Compressible Flow Through Nozzles, and the Vulcan 06 Valve

Most of our fluid mechanics offerings are on our companion site, Chet Aero Marine.  This topic, and the way we plan to treat it, is so intertwined with the history of Vulcan’s product line that we’re posting it here.  Hopefully it will be useful in understanding both.  It’s a offshoot of Vulcan’s valve loss study in the late 1970’s and early 1980’s, and it led to an important decision in that effort.  I am indebted to Bob Daniel at Georgia Tech for this presentation.

Basics of Compressible Flow Through Nozzles and Other Orifices

The basics of incompressible flow through nozzles, and the losses that take place, is discussed here in detail.  The first complicating factor when adding compressibility is the density change in the fluid.  For this study we will consider only ideal gases.

Consider a simple orifice configuration such as is shown below.


The mass flow…

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Installing OpenWatcom Fortran 77

We offer the OpenWatcom Fortran 77 on our Fortran 77 Resources page.  Below is a brief video on how to install it.

The installation is pretty straightforward.  If you have a purely 64-bit version of Windows, it won’t install; it doesn’t go beyond 32-bit.  The installation in Wine for Linux is similar and it runs fine there too.

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.


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


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.


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.


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,

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.


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.


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.


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.