30079_40.pdf

CHAPTER 40

FLUID MECHANICS

Reuben M. Olson

College of Engineering and Technology

Ohio University

Athens, Ohio

40.1 DEFINITION OF A FLUID 12 90

40.9.1 Laminar and Turbulent

Flow 13 07

40.9.2 Boundary Layers 13 07

40.2 IMPORTANT FLUID

PROPERTIES

12 90

40.10 GAS DYNAMICS 13 10

40.10.1 Adiabatic and Isentropic

Flow 13 10

40.10.2 Duct Flow 13 11

40.10.3 Normal Shocks 13 11

40.10.4 Oblique Shocks 13 13

40.3 FLUID STATICS

12 90

40.3.1 Manometers

12 91

40.3.2 Liquid Forces on

Submerged Surfaces 12 91

40.3.3 Aerostatics 12 93

40.3.4 Static Stability 12 93

40.11 VISCOUS FLUID FLOW IN

DUCTS

40.4 FLUID KINEMATICS 12 94

40.4.1 Velocity and Acceleration 1295

40.4.2 Streamlines 12 95

40.4.3 Deformation of a Fluid

Element 12 95

40.4.4 Vorticity and Circulation 1297

40.4.5 Continuity Equations 12 98

13 13

40.11.1 Fully Developed

Incompressible Flow 1315

40.11.2 Fully Developed

Laminar Flow in Ducts 1315

40.11.3 Fully Developed

Turbulent Flow in

Ducts 13 16

40. 1 1 .4 Steady Incompressible

Flow in Entrances of

Ducts

40.5 FLUID MOMENTUM 12 98

40.5 . 1 The Momentum Theorem 1 299

40.5.2 Equations of Motion 1300

13 19

40.11.5 Local Losses in

Contractions,

Expansions, and Pipe

Fittings; Turbulent

Flow 13 19

40. 11.6 Flow of Compressible

Gases in Pipes with

Friction

40.6 FLUID ENERGY 13 01

40.6.1 Energy Equations 13 01

40.6.2 Work and Power 13 02

40.6.3 Viscous Dissipation 13 02

40.7 CONTRACTION

COEFFICIENTS FROM

POTENTIAL

FLOW THEORY

13 20

13 03

40.12 DYNAMIC DRAG AND LIFT 1323

40.12.1 Drag

13 23

40.12.2 Lift

13 23

40.8 DIMENSIONLESS NUMBERS

AND DYNAMIC SIMILARITY 1304

40.8.1 Dimensionless Numbers 1304

40.8.2 Dynamic Similitude 13 05

40.13 FLOW MEASUREMENTS 13 24

40. 1 3. 1 Pressure Measurement s 1 324

40.13.2 Velocity Measurements 1325

40.13.3 Volumetric and Mass

Flow Fluid

Measurements 13 26

40.9 VISCOUS FLOW AND

INCOMPRESSIBLE

BOUNDARY LAYERS 13 07

All figures and tables produced, with permission, from Essentials of Engineering Fluid Mechanics,

Mechanical Engineers' Handbook, 2nd ed., Edited by Myer Kutz.

40.1 DEFINITION OF A FLUID

A solid generally has a definite shape; a fluid has a shape determined by its container. Fluids include

liquids, gases, and vapors, or mixtures of these. A fluid continuously deforms when shear stresses

are present; it cannot sustain shear stresses at rest. This is characteristic of all real fluids, which are

viscous. Ideal fluids are nonviscous (and nonexistent), but have been studied in great detail because

in many instances viscous effects in real fluids are very small and the fluid acts essentially as a

nonviscous fluid. Shear stresses are set up as a result of relative motion between a fluid and its

boundaries or between adjacent layers of fluid.

40.2 IMPORTANT FLUID PROPERTIES

Density p and surface tension or are the most important fluid properties for liquids at rest. Density

and viscosity JJL are significant for all fluids in motion; surface tension and vapor pressure are sig-

nificant for cavitating liquids; and bulk elastic modulus K is significant for compressible gases at

high subsonic, sonic, and supersonic speeds.

Sonic speed in fluids is c = VtfVp. Thus, for water at 15°C, c = V2.18 X 109/999 = 1480 m/

sec. For a mixture of a liquid and gas bubbles at nonresonant frequencies, cm = VATm/pw, where m

refers to the mixture. This becomes

c - / P*K> ^H

m V № + (1 - x)pg][xp8 + (1 - JC)P/]

where the subscript / is for the liquid phase and g is for the gas phase. Thus, for water at 20°C

containing 0.1% gas nuclei by volume at atmospheric pressure, cm = 312 m/sec. For a gas or a

mixture of gases (such as air), c = VkRT, where k = cp/cv, R is the gas constant, and T is the

absolute temperature. For air at 15°C, c = V(1.4)(287.1)(288) = 340 m/sec. This sonic property is

thus a combination of two properties, density and elastic modulus.

Kinematic viscosity is the ratio of dynamic viscosity and density. In a Newtonian fluid, simple

laminar flow in a direction x at a speed of w, the shearing stress parallel to x is TL — jji(du/dy) =

pv(du/dy), the product of dynamic viscosity and velocity gradient. In the more general case, TL —

IJi(du/dy + dv/dx) when there is also a y component of velocity v. In turbulent flows the shear stress

resulting from lateral mixing is TT = —pu'v', a Reynolds stress, where u' and v' are instantaneous

and simultaneous departures from mean values u and iJ. This is also written as TT = pe(du/dy), where

e is called the turbulent eddy viscosity or diffusivity, an indirectly measurable flow parameter and

not a fluid property. The eddy viscosity may be orders of magnitude larger than the kinematic

viscosity. The total shear stress in a turbulent flow is the sum of that from laminar and from turbulent

motion: T = TL + TT = p(v + e)du/dy after Boussinesq.

40.3 FLUID STATICS

The differential equation relating pressure changes dp with elevation changes dz (positive upward

parallel to gravity) is dp = -pg dz. For a constant-density liquid, this integrates to p2 ~ P\ = ~pg

(z2 - Zi) or A/? = y/z, where y is in N/m3 and h is in m. Also (pi/y) + Zi = (p2/7) + Z2; a constant

piezometric head exists in a homogeneous liquid at rest, and sincep1/y — p2ly = z2 ~ Zi, a change

in pressure head equals the change in potential head. Thus, horizontal planes are at constant pressure

when body forces due to gravity act. If body forces are due to uniform linear accelerations or to

centrifugal effects in rigid-body rotations, points equidistant below the free liquid surface are all at

the same pressure. Dashed lines in Figs. 40.1 and 40.2 are lines of constant pressure.

Pressure differences are the same whether all pressures are expressed as gage pressure or as

absolute pressure.

Fig. 40.1 Constant linear acceleration. Fig. 40.2 Constant centrifugal acceleration.

Fig. 40.3 Barometer.

Fig. 40.4 Open manometer.

40.3.1 Manometers

Pressure differences measured by barometers and manometers may be determined from the relation

Ap = yh. In a barometer, Fig. 40.3, hb — (pa - pv)/yb m.

An open manometer, Fig. 40.4, indicates the inlet pressure for a pump by pinlet = -ymhm — yy

Pa gage. A differential manometer, Fig. 40.5, indicates the pressure drop across an orifice, for ex-

ample, by pl - p2 = hm(ym - y0) Pa.

Manometers shown in Figs. 40.3 and 40.4 are a type used to measure medium or large pressure

differences with relatively small manometer deflections. Micromanometers can be designed to pro-

duce relatively large manometer deflections for very small pressure differences. The relation Ap =

ykh may be applied to the many commercial instruments available to obtain pressure differences

from the manometer deflections.

40.3.2 Liquid Forces on Submerged Surfaces

The liquid force on any flat surface submerged in the liquid equals the product of the gage pressure

at the centroid of the surface and the surface area, or F = pA. The force F is not applied at the

centroid for an inclined surface, but is always below it by an amount that diminishes with depth.

Measured parallel to the inclined surface, y is the distance from 0 in Fig. 40.6 to the centroid and

yF = y + ICG/Ay, where ICG is the moment of inertia of the flat surface with respect to its centroid.

Values for some surfaces are listed in Table 40.1.

For curved surfaces, the horizontal component of the force is equal in magnitude and point of

application to the force on a projection of the curved surface on a vertical plane, determined as above.

The vertical component of force equals the weight of liquid above the curved surface and is applied

at the centroid of this liquid, as in Fig. 40.7. The liquid forces on opposite sides of a submerged

surface are equal in magnitude but opposite in direction. These statements for curved surfaces are

also valid for flat surfaces.

Buoyancy is the resultant of the surface forces on a submerged body and equals the weight of

fluid (liquid or gas) displaced.

Fig. 40.5 Differential manometer.

Fig. 40.6 Flat inclined surface submerged in

a liquid.

Table 40.1 Moments of Inertia for Various Plane Surfaces about Their Center of

Gravity

Surface

ICG

Rectangle or square

Triangle

Quadrant of circle

(or semicircle)

(j-^)^ = a06" Af2

Quadrant of ellipse

(or semiellipse)

(^)^=0.0699^

Parabola

/3 9\

I-- — U/i2 = 0.0686/l/i2

Circle

Ellipse

*»

Fig. 40.7 Curved surfaces submerged in a liquid.

40.3.3 Aerostatics

The U.S. standard atmosphere is considered to be dry air and to be a perfect gas. It is defined in

terms of the temperature variation with altitude (Fig. 40.8), and consists of isothermal regions and

polytropic regions in which the polytropic exponent n depends on the lapse rate (temperature

gradient).

Conditions at an upper altitude z2 and at a lower one zl in an isothermal atmosphere are obtained

by integrating the expression dp — -pg dz to get

P2 -gfe ~ fr)

P^^^T-

In a polytropic atmosphere where plpl = (p/p^",

?! = \i _ ("- *) fe ~ fr)]"'01"1*

Pi L 8 n RT, \

from which the lapse rate is (T2 — Tl)/(z2 — Zi) — —g(n — l)/nR and thus n is obtained from

l/n = 1 + (R/g)(dt/dz). Defining properties of the U.S. standard atmosphere are listed in Table

40.2. The U.S. standard atmosphere is used in measuring altitudes with altimeters (pressure gages) and,

because the altimeters themselves do not account for variations in the air temperature beneath an

aircraft, they read too high in cold weather and too low in warm weather.

40.3.4 Static Stability

For the atmosphere at rest, if an air mass moves very slowly vertically and remains there, the

atmosphere is neutral. If vertical motion continues, it is unstable; if the air mass moves to return to

its initial position, it is stable. It can be shown that atmospheric stability may be defined in terms of

the polytropic exponent. If n < k, the atmosphere is stable (see Table 40.2); if n = k, it is neutral

(adiabatic); and if n > k, it is unstable.

The stability of a body submerged in a fluid at rest depends on its response to forces which tend

to tip it. If it returns to its original position, it is stable; if it continues to tip, it is unstable; and if it

remains at rest in its tipped position, it is neutral. In Fig. 40.9 G is the center of gravity and B is

the center of buoyancy. If the body in (a) is tipped to the position in (b), a couple Wd restores the

body toward position (a) and thus the body is stable. If B were below G and the body displaced, it

would move until B becomes above G. Thus stability requires that G is below B.

Fig. 40.8 U.S. standard atmosphere.

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