Harmonic balance method for FEM analysis of fluid flow in a ...

COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERINGCommun. Numer. Meth. Engng 2006; 22:347–356Published online 21 September 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/cnm.816Harmonic balance method for FEM analysis of uidow in a vibrating pipeM. Ragulskis 1; ∗; † , A. Fedaravicius 2 and K. Ragulskis 31 Department of Mathematical Research in Systems; Kaunas University of Technology, Kaunas; Lithuania2 Department of Mechanical Engineering; Kaunas University of Technology; Kaunas; Lithuania3 Lithuanian Academy of Sciences; Kaunas; LithuaniaSUMMARYA numerical procedure for the analysis of non-Newtonian uid ow in a longitudinally vibrating tube isdeveloped. The formulation of the problem is presented in dierential equation form and nite elementmodel is developed leading to the rst-order matrix dierential equation. A special modication ofthe harmonic balance procedure is proposed for this non-linear problem. Numerical validation of theharmonic balance procedure was performed by comparison of the average mass ow rate with theresults of direct time integration. Copyright ? 2005 John Wiley & Sons, Ltd.KEY WORDS:nite element method; harmonic balance; uid owINTRODUCTIONThe problem of uid ow control exploiting the vibrations of the boundary is importantin the process of design of various engineering devices and optimization of processes ofconveyance [1–4]. Such applications cover the transport of various suspensions, industrialwastes and many types of crude oils, especially at low temperatures. Analysis of such nonlineardynamical systems requires the development of adequate mathematical models andappropriate strategies for numerical modelling [5–8].Tube vibrations induced by internal or external ow is an important engineering problem [9].Nevertheless, analysis of an inverse problem—uid ow control by forced vibrations of thetube itself is also of great interest. Such a vibration-based ow control methodology couldenrich existing ow control techniques and provide background data for the development ofnew types of liquid material dosing equipment. It is assumed that the boundary (the tube) isa non-deformable rigid body.∗ Correspondence to: M. Ragulskis, Department of Mathematical Research in Systems, Kaunas University ofTechnology, Studentu 50-222, Kaunas, LT-51368, Lithuania.† E-mail: minvydas.ragulskis@ktu.LTReceived 28 June 2004Revised 5 January 2005Copyright ? 2005 John Wiley & Sons, Ltd. Accepted 27 June 2005

348 M. RAGULSKIS, A. FEDARAVICIUS AND K. RAGULSKISThe nite element formulation is used together with the approximate procedure for theanalysis of non-linear steady-state vibrations which consists from two basic steps. First anite element model is developed leading to the rst-order matrix dierential equation. Thenan approximate steady-state solution is sought using the proposed harmonic balance procedurewhich consists from the following main steps:(1) problem formulation in dierential equation form;(2) development of the nite element problem formulation;(3) modal decomposition of the solution;(4) development and application of the harmonic balance method.The harmonic balance method in general is one of the most straightforward and practicalmethods for estimating periodic steady-state solutions and can be applied to non-linear equationswith a periodic forcing term, thus providing better understanding of the reasons for thenon-linear eects taking place in the system [10].A semi-analytical method for calculating a non-Newtonian uid ow in a vibrating pipe ispresented. The method uses an ecient harmonic balance approach to evaluate steady-stateuid ow eliminating the time integration.FINITE ELEMENT MODELA Cartesian coordinate system is dened where the z-axis is parallel to the axis of the tube(Figure 1). It is assumed that the cross-section of the tube does not vary with the z coordinateand the velocity of uid ow does not depend on z, i.e. w = w(x; y; t), where w denotes thevelocity and t the time. The other components of velocity are assumed to be equal to zero.Thus the condition of incompressibility is satised identically. The stresses take the form x = y = z = −p; xy =0; yz = @w@y ;where p denotes the pressure and the viscosity of the uid. zx = @w@x(1)Figure 1. The principal model of the system.Copyright ? 2005 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng 2006; 22:347–356

HARMONIC BALANCE METHOD FOR FLUID FLOW 349The dynamic equilibrium equation in the direction of the z-axis can be written as(@ @w )+ @ ( @w )− @p + g = @w(2)@x @x @y @y @z @twhere is the density of the uid, g is the acceleration of gravity, and @p=@z the gradientof the pressure in the direction of the z-axis, is assumed constant along the tube. It is alsoassumed that the uid is non-Newtonian and that the viscosity is expressed as√ (@w ) 2 = 1 + 2 +@x( ) 2 @w(3)@ywhere 1 and 2 are constants. The non-Newtonian model given by Equation (3) presentsan approximation for the pseudoplastic model often used for the description of the non-linearfeatures of suspensions [11].The wall boundary condition− @w@n = (w − w∗ ) (4)takes into account the longitudinal vibration of the tube, where n is the outward normaldirection to the boundary of the cross-section of the ow inside the tube, is the coecientof slippage (sliding friction between the uid and the surface of the tube) and w ∗ is thevelocity of the wall in the direction of the z-axis. It is assumed that the boundary performsharmonic oscillations in the direction of the z-axis, so thatw ∗ = a sin !t (5)where a and ! denote the amplitude and the frequency of oscillations. The amplitude ofkinematic excitation of the boundary is a=!.It can be noted that, for the pipe of circular cross-section, the problem is axi-symmetricbut, bearing in mind that the results are applicable also for other tube shapes (to the tubesof other cross-sections), the 2D problem is solved. The cross-section of the ow is meshedusing the nite element approximation. The resulting matrix dierential equation:[C]{ ˙} +[K]{} = {F} (6)is obtained on the basis of the Galerkin method of weighted residuals [11, 12]. FEM matricestake the form:∫∫[C]= [N ] T [N ]dx dy∫∫[K]=∮[B] T 1 [B]dx dy + [M] T [M]ds∫∫ ({F}= [N] T∮+ [M] T w ∗ dsg − @p@z) ∫∫dx dy −√ (@w ) 2[B] T 2 +@x( ) 2 @w[B]{} dx dy@y(7)Copyright ? 2005 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng 2006; 22:347–356

350 M. RAGULSKIS, A. FEDARAVICIUS AND K. RAGULSKISwhere {} is the vector of nodal velocities. The upper dot in Equation (6) denotes dierentiationwith respect to time; s is the boundary line of the cross-section of the ow;[N]=[N 1 N 2 ··· N n ]⎡@N 1 @N 2···@x @x[B]=⎢⎣@N 1 @N 2···@y @y[M]=[M 1 M 2 ··· M m ]@N n@x@N n@y⎤⎥⎦(8)where N 1 ;N 2 ::: are the shape functions of the nite element in the cross-section of the ow,M 1 ;M 2 ::: are the shape functions of the nite element on the boundary of the cross-sectionof the ow.The nite element equations could be integrated in time and transient ow processes canbe analysed. Instead approximate methods of non-linear vibration theory are applied to thisproblem.HARMONIC BALANCE PROCEDUREFirst the eigenpairs of the homogeneous system [C]{ ˙}+[K]{} = 0 are determined and modaldecomposition of Equation (6) is performed [12]:[] T [C][]{Ż} + [] T [K][]{Z} = [] T {F} (9)where [] = [{ 1 } { 2 } :::];{ i } is the ith exponential decay eigenmode with correspondingeigenvalue i . Then the vector of velocities is expressed as⎧ ⎫z 1 ⎪⎨ ⎪⎬{} = []{Z} = [] z 2⎪⎩ ⎪⎭···where z i denote modal coecients for mode i.The determination of the averaged surface of velocities is performed by solving the followingfour successive problems.A. Steady-state harmonic solution. The sinusoidal variation of the wall velocity is assumedwhile the condition −(@p=@z)+g = 0 is being satised, and the steady-state harmonic motionof the linear system is obtained by solving the modal equations:(10)ddt (z i)+ i z i = f i (11)Copyright ? 2005 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng 2006; 22:347–356

HARMONIC BALANCE METHOD FOR FLUID FLOW 351where f i is the modal ⎧loading ⎫ for mode i. It is assumed that {F} = {F s } sin !t. Then,{F s } = ∮ ⎪⎨ f[M] T 1s ⎪⎬a ds and f2⎪⎩s ···⎪⎭ = []T {F s }. Thus f i = fis sin !t, producingz i = zi s sin !t + zi c cos !t (12)The vectors of the sine and cosine components of the steady-state motion { s } and { c } arethen expressed like:⎧ ⎫⎧ ⎫z1⎪⎨s z1⎪⎬⎪⎨c ⎪⎬{ s } = [] z2s ; { c } = [] z2c (13)⎪⎩ ⎪⎭⎪⎩ ⎪⎭······The following system of linear algebraic equations is solved for each eigenmode:[ ]{ } { }i −! zsi fsi=! i zic 0(14)B. Static solution. The static (constant in time) solution is obtained assuming that the term−(@p=@z)+g is constant in time and the wall velocity is equal to zero, that is:∫∫ ({F 0 } = [N] T g − @p )dx dy (15)@zThis requires the solution of the system of linear algebraic equations:[K]{ 0 } = {F 0 } (16)⎧ ⎫⎪⎨ f10 ⎪⎬This solution again is sought using modal decomposition. Denoting f2⎪⎩0⎪⎭ = []T {F 0 } and⎧ ⎫···⎪⎨ z10 ⎪⎬{ 0 } = [] z2⎪⎩0 , for each eigenmode we have:⎪⎭···zi 0 = f0 i(17) iUp to this point the Newtonian model of the uid is used as an approximation. The totalvelocities are:{(t)} = { 0 } + { s } sin !t + { c } cos !t (18)C. Averaged transverse velocities. The load vector is obtained in the process of assembly ofthe following loads:(a) the load occurring from the constant term −(@p=@z)+g;(b) the non-linear term due to the static solution and the harmonic motion.Copyright ? 2005 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng 2006; 22:347–356

352 M. RAGULSKIS, A. FEDARAVICIUS AND K. RAGULSKISThose calculations are performed for a number of discrete moments equally spaced duringthe period. Time average of load vector takes the form:√∫∫ ({F 1 } = [N] T g − @p ) ∫∫ (@w ) 2dx dy − [B]@zT 2 +@x⎛ √∫∫ (= [N] T g − @p )dx dy − 1 ∫∫ √√√ (m∑ ⎜@w⎝ [B] T 2@z m i=1@x( ) 2m (i − 1){× [B] { 0 } + { s } sin( ) 2 @w[B]{} dx dy@y) 2 ( ) 2@w∣ +t=ti@y ∣t=ti⎞( )}2+ { c } cosm (i − 1) dx dy⎠ (19)where the upper dash denotes time average; !t =(2=m)(i − 1);⎧@w⎪⎨ @x ∣⎫⎪t=ti⎬ {( ) ( )}2 2 =[B] {@w0 } + { s } sinm (i − 1) + { c } cosm (i − 1) ⎪⎩⎪@y ∣ ⎭t=ti(20)m is the number of discrete time moments in a period.Finally, the non-linear problem is solved obtaining the averaged surface of velocities usingpreviously described technique of modal decomposition for the solution of the system of linearalgebraic equations:[K]{ 1 } = {F 1 } (21)D. Mass ow rate. Finally, the average mass ow rate is found by integrating over thecross-sectional area:∫∫Q = w(x; y)dx dy (22)where w(x; y) are the averaged transverse velocities which are calculated from { 1 } by usingthe shape functions of the appropriate nite elements. It is assumed that the averaging intervalis much longer than the period of oscillations.NUMERICAL RESULTSCross-section of the tube is assumed to be a circle and one-fourth of it is analysed (Figure 2).The characteristics of the non-Newtonian uid represent a liquid-type suspension ( 1 =0:004g=mm s = 4 cP, 2 = − 0:0001 g=mm, =0:04 g=mm 2 s, =0:001 g=mm 3 , −(@p=@z) +g =0:003 g=mm 2 s 2 ). Radius of the tube is R = 10 mm; number of moments in the period used forCopyright ? 2005 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng 2006; 22:347–356

354 M. RAGULSKIS, A. FEDARAVICIUS AND K. RAGULSKISFigure 3. Numerical validation of the harmonic balance procedure.Figure 4. Relationship between the mass ow rate Q(g=s) and the amplitude of excitation a=! (mm) atvarious frequencies of excitation ! i =(i − 1) · 0; 2; i =1;:::;21.the cost of numerical integration over one time step is W; the calculation of the mass owrate is performed for each time step at a cost w; m time steps in a period of excitation areused; 10 periods are required to achieve steady-state motion, the cost of numerical integrationcan be approximated as 10m(W + w).Copyright ? 2005 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng 2006; 22:347–356

HARMONIC BALANCE METHOD FOR FLUID FLOW 355The costs of the stages of the harmonic balance procedure for the same problem can beapproximated:A. Steady-state harmonic solution—cost is W (comparable with one time step for the samemeshing).B. Static solution—cost is W=2 (about 50% of the cost of harmonic solution).C. Determination of the averaged transverse velocities—cost is mW=2 (m is the number ofdiscrete time moments in a period of excitation);D. Determination of the average mass ow rate—cost is w.Thus the total cost of the proposed harmonic balance procedure is ((3 + m)=2) · W + w. Forrealistic meshings w can be approximated as 0:1 W . So the cost of the proposed harmonicbalance procedure is about twenty times lower than that of numerical time integration. Numericalexperiments indicate that the proposed harmonic balance procedure produces resultswhich in some cases can dier by about 10% from the ‘exact’ ones obtained from numericalintegration.The problem for which the harmonic balance procedure was developed is typical for thetransportation of uid and therefore should be regarded as a demonstrational problem fordeveloping the algorithm. The potential of the method lies in its possible applications to thethree dimensional non-linear problems of computational uid dynamics.CONCLUSIONSNumerical model describing ow of non-Newtonian uid in a tube performing longitudinalvibrations is developed. It must be noted that this model incorporates non-linearity and representsthe dynamic behaviour of the liquid suspension. The inertia of the uid is taken intoaccount and FEM damping matrix is obtained. The velocity of the longitudinal motion ofthe walls is taken into account through the boundary condition and thus is directly incorporatedinto the nite element formulation. Harmonic balance method for determination ofaveraged transverse velocities of non-Newtonian uid is proposed. The results of the analysisshow that the longitudinal tube vibrations can be eectively applied for uid ow control andincorporated into the design of vibration spraying and dosing devices of various substances.REFERENCES1. Hui L, Tomita Y. A numerical simulation of swirling ow pneumatic conveying in a horizontal pipeline.Particulate Science and Technology: An International Journal 2000; 18(4):275–292.2. Hauser G, Sommer K. Plug ow type pneumatic conveying of abrasive products in food industry. Journal ofPowder and Bulk Solids Technology 1998; 12:511–519.3. Bubulis A, Ragulskis L. Dry substances transportation and dosing using vibrations. 41 InternationalesWissenchaftliches Kolloquium, Ilmenau, 1996; 492–496.4. Ragulskis M, Palevicius A. Theoretical and experimental studies of tubular valve dynamics. InternationalJournal of Acoustics and Vibration 2002; 7(1):53–57.5. Morgan K, Peraire J. Unstructured grid nite element methods for uid dynamics. Reports on Progress inPhysics 1998; 61(6):569–638.6. Thomasset F. Implementation of Finite Element Methods for Navier–Stokes Equations. Springer: Germany,1981; 159.7. Blawzdziewicz J, Feuillebois F. Calculation of an eective slip in a settling suspension at a vertical wall.International Journal of Fluid Mechanics Research 1995; 22(1):164–169.8. Baker AJ. Finite Element Computational Fluid Mechanics. Hemisphere PC: U.S.A., 1983; 510.Copyright ? 2005 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng 2006; 22:347–356

356 M. RAGULSKIS, A. FEDARAVICIUS AND K. RAGULSKIS9. Huang Y, Hsu C. Dynamic behaviour of tubes subjected to internal and external cross ows. Journal of Shockand Vibration 1997; 4(2):77–91.10. Jordan DW, Smith P. Nonlinear Ordinary Dierential Equations. Oxford University Press: U.K., 1999; 550.11. Rao SS. The Finite Element Method in Engineering. Pergamon Press: U.K., 1982; 625.12. Zienkiewicz OC, Morgan K. Finite Elements and Approximation. Mir: Moscow, 1986; 320.13. Hearn D, Baker MP. Computer Graphics. Prentice-Hall International: U.S.A., 1986; 351.Copyright ? 2005 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng 2006; 22:347–356