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The Research Page


Research and experimentation

High speed video film of water rocket launches 

2L 8b Filmed at 500 images /sec Leeds University.

1L 8b


Experiments and Research

From a strictly professional view point, there has been a considerable amount written about how water rockets work but little if no research data to substantiate  them.

In general it would appear as though a large part of the available literature on the subject  rests scientifically unproven.

Test repeatability and data to support conclusions are conspicuous by there absence.


 There is a real need to produce repeatable  test data and eliminate some of the commonly accepted oversimplifications.

As an engineer very few of the literature/ computer models available appear to represent what we have recorded.

Consequently our research continues!


 In search of the parameters that influence acceleration

During the power water jet phase the accelerations seen for both  bluff Badoit and Coke bottle forms are none linear.

That beyond the Tsiolkovski point very high accelerations have been filmed. With  in the case of the Badoit acceleration is in excess of 4300 m/s2.

These levels of acceleration are beyond the maximum of those expected by existing models and theory. WHY?

This is our quest to discover the reason behind the performance of certain water rockets.

General Relativity and rockets


Gravity research


Rate of change in mass ratio comparison..

Based on Arianespace A5 Flight data A167pdf


Pressure and volume test curves 2005

Click on the tabs at the bottom of the Excel spreadsheet to see the curves


Click on this title for My analysis of the pressure curves at 8bar can be seen in the attached spreadsheet

A comparison of the different thrust force parameters used to model performance are compared Some of the standard approximations for thrust do not take into account  either nozzle coefficient cf  or variable fluid mass flow rates.

An introduction of specific impulse Ispec is encouraged to compare thrust efficiency using data taken from the pressure and volume curves.

Models for the fluid thrust produced when the reservoir is static and the modification required for both the nozzle design coefficient  Cf and that for the dynamic case using the Tsiol parameter are compared. 

The profile of the volume (m3) against time (s) curves indicate the rate of mass flow. So could be used to indicate the Design Point for the nozzle. Note however these are produced for a static nozzle/bottle


Comments by JG


Pressure was recorded at a 5000times /sec whilst fluid volume was recorded on film at 500 frames per second.

One of the principle objectives was to verify whether adiabatic or  an isothermal model was more appropriate for modelling expansion process. As can be seen from the results this depends on the type of duct profile or bottle neck.

Different water volumes where used in the two types of bottle analysed

1L Badoit 320mls water

2L Coke  640mls water

The bottles where held securely in a test frame.

Mass flow-rate of fluid from the Badoit bottle was higher than that produced from the Coke bottle. At 8bar there was a 27% mass flow-rate difference whilst at 4bar there was a 30% improvement in  mass flowrate. Since Fth= qv where q is the mass flow-rate (Kg/s )and v the velocity of the jet (m/s). Then this brings into question the validity of using  Fth= 2P*An  .in our mathematical model. This expression needs to be modified to be more representative of the flow rate curve for the fluid flow through a certain  duct/nozzle profile.



Mixed fluid phase in considered during the creation of the jet.

A simple conical duct model is adopted for the Badoit nozzle. Whilst a square step shoulder is used for the Coke duct.

Attempts at measuring the aerodynamic drag force for the different bottle types was limited by the maximum velocity of the wind tunnel used for the experiment.

The excellent pressure results are made available thanks to Dr.Johannes Knapp and Chris Holland.

Analysis of a water propelled rocket by Chris Holland 2005 pdf

Note: Temporary problem with pdf file Service will be resumed as quickly as possible

Examination of the volume curve produced from film images recorded at 500 images per second 0.002s. Indicates that there could possibly be a superimposed  frequency of 1.5 cycles. B 50c/s C 20c/s Expansion compression cycle.


Comparing bottle types

Simple comparison of bottle emptying times without pressure. Demonstration of the 'appel d'air'. Low pressure above liquid volume induces a need for air to enter the bottle from the bottle outlet /neck to allow the liquid to leave the bottle. The opposing directions of the fluid flows sets up a fluid chugging oscillation within the bottle as bubbles of air enter the neck to allow the water to leave. The weak structure of the bottle buckles in and out as the pressure within the bottle reduces and is restored by the arrival of a new bubble of air.

  • This is a static test where forces introduced by the rocket accelerating upwards are not experiened.

  • The effect of a pressurised air volume is also removed.

This phenomena is similar to that found in static pressurised bottles (see below) when the air water interface breaches the nozzle (neck of the bottle) or the flow becomes choked. When a similar chugging oscillatiing fliud flow can be seen reflecting back into the bottle.


Badoit1litre completely filled with no air above liquid at start.  9.72 sec video

2L Coke filled with same 1liter water volume. Filled from Badoit volume.6.104sec video

2L Coke filled with same 1liter water volume With swirl 5.32sec  video

A vortex at the center of the fluid system can help the air pass through the liquid to the top of the inverted  bottle and reduce chugging.



Leeds University Water Rocket Project 2003


Dr  Johannes Knapp



Department of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK

 Students Martin Mansfield


                Stuart Weston  University Technician.

The effect of bottle design on water rocket performance


Coke 2143ml


Bottle 'A' is a standard 2145ml Coke bottle. 8bar 500mls water, Launch images recorded to a height of 2.4m

Comparisons of the two different bottle types identified differences between the mathematical model predictions and the actual rocket performance.  

Simple thrust force                 T = 2PAn

That is derived from the basic Bernoulli fluid flow equation.

Fluid flow dynamics

The Bernoulli flow equation compares the energy distribution within a fluid flow system. By generating fluid pressure equivalents for dynamic pressure 1/2rv2 due to the fluid flow velocity, potential pressure rgh  due to the hydraulic head and static pressure P forcing the fluid flow.



^ Click on image to load film 4.3Mb

The  thermodynamic expansion assumes adiabatic expansion for an ideal gas are used for the model of the cycle.

                                                   P0.V0g = P.V.



Conventional height and velocity  against time curves generated from the  high speed  film images produced some unexpected acceleration curves.

A clearly  none linear acceleration characteristic was identified  during the jet pulse event.

These experiments where repeated for a range of launch pressures ranging from 5 to 8 bar with 500mls of water.

The importance of efficient energy conversion  during the first 0.1secs after launch is identified


Notes by John Gwynn:

Recognising the limited resource available on such a project means that some of the questions that are raised during the first phase of the project cannot be answered within the time available.

Below I have tried to identify some of my thoughts:


A full interface model of the projectile with its environment , the feedback forces involved  and how they effect the overall projectile system efficiency .has not been achieved.

How these variable parameters affect  both the external and internal  fluid flow systems.

How the acceleration loads created during the initial jet phase effect, the nozzle efficiency, flight stability dynamics and changes in  aerodynamic drag force. 

Could well contribute to the nonlinear character of the acceleration values recorded during each jet phase.

Whilst the existing mathematical model predicted the performance of bottle 'A' fairly accurately at lower pressures. It fails to predict the none linear  acceleration change in  performance at higher launch pressures. Or identify the differences in initial jet impulse conversion efficiency seen with the two different bottle types.

The corresponding full flight data for the rocket after the initial launch phase could be used to quantify the overall flight performance, correlated to a specific high speed launch film.

The effect of different aerodynamic drag coefficients are recognised but not quantified. Due to the lack of time and resources available.

A good effort and a good base for the next research team.. 

A suivre  ......

Compares the initial launch phase of  two different types of bottle filmed at 500 images per second to a height of 2.4m.

Here the image sequence is run for bottle A and is made up by stringing together some of the images.

The background scale graduations are at100mm intervals.


Reference Leeds 2004 high speed films database.

Coke 8 bar  dynamic launch film1 image stream,  film2 image stream and static frame test 5b,6b, 7b

Badoit 8 bar dynamic and static frame image stream  film3

Badoit and Coke 5bar, 6bar, 7bar. and 8 bar comparison.

Badoit 5,6,7 and 8 bar gif  fixed bottle with 500mls. Plus 8b flight.

Coke  5,6,7, and 8 bar gif for fixed bottle  with 500mls Plus 8b flight.

Extracts from these are featured below.

Rocket launch of bottle 'A' 2145ml

Images from film taken with bottles held in a static test frame.

Notes: by John Gwynn

April 2004.

Note: Film Image frame numbers descend with time

C7bar Frame181 Plug removed and jet stream released.

C7bar Frame177 Parallel jet stream. Liquid surface deformed by fluid displacement and bottle vibration.

C7bar Frame169 Parallel water jet stream with exterior water level just above first black ring.

C7bar Frame168 Coaxial air pocket enters nozzle. Whilst external liquid is left near the wall of the bottle forming a liquid funnel for the air to pass at its center


C7bar Frame165 Start of mixed fluid air -water  jet stream

C7bar Frame181Start of parallel water jet.

Tjet = 0 

0.002s / image frame

C7bar Frame168 Coaxial air pocket is about to break nozzle.

Tjet = 13 frames > 0.026s


C7bar Frame167 Start of mixed fluid jet 'White Out'.

Tjet = 14 frames >  0.028s

C7bar Frame 146 Final Purge of mixed fluid white jet stream.

Tjet = 35 frames  >  0.07s

C7bar Frame 145. Angle of jet at nozzle outlet reduced.

Tjet = 36 frames  >  0.072s

Coke 8 bar Static Test

C8b 1   163 Plug removed  Tjet = 0

C8b  160  Parallel jet stream Tjet = 0.06s

C8b 152 Tjet = 0.022s

C8b 2 151 Start of mixed fluid white plume Tjet = 0.024s

C8b 150 Tjet = 0.026s

C8b 3  104 End of white plume Tjet = 0.112s


Badoit 1000ml

Bottle 'B' is a standard    1000ml Badoit bottle. 8bar 500mls water  Launch images recorded to a height of 2.4m.

Click on image to load film 4.7Mb

Rocket launch of bottle 'B' 1000ml

Images extracted from static frame test films recorded Dec2003 by Martin Mansfield Student at Leeds  University

Notes: by John Gwynn

April 2004.

Static B7barFrame 314 Parabola of air pocket indent reaches second  black ring.

Static B7barFrame 312 Stretched parabola of air pocket approaches first ring.

Static B7barFrame 310 Air pocket forms point and heads for the nozzle outlet.

Static B7barFrame 308 Coaxial air pocket is about to breach the nozzle outlet.

B7bar  Static Frame318 Plug withdrawn and deflected jet spray.

Tjet = 0

0.002s / image frame


B7bar Static Frame317 Start of parallel water jet stream 

Tjet = 0.002

B7bar Static Frame308 coaxial air stream reaches nozzle.

 Tjet = 10 frames > 0.02s

B7bar Static Frame 307 Start of mixed fluid jet 'white out transition'.

Tjet = 11 frames > 0.022s

B7bar Static Frame258 End of mixed fluid jet 'white out'

Water spill with unknown mass flowrate.

Tjet = 60 frames > 0.120s


Jet Sequence at 8bar

B8b 1 Start

B8 bar Static Frame 264 Plug release with deflected spray cone

Tjet = 0s

B8 bar Static Frame 262 Parallel water jet

Tjet = 0.004s

B8 bar Static Frame 253 Compressed air pocket approaches the nozzle outlet ( Just to the right of the white ring)

Boundary layer of liquid left at the interior wall as air pocket passes at its center.Jet stream parallel  Tjet = 0.022s

B8b 2

B8 bar Static Frame 252  Mixed fluid jet stream 'White transition'.

Tjet = 0.024s

B8 bar Static Frame 251 There appears to be some white froth inside the neck of the nozzle. Possibly a reflected shock wave?

Increase in the cone angle of jet stream leaving the nozzle

Tjet =  0.026s

B8b 3 Tjet =  0.112s


B8b Static Frame208

B8b Static Frame206 Jet cone begins to break up at edges and oscillate from side to side.

B8b Static Frame202 Tjet =  0.124s

B8 bar Static Frame 200 Reduction in the white intensity  of jet stream.

Cone angle of jet stream increases to


Tjet = 0.128s

B8 bar Static Frame 195

Tjet = 0.138s

A comparison  between the static and dynamic launch tests carried out at a pressure of 8bar and 500mls water for the two projectiles/bottles indicates that: 

 1.The onset of the white mixed fluid plume occurs earlier in the static test than is seen in the dynamic launch.

StatTjet 2 = 0,75 * DynamicTjet2

 2. That the duration of the mixed fluid plume continues longer during the static test than is seen with the dynamic launch. 

     By a factor   of  2.8 > 3.

Although the exact termination is difficult to determine an event window of earliest and latest has been adopted.

Measurement of the mass flow, the white intensity  and opaque nature of plume combined with the stability/ fluctuation of the jet output cone angle can be used to indicate more precisely this end point.

StatTjet 3 = 2,8 * DynamicTjet3

3.There is possibly  a common relationship between  the static and dynamic trial results for the two different bottle types.

Experimental apparatus for the static frame tests needs to be improved to eliminate any bottle movement during the experiment. 

It can also be seen from the films that the longitudinal axis of the test bottles is not controlled vertically.

Sychronised pressure / time histories could be recorded and thrust force measured.

Abstract : 

    Observation based on results taken at LU 2003  

    For the dynamic launched rocket  case  :

   The mixed white fluid plume starts later and has a shorter duration than the static frame test fluid jet 

   from the same bottle. Using the same water volume and air charge pressure.



Note: Based on the above research data gives the following approximations:

The mixed white fluid plume for the dynamic case starts later (1.33* StatTjet 3) and has a shorter duration than the static frame test fluid jet from the same bottle.( 0.357* StatTjet 3


  • From this comparison we could assume that the mixed fluid flow phase occurs later in the dynamic flight nozzle.



Click for Excel Table Comparison of static and dynamic tests at 8bar.

Photographs and images contained in the report are displayed with the kind permission of Dr. J..Knapp

Department of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK.




Indentified physical dissimilarities :

  • Body length. L mm

Rocket A = 336 mm

Rocket B = 320 mm

  • Bottle cross sectional diameter.

             BA D1= 98mm

             BB D2= 80mm.

  • Bottle volume

Vol A 2145mls

Vol B 1000mls

Note : That both bottles are unstable in flight due to the centre of mass being lower than the aerodynamic center. So there is no stabilising moment to govern the true vertical trajectory desired by the experiment.

The effect of this can be seen in the launch photographs.

A  comparison  of the centers of mass (top) and of aerodynamic cd (Black point in middle is the center of inertia  of an exact profile card cutout). Shows  that the aerodynamic center is above the center of mass.

Indicating that it needs to be lowered by approximately90mm.

Consequently as a projectile  the bottle will be unstable and veer away from the intended vertical flight path..

In addition to which they are both effectively blunt nose projectiles. Due to no nose cones being added.

Bottle A has a hemispherical form and bottle B an indented flat profile.


Rocket launch of bottle 'A' 2145mls

Scaling image for the 2Lcoke  bottle recorded at 500 images per second. For launch at 8bar

Mixed fluid spray plume approximate length L plume = 32.3 cm

100mm =13mm Scale factor 100/13= 7.6923

For image size 42mm scaled equivalent 323mm.

Number of visible pulse ripples contained in the white plume wake 10.


Rocket launch of bottle 'B' 1000mls.

Notes by John Gwynn:


Objective: To understand why some of our Badoit rockets fly better than predicted by existing 

                 mathematical models.



Comparison with some of the standard formula helps to focus on relevant parameters.

Systematically regarding each term and asking the question are they constant during the water jet impulse.


Thrust Force Fjet = mf * V1 + ( Pe-Pa)*Se                 N

Where (mf ) is mass flow rate leaving the nozzle in Kgs/sec

V1 Is the jet stream velocity m/s

Pe Is the pressure of the fluid stream as it leaves the nozzle N/m2

Pa Is the atmospheric pressure outside the nozzle into which the jet stream passes. N/m2

Se Nozzle matching parameter.

Note : V2 Is the actual velocity of the rocket m/s

Due to the ‘Tsiolkovski’ factor V2 can be greater than V1. Looking at the formula for determining the jet thrust force that propels the rocket.

Initially we can say that Fjet is directly proportional to the value of V1


Looking at Graph 2 for bottle B 1 liter.


What I would expect to see is some form of ‘Tsiolkovski’ affect during the water jet phase. In a jet system which is approaching a high fluid flow efficiency system.

This would be influenced by being able to identify the Tsiolkovski break point.

And also the rate at which this point is reached.

So we would need to identify some appropriate parameter like mf / Ts linked to a performance parameter like specific impulse. Fth/ (g* mf).


Mf = Mean fluid mass flow Kg/sec

Ideally we should be able to identify the actual mass flow at any point in time. During the fluid jet phase.

Ts = Time to reach the Tsiolkovski point sec.

Subsequent to this point we should be able to define the final phase of the mass flowrate.

And whether there is a stable fliud flow system or not.


The shape of these graphs could be explained if we can identify the Tsiolkovski break point for each curve.

So breaking the curve into two distinct phases. Before and after.

This I define as the point on the curve where the mass ratio M1/ Mn = 2.71828.

So beyond this point the LogN ( M1/Mn) term will be greater than one for any remaining fluid mass and V2 becomes >V1

Whilst before this point the fluid mass left in the bottle would effectively have a decelerating effect.

If we accept that the curves are sufficiently accurate to indicate a trend. Then we can look at the relative positions on the curves at which Ts occurs and compare the values of mf / Ts for each pressure.

As would be expected as the launch pressure increases the Ts point moves towards the origin.

For values of mf / Ts < 140 the gradient of the tail of the graph is negative ( Falls away) for both bottle types.

This we see exhibited at lower launch pressures

For bottle A the values of mf / Ts at the threshold point would appear to be lower at all pressures relative to the equivalent values found for bottle B .

At a pressure of 5bar the value of mf / Ts at which the graphs fall away after the threshold point are :

                     Threshold A mf / Ts < 120

                     Threshold B mf / Ts < 140


Table1 Launch pressure Bar Bottle ‘A’ mf / Ts Bottle ‘B’ mf / Ts
5 105.8 137.65
6 145.14 172.4
7 163.3 193.42
8 192.31 202.4
Note : These figures have been taken or calculated from results presented in the report .

If we assume that the Tsiolkovski mass effect becomes significant above specific values of mf / Ts for a specific projectile form Cd , nozzle design effy and mass. Then we might well have something interesting.

The characteristic curve shape will be defined by the change in rate of mass flow of the fluid jet and the resulting external aerodynamic drag.

Possibly we have identified the conditions when the mass effect becomes predominant


The initial dip in the acceleration / time curve is due to the initial value of mass flow mf being reduced due to the close fitting internal launch tube effectively blocking the nozzle . Causing the gas to expand and initially lift the rocket above the tube. Subsequently releasing the water jet thrust force.


So what are the possible causes for this effect of varying acceleration during the jet phase:

The principle candidates are



1.Change in mf mass flowrate leaving the rocket.

Thrust Force Fjet = mf * V1 + ( Pe-Pa)*Se               N

Where (mf ) is mass flow rate leaving the nozzle in Kgs/sec

V1 Is the jet stream velocity m/s

Pe Is the pressure of the fluid stream as it leaves the nozzle N/m2

Pa Is the atmospheric pressure outside the nozzle into which the jet stream passes. N/m2

Se Nozzle matching parameter.


Note : V2 Is the actual velocity of the rocket m/s

Due to the ‘Tsiolkovski’ factor V2 can be greater than V1. Looking at the formula for determining the jet thrust force that propels the rocket.

Initially we can say that Fjet is directly proportional to the value of V1.


Linked to Tsiolkovski’s formula.


                                        V2= V1 * LN M1/M2



2. A modification of this formula.

Adapted to higher accelerations over a short period of time.



3.Nozzle design efficiency at varying fluid flow rates and accelerations.


Variations in the fluid properties at high energy transfer rates.

Marking the difference between quasi static flow and dynamic flow characteristics.

Similar to that found with the performance of materials subject to either a slow loaded deflection or a high energy impact/impulse.

For bottle B The rate of change in volume can be defined by simple cone analogy dV/dx= p( r2/ i2).x2

Where x is the vertical displacement of the fluid surface along the axis of the convergent nozzle/bottle neck.

The term dV/dx can also be expressed as dV/dx = p *(Tanf.)2.x2

Where 2f is the included angle of a cone generated by rotating a line of gradient  about the x axis. That represents the convergent nozzle.

However for a curved duct like bottle A its a little more difficult .


Operation at the convergent nozzle design point.

The physical form of the bottle neck can be taken to be the profile of a convergent nozzle.

Optimum mass flow of  fluid is achieved when the pressure in the fluid at the nozzle has expanded down to the design pressure. For there to be optimum  mass flow on leaving the nozzle then the design pressure needs to be matched to the atmosphere into which the flow is released. So for example the pressure outside the nozzle needs to be the same as the design pressure of the nozzle. This  pressure into which the nozzle discharges  is referred to as the Back Pressure .


Under expansion

If the back pressure Pb is below the design pressure Pc the nozzle is said to be 'under expanded'.

In the case of an under expanded convergent nozzle the fluid flow expands down to the design pressure in the nozzle. An then expands violently and irreversibly down to the lower discharge or ambient ( back pressure).This is usually achieved by a series of alternating compression and expansion shock waves until the ambient back pressure is reached

A good example of this can be seen in the 8b Coke bottle launch image where a sequence of 10 shock waves can be identified in the mixed fluid plume.


Over expansion.

This occurs when the design pressure Pc is lower than the back pressure Pb.

For a convergent nozzle having a back pressure higher than the critical pressure for the nozzle means that the flowrate through the nozzle is reduced.


Note: This is why a convergent divergent nozzle is useful in that it allows the fluid flow to re   

           compress in a more controlled  process to approach  and match the higher back pressure. into which it 



Curves a and b are both both cases of fluid flow experiencing overexpansion within the nozzle.

Curve a shows the case were the mass flow of fluid through the nozzle is very low  and subsequently the pressure  achieved at the nozzle throat is well above the critical design pressure.

Taking a specific case where the design pressure Pc  is below the back pressure Px. Then the flow in the nozzle initially expands down from the initial pressure P1 to point Q  and then undergoes a violent re-compression along b to return to the background pressure.

In the case of end curve c  the back pressure is lower than the design pressure PR. So the fluid  flow has to expand down to the background pressure. This is achieved by a series of violent alternating irreversible  expansion and compression shock waves until the background pressure is reached.


Why irreversible?

In this illustration from Anderson Fundemental Aerodynamics


Pe6 is the Exhausts pressure.

Pis the boundary or back pressure into which the exhaust jet is released. on leaving the nozzle


If our objective is to maximise the mass flow passing through a nozzle. These characteristics need to be identified for both the quasi static flow condition and for the nozzle and fluid system exposed to high accelerations. 

Simply this means that for each duct profile there is a design point (or critical pressure) at which optimum mass flow can be achieved. Either side of this point and flowrate will  be either reduced or worse still become violently unstable.



Note: A series of static nozzle tests have been filmed with the high speed camera. To permit analysis of the fluid nozzle flow for the different bottles  over the same range of pressures.

Indicating that there is both a  progressive penetration of the air volume along the central axis of the fluid stream  as the fluid level descends   and reflected fluid from the shoulder of the bottle near the end of the fluid purge cycle. Some of the selected images from these films will be put on the site. 

The bell shaped intrusion  at the air /water interface probably  indicates the distribution of  fluid shear flow from the center towards the pressure vessel wall .


Because of the difference in jet events times recorded for the static and dynamic trials at the same test pressure. It is currently not possible to confirm whether the air pocket breaches the water volume contained in the base of the rocket during dynamic trials in a similar way  to that recorded for the static tests.

It is thought that the high acceleration loads just after launch and during water jet spike phase could radically alter both the thickness of the liquid boundary layer  and the form of the profile retained at the interior walls of the duct.



Dec 2002 Initial test flight of the heavy lift Ariane 5 ECA 

Vulcan 2  nozzle failed  at altitude due to low atmospheric pressure outside the nozzle causing the bell shaped expansion nozzle to deform. The resulting reduced thrust and localised heat transfer problems triggered the abortion of the inaugural flight.

Inadequate development testing was identified as one of the principle causes for this flight failure.

Improved cooling and stiffening of the nozzle that was then proved during  numerous static engine test fiirngs resolved the problem.






4.System frequency analysis.


Analogy with a step pressure impulse to the system ht producing a variably damped response function of the type


                               ( 1/C )* e - Kt/C.

                           C y’ + Ky = x

Where C is proportional to the dynnamic and kinematic viscous properties of the fluid.

K is proportional to the gas spring pressure or system stiffness.

For rocket ‘B’ I have identified a good match for the damping coefficient using LN t/fth

Where the rate of change of volume is represented by the form of the nozzle shape

dV/dx = p*(r2/ i 2)*x2 is assumed to be directly related to fth the thrust force

and t is the elapsed jet time in seconds

This matches the final positive gradient of the curve.

For the 8bar curve for rocket B there is what looks like either an error or a super imposed damping function of the order of 25c/s .



5.Aerodynamic drag resistance parameters.


Due to the high acceleration levels to which the blunt bottles are exposed during launch.

Surface air shear flow, aerodynamic resistance  / drag force and dynamic flight behaviour could change significantly for the different launch pressures.

If the rocket is dynamically unstable after launch this can increase the proportion of the energy absorbed by air resistance.

Effectively retarding the rockets upward acceleration.

The effect of different nose cone shapes see Aerodynamics section.


Fd= 0,5(de)(Cd)A(V2+Vair) 2

If this is applied to A and B the Fd or air drag force ratio is

Csa A/ Csa B = 1.5


The comparative values of Cd are not defined.

That there is a sudden change could indicate that a certain external flow regime has changed.

Where ever there are shock waves seen in the exhaust jet plume then there is likely to be an equivalent pulsing surge on the air flow passing over the form of the projectile. This could make the external air flow unstable and generate boundary layer separation.( and increased drag)



6. Natural acoustic frequencies of the pressure vessel

     Several natural frequency modes exist :

  • Longitudinal or organ pipe chugging

  • Tranverse tangential or spinning.

  • Tangential 1st harmonic or Sloshing.

  • Tangential radial fundamental or radial pulsating.

  • Where ever there are shock waves leaving the nozzle then there is likely to be a reflected shock wave that  returns inside the pressure vessel/bottle.


7. Compressed air volume/capacity and its effect on jet thrust and the thermodynamic 


  • The mixed fluid phase.

  • Water jet phase.

  • Air jet burst.

  • Convergent duct design efficiency for the three phases.



8. A combination of the above.


9. Experimental error. In recording the height and velocity time histories.


    Test Sample size.

    Any small errors in height measurement will be amplified when the velocity and acceleration 

    figures are calculated from the same data.




10. Gravitational field effect


The lateral thinking zone...

This is a what if hypothesis

What if we imagine that a form of gravitational field is created at high accelerations per unit mass.

Then at certain conditions we could imagine that the type gravitational field created ( similar in nature to that of electromagnetic field )  around the accelerated mass could deform. If so then we could also imagine it would be possible to pass through an equivalent to a gravitational sound barrier as the created gravitational field deforms around the moving mass. And the acceleration /mass ratio reaches a critical value

This would apply to ultra lightweight particles or projectiles that have high acceleration to mass ratio and experience high orders of acceleration .

So we have an equivalent to the Mach effect where the boundary layer deforms as the projectile approaches the critical velocity of sound. But in our case it is gravitational field flow and the critical point is Tsilokovski point . 


f we imagined that a vertcally accelerating mass under extreme conditions, was capable of creating its own gravitational field then time would be altered relative to a static observer.

We would find a relationship after the Tsiolkovski point for the end of the impulse as derived below.

The derivation would also help explain the exponential change in acceleration seen in the 8bar curve for the Badoit after the Tsiol point 

We could also say that the effective mass is modified by the degree to which it is accelerated.


 Up to the limit when  LN M1/M= 1 

 the limits of integration are respected but after this point we need to find another end function.


Because the limits of   u  <<<  v  

and                    dM <<<  M

.are no longer true. So eliminating u and dM  by saying that they tend to 0   u = 0  and  dM = 0 is also no longer acceptable.


Where  is the velocity of the light projectile relative to a stationary observer

              is the velocity of the the exhaust jet stream relative to the same stationary observer.

         dM is the incremental loss of fuel or in our case water mass from the projectile over elemental time dt.

   and M the mass of the rocket or projectile


So u tends to v    


Effective Mass Meff =  m g/Acceleration in g


Or gravitation effect =  1/Acceleration in g


Another way of trying to understand what could be happening is to try and imagine that for  a fraction of a second the projectile experiences a form of micro gravity.

Standard Tsiolkovski Derivation.

Modified Tsiolkovski derivation for the end impulse based on experimental results

Excel spreadsheet with worked example of end function modification based on high speed film imagery. Impulse end functionExcel

For the parameter      with dimensions  

The factors which could alter  its  value are:

An increase in the value M1/M2 that we already use in the Tsiolkovski formula.

An increase in the ratio a2/a1

An increase in the relative time ratio  T1/T2

John Gwynn


Understanding the gravitational attraction force between two masses.

Do we know what creates the weak gravitational force used by Newton in his gravitational mechanics formula and  accepted up to present.?  In short. No we don't. 

We have all the formula but we don't know how the weak gravitational force is created.

Does gravity exist in wave form and travel at the speed of light ?

Understanding the weak gravitational force between two masses 1N = 105 dynes (cm/s2)





Einstein's General theory of relativity 1915

Following  Einstein's earlier theory on time and space entitled Special Relativity in1905. His theory on General Relativity in 1915 was a logical development concerning the equivalence of gravity and acceleration . That states that neither time or distance can be  given absolute values but vary according to the state or time frame from which they are observed.

In general relativity both time and space can be altered by the gravity and motion of a mass.

The equivalence of gravity and acceleration

Gravity pulling in one direction on any mass is balanced by the same mass sensing an acceleration in the opposite direction.

Here are two examples

  • Vertical acceleration in a lift produces an equivalent downwards acceleration on the people in the lift sensing they are increasing in mass. As they sense through there body that there mass is being pressed against the floor of the lift.

  • A driver accelerating away from a standing start in  a sports car will experience a sensation of his own mass being pressed hard against the back of his seat.

Changes in a local gravitational field will affect local time relative to a static observer.


Now taking the example of our water rocket accelerating vertically upwards, like the example in the lift , the water in the bottle will think it's mass has increased in direct proportion to the upwards acceleration of the rocket.

This will be equivalent to an increase in the local gravitational field affecting the rocket.  Locally the magnitude and direction of the acceleration affecting the mass being equivalent and opposite to the upwards acceleration. Pulling downwards  in direct relation with the magnitude at which the rocket is accelerated upwards.

Relativity What is it?



Inertial frames of reference. What are they?


Investigation into the Relative Time fields with respect to  Tsiolkovski  mass transfer


Just in case you think this is a bit far fetched

March 2006

 Gravitational field effect generated by moving mass discovered.

A 'Gravitational  field effect' of a rotational mass has been discovered, by researchers financed by the esa,  and could help explain the difference with that predicted by quatum physics.

Although only 1*10-8 times the value of g (9.81m/s2) this makes it 1*1024 larger than the value predicted by quatum theory.


Here an isolated super conductor loop is spun and accelerated at 1000 to 1500 Rads/s2  producing a gravitational field that  opposes the direction  rotation and effectively increases the mass of the super conducting coil.

There is a gravitational field created whilst accelerating and decelerating the  super conducting coil mass in a direction opposing that of the acceleration.

This occurs when a Niobium superconductor (with a radius of 72mm) is accelerated at 21g. 


pdf Experiment into the detection of Gravitomagnetic field March 2006

Video of interview with Dr Martin Tajmar STAIF 2006


Dr. Martin Tajmar is a research scientist and project manager in the Space Propulsion group at ARC Seibersdorf research, Austria's largest research institution,  STAIF 2006 interview to discuss a variety of topics, including gravito-magnetic fields in superconductors & applications for advanced propulsion...




Earlier paper by Tajmar and de Matos comparing Maxwell electromagnetic wave formula and there possible similarity with gravity waves.



2002 Paper by Y. Chiao Berkley University On superconductor gravitational effects.



Looking for proof of the weak force gravity waves in space


LISA esa satellite group to be launched in 2015 to detect gravitational waves in space


This research is not to be confused with that of Eugene Podkletnov

1992 Original controversial research by Russian Eugene Podkletnov  at Tampere University in Finland that claimed a possible 0.3 to 2% reduction in mass seen in  materials placed above a static or rotating superconductor. Which he claims produce some type of gravitational shielding effect above a TypeII YBCO  superconductor spun at 5000rpm.

Although possesing a docterate in materials science specialising in the production of superconductors. 

The credibility of his results produced at Tampere University as a superconductor materials engineer are challenged because of him not being a  pure research physicist and the inability of others to reproduce his experiments. Both factors have combined to create his lack of credibility within the scientific community.

As a direct consequence of an article published in the 'Daily Telegraph' in 1992 prior to the actual publication of his results in a scientific journal he lost his post at his Finish University.

Here is a copy of  his original paper that was withdrawn after the publication of the article in 'The Daily Telegraph.'



Timeline in gravity related research





The order of acceleration seen with a Badoit nozzle is 450 to 500g.

Accepting that gravity is a very weak force and that the findings by Tajmar are 1*10-8g when the superconductor  is accelerated at 21g.

These changes are seen as the super conductor ring is accelerated and deccelerated.


Could a changing gravitomagnetic field affect :

  •  The Molecular linking within The fluid dynamics of boundary layers. Affecting the properties of external and Internal fluid boundary layers.

  •  The effective local  mass  of a light  projectile. Changing the Mass  time relationship.

  •  Help explain the Gallileo gravity equivalence of two different, free falling masses contacting the ground at the same time. When dropped from the tower of Pisa.

  • Gravitational field reversal between accelerating and free fall states.


With this in mind it is possible that at a certain level of projectile acceleration the local mass of the rocket plus water  increases  due to a  locally increased gravitational field from which it is then released as retardation starts and the acceleration reduces at the end of the water spike  returning back to gearth.

A sort of local gravitational sling shot or cannon. As the mass first increases as the gravitational field increases and then reduces as the local gravitational field reverses due to deceleration and then returns to normal gearth .  A sort of local gravitational elastic membrane which at first holds the projectile back and then catapults it upward.

Try and imagine two elongated cones aligned on the same axis with there tips facing each other. The lower cone represents the local gravitational field increase (becoming more concentrated) whilst the upper represents the decrease. in the local gravitational field. If this where true then local time would also alter.

Is this the cause of the dip ,  point of inflexion ( at the Tsiol Point)  and then an exponential increase in gradient in the acceleration curve.




 In search of the parameters that influence acceleration

Indentification of several parameters  following analysis of high speed launch film 2004.

Acceleration related derivatives.

The event of the jet impulse is defined by a time window which has a start and a finish. During this period a fluid is discharged vertically through a reduction nozzle to create a reactive jet.

The forcing system is the energy introduced by a compressed gas above the working fluid. And the potential energy of the fluid level in the reservoir. Once the projectile is airborne then a six degree of freedom flight model for the rocket determines additional inputs to the inertial mass and the fluid flow dynamics.

The density of the fluid being discharged during the impulse could change due to the presence of a mixed working fluid.

Based on high speed film  of  basic bluff projectile launches recorded at 500 images /sec

We can say that for a specific type of elongated conical nozzle there is an overall dynamic model which involves both internal and external fluid flow regimes.

That the acceleration during the power water jet phase is not linear with time.

That the acceleration of the rocket during the initial impulse phase is related to

If  y  represents acceleration then

                                      y = e                                      1

y = e


is the elapsed time during the jet impulse.

is the rate of change of volume related to the displacement along an axis of the   volume of rotation created by profile curve of the nozzle entry.

For the Badoit duct I have approximated this to  

dV/dx = p*(r2/ i 2)*x2   


by using a pure conical volume of the same angle.


      Where r is the cone radius at the water level

       I is the height of the water cone 

      r/I is the gradient of the surface of the cone about the 'x' axis

       x the water level displacement

Which produces the same family of end curve beyond the Tsiolkovski point as seen in the acceleration time curve recorded at 8bar.


An alternative form would be to use the change in mass :

Related by the density of the fluid in the jet      r = M/V

V  =  M/r

So we could express the rate of change of volume in terms of mass flow. By simply substituting the rate of mass change divided by the density of the working fluid.

= =

y e                                                                                2

Here M is the rate of change of mass/density ratio

John Gwynn All Rights Reserved April 2004 The Water Rocket Explorer

Oroginal Excel spreadsheet whilst looking for parameters that could match the acceleration curve after TSIOL POINT


Next acceleration related parameter to appear that produces a similar curve to the acceleration time curve produced is

y e                                                                                                    3

is the elapsed time during the jet impulse.

total time of the jet impulse.






OK this is where things start to get interesting

All three can be related to the Specific impulse parameter, since they can be directly linked to the rate of change in mass flow and impulse time duration.it

Specific Impulse Isp= F/ g.m* Sec   Dimensional analysis  Isp  =  F /  ( g*mdt)    


Comparison of the rates of change in mass ratio with Ariane5

Based on Arianespace A5 Flight data A167pdf


Specific Impulse

Specific impulse is the parameter used to measure the efficiency of a rocket jet engine, and is basically the ratio of the useful jet impulse Ft (Ns) produced divided by  the fuel  flow-rate (mass per second in Kg/sec) required to produce that impulse .

Specific impulse Isp is the index of performance for rocket engines and is defined by the following ratio :



Where the numerator is the total impulse obtained over the operating time of the rocket engine. Usually referred to as the ‘Burn time’. Defined by the time period from t0 to t1.seconds.

Accompanied by a denominator which represents the total fuel mass flow necessary to produce the impulse during the same time period.

‘g’ is the acceleration due to gravity gearth at sea level m/s2

The above expression can be simplified to the ratio given below if the flow rate and thrust are constant over the burn time.

For liquid fueled rocket engines this is usually the case,  so they can be represented  by the ratio of the thrust force produced  at the exit of the nozzle tearth  divided by the product of  gearth  at sea level  multiplied by the mass flow rate of fuel used to produce the thrust. 

Specfic  impulse       Isp =      sec


Where F is the Jet Thrust force at the outlet of the nozzle F = q * v2         N   Newtons   

‘q’             is the volumetric flowrate * density r  to give mass flowrate of fluid leaving the nozzle Kgs/sec

v2               is the exhaust jet velocity at the outlet of the nozzle m/sec

Mdt          is the mass flowrate of fuel

So in the first instance it is possible to use this simplified ratio to model the water rocket. 

To create an actual impulse and mass flow time history in the above integral ratio.

A more  detailed model can be obtained by using the flight film data determine the mass flow with time, height, velocity and acceleration. 

Sensitive accelerometers can be used to record flight data but the problem is the response time of the accelerometer immediately after launch.


Dimensional analysis of Is   Assume

Isp  = Sec


Isp = Sec


Tsiolkovski Impulse Function dimensional analysis


© All rights reserved by John Gwynn

The efficiency of the external and internal fluid dynamics of the projectile will have a significant affect the exhaust stream velocity and subsequently Is.

Specific Impulse is related to  relative time t local  with t earth and is also dependent on the local gravitational field glocal  relative to external  gearth 

Whilst the normal levels of acceleration are experienced then standard Tsiolkovski and specific impulse formula apply.

But after a certain threshold of gravitational acceleration levels then both the Tsiolkovski formula and the specific impulse factor need to be modified.





Possible parameters that could improve   Isp 

  1. An increase in thrust force for the same fuel mass flow rate.

          Induced by an improved fluid flow regime either inside or outside the projectile.

          Also combination of both


Note: The efficiency of the external and internal fluid dynamics of the projectile will have a significant affect  the exhaust stream velocity and subsequently the value of  Is.


So boundary layer disruption needs to be minimised to reduce drag. Flight dynamics need to be smooth and true to the intended flight trajectory to minimise this boundary seperation causing increased turbulence.

You can imagine the chaos set up in the boundary layer if the projectile has a twisting zig-zag flight pattern. Not to mention the disturbance to the internal fluid flow.

Remember that the boundary layer is like an invisible skin of molecules that covers the surface of the projectile. Where the boundary layer is stable and stays attached to the projectile then the flow adjacent will be lamina. We also need to minimise 'flutter' and vibration created by turbulent fluid flow

 If we imagine that the external flow regime is stable then for the flow inside we need to optimise the massflow though the nozzle to produce the maximum impulse per kilogram of fuel.

In the case of a water rocket this would be equivalent to the mass flowrate  of  water in Kg/sec.


     1.  A change in the fluid flow parameters that would improve the fluid flow  

        efficiency per Joule of energy input to  the system. 

       That means that physical characteristics like the critical Reynolds Number Recritical , q the   

         coefficient of heat transfer, Pr, Kn etc. changes as the state of the boundary layer changes. 


This could involve changes in the boundary layer  fluid properties.

Changes to the physical boundary layer properties that affect the coefficient of  heat transfer, viscosity, Re, Pr, Ma, Kn, Altering the nature and make up of the fluid boundary film.


The Knudsen Number Kn relates the number of  impacts encountered by a molecule as it travels along the length of the projectile. And is used to determine the relative fluid viscosity and the speed at which the fluid is capable of communicating changes in temperature and pressure.


In a compressible fluid the ratio of Mach No  Ma to Knudsen Number Kn  is important.

Kn is the ratio  of the average free distance travelled  by a molecule of fluid before it impacts with another and the characteristic length of the fluid flow L. Through which the fluid molecules exhibit a rapid relaxation by exchanging energy/ communicating at the speed of sound.


The viscosity of a perfect gas is proportional to the ratio Kn/Ma.


Compressibility of a fluid is directly related to the Mach No Ma.


For a compressible working fluid or gas. Where the value of Ma>1  is large the pressure waves propagate slowly but with a large amplitude.

Conversely for an incompressible fluid or gas. Where Ma < 0.25.Equivalent at sea level to 320/4= 80m/s. The velocity of the thermal heat transfer through the working fluid is faster than the fluid flow around the object. So pressure changes in the fluid flow external to the fluid boundary layer have little effect. A change in state of the fluid flow around a projectile is communicated faster by the  intercommunications between molecules  than can be transmitted  by external fluid flow velocity. 


For the application of perfect gas laws to be valid the value of Kn < 1/100.


   2.     A change in the density of the working fluid.

           The external and internal fluid  density


   3.   Modification to the relative time between  tearth  and  t local.

          Shift in relative time between tearth   for the thrust impulse Ft   mass flowrate Mdt t local..

              This will depend on the degree of acceleration seen by the mass and how it influences the local gravitational field

Investigation into the Relative Time fields with respect to  Tsiolkovski  mass transfer  


     4.   A modification to the local gravitational field to which the projectile is influenced.

        glocal   >>>  gearth

        If the gravitational field  around the projectile was altered then items1, 2 and 3 would be all  



Applied to compressed air over water 'Badoit' rockets  at 8bar

This would not apply to all water rockets but to those with efficient high Isp conformity

        a1 = 135.9g

         a2 = 448.22g

Change in acceleration after tTsiol              Da2 = a2 - a1     Da2 =  310g   ( 3063m/s2 )

Remaining Impulse time                            Dit2 = t2 - t1        Dit2 = 0.0173 sec

  Da2 / Dit2 = 17920g/sec

Ariane 4      2g/sec

Difference   8959 * A4   or  9.103

These light projectiles experience , for a fraction of a second,  levels of acceleration greatly superior to  those of normal space launchers like A4 or Ariane 5.

What happens when the projectile experiences a form of micro-gravity for a fraction of a second.

Excel spreadsheet

This raises the following possibility.

If at 135 glocal   the gravitational field change caused a change in   relative  time t local.

Then we would expect to see a change in the value of  Isp and an exponential increase in acceleration after the Tsiolkovski point TTsiol as seen in the acceleration curve produced from high speed film seen below.

......and the modified end function identified below could be a solution

         Modified end function            emg1/mg2 

         Original Tsiolkovski  formula    eM1/ M2 


Dimensional Analysis

it2 . e = a2. g


Assumptions :    

Were there a gravitational or relative time difference then this interface would occur at the exterior of the nozzle as the projectile fluid system meets the external world environmental  parameters 

PEarth, VEarth , gEarth, tEarth etc

it2 Term is the impulse of the remaining jet pulse.

a2 Is the change in acceleration between tTsiol or (1 ) and t2.

e Is the rate of change of mass ratio relative to the local time.

If the local time changes relative to then this will change the local gravitational constant glocal.

Giving a dimensional parameter of the form

it2 . e = a2. g


For the rate of change of mass the dimensions cancel  = 1


e Represents the rate of change of acceleration equivalent to the gravitational change of a rapidly accelerated mass.


As a comparison the rate of acceleration of a Badoit water rocket  (although only over a fraction of a second) is of the order of more than 400 times that for the Ariane 4 Refer to Excel spreadsheet

© All rights reserved by John Gwynn April 2004  The Water Rocket Explorer



Bottle ‘A’ mf / Ts

Bottle ‘B’ mf / Ts with nose













Phase 1


Jet Thrust Force Fth

305 N

424 N

Nett Specific Impulse Isp. With V1

3.571 secs

3.728 secs


Phase 2 with Tsiolkovski effect

Assuming the Tsiolkovski hypothesis we would expect to see


Jet Thrust Force2 Fth

705 N

1075 N

V2 Isp2 Tsiol factor

8.62 secs

9.654 secs




Note :Our rockets seem to kick in to a higher gear at about 6.5>7 bar. The mf / Ts equivalence would appear to be at the upper limit of the trial launches carried out 


Rocket launch of bottle 'B' 1000ml

Velocity Calculated from area under

Acceleration time curve at 8 bar


1st Stage WR ‘B’ 1000ml  Blunt Projectile

V 92.1m/s

Maximum acceleration of projectile B

Limited by height of image recorded h=2.4m

4397 m/s2

Tsiol Ts Point for B  8bar 0.0467 s
Tsiol  B mf / Ts 202.43


Tsiol  B  mf / (Ts*Ne) 4265

1st Stage WR ‘A’ 2145ml  Hemi-spherical nose

V 88.9 m/s

Maximum acceleration of projectile A

Limited by height of image recorded h=2.4m

3161.2 m/s2

Tsiol Ts Point for  A 8bar 0.047 s
Tsiol  A  mf / Ts 192.31


John Gwynn © 7.04.2004


Note: Because of the different volumes of bottles used the forcing compressed  air spring energy is not the same for both bottles. So the reservoir of J/s available to do work on the fluid will not be the same.

Refer to launch images and video with 10m frame height.


Nozzle acoustic.

The following images where recorded at 500frames per second. 0.002secs per image for the 2.145L bottle.2003



From analysis of the images above a combine dynamic rocket pulse frequency of approximately 1500c/s. can be seen. That is produced by the combination of the external rocket flight aerodynamic flow and that of the fluid flow through the nozzle.

Interestingly the spray plume seems to be symmetrical in form with that of the bottle.

The plume signature produced could be caused by:

  • Alternative compression expansion waves as the pressure leaving the nozzle collapses back to atmospheric.

  • Axial Chugging frequency in the bottle as the fluid is driven out.

  • External pressure fluctuations due to the separation of the external aerodynamic flow regime of the rocket.

  • A combination of the above.

These multiple pulses are not seen in the jet wake signature of the Badoit bottle.

The force input to the system can be represented by imagining a forcing step pressure impulse to the system ht producing a variably damped response function of the type:

 ( 1/C )* e - Kt/C.

.The damping function provided by the gas charge density left in the bottle will decrease with time. Whilst the generation of turbulent fluid flow and mixed fluid flow will encourage an increasing or cyclic damping effect.

High energy liquid fracture

Materials and fluids behave differently under high energy impact conditions.

So what could  happen to a high energy water jet?


Featured above is a photograph featured in Pour La Science ( Physics review letters 17 Dec2004)

The image on the left show  a low energy water jet impacting a bath of water. The water jet impinging  and deforming the surface

Whilst the image on the right shows a high energy water jet impacting the surface and entraining a very thin layer of air like a skin  trapped at the interface between the jet cord and the static water of the bath . Forming a sought of  trumpet that becomes unstable at its extremity where it begins to break up or fracture.

Why does it decay or dissipate its energy in this fashion.?



Fluid flow dynamics

Bernoulli flow equation compares the energy distribution within a fluid flow system. By generating fluid pressure equivalents for dynamic pressure 1/2rv2 due to the fluid flow velocity, potential pressure rgh, due to the hydraulic head and static pressure P forcing the fluid flow. .r



Understanding The Gravitational attraction force between two masses.

Note: Keep in mind that the Earth has a mean velocity of 29.78 Km/sec around our sun and depending on what you take as a reference point in the universe. Our solar system will have a relative motion within our galaxy ( say about its centre) and our galaxy will have a relative motion to some reference point in the universe.



The Scientist  Cold Play


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 ©John Gwynn and sons2003 

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