. 1.1 1 . . 1 • I OF L ORNLP i * 1 . . . .. . I : . . ... - - - - . 1 . . . L . . wie - . . - 2740 - 1. . - i . . ES SO 1:I- 1.1.4 16 . MICROCOPY RESOLUTION TEST CHART NATIONAL BUREAU OF STANDARDS - 1913 . . .. ORNG-pu 2140 OⓇNH- P-2740 Conf. 661141-2 AF317 PRICES MAGNETOHYDRODYNAMIC STABILITY ANALYSIS OF JIT-INJECTION FLOW OVER A CONCAVE WALL *a • _3.00; www.65 T. S. Chang and W. K. Sartory Oak Ridge National Laboratory Oak Ridge, Tennessee RELEASED FOR ANNOUNCEMENT DEC 1 5 1966 IN NUJIIEAR SCIENCE ABSTRACTS (Introduced by O. Laporte) The theoretical portion of this investigation consists of two parts: a calculation of the primary velocity profile in a jet-driven vortex tube, and an MHD stability analysis of the resulting flow. The primary 210w is observed experimentally to consist of a thin boundary layer near the outer tube wail in which the influence of the driving jets and viscous wall drag are felt, and an interior region in which angular momentum is conserved. We are concerned mainly with the outer boundary layer which the flow is essentially two dimensional and unaffected by the magnetic field. Assuming the boundary-layer thickness to be much less than the radius of the vortex tube, the wall curvature can be neglected and the ordinary laminar boundary-layer equations apply. The corresponding flow geometry is shown on the first slide. It is obtained by "unrolling" the vortex tube, and consists of a periodic series of flat plates separated by tangential injection slits. The injection velocity profile at the exit of each slit is assumed to be a fully developed parabola. X and Y are the dimensionless coordinates parallel and normal to the wall. The second slide shows a typical set of calculated boundary-layer velocity profiles. Here 7 is the dimensionless tangential velocity, and W is the ratio of slit width to boundary-layer thickness, a parameter in the analysis. Research sponsored by the U. S. Atomic Energy Commission under contract with the Union Carbide Corporation. The injected parabolic velocity profile spreads and decays as we move downstream. The final velocity profile, just upstream of the next injection blit, reserbles the ordinary flat-plate Blasius proflle. The boundary condition applied far from the surface is that the slope of these profiles should vanish. The velocity itself is determined from the cal- culations. The ratio of the velocity outside of the boundary layer to the average jet velocity, we call the recovery ratio. The only experimental quantity available for comparison with these re- sults is the recovery ratio. Experimental and theoretical' recovery ratios are shown on the third slide as a function of che dimensionless jet width, W. The solid curve shown here was obtained from the boundary-layer equa.. tions. The open points are experimental vulues obtained with tangential injection slits when a sufficient magnetic field was applied to give an ob- served laminar flow. They agree with the theoretical curve within about 5%. The sol.id points below the curve were obtained vithout magnetic stabilization, and the flow was observed to be turbulent. The solid points above the curve are experimental results obtained by non-tangential injection through round nozzles, and are not relevant to the theoretical work. The stability analysis is based on a linearized perturbation technique applied to the MHD flow of an incompressible viscous fluid of finite electrical conductivity in the presence of an axial magnetic field. The confining cylin- drical wall is assumed to be perfectly conducting. The velocity and magnetic field perturbations admitted are independent of 9 (or ) and sinusoidal in the axial coordinate 2, and are of the type used by Görtler to determine the sta- bility of a boundary layer on a concave wall. . As is usual in theoretical analyses of the stability of boundary-layer flow, we treat the stability as a local property of the boundary layer at a ------------- ---------7-------- --- LEGAL NOTICE The report me prepared as an accoint of Government sponsored work. Neither the United sutos, oor la Commission, nor any l o cting on behalf of the Commission: A. Makes any warranty or representation, expressed or implied, with respect to the accu- racy, completeness, Os' jsefulness of the laſormation contained in this report, or that the use of any information, apparalus, method, or process disclosed lu this sport may not infriage privaloly owned rigau; or B. Assumes any llabilities with rospect to the use of, ur for damages resulung from the Um of any information, apparitus, method, or procco& disclosed in this report, As usod to the above, person acting na beball of the Commission" Includes any em. ploylo or contractor of the Commission, or emaployce of such contractor, to the oxteat that swid om mioyee or zoniracicr of the Commission, or employce of such contractor prepares, diotominater, or provides access to, ny information pursuant to his employment or coutract wted the Commission, or ho employment with such contractor, given downstream distance, X (or a given angular position in the vortex tube, ). However, we have included normal flow terms and their derivaties, which result from the growth of the boundary layer. The next slide shows some of the theoretical stability results in a graph of the critical wave number of the disturbance versus Hartmann modulus. At large values of the Hartmann modulus, the wave number is inversely propor- tional to the Hartmann modulus. As the Hartmanr. modulus approaches zero, the wave riumber also approaches zero. The next slide shows a graph of the critical Görtler modulus versus Harta minn modulus. At large Hartmann moduli, the Görtlar mcdulus is di:ectly pro- portional to the Hartmann modulus. As the Hartmann modulus approaches zero, the Görtler modulus becomes very small and apparently approaches zero. The qualitative behavior of these curves in the limit of large Hartrari. modulus is consistent with the results of Chandrasekhar for the hydromagnetic stabilidation of cylindrical Couette flow. In the limit of zero Hartmann modulus, these results should be consistent with the earlier investigations of non-magnetic Görtler stability. In the case of the wave number, it has in .. . . fact been shown by Hämmerlin that the Görtler stability problein leads to the (rather pecular) result of a zero critical wave number. All of the earlier investigators, however, have fourid' non-zero values of the critical Görtler modulus, the best value being about 0.31 for the Blasius profile. To determine the cause of this discrepancy, a careful study was made of the stability of the ze: Wit he how, t Blasius profile. It was found that when the primary normal flow terms wire . omitted, a critical Gortler modulus of 0.31 was obtained at zero Hartmann modulus. When the normal flow terms were included, however, the non-magnetic Görtler modulus went to zero. The flow normal to the surface therefore seems to have a severely dest=b111zing effect on the flow. . ! ILL...!.!..MILIT!! TULVILNIUTTOS T OPYYTETTE" -- T' ISTITYSE FETAR: "*EXT TEXT TE- ET- Of course, we do not meen to suggest that a non-magnetic boundary layer on a concave surface is always unstable and that disturbances of infinite wavelength will appear in it. The maximum size of the disturbance as always limited because of the finite radiue of curvature and other dimensions of a ero physical apparatus, and a disturbance of finite wavelength will always have a non-zero critical Görtler modulus. If the validity of including the primary flow terms is accepted, however, then our work shows the necessity of including other effects such as a finite radius of curvature, or the distance or time available for amplification of a disturbance, in order to get a meaningful critical Görtler modulus. (These effects have been considered by other in- vestigators.) For hydromagnetic flow with & finite Hartmann modulus, the disturbance is limited by magnetic damping to a size on the order of the boundary-layer thickness so a finite wave number and Görtler modulus are obtained without further complications. The next slide shows a comparison between theoretical and experimental stability results. The lower curve gives the theoretical results for a dimensionless jet width of W = 0.6. The upper curve is a least-squares fit to the data. The value of ū corresponding to the data points ranges from 0.35 to 1.2. The theoretical curve lies about 20% below the data. The use of these parameters, which are based on the boundary-layer thickness, is some- . what deceptive, however; and if an attempt is made to calculate the critical velocity in a given fl.ow geometry, the theoretical prediction may be low by a factor of 2 or more. There are several possible causes for the low theoretical precition: 1. The boundary-layer thickness in the experimental measurements was on the order of 10% of the radius of curvature of the surface and also on the order of 10% of the length of the boundary layer (1.e., the peripheral distance between slita). Under these conditions, the flat-plate boundary-layer approximations may not be adequate for stability calculations, although they did give good results for the primary flow recovery ratio. 2. We have treated the stability as a local rather than a global property of the flow. In all cases the velocity profile analyzed for stability was the one believed to be most unstable. Roughly speaking, this might be interpreted as meaning that, at the critical Görtier modulus which we calculate, the energy loss by the disturbance due to dissipation exceeds the energy gain from the primary flow for all values of 0 except one, where the energy loss and gain just balance. The true critical Görtler modulus should then have sone higher value, at which the energy loss and gain through the entire flow regine balance. 3. In work on boundary-layer stability, the usual reason given for low theoretical results is that the Reynolds number must be well above the critical value so that the disturbance can be amplified to an observable intensity. The same explanation might be given here to the extent that fresh undisturbed fluid is being continuously injected, and the disturbance must grow in that, fluid. By considering the stability of a profile away from the injection slit, we have ignored that effect. : : ! - . " '! . . .. :: !!! : : \ -'* *' : . " 11. ..-1't . r . - . ORNL-DWG 66-7254 Slide 1 .8*10° =0.1 .6x10° X=0.3 V, TANGENTIRIL VELOCITY .4u10° R=1.0 .210 0210 L 10% Gu10' 1.Cow 10' 110' 2-10% .3x10' 4w10? Sulo? •6*10' 7*10' •8110 P, NORMAL DISTANCE FROM SURFACE · JET BOUNDARY LAYER PROFILES. W=0.6 slide 2 www. .. .... . .. ............ ............. .. .... .... .st .... ...... 0.7 ORNL-DWG 66-1196R CALCULATED CURVE BASED ON CAMINAR BOUNDARY-LAYER THEORY ROUND NOZZLE INJECTION - MAGNETIC STABILIZATION (2.8-cm TUBE) ORO: RECOVERY RATIO EXTRAPOLATED TO WALL O I SLIT ) A 2 SLITS MAGNETIC STABILIZATION Pof SLITS (10-cm TUBE) • 1 SLIT A2 S NO MAGNETIC STABILIZATION (10-cm TUBE) 0 0.1 02 0.3 0.4 0.5 0.6 0.7 08 09 10 11 12 1.3 1.4 5.5 .. Win NRO NRBEITZ CORRELATION OF JET VELOCITY RECOVERY RATIO FOR MAGNETICALLY STABILIZED AND UNSTABILIZED VORTEX FLOW Slide 3 BA, WAVE NUMBER BASED ON MOMENTUM THICKNESS 101 102 10 102 101 No, , HARTMANN MODULUS BASED ON MOMENTUM THICKNESS CRITICAL WAVE NUMBER, THEORY, W=0.6 Slide 4 NGU, GÖRTLER MODULUS AT TRANSITION 10-2 103 102 10 10° MARA, HARTMANN MODULUS BASED ON MOMENTUM THICKNESS CRITICAL GÖRTLER MODULUS, THEORY, W=0.6 :. lslide 5 7 ORNL-DWG 68-1194R1 AR A SLIT t ', * GÖRTLER MODULUS AT TRANSITION TO INSTABILITY 4th DEGREE LEAST-SQUARES CURVE- 2 SLITSU . ... .. LAMINAR BOUNDARY-LAYER THEORY . . IT 4 SLITS 0.4 0.8 12 16 20 24 28 32 HARTMANN MODULUS BASED ON MOMENTUM THICKNESS OF BOUNDARY LAYER 3.6 NHO . Un · SUMMARY OF HYDROMAGNETIC STABILIZATION EXPERIMENTS WITH A 10-cm DIAMETER JET-DRIVEN VORTEX TUBE (GÖRTLER MODULUS VS. HARTMANN MODULUS) Liwa. 1 Slide 6 ---- --- - - . -. . - -.. SO END DATE FILMED 2 / 6 / 67 *+,- V A .