Revival of water table experiment in fluid mechanics courses,part II

Abstract

In an effort to find a simple and efficient method for observing and predicting compressible flows in an instructional environment, we revisit the classical hydraulic analogy. The focus is made on using a water table apparatus to visualize the formation of oblique surface waves in a flow around a wedge. These waves are analogous to the oblique shock waves seen in a compressible flow. A mathematical derivation of this analogy is presented together with water table experiment data collected by graduate students. The comparison between the predicted values and measured values is made and shows a remarkable correlation between the two for simulated Mach numbers.

Keywords

Water table hydraulic analogy compressible flow visualization oblique shock relation

Introduction

Upper division and graduate level courses such as aerodynamics, aircraft design, propulsion, and flight mechanics spend a good portion of the curriculum addressing oblique shocks in compressible flows. This subject is discussed in the context of lift and drag forces generated over the supersonic airfoil, shock formation inside the air-breathing engine, the shock patterns of the rocket engine exhaust, and other applications. The derivation of fundamental equations are quite laborious especially considering the $β - θ - M$ relation, which describes oblique shock angle $β$ for given deflection angle $θ$ and a flow Mach number $M$ . Following these derivations, a classical example of a flat plate or a diamond-shaped airfoil at an angle of attack in a supersonic flow is usually discussed. However, to promote active learning, it is beneficial for a student to use the derived equations in an applied experiment. The experimental facility required for a demonstration of an oblique shock and the associated maintenance of such facility is usually expensive. For example, a Gunt Humburg HM 172 Supersonic wind tunnel costs about $400,000.¹ The cost is often prohibitive for a teaching college or university that does not have research funding for such equipment.

The water table is a much cheaper device that can be used to simulate the supersonic flow by making an analogy between compressible waves in a gas to surface waves in an incompressible fluid.^2,3 The mathematical formulation for the analogy has been developed by Jouget⁴ and Riabouchinsky⁵ and mentioned earlier by Mach.⁶ Since then, the analogy was applied in studied of supersonic nozzles,^7–9 channels.^10,11 and diffusers.¹² More recently, Matthews used the water table to study the compressible effects on the airfoil cascade.¹³ Lavicka used the method to study compressible flow outside a nozzle.⁹ The hydraulic analogy was also given attention in studies of oblique shocks. Arendze¹⁴ used the analogy to predict the oblique shock formation around stationary and accelerating wedges. Elward¹⁵ predicted the formation of oblique shock waves in the context of studying the efficiency of a gas turbine over the lifetime of the turbine blade. Hatch presented a study of oblique shocks over airfoils.¹⁶

The cost effectiveness of the hydraulic analogy using a water table is an attractive option for instructional use in academia.^14,16–18 Linke et al. present a comprehensive description of an entire undergraduate course on airfoil and nozzle design and manufacturing utilizing a water table.^19.20 Morishita provides a derivation for the hydraulic analogy for the oblique shock waves to be used in a cost-effective student demonstration²¹; however, the comparison between the experimental data and theoretical predictions seems to be lacking. Hafez et al.²² show pictures from flow visualization pictures over various shapes that can be used in schools to illustrate compressible flow phenomena.

This article describes an experiment that is implemented in graduate fluid mechanics courses at Manhattan College designed to promote active learning during lectures on oblique shocks. In this article, we revisit the classical derivation of the hydraulic analogy of the oblique shocks. Then, student data from an experiment are presented that shows comparison between the predictions of shock angles for various deflection angles using this theory and the measured values from the water table.

Theoretical considerations

Here, we derive the expression for the oblique hydraulic jump deflection angle θ as a function of upstream Froude number $F r_{1}$ and oblique hydraulic jump angle β. This is similar to the derivation of Elward¹⁵ except for two items: Elward states that the momentum equation in the tangential direction yields $V_{1 t} = V_{2 t},$ whereas in the current paper, we use a coordinate-shift method to obviate the need for the tangential momentum equation; we have also derived a simpler formula for the deflection angle $θ$ . For these two reasons, we believe that it is pertinent to re-derive the final, simpler, equation. Figure 1 shows a schematic of a classical compressible flow over a wedge illustrated by a thick black line forming an angle $θ$ with the free stream flow $V_{1}$ from left to right. An oblique shock is formed at an angle $β$ . Free steam velocity $V_{1}$ can be decomposed into velocity components normal and tangential to the shock wave denoted with subscript $n$ and $t$ , respectively. Similarly, velocity downstream of the shock wave $V_{2}$ can be decomposed into $V_{2 n}$ and $V_{2 t}$ .

Figure 1.

Sketch of velocity components upstream and downstream of a hydraulic jump produced by a deflection.

In this paper, we start with the normal hydraulic jump, using the well-known results for conditions before and after the jump. To simulate an oblique jump as in Figure 1, we do a “thought experiment” in which we observe the normal jump, except we are in inertial coordinates moving tangential to the jump at arbitrary speed V_p.

For a normal hydraulic jump it is known²³ that the depth ratio is

\frac{y_{2}}{y_{1}} = \frac{1}{2} [\sqrt{1 + 8 F r_{1}^{2}} - 1]

(1)

and the velocity ratio is given by

\frac{V_{2}}{V_{1}} = \frac{y_{1}}{y_{2}}

(2)

where

y

and

V

are fluid height with velocity upstream and downstream of the shock represented by subscripts 1 and 2 and

F r

is Froude number. Hence, we have for the velocity change (using subscript _n to indicate velocity normal to the jump)

\frac{V_{2_{n}}}{V_{1_{n}}} = \frac{2}{[\sqrt{1 + 8 F r_{1_{n}}^{2}} - 1]}

(3)

If we now change coordinates (as described above), choosing ones moving parallel to the jump at arbitrary speed $V_{p} (= V_{1_{t}} or = V_{2_{t}})$ , we can derive from geometry the velocity components before and after the jump

\tan β = \frac{V_{1_{n}}}{V_{}_{p}}

(4)

and

\tan (β - θ) = \frac{V_{2_{n}}}{V_{}_{p}}

(5)

Here, $β$ is the angle the incoming flow makes with the now oblique jump, and ( $θ$ – $β$ ) is the angle between the downstream flow and the jump. Eliminating $V_{p}$ from the above equations and using the equation for the velocity change leads to

\frac{V_{2_{n}}}{V_{1_{n}}} = \frac{2}{[\sqrt{1 + 8 F r_{1_{n}}^{2}} - 1]} = \frac{\tan (β - θ)}{\tan β}

(6)

We next need to rephrase the equation in terms of $F r_{1}$ the incoming flow Froude number, using

\sin β = \frac{V_{1_{n}}}{V_{1}} = \frac{F r_{1_{n}}}{F r_{1}}

(7)

Hence, after simplifying

\frac{\tan (β - θ)}{\tan β} = \frac{2}{[\sqrt{1 + 8 F r_{1}^{2} \sin β^{2}} - 1]}

(8)

This equation relates the flow deflection angle $θ$ and the oblique hydraulic jump angle $β$ in terms of the incoming Froude number.

It can be solved directly for $θ$ given $β$ and $F r_{1}$

θ = β - \tan^{- a} [\frac{2 \tan β}{\sqrt{1 + 8 F r_{1}^{2} \sin β^{2}} - 1}]

(9)

and iteratively for

β

given

θ

and

F r_{1}

. Note that this formula is significantly more compact than the equivalent result presented by Elward,¹⁵ although it yields the same results; in addition, it indicates explicitly the difference between the deflection angle θ and the jump angle β. Elward’s formula is

θ = \tan^{- a} [\frac{\tan β (\sqrt{1 + 8 F r_{1}^{2} \sin β^{2}} - 3)}{2 \tan β^{2} + \sqrt{1 + 8 F r_{1}^{2} \sin β^{2}} - 1}]

(10)

Note also that Morishita²¹ recently presented the same simplified formula; however, the derivation was not presented, so the above derivation is included here for completeness, as well as pedagogical reasons. Equation 9 is implicit in jump angle β but is easy to solve on the PC (e.g., using Excel’s Solver, Goal Seek function, or iterative methods in MATLAB).

Experimental setup

The experiment described in this article was performed on a commercial Aerodynalog bench-top water channel manufactured by AMRAD INC, shown in Figure 2 and described in more detail in Part I.¹⁸ The device is about 1.0 m × 0.5 m × 0.3 m in overall dimensions and consists of a closed loop channel, where water is driven by an electric impeller. The flowrate of the water can be adjusted from a maximum of about 1.5 L/s to 0.0 L/s by a valve that restricts the flow. After the impeller the water passed through flow-straighteners and was accumulated in a reservoir with depth $h_{o}$ shown on the left side of Figure 2. A water reservoir drains into the test section approximately 0.15 m and 0.10 m in length and width through a converging–diverging channel. The width of the cannel can be adjusted from about 100 to 30 mm resulting in adjustable water height $h$ in the test section. Based on the ratio of water height in the reservoir and the test section, the flow Froude number can be found, which is analogous to Mach number in compressible flow¹⁸

M = F r = \sqrt{2 (\frac{h_{o}}{h} - 1)}

(11)

Figure 2.

A water table manufactured by AMRAD, Inc. used in the experiment.

Two micrometer gauges were used to measure the reservoir water height $h_{o}$ and local water height $h$ . The device has an electric circuit that includes water, micrometers, and an indicator light. An indicator light was activated once the micrometers touched the water hence completing the circuit, allowing for accurate measurement of water level. Once the measurement was recorded, the micrometers were retracted to avoid disturbing the flow over the model.

To produce an oblique shock wave, several wedges were 3 D printed from ABS material and sanded using a fine-grit sandpaper to produce a sharp edge. The angle formed between two sides of the model is twice the deflection angle, as defined in Figure 1. Eight models were used: $θ = 5 °, 7.5 °, 10 °, 12.5 °, 15 °, 17.5 °, 20 °, 22.5 °$ can be seen in Figure 3. The smallest angle wedge produced consistently poor results, resulting in error of greater than 22%, which is greater than the desired $10 %$ error, and was therefore excluded from the results. The poor performance may be due to difficulty to sharpen the edge to indeed have a five-degree tip.

Figure 3.

Wedge models used in the experiment.

Several patterns were printed in color and placed underneath the test section to better visualize the deformation of water surface and identify the location of water waves. Some of the patterns used are seen in Figure 4. Students used their smart phones to capture a still image of the pattern through the flowing water. The pattern seen though the water was distorted due to refraction caused by uneven water surface associated with the waves. The location of the most distorted pattern was associated with the steepest wave front on the water and therefore was identified as the location of the wave. The bullseye pattern seemed to work best for visualizing the oblique shock waves. It is evident, however, that a more quantitative approach may be implemented to better estimate the wave location. Garretson et al. present a more systematic approach to measuring the distortions due to refraction through the water wave and can be used to better estimate the location of the wave.²⁴

Figure 4.

Patterned templates used to visualize surface waves.

Procedure

The width of the throat in the converging–diverging section was varied to give a range of $h_{o} / h$ values resulting in Froude numbers ranging from 1.60 to 3.33. Once the water height ratio was set resulting in a desired flow condition, one of the several wedge models was secured in the holder. Students then used their smartphones to capture the image of the pattern seen from above the test section. A sample image captured is in Figure 4(a). On the right-hand side of the image, a blue wedge model holder can be seen. The white wedge model is shown attached to the holder. Oblique shock waves are formed on both sides of the wedge. Each oblique shock wave is not well defined as it is supplemented by a cascade of smaller capillary waves. The presence of the capillary waves was slightly inhibited by adding several drops of dishwasher soap to the water to reduce the surface tension. After the image was captured, students installed another model and capture the associated image. Students installed models with increasing deflection angle until they tested all models or until they saw a detached shock.

Students uploaded their captured images onto a laptop computer and used a MATLAB program to measure the angles of the oblique shock. This was done by visualizing an image in MATLAB and selecting points A, B, and C shown in Figure 5(a) using ginput built-in function, which returns coordinates of each point. Point C was identified as the location associated with largest distortion of the pattern. In Figure 5, the outer black circle in the pattern appears as a cascade of peaks. The largest peak represents the location of highest curvature of the water, which was a good estimate for the location of the dominant wave front. Based on coordinates of point A and B, the reference horizontal line was established. Based on coordinates A and C, the oblique shock line was established. The measured angle $β$ was calculated using the coordinates of the horizontal reference line and the oblique shock line. The resulting values were then recorded and compared to values predicted by equation (9). Measured and predicted angles $θ$ were also visualized by MATLAB program and can be seen in Figure 5(b). To quantify the uncertainty of this method, same MATLAB code was used to measure the angles of photographs of two drawn lines forming a known angle between them. The difference between known and the measured angles was at most $\pm {0.94}^{o}$ .

Figure 5.

(a) Raw sample image of the oblique shock at $F r = 2.60$ and $θ$ = 7.5. (b) Angles computed using MATLAB program from selected points A, B, and C.

The experiment was designed to be performed by graduate students, who usually have a 3-h weekly class session. The first half of the session was spent deriving the hydraulic analogy and any theoretical considerations that are needed to perform the experiment. In the second half of the session, a class consisting typically of 15–20 students was divided into groups of two students. While the rest of the class started coding the theoretical prediction curves, one group was selected to collect data for one Froude number. Each group was given 10 min to adjust the apparatus to the desired Froude number and perform experiment for all wedge models. Allowing each group to set the apparatus to a unique Froude number allowed all students to familiarize themselves with other concepts such as back pressure as for each unique setting of the converging–diverging channel, the flow rate had to be adjusted to eliminate any waves in the channel, resembling concept of back pressure in the compressible flow.¹⁸ As their homework, students were asked to compare the measured $β$ values to the predicted ones. Results were discussed in the following lecture. Allowing each group to change the Froude number resulted in a complete $θ - β - M$ map representing wide range of Mach numbers and deflection angles. Results for only three unique Froude numbers are discussed in the next section.

Results

Table 1 shows a summary of results obtained by students in the experiment. For all Froude numbers, the wedge with a $5 °$ deflection angle consistently produced poor results with experimentally measured angle significantly deviating from the predicted one. Therefore, results for $θ = 5^{o}$ were omitted. In the case of $F r = 1.91,$ a detached shock was observed for a $θ = 20^{o}$ wedge, so a model with $θ = {22.5}^{o}$ deflection angle was not studied. A sample image of the detached sock for $F r = 1.91$ case is seen in Figure 6. For $F r = 2.31$ and $F r = 3.33,$ no detached shocks were observed or predicted, and all models were tested successfully.

Table 1.

Sample results of oblique shock study.

Deflection angle $θ (^{o})$	Shock angle $β (^{o})$ Measured on water table	Shock angle $β (^{o})$ Predicted by equation (9)	% Error
$M = Fr = 1.91$
7.5	40.2	40.7	1.2
10	43.6	40.0	8.9
12.5	47.4	37.8	25.4
15	52.1	53.6	2.9
17.5	59.1	54.3	8.9
20	DS	DS	DS
$M = Fr = 2.31$
7.5	33.2	32.3	2.9
10	36.0	32.8	9.8
12.5	39.0	38.0	2.7
15	42.2	43.9	3.8
17.5	45.8	46.2	0.8
20	49.9	48.9	2.1
22.5	55.0	54.6	0.8
$M = Fr = 3.33$
7.5	24.2	24.2	0.1
10	26.6	27.1	1.9
12.5	29.2	30.8	5.4
15	31.8	31.9	0.2
17.5	34.6	35.0	1.3
20	37.5	34.3	9.2
22.5	40.5	37.0	9.2

DS: Detached shock.

Figure 6.

Observed detached shock at $F r$ = 1.91 and $θ$ = 20.

Table 1 also shows the error associated with each trail between the predicted and measured values of the angle. Results show a remarkable agreement with predicted angles with error less than 10% in all but one trial. In the case of $F r = 1.91$ and a deflection angle of $θ = 12.5$ , the error exceeded 25%. This trial was considered an outlier. Overall, error valued proved this method sufficiently accurate for a good laboratory demonstration. However, it is worth mentioning that the success of this demonstration strongly depends on student’s ability to accurately identify the location of a surface wave from the image. In the present study, choosing systematically the location associated with the location of maximum template circle distortion worked well.

Figure 7 resents a graphical view of $β$ angle predicted by equation (9) and the associated experimental angles. The experimental data and the curves representing predicted values align nicely confirming the validity of the method.

Figure 7.

$θ - β - M$ with lines representing the solution obtained using equation (9) for $F r = M = 1.91$ (solid line), $F r = M = 2.31$ (dashed line), $F r = M = 3.33$ (dash-dotted line), and associated experimental data for $F r = M = 1.91$ (●), $F r = M = 2.31$ (■), and $F r = M = 3.33$ (▸).

Upon completion of the experiment, 18 students enrolled in the class were asked to fill out the survey evaluating this experience in the laboratory as pertaining to understanding the oblique shock theory and interest in the topic. The survey was a multiple choice questions type which evaluated if students strongly agree, agree, disagree, or strongly disagree with the effectiveness of the experiment. One hundred percent of the participants in the survey confirmed that the experiment helped them better understand the subject though visualization and that active participation in the experiment helped them gain interest in the subject. Moreover, students noted that they better understand compressible flow concepts, such as back pressure, flow Mach number, and other flow features simulated by the water table. The results from this survey were representative to results from students’ evaluations in other classes where this experiment was performed.

Conclusion

In this paper, we revisit the hydraulic analogy of the oblique shock waves. Theoretical considerations were re-derived. Data from a student experiment were presented and compared to the theoretical predictions. It was shown that for higher Froude number, the experimental results had a good agreement with theoretical predictions, with error of less than 10%. Further evaluation of the size of the model and the blockage of the test section on the overall results is recommended. For larger wedge angle models, the volume of the displaced liquid in the water table was significant and may have affected the flow speed around the model resulting is a local Froude number significantly different from the one measure using micrometers. The upper limit of size of the model for the existing test section size should be further evaluated that allows for accurate measurement of compressible waves.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Oleg Goushcha

References

USDidactic, Inc., Supersonic wind tunnel with schlieren optics. Celebration, Florida, 2018.

Orlin

Lindner

Bitterly

JG.

Report No. 875: application of the analogy between water flow with a free surface and two-dimensional compressible gas flow. National Advisory Committee for Aeronautics, 1947, pp. 311–328.

Calares

Hankey

Self-excited wave oscillations in a water table. AIAA J 1983; 21: 372–378.

Joughet

Quelque probleme d'hydrodynamic generale. J Math Pures Appl 1920; 3.1: 1.

Riabouchinsky

On the hydraulic analogy to flow of a compressible fluid. Comptes Rendus L'Academie Sci 1932; 195: 998.

Mach

Photography of the projectile phenomena in air. Sitzungber Wiener Akad 1887; 95: 164.

Preiswerk

Application of the methods of gas dynamics to water flows with free surface I: flows with no energy dissipation. Washington: National Advisory Committee for Aeronautics, 1940.

Preiswerk

No 935: application of the methods of gas dynamics to water flows with free surface. Technical Memorandum National Advisory Committee for Aeronautics, March 1940.

Lavicka

Polansky

Boiron

Study and comparison of results of experimental and numerical solution for hydraulic analogy. In: 5th international conference on heat transfer, fluid mechanics and thermodynamics, Sun City, South Africa, 2007.

10.

Ippen

AI.

An analytical and experimental study of high velocity flow in curved sections of open channels. PhD Thesis. California Institute of Technology, Pasadena, CA, 1936.

11.

Pouderoux

Water table analogue for compressible flow studies. Master of Science Thesis. Oregon State College, Corvallis, OR, 1953.

12.

Murphy

CL.

Supersonic diffusion using hydraulic abalogy. Master of Science Thesis. McGill University, Montreal, 1960.

13.

Matthews

CW.

Technical Note 2008: the design, operation, and uses of the water channel as an instrument for the investigation of compressible-flow phenomena, Natoinal Advirosy Committee for Aeronautics, 1960.

14.

Arendze

Quantitative water surface fow aisuaization by hydraulic analogy. Master of Science Thesis. University of Witwatersrand, Johannesburg, 2006.

15.

Elward

KM.

Shock formation in overexpanded Flow-A study using the hydraulic analogy. Virginia Polytechnic Institute and State University, Blacksburg, VA, 1989.

16.

Hatch

JE.

The application of the hydraulic analogies to problems of two-dimensional compressible gas flow. Georgia School of Technology, Atlanta, GA, 1949.

17.

Skews

BW.

Colour encoded visualization applied to the hydraulic analogy for supersonic flow. In: 8th international symposium on flow visualization, Sorrento, 1998.

18.

Goushcha

Revival of water table experiments in fluid mechanics courses, part I. Int J Mech Eng Educ. Epub ahead of print 2 April 2019. DOI: 10.1177/0306419019831393.

19.

Linke

Garreston

Jan

et al., Integrated design, manufacturing and analysis of aifoil and nozzle shapes in an undergraduate course. In: 45th SME North American manufacturing research conference, Los Angeles, CA, 2017.

20.

Linke

Garretson

Jan

et al., Design and manufacturing of nozzles and airfoil shapes for compressible flow visualizations in a new engineering course. In: 2017 ASEE annual conference & exposition, Columbus, OH, 2017.

21.

Morishita

Desktop high-speed aerodynamics by shallow water analogy in a tin box for engineering students. Int J Aerospace Mech Eng 2018; 12: 1092–1105.

22.

Hafez

Jan

Linke

et al. Hydraulic analogy and visualisation of two-dimensional compressible fluid flows: part 2: water table experiments. IJAD. 2018; 6: 67–82.

23.

Pritchard

Mitchell

JW.

Fox and McDonald's introduction to fluid mechanics. Hoboken, NJ: Wiley, 2015.

24.

Garretson

Torner

Seewig

et al. An algorithm for measuring the hydraulic jump height of an airfoil in a water table. Measurement 2018; 148: 1–11.