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This project covers the observation of the different regimes of flow in a converging nozzle and in a converging-diverging nozzle. The parameters that are being observed are the relationships between the pressure ratio vs the distance along the nozzle, the mass flow rate parameter vs the backpressure ratio, and the exit and throat pressure ratio vs the backpressure ratio.
Nomenclature
I. Introduction
Nozzles are hollow cylinders with varying cross sectional-area from large to small. Nozzles are designed to increase the velocity of the medium traveling through them. Nozzles can also be designed to change the direction and flow shape of the flow. While increasing the flow velocity by decreasing its cross sectional-area, nozzles encounter an increase in momentum that is in the opposite direction of flow. For a first-order analysis of nozzles, using isentropic flow throughout the nozzle is adequate because the temperature of the flow is constant and the friction loss along the nozzle is minimal. However, nozzles can only accelerate the flow of substances until the compressibility effects start to take place where the speed of the medium is equal to the speed of sound through the medium. After that point, if the nozzle converges any further the velocity of the medium will decrease. Consider that there exists constant pressure and constant temperature at the inlet of a nozzle. If the pressure at the exit of the nozzle is reduced then the flow rate through the nozzle will increase. This phenomenon occurs until the reduced pressure at the exit is enough to cause the flow to become sonic before the nozzle exit. Creating a ratio between the total pressure and the pressure at the back of the nozzle can indicate what regime of flow is occurring within the nozzle. The same thing can be said about the ratio between the total temperature and the temperature at the back of the nozzle. The pressure and temperature ratio for the choked condition with a specific heat ratio of 1.4 for dry air is:
From this ratio there can exist 2 regimes; non-choked, and choked. From these two regimes, can exist several conditions.
>> Condition 1: P_B/P_0 =1 so no flow occurs
>> Condition 2: There is a slight pressure drop across cross the nozzle creating subsonic flow with decreasing area
>> Condition 3: There is a substantial pressure drop across cross the nozzle creating subsonic flow with decreasing area
>> Condition 4: The flow is choked and the pressure drop and flow velocity is sonic
>> Condition 5: The flow is choked and the pressure drops to where it can no longer accelerate the flow
Nozzle design revolves around 3 parameters: mass flow rate, nozzle flow exit velocity, and nozzle exit pressure. For converging-diverging nozzles under ideal conditions, the flow is supersonic and isentropically expands after the throat which decreases the static pressure and static temperature. 4 regimes and 7 conditions can occur with converging-diverging nozzles. The regimes are un-choked flow, decelerated isentropic flow, supersonic to shock to subsonic non-isentropic flow, and full supersonic flow throughout the nozzle after the shock. The 7 conditions are:
>> Condition 1: The flow is subsonic throughout the nozzle
>> Condition 2: The flow is sonic at the throat and subsonic after the throat
>> Condition 3: The flow is sonic at the throat, speeds up to supersonic but a normal shock wave insetropically decelerates the flow
>> Condition 4: The flow is sonic at the throat, speeds up to supersonic but a normal shock wave forms after the nozzle
>> Condition 5: The flow over expanded
>> Condition 6: The flow is sonic at the throat, speeds up to supersonic
>> Condition 7: The flow is under expanded
Using this relation:
The mach number throughout the nozzle can be determined isentropically.
II. Experimental Setup
An experiment can be created where the inlet of the nozzle is supplied pressure while the outlet of the nozzle is kept at atmospheric pressure or vice versa. The pressure can be measured along the nozzle using pressure probes. The mass flow rate can be measured using a rotameter. Data using this experimental setup has already been acquired and is tabled below.
III. Simulation Setup
This same experiment can be compared to an extrapolated solution using Ansys Fluent. This can increase the amount of flow information throughout the nozzles to provide a in depth picture of the flow instead of pressure information from probes. However it always necessary to verify simulated results with experimental results to provide details on how accurate the simulation is from the physical phenomena. The same boundary conditions of the inlet and outlet pressures as well as the operating temperature from the physical experiment will be used to create the simulation’s boundary conditions.
Several tools can be used to further define a quadrilateral mesh. These tools are edge sizing implementing number of divisions, face sizing implanting element size, face meshing with a quadrilateral method, and refinement. Proper usage of these tools depends on the context but as a general rule the mesh should be further refined with a decrease in element size in high areas of flow activity. The mesh’s element size around the throat should be further refined in order to increase simulation accuracy and fidelity in that region.
Next, the procedure that was used to simulate nozzle geometries and boundary conditions presented above will be discussed. The k-epsilon model with scalable wall functions was used as the viscosity model that was to generate the solutions for each run as well as the energy equation. The fluid material that was used in the domain was air that is being treated as an ideal gas so that density can be solved at each point using the ideal gas law. This material is used as a fluid in the cell zone conditions for the geometry. The operating pressure was set to 101,325 Pascals in the operating conditions. Gravity was not used in any of the simulations. The inlet and outlet boundary conditions were set to pressure-inlet and pressure-outlet. These boundary conditions changed in accordance with the table listed above. The convergence criteria in the residual monitors were all set to 1e-06 in order to give a precise solution. Any convergence criteria that is any smaller is too precise and any solution with residuals that is bigger than 1e-04 needs to be further solved. For this particular set of simulations, the number of iterations it takes for convergence revolved around 360 iterations.
IV. Results
Runs 1 through 3 are in regime 1 (unchoked). Run 3 is the closest to being in regime 2 (choked decelerated flow) but the Mach number at the throat is 0.8 instead of 1. Runs 4 through 10 are in regime 3 (supersonic to shock to subsonic non-isentropic flow). Runs 11 through 12 are in regime 4 (full supersonic flow throughout the nozzle after the shock). Runs 1 through 2 are in condition 1. Run 3 is in condition 2. Runs 4 through 10 are in condition 3. Run Run 11 is in condition 4. Run 12 is condition 6. None of the runs in this data set exhibit signs of condition 5 or 7.
V. Conclusion and Observations
For the converging nozzle, the exit flow at some pressure iterations cycle between sweeping upward and downward respective of the y axis an. This is probably due to the instability of the choked pressure profile at the throat. The expansion fans at the throat become more apparent at higher pressures with a converging nozzle. At higher pressures, the Mach number at the throat remains 1 but further increase after the throat. Increasing the pressure for a converging nozzle dramatically changes the velocity profile at the throat. The boundary layer remains at Mach or higher than Mach while the velocity at the center of the parabolically decreases. This effect increases with higher pressures. The linearity of the static pressure across the taps increases with increasing pressure except for taps 1 and 10. The total pressure at the entrance of the converging nozzle shrinks with increasing pressure. Shock diamonds become more apparent due to the increasing expansion fans from the increasing pressure.
For the converging-diverging nozzle, the static pressure and the Mach number across the x-axis seem to be inverse of each other. There are points in the runs where the total pressure and Mach number along the x-direction no longer increase after the converging section along the diverging part of the nozzle. This is also the point where the static pressure is at i’s most negative. This point gets moved further down the x-axis with increasing pressure The total pressure drop across the x-axis is like that of the converging nozzle.
In this project, various flow regimes and conditions were observed through experimental and computational resources. Concepts and data from this project could be used to further explore converging and converging-diverging nozzle designs.
References
Narsipur, R., “ASE4343_Project-2_Handout.pdf”, ASE4343 Compressible Aerodynamics, 2021 (unpublished) 2
MATLAB and Statistics Toolbox Release 2020b, The MathWorks, Inc., Natick, Massachusetts, United States. 3
MATLAB. (2010). version 7.10.0 (R2010a). Natick, Massachusetts: The MathWorks Inc. 4
ANSYS (2016). ANSYS Fluent - CFD Software | ANSYS 5
Anderson. (2016). Fundamentals of aerodynamics (6th ed.). McGraw-Hill Education. 6
Appendix
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