Quasi-one-dimensional flow A=A(x), p=p(x), r=r(x), T=T(x), u=u(x)

Nozzle 3.6 INSTRUCTION MANUAL A DeLaval Nozzle Analysis Program for Microsoft Windows By AeroRocket | Buy On-Line | Add To Cart | Nozzle Instructions

Nozzle 3.6 is a one-dimensional with cross-sectional area variation, compressible flow computer program for the analysis of converging-diverging nozzles. Nozzle internal flow may be entirely subsonic, entirely supersonic or a combination of subsonic and supersonic including shock waves in the diverging part of the nozzle. Shock waves are clearly identified as vertical red lines on all plots. The cross-sectional shape in the axial direction of the nozzle is specified by selecting from five standard nozzle types or by defining nozzle geometry using the Free-Form nozzle geometry method. Nozzle plots color contours of pressure ratio, temperature ratio, density ratio, and Mach number and has a slider bar that displays real-time values of all nozzle flow properties. New in this version is the ability to determine shock-angle, jet-angle (plume-angle) and Mach number for axisymmetric and two-dimensional nozzles in the region near the lip for underexpanded and overexpanded flow.

NOZZLE SUMMARY OF FEATURES
(1) Specify nozzle geometry as either Parabolic, Conical, Bell, Imported, or Free-Form. Free-Form nozzle shapes may use up to 30 points to define nozzle geometry.
(2) Standard and Import nozzle shapes may have up to 1000 axial points defining the cross-sectional area distribution of the nozzle.
(3) Select either the classical isentropic and normal-shock relations method or the MacCormack backward-predictor forward-corrector finite difference method to determine characteristics of nozzle internal flow.
(4) Locate internal shock waves quickly using the slider bar that displays nozzle property verses axial location in real time.
(5) Determine Mach number (V/c), pressure ratio (P/P0), density ratio (R/R0) and temperature ratio (T/T0) at each axial location in the nozzle.
(6) Determine shock wave location, Mach number before the shock wave, Mach number after the shock wave and nozzle area at the shock wave location.
(7) Specify fluid properties for a large number of inert gases, liquid fluid rocket propellants and solid fuel rocket propellants or specify your own.
(8) Specify the units of analysis as MKS (meter-newton-seconds), CGS (centimeter-newton-seconds), IPS (inch-pound-seconds) and FPS (foot-pound-seconds).
(9) Enlarge all plots for easy data reduction and output all data to a color printer.
(10) Easily select any plot for review and printout.
(11) Fast solution - most analyses completed in less than 15 seconds.
(12) Generate color contour plots for Mach number (Mn), Pressure ratio (P/P0), Temperature ratio (T/T0), and density ratio (R/R0).
(13) Determine exterior flow properties in the nozzle-lip region for underexpanded nozzles and overexpanded nozzles.
(14) Added a hybrid rocket motor propellant having the following fuel and oxidizer to the list of combustion gases: 85% Nitrous Oxide, 15% HTPB.
(15) Made the SSME example (shown below) the start-up data for Nozzle program analyses. Data easily cleared for new data entries.
(16) Two-dimensional plume analysis using the method of characteristics for underexpanded (Pexit > Patm) flow.
(17) Nozzle_Examples.zip in the Nozzle directory includes 34 nozzle examples used for validation purposes.

Nozzle Minimum System Requirements
(1) Screen resolution: 800 X 600
(2) System: Windows 98, 2000, XP, Vista, NT or Mac with emulation
(3) Processor Speed: Pentium 3 or 4
(4) Memory: 64 MB RAM
(5) English (United States) Language
(6) 256 colors

For more information about Nozzle please contact John Cipolla at aerocfd@aerorocket.com.

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NOZZLE OPERATING INSTRUCTIONS
A) BACKGROUND THEORY - NOZZLE INTERNAL FLOW
As the exit back pressure, Pe is reduced below Po, flow through the nozzle begins. If Pe is only slightly less than Po, the flow throughout the nozzle is subsonic and the pressure profile along the axis would be like curve A in Figure 1. Reducing Pe increases the mass flow rate through the nozzle. As the flow rate increases, the pressure at the throat decreases until it reaches the critical pressure as indicated by curve B (PCRIT-1). The exit pressure Pe which exactly corresponds to sonic conditions at the throat can be easily determined from isentropic flow relations. The flow is subsonic everywhere in the nozzle except at the throat, and mass flow is the maximum possible for the given nozzle and the reservoir conditions.

Suppose the exit pressure is now reduced to a value corresponding to curve F (PCRIT-3) in Figure 1 where no shocks are present in the nozzle. The exit pressure at F is such that the entire expansion is isentropic and the flow is supersonic in the diverging portion of the nozzle. The value for pressure is simply obtained from the isentropic relationships for Mach number, pressure, temperature and density and represents an optimal nozzle design. The pressure within the nozzle exit cannot be reduced further and when the external pressure is reduced to G the fluid leaving the nozzle changes its pressure through a complicated flow pattern outside the nozzle. Thus, the curves B and F represent the two limiting cases of exit pressure for isentropic flow in such a nozzle. For exit pressures below that at B, a shock wave forms within the diverging part of the nozzle, changing the flow from supersonic to subsonic and compressing the gas exactly enough to match the nozzle exit conditions. Because of the entropy rise across the shock, the overall flow through the nozzle is not isentropic, although the flow on either side of the shock can still be considered isentropic. The lower limit for this kind of flow pattern is given by a shock occurring exactly at the exit of the nozzle as indicated by curve D (PCRIT-2). The flow conditions for exit pressures between curves B and D may be computed with aid of the isentropic relationships and normal shock analysis. At still lower exit pressures the flow adjusts itself through a series of two-dimensional or three-dimensional shock waves and the average exhaust velocity is generally still supersonic.

The designer must chose an appropriate condition from the previous possibilities for his particular application. When the flow leaves the nozzle at supersonic speeds and its pressure exactly equals the surroundings (curve F) , the nozzle is called correctly expanded (PCRIT-3). If the exit area of the nozzle is less than the correctly expanded value for a given back pressure, the nozzle is underexpanded and the fluid leaving the nozzle has a pressure greater than the surroundings (curve G). On the other hand, if the exit area of the nozzle is too large, shock waves form within or just outside the nozzle and the flow is called overexpanded. The particular mode of operation of any nozzle can be quickly checked by first establishing the limiting pressure curves B and D and comparing them with the specified exit pressure. REFERENCE: Pages 299 and 300 Fluid Flow, Sabersky and Acosta.

 
Figure 1 and Figure 2.


B) STEP-BY-STEP NOZZLE ANALYSIS EXAMPLE (REFER TO FIG 2a and FIG 3)
BASIC DIMENSIONS ARE FOR THE SPACE SHUTTLE MAIN ENGINE (SSME)
1) Using the Units pull-down menu check Length(inch), Pressure(lb/in^2) to specify the units of the analysis.
2) Select Bell nozzle in the Nozzle Shapes section. The data entries for a Bell nozzle having a bell shape will appear. Please see notes below.
3) Enter an entrance temperature of 5400 degrees R.
4) Enter an entrance pressure of 3000 psia.
5) Enter an Atmospheric pressure of .0017 psia to simulate vacuum conditions in space. Please see Note-5 for other options, for example the optimal design condition where no shocks are present.
6) Using the Gases pull-down menu select OXYGEN, HYDROGEN as the working fluid in the nozzle. By selecting OXYGEN, HYDROGEN the value for the ratio of specific heats (g) and the Gas Constant (Rgas) are automatically specified in the appropriate spaces in thi s case having units, in^2 *sec^2/R.
7) Enter total nozzle length as 127 inches (The converging section is 6 inches long and the diverging section is 121 inches long).
8) Enter throat diameter as 10.3 inches.
9) Enter the throat location from the origin as 6 inches.
10) Enter the upstream throat radius as 7.725 inches (1.5 * Rthroat).
11) Enter the downstream throat radius as 1.967 inches (0.382 * Rthroat) .
12) Enter the entry angle of parabolic section as 32 degrees.
13) Enter the exit diameter as 90.7 inches.
14) Enter the total number of grids along the nozzle axis as 1000 points. A maximum of 1000 nozzle X,Y coordinates may be defined.
15) In the Solve Flow section select the Classical gasdynamics method option.
16) Click the SOLVE NOZZLE FLOW command button to determine nozzle flow characteristics using the method specified in step (16).

Figure 2a: Nozzle Dimension Locations

NOZZLE RESULTS (REFER TO FIGURE 4)
18) Use the slider-bar to see real-time results for Nozzle radius [Y(X)], Nozzle cross-sectional area [A(X)], Mach number [Mn], Pressure ratio [P/Po], Temperature ratio [T/To] and Density rio [R/Ro].
19) After selecting the desired plot variable option-button, enlarge the plot by clicking the ENLARGE PLOT command button. The plots can be printed from the enlarged plot screen.
20) Nozzle results may be sent directly to a printer in text form by clicking File and then Print Detailed Results.
21) Click the Show results option to display the Results section.
The results of the analysis are:
a) Mach number (Mn) at exit is 5.182
b) Pressure ratio (P/P0) at exit is .0007
c) Temperature ratio at exit is .2227
d) Thrust produced is 466,151.266 pounds.
e) Mass flow rate through nozzle is 1024.329 pounds per second.
f) Thrust coefficient (CF) is 1.865

SSME ACTUAL MEASUREMENT
Mass flow rate: 1035 pounds per second (1.0% difference)
Vacuum thrust: 470,000 pounds (0.80% difference)

Note-1: The Bell nozzle shape uses a parabolic curve approximation from the throat to the nozzle exit. For an approximate G.V.R. Rao Bell nozzle configuration the contour immediately upstream of the throat is a circular arc with radius 1.5*Rthroat. The divergent part of the nozzle immediately downstream of the throat is made up of a circular section with a radius of 0.382*Rthroat and then a parabola to the exit of the nozzle.

Note-2: If Free-Form Shape is selected in step (2) the Imported and Graphical Shapes entry box appears. Enter all required data and then bring up the Free-Form screen by double-clicking on the DEFINE FREE-FORM NOZZLE SHAPE icon. Generate the nozzle shape by dragging up to 30 points into position on the screen and then return to the main screen.

Note-3: In step (2) Import a list of X,Y nozzle coordinates by clicking on File and then Import Nozzle Shape. The data must be in the following format: First line: [Total number of nozzle X,Y coordinates]. Second and subsequent lines: [Point number], [X nozzle location], [Y nozzle location] and have a .TXT file delimiter. A maximum of 1000 nozzle X,Y coordinates may be defined.

Note-4: In step (15) the selection of the MacCormack finite difference method will allow Nozzle to use the forward-predictor backward-corrector finite difference CFD method to compute nozzle flow. This option computes curve F (PCRIT-3) which is the optimum design condition when no shocks are present in the nozzle (isentropic) and the flow is entirely supersonic in the diverging part of the nozzle. For optimum nozzle expansion the nozzle exit pressure, P2 is equal to the external pressure, Patm. Rocket nozzles are normally designed using the PCRIT-3 flow expansion condition for optimal performance. This method is only accurate if the residuals are reduced to at least 1.0E-6. In practice the number of nozzle points is usually less than 50, the CFL should be about 0.80, the starting Mach number should be around 0.001 and finally the total number of iterations should be at least 750 and sometimes up to 2000.

Note-5: To compute an optimal nozzle design when no shocks are present and If the Classical gasdynamics method is selected insert 0.0 for the Atmospheric (back) Pressure. SOLVE the flow and the value for PCRIT-3 and therefore atmospheric pressure is automatically determined and displayed in the Results section and reflected in the input data section. To compute the case where the flow is sonic (M=1) at the throat and subsonic everywhere else (PCRIT-1) insert a value for Exit Pressure just slightly smaller than the Entrance Pressure. SOLVE the flow and an estimate for PCRIT-1 appears in the status bar at the bottom of the screen. Insert this estimate for PCRIT-1 into the value for Exit Pressure and SOLVE again. The new value for the Exit Pressure in the Results section is the new value for PCRIT-1.

Note-6: Please remember to "Click" back using the Return icon. Using the [X] box will kill the results and delete the modifications or may hang the application.


Figure 3: Input data for ideal expansion, no shock in nozzle.


Figure 4: Results for ideal expansion, no shock in nozzle.


Figure 5: Mach number contour plot for ideal expansion, no shocks in nozzle.


Figure 6: Results where back pressure is 100 psig causing a shock in the diverging part of the nozzle. This figure is not part of the SSME example.

This part of the description is to illustrate the Free-Form screen and is not part of the SSME nozzle example.

Figure 7: Free-Form screen for generating nozzle geometry and is not part of the SSME nozzle example.


C) NOZZLE EXTERNAL FLOW
This treatment of nozzle external flow uses two-dimensional oblique shock and Prandtl-Meyer expansion theory to predict shock-angle (Beta or b), jet-angle (Theta or q) and Mach number (Mjet) in the jet. If the nozzle is axisymmetric, as is true in most nozzles, the present solution is valid in the immediate vicinity of the nozzle-lip region. Far away from the nozzle-lip, the expansions and compressions (shocks) are not defined by two-dimensional oblique shock and Prandtl-Meyer theory and are valid for two-dimensional flow. For this reason the external nozzle analysis is limited to the region near the nozzle-lip where the analysis is valid for two-dimensional and axisymmetric flow. A nozzle is underexpanded when Pa/Pc < Pe/Pc and is characterized as a nozzle that experiences a series of external Prandtl-Meyer expansion waves starting from the lip of the nozzle. Similarly, a nozzle is overexpanded when Pa/Pc > Pe/Pc and is characterized as a nozzle that experiences a series of oblique shocks and compressions starting from the lip of the nozzle. In this analysis, Pa/Pc is the ratio of the atmospheric (Pa) or back pressure to the chamber pressure (Pc) and Pe/Pc is the ratio of the nozzle exit pressure (Pe) to the chamber pressure (Pc). Variables with subscript (c) are related to the entrance of the nozzle and variables with subscript (jet) are related to the exterior-region adjacent to the nozzle-lip. Finally, variables with subscript (a) are related to the atmospheric pressure or back pressure of the environment.

Nozzle will determine the oblique shock angle (Beta or b), and the outer boundary or jet-angle (Theta or q) depending on whether the flow is overexpanded or underexpanded using the Pa/Pc and Pe/Pc pressure ratio criteria. For underexpanded flow, Nozzle uses Me, the exit Mach number to determine the Prandtl-Meyer function in region (e) of the flow. Mjet is then computed using the isentropic expansion equation by assuming Pjet = Pa. Then, using Mjet the Prandtl-Meyer function is determined in region (jet) of the flow. Finally, the outer boundary or jet-angle is determined using the relationship, Theta(q) = n(Mjet) - n(Me). Theta (q) is defined as the angle the jet boundary makes with the horizontal axis of the nozzle-lip.

For overexpanded flow, Nozzle determines Pe/Pc and Pjet/Pc to compute the pressure ratio across a possible oblique shock using the relationship, Pjet/Pe = Pjet/Pc * Pc/Pe. Using the pressure ratio across the oblique shock (Pjet/Pe), the shock angle Beta (b) and the jet-boundary Theta (q) is computed using oblique shock theory. Finally, The Mach number in region (e) normal to the shock is determined from, Mn = Me sin(Beta) and from this the Mach number in the (jet) region of the flow is determined using the normal shock relationship. Figure-8 and Figure-9 define the variables used for overexpanded flow and underexpanded flow, respectively. Figure-10 illustrates the SSME nozzle having an atmospheric pressure of 14.7 psia and the resulting overexpanded external flow with shock and jet boundary. Finally, Figure-11 illustrates the SSME nozzle having an atmospheric pressure of 2.0692 psia with a slightly underexpanded external flow and a jet boundary set at almost 0.0 degrees. This condition can be understood to mean optimal expansion when no shocks are present and the nozzle is exhausting directly into the atmosphere.

Figure 8: Overexpanded nozzle (Pa/Pc > Pe/Pc)


Figure 9: Underexpanded nozzle (Pa/Pc < Pe/Pc)


Figure 10: Overexpanded nozzle where Patm/Pc > Pe/Pc and external shocks occur.


Figure 11: Properly Expanded nozzle (Slightly Underexpanded) where Patm/Pc < Pe/Pc.


TWO-DIMENSIONAL PLUME ANALYSIS
For underexpanded (Pexit > Patm) nozzles the external flow must adjust itself through a series of expansion and compression waves. Initially, at the nozzle exit the flow goes through a Prandtl-Meyer expansion wave to adjust the flow to ambient pressure. Then, to maintain the constant-pressure boundary condition on the outer boundary of the plume, the expansion wave reflects off the jet boundary as a compression wave. This process is repeated through several cycles of expansion and compression waves that reflect off the boundary of the plume.

The routine in NOZZLE that models the external plume assumes the exit flow is supersonic, two-dimensional and underexpanded. The user may run the plume analysis by clicking FILE and then clicking 2-D Plume Analysis. The plume analysis is used by first inserting the ratio of ambient pressure to exit pressure (Pa/Pe). If results for the flow on the main screen are for an underexpanded nozzle,  the value for Pa/Pe is inserted and the plume results are displayed automatically. Otherwise, the user can over ride the input and insert another value for Pa/Pe. Likewise, the other plume input values including, Nozzle exit plane Mach number (Me), Specific heat ratio (Gamma) and Nozzle exit diameter are entered automatically and the external plume flow computed if the flow is underexpanded. The user may step though the flow field by clicking the LOCATION button to display the physical location in the plume and the properties at that location.  Finally, the screen may be enlarged and screen results printed by the click of a button.

NOZZLE REVISIONS
Nozzle 2.7 and Nozzle 2.8 Features
1) Nozzle outputs nozzle shapes in X,Y format. First, the user must run the program or click Plot Shape to generate the points describing the nozzle. The user may output X,Y nozzle coordinates and all axially varying nozzle parameters using the Save Data File As command. The data file created using Save Data File As has the .CSV extension to distinguish it from the imported shape file that has the .TXT extension.
2) Nozzle shows up on the Status Bar. The program may be minimized, maximized or terminated using the window controls.
3) Nozzle can model ultra-small nozzle shapes. Very small nozzles use scientific notation while larger (Greater than .001 diameter) nozzles use standard output format.
4) Mass flow rate in kg/sec or lbm/sec added to the output.

Nozzle 2.9 Features
1) Fixed a few minor problems involving display of very small nozzle dimensions and output results.

Nozzle 3.0 Features
1) Fixed error in the computation of thrust and mass flow rate.
2) Fixed a few spelling errors.

Nozzle 3.1 Features
1) Fixed error in the computation of thrust and mass flow rate.

Nozzle 3.2 Error Fix
1) Fixed an error in Nozzle that manifested itself when analyzing nozzles with area ratios (Ae/At) greater than 60. Specifically, for larger area ratios and for pressure ratios (Pe/Pc) sufficient for a shock to form in the diverging portion of the nozzle, Nozzle would incorrectly determine that the flow was sonic at the throat and subsonic everywhere else in the nozzle. The problem was that the constant PCRIT was miss-dimensioned as a string variable when it should have been defined as a single precision variable. This was an intermittent precision problem because sometimes the results were correct and sometimes incorrect depending on the value of the pressure ratio and area ratio.

Nozzle 3.3 Features and Error Fixes
1) Added color contour plots for Mach number (Mn), Pressure ratio (P/P0), Temperature ratio (T/T0), and density ratio (R/R0).
2) Fixed a few FREE-FORM screen nozzle geometry errors. Nozzle would occasionally fail to analyze some FREE-FORM nozzle geometries when the ratio of specific heats were less than 1.4.
3) Cleaned up a few presentation errors and enhanced results display.

Nozzle 3.4 Features
1) Added the ability to specify the upstream radius and downstream radius on either side of the throat for Conical and Bell nozzle shapes.

Nozzle 3.5 Features (12/14/03)
1) Added the ability to interrupt the nozzle analysis or to Stop the nozzle analysis.
2) Improved the initial slope of the parabolic portion of the Bell nozzle shape.

Nozzle 3.6.2 Features and Error Fix (06/02/04)
1) Added the ability to determine underexpanded and overexpanded external flow in the vicinity of the nozzle-lip region.
2) Fixed confusion concerning Nozzle exit (back) pressure and Atmospheric pressure (Patm). These two quantities should always be identical, but confusion about these entries caused thrust to be computed incorrectly. Now, the user enters only the Atmospheric (back) pressure. Previously, this entry did not accept pressures less than the optimal design condition (Pdesign) where no shocks are present in the nozzle and the flow exhausts directly into the atmosphere. However, to allow for underexpanded and overexpanded nozzles this constraint needed to be lifted. Now, a small non-zero value may be specified for the atmospheric pressure (Patm) corresponding to near-vacuum conditions.

Nozzle 3.6.3 Error Fix (02/22/05)
1) Under certain conditions when the maximum velocity exceeded Mach 7 in the diverging part of the nozzle, Nozzle would erroneously insert a shock wave. This condition has been fixed by increasing the upper limit of the maximum exit velocity (Me) to Mach 20 which increases the upper limit of the area ratio (Ae/At) to over 15,000.
2) Under certain conditions when the user decided to Cancel a nozzle geometry Import, Nozzle would repeatedly provide an error message. The user would have to perform a CTR-ALT-DEL to exit the program.

Nozzle 3.6.4 Features and Error Fix (07/25/05)
1) When displaying external flow contour plots for overexpanded nozzles the value of Mjet was inadvertently displaying the normal component of Mach number across the oblique shock emanating from the nozzle lip. Instead, the total Mach number in the jet region behind the oblique shock wave should have been displayed.
2) Added a hybrid rocket motor propellant having the following fuel and oxidizer to the list of combustion gases: 85% Nitrous Oxide, 15% HTPB.

Nozzle 3.6.5, 3.6.6 Feature (10/31/05)
1) Added a plume analysis for supersonic, two-dimensional and underexpanded nozzles.

Nozzle 3.6.7 Features and Error Fix (11/26/06)
1) Included Nozzle_Examples.zip in the Nozzle directory which includes 34 nozzle examples used for validation purposes.
2) The gas Nitrogen Dioxide in the Gases pull-down menu should be labeled Nitrous Oxide (N2O). (Fixed)
3) Program terminated if attempting to read a misspelled or non-existent file using the Open Project command. (Fixed)

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For more information about Nozzle 3.6 please contact AeroRocket at aerocfd@aerorocket.com.

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