Modeling of Fluid Motions

199

MODELING OF FLUID MOTIONS Dimensional analysis and similarity theory have long been used to study hydraulics and fluid flow in pipes and channels1. Many parts of fluid dynamics are still too complicated to analyze accurately by mathematics alone. Thus, through the years, a lot of effort has been devoted to developing similarity methods for fluid dynamics, hydraulics, aerodynamics, and naval hydrodynamics. Certain dimensionless groups, which express ratios of important physical forces or effects, occur so commonly in these methods that they have been given standard names. For example, the ratio of inertial forces in the flow to friction (i.e., viscous) forces is called the Reynolds number, and the ratio of inertial forces to gravitational forces in a flow that has a free surface is called the Froude number. Compilations of over 200 such fluid-flow dimensionless groups are available for guidance2. Forming the governing pi-terms in a specific fluid flow study can be simplified by first listing the important kinds of forces that are thought to govern the flow. Representative or typical lengths, velocities, and fluid properties are used to define the forces. Some common forces, along with the combination of velocities, lengths, and fluid properties that characterize them, are listed in Table 9.1, as are the definition of the parameters and their dimensions. Once the important forces have been identified, the pi-terms that express the flow physics can be found by forming ratios of the forces. For example, in the flow of air around a wing, one might expect that aerodynamic friction or drag on the wing will be important. Since the flow has to accelerate around the wing, inertia forces will also be important. From Table 9.1, inertia and viscous forces are therefore selected as

TABLE 9.1 Common "Forces" in Fluid Dynamics Type of Force Viscous Pressure Inertia Gravity Surface Compressibility ame ter g L Kb Ap V

μ P

σ

Characterization

Symbols Definition acceleration of gravity reference length bulk modulus pressure difference velocity dynamic viscosity density surface tension

\LVL

ApL·2 pV2L2 gL3 CL

KbL2 Dimensions LT'2 L ML'XT'2 MLXT'2 LT'1 MLlTl ML3 MT~2

200

Ί D

Figure 9.1 Incompressible Fluid Flow in a Circular Pipe important forces. (Note that the viscous force depends on fluid viscosity and the inertia force depends on fluid density. In fact, each force introduces another fluid property.) The ratio of these two forces gives pV2L2^VL = pVL/μ, defined as the Reynolds number, NRe. The Reynolds number is an important pi-term in almost all fluid dynamics problems. Unless the modeler has good reason to think that either inertia or viscous forces are unimportant, it should be included as a modeling parameter in all fluid dynamics studies. It is tempting to insert numbers in the nondimensional force ratios and then, if a force ratio is very large or very small, discard the force that seems to be much the smaller of the two. This can be an unreliable procedure. The forces in the ratios are only representative of the actual forces. Thus, a Reynolds number of, say, 100 does not necessarily mean that inertia forces are 100 times larger than viscous forces. In fact, for a fluid flow in a pipe, the Reynolds number based on pipe diameter as a characteristic length must be at least 10,000 before viscous forces can be considered small enough to be neglected. A force should only be discarded if there is a good physical reason to believe it to be small, or if preliminary experiments actually show it to be. Fluid Flow in a Circular Pipe The flow of an incompressible liquid in a pipe is a common engineering problem. The two questions of most interest are: What is the pressure drop along the pipe required to produce a desired flow rate, and what is the maximum flow rate that can be obtained by a given pressure drop? Typical answers could be the required pump horsepower or the pipeline slope. Most engineers are familiar with the use of "friction factors" to solve these kind of problems. Dimensional analysis allows us to understand the validity of this method as well as to make model experiments. Figure 9.1 shows the problem. We will assume that the pipe is long enough that any effects of the inlet or outlet can be neglected. Since the liquid is incompressible, the volumetric flow rate at any point along the pipe is the same; this allows us to characterize the flow by an average velocity u, which is the same at all cross-sections and equal to the flow rate divided by the cross-section area. The flow is steady so u does not change with time. One of the causes of the pressure drop is friction, so viscosity μ should be included in the list of parameters; alternatively, we could include a viscous force from Table 9.1. Since the pipe wall is not perfectly smooth, the liquid near the wall must accelerate around the "bumps." We will characterize the bumps by an average roughness height k. Liquid acceleration implies the presence of inertia forces, so we will also include the liquid density p, or alternatively, an inertia force from Table 9.1. Actually, inertial forces are usually caused more by turbulence than by wall roughness. It might be recalled that fluid flow is of two general kinds: laminar, in which each particle of liquid moves in a straight line parallel to the pipe axis, and turbulent,

201 TABLE 9.2 Parameters for Fluid Flow in a Pipe Parameter Pipe diameter Wall roughness Average velocity Viscosity Density Pressure drop per unit length

Type Geometric Geometric Flow Fluid property Fluid property Response

Symbol D k u μ P Apll

Dimension L L LTl MLlT~l ML~3 ML'2!-2

in which random, unsteady motions are superimposed on the straight-line flow. Laminar flow occurs only for relatively low velocities. The small, unsteady motions in turbulent flow represent large inertia forces3. For the moment, we will consider only the problem of determining the pressure drop. Since the pipe is long, and inlet and outlet effects are neglected as being small, the pressure drop along any given section of pipe must be the same as the pressure drop along any other section of the same length. Thus, the pressure drop per unit length is the relevant parameter rather than the total pressure drop. The six parameters that specify our physical picture of the flow are listed in Table 9.2. Since there are three fundamental dimensions, and the μ, p, D determinant is of rank three, there are 6 - 3 = 3 pi terms. From one of the methods described in Chapter 3, the general relation between the three pi terms is: Ap (pu2)(l/D)

- »[?■ i]

(1)

where ψ[...] represents a functional form. The first term on the right side, pwD/μ, is the Reynolds number, the ratio of inertia forces to viscous forces. This term could have been written down directly by referring to Table 9.1. The process of finding the other two pi terms would be simplified, since one of the parameters, say μ, could be eliminated from the parameter list. It is conventional to introduce a factor of one-half in the combination pu2 so that it can be identified with kinetic energy, and to call the function ψ the friction factor/. The common form of Eq. 1 is then:

where

= 4"*· i

(3)

The problem of computing the pressure drop has been reduced to determining a functional relation/ that depends on two parameters, NRe, and k/D. This function has been determined by many researchers. A convenient way of presenting their results is the log-log plot shown in Figure 9.2.

202

**" ^ ai O h-

Ο <

0.1 0.09 0.08 0.07

-^ΙΩ

ai

0.06 0.05

< ai

0.04

<

0.03

CL·

o

0.02

CO

ai

o 0.01 5

10°

2

5

104

2

5

105

2

oai

REYNOLDS NUMBER, N R e Figure 9.2 Stanton Diagram of /versus NRe (Ref. 4) Many of the features of Figure 9.2 can be understood by our dimensional analysis. First when the velocity is small, more properly when NRe is small, the flow is laminar, the particles move in straight line, unaccelerated motion, and fluid inertia can not influence the pressure drop. Consequently, the function/ can not depend on density. By examining Eq. 2, it is found that NRe must be a factor multiplying/, so Eq. 2 can be rearranged in the form:

^-(£> -2p«

(4)

or Ap

= Uiw

hM

(5)

for otherwise the density can not be eliminated; here \|/ is a new functional form. It seems reasonable that any small roughness at the wall will not affect the overall slow, straight line motion of the liquid. Hence, kID is probably an extraneous parameter in laminar flow. If true, one experiment would be sufficient to determine \|Λ since it is now a constant, independent of the other parameters. In fact, experiments do show that/ = \|/ does not depend on kid in laminar flow, and the value of the constant is 64. In the general function, Eq. 2, these results reduce t o / = 64/NRe, as can be seen by direct substitution. The region in Figure 9.2 for which f = 64/NRe is labeled laminar.

203 The region immediately to therightof the laminar zone in Figure 9.2 is a transition zone, where inertia forces begin to be important. To the right of this is the turbulent zone, where inertia forces are the most important force. As NRe is increased in the turbulent zone, the roughness at the wall gradually begins to cause most of the turbulence. Eventually, the friction factor becomes independent of NRe and depends solely on the roughness kID. The effect of viscosity disappears, and the pressure drop depends only on inertia forces. In this region, the friction factor curves should then be parallel to the Reynolds number axis, which as shown in Figure 9.2 does occur. The smoother the pipe, the larger the Reynolds number must be before the flow enters this wholly rough zone. Although dimensional analysis is not capable of establishing the numerical values, it is clear that it does explain the shape of the friction factor curves. The second problem mentioned earlier, namely calculating the flow rate for a specified pressure drop, has the same dimensional analysis solution as the first problem. Equation 2 merely has to be rearranged so that u, the response parameter now, appears only in the pi term on the left. One such rearrangement is:

■-(£

Ψ

pP3Ap

(6)

μ2/ '

In practice, these kinds of problems are solved by iteration, since the function ψ is not tabulated. The three pi terms displayed in Eq. 2 can obviously be used to conduct modeling studies of complicated piping networks. We will now develop this modeling law. The effects of valves and other flow constrictions are neglected in this model, which therefore is somewhat simplified. Suppose that the model is geometrically similar* to the prototype with a scaling parameter λ. As soon as the model liquid has been chosen, the scaling factors for density and viscosity are also known: λρ = pjpp and λμ = μρ/μ„,. Equality of NRe for the model and prototype (which insures that the inertia and viscous forces are in scale) requires that: puP}

_

μ I

(puP^

(7)

~ { μ )p

so that

λ,. =

Λ.μ™λ

u„

(D, 9m )

λρλ

(8)

The ratio μ/ρ is known as the kinematic viscosity v. Using this combination parameter, Eq. 8 can be rewritten as

K =

K

(9)

The response pi term determines the relation between the measured pressure drop in the model and the pressure drop in the prototype: Exact geometric similarity is not necessary if all forces are scaled; for example, if theflowis laminar, the surface roughness need not be scaled.

204 (ApA

=

ίΑρλ

(10)

This can be expressed as a scaling factor for pressure

K - \K

- -jT

(11)

Note that the velocity in the model is larger than in the prototype. If the geometric scale factor is small, the model velocity may become so large that the model pressure drop becomes unacceptably large for model test hardware. In this case, it can be argued that scahng of the Reynolds number is not needed if the flow is in the wholly rough zone. This simplification effectively eliminates Eqs. 8 and 9. The velocity scale is then undefined - that is, the model velocity can take on any convenient value that keeps the model pressure drop acceptable. Even when the flow is not wholly rough, it is sometimes argued, with less justification, that the modeling is satisfactory i£NRe for the model is of the same order of magnitude as the prototype NRe, assuming that both model and prototype flows are turbulent. This assumption eliminates the need to satisfy Eqs. 8 and 9, but the model velocity should still be as large as can be obtained practically to improve the scahng of turbulent forces.

Lift and Drag Forces on an Airfoil Determining the forces acting on a solid body in motion through a fluid is another important problem influiddynamics and one that usually must be investigated experimentally with scale models. As an example problem, we will consider the force exerted on an airplane wing moving at a constant velocity u through air. The problem is shown schematically in Figure 9.3, where, as is usual in fluid mechanics, the body is assumed to be at rest and the air to flow by it with the same constant velocity u in the opposite direction. This is legitimate because a constant velocity can be added to any dynamics problem without changing the forces. Making the air flow by the body transforms the problem into a steady one, and eliminates the time parameter from the list of modeling parameters.

u

Figure 9.3 Air Flow Past an Airfoil

205 TABLE 9.3 Parameters for Flow Around an Airfoil Parameter Airfoil Chord Airfoil span Other length ratios

Type Geometric Geometric Geometric

Angle of attack

Geometric

a

-

Air density

Fluid property

ML~3

Air viscosity

Fluid property

Air thermal conductivity

Fluid property

P μ k

Air specific heat

Fluid property

C

Ratio of specific heats

Fluid property

y

Speed of sound Freestream air temperature

Fluid property Thermal

c ta

Airfoil temperature

Thermal

tw

Air speed

Flow

Force exerted on airfoil

Response

u F

Symbol

c I

u

P

Dimension

L L L

ML-lT~x MLT~3Q-1 L2r2Q-1

LTX Θ Θ LTl MLT2

The geometry of the problem - airfoil shape and angle of attack - must be included in the modeling analysis. From the previous discussion on pipe flow, we expect that fluid viscosity and density will also play an important role in determining the forces. Since no restrictions have been placed on the magnitude of u, it may be supersonic (faster than sound velocity), so the speed of sound in air should be included in the list of parameters. For supersonic flow, various kinds of thermodynamic parameters must be considered to account for the interchange of kinetic and thermal energy. The relevant parameters are the specific heat of the fluid at constant pressure, the ratio of specific heats at constant pressure to that at constant volume, the thermal conductivity of the fluid, and the temperature of the fluid and the body. These parameters should cover all the physics of the problem. At least, we will use them to formulate a model analysis and see if it agrees with experiment. The list of parameters and their dimensions in an MLTQ system (where Θ is the fundamental dimension for temperature, which will be discussed further in Chapter 12) are given in Table 9.3. (Note that the weight of the airfoil and gravity are not included; this eliminates buoyancy forces from consideration.) The 14 parameters have 4 fundamental dimensions and a determinant rank of order 4. There are thus 10 pi terms. From Table 9.1, we see that 2 of the pi terms are NRe and NMa. By a slight extension of the table, we can infer that another one is the ratio of F to an inertia force. Four of the other pi terms are the geometric ratios (2), the specific heat ratio, and the angle of attack, all of which are already nondimensional. The ratio of the 2 temperatures is also a pi term. This leaves only 2 pi terms to be determined from the parameters not included in the first 8, such as thermal conductivity and specific heat. (All 10 pi terms can be determined by the methods given in Chapter 3.) A convenient form of the model law is:

-pu2lc

=

Ψ ot,

pu I

μ '

^μ k '

u^ cpta

(12)

The combination pul/μ is the Reynolds number; the combination ula is the Mach number, NMa, and the combination of fluid parameters cp\xJk is called the Prandtl number NPr after the German fluid dynamicist

206 L. Prandtl. The combination F/0.5pu2lc is invariably called a force coefficient. If we are measuring the force perpendicular to the velocity vector, the force is the lift force and the force coefficient is the lift coefficient CL. Likewise the drag coefficient CL is used for the force parallel to the velocity vector. The three thermal pi terms in Eq. 12 impose severe restrictions on the possibility of general scale model studies. Fortunately, experiments have shown that these terms are important primarily in supersonic flow, where they affect the heating of the thin boundary layer near the wing; otherwise, they are negligible for subsonic flow. Also, the Prandtl number is roughly the same for all common gases, so it can be neglected safely in many cases. Empirical methods can be used to account for the heating due to the conversion of kinetic energy to thermal energy, or the gas used to simulate the air can be heated or cooled to help make the scale model tests more realistic. We will neglect all these terms here, so the flow must be restricted to subsonic or low supersonic speeds. Consequently, our model law implies that the lift and drag coefficients depend primarily on five pi terms: geometry (2), a, NRe, and NMa. Geometry and angle of attack can easily be modeled. Much more difficulty is caused by attempting to scale the prototype NRe and NMa together. For example, suppose wind tunnel tests are to be conducted with air at the same temperature and pressure as the prototype; this means that the speed of sound for the model air is the same as the prototype's. Mach number equality requires that the model velocity is the same as the prototype. But then the model NRe will be too small because:

(NRX

ÎP-Y^Y^V \

(AW-

ΙΡΡΑ^Λ^

-

= λ

(13)

Similarly, if the NRe's are made equal, the NMa's will be out of scale. Mach number and Reynolds number similarity can be satisfied only if the model gas is different from the prototype or if the same gas is used at a drastically different pressure or temperature; either alternative can be accomplished over a limited range of NRe and NMa with certain closed-loop wind tunnels. If the prototype Mach number is less than about 0.3, the gas behaves as an incompressible fluid and Mach number scaling becomes irrelevant. Only the Reynolds number then needs to be scaled. The scale model velocity is made higher than the prototype to satisfy the NRe requirements. (The model NMa must still be kept smaller than 0.3 - 0.4, however.) Further simplification is possible if the NRe's are large enough, because then the effect of viscosity is small compared to the effect of inertia (density). Therefore, for small NMa and large NReJ Eq. 12 simplifies to:

1

2»

=

Ψ|α,

7

(14)

-2pu lc Wind tunnel tests with a scale model can be used to determine the variation of CL and CD with a. Figure 9.4 shows such a typical determination, in the form of a "polar curve" in which CL is plotted against CD for various values of a. In Fig. 9.4, CL begins to decrease as a increases to values larger than about 16°; that is, the wing begins to stall. The angle at which stall occurs is related to viscous shear stresses in the thin fluid boundary on the wing, which cause the flow to separate from the wing surface. Model values of the stall angle cannot always be scaled up to prototype, unless NRe is simulated. This is usually a minor drawback except when the flow is transonic (i.e., near NMa = 1).

207

1-2

O

/*xf ~~~~^A

1-0

f\z°

0-8

—

O

O

0-6

Ίο°

A°

0-4 0-2

0-04

008

0-12

0-16

DRAG COEFFICIENT, C D x 100 Figure 9.4 Typical CL versus CD Polar Curve for an Airfoil (Ref.4) For higher air speeds, Eq. 14 can be extended to include compressibility effects:

-

-pu2lc

= ψ[α, £ /, NM]

(15)

The prototype Mach number can be duplicated in the model tests by using the same velocity and fluid as the prototype. But when the prototype is large, such as a Saturn V rocket or Boeing 747, the Reynolds number may be underestimated by a power of ten in even the largest wind tunnels. For high-lift devices, which operate near the stall angle, or for transonic speeds, where the exact location of the boundary layer separation point is important, the discrepancy between the prototype and model NRe can underestimate flight values of CL by a significant amount. Figure 9.5 shows, for example, the effect of NRe on CL on several high-lift devices. To help alleviate these kind of scaling problems, NASA has constructed a very large wind tunnel, the National Transonic Facility, whose operating envelope is shown in Figure 9.6. Even in standard subsonic wind tunnel testing, corrections are needed to compensate for the influence of the tunnel walls. Another possible discrepancy is that the prototype is flying into still air of a low turbulence level, while the model is tested in a system where the air is flowing at a turbulence level that can be high unless it is reduced by baffles and flow straighteners5.

208

.4

AOTUNNEL ▲■ FLIGHT

-f

(A) J Ü _J <

wω

Ü

0.4

(B)

40° FLAP 96° INBOARD LEADING EDGE KRUEGER FLAP THREE-SEGMENT OUTBOARD LEADING EDGE SLATS, WITH NONE ON OUTER VJ

50° FLAP PLUS LEADING EDGE SLAT

0.2

1

2

4

6

10

15

1

2 6

REYNOLDS NUMBER, 10

Figure 9.5 Effect of Reynolds Number on Maximum Low-Speed Lift (Ref. 6) Propellant Sloshing in Spacecraft Tanks The next problem discussed in this survey of the modeling of fluid motions is one that is important in the design of spacecraft and rockets containing large quantities of liquid in their tanks. In addition, it is a problem that introduces several new dimensionless parameters. Perturbations to the motion of the spacecraft, perhaps due to guidance corrections, cause the liquid in the tanks to slosh - that is, to form waves on the free surface. The waves oscillate at a definite natural (slosh) frequency. The situation is shown schematically

CHORD REYNOLDS NUMBER AT CRUISE

.5 MACH NUMBER

1.0

Figure 9.6 Operating Regime of National Transonic Facility

209

SLOSH WAVE

UNDISTURBED FREE SURFACE LEVEL

LIQUID density P viscosity μ surface tension o/

Ö

ACCELERATION OF ROCKET, g e f f

Figure 9.7 Liquid Sloshing in a Propellant Tank in Figure 9.7. The oscillations of the liquid can exert large forces and moments on the spacecraft. Therefore, the characteristics of the sloshing liquid must be predicted in advance so the guidance system can be designed properly. The tank must also be made structurally strong enough to withstand the slosh loads. Since the slosh wave length must be compatible with the dimensions of the free surface, the problem geometry, such as tank radius R0 and semi-height /, and liquid height h (Fig. 9.7) should be included in the list of modeling parameters. Free surface motions of a liquid can be greatly influenced by gravitational forces, since the potential energy of a particle on the surface changes as it moves up and down. In this problem, an effective gravitational acceleration g# is exerted on the spacecraft by any constant linear acceleration, where gelt is equal in magnitude to the acceleration but opposite in direction; g^ can result from thrusting, for an accelerating rocket, or from aerodynamic drag, for an orbiting satellite. (Note that the centrifugal acceleration of an orbiting satellite just balances the earth's gravitational acceleration, to give a condition of nearly "zero gravity.") Since gravity is important, fluid density must also be important. As always, viscosity should be included to account for friction. Here, the liquid has a free surface, so surface tension forces may also influence the wave motion; these forces are characterized by the surface tension σ of the liquid. Since we are interested in the sloshing that occurs in response to perturbations of the spacecraft motion, the unsteady motion of the spacecraft must be included in the parameters. Here, we will assume the motion is oscillatory and will characterize it by a frequency / and a translational amplitude x0. The response parameter is chosen to be the amplitude F of the oscillating force exerted on the tank by the liquid. The complete list of parameters is shown in Table 9.4.

210 TABLE 9.4 Parameters for Liquid Sloshing Parameter Tank Radius Tank semi-height Liquid depth Density Viscosity Surface tension Steady acceleration Linear amplitude

Type Geometric Geometric Geometric Liquid property Liquid property Liquid property Effective gravity Tank motion

Forcing frequency Slosh force

Tank motion Response

Symbol Ro

I h P μ

σ 8eff Xo

f F

Dimension L L L ML'3 ML~lT-1 MT~2 LT~2 L

r1

MUT1

There are ten parameters and three dimensions, so there should be a total of seven pi terms. By referring to Table 9.4, we can see that five forces are involved in this problem: viscous force, gravitational force, inertia force, surface force, and slosh force. These five forces can be arranged into four pi terms. It is almost self-evident that the four parameters with dimensions of length can be arranged to give the final three pi terms. Of course, all the pi terms could be found by the methods described in Chapter 3. The form of the modeling law chosen here was deduced by Abramson and Ransleben7:

- = JA L * ~[Κ \R;

f

R; R; yg0>

^

J*JL V

μ

'

(16)

σ

The fifth term in Eq. 16, ^g^R^/μ, is a ratio of gravitational forces to viscous forces, which from Table 9.4 is called theGalileo number NGa. The last term, g^R^/G, a ratio of gravity to surface tension forces, is called the Bond number NBo after the English physicist W. N. Bond, who used it to indicate the importance of surface tension in the rise rate of bubbles in a liquid. The fourth term in Eq. 16, fsJRJg^, is a kind of Froude number NFr1 which is a ratio of inertia to gravitational forces. Here, NFr indicates the way that frequency and length should be related to g^. When the effective gravity is small, as it is for a satellite in orbit or during a coasting phase of a rocket, the Bond number can be an important scaling parameter. Letting λ = (R0)J(R0)P t>e a s usual the geometric scale factor, scaling of the Bond number requires that:

λ = \hP

(17)

or alternatively that: λ.

=

λ2

(18)

211 where λ^ = (g^)J{g^)p and λ^ = (σ/ρ)„/(σ/ρ)ρ. For tests run at standard gravity, λ, is large, since (g^)p is small. Equation 17 implies, then, that λ must be made much less than 1, unless λ^/ρ can be made very large, which is generally not possible because of the small range over which σ/ρ can be varied from liquid to liquid. Tests using very small model tanks have been conducted8, although the lower limit on NBo that could be simulated was about twenty, for practical lessons . The main disadvantage to this method is that the Galileo number of the model is much larger than the prototype's, so viscous forces are substantially out of scale. An alternative method, which satisfies Eq. 18, is to let the model tank drop freely in drop-tower9, thereby obtaining the correct scaled value of g^ even down to g^ ~ 0. Magnetic fluids have even been used for tests conducted inside large magnets10, in which a magnetic force is used to cancel the gravitational body force. For the examples discussed in the remainder of this section, it will be assumed that the prototype NBo is large enough that surface tension forces can be neglected. (Note that NBo is quite large for large rocket tanks even when geg is as small as 10"6 of standard gravity.) If the prototype NBo » 1, the primary scaling factor is the Froude number. Hence, when the model tests are conducted on earth with g^ = g, the scaling factor λ^ is glg^. For a rocket under thrust as the prototype, a typical range for Xg = 1/4 to 1/3. Froude scaling requires that:

v-l-V?

(19)

Equation 19 is the scaling factor for the frequency of the scale model motion in terms of the prototype. We can also use it to calculate the prototype natural frequency once we have measured the model natural frequency. Since all frequencies must scale as dictated by Eq. 19, the prototype natural frequency (fn)p is related to the model natural frequency (fn)m by:

■Vf


(20)

So, if the model natural frequency is determined from a model test run at standard gravity, the prototype natural frequency can be calculated ("scaled up") to any level of thrust acceleration from Eq. 20, provided that NGa is also in scale. It turns out, however, that small viscous forces have only a negligible influence on /„, so NGa does not have to be accurately simulated for liquids of small viscosity in order to determine (fn) accurately. This is definitely not the case, however, when the peak slosh force of the model is to be scaled up. The peak force at resonance is a strong function of the viscous damping; thus, for accurate peak force scaling, NGa must be in scale. From the relation (NGa)m = (NGa) , we find:

λ, = V V ?

(21)

In practice, Eq. 21 is almost impossible to satisfy because of the small values of v for most liquid propellants and the large value of RJor most prototypes. As an example, suppose (R0)p = 10 ft, (R„)m = 1 ft, and λ, = 1/4. Then, from Eq. 21, λ,, must be 0.0226. The kinematic viscosity of liquid oxygen, a typical propellant, is about 0.002 cm2/sec; the model fluid must therefore have a kinematic viscosity of 0.000045 cm2/sec, which

212 is several powers of ten less than any existing fluid (with the possible exception of liquid helium near 0 °K). If the propellant is kerosene, which has a kinematic viscosity of 0.02 cm2/sec, the situation is slightly more promising, for the model liquid can now have a kinematic viscosity of 0.00045 cm2/sec, but even here a rather exotic model fluid would have to be used. We must either allow NGa to go out of scale or use a much larger model. When the prototype itself is not too large, it is possible to achieve complete scaling. Abramson and Ransleben7 have proved the validity of Eq. 16 (when N^ can be neglected) by using a 8.75-ft diameter cylindrical tank containing water as the prototype, and a 1.18-ft diameter cylindrical tank containing méthylène bromide as the model. The comparison of the prototype and model slosh forces is shown in Figure 9.8. As can be seen, the comparison is very close, even near the natural frequency where scaling of viscous effects is critical. Note that the results are linear over the tested range, since the force parameter, which is proportional to F/x0, is practically independent of the excitation amplitude parameter x0IR0. (In the linear range, F is proportional to xoy so the force response term should remain unchanged when only x0 is changed. Since the pi term xJRQ does change when x0 changes, it clear that this pi term can not influence the force in the linear range. Conversely, if the response term changes when x0IR0 changes, the response is nonlinear, and the prototype value of x0IR0 must be simulated in the model tests.) In the linear range, the difficulty of scaling viscous effects can be overcome by noting that the response shown in Figure 9.8 resembles the forced response of a damped oscillator (a pendulum-dashpot or alternatively

•

FULL-SCALE TESTS, x 0 / d - 0.00475

O MODELTESTS WITH WATER, x 0 / d = 0.00833 ü MODEL TESTS WITH METHYLENE BROMIDE, x 0 / d = 0.00417 O MODEL TESTS WITH METHYLENE BROMIDE,

_o

48

q 1 ^NATURAL

X

N O OH σ> Q. 1_1_

OH

If

40

^ Û£ <· QL < LJJ C_> Q£

o

si1

n 16

O X



8

o1 CO

1

é

32

UJ

ft

FREQUENCY

U

4r

\

m

-i-rti

,fF

0 1 2 3 4 5 EXCITATION FREQUENCY PARAMETER , R 0 ( 2πί ) 2 /g

Figure 9.8 Model and Prototype Tests of Sloshing in a Cylindrical Tank (Ref. 7)

213 a mass-spring-dashpot). In fact, the slosh response of an inviscid liquid and an undamped oscillator are exactly equivalent mathematically10. From this observation, we can postulate that the response of the set of oscillators and rigidly-attached mass shown in Figure 9.9 will duplicate the force response of the sloshing liquid if the pendulum lengths, masses, and dashpots are chosen properly; one oscillator is used to represent each slosh mode. (Figure 9.8 shows only the response of the first, fundamental slosh mode, but in general there may be several important modes, each with its own natural frequency.) So let us determine the pi terms for this problem. Table 9.5 shows one possible set of parameters. The pi terms for this set can also be arranged into a scaling law:

PgeffRÏ

= Ψ

L Jl h. 3-

R:

RO RO pR?

ÜL·. 3

SÎL

pR o μ*/

^8^*

μ

(22)

Note that the NGa has not been eliminated from this new problem. We still have to be concerned with viscous forces. But the problem of determining the viscous effects has been simplified by lumping them into the damping constant of the dashpot. The procedure is to conduct tests with several different liquids, each time examining the peak slosh force to determine the damping. In that way, an empirical correlation can be

Figure 9.9 Equivalent Mechanical Model of Propellant Sloshing

214 TABLE 9.5 Parameters for Mechanical Model of Liquid Sloshing Parameter Tank Radius Tank semi-height Liquid depth Thrust acceleration Slosh mass Rigid mass Pendulum length

Type Geometric Geometric Geometric Effective gravity Model property Model property Model property

Symbol

U

Dimension L L L LT~2 M M L

Dashpot constant

Model property

Q

MT~l

Density Viscosity

Liquid property Liquid property

P μ

ML~3 ML-'T~l

Ro

I h 8
nti

m0

developed that can be used to predict the dashpot constant as a function of the other parameters. It is more common, however, to correlate the damping ratio γ defined as C,/4m,n/· than the actual dashpot constant. As an example, the damping ratio of the first slosh mode for a cylindrical tank has been found to be:

ϊι =

a79

A/5r

(23)

This correlation can be used to predict the damping ratio of the prototype for the prototype values of g^ and v. In other words, we will compute the slosh force, rather than scale it up from model force responses. References 11 and 12 give a more complete discussion of the derivation and useof these equivalentmechanical models. Spread of Oil Slicks on a Calm Sea When oil is spilled on water, it spreads outward to form an ever enlarging thin sheet or slick on the surface of the water. The rate that the oil spreads on a calm sea, where the slick is not broken up into several smaller slicks by waves or currents, is a result of the balance between spreading and retarding forces. Spreading is promoted by the force of gravity as it tries to make the slick thinner atop the heavier water just it would on a horizontal solid surface. Surface tension at the edges of the slick also spread the oil by "pulling" on it all around its perimeter. As the oil spreads outward, it drags with it a thin layer of the water, and this drag of the water retards the spreading. Inertial forces also retard the spreading, since both the oil and the water are being accelerated from rest. Using these physical ideas, Fay13 and others have analyzed the forces and estimated the spreading rate. Most of their results can also be derived by dimensional analysis and model experiments. Figure 9.10 shows an idealized sketch of an oil slick spreading on calm water. Based on the remarks given above, we expect that parameters must be included in the analysis that characterize geometry, gravity forces, surface tension forces, viscous forces, inertial forces, and, since the flow is transient, time. Table 9.6 shows the selected parameters that describe these effects. We are interested in the diameter D as the time-dependent response

215

D ■ diameter of slick

WATER

Figure 9.10 Geometry of Oil Slick on a Calm Sea parameter. In order to have a parameter that can serve as a characteristic length in the analysis, we also include the volume V of the oil spill; the corresponding length parameter is Vm. By reference to the table of forces or by using any of the other methods already discussed, the model law can be found to be:

V s - \*pg,*

p \ν·

σ Λ/ί"

(24)

pj

The first term on the right hand side of Eq. (24) is the ratio of inertia forces to gravitational forces, a Froude number. The second term is the ratio of gravitational forces to viscous forces, a Galileo number. The third term is the ratio of viscous forces to surface tension forces, and is sometimes called the Ohnesorge number. Fay's main contribution to this problem was his realization that the spreading has three distinct phases, and each phase is controlled by one of these three dimensionless pi terms. In the beginning, the slick is relatively thick and gravity is a much stronger spreading agent than surface tension. The inertial force due to fluid acceleration is the most important retarding force because viscous forces have not yet had time to develop. Thus in this phase, the first pi term controls the spreading, and the slick diameter is: TABLE 9.6 Parameters for Spreading of Oil on a Calm Sea Parameter Volume of oil Slick diameter Slick thickness Density of water Kinematic viscosity of water Surface tension between water and oil Density difference between water and oil

Type Geometric Geometric and response Geometric Liquid property Liquid property Liquid property Liquid property

Gravity Time after spreading begins

Force of gravity time duration

Symbol V D h P V

Dimension L3 L L ML'3 L*TX

m1 3

σ Ρ-Ρ*=Δρ

ML~

8 t

LT1 T

216

,/

Grovity-lnertîal Regime Gravîty-Vîscous Regime

5

10

50

100

TIME AFTER SPILL, SECONDS Figure 9.11 Comparison of Model Oil Spills and Theory (Ref. 14)

D 1/1/3

pVh Apgt2

ißi

(25)

where ßt is a constant to be determined, and "oc" means "is proportional to.". (ßt may depend on Δρ/ρ but not on the other pi terms, all of which represent forces that are negligible during this first phase of the spreading.) It seems clear that ßj should be negative, since the greater g is or the smaller p is, the faster D should increase. Fay's analysis and model experiments show that ßj = -1/4. So, for the beginning phase, the spreading law is:

D

K

(26)

Figure 9.11, which shows the results of some small scale spills14, demonstrates the validity of Eq. (26). The value ft = -1/4 could be determined from these results. Later in the spreading, gravity is still the dominant spreading force, but the acceleration of the slick has decreased almost to zero, and viscous forces are the primary retarding agents. For this phase Eq. (24) is approximated by:

D

y\l3

:*VT

(27)

217

10

5

1

4

:

X 2 x ΙΟ TONS OIL Δ 80 BBL OIL O 30 BBL OIL V 110 BBL OIL/WATER t> 90 BBL OIL/WATER 106 U ü 25 x 10''BBL OIL/WATER σ

D = 10

P y

IT 1v

l 2

/

'

'o

10 h

D>

l(f 1013

1 y

LE

D>

X

y

-J 5

/I "" D = p J z/ J σ

y/^^

1 2

L' '

LTD

DAY 10 4

xx x

W£7"v D

HR

/x#«x

y

/ ^

<

Π7

Γ

WEEK

MONTH

1 ■ 1

10 s 10 6 TIME AFTER SPILL, t ( sec )

107

Figure 9.12 Comparison of Measured Oil Slick Size and Theoretical Estimates for Surface Tension-Viscous Spreading Regime Fay's analysis and the spill tests shown in Figure 9.11 indicate that β2 = 1/6, so the slick diameter during this phase increases as:

D

[gAPy:2 V / 6

(28)

In the final phase, the slick is so thin that gravity no longer is effective in the spreading dynamics. Surface tension forces now promote the spreading and are balanced by viscous forces. Fay's analysis for this phase predicts that:

D

&

(29)

The rate of spreading in the final phase is independent of the initial volume of the slick. This phase is difficult to investigate in model tests because of the extreme thinness of the slick. The simplified physical picture of the spreading may be incorrect, also, because of chemical decomposition of the slick after being exposed to air and water for extended periods of time. Nonetheless, data from the 1968 Torrey Canyon accidental spill (« 10,000 barrels) and some smaller spills conducted by the Coast Guard at least do not contradict Eq. (29), as shown in Figure 9.12. These tests showed a rapid spread to a size that depends on the volume of the spill, followed by a period of little or no growth. Unfortunately, data for the early periods are not available. It can be seen, however, that the data for the final diameters can be bounded by choosing a constant of proportionality for Eq. (28) somewhere between 1 and 10.

218

Figure 9.13 A 6-ft Model of a Small Prototype Liquid-Metal Journal Bearing Our model of spreading is obviously simplified. More than likely, physicochemical processes, and the action of waves and currents must be included to make a more accurate model.

Turbulent Flow in a Bearing This final example shows how ingenuity can be used to overcome modeling restrictions. In order to study turbulent flow in a small, high-speed, liquid-metal journal bearing (about an inch in diameter), Burton and Carper15 found it necessary to make measurements of velocity profiles, turbulent fluctuations, and pressures in bearing clearances as small as 0.01 inch. To make the measurements, they constructed a large model (about 6-ft diameter) and used air as the model fluid - a dissimilar material model. The model and prototype are pictured in Figure 9.13; note that the term scale model does not always mean a model that is smaller than prototypical. The model rotated at a very small speed, in accordance with Reynolds number scaling. The bearing clearance of this model was 1.0 inch, easily allowing the necessary instrumentation to be inserted. These studies provided some truly fundamental data on turbulent flow in bearings that probably could not have been obtained in any other way. Conclusions In this brief discussion of modeling fluid motions, we have introduced several important dimensionless numbers. The Reynolds number relates fluid inertial forces to viscous forces, and is an important parameter in nearly every fluid flow problem. Special knowledge of the problem is required to assume that NRe is

219 unimportant. The Mach number relates inertial forces to compressibility forces, and is an important parameter when the flow velocity is greater than about 0.4 times the speed of sound in the fluid. Compressibility effects can be dominant in such cases. The Froude number relates inertial forces to gravitational forces, and is an important parameter whenever the fluid has a free surface. For wave motions, ship resistance studies, and forces on submerged bodies, the Froude number can be the most important scaling parameter. Every fluid motion will involve either one of these nondimensional force ratios or some other force ratio that can be derived from Table 9.1 References 1.

H. Rouse and S. Ince, History of Hydraulics, Dover, New York, 1963.

2.

J. P. Catchpole and G. D. Fulford, Table 10-51. Dimensionless Groups, in Handbook of Tables for Applied Engineering Science, ed. R. C. Weast, CRC Press, 1973, pp. 1027-1037. H. Schlichting, Boundary Layer Theory, 4th ed., McGraw-Hill, New York, 1960. J. K. Vennard, Elementary Fluid Mechanics, 4th ed., John Wiley & Sons, New York, 1961. L. M. Milne-Thomson, Theoretical Aerodynamics, 4th ed., Macmillan, New York, 1966. J. Lukasiewicz, The Need for Developing a High Reynolds Number Transonic Wind Tunnel in the U.S., Astronautics and Aeronautics, 9, April 1971, pp. 64-71. H. N. Abramson and G. E. Ranslebem, Simulation of Fuel Sloshing Characteristics in Missile Tanks by Use of Small Models, Amer. Rocket Society J., 30,1960, pp. 603-613. F. T. Dodge and L. R. Garza, Experimental and Theoretical Studies of Liquid Sloshing at Simulated Low Gravity, Trans. ASME, J. Applied Mechanics, 34,1967, pp. 555-562. J. A. Salzman and W. J. Masica, An Experimental Investigation of the Frequency and Viscous Damping of Liquids During Weightlessness, NASA TN D-5058, 1969. F. T. Dodge and L. R. Garza, Free-Surface Vibrations of a Magnetic Liquid, Trans. ASME, J. Engineering for Industry, 94,1972, pp. 103-108. F. T. Dodge, Chapter 6: Analytical Representation of Lateral Sloshing by Equivalent Mechanical Models, in The Dynamic Behavior of Liquids in Moving Containers, éd. H. N. Abramson, NASA SP-106,1966.

3. 4. 5. 6. 7. 8. 9. 10. 11.

12. 13. 14.

15.

F. T. Dodge and L. R. Garza, Equivalent Mechanical Model of Propellant Sloshing in the SATURN S-IVB Workshop Configuration, 1970 Shock and Vibration Bulletin, part 7,1970, pp. 169-180. J. A. Fay, The Spread of Oil Slicks on a Calm Sea, Mass. Inst. Tech., Fluid Mechanics Laboratory Pub. 69-6, Cambridege, August 1969. F. T. Dodge, J. T. Park, J. C. Buckingham, and R. J. Magott, Revision and Experimental Verification of the Hazard Assessment Computer System Models for Spreading, Movement, Dissolution, and Dissipation of Insoluble Chemicals Spilled on Water, U. S. Coast Guard Report CG-D-35-83, June 1983. R. A. Burton and H. J. Carper, An Experimental Study of Annular Flows with Application to Turbulent Film Lubrication, Trans. ASME, J. Lubrication Technology, 89,1987, pp. 381-391.