Types of Shock Waves in Divergent Nozzle

This lab experiment was based on determining the various kinds of shock waves produced in a convergent-divergent nozzle by altering the effect of the pressure ratio respectively. The experiment was conducted using computational fluid dynamics and the results produced were compared with the theoretical results to present the accuracy of the experiment. Further, the three different kinds of shock were also demonstrated and discussed.

Introduction

A Convergent-Divergent nozzle is a tube that is used for accelerating the fluid to the required velocity. Since it is a commonly used geometry, it requires highly accurate analysis in order to optimize fluid flow. Using computer oriented softwares such as computational fluid dynamics to analyze supersonic and subsonic flows in the convergent-divergent nozzle can yield in high precision and faster results.

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Convergent-Divergent nozzles are applicable for a variety of industrial applications, one of which includes the use of biological organisms. The function of the nozzle is to produce a supersonic air stream for optimizing the temperature in order to make it harmless to the biological organisms. [1]

Another engineering application that utilizes converging-diverging nozzles is the industrial steam turbine. The nozzles are used as accelerators with compressible fluids e.g. water to increase their velocity to attain velocities that are supersonic before hitting the turbine blades.

This lab experiment is based on analyzing the various types of shock waves created in the convergent-divergent nozzle. The inlet and outlet pressures are varied to display the curved, straight and lambda shock.

Theory

‘The flow through a converging-diverging nozzle is one of the benchmark problems used for modeling the compressible flow through computational fluid dynamics (CFD). Occurrence of shock in the flow field displays one of the most prominent effects of compressibility over fluid flow.’ [3]

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Shock waves are a type of discontinuity. Across a shock, there is a significant increase in pressure, temperature and density of the flow. Shock waves depend on Mach number both upstream and downstream of the flow. When the upstream Mach number is subsonic (Ma < 1), the downstream Mach number is supersonic (Ma > 1). The pressure ratio of the fluid flow can be determined using the below mentioned equation. Figure 1 displays the schematic of the convergent-divergent nozzle used for conducting the experiment.

The dimensions are as follows: (L) = 0.6m

(r1) = 0.1m

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(r2) = 0.12m

Using equation (1) the pressure ratio can be determined theoretically based on the Mach No. and the ratio of the specific gas constant.

……………………………………. (1)

Taking a general case from Table 1 as an example, the inlet and outlet pressures were 220,000 Pa and 100,000 Pa correspondingly. Hence, following the initialization of the boundary conditions, the pressure contour was plotted and the pressure ratio was computed to be 0.102 as displayed in equation (2). Using equation 1, similar pressure ratio is recorded by using the Mach. No. respectively. Table 1 displays the summarized results obtained. The ratio of specific gas constant is taken as 0.14.

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…………….……….. (2)

Equipment

Computational fluid dynamics is an advanced technology used to simulate the flow using simultaneous fluid properties inside a given control volume. It uses computer based modeling to analyze the fluid flow. The volume occupied by the fluid is divided into discrete cells in order to produce highly accurate results.

CFD’s can be used in a wide variety of applications which are usually complicated to work on. Such examples include blood flow inside the veins and the heart of a human body and the simulation of the air flow over a cyclist in order to increase the overall efficiency. ‘CFD is attractive to industry since it is more cost-effective than physical testing.’ [2]

The CFD package used for conducting the experiment was Ansys 12.1. The mesh model of the convergent-divergent nozzle was the input to the FLUENT software inbuilt inside Ansys 12.1. The following section outlines in depth about the procedure and the results discussed from the experiment.

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Procedure

The procedure for performing CFD analysis on convergent-divergent nozzles is described below:

Read the mesh file from the file menu.

Check and scale the mesh as per requirement from the mesh menu.

Display the grid from the display menu and change the Colors option to Color by ID.

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Define the models and enable the k-epsilon option from the define menu.

Define the material as ideal gas from the define menu.

Define the operating conditions and change the operating pressure to zero from the define menu.

Define the boundary conditions and set the inlet pressure and temperature as per requirement from the define menu.

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Repeat the previous to set the boundary conditions at outlet respectively.

Solve and initialize the mesh from the solve menu.

Solve and monitor the residual from the solve menu. Select print and plot options.

Save the case file from the file menu.

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Solve the calculation by setting the number of iterations as per requirement.

Save the data file from the file menu.

Compute and display filled contours of static pressure from the display menu.

Compute and display velocity vectors from the display menu. Set the Scale and Skip to 5.

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Observe the flow and zoom the view for better display.

Repeat the above process with a different set of inlet and outlet pressures to draw comparisons.

Results

Displayed below are the series of results of shock waves based on pressure contours and velocity vectors that were obtained by altering the pressure ratio. The results in Table 1 are based on turbulent fluid flow for an ideal gas.

6. Discussions

The relationship between the inlet pressure and the pressure ratio is displayed in Figure 7. As observed from the graph, as the inlet pressure increases, the pressure ratio decreases, thus confirming the inverse relationship between the 2 quantities. Further discussions are based on the shock observation from the varying pressure ratio. The three different types of shock are displayed in figures 2, 3 and 4.

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Figure 2 displays a straight shock as the light blue color-coded contour is a roughly straight line prior to the dark blue contour. This is observed when the inlet pressure is 300,000 Pa and the outlet pressure is 100,000 Pa.

With reference to figure 3, a lambda shock is observed as the light blue color-coded contour is in the shape of lambda. The yellow contour shows the area of higher pressure than the light-blue contour. The inlet pressure was 250,000 Pa and the outlet pressure remains unchanged.

Figure 4 displays a curve shock. This is confirmed as the pressure color-coded light blue contour is curved as it approaches the boundary of the nozzle after the dark-blue contour. This was recorded at an inlet pressure of 220,000 Pa and the outlet pressure remained constant throughout the experiment at 100,000 Pa respectively.

In addition, figure 8 displays the error percentage between the theoretical and experimental pressure ratio for each case. The highest error of 4% was produced at the inlet pressure of 220,000 Pa. The minimum error percentage recorded were for inlet pressures of 120,000 Pa and 250,000 Pa respectively.

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Conclusion

The above mentioned results and discussions confirm that the experiment conducted using CFD produced highly accurate results when compared with the theoretical study.

Further, the experiment also demonstrated the ability of using computational fluid dynamics to demonstrate various complicated fluid flow parameters, in this case being the shock production in a convergent-divergent nozzle.

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