Compressible Flow through a Convergent-Divergent Nozzle Report PDF

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Compressible Flow through a Convergent-Divergent Nozzle Report...

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University of Salford School of Computing, Science and Engineering Aerodynamics Laboratory Compressible Flow through a Convergent-Divergent Nozzle

Summary The main aim of the report is to investigate the behaviour of a compressible flow in a supersonic wind tunnel in different circumstances. It is found out that the flow is influenced by the injector pressure, the area throat ratio and the difference between the inlet and exit pressure. A spread sheet is used to plot a graph for each run which is used to clearly explain and discuss all the changes that occur to the flow along the convergent-divergent nozzle.

Contents 1. 2. 3. 4. 5. 6. 7. 8.

Introduction Aims and objectives Theory Description of apparatus Experimental procedure Results Discussion Conclusion References Appendixes

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1. Introduction All matter is compressible to some extent. The volume of an element of matter decreases under pressure and the density is increasing. Pressure changes in gases do produce significant volume and density changes. If the percentage change in pressure is small, the effect of the density change on the forces involved may be very small. This applies to most flows over wings, aerofoils or bodies at speeds below 40% of the speed of sound. There are two classes of aerodynamics flows: incompressible flow and compressible flow. Incompressible flow. “An incompressible flow is one in which the density of the fluid elements can be assumed to be constant. Incompressible flow is an idealization that never actually occurs in nature. However, for those flows where the actual variation of density is negligibly small, it is convenient, to simplify our analysis, to make the assumption that density is constant. The assumption of incompressible flow is an excellent approximation for the flow of liquids, such as water or oil, and for the low-speed flow of air, where the speed is less than 200 to 300 mph.” (Shevell, Fundamentals of Flight, 1989) Compressible flow. “A compressible flow is one in which the density of fluid elements must be considered variable from point to point along the streamlines. In such cases, the equations must treat density as a variable. Since most current aircraft operate at speeds at which compressibility effects are significant, we must now develop the equations that relate pressure, density and speed at various points along a stream tube without the simplifying assumption that density is constant. The impact of compressibility will be shown to be a function of the Mach number M, the ratio of the air velocity to the speed of sound in air. ” (Shevell, Fundamentals of Flight, 1989)

2. Aims and Objectives The main aims and objectives of this laboratory are to demonstrate the different flow regimes which occur within a convergent-divergent nozzle. This will be done by operating a small supersonic wind tunnel at a series of different injector pressures. Furthermore, stagnation and static pressure will be measured along the length of the nozzle, from which values of Mach number and theoretical values will be calculated. The experimental and theoretical values will be compared and the approximate location of any shock waves will be identified.

3. Theory Incompressible flow behaves in a natural manner by accelerating as duct area decreases and decelerating as area increases. Compressible flow depends on the local Mach number of the flow as well as any change in cross-section. The equation below shows the relation between a simple compressible flow and area (A), speed (V) and Mach number (M): dA dV =( M 2−1 ) (1) A V Equation (1) is very important and has to be studied closely. It reveals the following information: 2

1. If 0≤M≤1 (subsonic flow) the quantity in parentheses in eq. (1) is negative. Therefore, the velocity gradient increases (positive du) and there is a decrease in area (negative dA). Likewise, a decrease in velocity (negative du) is associated with an increase in area (positive dA). For a subsonic compressible flow, to increase the velocity, a convergent duct must be used and to decrease the velocity, a divergent duct is used. 2. For M>1 (supersonic flow) the quantity in parentheses in eq. (1) is positive. Hence, dA and dV are directly proportional to one another. A decrease or increase in one corresponds to the same in other. For a supersonic flow, to increase the velocity, a divergent duct must be used, and to decrease the velocity a convergent duct is used. 3. When dA=0, one of the possible solution is that M=1. At this point the mass flow rate of air through the duct is at its maximum value, and the duct is said to be choked. Figure 1 shows a typical convergent-divergent nozzle or duct which will accelerate subsonic flow to supersonic speed. The point where the duct walls are locally parallel is called the throat. In order to accelerate the flow to supersonic speed the Mach number at throat should be one.

Figure 1: Convergent- Divergent Nozzle

Stagnation pressure ( p0 ) is related to static pressure (p) and Mach number (M) as shown below in equation (2): (2)

γ p0 γ −1 2 γ−1 M ) =(1+ 2 p

, where γ is the ratio of specific heats; for air γ=1.4

The ratio of the cross-sectional areas, at any point and at throat is illustrated below: γ +1 γ −1 2 2 (γ−1) M 1+ 2 A MT ( , where A T is the area at the throat and A is (3) ) = γ −1 AT M 2 MT 1+ 2 the area at any point. Equations (2) and (3) have a simplified form for air as shown below: (2.1) (3.1)

p0 =( 1+0.2 M 2)3.5 p 2 3 1 1+0.2 M A ) = ( 1.2 AT M

Further explanation of the equations stated above can be found on Appendix II: Equations for Isentropic Flow in Air – Aerodynamics, PA, 2014. 3

For a convergent-divergent nozzle of known dimensions, there can only be two possible exit pressure for which the flow thorough the nozzle will be isentropic. For exit pressures other than those found from equation (3), no isentropic solution exists. This involves the presence of a Normal Shock Wave in the divergent part of the duct. A shock wave happens when there is an instantaneous increase in static pressure, a decrease in stagnation pressure and a sudden decrease in flow speed and Mach number. A normal shock wave lies perpendicular to the oncoming fluid and causes a supersonic flow to revert to subsonic speed downstream of the wave. It is possible to identify different flow patterns that occur along the length of a con-di nozzle, especially in the divergent section. The detailed descriptions of these patterns are stated on Appendix I: Pressure Distributions Along a Con-Di Nozzle According to Exit Pressure – Aerodynamics, PA, 2014.

4. Description of Apparatus The experimental apparatus for this laboratory was made of a supersonic wind tunnel, a barometer to measure the atmospheric pressure (in mmHg), a thermometer to measure the ambient pressure (in °C), a small inclined manometer (Fig. 2), a multi-tube mercury manometer, a computer to insert the data collected from apparatus and noise protection earmuffs. Figure 3 shows the tunnel used to do the laboratory. It is mainly used to carry on experiments and demonstrations, primarily in two-dimensional supersonic flow. It has a working section of 10cm x 2.5cm and it can operate a 1.8 maximum Mach number. The blowing pressure at M=1.8 is 7 bar gauge. The tunnel becomes uneconomical in air consumption above a Mach number of about 1.8.

Figure 2: Small inclined manometer

Figure 3: Supersonic wind tunnel

The general arrangement of the tunnel can be seen in Figure 4. The working section is made of two cast steel side panels. Between them there is a fixed flat liner of Hydulignum. There are different liners for Mach 1.4 and 1.8. Pressure tappings are provided at 2.5cm centres along the length of the flat liner. The air then enters the fibreglass contraction cone at the entrance to which there is a pressure tapping for determining the stagnation pressure in the tunnel. The throat has a few inches along the working section and it is the point that sonic velocity occurs. The walls then diverge allowing the flow to go further supersonic.

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The tunnel is fed from a receiver that is charged to a pressure higher than the required tunnel injection pressure. The valve has to be continually adjusted to maintain a constant pressure in the injector. The tunnel can run about 12 seconds at a Mach number of 1.8 at a pressure of 6.9 bar gauge. The tunnel is fitted with a compressor having a driving motor of 200HP. The tunnel control valve is used to maintain the correct pressure at the injector and operates on the induction principle. The principal particulars of the tunnel are showed below:  Overall dimensions: 310cm long x 109cm high x 36cm wide;  Working section: 10cm x 2.5cm;  Maximum Mach number: 1.8;  Blowing pressure at M = 1.8: 7 bar gauge;  Approximate air consumption: 0.5 kg/sec;  Blowing pressure at M=1.4: 5.2 bar gauge;  Approximate air consumption: 0.45 kg/sec.

Figure 4: General arrangement of the wind tunnel

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5. Experimental procedure First step is to turn on the wind tunnel switch and allow enough time for the Ingersoll Rand compressor to charge. Laboratory ambient pressure and temperature is recorded in order to be used in further calculations to get accurate results. The tunnel is designed to produce flow at a specific Mach number in the working section: 1.4 and 1.8. The tests will be undertaken with injector pressures of 2, 5 and 7 bar. In total there are 6 runs and data will be measured, recorded and inserted on a spreadsheet. This will carry out the necessary calculations to produce distributions of pressure ratio, Mach number and area ratio along the length of the nozzle. The resulted graphs will be analysed later. The experiment started with a liner Mach number of 1.4 with an injection pressure of 2 bar gauge. During every run, a student took the value of the inclined manometer and one took the readings from the multi tube manometer. Noise protection earmuffs are used for every run. The pressure from the injector is adjusted by the lecturer. To achieve the best results, the pressure must be maintained constant as long as possible. The procedure is to charge the air receiver to its maximum pressure and then to open the tunnel control valve very rapidly until the correct injection pressure is indicated on the tunnel pressure gauge. The control valve must be opened slowly to keep the pressure constant. Finally, when the receiver pressure is just above the injection pressure and the control valve is fully open, the run is terminated. The valve is closed and then the receiver will start to recharge. The recharging takes about 4 minutes.

6. Results Table 1 shows the data recorded from the apparatus at the first run at an injector pressure of 2 bar with the Liner Mach number 1.4. The proven calculation for the spreadsheet from Table 1 is showed below. The data in yellow is taken from the apparatus and the rest is calculated by the spreadsheet. Atmospheric pressure is read from the barometer in mmHg and is converted below in S.I units, Pascals. The recorded temperature is also converted from ℃ to Kelvin. N Patm =ρmercury gh=13.530 x 9.80665 x 748.70=99340.49 Pa( 2 ) m T ( K ) =273.15+℃=273.15+20.7=293.85 K All pressure recordings are gauge, therefore it has to be subtracted from the atmospheric pressure. Because the static pressures at every station are recorded in cm Hg, all the calculation will be done with respect to this. The small inclined manometer is used to calculate the stagnation pressure at entry point. P −α∗0.2∗760 , where α is the inclined manometer angle; P01= a 1013.25 748.70−13.6∗0.2∗760 = 746.658 mmHg Therefore, P01 = 1013.25 6

The static pressure at entry

P1 is calculated in function of the datum point (26).

P (¿ ¿ cmHg station 1−P cmHg datum station 26 )∗10 P1=P a−¿ P1=748.70− (16.9 −7.4 )∗10=653.7 mm Hg

Table 1: Run #1 Pressure 2bar Liner Mach number 1.4

P 01 746.6598 =1.142206 = 653.7 P1 Equation (2) is used to calculate the entry Mach number. It has to be organised in terms of Mach number as showed below: 1 P 3.5 1 ( 01 ) −1 3.5 P1 1.142206 −1 M 1= = =0.43998 0.2 0.2

The ratio of stagnation/static pressure at entry is:

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The spreadsheet does the calculation for every station from 1 to 26. Using the resulted data, graphs can be plotted in terms of Mach number, Area ratio and pressure ratio. An example calculation for a random tapping (12) is showed below. The ratio of static pressure is: P12 P a− ( PcmHg station 12−PcmHg datum station 26)∗10 = =0.860792 ; P1 653.7 To make the calculation easier, the spreadsheet is using the X parameter as a function of entry Mach number, area and pressure ratio. 1+0.2 x M 1 1

P12 x (¿¿ 2) 2 x M 1=0.4484 P1 1 X= x ¿ Area ratio The local Mach number can now be found. 1

1+0.8 X 2 ¿ 2 −1 1

= 0.6537 (¿¿ 0.4 )2 ¿ M station 12 =¿ Equation (2) is now used to calculate the stagnation pressure. 2 3.5 P0 12= P12(1+ 0.2 M station12 ) =749.8 Pa The speed of sound is found using the equation of state for a perfect gas. Then it can be used to find the air velocity. v a2=γRT ∧M = , where γ air=1.4, R is a gas constant∧for air is 287.053 a ∴ v=M √ γRT =0.6537 x √ 1.4 x 287.053 x 293.5=215.6 m /s The local Mach number column is extracted from Table 1 to produce a new p ¿ , Mach number spreadsheet that is then used to plot the variation of the pressure ratio ( p0 and area throat ratio. An example of spreadsheet is illustrated below. The new spreadsheets created are showed in Ta the variation of different p rent run, starting with 1.4 M er and a pressure of 7 bar. A Th ence between the given geo bar with the 1.4 Liner Mac low achieves the highest M st run. Also, the pressure r

Table 2: Run#1 Table 2:Pressure Run #1 and Spread Area sheet Throat Ratio

The area throat ratio is calculated using equation (3), where γ for air is 1.4 and Mach number at throat is 0.75974. 2 3 A M T 1+0.2 M ) ( = A T M 1+0.2 M T2 Equation (2) can be organised, so that the stagnation pressure can be found very easily. P0=P(1+ 0.2 M 2)3.5 Pa Graph 1 was plotted using Table 2 and is used to investigate the flow behaviours across the nozzle.

Run #1, Liner Mach number 1.4, Injector Pressure 2 bar 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

M P/p0 A/At A/At original geometric distribution

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10 15 20 25 30 35 40 45 50 55 60 65 Distance x (cm) Graph 1: Run #1 Pressure and Area Throat Ratio Distributions

The resulted graphs for the other runs are showed below.

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Run #2 Liner Mach number 1.4 Pressure 5 bar 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

M P/p0 A/At

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Distance x (cm) Graph 3: Run #3 Pressure and Area Throat Ratio Distributions

Graph 2: Run #2 Pressure and Area Throat Ratio Distributions

Run #3 Liner Mach Number 1.4 Pressure 7 bar 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

M P/p0 A/At

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Graph 3: Run #3 Pressure and Area Throat Ratio Distributions

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Run #4 Liner Mach number 1.8 Pressure 2 bar 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

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Graph 4: Run #4 Pressure and Area Throat Ratio Distributions

Run #5 Liner Mach number 1.8 Pressure 5 bar 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

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Graph 5: Run #5 Pressure and Area Throat Ratio Distributions

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Run #6 Liner Mach number 1.8 Pressure 7 bar 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

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Graph 6: Run #5 Pressure and Area Throat Ratio Distributions

7. Discussion “For the isentropic expansion of a gas through a convergent-divergent nozzle, the Mach number monotonically increases from near 0 at the inlet to M=1 at the throat, and to the supersonic value at the exit. The pressure monotonically decreases from po at the inlet to 0.528 p0 at the throat and to the lower value at the exit. The distribution of M, and hence the resulting distribution of p, through the nozzle depends only on the local area ratio A/At. This is the key to the analysis of isentropic, supersonic, quasi-one-dimensional nozzle flows.” (Anderson, Fundamentals of Aerodynamics, 1991) Air will start to flow through the nozzle only there will be a difference of pressure between the inlet and exit. The inlet pressure must be bigger than the exit pressure. Graph 1 shows the changes in flow that can be related to those along a Venturi tube. The small pressure difference will produce a very low-speed subsonic flow inside the nozzleessentially a gentle wind. The local Mach number increases slightly through the convergent portion, reaching a maximum value at the throat, 0.75974. The local Mach number will decrease in the divergent section, reaching a very small but finite value. The pressure in the convergent section will decrease gradually to a minimum value at the throat, and then will increase to the exit. At the point where the Mach number is the highest, the area ratio is the lowest. At Run #2, where the injector pressure is increased to 5 bar, the flow reaches Mach number 1 as shown in Graph 2. There is no change in pressures in the convergent section of the duct; therefore the flow goes supersonic past the throat. The exit pressure is greater than 12

that for the isentropic flow in the divergent section, so that a shock wave will be present. The flow is going subsonic after the shockwave, continuing to decelerate in the divergent section. “Between the throat and the normal shock wave, the flow is given by the supersonic isentropic solution, as shown in Figure 5, b and c. Behind the shock wave, the flow is subsonic. This subsonic flow sees the divergent duct and isentropically slows down further as it moves to the exit. The flow on both the left and right sides of the shockwave is isentropic; however, the entropy increases across the shock wave.”(Anderson, Fundamentals of Aerodynamics, 1991)

Figure 5: Supersonic isentropic solution (Anderson, Fundamental of Aerodynamics, 1991)

Graph 3 shows a supersonic flow which pushes the shock wave to move out of the nozzle. This is caused by lowering the exit pressure below the back pressure. The back pressure is defined as the pressure of the surroundings downstream of the exit. The nozzle is said to be over-expanded, because the pressure at the exit has expanded below the back pressure ( pexit < p back ¿ . In flowing out the surroundings, the jet of gas from the nozzle must somehow be compressed s...