Drag Measurement in Unsteady Compressible Flow Part 1: An Unsteady now Facility and Stress Wave Drag Balance

 

 

M.S. Bredin^{I, II}; B.W. Skews^I

^IProfessor, FSAIMechE, Director Flow Research Unit, University of the Witwatersrand. E-mail: beric.skews@wits.ac.za
^IINow at Pebble Bed Modular Reactor Pty. Ltd, South Africa

 

 

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ABSTRACT

The determination of drag forces acting on a body positioned in an unsteady compressible flow is examined. There is very limited data available and the situation is particularly difficult for very fast flow transients such as is encountered by the passage of a shock wave over the body. The development of a rig for producing a range of fast transient flows is described as well as the development of a stress wave drag balance having a response time approximately an order of magnitude better than other balances of this type and able to resolve drag variations in the sub-millisecond range. Results obtained from measurements on spheres and cones are treated in a companion paper.

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Nomenclature

Roman

a speed of sound

C_d drag coefficient

P pressure

R gas constant

T absolute temperature

Greek

γ specific heat ratio

ν velocity

ρ density

 

1. Introduction

A substantial amount of data exists on the drag of bodies. These results are generally presented as a function of Reynolds number, and for a lesser number of cases as a function of Mach number. Good sources of data for bodies considered in this work are those of Hoerner¹, Clift², and Morrison³.

However, nearly all the data referred to are for steady flow situations, these generally being obtained from wind tunnel tests. The first work on unsteady sphere drag was done by Stokes⁴ who derived a theoretical prediction for the force on a sphere oscillating in a fluid for the case of creeping flow. He showed that the unsteady drag was the sum of the steady drag and what was called an added mass force, where this latter term is the additional force required to accelerate the fluid. This was followed by the work of Boussinesq⁵ and Basset⁶. They independently added a further term called the history integral force, sometimes referred to as the Basset history force. This accounts for the unsteady viscous diffusion of vorticity around the sphere. The bulk of subsequent work has been in the determination of empirical correlations for flows at these low Reynolds numbers.

Generally, experimental work on unsteady drag falls into two categories and is done on spheres. The first subjects a sphere to forced simple harmonic motion at Reynolds numbers up to 50 000. The period of oscillation is the order of seconds. Compressibility effects are negligible. The second involves placing a sphere in a shock tube and passing a shock wave over it. The sphere is unconstrained and is accelerated along the tube. Its trajectory is recorded and differentiated twice to yield acceleration and hence the drag. The post-shock gas velocity in the tube is constant and the sphere accelerates in the direction of gas flow. Therefore, once the shock has passed, the sphere is decelerating relative to the gas flow, which limits the tests that can be done. Measurable displacement of the sphere only takes place once the shock has passed so data can not be obtained during the shock diffraction phase when the shock is traversing the body. Maximum Reynolds numbers are of the order of 120000.

The current research involves constraining a body in a special shock tube and passing a compression wave, a shock wave, or other variable flow field over it. The drag is to be measured directly through the use of a stress wave drag balance. The pressure wave can be used to generate flow up to a Reynolds number of 600 000 in less than 10 ms and the instrumentation should be fast enough to follow the force as the wave passes over the sphere.

No previous work, except one, be it experimental or theoretical, covers the flow conditions of the current work. The one exception is a shock wave study⁷ on sphere drag using an onboard accelerometer to determine the force and is a useful comparison for the current studies using the stress wave drag balance. It restricts itself to the early shock diffraction phase and thus is different from most other previous work which tracks the motion of an unconstrained sphere well after the shock has passed. Nevertheless it is useful to briefly review the results of shock tube tests using an unconstrained sphere. Tyler and Salt⁸ measured trajectories of spheres accelerated by shock waves with Reynolds numbers from 1 x 10⁴ to 5 x 10⁵. A sphere was held by an electromagnet at the top wall of a shock tube and released just before shock arrival. The drag exhibited periodic discontinuities. The initial drag coefficient was 10 to 12% higher than the steady state value and then as the body decelerated rose to between 20 to 30 % higher. The C_d then dropped abruptly with this process repeating with a period of about 2 ms. It is presumed that this effect is due to sequential vortex shedding although this has not been proven.

Igra and Takayama⁹ also tracked sphere trajectories for spheres originally lying on the shock tube floor. The proximity of the floor may have an effect and the post-shock flow properties were not constant. Drag coefficients were 50 % higher than steady state values for 1 x 10⁴ < Re < 1 x 10⁵, and 100 % higher for lower Reynolds numbers.

Rodriguez et al.¹⁰ used a vertical shock tube to study the unrestrained motion of spheres originally in free fall impacted by an upward moving shock wave. Drag coefficients were 25 % higher than steady state values for 5 000 < Re < 1.2 x 10⁵.

Britan et al.¹¹ used a horizontal tube with the sphere initially resting on a light wire support. They took into account the ratio of the shock tube cross-sectional area to the frontal area of the sphere and found marginal influence for their tests with ratios between 3 and 8.5. They examined the wave pattern of the shock diffracting around the sphere. The wave is well downstream by the time the body displacement can be measured so drag for the early stage of the motion is thus not determinable. However, by examining the wave pattern it was suggested that the maximum drag would occur just before the shock passed the equator causing an initial acceleration and that a high pressure could be developed behind the sphere as the waves diffracted to the rear and that this could result in a short period of negative drag. They concluded that the sphere decelerates for some time after the initial acceleration and then accelerates again.

There have been a number of investigations to determine the magnitude of the history force. The general view (see Liang and Michaelides¹², for example) indicates that the history force is negligible when the ratio of particle to fluid densities is greater than 1 000. Vojir and Michaelides¹³ conducted a theoretical study in which a sphere was subjected to a step increase in velocity. The sphere's velocity without the history force lagged significantly behind that with it. This indicates that the history force causes a significant increase in drag when the relative velocity increases rapidly.

In the current study the main objective is to measure the drag forces on spheres and cones subjected to very rapid transients such as in shock and blast wave loading. The loads experienced during the shock diffraction phase are of particular interest. As indicated above using unconstrained spheres with the drag inferred from the body acceleration is limited since the body inertia limits its early displacement and the response of the measurement system is severely limited since it is dependant on body displacement. To overcome this problem the body needs to be held on a sting which can be used to measure the force directly. A suitable facility also needs to be developed to produce reproducible transient compressible flows.

 

2. The Flow Facility

A simple shock tube consists of two ducts initially at different pressures and separated by a frangible diaphragm as shown in the time-distance plot of figure 1.

 

 

Bursting of the diaphragm causes a shock wave to move down the low pressure section and an expansion wave up the other. This will form the basis of the current facility. However, in order to develop a compression wave rather than a shock, rapid removal of a diaphragm is unsuitable and needs to be replaced by a valve whose opening time can be controlled so as to generate waves of different rise times and thus different flow acceleration around the body.

The most useful tool for representing such flows is known as the x -t, or wave diagram. The mathematical technique used to construct them is known as the method of characteristics (Thompson¹⁴). The underlying principle is that any flow perturbation will travel at the speed of sound in the flow. When the frame of reference is fixed relative to the wall of the duct then the perturbation will travel at the local speed of sound plus the local gas velocity. The flow resulting from an opening valve can be represented as a piston starting from rest and accelerating down a tube driving gas ahead of it as shown in figure 2.

 

 

As the piston starts moving a weak pressure wave is transmitted down the duct as the first characteristic shown. As the piston accelerates further more weak waves are transmitted. Each of these weak compression waves pressurises and heats the gas slightly. Thus the sound speed behind each wave is increased. Additionally each wave is moving into gas which is moving and has been pressurised and heated by the preceding wave and thus subsequent waves will catch up with the earlier ones. As the waves combine they result in a shock wave of increasing strength. If the piston reaches a steady velocity, so will the shock wave. In the valve case the piston path simply represents the contact surface between the gases that were originally on either side of the valve.

In order to record as much drag data as possible it is desirable to have as long a testing time as possible. This means maximising the tube length as well as reaching a suitable compromise between driver length, driven length, and test section position. By extending the tube across a number of rooms a length of 50 m was available and the tube was made to extend over this entire length. A number of wave diagrams were generated for different driver pressures using shock tube wave diagram software¹⁵. The test section position was selected so that the available flow testing time was comparable to the force balance testing time. The final lengths selected were a 15 m long driver and a 35 m long driven section. The tube is made from PVC piping of 135 mm internal diameter. The final overall layout is shown in figure 3. Some of the features identified in this figure will be dealt with later.

 

 

Figure 4 is a schematic of the initial valve arrangement. The driver and driven sections of the shock tube are separated by a piston which is held in place by compressed air, which in turn is confined by a frangible diaphragm. To open the valve the diaphragm is burst thereby venting the air. A variable orifice is incorporated to modify the venting time and thus the pressure difference across the piston and thereby the valve opening time. A T configuration between the driver and driven section is used to ensure easy diaphragm replacement and maintenance access.

 

 

Many modifications of the valve were made before satisfactory operation could be obtained. The prime difficulty is that very high speed operation is needed, of the order of milliseconds, with the necessity of light weight components. In the early stage of development component failures were common, the primary issue being to stop the moving piston after valve opening. Earlier tests with PVC and Delrin components were found to be unsatisfactory and a thin-walled steel piston had to be used. A number of piston stoppers were destroyed before satisfactory performance could be obtained. The final version consisted of a compound element made from 5 mm nitrile sheet, a 3 mm steel plate to distribute the load and a 20 mm foam rubber layer. Even then the steel plate needed to be replaced after every 20 or so tests. The valve opening times could be adjusted to give compression wave rise times between 5 and 20 ms. These waves would then steepen up as they propagated down the tube.

In order to overcome issues relating to changing of diaphragms and irregular bursting, the initial venting of the chamber was accomplished by adding a novel design incorporating two additional pistons. The assembly shown in figure 5 is attached to the end flange which originally clamped the diaphragm.

 

 

 

 

 

 

Cavity 1 is that which controls the main piston motion which vents the driver as shown in figure 4. The second piston is initially held forward by pressurising cavity 2. This cavity is decompressed, relatively slowly, through port 1, through a small ball valve. This reduces the pressure holding the secondary piston so it starts to move backward. Once the sealing o-rings disengage, the air holding the secondary piston vents rapidly and the piston moves back very quickly, dropping the pressure in cavity 1 and allowing the tube to fire. Thereafter the tertiary piston is forced forward to reposition the secondary piston by supplying air to port 3. The tertiary piston is then withdrawn by venting port 3 and pressurising ports 1 and 2 in readiness for the next firing. This design resulted in quiet operation. In order to control the outflow so as to adjust the wave rise time the original internal variable orifice was replaced by a movable steel ring which could be adjusted over the outlet ports. The final assembly showing the outlet ports and adjacent adjustment ring is given in figure 9 with the tertiary piston and supply lines on the right.

 

 

 

 

This design allowed a turnaround time between tests of less than 5 seconds and since the control is through small valves the whole operation could very easily be automated.

Further improvements to the main piston stopper unit have subsequently been made by increasing the distance over which it is brought to rest, through incorporating a 150 mm long rubber stopper. This has also allowed the re-design of a lighter main piston made of PVC and aluminium. At this stage, satisfactory operation was achieved to enable a large number of tests to be undertaken.

 

3. Flow Instrumentation

Two methods are considered. The first involves the simultaneous measurement of the variation of temperature, static pressure and stagnation pressure during the test, and the second involves the measurement of the initial temperature and pressure before wave arrival and then recording the static pressure during the test. The flow properties can then be calculated using the isentropic compression assumption in the case of the compression wave or the Rankine-Hugoniot relations in the case of a shock. The problem with the second approach is that it assumes all waves are travelling in the direction of the flow. A reflected wave from the end of the shock tube would thus be interpreted as an additional acceleration whereas the flow is actually decelerated. Measurement using the first method was thus initially selected. For static pressure measurement a flush mounted quartz-based transducer with a rise time of 1 μs was mounted in the tube wall. For stagnation pressure a similar transducer was mounted with its sensing face pointing upstream. A short pipe was attached to the active face to ensure full stagnation and was positioned away from the pipe wall outside expected boundary layer growth. A thermocouple with an advertised 3 us response time was mounted with its active face flush with the inside wall of the tube. The three transducers are positioned at the same axial position. They should ideally be placed to measure free-stream conditions at the position of the body to be tested but the body itself affects the flow at that position. Thus two test sections were made; one at the position of the body to be tested and one 626 mm upstream where the body influence would be minimal as shown in figure 3. At that stage the shock reflected back upstream off the body has attenuated to such an extent that it cannot be discerned on a pressure trace. (Some tests were also run with the two test sections adjacent to each other so the distance between the transducers was then 126 mm). The conditions at the downstream test section were calculated from those at the upstream section using the method of characteristics. This method was then verified by running tests with no body in the downstream test section.

 

 

The static pressure measurement proved acceptable although the transit time of the shock across the sensing face of about 10 μs should not be confused with the rise time of a finite compression wave. The stagnation pressure measurement proved adequate for compression waves but with shock waves the wave reflection and resonance in the short tube caused problems. The thermocouple proved to be totally inadequate with a response very much longer than advertised. It was replaced with strain gauge based temperature sensors. To compensate for strain sensitivity two gauges were installed to experience the same strain but with one insulated from the temperature variations in the flow. This arrangement gave an excellent signal to noise ratio but the frequency response, although better than the thermocouple, was still too slow, making deconvolution of the signal a possibility. Most tests reported later used the second method of measurement with limited testing time so that neither the contact surface (the front of the gas initially in the driver which had passed through the valve, see figure 1) nor the reflected shock, arrived at the test section to bias the results.

The temperature, T₀, and pressure, P₀, at the upstream test section were measured before wave arrival in the initially stagnant air, and the sound speed, a₀²= γRT₀, calculated. The static pressure variation with time was then used with the isentropic relationship to determine the variation in sound speed.

The velocity variation may then be determined from (see Thompson¹⁴).

The values of v and a were used to construct a wave diagram as far as the downstream test section. Initially shock waves were treated separately with its speed determined from the pressure rise across it. However for the weak shocks developed in this work there is very little difference between the Rankine-Hugoniot and the isentropic relations, so the latter may be used with very little error. Once the downstream pressure trace is established the local variation in temperature and density can be determined.

The viscosity may then be established from the Sutherland Law¹⁶.

Once the drag force, D, is measured there is then sufficient information to determine both the instantaneous variation of Reynolds number and drag coefficient

where φ is the body diameter and A the frontal area.

 

4. The Stress Wave Drag Balance

In terms of the required response time the most demanding case is when the flow is accelerated by means of a shock wave. It was estimated that the force would rise to a maximum within 60 μs. The balance should thus have a response time of a few microseconds. Typical wind tunnel force balances rely on the aerodynamic forces being in static equilibrium with the reaction force of the balance. Typically these balances achieve a response time of 200 ms¹⁷.

The principle of operation of the stress wave drag balance (SWDB) is derived from the Hopkinson Bar¹⁸. A cylindrical bar is suspended horizontally from threads and struck at one end with an object such as a bullet. By observing the response of the bar information such as the original bullet speed and its impact time could be determined. The impact would also generate a compressive stress wave in the bar. By placing strain gauges on the surface of the bar the stress-time history can be determined. Various implementations of this arrangement have been used extensively for the testing of materials under dynamic loading conditions.

Sanderson and Simmons¹⁷ were responsible for converting a Hopkinson bar into an aerodynamic balance. The SWDB is a slender rod with an aerodynamic model attached to the one end. Adding the model constitutes adding a lumped mass and changes the response of the balance. An unsteady drag force acting on the model results in stress variation within the model. The stress waves then encounter the interface between the model and the rod, some are transmitted into the rod and some are reflected back into the model, and then again back to the interface. This continues until the entire wave has been transmitted into the rod. This is known as the mechanical response time. The rod is suspended on threads so that the supports do not interfere with the stress wave propagation and the drag force is determined by interpreting the stress wave using the unit impulse response as a reference. This process is called deconvolution and allows a transducer's response time to be reduced below its mechanical time constant. The testing time for the SWDB is determined by the time it takes for a stress wave to propagate down its length, reflect at the end, and return to the strain gauges. This can be maximised by increasing the length of the rod and selecting a material with a low speed of sound. It is important to evaluate different model/rod materials and sizes. The SWDB technique has particularly been developed for measurements in short-duration hypersonic facilities¹⁹ where testing times may be of the order of milliseconds.

Sanderson and Simmons¹⁷ approximated their test model as a cylinder. They analysed the wave transmissions for a step force input to determine the stress in the rod just behind the model where the strain gauges would be positioned.

and subscript 1 refers to the model and subscript 2 to the rod.

Analysis of these equations for practical size models and rods reveals that typical mechanical response times are of the order of a few hundred microseconds up to a few milliseconds. Since the required response time is 10 us the balance output needs to be interpreted through deconvolution before the steady state is reached, using the unit impulse response as a reference. The balance output y(t) as a function of the applied load u(t) and the unit impulse response of the balance g(t) is given by¹⁷

This is an inverse problem since g_i-j and y_j are known and u_j needs to be determined. Unfortunately real data always has noise and is recorded using an analogue to digital converter. Solving the inverse problem amplifies noise and the steps introduced by the converter. Thus a polynomial is first fitted to the data and the smoothed data deconvoluted. The equations may be rearranged to solve for u_i as follows:

The solution of the first of these is used to solve the second, which is then used to solve the third and so on. Deconvolution is thus computationally expensive.

The implementation of these techniques for the current shock tube application presents two major challenges, which required a significant number of options to be tested before satisfactory results could be obtained. Firstly the deconvolution process requires that the signal to noise ratio be as large as possible at all time steps. To achieve high strain a compromise has to be made between two conflicting design parameters, the sensitivity and the mechanical time constant. High sensitivity is achieved by selecting a material with a low modulus of elasticity and minimising the rod cross-sectional area. The mechanical time constant determines how long it takes for the strain to reach the steady state strain level, which is a function of both the model and the rod. In terms of the rod, its density, speed of sound, and cross-sectional area should be as large as possible. A further constraint is that the cross-section should be sufficiently low so as not to cause too much flow blockage downstream of the model. A further complication is that the modulus of elasticity, E, density, ρ, and sound speed, c, are related by E = c²p. Thus maximising the density and sound speed will also maximise the modulus. The requirement for high sensitivity is thus the opposite of the requirements for low mechanical response.

The second challenge is ensuring that the testing time is adequate. When the stress wave propagates to the end of the balance it reflects and returns to the front end of the balance where it reflects again. It can oscillate a number of times before decaying. Every time it passes the strain gauges it causes a response. As long as the impulse response included this response, the reflections can be deconvoluted out of the signal. For the current shock tube application it was found necessary to constrain the end of the balance in the shock tube. This resulted in vibrations from the valve closure entering the rod from the rear, although as described later, methods to overcome this problem have also been implemented. What is crucial, however, is to ensure that the boundary conditions during calibration are identical to that during operation, even if this means reducing the testing time. This is not a serious limitation for the present case where the phenomenon of shock diffraction, i.e. shock propagation over the body, is of very limited duration.

The development of suitable calibration methods is in itself a significant exercise and of absolute importance. A number of techniques are described by Mee²⁰ for the hypersonic application. The suspension method described by him is similar to that developed independently for this work. The major difference is that the shock tube application requires a very much higher response time (microseconds) than that for the hypersonic facilities (milliseconds).

The ideal way to perform the calibration would be to apply a known impulse and record the balance response. The response would then be normalised to obtain the unit impulse response. The problem is to be able to apply a known impulse. However, the derivative of a step input is an impulse. For this to work the step input force application has to be instantaneous compared to the response time of the balance. Secondly the response has to be differentiated which amplifies any noise in the signal. It is practically impossible to apply a step input through an impact hammer or similar device, so the technique which has been evolved is to remove a force in a step-like manner. This is done by suspending the balance vertically from a thin thread attached to the model, and then the thread is cut as rapidly as possible. As long as the thread releases the load far faster than the mechanical response time of the balance, then the calibration is valid.

Initial testing using 20 mm diameter copper tubing as the rod showed significant noise levels. To overcome this problem and instead of using the balance's own mass as the calibration mass, the balance was suspended vertically with the model at the bottom. A mass, much larger than the balance mass was then suspended from the model using a thin thread as shown in figure 11. This resulted in measurable and meaningful output for the first 2 ms and also allowed different loading conditions to be used to check the balance linearity.

 

 

This change in calibration procedure with the rod end rigidly attached precluded allowing the balance to swing freely in the shock tube because the change in end conditions as the wave reflected would then be different. The shock tube mount was modified accordingly.

During the development of the balance a number of interesting issues came to light. For example, when calibrations were attempted with various rods alone and thus with no model, the weights were attached via a thread attached through a hole drilled diametrically through the end of the rods. Significant oscillations were noted. When holes were drilled at right angles these oscillations were out of phase with the previous case, indicating that even the small hole at the end of the rod induces significant changes to the stress wave pattern. However when a large mass was attached to the end without the holes these oscillations were completely removed.

In order to achieve the initially planned testing time of 30 ms the rod used for the balance would need to have been some 15 m long depending on the material used. In view of available material this would mean that two or more sections would have to be joined, with the joint expected to produce a reflected wave. In order to test this, a copper pipe balance was cut in half and rejoined using silver solder. Even with a carefully cleaned joint the reflected stress wave was larger than the noise band. Although in principle the deconvolution algorithm would take care of this wave it was decided that in view of the very much shorter tests times required for the shock loading measurements that joining would be avoided and shorter lengths of rod used.

Once deconvolution tests were started it was found that the copper tube balance was not sensitive enough. A new balance was made from a 10 mm diameter, 2 m long PVC rod. To keep the mechanical response time low a hollow PVC sphere was made with a wall thickness of 4 mm. The balance was calibrated and when it was deconvoluted a step input was recovered. To improve the sensitivity still further an 8 mm diameter PVC balance was made. The balance and deconvolution were found to be working satisfactorily and installation in the shock tube could proceed.

To eliminate air friction between the balance and the airflow the balance was fitted with an aerodynamic shield. This consisted of a brass pipe just large enough to accommodate the balance as well as the strain gauge leads. A bush was made from PTFE for the front of the shield to ensure that little air could pass between the shield and the balance. The bending stiffness of the balance was low, so to support it along its length some short strips of foam were glued to it at intervals to avoid it touching the shield. The balance was then recalibrated to ensure the shield assembly did not interfere with operation.

A special experimental arrangement was implemented in order to verify the balance performance under shock wave loading conditions. A cylindrical flat-faced model was made and calibrated and fitted, with a very slight clearance into a rigid plate attached to the shock tube wall as indicated in figure 12. A fast response pressure transducer was inserted into the plate. The process thus represents shock wave reflection off a rigid wall where the transducer and balance response could be directly compared. The pressure will stay constant for a period well beyond the test time required of the balance. To make the response similar to what would be expected of the sphere the cylinder was made of the same diameter and mass of the sphere, and will be referred to as the verification model.

 

 

The initial unfiltered results indicated that the hollow sphere was oscillating. This was then replaced with a solid sphere but this has the effect of increasing the mechanical response time of the balance. It was thus clear that the signal to noise ratio had to be improved as well as reducing the response time. One option was to improve the strain gauging. The conventional gauges were thus replaced with semi-conductor gauges with gauge factors of 150, giving a signal to noise ratio improvement of 75 times. Another issue to consider is gauge length. It is clear that gauge length should be kept short in the direction of stress wave propagation in view of its variation in time. In the present application gauge lengths of 1 mm were chosen. A second method of increasing the output is to increase the bridge voltage. In order to limit the current, higher resistance gauges should be used. 350 Ωgauges were selected. The effect of these changes was to improve the signal to noise ratio by a factor of 219.

These changes allowed the choice of rod material to be revisited to produce a stiffer balance. The mechanical response time would be reduced and the signal to noise ratio increased when compared to the PVC balance with conventional gauges. Mild steel, aluminium, brass and PVC were compared. A rod diameter of 8 mm was retained for all cases as was the cylindrical model used previously, as well as the signal amplification arrangements. The comparison is based on the theory given by Sanderson and Simmons¹⁷, for the first 0.5 ms of operation, since this is the main region of interest. The results are given in figure 13. The fast mechanical response time of the brass would result in less noise from deconvolution and was thus chosen for the final balance. A 3 m length of 8 mm diameter brass rod was obtained and instrumented with semi-conductor gauges.

 

 

During dynamic calibration, however, some new issues became apparent which were not previously evident because of the then lower sensitivity. Firstly, the balance was now so sensitive and its mechanical response so short that it measured vibrations induced in the wire used to suspend the calibration weights when it was cut. These vibrations are an artefact of the calibration and have nothing to do with the impulse response of the balance. Secondly, the mechanical response time was faster than predicted by theory. This resulted in an overshoot similar to that for mild steel, as shown in figure 13.

In order to reduce the overshoot the stiffness of the sphere was modified by changing material, even though this could slightly increase the response time of the balance. New spheres, and verification models, were made from aluminium, mild steel and brass. None of these exhibited overshoot but the rise time for brass and steel was considerably longer than the aluminium, although it still showed some sensitivity to vibrations, and thus aluminium was chosen for the final model design.

An additional step was to reduce the transmission of vibrations from the wire into the balance. Figure 14 shows the original arrangement and the modifications made. In the original set-up the wire is in direct contact with the rod allowing direct transmission of wire vibrations. The approach was then to pass the vibrations into the model itself which would attenuate these perturbations. A short grub screw was inserted into the model and tightened up against the knot in the wire. A second grub screw was inserted with a small air gap between it and the first screw and the rod was then tightened such that it locked against this screw. This resulted in the stress travelling through the body of the model. In addition a 5 mm thick neoprene disk was inserted between the wire knot and the first grub screw. These actions reduced the vibrations to an acceptable level, and since there was no overshoot a filter could be used that would remove the remaining vibrations without affecting the significant features of the impulse response.

 

 

Various support wires were also investigated during this development phase. Initial tests were done using high tensile wire to support the weights. The wire was cut using side cutters. Attempts were also made to remove the load by melting the wire with an oxy-acetylene torch. Although the melting was rapid the material still yielded over a long length of time relative to the mechanical response time. In another set of tests the wire was replaced by string, which was either burned or cut. However the individual fibres did not fail simultaneously thus causing large spikes in the applied load. Initial tests with nylon gut looked very promising as the vibrations were significantly reduced as long as a suitable length was chosen. If it was shorter than 600 mm the vibrations remained, and if it was longer than 1 200 mm the step response appeared to be damped. However the main problem was that the rise time of the force was too slow and the requirements of the deconvolution could not be met. The gut tests highlighted the influence of length and further tests were then done for the high-tensile wire. It was found that the response was independent of length for lengths of 35, 56, and 87 cm but was affected for very short lengths as shown in figure 15. The results for the 56 and 87 cm lengths are almost indistinguishable. It is suspected that for very short lengths of wire the stress ramp induced in the wire at cutting cannot steepen up into a step input. Tests were also done to find if there was an influence of test weight. Ideally the wire should be loaded very close to its failure load in order to get very rapid separation. However, various loads have to be used in order to determine the linearity of the balance. It was found that a 30% variation in applied weight has no effect on the rate of loading and hence the impulse response. Within these constraints and with the modifications made, the calibration and deconvolution procedures met all the requirements for shock tube implementation and testing.

 

 

The verification rig described previously and shown in figure 12 was the main facility to test the balance behaviour under shock wave loading. The reflected pressure was monitored by the transducer, which when multiplied by the area of the balance face gives the load. In order to ensure essentially one-dimensional loading a very small gap was left between the cylinder and the back plate. This unfortunately resulted in some friction causing the actual loads recorded after deconvolution being slightly lower than those determined from the transducer, as shown in Fig 16. It was deduced that vibration between the cylinder and the plate was also responsible for the peaks at 0.9 and 1.2 ms. These effects would not be a problem for the main tests with spheres and cones. As will be shown in a companion paper the drag tests on spheres and cones give much smoother results. Some tests with very weak shock waves showed that the balance could respond to a load of a few Newtons even though the friction between the cylinder and the sleeve stopped the balance from following the applied force exactly. Despite the error induced by the verification arrangement there is satisfactory agreement between the balance and the pressure transducer. The results also indicate that the resolution of the balance is somewhat better than 20 us.

 

 

A number of factors have to be borne in mind in treating the raw data. Due to strain gauge drift the recorded output was not exactly zero before shock arrival so this was first adjusted. The data then needed to be filtered because the deconvolution process amplified any noise. The analogue signal from the strain gauges was recorded digitally at 10 million samples per second. This also introduces steps at the different output levels. The filtering was critical to ensure that the noise was reduced to a satisfactory level without unduly slowing the response time of the balance. The 20 μs response obtained is entirely due to the filtering applied.

A number of smoothing techniques were tried. In order not to lose small variations a polynomial was fitted to each data point as well as a number of preceding and subsequent points (usually 100) with the ability to change polynomial order depending on curve complexity. This process is computationally expensive as some 20 000 points needed to be filtered for each test. This results in 20 000 polynomials being fitted, each one covering 201 data points. This process delivered very satisfactory results. When the calibration curves required differentiation to determine the impulse response the polynomials themselves were differentiated. Once the filtering was done every tenth data point was selected and the reduced data deconvoluted using the impulse response. Reducing the data set before filtration was found not to be as effective as the rise time was increased for a given noise level.

When integrating the drag balance into the shock tube a number of negative interactions arose that required some modifications to be made to the facility. As indicated earlier the balance was rigidly supported at the rear end. This obviated the need to have an additional measurement facility to account for the test piece moving down the tube during a test if it had been supported on threads as done by Sanderson and Simmons¹⁷. When the shock tube was fired the recoil was severe and vibrations were transmitted down the tube walls and back up into the balance through the rear support. This would appear as an additional drag in the results. To overcome this effect the shock tube was unbolted from the valve assembly leaving a narrow gap as indicated in figure 3. Some air escaped through this gap but it was confirmed that this had a negligible effect on the pressure trace measured at the test section. The tube had also to be securely fastened to the floor. Although some vibrations remained it was shown from special tests that by isolating the balance from the flow and firing the tube these vibrations were negligible.

 

5. Implementation

Although the tests described in the companion paper are for shock wave impact on cones and spheres the facility has been designed to cater for a wider range of transient flows so that both accelerating and decelerating flows may be produced rather than just the step change to a uniform flow induced behind a plane shock wave.

Because of the finite opening time of the valve a compression wave is produced just downstream of it as indicated in figure 2. This steepens up as it propagates down the tube. For pressure measurements taken at 3, 5, 7, and 9 m from the valve the initial section of the pressure wave contained a shock which grew stronger as the wave propagated down the tube. By the time it reached the test section at 14.2 m from the valve it had completely formed into a shock. The range of shock wave flows achieved is shown in figure 17. Although the post-shock flow is a bit noisy, presumably due to the very complex flow at the valve, it is reasonably uniform and acceptable for testing. Reynolds numbers from 100 000 to 600 000 can be obtained. These results were obtained with the valve opening as rapidly as possible, with the variable orifice fully open, and with the variation in Reynolds number achieved by simply varying the driver pressure.

 

 

To obtain compression waves, the test section was moved to be 4.55 m from the valve. A typical result is given in figure 18. The precursor shock has already formed. This pressure trace was obtained from the downstream transducer at the position of the sphere so the reflected shock from the sphere appears as a spike just before 0.3 ms.

 

 

In order to generate a wider range of flow variation, experiments were conducted simply by making larger gaps in the tube such as that used to prevent vibration transmission. If the gap is made large enough decelerating flow could be induced at the test section as well us other interesting wave profiles which give scope for future work. Thus a 28 mm gap produces the flows shown in figure 19 for two different driver pressures. Gaps of up to 1 100 mm have been tested to give decelerating flows suitable for comparison with data generated by other workers using unconstrained spheres as described earlier.

 

 

The facility thus has a lot of potential for a range of tests to be conducted under a variety of flow conditions. This can be coupled with flow simulations in order to establish suitable hardware arrangements to achieve particular flow variations. For smoother and more controlled pressure traces to be produced the valve or other methods of flow control would need to be revisited. However, for the present study on spheres and cones the basic shock tube arrangement with a following pseudo-steady flow will be used.

 

6. Conclusion

A valve operated shock tube facility has been developed for the production of short duration transient flows with the potential for generating a variety of transient flows ranging from the step input across a plane shock wave to complex accelerating and decelerating flows. A stress wave drag balance has been produced which can follow very fast transient drag variations in the sub-millisecond range. As far as is known it is the only balance of its kind with this capability.

 

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