STOCKHOLM SWEDEN 2020,
Laboratory exercise - compressible flow
Oblique shock waves
JOSEFINE GESSL
MATHEUS SVEDHOLM
KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES
We would like to start with thanking our supervisor Michael Liverts. He has been very helpful in making the project a reality and giving us feedback. When the school closed and we had to rethink the project he helped us in sorting everything out.
We would also like to thank Yushi Murai, a PhD student. He helped us a lot with feedback on the PM and in understanding the equipment.
Lastly we would like to thank Sembian Sundarapandian, a post doctorate at the de- partment. He helped us with the lab setup and organizing and understanding everything when we were in the lab.
When studying to become an engineer the education is mainly theoretical and to confirm the theory laboratory exercises are incorporated in the curriculum. Being able to visualize the phenomenons studied helps in giving the students a deeper understanding. The purpose of this report is to aid the department of fluid mechanics at KTH in designing an experiment to help the students understand the theory of oblique shock waves.
The project is divided in two parts. Part one the design of the wedge and base plate and part two the creation of the actual experiment and lab PM.
The design of the wedge is quite simple. It is a sharp wedge with two angles mea- sured from the horizontal plane, 8° at the top and 4° at the bottom, this is so that two measurements can be made per experiment.
Writing the PM and the design of the exercise is the main part of this project. The result is a laboratory exercise where the students compare the flow around a wedge in a shock tube for three different Mach numbers, subsonic, transonic and supersonic. Using shadowgraph optics and a high speed camera the image of the shock is captured. The different regimes are then discussed and compared with the help of theoretical calculations and the measured values.
Keywords: Oblique shock wave, shadowgraph, shock tube
Inom ingenjörsutbildningar är utbildningen huvudsakligen teoretisk och för att bekräfta teorin integreras laborationer i läroplanen. Möjligheten att visualisera de studerade fenomenen hjälper studenterna att få en djupare förståelse. Syftet med denna rapport är att hjälpa den strömningsmekaniska avdelningen på KTH att ta fram ett experiment som hjälper studenterna att förstå teorin om sneda stötvågor.
Projektet är uppdelat i två delar. Del ett, design av kilen och fästet och del två framtagningen av själva experimentet och lab PMet.
Kilens design är ganska simpel. Det är en vass kil med två vinklar mätta från ho- risontalplanet , 8° på ovansidan och 4° på undersidan, detta så att man kan göra två mätningar per experiment.
Utformandet av laborationen och labpeket är huvuddelen av detta projekt. Resultatet är en laboration där studenterna jämför flödet runt en kil i ett stötrör för tre olika machtal, subsoniskt, transoniskt och supersoniskt. Med hjälp av en höghastighetskamera och shadowgraph optik kan stötvågen fångas på bild. De olika scenarierna diskuteras och jämförs med hjälp av teoretiska beräkningar och de uppmätta värdena.
Nyckelord: sneda stötvågor, shadowgraph, stötrör
1 Introduction 1
2 Background 3
2.1 Oblique shock wave theory . . . 3
2.2 Shock tube. . . 9
2.3 Schlieren photography . . . 11
2.4 Shadowgraph . . . 13
3 Procedures and result 15 3.1 Design and Blueprints . . . 15
3.2 The experiment . . . 18
4 Oblique shock wave experiment, lab-pm 19 4.1 Summary. . . 19
4.2 Wave motion in a shock tube . . . 19
4.3 Experimental rig and test equipment . . . 22
4.4 Experiments . . . 25
4.4.1 Experiment 1: Subsonic M2 . . . 25
4.4.2 Experiment 2: Supersonic M2, θ > θmax . . . 26
4.4.3 Experiment 3: Supersonic M2, θ < θmax . . . 26
4.5 Evaluation using experimental data . . . 26
4.5.1 Experiment 1 . . . 26
4.5.2 Experiment 2 . . . 27
4.5.3 Experiment 3 . . . 27
4.6 Discussing the assumptions . . . 28
5 Preparatory Questions 31
6 Discussion and Conclusion 33
1 Introduction
As an engineering student the majority of the education is purely theoretical and many of the phenomenons are difficult to understand and sometimes even harder to visualise. It is therefore important to include laboratory exercises in the education so that students get the opportunity to learn how to use their theoretical knowledge in real world experiments.
One of the big areas of interest in fluid dynamics is compressible flow and particularly the flow around different objects. When the flow hits a sharp object, e.g a wing profile or a wedge, at a high enough speed a shock wave will form. This is something that might be difficult to understand and to facilitate it can be visualised in a laboratory environment.
This experiment is not new and similar ones have previously been conducted at KTH but the department of fluid mechanics have purchased a new square cross section shock tube that uses a fast opening valve instead of a membrane and the old setup no longer works. Therefore the purpose of this project is to assist the department of fluid mechanics at KTH to design, construct and test an experimental setup for demonstrating the theory of oblique shock waves for students of the course Compressible Flow SG2215. The main objectives for the lab are:
1. Observe oblique shock waves in supersonic flows.
2. Test theoretical knowledge.
In order for the students to perform the exercise properly a PM is written and that PM is the main focus of this project.
2 Background
2.1 Oblique shock wave theory
In difference to the normal plane shock wave, that only occurs in one-dimensional super- sonic flows, the oblique shock waves are prevalent in two- and three-dimensional super- sonic flows. In nature these waves are inherently two-dimensional and therefore in this report we will regard them as such, i.e flow field properties are functions of x and y.
Figure 1: Flow relations over a corner. Image taken from [1].
As a supersonic flow passes over a corner or wedge with angleθan oblique shock wave occur and with that an instantaneous change in the air properties. This results in two sections, section 1 upstream of the shock and section 2 downstream of the shock, which is visible in figure: 1. The sections will henceforth be indicated using indices 1 and 2. As the picture shows the Mach number, M, decreases and the pressure, p, temperature, T, and density, ρ, increases.
Figure 2: Oblique shock wave geometry. Image taken from [1].
Looking at the bottom part of figure: 2 it is clear that the velocity vector, V, can be divided into two components, one that is normal to the shock, u, and one that is tangential, w. In order to calculate the downstream Mach number and the other air property relations some calculations has to be made. These are similar to the normal shock calculations and depend on three fundamental physical principles, the conservation of mass, momentum and energy.
Beginning with mass conservation and using the control volume at the top of figure:
2. The integral form of the mass conservation is called the continuity equation and is
Ï
SρV· dS= 0. (1)
The faces b, c, e and f in figure: 2 are all parallel to the velocity and therefore they will not contribute to the integral. Evaluating the integral 1over faces a and d yields
−ρ1u1A1+ ρ2u2A2= 0,
where A1, A2are the areas of faces a and d and equal. This gives the continuity equation for an oblique shock wave as
ρ1u1= ρ2u2. (2)
Applying the momentum equation, Ï
Sρ ¯v( ¯v d ¯s) + Ï
S
P d ¯s = 0, (3)
to the control volume and only regarding the tangential component it is clear that the pressure terms of faces b and f cancel each other out (similarly for c and e) and we are left with
−ρ1u1w1A1+ ρ2u2w2A2= 0.
Remembering that A1= A2 and equation 2 the only remaining part is
w1= w2. (4)
This result is very important and shows that the tangential velocity components are conserved across the oblique shock wave. Again applying the momentum equation 3 to the control volume but this time to the normal component gives
(−ρ1u1)u1+ (ρ2u2)u2= −(−p1+ p2),
which can be simplified to
p1+ ρ1u21= p2+ ρ2u22. (5) The last step in obtaining the relations is to consider the energy conservation. This is made by applying the integral form of the energy equation,
Ï
Sρ(e +1
2v¯2( ¯vd ¯s) + Ï
S
P ( ¯vd ¯s),
to the control volume. This can be rewritten as,
e1+1 2u21+1
2w12+P1
ρ1
= e2+1 2u22+1
2w22+P2
ρ2
.
To simplify the expression further the definition of enthalpy,
h = e +P ρ, is used and the final expression that we are left with is
h1+1
2u21= h2+1
2u22. (6)
The interesting thing to note about these relations is that if the upstream Mach number is known the variations across the oblique shock can be expressed by using the relations for the normal shock. It is important to note that in order to do this the upstream Mach number will need to be rewritten in the same way as the velocity into a Mach number normal to the shock,
sinβ =Mn1
M1 =⇒ Mn1= M1sinβ. (7) By doing this we can formulate the well known equations that relates to all the variations across a shock wave. These relations are,
Mn22 = Mn12 + [2/(γ − 1)]
[2γ/(γ − 1)]Mn12 − 1, (8) ρ2
ρ1= (γ + 1)Mn12
2 + (γ − 1)Mn12 , (9)
u2
u1=2 + (γ − 1)Mn12
(γ + 1)Mn12 , (10)
P2
P1= 1 + 2γ
γ + 1(Mn12 − 1) (11)
and T2
T1= (P2 P1)(ρ1
ρ2
) . (12)
Many of these relations can be found in compressible flow textbooks, e.g [1], and are then tabulated for γ = 1.4. The Mach number behind an oblique shock can be found with the angles θ and β from figure: 2. This yields the expression
M2= Mn2
sin(β − θ). (13)
One major difference between a normal and an oblique shock wave, that is made clear by equation 7 through 12, is that the changes are now dependent on two variables, M1
and β, instead of just one variable. It is also clear, from equation 13, that M2 cannot be found until θ is found. But θ is also a function depending on M1 and β. Therefore, in order to calculate M2 we have to start with θ. From the geometry in figure: 2 the relations
tanβ = u1 w1
(14) and
tan(β − θ) = u2
w2 (15)
can be found. Adding the result from the momentum equation with regards to the tangential component, equation 4, it can be rewritten as
tan(β − θ) tanβ =u2
u1. (16)
Solving this for uu21 and plugging it into the normal shock relations, equation 2, 7 and 9, we obtain
tan(β − θ)
tanβ =2 + (γ − 1)M12sin2β
(γ + 1)M12sin2β . (17) Solving this for θ and with some trigonometric manipulations it can be expressed as
tanθ = 2cotβ M12sin2β − 1
M12(γ + cos2β) + 2. (18)
Equation 18 is called the θ − β − M relation and specifies θ as a unique function of M1
and β. This equation is very important in analysis of oblique shock waves. Its use can be simplified by plotting the results in a graph, see image below.
Figure 3: θ − β- M curves for γ = 1.4. Wave angle versus deflection angle with Mach number as a parameter.
The properties and relations described in this section represent the exact solutions for the flow over a wedge or a two-dimensional compression corner. As the purpose of this lab is to demonstrate the occurrence of oblique shock waves it is reasonable to compare these relations to the experimental results when using a wedge in a shock tube.
2.2 Shock tube
Shock tubes are tubes constructed generate shock waves and observe flow in different regimes. They are usually built out of metal with a rectangular or circular cross-section and consists of two sections, the driver section and the driven section as can be seen in figure: 4. The driver section contains highly pressurised gas while the driven section has a partial vacuum, the two sections are separated by either a valve or a diaphragm.
Valves can be opened when the desired pressure ratio is attained in the tube while a diaphragm is usually made of a thin metal sheet designed to burst at a precise pressure.
Using a valve will reduce the downtime in between experiments but will and make it easier to maintain a precise pressure ratio, in difference to the diaphragm which will have have longer downtime but and the pressure settings aren’t as precise. When the valve is opened or the diaphragm bursts the pressure difference between the sections will result in a rapid depressurisation of the driver section resulting in the creation of a shock wave propagating through the driven section. The driven section contains the test section of the tube with measurement equipment to analyze the shock wave.
Figure 4: Illustration of shock tube in laboratory exercise.
The propagation of the shock wave through the shock tube is shown in figure: 5.
The rapid depressurisation of the shock tube when the valve is opened or the diaphragm bursts will result in a shock wave propagating through the driven section. This wave will eventually reflect at the end of the shock tube. Behind the shock wave a secondary wave is created at the interface between the driving and the driven gas that is called the contact surface. Because it is created the initial shock wave increases the pressure of the gas behind it, creating a secondary shock wave. It propagates in the same direction at a
lower speed. At the same time as the shock wave is created and propagates to the right an expansion fan is created that moves to the left. The difference in pressure between the shock wave, contact surface and the expansion fan can be seen in figure: 6.
Figure 5: Propagation of shock wave in shock tube. Image taken from [2].
Figure 6: Flow in a shock tube after the diaphragm is broken. Image taken from [1].
2.3 Schlieren photography
Schlieren photography is a method to capture images of density changes in translucent mediums for example gases. This makes it possible to photograph the propagation of shock waves in air and how they interact with objects. Schlieren photography is based on the principle of refraction in accordance to Snell’s law.
Snell’s law states that the light will travel at different speeds depending on the medium it travels through. When light enters a medium it will refract with an angle and it is this angle that schlieren photography is based on. Snell’s law :
sinθ2
sinθ1
= v2 v1=n1
n2 (19)
Figure 7: Snell’s law. Image taken from Wikipedia. [3]
where θ1 is the light angle in the first medium, θ2 is the refraction angle in the second medium, v1 and v2 is the speed of light in the two mediums and n1 and n2 are the different refractive indexes of the two mediums. The refractive index of air varies with changes in density and these variations allows us to capture an image of the shock wave.
The refractive index n is calculated in equation: 20.
n = c
v (20)
Where c is the speed of light in vacuum and v is the speed of light in the medium.
Figure 8: Setup for schlieren photography. Image taken from NASA [4]
.
Schlieren photography works by having a point source of light, for example a laser or a white LED. The light is then passed through a slit to get the light to move in parallel rays through the system. In the setup shown in figure: 8two concave mirrors are used on each side of the test section, the first mirror is located so that the light passing through the slit moves in parallel rays through the test section in the shock tube. As the light passes through the shock wave, in the test section, the light will refract in accordance to Snell’s law, equation: 19, due to the change in density across the shock wave. The refracted light is shown as a dotted line in figure: 8. The light is then focused with the second mirror with the focal point located at the edge of a knife so only the parallel light rays pass the knife edge and get recorded by the camera. The light that is refracted by the shock wave will get blocked out by the knife edge and therefore appear black in the image. Together with the unaffected light it forms a black and grey scale image of the shock. This way of capturing density variations works for any kind of density variation, e.g if a lighter is held in front of the light source the gas seeping out can be seen.
2.4 Shadowgraph
Shadowgraph photography has a simpler setup than schlieren photography. The optical sensitivity of the shadowgraph is an order of magnitude lower than that of schlieren photography. In experimental setups it works in practically the same way as schlieren.
The difference is that it does not use the knife edge to filter out the refracted light. By not having the knife edge you get less of a contrast in the image, since the refracted light will still pass through it and result in a grey shadow. Schlieren imaging will result in higher contrast than shadowgraph photography but is harder to setup because of the positioning of the knife edge at the focal point of the mirror/lens. The difference in experimental setups can be seen by comparing figures: 8 and 9.
In shadow photography when the light passes the shock wave in the test section it casts a shadow on a reference plane that is then focused with the second lens on to the camera. The greater the distance l of the recording plane the higher sensitivity of the flow visualisation you get on the shadowgram, however by increasing the distance l we also lose focus in the image. So a compromise has to be made between contrast and focus.
Figure 9: Experimental setup of shadowgraph photography. Image taken from [5].
3 Procedures and result
3.1 Design and Blueprints
In order to generate an oblique shock wave the shock tube has to be fitted with a wedge.
This meant that we had to design a wedge and plate so that the wedge could be mounted inside the tube. The design of the wedge is quite simple. It is constructed to reach into the middle of the test section of the shock tube. To save timr when performing the exercise the wedge has two angles measured from the horizontal plane, 4° at the bottom and 8° at the top, this will allow the students to get two measurements of the deflection of the shock wave per experiment. Figure: 10shows the blueprint of the wedge.
As the function of the shock tube is depending on its ability to build pressure it is vital that it is completely sealed. Therefore the base plate for the wedge has to form a tight seal with the other parts of the tube. To achieve this the plate has two mating surfaces matching those of the pipe. On one side there is a milled groove and on the other side an elevated square. These have the same dimensions as the corresponding part on the tube. When mounting the base gaskets are inserted to ensure no leakage. The wedge is mounted on the base plate with two M8 screws, and everything is then tightly screwed together with 8 M18 screws. Figure: 11shows the blueprint for the wedge base plate.
DRAWN CHECKED ENG APPR MGR APPR UNLESS OTHERWISE SPECIFIED DIMENSIONS ARE IN MILLIMETERS ANGLES ±X.X° 2 PL ±X.XX 3 PL ±X.XXX NAME HobbitDATE 01/31/20Solid Edge TITLE SIZE A2DWG NOREV FILE NAME: wedge_drawing.dft SCALE:WEIGHT:SHEET 1 OF 1
REVISION HISTORY REVDESCRIPTIONDATEAPPROVED 230
50,8
9 O
threaded hole for M8 screw 49,8
4°
8°
center line
Figure 10: Blueprint of wedge
DRAWN CHECKED ENG APPR MGR APPR UNLESS OTHERWISE SPECIFIED DIMENSIONS ARE IN MILLIMETERS ANGLES ±X.X° 2 PL ±X.XX 3 PL ±X.XXX NAME HobbitDATE 01/31/20Solid Edge TITLE SIZE A2DWG NOREV FILE NAME: Base_drawing.dft SCALE:WEIGHT:SHEET 1 OF 1 REVISION HISTORY REVDESCRIPTIONDATEAPPROVED
170
240 151
81
50
50
4 4
150
O 18
9 O
10 4
4
151
81
150
80
A
DE TA IL A
2,5
2,3 3,5
Figure 11: Blueprint of base
3.2 The experiment
The design of the actual experiments builds on the theory of section 2. The goal of the lab is to make it interesting and to facilitate the understanding of the theory. This combination is not easy and several ideas on what would show the theory in the best way was discussed to later be tried out in the laboratory. The main ideas were:
Potential experiments 1. Measure angles β and θ.
2. Temperature on the wedge (interesting for heat shields in supersonic flight).
3. Mach number ahead of and behind the shock wave.
4. Difference in density.
5. Secondary shock wave.
6. Try to detect the reflection of the expansion fan.
7. Calculate the difference in refractive indices.
8. Show detached shock formation.
Previous years the experiment was to calculate the the Mach number before and after the shock wave using the values from the measurements and compare it with what you get when you instead look at the shock wave angle in the video and use the θ−β−M-relation.
4 Oblique shock wave experiment, lab-pm
This is the final PM made out of the project objectives. The plan is for it to be handed out to the students and to be used as preparation before the exercise as well as instructions during the exercise. It is inspired by the old lab-pm.
4.1 Summary
In this lab the objective is to study different flow regimes in a shock tube around a wedge. Three different Mach numbers will be studied and they are determined for different pressure ratios between the high and low pressure side of the fast opening valve. The speed of the shock is determined by measuring the time needed for the shock to move a certain distance along the tube. Experimental values are then compared with theoretical values based on idealized shock tube flow and the flow Mach number is verified by measuring the shock wave angle when oblique shocks are formed.
4.2 Wave motion in a shock tube
The theoretical calculations are based on idealized shock tube flow. This incorporates the following assumptions:
• Ideal gases in driver and driven section
• Instant opening of the valve
• Flow without friction and heat conduction
The shock tube is initially split into a driver (high pressure) and driven (low pressure) sections by a Fast Opening Valve (FOV). Figure: 12 shows the wave motion inside the shock tube after the FOV opening.
Figure 12: Wave motion inside the shock tube after the Fast Opening Valve (FOV) opening. Adapted from Ref. [1]
.
In figure: 13the different states inside the shock tube are shown.
a) State 1, is when the FOV opens resulting in a rapid depressurisation of the driver section, leading to the formation of a shock wave propagating through the driven section. At the same time en expansion fan is formed propagating in the opposite direction through the driver section.
b) The interface between the driver gas and the driven gas is called the contact surface.
It separates states 2 and 3. State 2 contains the driven gas creating an oblique shock wave if the flow is supersonic.
c) State 3, contains the driving gas moving at the same speed as the driven gas. The driving gas acts as a piston on the driven gas.
d) State 4, is when the shock waves reflect against the end walls. Interference between the shock waves is initiated, making it difficult to analyze the gas properties.
e) State 5, is when the flow inside the shock tube has settled.
Figure 13: x-t diagram of the flow.
To study the flow around the wedge for different Mach numbers, the Mach number in state 2, M2, is varied by changing the pressure ratio p4/p1. The two can be related through the incident shock Mach number, Ms.
The continuity, momentum and energy equations across the incident shock wave can be written as:
ρ1cs= ρ2(cs− u2), p1+ ρ1c2s= p2+ ρ2(cs− u2)2,
h1+cs2
2 = h2+cs− u22 2 ,
where ρ,u,p and h are the density, velocity, pressure and enthalpy, respectively and cs
is the velocity of the incident shock wave. The numbers in subscripts correspond to the states in the shock tube. Together with the normal shock jump relations, M2 can be derived as a function of Ms
M2= 2(Ms2− 1)
[2γ1Ms2− (γ1− 1)]12[(γ1− 1)Ms2+ 2]12 .
Considering the boundary conditions of each wave mentioned in the points a to d previ- ously, the relationship between p4/p1 and Ms is expressed as
p4
p1=2γ1Ms2− (γ1− 1) γ1+ 1
·
1 −γ4− 1 γ1+ 1
a1 a4 µ
Ms− 1 Ms
¶¸−2γ4
γ4−1,
where γ and a are the heat capacity ratio and speed of sound.
4.3 Experimental rig and test equipment
The set-up of the experiment is sketched in figure: 15.
The total length of the shock tube is approximately 4 m and it has a rectangular cross-section, 50 x 120 mm. The driver section where the pressure is high, can reach 32 bar (limited for safety reasons), and the driven section where the pressure is low, down to 0.01 bar, are separated by a fast opening valve (FOV). In most shock tube setups a membrane is used instead of the FOV. This method is more time consuming as the membrane has to be replaced after every experiment. The rupture of the membrane can be inconsistent and it causes debris that can damage the wedge. This will affect the measurements and cause inconsistency between experiments. Therefore in order to get more precise measurements the membrane has been replaced with a FOV. The opening time of the FOV is very short 3 − 5 milliseconds. At a predetermined pressure ratio the FOV opens and a shock wave is formed which moves into the low-pressure gas.
The downside of using an FOV is that it takes a longer time for the shock wave to form compared to membranes. This time delay causes a discrepancy in thep4/p1toMsrelation from the theoretical value where instant formation is assumed. The theoretical value only holds from when the incident shock wave has fully formed. The experimental volume for the driver section will increase as a result of the FOV not opening instantaneous. To reach the theoretical value for Ms a larger volume of gas has to be pumped into the driver section to match the effective p4 as in figure: 14. To find the effective p4 a calibration curve of p4/p1 versus Ms is utilized. The calibration curve for KTH shock
tube can be seen in figure: 17.
Figure 14: Shock formation in ideal case (1) and actual case (2).
The test section is located approximately 2.5m downstream of the valve so that the shock wave has time to fully form before reaching the wedge. By placing the test section in the middle of the shock tube the time it takes for the reflection to affect the wedge is delayed, increasing the test time. This makes it possible to analyze the flow behind the initial shock wave.
The gas supplied to the driver section comes from a high-pressure gas bottle with a pressure regulating valve. In this experiment air will be used. The pressure in the driver section is limited to 32 bar, for safety reasons, and in the driven section the pressure is kept below atmospheric pressure before the shot to effectively increase the pressure ratio. In order to regulate and control the pressure a computer program is used. This computer is connected to the driven and driver section of the tube and is used to regulate the pressure levels and set the valve opening. The gas in the driven section is evacuated by a two-stage vacuum pump and the lowest pressure possible is about 0.01 bar.
Figure 15: Shock tube and measurement equipment.
After evacuating the air in the driven section, ambient air is jet in through a small valve and the pressure is adjusted to the right level.
The driven section of the tube is equipped with a number of sensor ports mounted in the side wall and in two of these ports piezoelectric pressure sensors (PCB 113B28) are placed, 36 cm apart. The sensor is built up by a diaphragm and quartz plates. When these components are pushed together they will produce an electric voltage, this is called the piezoelectric effect. The surface of the diaphragm is mounted flush with the inner side of the tube and as the shock passes over the sensor the pressure deforms the diaphragm which will push the quartz plates together generating a corresponding voltage. The sensor is very fast and the response time is less than 1 µs. The produced signal is passed to the oscilloscope via the signal conditioner. As the shock passes, a step rise in pressure is observed and ∆t is obtained by measuring the time between the two rising edges. Since the distance between the sensors is known and ∆t is obtained the velocity of the shock can be calculated.
The test section is located downstream of the sensors and contains a wedge with two angles measured from the horizontal plane, 8°at the top and 4° at the bottom. The test section has vertical glass windows to facilitate optical access and the edge of the wedge is mounted in the middle of the cross-section and aligned horizontal and normal to the glass surfaces. To visualize the oblique shock at the wedge shadowgraph technique is used. A
fibre coupled LED light is used to create the continuous, single point light source needed.
This light passes through a converging lens, so that the beams become parallel, the light is then directed at a mirror to steer it through the test section, through a second lens and at the focus of this lens a third smaller lens to focus the light into the high speed camera, as previously shown in figure: 15. The camera is triggered by the pressure sensors signal.
4.4 Experiments
In this laboratory exercise the students will perform three experiments. The idea is to compare three different flow Mach numbers around the wedge, i.e. the Mach number in state 2, M2. In all the experiments, the temperatures of the different gases in driver and driven sections are assumed to be 300 K (room temperature).
It is important to come prepared to the lab therefore before partaking in the lab each student will have to correctly answer a quiz with questions ensuring that the assignment is understood. The quiz will be handed out in the beginning of the lab exercise. Students will also have to bring a calculator and are recommended to bring the course book and a protractor.
4.4.1 Experiment 1: Subsonic M2
The time trace of the measured pressure signals can be measured from the oscilloscope.
The measured pressure value in state 2 after the incident shock arrival at the wedge (tw in figure: 13),p2, as well as the initial driven section pressure, p1, are indicated on the plot.
From the pressure ratio p2/p1, the incident shock Mach number, Ms can be obtained as
Ms= cs a1=
·γ1+ 1 2γ1
µp2 p1− 1
¶ + 1
¸12 .
The equation is derived from the three governing equations mentioned in the Section: 4.2 and the ideal gas law.
The Mach number behind the incident shock, M2= u2/a2, is determined using the normal shock relations. The continuity equation gives the velocity behind the incident shock, u2, i.e. the velocity in state 2 as
u2= µ
1 −ρ1
ρ2
¶ cs.
Assuming ideal gas, the speed of sound is given as a =p
γRT. Hence in state 2, a2 is calculated as
a2= sT2
T1a1.
Using the normal shock jump relations for temperature and density (a table of normal shock properties may be consulted), M2 is obtained experimentally from the pressure measurement of p2.
4.4.2 Experiment 2: Supersonic M2, θ > θmax
The same procedure as in Experiment 1 should be performed to confirm the value of M2
experimentally.
4.4.3 Experiment 3: Supersonic M2, θ < θmax
The same procedure as in Experiment 1 and 2 should be performed to confirm the value of M2 experimentally.
Furthermore, M2 can be verified from the shadowgraph snapshot using the θ − β − M curves shown in figure: 16 where θ is the wedge angle, β is the shock angle and can be measured from the snapshot. To improve accuracy, it is recommended to measure the angle between an oblique shock and the surface of the wedge, then addθ to the measured value. When using the curve to obtain M2, careful attention needs to be paid to the subscripts: the Mach number ahead of the oblique shock, M1, in theθ −β−M relation is the Mach number of state 2, M2.
4.5 Evaluation using experimental data
4.5.1 Experiment 1
Target M2= 0.46
1 (a) Calculate the Ms required to achieve the target M2.
(b) Calculate the theoretical p4/p1 to achieve the obtained Ms. (c) Calculate the actual p4/p1 required to achieve the obtained Ms.
Explain why there is a discrepancy between the theoretical and actual values.
2 (a) Calculate the Ms from the pressure measurement on oscilloscope.
(b) Calculate the M2 from the Ms obtained experimentally.
How well does it match the target M2?
(c) Watch recording of shock.
Explain your observation referring to the obtained M2 value.
4.5.2 Experiment 2
Target M2= 1.04
1 (a) Calculate the Ms required to achieve the target M2.
(b) Calculate the theoretical p4/p1 to achieve the obtained Ms. (c) Calculate the actual p4/p1 required to achieve the obtained Ms. 2 (a) Calculate the Ms from the pressure measurement on oscilloscope.
(b) Calculate the M2 from the Ms obtained experimentally.
How well does it match the target M2? (c) Watch recording of shock.
Explain your observation referring to the obtained M2 value.
θ − β − M curve should also be referred.
4.5.3 Experiment 3
Target M2= 1.34
1 (a) Calculate the Ms required to achieve the target M2.
(b) Calculate the theoretical p4/p1 to achieve the obtained Ms. (c) Calculate the actual p4/p1 required to achieve the obtained Ms. 2 (a) Calculate the Ms from the pressure measurement on oscilloscope.
(b) Calculate the M2 from the Ms obtained experimentally
(c) Calculate the M2 by measuring the shock angle from the shadowgraph snap- shot. ObtainM2 from both top and bottom wedge angles.
(d) Watch recording of shock.
Explain your observation referring to the obtained M2 values.
θ − β − M curve should also be referred.
(e) Compare theM2values obtained in 3 different ways and the target value. How well do they match? Which value do you think is the most reliable one?
4.6 Discussing the assumptions
Ideal gas
The assumption of ideal gases is probably relevant at the low Mach numbers, Ms< 3, used in the lab exercise, because only minor variations in the heat capacity will occur.
Instant valve opening
The time it takes until stationary shock speed is reached depends on the valve opening.
The valve opening also influences the character of the contact surface. In the lab exercise, the shock wave is fully developed and plane when it passes the sensors and reaches the wedge. But the time delay due to the finite opening time causes the shock curve in the x -t diagram to be shifted upwards. We also have to consider the mixing in the contact zone due to an imperfect valve opening and thermal gradients which decrease the time of stationary flow around the wedge.
Boundary layer formation
Behind the shock, the gas temperature and velocity are different compared to the situation along the walls and hence boundary layers start to develop as soon as the valve opens. The boundary layer development causes the shock to attenuate, the contact surface to accelerate and the available time for which the flow is constant to diminish.
However, these effects are of minor importance as the shock Mach number is low and the tube is short in our case.
Figure 16: θ − β − M graph
1 1.5 2 2.5 3 3.5 Incident shock Mach number, M s [-] 0 1 2 3 4 5 6 7 8 9 ln(p
4 /p
1 )
He-Air: a
1/a
4=0.34, T
1/T
4=1 He-Air, KTH Shock tube: ln(p
4/p
1) = 0.31M
s 3- 2.6M
s 2+ 8.1M
s- 5.5 Air-Air: a
1/a
4=1
Figure 17: Theoretical and calibration curve forp4/p1 vs. Ms relation.
5 Preparatory Questions
Before the students are allowed to begin using the lab equipment they must answer a few questions just to make sure they have prepared, read and understood the laboratory in- structions. These questions are inspired by the old theory questions.
Mark the correct alternative, 7 out of 10 correct answers required in order to partake in the lab
1. The pressure sensors used to measure the shock speed consists of (a) thin platinum film
(b) quartz plates (c) thermo-elements
2. The speed of sound can be expressed as (a) γRT
(b) pγRT (c) pγR/T
3. Before the valve opens, the speed of sound is (a) much larger in the high pressure part (b) much lower in the high pressure part
(c) about equal in both parts 4. θ is the angle for the
(a) wedge
(b) oblique shock (c) contact surface
5. When the fast opening valve opens this is true for the expansion fan, (a) it is the first to interact with the wedge
(b) it propagates in the opposite direction to the contact surface
(c) it is created at the boundary layer between the driving and the driven gas 6. The technique used to display the picture is called
(a) shadowgraph (b) schlieren
(c) laser spectrometry
7. What are the advantages of using FOV instead of membrane
(a) no debris and therefore smaller risk of damaging the equipment and increased consistency between measurements
(b) precise pressure settings (c) all of the above
8. What are the three assumptions that the theory is based upon (a) Wedge is blunt, ideal gases and temperatures below 0°C
(b) Slow opening of valve to build pressure, high friction flow and ideal gases (c) Ideal gases, instant opening of valve and flow without friction and heat con-
duction
9. The oscilloscope is used to measure
(a) the time it takes for the shock wave to pass the two sensors (b) the density of the shock wave
(c) the temperature of the shock wave 10. What gas will be used in the experiment?
(a) Air (b) Argon
(c) Helium
6 Discussion and Conclusion
The goal was to create a laboratory exercise that shows the students different flow regimes around a wedge and by that giving them a deeper understanding of the theory. Previous years the exercise was very similar to that of our experiment 3. There weren’t enough time to do several shocks since the membrane had to be replaced and the process of unscrewing and replacing it took up most of the time. With the new shock tube and the FOV the downtime has been drastically reduced and therefore there is more time for the students to test different scenarios.
One of the most vital relations in the analysis of oblique shock waves is the θ −β−M- relation but this relation can sometimes be difficult to understand and the plot, figure 3, is at first glance confusing. Since this relation is very important the exercise is built around it. By doing the experiments at three different flow Mach numbers we cover the most important aspects. Experiment one, where the Mach number is subsonic, might for some be a bit redundant since shock waves only occur in supersonic flows but it is still helpful as a confirmation of the theory. Moving on to the second experiment, where the flow is transonic an interesting phenomena occur. This part of the experiments weren’t possible in the old setup. But with the new tube and FOV the students can now observe all the stages. The transonic regime is loosely defined, approximately when M > 1.2, and since the mach number is bigger than one it might be expected that we would get the oblique shock formation. This is where the θ − β − M-relation is important. Students will discover that for this Mach number, due to that the angle of the wedge, θ,is larger than θmax, they wont get a oblique shock solution but instead a detached shock occurs.
To understand this completely it is necessary to understand the θ − β − M-relation and therefore this part of the experiments will be very helpful for the students. The final experiment is when the flow Mach number is supersonic and the oblique shock wave occurs. The results are used to compare the theoretical values with the actual measured values and once again a strong relation to theθ − β − M-graph.
All these experiments are the results from the PM we have written and they have all been tested and verified in the lab by the department. After doing the experiments the students will have gained a deeper understanding in the formation and special conditions for the oblique shock wave.
With regards to the design of the wedge and the base plate. Even though the design itself is quite simple and it works fine as it is now, there are still a few changes we would
make if given the opportunity to remake it.
Firstly, the small plate on which the wedge is fastened. This could be made the same size as the back of the wedge to make it more secure. But since everything works as it is now this is not a necessary change only a "wish we thought of this before sending it to the workshop".
Also, a change that would not require making a new piece and that we think should be made when removing the wedge from the tube is simply writing the angles on the side of the wedge. It is a very simple thing but it would help and make sure that there is no confusion about which angle is up once the wedge is mounted inside the tube. Otherwise if you forget how you mounted the wedge you would either have to remove the test section or setup the shadowgraph system and measure the wave angle. Both these options are time consuming and can easily be solved by just marking the wedge with a pen.
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[4] Schlieren System, NASA, Editor: Nancy Hall, Latest update: 2015-05- 05, Accessed: 2020-02-19,https://www.grc.nasa.gov/www/k-12/airplane/
tunvschlrn.html
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