New technologies for collecting and processing velocity data make it possible to obtain a detailed estimate of the velocity head correction factor. This factor traditionally has been either approximated or ignored in turbine efficiency measurements. Results of hydraulic laboratory tests suggest these technologies offer potential for significant improvement of the testing protocol for any reaction turbine, but especially for lowhead turbines with nonuniform exit velocities.
By Lee H. Sheldon and Rodney J. Wittinger
A hydraulic turbine’s power output is typically computed as the product of the flow rate, the head, a conversion constant, and the unit efficiency. Conversely, to determine the efficiency of a hydraulic turbine, four parameters must be known:
– Flow rate;
– Head;
– Power output; and
– The specific weight of water.
One important consideration in determining efficiency is head measurement, including the effect of customary simplifications on determining the velocity head at a turbine draft tube exit.
Head is commonly characterized as the difference in the upstream and downstream water surface elevations. In actuality, head (now more often called “specific energy”), is the difference between the upstream energy level of the fluid entering the turbine and the downstream energy level. The velocity head is a component of the energy level at any measuring section. Any error in the velocity head introduces an error in the head and consequently in the efficiency.
Some hydraulic history
Calculations of the energy in flowing water rely upon the Bernoulli equation, which represents the total energy as the sum of the velocity head, the pressure head, and an elevation relative to some fixed datum. The change in energy is equal to the change in this sum. In developing the complete equation in the 18th century, Leonhard Euler (who named the equation for the significant contributions of his friend, Daniel Bernoulli) derived one of the three familiar expressions of energy as:
Equation 1:

H
_{v} = V
^{2}/2g
where:
– H _{v} is velocity head;
– V is velocity; and
– g is the acceleration of gravity.
Velocity head is expressed as energy per unit weight, with the English units being footpounds per pound, or simply feet after canceling the units of weight.
Early hydraulic experts found the velocity head term to be problematic, because using average velocity to compute the velocity head yielded an incorrect estimate of the total kinetic energy in a volume of water. Gaspard Gustave de Coriolis formulated the expression:
Equation 2:

H
_{v} = aV
_{avg}^{2}/2g
where the correction coefficient a was estimated to be a constant between 1.40 and 1.47.
However, a rival of Coriolis’s, hydraulic theoretician Pierre Vauthier, later recognized that a was a variable whose value depended upon the velocity profile in the fluid being measured.^{1}
Vauthier’s findings have significance for modern turbine performance testing. When developing performance requirements for replacement turbines at its aging hydroelectric facilities, the U.S. Army Corps of Engineers found it useful to revisit the question of the velocity head correction factor. The Corps applied modern technology to compute velocity heads from detailed measurements of the flow profile in a laboratory model of a complete turbine draft tube. Although the results are qualified by limited data sets and the difficulty in transferring laboratory results to prototype performance, the Corps’ investigations suggest the potential for significant improvements in performance test procedures.
Applying the correction factor in hydraulic testing
Today, hydraulic engineers understand that the correction coefficient a is equal to 1.0 only when the flow profile is completely uniform. In sections having nonuniform flow, it is always greater than 1.0. The velocity head correction factor at a given nonuniform flow section may be calculated as the ratio of the actual kinetic energy in the fluid to the kinetic energy if the flow were completely uniform, leading to the following equation:
Equation 3:

a = [Sv
^{3}a ]/[V
_{avg}^{3}A]
where:
– v is the velocity at an infinitesimal stream tube;
– a is the cross sectional area of the stream tube;
– V _{avg} is the average velocity in the whole cross section; and
– A is the area of the whole cross section.
Although Equation 3 represents an advance in the understanding of the velocity head correction coefficient, in practice it has been applied only rarely because of the difficulty in determining velocity profiles. However, it is possible to use Equation 3 to compute a numerically, if a matrix of point velocity measurements is available.
Measuring velocities in a laboratory model
To evaluate the effect of draft tube velocity profiles on the velocity head, researchers at the Corps’s Engineering Research and Development Center (ERDC) in Vicksburg, Miss., measured point velocities in a 1:25 scale model of the existing turbines at the Corps’s 810MW Lower Granite hydroelectric project. Lower Granite, on the Snake River in eastern Washington, contains six 135MW generating units.
In the laboratory tests, the horizontal components of the velocities upstream of the draft tube exit were measured using a laser Doppler velocity meter. The laser Doppler velocity meter uses precisely controlled light frequencies to measure water velocity; laboratory reports indicate an accuracy of better than 0.15 percent.
At the upper velocity range encountered in the Lower Granite experiments, the meter’s accuracy corresponds to a difference of plus or minus 0.01 foot per second in the prototype. The meter does not require calibration, as there is no drift in the light frequencies emitted regardless of air temperature, water temperature, or equipment aging. The laser Doppler velocity meter is also non intrusive, which allows the flow field to be measured without introducing disturbances from the meter.
Velocity measurements were taken sequentially at points over a uniformly spaced grid, or matrix, within each of two draft tube barrels. The use of an evenly spaced grid, including edges, greatly simplifies spreadsheet manipulation of the data, as the crosssectional areas associated with all of the point measurements are identical. It is not necessary for the grid to be completely filled in for the calculation methodology to work. If points are missing, for whatever reason, the user needs to alter the number of grid spaces in the formulas. With this adjustment, the user can expect some increase – albeit minute – in uncertainty.
Sample data sets for two different operating conditions, illustrated in Figure 1 and Figure 2 on page 58, show substantial variation in point velocities across the model draft tube, including some areas of reverse flow.
Processing the velocity data
The measured velocity grid (see Table 1 on page 58) is processed in a Microsoft Excel spreadsheet to develop correction factor a. Table 1 is a record of the point velocity measurements in the left barrel (looking upstream) of the Lower Granite scale model’s draft tube. (The Lower Granite draft tube has two barrels.)
In prototype terms, the grid consists of 121 velocity measurements, taken 6.25 feet upstream of the draft tube exit at a discharge of 22,750 cubic feet per second (cfs). From the 121 point values in Table 1, we can calculate:
Equation 4:

V
_{avg} = 9.6 feet/second
Equation 5:

V
_{avg}^{2}/2g = 1.431 feet
Equation 6:

V
_{avg}^{3} A = 106,804 feet
^{5}/second
^{3}
In these equations, the variables V_{avg} and A are defined as in Equation 3 (see page 54).
Table 2 on page 60 gives the computed point values of v3a, the summation of which will be the numerator in the velocity head correction factor a (Equation 3). The denominator is given in Equation 6.
Finally, we compute a by:
Equation 7:

a = [Sv
^{3}a]/[V
_{avg}^{3}A] = 208,909/106,804 = 1.956
and the velocity head, H_{v}, by Equation 2.
Results of these computations show that the corrected velocity head (2.80 feet) is almost two times the value that was computed directly from the average velocity (1.43 feet), as is customarily used.
The same analysis was performed for the right draft tube barrel.
Although the velocity distributions in the two barrels were quite different (see Figure 2), the computed value of a was 1.924, a value very similar to the correction factor computed for the left barrel.
Note that some of the point velocity values are negative, indicating reverse flow. In the authors’ opinion, this poses no theoretical problem in calculating the average velocity. However, the subject of calculating a velocity head correction factor at a section having zones of reverse flow has not, to our knowledge, been addressed in the literature.
The formula for the velocity head correction factor actually accounts for negative velocities, as can be shown through the following deductive reasoning: Assume a situation where the reverse flow approaches the exact outflow over an equal crosssectional area. In this case, the average velocity and average velocity head approach zero, so the denominator of the correction factor also approaches zero. However, the correction factor cannot become infinite, because that represents a physical impossibility. Therefore, the numerator must also approach zero, no more slowly than the denominator. We also know that the correction factor is greater than or equal to 1.0, so the numerator must approach zero no faster than the denominator. Since the numerator must approach zero at the same rate as the denominator, the negative velocities in the numerator must retain their algebraic negative sign in the summation.
Practical implications for testing and test codes
The laboratory results raise some important questions about traditional methods of calculating velocity head. Historically, velocity head has been calculated from the average velocity with the correction factor being neglected. But, in the laboratory draft tube barrels, the highly nonuniform flow resulted in a velocity head correction factor of almost 2.0.
Table 1: Grid of Point Velocities Measured at Various Locations in the Draft Tube of the Lower Granite 1:25Scale Turbine Model1 
The high correction factor indicates that considerably more energy is being discharged from the draft tube, and hence not converted to mechanical power, than would be calculated based on the average velocity. In an efficiency test, any error in calculating the velocity head exiting a hydraulic turbine would lead to a proportional error in calculating the true efficiency of the turbine. As the velocity at the draft tube discharge is always higher and less uniform than at the intake, underestimating the velocity head correction factor results in overestimating the difference between upstream energy and downstream energy – and hence overestimating total specific energy at the turbine. Consequently, the actual efficiency is higher than calculated by the present method.
Table 2: Computed Values of v^{3} for each Velocity Point Measured in the Draft Tube of the Lower Granite 1:25 Scale Turbine Model 
The current test codes of both the American Society of Mechanical Engineers (ASME) and International Electrotechnical Commission (IEC) stipulate that downstream energy is to be calculated at a point before the draft tube exit. An accurate estimate of the velocity head correction factor is especially important for testing under these codes, because velocity profiles tend to be highly nonuniform near the draft tube exit. In contrast, a 1949 version of ASME’s PTC18 test code stipulated that downstream energy was to be evaluated in the tailrace downstream of the draft tube exit. Under these conditions, the error introduced by using an average velocity head would be much less, because flow in the tailrace is typically much slower and more uniform than flow in the draft tube.
Improving current practices
Accurately estimating the velocity head correction factor a could be beneficial in identifying performance improvements in turbine rehabilitation. The correction factor also offers potential for improving turbinedraft tube designs or modifications for fish passage, both in terms of performance and environmental conditions for fish.
Under current ASME and IEC test codes, the magnitude of the velocity head correction factor at the draft tube exit may introduce a significant error in determining the actual efficiency of a hydraulic turbine. The 1949 version of the ASME code did not introduce this error because, as stated earlier, its measuring station was downstream of the draft tube exit, where the true energy level could be more accurately determined. However, current ASME and IEC codes call for moving the measurement location for downstream energy from the tailrace, one monolith downstream of the draft tube exit, to a point inside the draft tube. This change was made to reduce any bias errors in laboratory model testing. However, it seems this change may introduce more error under some conditions than was eliminated in using the older method.
We contend that ignoring the velocity head correction factor in the current codes will cause errors in measuring the true efficiency of a turbine – errors that become proportionally larger as the head becomes smaller.
If a magnitude of error is confirmed, we recommend implementing one of two correction procedures: 1) use the older procedure where the downstream energy level is measured in the tailrace; or 2) specify that the model test includes measuring and accounting for the velocity head correction factors at the different flow rates. n
Messrs. Sheldon and Wittinger may be contacted at the U. S. Army Corps of Engineers, Hydroelectric Design Center, P.O. Box 2946, Portland, OR 972082946; (1) 5038084298 (Sheldon) or (1) 5038084280 (Wittinger); Email: [email protected] or [email protected]
Note
 Rouse, H., and S. Ince, History of Hydraulics, Iowa Institute of Hydraulic Research, Iowa City, Iowa, 1957.
Lee Sheldon, P.E., and Rod Wittinger, P.E., hydraulic turbine engineers in the mechanical section of the U.S. Army Corps of Engineers’ Hydroelectric Design Center, have a combined total of more than 70 years’ experience in hydraulic turbines. They collected the laboratory data in this article using a fish guidance efficiency model built in connection with the Corps’ turbine survival program.
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