Munson B.R. Fundamentals of Fluid Mechanics

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A.5 Basic Representative Examples 707

For most CFD problems the governing equations to be solved are partial differential equa-

tions [rather than an ordinary differential equation as in the above example (Eq. A.1)] and the

finite difference method becomes considerably more involved. The following example illustrates

some of the concepts involved.

(7)

We cannot use Eq. 7 for since it would involve the non-

existing Rather we use the initial condition 1Eq. 42, which gives

The result is the following set of N algebraic equations for the N ap-

proximate values of hat times

For most problems the corresponding equations would be more

complicated than those just given, and a computer would be used to

solve for the For this problem the solution is simply

. . .

⫽ H

Ⲑ

11 ⫹ ¢t2

⫽ H

Ⲑ

11 ⫹ ¢t2

⫽ h

N⫺1

Ⲑ

11 ⫹ ¢t2

. . .

⫽ h

Ⲑ

11 ⫹ ¢t2

⫽ h

Ⲑ

11 ⫹ ¢t2

⫽ H

1N ⫺ 12¢t.

⫽ ¢t, . . . , t

⫽t

⫽ 0, t

⫽ H

i ⫽ 1

⫽

i⫺1

11 ⫹ ¢t2

or in general

The results for are shown in Fig. EA.1c. Tabulated

values of the depth for are listed in the table below.t ⫽ 1

0 6 t 6 1

⫽ H

Ⲑ

11 ⫹ ¢t2

i⫺1

It is seen that the approximate results compare quite favorably

with the exact solution given by Eq. 5. It is expected that the finite

difference results would more closely approximate the exact re-

sults as is decreased since in the limit of the finite dif-

ference approximation for the derivatives 1Eq. 62approaches the

actual definition of the derivative.

¢t S 0¢t

i for

0.2 6 0.4019H

0.1 11 0.3855H

0.01 101 0.3697H

0.001 1001 0.3681H

Exact 1Eq. 52 — 0.3678H

for t ⴝ 1t ⴝ 1⌬t

Consider steady, incompressible flow of an inviscid fluid past a

circular cylinder as shown in Fig. EA.2a. The stream function,

for this flow is governed by the Laplace equation 1see Section 6.52

(1)

⫹

⫽ 0

The exact analytical solution is given in Section 6.6.3.

Describe a simple finite difference technique that can be used

to solve this problem.

OLUTION

Flow Past a Cylinder

artificial, uniform flow conditions at a location where the actual

flow is not uniform. If these boundaries are farther than neces-

sary from the object, the flow domain will be larger than neces-

sary and excessive computer time and storage will be required.

Experience in solving such problems is invaluable!

Once the flow domain has been selected, an appropriate grid is

imposed on this domain 1see Fig. EA.2b2. Various grid structures

can be used. If the grid is too coarse, the numerical solution may

not be capable of capturing the fine scale structure of the actual

flow field. If the grid is too fine, excessive computer time and

XAMPLE A.2

The first step is to define a flow domain and set up an appropri-

ate grid for the finite difference scheme. Since we expect the

flow field to be symmetrical both above and below and in front

of and behind the cylinder, we consider only one-quarter of the

entire flow domain as indicated in Fig. EA.2b. We locate the up-

per boundary and right-hand boundary far enough from the

cylinder so that we expect the flow to be essentially uniform at

these locations. It is not always clear how far from the object

these boundaries must be located. If they are not far enough, the

solution obtained will be incorrect because we have imposed

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708 Appendix A ■ Computational Fluid Dynamics and FlowLab

storage may be required. Considerable work has gone into form-

ing appropriate grids 1Ref. 62. We consider a grid that is uniformly

spaced in the x and y directions, as shown in Fig. EA.2b.

As shown in Eq. 6.112, the exact solution to Eq. 1 1in terms

of polar coordinates r, rather than Cartesian coordinates x, y2

is The finite difference solution ap-

proximates these stream function values at a discrete 1finite2

number of locations 1the grid points2as where the i and j in-

dices refer to the corresponding and locations.

The derivatives of can be approximated as follows:

and

This particular approximation is called a forward-difference ap-

proximation. Other approximations are possible. By similar rea-

soning, it is possible to show that the second derivatives of can

be written as follows:

(2)

and

(3)

⬇

1¢y2

i, j⫹1

⫺ 2c

i, j

⫹ c

i, j⫺1

⬇

1¢x2

i⫹1, j

⫺ 2c

i, j

⫹ c

i⫺1, j

⬇

¢y

i, j⫹1

⫺ c

i, j

⬇

¢x

i⫹1, j

⫺ c

i, j

c ⫽ Ur11 ⫺ a

Ⲑ

2 sin u.

Thus, by combining Eqs. 1, 2, and 3 we obtain

(4)

Equation 4 can be solved for the stream function at and to give

(5)

Note that the value of depends on the values of the stream

function at neighboring grid points on either side and above and

below the point of interest 1see Eq. 5 and Fig. EA. 2c2.

To solve the problem 1either exactly or by the finite difference

technique2it is necessary to specify boundary conditions for

points located on the boundary of the flow domain 1see Section

6.6.32. For example, we may specify that on the lower

boundary of the domain 1see Fig. EA.2b2and a constant,

on the upper boundary of the domain. Appropriate boundary con-

ditions on the two vertical ends of the flow domain can also be

specified. Thus, for points interior to the boundary Eq. 5 is valid;

similar equations or specified values of are valid for boundary

points. The result is an equal number of equations and unknowns,

one for every grid point. For this problem, these equations

represent a set of linear algebraic equations for the solutionc

i, j

c ⫽ C,

c ⫽ 0

i, j

⫹ 1¢x2

i, j⫹1

⫹ c

i, j⫺1

i, j

⫽

231¢x2

⫹ 1¢y2

31¢y2

i⫹1, j

⫹ c

i⫺1, j

⫹ c

i, j⫺1

2⫺ 2 a

1¢x2

⫹

1¢y2

b c

i, j

⫽ 0

⫹

⬇

1¢x2

i⫹1, j

⫹ c

i⫺1, j

2⫹

1¢y2

i, j⫹1

i, j + 1

i, j

– 1

i – 1, j i + 1, j

ψψ

(a)

(b)

(c)

F I G U R E EA.2

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A.6 Methodology

In general, most applications of CFD take the same basic approach. Some of the differences

include problem complexity, available computer resources, available expertise in CFD, and

whether a commercially available CFD package is used, or a problem-specific CFD algorithm is

developed. In today’s market, there are many commercial CFD codes available to solve a wide

variety of problems. However, if the intent is to conduct a thorough investigation of a specific

fluid flow problem such as in a research environment, it is possible that taking the time to develop

a problem-specific algorithm may be most efficient in the long run. The features common to most

CFD applications can be summarized in the flow chart shown in Fig. A.6. A complete, detailed

CFD solution for a viscous flow obtained by using the steps summarized in the flow chart can

be accessed from the book’s website at www.wiley.com/college/munson.

A.6 Methodology 709

The preceding two examples are rather simple because the governing equations are not too

complex. A finite difference solution of the more complicated, nonlinear Navier–Stokes equation

(Eq. 6.127) requires considerably more effort and insight and larger and faster computers. A typ-

ical finite difference grid for a more complex flow, the flow past a turbine blade, is shown in Fig.

A.5. Note that the mesh is much finer in regions where large gradients are to be expected (i.e.,

near the leading and trailing edges of the blade) and more coarse away from the blade.

of which provides the finite difference approximation for the

stream function at discrete grid points in the flow field. Stream-

lines 1lines of constant 2can be obtained by interpolating values

of between the grid points and “connecting the dots” of

The velocity field can be obtained from the deriva-

tives of the stream function according to Eq. 6.74. That is,

u ⫽

⬇

¢y

i, j⫹1

⫺ c

i, j

c ⫽ constant.

i, j

and

Further details of the finite difference technique can be found in

standard references on the topic 1Refs. 5, 7, 82. Also, see the com-

pletely solved viscous flow CFD problem in Section A6.

v ⫽⫺

⬇ ⫺

¢x

i⫹1, j

⫺ c

i, j

F I G U R E A.5 Finite difference

grid for flow past a turbine blade. (From Ref. 9,

used by permission.)

CFD Methodology

Problem

Models

Governing

Equations

Assumptions &

Simplifications

Algorithm

Development

Run Simulation

Steady/

Unsteady

Convergence

Geometry

Special

Requirements

Structured or

Unstructured

Discretization

Method

Implicit or

Explicit

Accuracy

Verification

& Validation

Visualize

Flow Field

Postprocess

Values

Interpret

Results

Physics Discretize Solve AnalyzeGrid

F I G U R E A.6 Flow chart of general CFD methodology.

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710 Appendix A ■ Computational Fluid Dynamics and FlowLab

The Algorithm Development box is grayed because this step is required only when devel-

oping your own CFD code. When using a commercial CFD code, this step is not necessary. This

chart represents a generalized methodology to CFD. There are other more complex components

that are hidden in the above steps, which are beyond the scope of a brief introduction to CFD.

A.7 Application of CFD

In the early stages of CFD, research and development was primarily driven by the aerospace

industry. Today, CFD is still used as a research tool, but it also has found a place in industry as

a design tool. There is now a wide variety of industries that make at least some use of CFD,

including automotive, industrial, HVAC, naval, civil, chemical, biological, and others. Industries

are using CFD as an added engineering tool that complements the experimental and theoretical

work in fluid dynamics.

A.7.1 Advantages of CFD

There are many advantages to using CFD for simulation of fluid flow. One of the most important

advantages is the realizable savings in time and cost for engineering design. In the past, coming up

with a new engineering design meant somewhat of a trial-and-error method of building and testing

multiple prototypes prior to finalizing the design. With CFD, many of the issues dealing with fluid

flow can be flushed out prior to building the actual prototype. This translates to a significant sav-

ings in time and cost. It should be noted that CFD is not meant to replace experimental testing, but

rather to work in conjunction with it. Experimental testing will always be a necessary component

of engineering design. Other advantages include the ability of CFD to: (1) obtain flow information

in regions that would be difficult to test experimentally, (2) simulate real flow conditions, (3) con-

duct large parametric tests on new designs in a shorter time, and (4) enhance visualization of com-

plex flow phenomena.

A good example of the advantages of CFD is shown in Figure A.7. Researchers use a type of

CFD approach called “large-eddy simulation” or LES to simulate the fluid dynamics of a tornado as

it encounters a debris field and begins to pick up sand-sized particles. A full animation of this tor-

nado simulation can be accessed by visiting the book website. The motivation for this work is to

investigate whether there are significant differences in the fluid mechanics when debris particles are

present. Historically it has been difficult to get comprehensive experimental data throughout a tor-

nado so CFD is helping to shine some light on the complex fluid dynamics involved in such a flow.

A.7.2 Difficulties in CFD

One of the key points that a beginning CFD student should understand is that one cannot treat the

computer as a “magic black box” when performing flow simulations. It is quite possible to obtain a

fully converged solution for the CFD simulation, but this is no guarantee that the results are physi-

cally correct. This is why it is important to have a good understanding of the flow physics and how

they are modeled. Any numerical technique (including those discussed above), no matter how sim-

ple in concept, contains many hidden subtleties and potential problems. For example, it may seem

reasonable that a finer grid would ensure a more accurate numerical solution. While this may be true

(as Example A.1), it is not always so straightforward; a variety of stability or convergence problems

may occur. In such cases the numerical “solution” obtained may exhibit unreasonable oscillations or

the numerical result may “diverge” to an unreasonable (and incorrect) result. Other problems that

may arise include (but are not limited to): (1) difficulties in dealing with the nonlinear terms of the

Navier–Stokes equations, (2) difficulties in modeling or capturing turbulent flows, (3) convergence

issues, (4) difficulties in obtaining a quality grid for complex geometries, and (5) managing resources,

both time and computational, for complex problems such as unsteady three-dimensional flows.

A.7.3 Verification and Validation

Verification and validation of the simulation are critical steps in the CFD process. This is a neces-

sary requirement for CFD, particularly since it is possible to have a converged solution that is non-

physical. Figure A.8 shows the streamlines for viscous flow past a circular cylinder at a given instant

VA.3 Tornado

simulation

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A.7 Application of CFD 711

after it was impulsively started from rest. The lower half of the figure represents the results of a finite

difference calculation; the upper half of the figure represents the photograph from an experiment of

the same flow situation. It is clear that the numerical and experimental results agree quite well. For

any CFD simulation, there are several levels of testing that need to be accomplished before one can

have confidence in the solution. The most important verification to be performed is grid convergence

testing. In its simplest form, it consists of proving that further refinement of the grid (i.e., increasing

the number of grid points) does not alter the final solution. When this has been achieved, you have a

grid-independent solution. Other verification factors that need to be investigated include the suitability

F I G U R E A.7 Results from a large-eddy simulation showing the visual appear-

ance of the debris and funnel cloud from a simulated medium swirl F3-F4 tornado. The fun-

nel cloud is translating at 15 m/s and is ingesting 1-mm-diameter “sand” from the surface as

it encounters a debris field. Please visit the book website to access a full animation of this

tornado simulation. (Photographs and animation courtesy of Dr. David Lewellen, Ref. 10, and

Paul Lewellen, West Virginia University.)

F I G U R E A.8 Streamlines for flow past

a circular cylinder at a short time after the flow was

impulsively started. The upper half is a photograph

from a flow visualization experiment. The lower half is

from a finite difference calculation. (See the photo-

graph at the beginning of Chapter 9.) (From Ref. 9,

used by permission.)

JWCL068_AppA_701-713.qxd 9/23/08 12:07 PM Page 711

712 Appendix A ■ Computational Fluid Dynamics and FlowLab

of the convergence criterion, whether the time step is adequate for the time scale of the problem, and

comparison of CFD solutions to existing data, at least for baseline cases. Even when using a com-

mercial CFD code that has been validated on many problems in the past, the CFD practitioner still

needs to verify the results through such measures as grid-dependence testing.

A.7.4 Summary

In CFD, there are many different numerical schemes, grid techniques, etc. They all have their

advantages and disadvantages. A great deal of care must be used in obtaining approximate numer-

ical solutions to the governing equations of fluid motion. The process is not as simple as the

often-heard “just let the computer do it.” Remember that CFD is a tool and as such needs to be

used appropriately to produce meaningful results. The general field of computational fluid dynam-

ics, in which computers and numerical analysis are combined to solve fluid flow problems, rep-

resents an extremely important subject area in advanced fluid mechanics. Considerable progress

has been made in the past relatively few years, but much remains to be done. The reader is encour-

aged to consult some of the available literature.

A.8 FlowLab

The authors of this textbook are working in collaboration with Fluent, Inc., the largest provider

of commercial CFD software (www.fluent.com), to offer students the opportunity to use a new

CFD tool called FlowLab. FlowLab is designed to be a virtual fluids laboratory to help enhance

the educational experience in fluids courses. It uses computational fluid dynamics to help the stu-

dent grasp various concepts in fluid dynamics and introduces the student to the use of CFD in

solving fluid flow problems. Go to the book’s website at www.wiley.com/college/munson to

access FlowLab resources for this textbook.

The motivation behind incorporating FlowLab with a fundamental fluid mechanics textbook is

twofold: (1) expose the student to computational fluid dynamics and (2) offer a mechanism for stu-

dents to conduct experiments in fluid dynamics, numerically in this case. This educational software

allows students to reinforce basic concepts covered in class, conduct parametric studies to gain a bet-

ter understanding of the interaction between geometry, fluid properties, and flow conditions, and pro-

vides the student a visualization tool for various flow phenomena.

One of the strengths of FlowLab is the ease-of-use. The CFD simulations are based on pre-

viously developed templates which allow the user to start using CFD to solve flow problems with-

out requiring an extensive background in the subject. FlowLab provides the student the opportu-

nity to focus on the results of the simulation rather than the development of the simulation. Typical

results showing the developing velocity profile in the entrance region of a pipe are shown in the

solution window of Fig. A.9.

Axial Velocity (m/s)

0.0442

0.0395

0.0347

0.03

0.0253

0.0205

0.0158

0.0111

0.00631

0.00157

Axial Velocity

inlet

Legend

x = 0.5d

x = 1d

x = 5d

x = 10d

x = 25d

outlet

Position (n)

0.10

F I G U R E A.2 Entrance

flow in a pipe. Velocity profiles as a

function of radial position for various

locations along the pipe length.

JWCL068_AppA_701-713.qxd 9/23/08 12:07 PM Page 712

References 713

Problems have been developed that take advantage of the FlowLab capability of this text-

book. Go to the book’s website, www.wiley.com/college/munson, to access these problems (con-

tained in Chapters 7, 8, and 9) as well as a basic tutorial on using FlowLab. The course instructor

can provide information on accessing the FlowLab software. The book’s website also has a brief

example using FlowLab.

References

1. Baker, A. J., Finite Element Computational Fluid Mechanics, McGraw-Hill, New York, 1983.

2. Carey, G. F., and Oden, J. T., Finite Elements: Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J.,

1986.

3. Brebbia, C. A., and Dominguez, J., Boundary Elements: An Introductory Course, McGraw-Hill, New

York, 1989.

4. Moran, J., An Introduction to Theoretical and Computational Aerodynamics, Wiley, New York, 1984.

5. Anderson, J.D., Computational Fluid Dynamics: The Basics with Applications, McGraw-Hill, New

York, 1995.

6. Thompson, J. F., Warsi, Z. U. A., and Mastin, C. W., Numerical Grid Generation: Foundations and

Applications, North-Holland, New York, 1985.

7. Peyret, R., and Taylor, T. D., Computational Methods for Fluid Flow, Springer-Verlag, New York,

1983.

8. Tannehill, J. C., Anderson, D. A., and Pletcher, R. H., Computational Fluid Mechanics and Heat

Transfer, 2nd Ed., Taylor and Francis, Washington, D.C., 1997.

9. Hall, E. J., and Pletcher, R. H., Simulation of Time Dependent, Compressible Viscous Flow Using Cen-

tral and Upwind-Biased Finite-Difference Techniques, Technical Report HTL-52, CFD-22, College of

Engineering, lowa State University, 1990.

10. Lewellen, D. C., Gong, B., and Lewellen, W. S., Effects of Debris on Near-Surface Tornado Dynamics,

22nd Conference on Severe Local Storms, Paper 15.5, American Meteorological Society, 2004.

JWCL068_AppA_701-713.qxd 9/23/08 12:07 PM Page 713

ppendix B

Physical Properties of Fluids

Methane

Carbon dioxide

Hydrogen

Helium

Heptane

Octane

Water

Carbon tetrachloride

Mercury

Kerosene

SAE 10W oil

SAE 10W-30 oil

SAE 30W oil

Castor oil

Glycerin

–20 0 20 40 60 80 100 120

Temperature, °C

4.0

2.0

1.0

1 × 10

–1

1 × 10

–2

1 × 10

–3

1 × 10

–4

1 × 10

–5

, Dynamic viscosity, N

•

s/m

Air

F I G U R E B.1 Dynamic (absolute) viscosity of common fluids as a function

of temperature. To convert to BG units of multiply by

(Curves from R. W. Fox and A. T. McDonald, Introduction to Fluid Mechanics, 3rd Ed.,

Wiley, New York, 1985. Used by permission.)

2.089 ⴛ 10

ⴚ2

ⴢ

Ⲑ

ⴢ

Ⲑ

714

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Appendix B ■ Physical Properties of Fluids 715

Carbon tetrachloride

Mercury

Octane

Water

Kerosene

Air

Carbon dioxide

Methane

Hydrogen

Helium

SAE 10W oil

SAE 10W–30 oil

SAE 30W oil

Glycerin

1 × 10

–2

1 × 10

–3

1 × 10

–4

1 × 10

–5

1 × 10

–6

1 × 10

–7

, Kinematic viscosity, m

–20 0 20 40 60 80 100 120

Temperature, °C

Heptane

F I G U R E B.2 Kinematic viscosity of common fluids (at atmospheric pres-

sure) as a function of temperature. To convert to BG units of multiply by 10.76.

(Curves from R. W. Fox and A. T. McDonald, Introduction to Fluid Mechanics, 3rd Ed.,

Wiley, New York, 1985. Used by permission.)

Ⲑ

sft

Ⲑ

JWCL068_AppB_714-718.qxd 9/23/08 12:07 PM Page 715

716 Appendix B ■ Physical Properties of Fluids

TABLE B.1

Physical Properties of Water ( )

Specific Dynamic Kinematic Surface Vapor Speed of

Density, Viscosity, Viscosity, Pressure,

Temperature c

( ) ( ) ( ) ( ) ( ) (lb兾ft) [ (abs)] (ft兾s)

32 1.940 62.42 4603

40 1.940 62.43 4672

50 1.940 62.41 4748

60 1.938 62.37 4814

70 1.936 62.30 4871

80 1.934 62.22 4819

90 1.931 62.11 4960

100 1.927 62.00 4995

120 1.918 61.71 5049

140 1.908 61.38 5091

160 1.896 61.00 5101

180 1.883 60.58 5195

200 1.869 60.12 5089

212 1.860 59.83 50621.469 E ⫹ 14.04 E ⫺ 33.165 E ⫺ 65.886 E ⫺ 6

1.152 E ⫹ 14.12 E ⫺ 33.393 E ⫺ 66.342 E ⫺ 6

7.507 E ⫹ 04.26 E ⫺ 33.827 E ⫺ 67.207 E ⫺ 6

4.736 E ⫹ 04.40 E ⫺ 34.385 E ⫺ 68.315 E ⫺ 6

2.888 E ⫹ 04.53 E ⫺ 35.106 E ⫺ 69.743 E ⫺ 6

1.692 E ⫹ 04.67 E ⫺ 36.067 E ⫺ 61.164 E ⫺ 5

9.493 E ⫺ 14.79 E ⫺ 37.383 E ⫺ 61.423 E ⫺ 5

6.979 E ⫺ 14.86 E ⫺ 38.233 E ⫺ 61.500 E ⫺ 5

5.069 E ⫺ 14.91 E ⫺ 39.262 E ⫺ 61.791 E ⫺ 5

3.631 E ⫺ 14.97 E ⫺ 31.052 E ⫺ 52.037 E ⫺ 5

2.563 E ⫺ 15.03 E ⫺ 31.210 E ⫺ 52.344 E ⫺ 5

1.781 E ⫺ 15.09 E ⫺ 31.407 E ⫺ 52.730 E ⫺ 5

1.217 E ⫺ 15.13 E ⫺ 31.664 E ⫺ 53.228 E ⫺ 5

8.854 E ⫺ 25.18 E ⫺ 31.924 E ⫺ 53.732 E ⫺ 5

Ⲑ

.ft

Ⲑ

slbⴢs

Ⲑ

slugs

Ⲑ

ⴗF

SNMGR

Sound

,Tension

,Weight

BG Units

Based on data from Handbook of Chemistry and Physics, 69th Ed., CRC Press, 1988. Where necessary, values obtained by interpolation.

Density and specific weight are related through the equation For this table,

In contact with air.

From R. D. Blevins, Applied Fluid Dynamics Handbook, Van Nostrand Reinhold Co., Inc., New York, 1984.

g ⫽ 32.174 ft

Ⲑ

.g ⫽ rg.

TABLE B.2

Physical Properties of Water ( )

Specific Dynamic Kinematic Surface Vapor Speed of

Density, Viscosity, Viscosity, Pressure,

Temperature c

() (

)

(

)

(

)

(

)

(N兾m) [

(abs)]

(m兾s)

0 999.9 9.806 1403

5 1000.0 9.807 1427

10 999.7 9.804 1447

20 998.2 9.789 1481

30 995.7 9.765 1507

40 992.2 9.731 1526

50 988.1 9.690 1541

60 983.2 9.642 1552

70 977.8 9.589 1555

80 971.8 9.530 1555

90 965.3 9.467 1550

100 958.4 9.399 15431.013 E ⫹ 55.89 E ⫺ 22.940 E ⫺ 72.818 E ⫺ 4

7.010 E ⫹ 46.08 E ⫺ 23.260 E ⫺ 73.147 E ⫺ 4

4.734 E ⫹ 46.26 E ⫺ 23.650 E ⫺ 73.547 E ⫺ 4

3.116 E ⫹ 46.44 E ⫺ 24.134 E ⫺ 74.042 E ⫺ 4

1.992 E ⫹ 46.62 E ⫺ 24.745 E ⫺ 74.665 E ⫺ 4

1.233 E ⫹ 46.79 E ⫺ 25.534 E ⫺ 75.468 E ⫺ 4

7.376 E ⫹ 36.96 E ⫺ 26.580 E ⫺ 76.529 E ⫺ 4

4.243 E ⫹ 37.12 E ⫺ 28.009 E ⫺ 77.975 E ⫺ 4

2.338 E ⫹ 37.28 E ⫺ 21.004 E ⫺ 61.002 E ⫺ 3

1.228 E ⫹ 37.42 E ⫺ 21.307 E ⫺ 61.307 E ⫺ 3

8.722 E ⫹ 27.49 E ⫺ 21.519 E ⫺ 61.519 E ⫺ 3

6.105 E ⫹ 27.56 E ⫺ 21.787 E ⫺ 61.787 E ⫺ 3

Ⲑ

sNⴢs

Ⲑ

ⴗC

SNMGR

Sound

,Tension

,Weight

SI Units

Based on data from Handbook of Chemistry and Physics, 69th Ed., CRC Press, 1988.

Density and specific weight are related through the equation For this table,

In contact with air.

From R. D. Blevins, Applied Fluid Dynamics Handbook, Van Nostrand Reinhold Co., Inc., New York, 1984.

g ⫽ 9.807 m

Ⲑ

.g ⫽ rg.

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