templates for Ed. 2Examples: New homework & templates, Ed. 2Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Examples: New homework & templates, Ed. 2Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29Slide Number 30Slide Number 31Slide Number 32Slide Number 33SummarySlide Number 35Slide Number 37Slide Number 38Slide Number 39Slide Number 40Slide Number 41Slide Number 42Slide Number 43Slide Number 44Slide Number 45Slide Number 46Slide Number 47Slide Number 48Slide Number 49Still-Slide Back-Up toLive Demonstration of FlowLab templateDiffuser_angleExample: Flow through a conical diffuserFlow through a conical diffuser User Interface for FlowLabFlow through a conical diffuserFlow through a conical diffuser (continued)Flow through a conical diffuserFlow through a conical diffuserFlow through a conical diffuserFlow through a conical diffuserFlow through a conical diffuserStill-Slide Back-Up toLive Demonstration of FlowLab templateBlock_meshExample: Flow over a rectangular blockSlide Number 63Slide Number 64Slide Number 65Slide Number 66Slide Number 67Slide Number 68Slide Number 69Slide Number 70Slide Number 71Slide Number 72Slide Number 73Slide Number 74Slide Number 75Slide Number 76Slide Number 77Slide Number 78