Basset Force

8/3/2019 Basset Force

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BRIEF COMMUNICATIONSThe purpose of this Brief Communications section is to present important research results of more limited scope than regulararticles appearing in Physics of Fluids A. Submi ssion of material of a peripheral or cursory nature is strongly discouraged. BriefCommunications cannot exceed three printed pages in length, incl uding space allowed for title, figures, tables, references, andan abstract limited to about 100 words.

On the influence of the Basset history force on the motion of a particlethrough a fluidP. J. Thomasa)Deutsche Forschungsanstalt ftir Luft- und Raumfahrt, Institut ftir Experimentelle Striimungsmechanik,Bunsenstrasse IO, D-3400 Giittingen, Germany

(Received 27 December 1991; accepted 6 May 1992)A parameter study on the development of the Basset integral appearing in the equation ofmotion of a particle moving through a fluid is presented. The particle motion isinvestigated numerically for the flow across an aerodynamic shock. It is found that, for thistype of flow, the Basset history force acting upon the particle described by the Bassetintegral can be many times larger than the viscous drag in the immediate shock region. It isdemonstrated how this force affects the particle motion across the shock for differentparticles. Nevertheless, the results obtained show that it is justified to neglect the Bassetintegral for the theoretical description of the motion of the types of particlescommonly used in flow measurement tracer techniques for the type of flow considered here.

Measuring techniques like laser Doppler anemometry of the Basset integral and the influence of the Basset force(LDA) or particle image velocimetry (PIV) rely on the on the motion of a particle in the presence of a large ve-velocity information obtained from micron sized particles locity gradient in greater detail. The study will show howtransported with a fluid. The ability of these so-called seed- the Basset force affects the particle motion across an aero-ing particles to faithfully follow any velocity changes oc- dynamic shock and it will give stronger support to thecurring in a flow field is one of the key assumptions of such common simplification of neglecting the Basset integraltechniques. This assumption, however, is not necessarily when the motion of seeding particles under such flow con-satisfied under the influence of a large velocity gradient. ditions is considered.Under certain assumptions made later on, the motionof a particle through a fluid can be described by the Basset-Boussinesq-Oseen (BBO) equation (see, for instance,Soo). The integral term appearing in this equation [Eq.( 1 )] takes into account deviations of the flow pattern fromthe steady state. This term is interpreted as an additionalflow resistance and is usually referred to as the Basset his-tory integral or Basset integral.

Consider a small (order of size:pm), spherical, nonde-formable particle not disturbing the flow. Neglecting walleffects, the motion of that particle in a fluid can be de-scribed by the BBO equation, which is given in the nota-tion of Soo as

There are numerous publications that numericallystudy the motion of particles across aerodynamic shocks(e.g., Walsh2 and Nichols3). All of these publications ac-count only for the viscous drag acting upon the particle.Nevertheless, it was shown by Hughes and Gilliand4 that,in the case of a flow accelerated at high rates, the Basset(history) force, described by the Basset integral, can bemany times as large as the stationary viscous drag. OnlyTedesch? discusses the Basset integral for the motion of aparticle moving across an aerodynamic shock. He arrivesat the conclusion that the Basset force can be neglected forthe special case he examined. He does not, on the otherhand, show how it actually affects the particle motion.

~~~~~=~r~~~~(~~-S~-~r~~P+;;13ppF; curup>P+6$ &iii

Px s(d/d71 w,-Up) dr+Fa

PO

with

The purpose of this work is to study the development

(1)

(2)2090090 Phys. Fluids A 4 (9), September 1992 0899-8213/92/092090-04$04.00 @ 1992 American Institute of Physics

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Shock

FIG. 1. Oblique shock and measuring traverse %.

where Up and U, denote particle and fluid velocity, respec-tively, and pp, pF are the density of the particle materialand the fluid. The raqius of the particle is rp, the viscosityof the fluid is denoted by PF, and the empirical drag coef-ficient of the particle is given by C,. The physical signifi-cance of the five terms on the right-hand side of Eq. ( 1) isas follows: The first term represents the force acting on theparticle due to a stationary viscous flow. The second termis due to the pressure gradient in the surrounding fluid.The third term is referred to as added or apparent massand represents the force required to accelerate the mass ofthe fluid surrounding the particle and moving with it; theincrement being one-half the mass of the fluid displaced(l/2 mF). The fourth term is the Basset history integral,which was already discussed above. The last term on theright-hand side of Eq. ( 1) denotes external forces such asgravity.The pressure term disappears for the flow across anoblique shock in the region of constant downstream flowconditions. It is discussed in Thomas6 that it is justified toneglect the added mass term for the type of flow consid-ered. Furthermore, the influence of external forces will beexcluded.The simplified equation of the particle motion acrossthe shock is integrated numerically by the use of theBulirsch-Stoer method using the routines suggested byPress et al. and the equation for the drag coefficient C, byHenderson. The velocity across the shock was modeled asa hyperbolic tangent profile.With respect to various earlier LDA experiments (e.g.,Krishnan et al. and dHumieres et al. lo) in which the flowvelocity was measured along a traverse EF (Fig. 1 ), ourresults are presented in the coordinate system X, y of Fig. 1.Because of the deflection of the flow, a particle detected atany location along % has originally crossed the shockfront at a position lying below the intersection of EF andthe shock front. A transformation of the particle motioninto the measuring system of the LDA is obtained by sim-ple geometrical considerations (see Thorna&.The theoretical results presented below refer to thecomponent of the tlow velocity in the main flow directionand to the following upstream flow conditions: Mach num-ber M= 1.94, total pressure PO=97 400 Pa, static pressureP, = 13 650 Pa, total temperature T,=291.4 K. The shockangle is 0=47X and a shock thickness of about 7 pmperpendicular to the shock was assumed.

Figures 2(a) and 2(b) show the development of theBasset force and the viscous drag, both per unit mass,across the shock. It can be seen from Fig. 2(a) that, withinthe shock region, the Basset force starts to increase towardlarger negative values farther upstream than the viscousdrag. The Basset force takes on its first maximum withinthe second half of the shock region and starts decreasingrapidly thereafter. The viscous drag reaches its negativemaximum slightly farther downstream and decreases thenat a slower rate than the Basset force. Figure 2(a) showsthat, just downstream of the shock (for x 2 10 pm), theBasset force is already about one order of magnitudesmaller than the viscous drag. Unlike the viscous drag, theBasset force changes its orientation at some location down-stream of the shock; this can be seen in Fig. 2(b), wherethe scale of the ordinate was magnified. The Basset forcetakes on another maximum at about ~~0.06 mm and thenapproaches zero. Figure 3 shows the ratio of the Bassetforce and the viscous drag in the immediate shock region.It can be seen that, for the largest particle shown (r,=2.5

(a)

a.ao1 0.005 0.01 0.05X [mm1(b)

: viscous drag-- ---- ---- ^__. : Basset force

FIG. 2. (a) and (b) show the development of viscous drag and Bassetforce across the shock (p,=O.874. lo3 kg/m).2091 Phys. Fluids A, Vol. 4, No. 9, September 1992 Brief Communications 2091



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- r, =o.o25p ,..' ,,,,----. rpzo.25 m ._ _ - - .rp:(J5 pm ;y"_ --..-.r ,=,,o p :.-rp=2,5 p ;:.:'I',:',:'

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FIG. 3. Ratio of Basset force and viscous drag across the shock.

pm), the Basset force takes on values of up to around 30times the value of the viscous drag. The ratio decreaseswith decreasing particle size. All graphs shown in Fig. 3slope steeply toward the end of the shock region represent-ing the rapid decrease of the Basset force after havingtaken on its negative maximum [Fig. 2(a)].The actual influence of the Basset force on the particlemotion is shown in Figs. 4(a) and 4(b) for two particles of

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380.0 1. " " " '. " " " "-0.05 0.00 0.05. D-10 0,15 0.20(a) x Emml

-0.50 0 .110 0.50 I.00 1 .a0 Z-00lb) x Cmml

FIG. 4. (a) and (b) show the influence of the Basset force on the particlemotion across theoblique shock, for a particle with a radius of r,=O.5 pmand varying density.

different density. From Figs. 4(a) and 4(b), it can be seenhow the influence of the Basset force on the particle motiondecreases with increasing particle density. Similar curvesare obtained for particles of constant density if the particlesize is varied, with the influence of the Basset force on theparticle motion decreasing for increasing particle size (seeThorna&. From Figs. 4(a) and 4(b), it can be seen thatthe curves obtained, if the Basset force is taken into ac-count or neglected, intersect each other at some locationdownstream of the shock. This behavior reflects the changeof the Basset forces orientation discussed above. Upstreamof the point of intersection, the Basset force acts as anadditional flow resistance, whereas downstream of thispoint, its overall effect is a higher particle velocity. Its neteffect therefore is an increased relaxation length. The in-terpretation of the Basset force as an additional flow resis-tance is thus not entirely correct and is misleading in thiscase.The seeding particles commonly used in LDA andother t racer techniques have a radius of around rpz0.5 ,umand a density equal to or higher than approximatelypPz 1. lo3 kg/m3. The calculations have shown that thelargest difference between the velocity obtained at any lo-cation downstream of the shock if the Basset force is ne-glected or taken into account is of the order of l-2 m/setfor these particles. This corresponds to l%-2% relative tothe velocity jump across the shock and is of the order of thevelocity difference that can be resolved by a LDA. Theinfluence of the Basset force on the particle motion canthus be neglected for tracer technique applications for thetype of flow studied here. The Basset force must, on theother hand, be taken into account as is suggested by Figs.4(a) and 4(b) and further results presented in Thomas6for particies having a radius of around r,zO.5 pm and adensity of pP 5 0.1. lo3 kg/m3 as well as for particles havinga density of pPz 1.0. lo3 kg/m3 and a radius of r,SO.O25pm.

Present address: Department of Applied Mathematics and TheoreticalPhysics, University of Cambridge, Silver Street, Cambridge CB3 9EW,England.S. L. Sob, Fluid Dynamics of Multiphase Systems (Blaisdell, Waltham,MA, 1967).M. J. Walsh, Influence of particle drag coefficient on particle motion inhigh-speed flow with typical laser velocimeter appl ications, NASATech. Note TND-8120, 1976.3R. IL Nichols, Calculation of particle dynamics effects on laser veloci-meter data, in Wind Tunnel Seeding Systems for Laser Velocimeters,NASA Conf. Publ. 2393. Workshop at Langley Research Center 19-20September 1985.4R. Rr Hughes and E. R. Gilliand, The mechanics of drops, Chem.Eng. Prog. 48, 497 (1952 j.G. Tedeschi, Analyse ThCorique du comportement de particules rela-tivemst a la traverste d un onde de choc, Diplome detudes appro-fondles de mecanique des Fluides, LUniversite D Aix-Marseil le II, In- s?tut de Mechanique statistique de la Turbulence, Marseille, France,1989.6P. Thomas, Experimentelle und theoretische Untersuchungen zumFolgeverhalten von Teilchen unter dem Einfluss grosser Geschwin-digkeitsgradienten in kompressibler Stromung, Deutsche Forschung-sanstalt ftlr Luft- und Raumfahrt, Germany, Research Report No.DLR:FB 91-25, 1991.

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H. W. Press, B. P. Flannery, S. A. Tenkolsky, and W. T. Vetterling,Numerical Recipes (Cambridge U. P., Cambridge, England, 1986).C. B. Henderson, Drag coefficients of spheres in continuum and rar-efied flows, AIAA J. 14,707 (1976).9V. Krlshnan, K.-A. Biltefisch, and K.-H. Sauerland, Velocity and tur-bulence. measurements in the shock redon usine the two comoonent

Optical Methods in Flow & Particle Diagnostics, edited by W. H.Stevenson (1987).

C. dHumieres, F. Micheli, and 0. Papimyk, Etude du comportementdes Aerosols pour la Mesure en Velocimetrie Laser, in Actes du 2 BmeCongres Franc0 phone de Velocimetrie Laser, ONERA, Meudon,

laser Doppler anemometer, in Proceedings of ICALEO 1987, Vol. 63, France, 25-27 September 1990.

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