Computing Open Channel Flow Using a Pitot Tube

In our last post we discussed the basics of hydraulic jump.  We showed how the flow across the jump could be estimated using the depths of the water before and after the jump.  In this post we will show another method, using a pitot tube, which is also used in a wide variety of other applications.

The calculations in our earlier post assumed the velocity of the water to be uniform.  In a flow channel with parallel straight sides and rectangular cross-section, we assumed the flow to be

Q = uwy

where the variables are as before.  For both open and closed channels, however, the velocity will not be uniform; the velocity we used was an average velocity.  If we actually measured the velocity at various points in the flow, we would get different results for different places, as illustrated below (from Open Channel Flow Measurement):

Flow-variance

The following presentation assumes a rectangular cross-section with parallel walls.

Pitot Tube Measurements

We discussed the concept behind the Pitot tube in Wind Tunnel Testing.  The simplest way to show this for open channel flow is to use the presentation of Vennard (1940):

Pitot-Tube

Integrating the Results

Separating the static pressure from the stagnation (dynamic) pressure is fairly simple in open channel flow, although reading the results can be tricky if the flow level varies.  We thus take several measurements at several heights and compute a velocity at each point, getting a velocity profile for the flow.  But how to integrate this relationship for a total flow?  There are several methods.

One would be to take a trend line of the results (as discussed in Least Squares Analysis and Curve Fitting) and integrate it in closed form.  For this to work requires that the trend line have a very good correlation with the original data.

Another would be to take the results and graphically integrate them.  For many engineering applications, graphical methods were popular for many years because they avoided many difficult calculations which strained the limited computational power of the time.  Although the results were approximate, with CAD a higher degree of precision can be obtained than was possible before.  The tricky part of a graphical method such as this is the scaling, which must be understood to properly interpret the results.

Yet another is the use of numerical integration, generally piecewise with methods such as the Trapezoidal Rule and Simpson’s Rule.  With a properly laid out spreadsheet, this can be done with minimal effort, although attention to detail is crucial to success.

An implementation of numerical integration can be used which simplifies the calculations, and is suggested in Open Channel Flow Measurement.  It involves a little pre-planning in that the points where the data is taken need to be pre-determined (they should be in any case.)  Since the more points of data the more accurate the result (all other things equal,) we’ll use six points.  Those points are at the surface, at the bottom of the channel, and at 20% (0.2), 40% (0.4), 60% (0.6) and 80% (0.8) of the total depth.  The mean velocity can be thus computed as follows:

u_{mean}=\frac{u_{surface}+0.2u_{0.2}+0.2u_{0.4}+0.2u_{0.6}+0.2u_{0.8}+u_{bottom}}{10}

Boundary layer considerations would indicate that the velocity at the bottom of the channel be zero, but if it is possible to take a measurement it would be better.

Once the u_{mean} is calculated, it can be used in the equation at the beginning of the post to estimate the flow in the channel.

References

Vennard, J.K. (1940) Elementary Fluid Mechanics.  New York: John Wiley & Sons, Inc.

 

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Getting the Jump on Hydraulic Jump

Hydraulic jump is one of the more interesting phenomena in open channel flow.  It is used, for example, in stilling basins, where it is necessary to reduce the fluid velocity coming out from a dam by dissipating the kinetic energy in the flow.  An example of what it looks like is below.

Figure 163.png

This is a very basic treatment of the subject.  It is based on Streeter (1966) and also Dodge and Thompson (1937), where many of the graphics come from.  To keep things simple we will assume that the jump takes place in an open channel with two vertical parallel walls.  But first we need to introduce a couple of concepts.

Froude’s Number

In James Warrington’s monograph about propulsive power, he mentions “the investigations of the elder Froude.”  The Froude Number, which is named after him, is basically the ratio of the inertia force to the gravity force, and is defined in Wind Tunnel Testing as

Fr = \frac{u^2}{g_cD}

Although it’s confusing, the Froude Number is frequently the square root of this value, and for this study will be designated by the equation

Fr = \frac{u}{\sqrt{g_cy}}

In this case u is the average velocity of the fluid in a one-dimensional channel; the fact that it’s average is very important.  The constant g_c is the gravitational constant and y is the depth of the fluid (usually water) in the channel.  This is the way we will use the Froude Number in this study.

Since we’re restricting ourselves to vertically-walled channels with rectangular cross-sections, we can make another modification to the Froude Number.  If we define the flow as

Q = u w y

where Q is the flow in volume per unit time, w is the width of the channel and y is defined as before, the Froude Number can be stated as

Fr = \frac{Q}{w{g_c}^{\frac{1}{2}}y^{\frac{3}{2}}}

Keep in mind this is only for channels with vertical walls and rectangular flow cross-sections.

Specific Energy

Let us consider a vertical cross section of a one-dimensional horizontal channel flow.  Bernoulli’s Equation states that the fluid energy/head at that cross section should be

E = y + \frac{u^2}{2 g_c}

What we have done here is to eliminate the pressure term in Bernoulli’s Equation, which does not apply for a surface.  We are left with the potential term (the depth of the fluid in the channel) and the kinetic term (due to the movement of the fluid.)  The energy E is thus in units of length, or is a head term.

Again assuming vertical, parallel walls and substituting the flow term, this equation can be stated as

E = y + \frac{Q^2}{2w^2 y^2 g_c}

The resulting curve looks like this:

Figure 165

Obviously this figure allows for non-parallel sides, but the basic theory is the same.  The depth which we call y is called d in this diagram.  At a depth of zero the specific energy is infinite, decreasing to a minimum at the point where the curve in (c) comes closest to the ordinate, and then decreases to the point where the second term on the right hand side vanishes and the specific energy equals to the depth.  We can find the location of this minimum by differentiating with respect to depth as follows:

\frac{dE}{dy}=0=1-\frac{Q^2}{w^2y^3g_c}

Solving this equation for y yields a cubic equation with one real and two complex roots.  Using the real root only, the depth at which the energy is minimised is given by the equation

y_c = \frac{Q\frac{2}{3}}{w^\frac{2}{3}g_c^\frac{1}{3}}

This is referred to as the critical depth, and flow at this point is referred to as critical flow.  For depths shallower than this, the flow is referred to as rapid or supercritical.  For depths above this, the flow is referred to as subcritical or tranquil.

Some algebra will also reveal the following:

  1. At critical flow, the Froude Number is unity.
  2. At critical flow, the specific energy is \frac{3}{2}y_c .
  3. Since the Froude Number is inversely proportional to y^\frac{3}{2} , for values of y greater than critical (subcritical/tranquil flow) the Froude Number is less than unity.
  4. Conversely, for values of y less than critical (supercritical/rapid flow) the Froude Number is greater than unity.

Solving for Hydraulic Jump

Now that we’ve laid the groundwork for some basics of flow in channels, it’s time to consider hydraulic jump itself, which is illustrated below.

The first thing worth noting is that the incoming flow (from the left) must be supercritical for hydraulic jump to take place.  From that the flow on the right must be subcritical.  In the process of the jump energy is dissipated, which is a big purpose of inducing hydraulic jump in, say, a stilling basin.  But this also means that conservation of energy does not apply here.  What does apply is a) conservation of momentum and b) conservation of mass flow.

Starting with conservation of momentum, after a little algebra and using our y notation for depth (as opposed to d in the figures) we have

g_cy_1^2 + u_1^2y_1 = g_cy_2^2+u^2_2y_2

After a great deal more algebra and application of the Froude Number,

\frac{1+2Fr_{1}^2}{Fr_1^\frac{4}{3}} = \frac{1+2Fr_{2}^2}{Fr_2^\frac{4}{3}}

Conservation of mass flow requires that

Q_1 = Q_2

or

Fr_1y_1^\frac{3}{2}=Fr_2y_2^\frac{3}{2}

There is more than one way to combine the conservation of momentum and mass flow equations.  One way results in determining the relationship between the two Froude Numbers, thus

Fr_2 = \frac{2\sqrt{2}Fr_1}{\left( \sqrt{1+8Fr_1^2}-1 \right)^\frac{3}{2}}

Pages-from-Hydraulic-Jump

Plot of the Froude Number before the jump (Fr1) and after (Fr2)

We can also reapply the conservation of mass flow equation and eliminate Fr_2 in this way:

\frac{y_1}{y_2} = \frac{2}{ \sqrt{1+8Fr_1^2}-1 } 

We now have a definite relationship between a) the two Froude Numbers and b) the ratio of the water depths on both sides of the jump to the first Froude Number (and thus the second.)

Determining the Flow

At this point we have the basics of hydraulic jump.  There are many ways we can apply this theory.  In this monograph we’ll look at two: using the results to estimate the flow, and using the results to estimate the energy loss.

We’ll start with presenting this graphic overview of hydraulic jump and the effects of varying the upstream Froude Number (and thus the downstream one, as we saw earlier) from Research Studies on Stilling Basins, Energy Dissapators and Associated Appertuances.

Pages-from-HYD-399-2

Let’s consider a worked example, taken from Streeter (1966).  A hydraulic jump takes place where the upstream depth is 5′ and the downstream depth is 31′.  The sluice is 50′ wide.  Determine the Froude Numbers on either side of the jump and the flow through the jump in cubic feet per second.

We start by rewriting the equation that relates the ratio of depths to the upstream Froude Number (1) as follows:

\frac{y_1}{y_2} - \frac{2}{ \sqrt{1+8Fr_1^2}-1 } = 0

We then note that \frac{y_1}{y_2} = \frac{5}{31} = 0.161.   We could go through more algebra and solve for Fr_1 , but it’s simpler to use goal seek and compute it by finding the zero of the above expression; it comes out to Fr_1 = 4.72.

We then substitute this result into this equation

Fr_2 = \frac{2\sqrt{2}Fr_1}{\left( \sqrt{1+8Fr_1^2}-1 \right)^\frac{3}{2}}

and compute Fr_2 = 0.306 .

Since we now know the Froude Numbers for both sides, we can compute the flow.  We defined the Froude Number earlier in terms of flow; rearranging that equation yields

Q = Fr w{g_c}^{\frac{1}{2}}y^{\frac{3}{2}}

Taking either the upstream Froude Number and depth or the downstream Froude Number and depth and substituting these, the sluice width (50′) and the gravitational constant, the result Q  = 14,986.4 \frac{ft^3}{sec} .

Computing the Energy Loss

One of the main purposes for inducing hydraulic jump is to dissipate kinetic energy in a fluid flow.  Earlier we saw that at any point in the flow there is a specific energy, or head, in the flow.  If we subtract the specific energy at the upstream (Point 1) from the downstream (Point 2) the difference between the two heads is

\Delta E = \frac{\left( y_2 - y_2 \right)^3}{4y_1y_2}

or strictly a function of the two water depths.  If we substitute the two water depths from the last example, this yields an energy/head change of 28.35′.  Although the units are of length, it is more accurate to say that this is an energy per unit weight of water, or ft-lb/lb.  The power dissipated during the hydraulic jump is thus

P = Q\gamma\Delta E

Substituting the head change we just computed with the flow we computed earlier and the unit weight of water yields P = 26,510,196 ft-lb/sec = 48,200 hp.

You are urged to set these equations up in a spreadsheet and substitute the values given to confirm the validity of your spreadsheet before using them for another case.

References (other than ones on this site)

  • Dodge, R.A., and Thompson, M.J. (1937) Fluid Mechanics. First Edition.  New York: McGraw-Hill Book Company
  • Streeter, V.L.  (1966) Fluid Mechanics.  Fourth Edition.  New York: McGraw-Hill Book Company.

The Experimental Determination of the Metastatic Height

In the earlier post on buoyancy and stability, methods for the determination of the metastatic height are discussed.  This is yet another method of doing this, taken (with a few modifications) from E.H. Lewitt, Hydraulics and Fluid Mechanics (Sir Isaac Pitman and Sons, 1923.)   This text also has a treatment on the “moment of inertia” method of computing metastatic height.

The metastatic height of a ship or pontoon may be found experimentally whilst the
vessel is floating, if the position of the centre of gravity is known.

Figure 21Let W be the weight of the ship, which is known, and let G be the centre of gravity. Let a known movable weight m be placed on one side of the ship.  A pendulum consisting of a weight suspended by a long cord is placed in the ship, and the position of the bob when at rest is marked. Let l be the length of the pendulum. The weight m is then moved across the deck through the distance x, the new position of m being denoted by m’. This will cause the ship to swing through a small angle \theta about its metacentre M. Then, as the pendulum inside the ship still remains vertical, the angle \theta may be measured by the apparent deflection of the pendulum.

Let the apparent horizontal displacement of the pendulum weight = y. Then,

\tan \theta = \frac{y}{l}

Referring to Fig. 21, the moment caused by W about M equals the moment about M caused by moving m to m’.  Or,

W \times GM \tan \theta = mx

from which

GM = \frac{mx}{W \tan\theta}

and, as all the quantities on the right of this equation are known, the metacentric height can be calculated.

This experiment is often carried out on a ship in order to determine the exact position of G which is difficult to estimate from the distribution of the ship’s weight.

Example 2

The Propulsive Power of Screw Ships

James N. Warrington
Associate, American Society of Naval Engineers
Presented at the Annual Meeting, 22 December 1893
Reprinted in the Naval Engineers Journal, July 1983

Since the investigations of the elder Froude, friction and wave making have been fully recognized as the leading elements in the resistance of ships. The resistance due to eddies is known to be quite inconsiderable in modern ships, while the variations in resistance which accompany change in trim, bodily rise or subsidence of the vessel, and wave interference. all occur at such speeds as produce waves of great magnitude relative to the displacement. Within the limit of speed at which these phenomena appear, friction and wave making may be regarded as supplying substantially the entire resistance, and it is here essayed to formulate the propulsive power absorbed by these two elements of resistance with with such a degree of approximation as may be attained by attributing the properties of a trochoidal wave to the waves of a ship.

Friction

The horse-power of the frictional resistance may be expressed by the formula

eq1

in which S represents the area of the wetted surface, V the speed in knots, and C_1 a constant to be derived from the coefficient of friction.  For a clean painted surface exceeding fifty feet on length Froude’s experiments show a resistance of 0.25 lb. per square foot at a speed of ten feet per second, and 1.83 as the power of the speed to which the resistance is proportional.  From these quantities are decided the values n = 2.83 and C_1 = 32,566 , the speed being expressed in knots of 6,080 feet and the wetted surface in square feet.

Wave Making

USS Cincinnati, New York, NY, June 1896.

Considering the wave generated by the ship in motion, it may be observed that similar ships produce similar waves, and that for a given ship the ratio of the magnitude of any series of waves to the magnitude of any other series, as for example of the bow waves to the stern waves, will be constant at all speeds so long as their generation is free from mutual interference.  Moreover the wave of any series will retain their similarity at all speeds, for the orbital velocity as well as the speed of advance of the wave form must both be proportional to the speed of the ship.

The aggregate energy of all the wave of the ship may therefore be conceived to be embodied in a single series of representative waves having a speed of advance equal to the speed of the ship.

As the particles of water in the longitudinal stream lines approach the ship they meet the component stream lines diverging from the apex of the bow wave and follow the resultant stream lines around the ship.  The velocity in the diverging stream lines must clearly be sufficient to impart the necessary transverse displacement during the time occupied in the passage from the bow to the midship section.  The diverging stream-line velocity will then be proportional to \frac{VA^{\frac{1}{2}}}{L} , in which V is the speed of the ship, A is the area of the midship section, and L the length of the combined entrance and run.  But this diverging stream-line velocity is closely proportional to, if not identical with, the orbital velocity of the bow waves.  Hence these orbital velocities will also be proportional to \frac{VA^{\frac{1}{2}}}{L} , and since the wave of each series grow in like proportion with increase of speed, the orbital velocity of the representative wave also will be proportional to the same ratio.

Assuming now that the waves of the ship possess the properties of trochoidal waves, it follows that the height of the representative wave will be proportional to \frac{lu}{V} , in which l represents the wave length, u the orbital velocity, and V the speed of advance.  Furthermore, the breadth of the wave will be proportional to its height, for as the height increases the orbital velocity increases throughout the entire breadth of the wave, and imparts its motion to more remote particles previously at rest.

The horse-power of a series of trochoidal waves, having a length l , a height h , and a breadth h , is thus expressed by Mr. Albert W. Stahl, U.S. Navy:

The ratio \frac{h^2}{l^2} is constant for similar ships and for all speeds of a given ship, and is proportional to \frac{A}{L^2} .  Since with such a range in this latter ratio as may be found between a torpedo boat and an armored cruiser, the quantity in parenthesis would vary by a fraction of one percent, it may reasonably be neglected.  The wave horse-power then becomes proportional to h^3l^\frac{1}{2} . But l^\frac{1}{2} is proportional to V , and h to \frac{VA^{\frac{1}{2}}}{L} , hence the horse-power absorbed in wave making may be expressed by the formula

in which C_2 is a constant to be derived from the performance of a model ship.

USS Montgomery, Tompkinsville (Staten Island), NY, June 1896

The imaginary representative wave is conceived as having a speed of advance equal to the speed of the ship, and in this respect is plainly in harmony with the transverse series.  The speed of the advance of the diverging waves is dependent not only upon the speed of the ship but also upon the angle of divergence of their crest lines, and this angle of divergence is dependent upon the bluntness of the ship, or in other words upon the ratio \frac{A^\frac{1}{2}}{L} , so long as the mode of distribution of the displacement remains the same.  The length of the diverging waves will then vary with the ratio, and the effect will be that in the blunter ship the diverging series will absorb a greater amount of energy relative to the transverse series than in the ships of finer form.  This relative increase in energy in the diverging series must, however, be accompanied by a corresponding diminution of the energy in the transverse series, for the total energy in both series is derived from the diverging component stream lines, the velocity of which is proportional to \frac{VA^{\frac{1}{2}}}{L} , and this total energy thus derived will be independent of the relative importance of the two waves.  The condition necessary to the preceding statement is that the model ship, whence the constant is derived, and the projected ship shall be similar in cross section and identical in mode of distribution of the displacement.  When this condition is fulfilled, therefore, the representative wave becomes a true measure of all the waves subject to the approximation involved in its trochoidal nature, and in the modification expression (2).

Concerning the limit of speed in the application of the formula, it should be observed that each wave-making feature is assumed to generate its appropriate wave without interference; hence the beginning of each interference marks the limit of its applicability.

At corresponding speeds the power absorbed in wave making becomes proportional to the square root of the seventh power of the scale of comparison, as should be expected.

The Wake Current

The sum of P_f and P_w is the effective horse-power required to tow the ship.  When propulsion is effected by a screw propeller at the stern, other varying elements are introduced.  The following current, by delivering momentum to the propeller, restores a portion of the energy already expended in overcoming the frictional and wave-making resistances.

The Pathfinder

Let V represent the speed of the ship, v the mean speed of the following current, and d the diameter of the propeller.  Then the energy imparted to the propeller per unit of time by the following current will be proportional to

but assuming a constant percentage of real slip, d^2 is proportional to

which being substituted in (4) reduces the increment of energy to the form

The following current has three components of independent origin, viz: the frictional wake, the orbital velocity of the stern wave, and the orbital velocity of the recurring transverse bow wave.  It has been shown that when the waves of the bow series interfere with the formation of the stern wave the limit to the application of this analysis has been reached.  Therefore the effect of the bow wave need not be considered.

The speed of the frictional wake will here be regarded as proportional to the speed of the ship; hence the energy derived from it, being a constant proportion of the total power, need not be considered here.

Still attributing the properties of the trochoidal wave to the waves of the ship, the orbital velocity of the stern wave at the center of the propeller will be proportional to

in which L_a is the length of the after boddy, Exp the modulus of the hyperbolic system of logarithms, h the depth of immersion of the axis of the propeller, and \epsilon a constant such that \epsilon V^2 equals the wave length.

Substituting (6) for v in expression (5) the increment of energy becomes

The value of constant C_3 as derived from the results of the experiment of Mr. George A. Calvert is C_3 = 1.66 ; but this value should be regarded as provisional, since C_3 like C_2 depends upon the mode of distribution of the displacement.

Summary

Let P_1 represent the total effective horse-power required at the propeller.  Then

C_2 and C_3 are constants, depending upon the form of the ship, C_3 having a provisional value of 1.66.

Analysis of Power

If from the indicated horsepower, the power required to overcome the constant resistance of the unloaded engine is deducted, the result will be the new horse-power applied to the propulsion; the the ratio of P_f to this net horse-power will represent the combined efficiency of the mechanism of the engine so far as the variable friction is involved, and of the propeller, including the effect of the frictional wake and of the augmentation of resistance by the propeller.  These latter two effects are thus assumed proportional to the power applied.  The ratio \frac{P_f}{Net\,H.P.} may then be expected to remain constant so long as the propeller efficiency remains constant.

Applications

U.S.S. Bancroft

The results of the progressive trial of the U.S.S. Bancroft given by Passed Assistant Engineer Robert S. Griffin, U.S. Navy, suffice for determination of the value of the constant C_2 of the vessel.  The Bancroft is a twin-screw practice ship of the following dimensions: length of immersed body = L = 189.5 ft.; displacement on trial = 832 tons; area of immersed midship section on trial = A = 277 sq.ft.; wetted surface on trial = S =6,150 sq.ft.

RMS Umbria. The original uploader was Aquitania at Thai Wikipedia – Transferred from th.wikipedia to Commons by Britannic., Public Domain, https://commons.wikimedia.org/w/index.php?curid=5546917

Having calculated the power absorbed by surface friction on the basis of Froude’s coefficient and exponent, and also the value of the factor involving the following current, the constant C_2 is given such a value as will maintain the ratio \frac{P_f}{Net\,H.P.} approximately constant.  The value of C_2 thus found is for the Bancroft 329, and ratio \frac{P_f}{Net\,H.P.} = 0.545 .  The resistance of the unloaded engine is assumed at 2 lbs. per square inch of the area of the low-pressure pistons.  Using these values, the power at the several speeds hsa been calculated and is given in Table I, where the observed powers are also given show the degree of approximation.

In order to illustrate the scope of the formula the ratio \frac{P_f}{Net\,H.P.} is given in Table II together with explanatory data for each of seven vessels, five of them being twin-screw ships of the U.S. Navy, and the two others the S.S. Umbria of the Cunard line and the torpedo boat Sunderland, built by Messrs. William Doxford and Sons, Sunderland, England.  With the exception of the Umbria these vessels may be regarded as approximations to the form of least resistance in smooth water.  The Umbria has a quasi-parallel middle body, the length of which has been approximated by ascribing to the combined entrance and run a prismatic coefficient of fineness of 0.56.  Table III contains the results of the progressive trial of the Umbria together with the calculated power.  The same comparison is shown in Figure I.

In making these calculations, the constants of the Bancroft have been employed, their values being C_1 = 32,566,\,C_2 = 329,\,C_3 = 1.66 .  If should be stated that the application of this value of C_2 to all of these vessels is not strictly warranted, for they are undoubtedly dissimilar in cross section and also in mode of distribution of displacement.  Nevertheless it is assumed that they possess a sufficient degree of similarity to serve the purpose of illustration.

The resistance of the unloaded engine is based on two pounds per square inch of the area of the low-pressure pistons in all cases.  The length L is the combined length of entrance and run, being the length of the immersed body in all cases except the Umbria.  The ratio \frac{L}{A^{\frac{1}{2}}} given as an index to the character of the ship form a propulsive point of view.  If all these vessels possessed the required degree of similarity, and had propellers and engines of equal mechanical efficiency, the ratio \frac{P_f}{Net\,H.P.} should be the same for all, provided the speeds were within the limit indicated.

Limit of Speed

Since the formula gives only the power absorbed in surface friction and in waves generated without interference, it is obvious that it may be used as a measure of the total power only when these elements constitute the entire resistance.  The earliest departure from the condition of applicability may be expected to accompany an interference by the recurring transverse bow wave in the generation of the stern wave.  Since the relative magnitude of the bow and stern waves is dependent upon the form of the ship, it is clearly impossible to formulate generally the limit at which such an interference will occur.  The limit must therefore be found form the performance of the model ship whence the constants in any case are derived.

The U.S.S. Bancroft, within a speed of 14.52 knots, gives no evidence of the limit, as may be seen by reference to Table I.  At this speed V^2 = 1.112L .

The S.S. Umbria gives no indication of the limit within the speed of 20 knots, as may be seen by Table III or Fig. I.  At 20 knots V^2 = 1.136L,\,L=0.8 \times length of ship.

On the other hand, the progressive trial of the torpedo boat Sunderland affords a convenient illustration of the inapplicability of the formula.  Figure 2 shows the power curves, both actual and calculated.

At the speed of 14 knots the value of the ratio \frac{P_f}{Net\,H.P.} is such as to warrant the conclusion that at this speed the formula gives the power with sufficient accuracy.  A glance at the curves, however, shows that this cannot be said of any higher speed until 20 knots is approached.  Fourteen knots may then be regarded as the limit of speed at which the formula may be applied.  At this speed V^2 = 1.43L .  Beyond this speed the familar hump appears.  It may be of interest to note that the curves intersect again in the region of 20 knots, at which V^2 = 2.9L .

It is not intended to question the existence of minor humps at speeds below the limits here noticed, but simply to observe that in the published curves of the Bancroft and Umbria they are not appreciable.  It is possible that such humps may be partially neutralized by the action of a simultaneous increase in the wake current upon the propeller.  In any event, their existence must affect the accuracy of the calculated power, although commonly to such as slight extent as to be inappreciable.

Since the magnitude of the waves of the ship varies with the ratio \frac{A^\frac{1}{2}}{L} it follows that both the size of the hump and the speed at which it appears will be influenced by this ratio.

Conclusion

The method of this analysis may be summarized thus: The resistance of ships is so largely due to surface friction and wave making that all other forms may be neglected except at very high speeds.  Within the limit of speed at which interference occurs in the generation of waves, the formula (eq. 8) enables the power absorbed by these two principal elements to be computed, provided the constants C_2 and C_3 have been experimentally determined by means of a model ship having the same mode of distribution of displacement as the projected ship.  The ratio of the power thus computed to the net power of the engine is constant so long as the propeller efficiency is constant.  Finally, the indicated horse power may be obtained by adding to the net power the power absorbed in the constant resistance of the engine.

Appendix

Fluid Flow in Pipes, Losses and Flow Metering

Channeling fluid flow through pipes is an everyday occurrence.  We see it in plumbing, industrial plants, and fluid power (hydraulic) systems.  A proper understanding of the basics of fluid flow in pipes–including measurement and losses–is essential for proper configuration of these systems.  In this monograph we’ll concentrate on incompressible fluids; compressibility introduces some complications.

Classical Flow Metering: Theory

Euler’s inviscid equation states that

X_i-\frac{1}{\rho}\frac{\partial p}{\partial x_i} = \frac{\partial v_i}{\partial t} + \left( v\cdot grad \right)v_i

The fact that it is inviscid is important; we will see why shortly.

Slater and Frank (1947) show that this can be transformed to Bernoulli’s Equation, thus

\rho V + p + \frac{ 1}{ 2} \rho v^2 = constant

or, for passage from State 1 to State 2,

\rho_1 V_1 + p_1 + \frac{ 1}{ 2} \rho_1 u_1^2 = \rho_2 V_2 + p_2 + \frac{ 1}{ 2} \rho_2 u_2^2 .

Bernoulli’s Equation is basically an expression of conservation of energy.  There are three terms on each side:

  1. The first term is the energy from an external force field.  For practical applications, this means gravity.
  2. The second is the pressure energy.
  3. The third is the velocity energy.  We use u for velocity as a borrowing from CFD, u being the velocity in the x-direction and v being the velocity in the y-direction.  We’ll stick with one-dimensional flow in this case.

For our purposes we will ignore the external force field/gravity term.  Leaving out the gravity term, Bernoulli’s Equation can be written thus:

p_1 + \frac{ 1}{ 2} \rho_1 u_1^2 = p_2 + \frac{ 1}{ 2} \rho_2 u_2^2

where p is the pressure and \rho is the density.

Let us now add the assumption that the fluid is incompressible, thus

\rho_1 = \rho_2 = \rho

Dividing through by the density yields

\frac{p_1}{\rho} + \frac{ 1}{ 2} u_1^2 = \frac{p_2}{\rho} + \frac{ 1}{ 2} u_2^2

At this point the business about being inviscid comes into play; both Euler’s and Bernoulli’s Equations assume no energy losses due to viscosity.  We could try something really fancy (like starting with the Navier-Stokes equations, which were actually first formulated by Saint-Venant) or we could do something simplistic like add a head (energy per unit mass) loss, like this:

\frac{p_1}{\gamma} + \frac{ 1}{ 2g_c} u_1^2 - h_L= \frac{p_2}{\gamma} + \frac{ 1}{ 2g_c} u_2^2

Note that we have also divided through by the acceleration due to gravity g_c ; this is so that the units of head loss will be in unit length, which will match them with a manometer.  \gamma is the unit weight of the fluid.

Rearranging terms yields

\frac{p_1 - p_2}{\gamma} - h_L= \frac{ 1}{ 2g_c} \left( u_2^2 - u_1^2 \right)

Now let us turn to continuity of mass flow, which (unless a leak is sprung somewhere) is not subject to loss, or

\rho_1 A_1 u_1 = \rho_2 A_2 u_2

Since the fluid is incompressible, the densities cancel out.  At this point we also make another simplifying assumption: the velocity of the fluid across the cross-section of the pipe or restriction is uniform, or at least that the variation across the cross-section is not significant.  Solving for u_1 ,

u_1 = \frac{A_2}{A_1} u_2

Substituting this into the last form of Bernoulli’s Equation yields

\frac{p_1 - p_2}{\gamma} - h_L= \frac{ u_2^2}{ 2g_c} \left( 1 - \left( \frac{A_2}{A_1} \right)^2 \right)

If we are reading a U-tube manometer, the difference in column heights will be

\Delta H = \frac{p_1 - p_2}{\gamma}

Substituting this bring us to

\Delta H - h_L= \frac{ u_2^2}{ 2g_c} \left( 1 - \left( \frac{A_2}{A_1} \right)^2 \right)

Assuming the uniform flow across the cross-section, the flow rate is

Q = u_1 A_1 = u_2 A_2

Substituting and solving for the flow rate, we have at last

Q = A_2 \sqrt{\frac{2g_c\left( \Delta H - h_L \right)}{\left( 1 - \left( \frac{A_2}{A_1} \right)^2 \right)}}

Up to now, it probably seems that we’ve pulled several “rabbits out of a hat.”  These are simplifying assumptions that are based on experimental experience, the physical reality of the experiment, or both.  But we have saved the biggest rabbit for last: we will dispense with the head loss term h_L (Mott (1994)) and restate the equation above as

Q = C_d A_2 \sqrt{\frac{2g_c\Delta H}{\left( 1 - \left( \frac{A_2}{A_1} \right)^2 \right)}}

C_d is referred to as the coefficient of discharge; 0 < C_d < 1 .  We can rearrange this equation to determine C_d experimentally as follows:

C_d = \frac{Q}{A_2 \sqrt{\frac{2g_c\Delta H}{\left( 1 - \left( \frac{A_2}{A_1} \right)^2 \right)}}}

Classical Flow Metering: Implementation

Now that we have all of this theory, we ask ourselves, “What’s it good for?”  There are two ways we can take this; the first thing we’ll consider is flow measurement.  In short, we can set up a place in the flow where we change the flow area and, by measuring the pressure difference, we can determine the flow.  This is the way flow has been measured in pipes for many years.

Let’s consider a couple of examples.  The first is a sharp-edged orifice, simply a plate placed in the flow stream as shown below.

Sharp-Edged Orifice. From Lewitt, E.H. (1923) Hydraulic and Fluid Mechanics. London: Sir Isaac Pitman & Sons, Ltd.

Here we see a U-tube manometer which measures pressure on the upstream and downstream side of the orifice.  The fluid velocity varies as we have shown.  The placement of the orifices is an important problem in using this type of device.

The major problem with this type of orifice is that it will measure fluid flow, but generates significant losses in use.  A more efficient measuring device from that standpoint is the venturi meter, shown below.

Venturi Meter. From O’Brien, M.P., and Hickox, G.H. (1937). Applied Fluid Mechanics. New York: McGraw-Hill Book Company. George Hickox worked for the Tennessee Valley Authority; the building of the original TVA system had many technological spin-offs.

The setup is similar except for the (God forbid) mercury manometer, which worked but, because of its hazards, is mercifully rare.  The practical drawback to a venturi flowmeter is the long pipe length it occupies, which can be difficult to include in a busy piping system.

The efficiency variation can be seen in the coefficients of discharge, which vary from around 0.6 for a sharp-edged orifice to just under unity for a venturi meter.  There are also flow meters which occupy a middle ground between the two, such as the rounded edge manometer.

As flow meters go, classical flow meters are not as common as they used to be to measure flow.  We think of other methods as “new” but the Rotameter was first patented in 1908.  At the right is a variation of the concept dating from 1914.  Rotameters are still very commonly used, although their accuracy can be highly variable.  Today we have devices such as paddle wheel and magnetic flowmeters to more accurately measure the flow and to report the results to computer controlled systems.

“Minor” Losses in Systems

Another important application of this theory is the prediction of “minor” losses in such restrictions as valves, elbows, enlargements or contractions in the piping and other flow constrictions in the system.  The use of the term “minor” is misleading; careless design can turn these into major losses very quickly, and the designer must be diligent in avoiding them if he or she wants a successful design.  (Unless, of course, you’re trying to restrict fluid flow…)

Picking up where we left off in the theory, dividing both sides by the small area yields

u_2 = C_d \sqrt{\frac{2g_c\Delta H}{\left( 1 - \left( \frac{A_2}{A_1} \right)^2 \right)}}

Eliminating the square root,

u_2^2 = C_d^2 \frac{2g_c\Delta H}{\left( 1 - \left( \frac{A_2}{A_1} \right)^2 \right)}

Solving for the head differential,

\Delta H = \frac{u_2^2}{2g_c} \frac{\left( 1 - \left( \frac{A_2}{A_1} \right)^2 \right)}{C_d^2}

Since we normally prefer to consider the velocity in the incoming pipe u_1 , we substitute to

\Delta H = \frac{u_1^2}{2g_c} \frac{\left( \frac{A_1}{A_2} \right)^2\left( 1 - \left( \frac{A_2}{A_1} \right)^2 \right)}{C_d^2}

and then

\Delta H = \frac{u_1^2}{2g_c} \frac{\left( \left( \frac{A_1}{A_2} \right)^2 - 1 \right)}{C_d^2}

We normally prefer to determine the second fraction on the right hand side experimentally as a loss or resistance coefficient K , so at last

\Delta H = K \frac{u_1^2}{2g_c}

The derivation indicates that, unlike the discharge coefficient, the loss or resistance coefficient can and frequently does exceed unity.  It is obviously possible to use this equation to determine K experimentally by solving for it.

Equivalent Length

One way to express the effect a restriction has on the system is to compute its equivalent length of pipe.  Thus, we can say that, for a given diameter of pipe, a restriction has the same effect on the pressure drop of a system as a certain length of pipe L_e would have.  That length of pipe is given by the equation

L_e = D_1 \frac{K}{f_t}

The friction factor f_t is determined from the well-known Moody chart in the turbulent region only.  A good correlation for this can be found with the equation (Mott (1994))

f_t = \frac{1}{4\left[ log\left( \frac{1}{3.7\frac{D_1}{\epsilon}}+\frac{5.74}{R_e^{0.9}} \right) \right]^2}

where R_e is the Reynolds Number and the fraction and \frac{D_1}{\epsilon} is the relative roughness of the pipe, or the ratio of the diameter to the height of the asperities along the wall of the pipe.

Equivalent length is a good practical way of looking at the losses in a section of pipe with all of its restrictions without getting into “apples and oranges” comparisons.  Since they are in series, the equivalent and actual lengths of the pipe are simply added.

Conclusion

Losses in fluid elements such as valves and pipes can be considered in two ways.  Orifices such as the sharp-edged orifice and the venturi can be used to measure the flow in the system.  The theory can be flipped to enable us to estimate losses in the system and thus avoid unwanted energy, pressure and flow degradation of the fluid system.