Viscous Compressible flow in Nozzles

2009

Computational Fluid Dynamics

Project 3

Viscous Compressible Flow in Nozzles

2/21/09

Shiva Naraharisetty

Robbie Driscoll

Sandeep Kumar

William Stoddard

Rajiv Kattekola

Table of Contents

Problem formulation3

Gambit and FLUENT software5

Results

Case 1 7

Case 211

Case 315

Comparison of laminar and turbulent flows18

References21

Work Report22

Problem Formulation

The objective of this project is to analyze the viscous compressible flow properties of air along the length of the nozzle. The variations of velocity, Mach number, density and pressure are studied for three cases which essentially depend on the back pressure of the nozzle. The cases all had an initial static pressure given as 780 kPa and static temperature of 270 K, with a target inlet velocity given as Vi = 70.869 m/s. The back pressure was then set to 690 kPa, 350 kPa, and 60 kPa for case 1, case 2, and case 3, respectively. The geometry is created in gambit and the simulation is run in the fluent for all the three cases and compared with the analytical values that are calculated for a quasi dimensional flow.

We were given that turbulent flow would be used in all cases, except in part of case 3, where a laminar and turbulent flow were to be compared. The converging–diverging axisymmetric nozzle’s geometry was given through a variation of the nozzle radius, given in the following equation. The length of the nozzle was given as 8 m.

Figure 1: Diagram of nozzle geometry

The fluid used for simulation is air, with the density changed from ‘constant’ to ideal gas. Viscosity was set to one of the turbulent models, as directed in the project instructions.

Theory of Nozzles:

A nozzle is a mechanical device designed to control the characteristics of a fluid flow as it exits (or enters) an enclosed chamber. A nozzle is often a pipe or tube of varying cross sectional area, and it can be used to direct or modify the flow of a fluid. Nozzles are frequently used to control the rate of flow, speed, direction, mass, shape, and/or the pressure of the stream that emerges from them. In our problem it is converging diverging nozzle is also known as DeLaval nozzle. Velocity always increases in the converging section of nozzle and may increase or decrease in diverging section depending upon the type of flow whether it is subsonic or super-sonic.

Figure 2: Diagram of a DeLaval nozzle

The above figure depicts the operation in a typical converging diverging nozzle. Fluid comes from a chamber which has a very large cross section area and velocity can be safely assumed to be low. The pressure at this condition is called as stagnation pressure. Gas flows from the chamber into the converging portion of the nozzle, past the throat, through the diverging portion and then exhausts into the ambient as a jet. The pressure of the ambient is referred to as the 'back pressure' and given the symbol.

When back pressure is equal to the chamber pressure, there is no flow through the nozzle. As the back pressure lowers, the mass flow will increase until choked flow, where Mach 1 is reached at the throat. Any further lowering of the back pressure can't accelerate the flow through the nozzle, because that would entail moving the point where M=1 away from the throat where the area is a minimum, and so the flow gets stuck. Increasing the nozzle pressure ratio further will not increase the throat Mach number beyond unity. Downstream (i.e. external to the nozzle) the flow is free to expand to supersonic velocities.

Divergent nozzles slow subsonic fluids, but accelerate sonic or supersonic gases. Convergent-divergent nozzles can therefore accelerate fluids that have choked in the convergent section to supersonic speeds. This process is an efficient way of expanding and accelerating a high pressure gas.

Gambit and FLUENT software

Gambit

Geometry of the mesh was generated using Gambit. A 2D mesh was generated with axis boundary at the bottom, so an axisymmetric modeling in FLUENT could be used, as the flow is assumed to be symmetrical. A very coarse mesh was generated using uniform cell size. Two clustered grids were generated for each of the cases – one coarse grid and one finer grid. The variables which were sensitive to grid refinement were plotted and these were studied carefully on both the grids to check the effect of the grid on the solution. Successive Ratio option was used for creating clustered edges in the geometry. This would be used to better resolve the turbulent boundary layer, as in this project, turbulent viscous flow is to be used. Both the wall (upper edge in the figures) and the inlet and outlet were clustered for the clustered cases, in order to resolve not only the boundary layer but the effects of the inlet and outlet. The refined grid was also tested for Laminar flow, neglecting the effect of turbulence on the flow equations for the 3rd case (60 kPa) and the results were plotted for both the Laminar and Turbulent cases.

Also, techniques such as finding distance on the geometry for an already generated mesh, with no selectable points, were learned and used while checking the mesh. This was done by creating points on the geometry with making use of Create Real/virtual vertex command from Create vertex drop down box. By creating edge between those two points and making use of "summarize" option in the create edge drop down box we can get the whole information of the edge. By creating edge like this, it does not make any changes to the mesh if it is already meshed.

Figure 3: Coarse unclustered mesh

Figure 3 shows the coarse unclustered mesh. Evenly spaced grid points 100 axially and 21 vertically were chosen to mesh the unclustered coarse mesh. Figure 4 shows the coarse clustered mesh. The number of points was set to 200 along the axis and 130 vertically. As one can see, it is clustered at the wall, the inlet, and the outlet.

Figure 4: Coarse clustered mesh

Figure 5 below shows the finer clustered mesh. For this, 450 points were taken axially, with 120 vertical points for each axial location. Similar clustering is seen in the figure as in the coarse clustered mesh.

Figure 5: Fine clustered mesh

FLUENT

In FLUENT, the density based, implicit, steady, axisymetric, Green- Gauss cell based solver was chosen for the clustered meshes. The earlier test of the very coarse unclustered mesh took place on an unsteady axisymmetric density based solver, and run with an animation to see when values became steady. In all cases, the energy equation was utilized, and density of the material (air) set to ideal gas rather than constant density. A second order upwind discretization was chosen. Operating conditions were set to 0 Pa pressure. The inlet was a pressure inlet set to the static pressure desired, and the total pressure was set to that plus 1/2ρV2, where ρ is the density and V was the desired inlet velocity. This came out to 820 kPa total pressure. The solution was initialized from the nozzle inlet conditions. The outlet was a pressure outlet set to the pressure corresponding to the particular case (690, 350, and 60 kPa for cases 1, 2, and 3).

In FLUENT several viscous models are available. Two of the most commonly used turbulent models are the Spallart Allmaras and k-epsilon method. They both work off of the Boussinesq hypothesis, relating the Reynolds stresses to the mean velocity gradients using the following equation.

The k-epsilon model solves for 2 variables (k and ε) using 2 equations. In these, k is the kinetic energy and ε is the specific dissipation rate. The Spallart Allmaras method a variable representing turbulent viscosity is solved for, and only one equation is needed, typically reducing computation time, though in our case, we found it quickly converged to a level where the residuals were not quite low enough to meet the convergence criteria, until a very long simulation time. Spallart Allmaras was chosen as the viscous model for the coarse clustered case, due to its simple one equation method. However, with Spallart Allmaras turbulence method it took lot of time for the convergence of the problem when compared to k-epsilon method, which had been seen to work well for the unsteady unclustered case. So the fine mesh is done using k-epsilon turbulence model.

Surface monitors were used to plot the values of y+ and the report type was chosen as facet average. Creating the fine mesh with approximately same grid point density on both X and Y axis tended to be most likely to produce y+ values in the desired range. As it is taking lot of time for convergence we have decreased the convergence criteria to 10-4 for the fine clustered mesh case, whereas it had been 10-6 for the coarse clustered mesh cases. We could not find much difference in the results for the case with y+ values not in the range and the case with y+ values in the range. For the coarse mesh it took nearly 5 hrs for the convergence and for the fine case it took approximately 15 hrs for convergence.

Results:

To observe the effect different mesh sizes have on the simulation, the pressure and Mach number along the centerline were plotted for each case on the same plot for each of the 3 different meshes. These are two main factors that determine if the boundary conditions are met, and how the flow behaves. Other factors such as temperature, density, or velocity act in similar ways to Mach or pressure. One could determine the temperature using the simple relation of a constant total temperature and the isentropic relations based on Mach number. From that and pressure, density can be calculated. From temperature, speed of sound, and then velocity can also be calculated. Therefore pressure and Mach are sufficient to characterize the flow for comparison on the graphs. Contours of pressure, density, velocity, and Mach number are all displayed to get a better idea of how the flow behaved.

Case1

For all 3 meshes in case 1, a relatively weak shock forms very near to the throat, providing the highest pressure rise of the three backpressure cases, but without rising back to the original 780 kPa, which is expected for flow where the static pressure is given for the inlet, rather than the total pressure. The placement of the shock is fairly consistent for all 3 cases. The resolution of the shock is best for the fine clustered mesh.

Oddly, though, the two best in agreement on the placement of the shock were the unclustered coarse mesh and the clustered fine mesh case, though the finest case lies between the other two cases. The pressure graphs in figure 7 give an idea of the cause, which is that the coarse mesh started at a lower pressure overall. If it had converged to a higher pressure it would likely have coincided with the coarse clustered mesh case.

Figure 6: Comparison of Mach number along the axis for case 1 (690 kPa)

Figure 7: Comparison of static pressure along the axis for case 1 (690 kPa)

Figure 8: Contours for case 1, coarse unclustered mesh (top left pressure, top right density, lower left velocity, lower right Mach number)

Figure 9: Contours for case 1, coarse clustered mesh (top left pressure, top right density, lower left velocity, lower right Mach number)

Figure 10: Contours for case 1, fine clustered mesh (top left pressure, top right density, lower left velocity, lower right Mach number)

Case 2

As one can see from plots ** , the greatest agreement for case 2 is between the fine and coarse clustered meshes, which take best account of the boundary layer. The coarse unclustered mesh generates a shock that is a bit further downstream from the shock as predicted by the clustered meshes.

When checking the y+ ( , where uτ is the friction velocity ) value, which for turbulent flow in fluent, should fall between 30 and 60 or so, the two clustered meshes managed to converge to a value within that range. The unsteady case run for the coarse unclustered mesh, on the other hand, had values over a couple orders of magnitude higher than the expected range. This is due to the inability to capture as many points in the boundary layer. Some attempts to refine the grid ended up with y+ too low, indicating it may have encountered the viscous sublayer, where turbulent modeling is not as valid.

The coarse unclustered mesh’s overestimation of drag seems to have increased the separation, making the equivalent cross-sectional area at which the shock should occur appear further downstream. After the strong shock wave, the flow is seen to separate a bit from the wall. This was not seen in the inviscid case of project 2. Because of this, the profile of pressure is different from predicted quasi-one dimensional flow, because the area through which flow is at the centerline speed is less than the total area of the nozzle at a given axial location.

Figure 11: Comparison of Mach number along the axis for case 2 (350 kPa)

Figure 12: Comparison of pressure along the axis for case 2 (350 kPa)

Figure 13: Contours for case 2, coarse unclustered mesh (top left pressure, top right density, lower left velocity, lower right Mach number)

Figure 14: Contours for case 2, coarse clustered mesh (top left pressure, top right density, lower left velocity, lower right Mach number)

Figure 15: Contours for case 2, fine clustered mesh (top left pressure, top right density, lower left velocity, lower right Mach number)

Case 3

For the third case, 60 kPa back pressure, a very high level of agreement is found between all three meshes. The two best in agreement are still the clustered meshes. The favorable pressure gradient at all points means no separation is seen, though a dip from maximum Mach number and a slight rise in pressure more than inviscid theory would predict is seen at the end for all 3 meshes where the viscous effects have reached the most developed point for the nozzle.

Figure 16: Comparison of Mach number along the axis for case 3 (60 kPa)

Figure 17: Comparison of pressure along the axis for case 3 (60 kPa)

Figure 18: Contours for case 3, coarse unclustered mesh (top left pressure, top right density, lower left velocity, lower right Mach number)

Figure 19: Contours for case 3, coarse clustered mesh (top left pressure, top right density, lower left velocity, lower right Mach number)

Figure 20: Contours for case 3, fine clustered mesh (with turbulent model) (top left pressure, top right density, lower left velocity, lower right Mach number)

Comparison With Laminar Flow

When comparing the laminar fine clustered case 3 to the turbulent fine clustered case 3, one can see that there is still relatively good agreement between the data. Turbulent flow modeling causes a slightly lower Mach nearest to the exit of the nozzle, and a slightly higher pressure overall for the same total pressure inlet, but otherwise are nearly indistinguishable from centerline data. The contours show a slightly higher amount of uniformity in Mach number across the cross section at the end of the nozzle, but again, mostly remains the same between the two cases.

Figure 21: Comparison of Mach number along axis for case 3, fine clustered mesh, with turbulent or laminar viscosity

Figure 22: Comparison of pressure along axis for case 3, fine clustered mesh, with turbulent or laminar viscosity

Figure 23: Contours for case 3, fine clustered mesh with laminar viscosity modeling (top left pressure, top right density, lower left velocity, lower right Mach number)

References

Anderson, John D. Jr., Computational Fluid Dynamics The Basics With Applications. McGraw-Hill, Inc. New York, 1995

Anderson, John D. Jr,. Modern Compressible Flow: With Historical Perspective McGraw-Hill, Inc. New York, 2004

University of Vermont, “Nozzle Applet” page. http://www.engapplets.vt.edu/fluids/CDnozzle/cdinfo.html

“Near-Wall Mesh Guidelines for Wall Functions” webpage, Fluent inc. 2007 http://web.njit.edu/topics/Prog_Lang_Docs/html/FLUENT/fluent/fluent5/ug/html/node373.htm

Work Report

Shiva Naraharisetty – Co-wrote report. Helped run FLUENT cases

Robbie Driscoll – Worked on Fluent cases, cowrote report

Sandeep Kumar – Generated the fine meshes, Ran the FLUENT cases, Co-wrote report.

William Stoddard – Did work on case 3 in Gambit/Fluent, cowrote report, did calculations.

Rajiv Kattekola – Generated the Coarse mesh, Worked on running the FLUENT cases for the fine and coarse meshes, Co-wrote report

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