Chang, C.T.

An investigation was made of the possible effect of the boundary layer on the lack of separation observed between the shock and the current sheath in a magnetically driven shock tube operating under discharge conditions of high voltage and low pressure. Since the calculated and the observed maximum displacement thickness in the pressure range investigated was around 1 mm only, it was concluded that the presence of a viscous layer at the wall cannot be of importance. Instead, the observed lack of separation is most likely due to the leakage of the compressed gas through the current-sheath, and to the diffusive nature of the driving current

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Downloaded from orbit.dtu.dk on: Dec 20, 2017The effect of boundary layer on the separation distance in a magnetically driven shocktubeForskningscenter Risø, RoskildePublication date:1975Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Chang, C. T. (1975). The effect of boundary layer on the separation distance in a magnetically driven shocktube.  (Denmark. Forskningscenter Risoe. Risoe-R; No. 321).™ Risø Report No. 321 m O z u O J* Danish Atomic Energy Commission 5 Research Establishment Risø The Effect of Boundary Layer on the Separation Distance in a Magnetically Driven Shock Tube by C. T. Chang February 1975 Sain dlåtribmori: Jul. Gjaltorup, 17, Sølvfad«, DK-1307 CopmlMfm K, Denmark Anllabk en txthangt from: Library, Daniab Atonic Energy Comaateio^ M M , DK-4000 KoakJM«, Dmaarfc UDC 533.6.011.72 February 1 975 Risø Report No. 321 The Effect of Boundary Layer on the Separation Distance in a Magnetically Driven Shock Tube by C.T. Chang Association Euratom - AEK Danish Atomic Energy Commission Research Establishment Risø Roskilde, Denmark Abstract An investigation was made of the possible effect of the boundary layer on the lack of separation observed between the shock and the current sheath in a magnetically driven shock tube operating under discharge conditions of high voltage and low pressure. Since the calculated and the observed maximum displacement thickness in the pressure range investigated was around 1 mm only, it was concluded that the presence of a viscous layer at the wall cannot be of importance. Instead, the observed lack of separ-ation i s most likely due to the leakage of the compressed gas through the current-sheath, and to the diffusive nature of the driving current. ISBN 87 550 031 S 2 CONTENTS Page 1. Introduction 5 2. Brief Review of the Boundary Layer Effect in Conventional Shock Tubes 5 3. Estimation of the Boundary Layer Eftect in Our Experiment . . 7 4. Conclusion 8 Notes and References 10 Figures }i - 5 -1. INTRODUCTION In magnetically driven shock tubes operating under discharge conditions of high voltages and low p res su re s , i. e. conditions where high speed shocks a re produced, a lack of separation between the shock and the current-sheath i s often observed ' . As a possible analogy a lack of sufficient separation between the shock and the contact surface was observed by previous inves-tigators in conventional diaphragm-type shock tubes operating at low initial 2 3) p res su res * ' . They attributed the observed lack of separation, and the associated non-uniformity of the flow, to the mass leakage of the flow through the viscous boundary layer at the wall. Since the reported phenom-ena show the same tendencies as those observed in the magnetically driven shock tubes, the possible influence of the boundary layer should not be over-looked. In this repor t , we first present a brief review of the boundary layer effect in conventional shock tubes; this is followed by an examination of i t s possible influence on our experiments . Based on the calculated as well as on the experimentally indicated displacement thickness , it id concluded that the effect of boundary layer cannot be of great importance under our exper-imental conditions. Instead, the observed lack of separat ion is to a great extent caused by the lerkage of the mass through the current -sheath and by the diffusive nature of the driving cur ren t . 2. BRIEF REVIEW OF THE BOUNDARY LAYER EFFECT IN CONVENTIONAL SHOCK TUBES In an idealized shock tube, one assumes all the gas initially occupying the region between the position x reached by the shock and the diaphragm to be compressed into region 2 between the shock and the contact aurface, see fig. 1. Accordingly, the separation distance, A , between the shock and the contact surface, and the corresponding test t ime, t , i s given by T* */u2 ' W 1 - 1 ) ' *!> where u„ i s the flow velocity of the compressed gas, V i s the shock speed, and V « P2fPy M - 6 -is the compression rat io. P9 and P . are the mass densities of the gas downstream and upstream of the shock respectively. For a given gas, n i s a function of V . Consequently, for a given shock speed, one should expect the test rime * to increase proportionally with the distance x travelled by the shock. In an experiment conducted in a shock tube operating at low initial pressures (~ I Torr) Duff first noticed that at a given initial pressure p^, contrary to expectations, there i s a limiting value of the test time * , i . e . 2) . . T cannot be increased by increasing x indefinitely . He attributed this eflect to the mass leakage through the viscous boundary layer at the wall. Subsequently, Roshko ' made some further experiments and obtained quite good agreement with Duffs ear l ie r results. He also showed analytically that fhe maximum test time T is related to the shock Mach number, M_, m s and tne initial state of the gas by Tn ~ P l D 7 G ( M s ) . (3) In t\ e above expression D = - A / L (4) is the hydraulic diameter with A as the c ro s s sectional area of the tube and L the wetted per imeter . The function G(M ) is given explicity by s ' T2 T , * G(M l . ± _2 l l ± L , (5, where T. and T 2 a re the temperature upstream and downstream of the shock respectively. Z 7 is the compressibil i ty factor for the downstream gas in region 2, thus , kT0 <VPI = Z; "nf ' ( 6 ) For helium, X., is related to the degree of first anu second ionization a and £ by y- = (' u Q + * )o • ( 7 > 2 Assuming the existence of an equilibriuon sta'.e in region 2 , using the result of i-'ueks .ind Artmann for o'» and t 2 » w e have calculated G(Mg) in - 7 -the range of shock Mach number M_ covered in our experiment. (The corresponding current-shock speed u_ i s from 1.76 to 7.30 cm/as ) . The result i s shown in fig. 2. It should be mentioned that average values of Z 2 , H and T 2 /Tj are used in calculating G(M ) by interpolating the data of reference 5 between p. = 0. 5 and 1 Torr. From eq. (2) one observes that the limiting test time * is proportional to the initial pressure p. and to the square of the tube diameter as indicated by Duffs earlier experiment. Furthermore, one notices that T decreases m rapidly with the shock Mach number M , since G(M ) increases rapidly with Mg . 3. ESTIMATION OF THE BOUNDARY LAYER EFFECT IN OUR EXPERIMENT The following phenomenon was observed in the magnetically driven shock tube used in our experiment. When the discharge voltag* is kept constant, the current-sheet speed increases as the discharge pressure decreases. According to the predication of eq. (3), one would expect the separation distance to be smaller at lower pressures than at higher press -ures. Since this trend was observed experimentally (see fig. 3, ref. 1), the possible influence of the boundary layer effect cannot be overlooked. A direct check of the theoretical prediction, as given by eq. (3), i s not feasible, because an experiment of this kind involves a considerable elong-ation of the electrodes and of the duration of the driving current. Instead, we shall resort to the following indirect means. The importance of the boundary layer depends on the ratio between the displacement thickness b* ' and the hydraulic diameter D, Since in our case D s 2 d, where d i s the gap between the electrodes, from eq. (4), we have this ratio as b*/2d. According to Mir els ', for a laminar flow« b* at a distance 1 behind the shock i s given by l ' / 2 H ) & # • (^-) f ( T , , M B ) , (8) where the function f(T1 , M0) can be shown to depend on the free stream property of the compressed gas and on two other tanetkma relate* to tfc* velocity and the static enthalpy profile of the boundary layer, Thea« latter two functions are tabulated by him with respect to the oomprepiejcn, raffr . Assuming the existence of an equilibrium state la the freo «tre«am r«gfal* behind the shock, and interpolating between the tabulated data of : - 8 -we can express f(T,, M ) explicitly as a function of the shock Mach number M , or, as shown in fig. 3, a function of the current-sheet speed u_. Once more it should be mentioned that in evaluating the property of the free stream flow in the compressed region, the same average process was applied as used in the computing of G(M ). The detailed calculation is rather involved and i s not worth presenting here , but it should be mentioned that a Prandtl number Pr = 0. 67 i s used, and that in evaluating the viscosity coefficient. »g, a relationship • » > , MT 2 / T , ) 0 - 6 4 7 (9) is used instead of the more accura te Sutherland formula . From the figure we notice that in the range of the current-sheet speed encountered, the maximum value of 6 " 6 x 1 0 " cm ' *". Since the observed separation g» distance in the pressure ra rge in question never exceeds 4 cm , even if we assume that it occurs ui t i e lowrest discharge p re s su re p. = 0.1 T o r r , we obtain (-1) & */'->,i ** ° - ' f r o m efl- (8)-The above calculation, however, i s based on a viscosity law that is only valid for a neutral gas, a more cor rec t way of evaluating the viscosity coef-ficient n., might be to assume the gas in region 2 to be fuily ionized and to take the Coulomb interaction into account. At the discharge conditio.! V - 13 kV and p. = I Torr from the spec-troscopic data, we may take the average value of T " 2.67 x 10 K and 17 " -5 e n * ?0 and obtain n9 = 8. 28 x 1 0 C .G.S . unit according to the Spitzer e 11) formula . Since i : is about two o rde r s of magnitude smaller than the calculated value based on eq. (9), we should expect that the maximum ratio (-1) 6*/2d will 'je reduced to about two per cent when the Coulomb interaction is present. The calculation presented here is certainly very crude, but it may give a reasonable order of magnitude estimate, as can be seen from the differential interferrogram (presented in fig. 4) that the indicated boundary layer thickness does not seem to exceed 1 mm. 4. CONCLUSION Phenomenologically, insufficient separation between the shock front and the boundary of the driving medium was observed both in magnetically driven shock tubes and in conventional diaphragm-type shock tubes running at low initial p ressures . - 9 -Since the displacement thickness is not sufficiently developed at the point of observation, both with regard to calculations and to experiments, the observed lack of separation in magnetically driven shock tubes cannot be caused by the presence of the viscous boundary layer at the wall. To a large extent the phenomenon can be attributed to the flow leakage in the compressed region downstream of the shock. However, the leakage is not through the wall boundary layer, but through the driving medium (the current sheath) itself. Furthermore, due to the diffusive nature of the driving current in most of our experiments (especially those corresponding to high speed shocks) the shock has insufficient time to emerge from the current sheath. - 10 -NOTES AND REFERENCES 1) C. T. Chang, Z. Naturforsch. 2j)a (1 974) 1 838-1 845. In the same work an extensive list of references i s given regarding the observed lack of separation in magnetically driven shock tubes as recorded by other investigators. 2) R . E . Duff. Phys. Fluids 2_ (1 959) 207-21 6. 3) A. Roshko, Phys. Fluids 3_ (1 960) 835-842. 4) S. P . Kalra, Atomic Collisional Rate and Ionization Relaxation Study in a Helium Shock Wave. AD 749553 (1 972) 98 pp. Extrapolating data from the report , we obtain the ionization relaxation time *~ 0.04 n s at an initial p r e s su re p. ~ 1 T o r r and T ~ 0. 06 u s at p. = 0. 5 Torr . 5) W. Fucks and J . Artmann, Z. Physik ? 72 (1 963) 118-1 24. 6) M. Popovic, Experimental Investigation of Magnetically Driven Shocks with an Energy Source of Constant Voltage. Risø Report No. 206 (1 969) 48 pp. 7) The displacement thickness & is defined as QO 6* = j" (, _ PJL, d v ~ ( 1 --^)b, o where o is the bounuary layer thickness. Flow parameters with sub-scripts w and e refer to those evaluated at the .vail and at the free stream respectively. 6 ( 0 implies a m a s s leakage through the boundary layer. 8) H. Mirels, In: Shock Tube research . Proceedings of the 8th International Shock Tube Symposium, Lone .., 5-8 July 1 971. Edited by J . L. Stolley, A. G. Gaydon and P . R. Owen (Chapman and Hall, London. 1 971) paper No. 6. 9) H. Mirels, Phys. Fluids 9 (1 966) 1 265-1 272. 10) James Jeans, An Introduction to the Kinetic Theory of Gases (Cambridge Univ. PreSE, Cambridge, 1 948) p. 174. 11) L. Spitzer, J r . , Physics of Fully Ionized Gases 2nd rev. ed. (Inter-soience, New York and London, 1 962) p. 146. - 11 -Fig. I. Idealized trajectories of the currant sheet OC, the luminous front oL and the shock 55 in the x -1 diagram. Their arrival time« at a fixed location are denoted by tt, t^ and tg respectively. A is the separation distance; t • tc - tm is the test time. In a pressure driven shock tube the diaphragm is assumed to be located at the origin. 0<M.) -»»» •i £?• Wig. 2. Variation of the tawttee 0(M.» vera* the M M t t a ^ M s * * « , . - 12 -f f f t W 60x10-5i5 so-Standard Condition p,= 1Torr T,:300d*gK u2,cm/ns 2J0 30 4.0 &0 60 70 Fig. 3. Variation of the function f(T,, Mg), eq. (8), with respect to the current sheet speed u 2 . p is a reference p r e s s u r e . A 69¥3* Fig. 4. Two-wavelength different ial- interferrograms of the plasma region behind the shock front. The tip of the probe is at an axial position x * 35 cm from the end insulator plate. Discharge condition: Voltage, V • 13 kV, p r e s su re p. » 0 . 5 T o r r . Light source, a Q-switched ruby laser, pulse duration 30 nsec . Directioi. of motion is indicated by the arrow. 2 cm gap between the e lec t rodes . 

The effect of boundary layer on the separation distance in a magnetically driven shock tube

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The effect of boundary layer on the separation distance in a magnetically driven shock tube

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