Figure 9.2 The relationship between pressure gradient and the four variables, velocity, pipe diameter, fluid density, and fluid viscosity. (Reprinted with permission of John Wiley and Sons Inc.)

Figure 9.3 A single dimension-less plot that also includes all of the information included in the four plots in Fig. 9.2. (Reprinted with permission of John Wiley and Sons Inc.)

Now it is possible to write a set of three equations, one for each basic dimension, and to solve them simultaneously for three values. Since we have five unknowns and only three equations, it is necessary to assume the values of two convenient variables.

L: -3a + b + c - 2e - 4f = 0 T: 0 - b - 0 + e + 2f = 0

For our example, we will assume a = 1 and e = 0 to produce two independent Pi terms. That will insure that AP/ L will appear in only one Pi term. Solving the three equations yields the following:

1 + f = 0 f = -1 -3 + b + c - 4f = 0 c = 1 -b + 2f = 0 b =-2

Therefore, by using the values of a = 1 and e = 0, one possible Pi term follows:

To generate the second independent Pi term let e = 1 and a = 0. This will ensure that every variable appears in one of the Pi terms and that AP/L does not appear in n2.

1 + f = 0 f = -1 b + c - 2 - 4f = 0 c = -1 -b + 1 + 2f = 0 b =-1

Note that n2 is the 1/Reynolds number, a well-known dimensionless parameter used in many fluid mechanics applications.

Therefore, if we make n2 for the model equal to n2 for the prototype, then n1 for the model will predict meaningful values for n1 of the prototype.

When we equate Pi terms involving length ratios only, the model satisfies the condition of geometric similarity. Scale models in which all three dimensions of the model (height, width, and length) have the same scale have geometric similarity. A common dimensionless Pi term involving geometry is the ratio of length to width or the ratio of two lengths.

If a model is a 1/10 scale model, this means that the ratio of lengths between the model and the prototype is 1/10. The flea in our example introduced above might be considered a 1/1000 scale model of a human, if it were similarly proportioned to the human. If we only know the flea's length and the man's height, we can consider the flea geometrically similar. We will call the flea a 1/1000 scale model for a first approximation.

But if we consider the overall effectiveness of the model, what about the other Pi terms that have forces and time in them? For a complete model, we also need some other types of similarity.

For a perfect model, we must be sure that we have included all important variables. Further, all Pi terms in the model must be equal to each Pi terms in the prototype. When we achieve geometric similarity by matching the geometric Pi terms, as with a scale model airplane, the model will look like a smaller version of the prototype but will not necessarily behave like the prototype. When we equate Pi terms involving force ratios we achieve dynamic similarity.

For example, density is a variable that contains force as one of its dimensions. A Pi term in some fluid mechanics problems that includes drag force and density could be the term that follows. For this Pi term the variable D represent drag force, p represents fluid density, V represents fluid velocity, and t1 and t2 are geometric dimensions.

To achieve dynamics similarity, n1 in the prototype must equal n1 in the model.

9.5 Kinematic Similarity

When we equate Pi terms involving velocity and/or acceleration ratios we obtain kinematic similarity. Many Pi terms that have a force dimension will also have a velocity or acceleration dimension. For example, the Reynolds number is a common Pi term in fluid mechanics that has both force dimensions and velocity dimensions. The force dimension shows up in both density and viscosity. To achieve kinematic similarity, and to achieve dynamics similarity, the model and the prototype must have Reynolds number similarity. That is, the Reynolds number of the model must equal the Reynolds number of the prototype.

To achieve similitude, we must have a model with geometric similarity, dynamic similarity, and kinematic similarity when compared to our prototype.

9.6 Common Dimensionless Parameters in Fluid Mechanics

Dimensionless Pi terms are used in many fluid mechanics applications. Some of the terms are used relatively frequently and have names. Some important dimensionless groups in fluid mechanics are shown in Table 9.1. In the table below p is density, V is fluid velocity, and L is some geometric length. Viscosity is represented by m and gravitational acceleration is represented by g. Pressure is denoted by p. The frequency of the changing flow is v. Surface tension in the Weber number is represented by s.

9.7 Modeling Example 1—Does the Flea Model the Man?

If we use a flea to model a man (or vice versa) we need to decide on a few important parameters. Let us first assume that we want to model the terminal velocity of the prototype using the model. The terminal

TABLE 9.1 Shows Some Common Dimensionless Parameters

Pi term Name Ratio pVL/m Reynolds number, Re Inertia/viscous

V/(gL)112 Froude number, Fr Inertia/gravity p/(pV2) Euler number, Eu Pressure/inertia

V/c Mach number, Ma Velocity ratio

L(«p/m)1/2 Wormersley number, a Inertia/surface tension pV2L/s Weber number, We Inertia/surface tension velocity is the maximum velocity that a falling object in free fall will reach. We could assume that the maximum velocity of the prototype will be a function of drag, D; the weight of the object, W; the geometry of the object (which we will designate by three dimensions L, t±, and t2); the density of the fluid in which the object is falling, p; and the viscosity of the fluid in which the object is falling, m. Mathematically, we can write the relationship as follows:

Therefore, for this modeling problem, there are eight variables and three basic dimensions that are required to describe the units on those variables. The three basic dimensions are time, length, and force. If our assumptions are valid, then the required number of Pi terms is n = 5.

n = k - r = 8 - 3 I will choose the following five independent Pi terms:

nx |
= PVmaxL/M |
Reynolds number |

n2 |
= D/(pVL t1h) |
coefficient of drag |

n3 |
= D/W |
force ratio |

n4 |
= t1/L |
geometrical scale |

n5 |
= t2/L |
geometrical scale |

For a first approximation for our model, we will take the flea weight to be 10~4 g. The flea length will be 1.5 mm and we will consider the flea and the man to have the same aspect ratio. That means that we will assume n4 and n5 to be identical in the model and prototype a priori. The man's length will be 1.5 m and the weight will be 100 kg.

Starting from this data, our geometric scale is 1000:1. The man is 1000 times longer and 1000 times wider than the flea. Because we are looking to model the terminal velocity of the falling man and/or the flea, the drag force will be equal to weight for n3 = 1.

For kinematic and dynamic similarity let n = n and let n2 =

" " xman J-flea "man n2w The resulting equations are shown below.

(Vman/Vflea)2 = (Wmfln/Wf1ea)(t1 X t2)flea/(t1 X t2) man

If the second Pi term is identical in the flea and the man, the terminal velocity of the man will be ~30 times that of the flea.

man ea

If the first Pi term, the Reynolds number, is identical in the flea and the man, then the velocity of the man will be 1000 times that of the flea. We can see that these two conditions are mutually exclusive and the man and the flea, falling through the same fluid are not a good model for each other. For Reynolds similarity, the man must fall 1000 times slower. For drag similarity, the man needs to fall 30 times faster.

If you want to allow the man and flea to fall at the same velocity and have Reynolds number similarity, it would also be theoretically possible to change fluids so that we have a different kinematic viscosity, n. We might ask ourselves whether it is practical. The kinematic viscosity of standard air is nair = 1.46 X 105 m2/s. Now if we set the velocity of the man and the flea equal, with the constraint of maintaining Reynolds number similarity, the equation is shown below.

am b

amb m> flea

Since viscosity divided by density is equivalent to kinematic viscosity; m/p = n, and if the flea is falling through air and the man is falling through an alternate fluid, it is also possible to write the equation as:

nair = 1 n alternate 1000

For Reynolds number similarity the man needs to fall through a fluid that has a kinematic viscosity 1000 times greater than that of standard air or ~1.5 X 102 m2/s. Glycerine at 20 °C has a kinematic viscosity of 1.2 X 103 m2/s, so we would need to use glycerine at a much cooler temperature to achieve that kinematic viscosity! Because of glycerine's very large density compared to air, now buoyancy would become a significant factor and we realize that the flea just is not a good model for the man (and vice versa).

Our original question was, "Why can a 1.5 mm flea fall a meter without injury but a 1.5 m man can't fall 1 km unhurt?" The answer is that the model and prototype are geometrically similar (1/1000 scale) but not dynamically and kinematically similar. The low Reynolds number and relatively high drag effect predict that the flea falls very slowly!

We would like to study the flow through a 5 mm diameter venous valve carrying blood at a flow rate of 120 mL/min. We will use water instead of blood, which is more difficult to obtain and more difficult to work with. Take the viscosity of blood to be 0.004 Ns/m2 and the viscosity of water to be 0.001 Ns/m2. Complete geometric similarity exists between the model and prototype. Assume a model inlet of 5 cm in diameter. Determine the required flow velocity in the model that would be required for Reynolds number similarity.

Begin by calculating the given mean velocity in the prototype. The mean velocity in the prototype can be calculated from the flow rate in the prototype divided by the cross-sectional area in the prototype.

cm3 m3

Next, recognize that for Reynolds number similarity, the following equation must be true, where the subscript P represent the Reynolds number in the prototype and the subscript m represents the Reynolds number in the model.

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