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The paper presents a numerical investigation of the 3D inviscid and incompressible swirling flow in a pipe with constant diameter. In order to evaluate the influence of different parameters (numerical schemes, grid refinement, velocity-pressure
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  Scientific Bulletin of thePolitehnica University of TimisoaraTransactions on Mechanics Special issueWorkshop onVortex Dominated Flows –Achievements and Open ProblemsTimisoara, Romania, June 10 - 11, 2005 A NUMERICAL INVESTIGATION OF THE 3D SWIRLING FLOWIN A PIPE WITH CONSTANT DIAMETER.PART 1: INVISCID COMPUTATION Sebastian MUNTEAN , Senior Researcher Center of Advanced Research in EngineeringSciencesRomanian Academy - Timisoara Branch Albert RUPRECHT , Head of Fluid MechanicsGroupInstitute of Fluid Mechanics and Hydraulic MachineryUniversity of Stuttgart Romeo SUSAN-RESIGA , Assoc. Prof.Department of Hydraulic Machinery“Politehnica” University of Timisoara * Corresponding author  : Bv. Mihai Viteazu 24, 300223, Timisoara, RomaniaTel.: (+40) 256 403692, Fax: (+40) 256 403700, Email: ABSTRACT The paper presents a numerical investigation of the3D inviscid and incompressible swirling flow in a pipe with constant diameter. In order to evaluate theinfluence of different parameters (numerical schemes,grid refinement, velocity-pressure coupling methods,outlet boundary conditions) the 3D Euler computationswere performed. The 3D computational domain corre-sponds to a pipe with constant diameter. The objectiveof this study is to assess the numerical set-up for swirling flows. In order to validate our methodologyand to estimate the accuracy of the numerical results,we have performed extensive comparisons withavailable experimental data. KEYWORDS swirling flow, 3D Euler investigation NOMENCLATURE U, V, W  [m/s]mean velocity components(axial, radial and tangential)R[m]radius R 2D  = [m]diameter L [m]length S   [-]swirl number   g  [m/s 2 ]gravity Subscripts and Superscripts r  radial direction θ  tangential direction  z  axial direction c characteristic ABBREVIATIONS in, out  inlet section, outlet section 1. INTRODUCTION In fluid dynamics research, swirling flows gainedan increasing interest in the last decade, because theswirl is an essential phenomenon for many technicalapplications. For example, cyclone separators makeuse of the body forces acting on a swirling fluid, and burners take benefit of the stabilizing effect that swirlimparts on the flow. In this instance, discrete-fre-quency noise and vibration may be undesirable sideeffects. Vibration due to excess swirl also representa problem in the draft tubes of water turbines and thefeatures of vortex flows can also adversely affect the performance of axial-radial outlet casing of axialturbines and compressors. The basic characteristicsof fluidic vortex valves, which range in size fromminiature fluidic devices to flood-dam control valves,are a consequence of the dissipative nature of vortexflows. As a result, investigation of the vortex flowdevelopment in pipes has become, in recent years,the subject of many experimental, numerical, andtheoretical studies. One of the reasons for this is theincreased scientific interest to understand the mecha-nisms that govern the stability of vortex flows andthose that lead to the vortex breakdown phenomenon.In order to find a solution, it is necessarily to bringtogether multidisciplinary teams of theoreticians,experimentalists and computationalists, in order totackle unresolved issues and enhance the understandingof the complex flow physics associated with swirlingflow phenomena.Consequently, with carefully chosen and experi-mentally validated assumptions, one can devise amethodology for computing the swirling flow, such thatvery good and engineering useful results are obtained.  Proceedings of the Workshop on VORTEX DOMINATED FLOWS. ACHIEVEMENTS AND OPEN PROBLEMS, Timisoara, Romania, June 10-11, 2005 78  Figure 1 . The experimental sections displaced along to the pipe, [2 1  ]. Figure 2. The computational domain of the pipe and the survey sections. The inlet section corresponds toexperimental section displaced at x/D=4.3 (see Figure 1  ). Other seven sections of measurements presented in Figure 1  are labelled: S  1  , S2, S3, S4, S5, S6 and S7 being displaced at 3.4, 7.2, 1 7, 23.8, 30.6, 44.2 and 7  1 .3diameters downstream relative to the inlet section. In section 2 the experimental data base is outlined.In order to validate our methodology and to assess theaccuracy of the numerical results, we performed ex-tensive comparisons with detailed experimental data.The equations and boundary conditions for Euler computation in a pipe are presented in section 3.Section 4 is devoted to investigate the numericalconditions of a swirling flow. Systematically, the solver  parameters, boundary conditions and mesh refinementhave been investigated to establish the best numericalconditions for swirling flow computation.Our best numerical results and available experi-mental data are compared in section 5. Comparisonof numerical results with available experimental datavalidates the methodology and evaluates its accuracy.The last section presents the main conclusions of this study. 2. EXPERIMENTAL DATA BASE The experimental data used for evaluation of thecomputation in this work has been provided by Dr.Wiendelt Steenbergen, Eindhoven University of Technology. The measurements are included in hisPhD Thesis [21], in which all details of experimentalsetup can be found. The experimental data is alsoincluded in the ERCOFTAC database ( Test case 72:Turbulent Pipe Flow with Swirl  ).The following experimental data are available:the distributions of mean velocities and Reynoldsstresses. Eight measurement sections are availableas follows: 4.3, 7.7, 11.5, 21.3, 28.1, 34.9, 48.5, 75.6(see Figure 1). The maximum allowed deviation of the mean velocity components from them nominalvalues was ±1%. All data are available in non-dimen-sional form: mean velocities are scaled with the bulk velocity, Reynolds stresses are scaled with the bulk velocity squared, radial positions (measured fromthe pipe axis) are scaled with the pipe radius R, whilethe axial positions of the measurement planes arescaled with the pipe diameter D. The data are expressedin a cylindrical coordinate system, with the velocitycomponents into the axial, circumferential and radialdirection denoted by U, W and V, respectively.The measurements were performed in a hydraulicallysmooth pipe with diameter  D =70 [ mm ] for three meas-urement series: the wall jet   (WJ) with initial swirlintensity S  0 ≈  0.18; the concentrated vortex  (CV)with initial swirl intensity S  0 ≈  0.18 and a solid bodywith S  0 ≈  0.05. All measurement series were performedfor Reynolds number (based on the diameter and onthe    bulk    velocity   U   =   4 . 3   [ m/s ]   and   water    cinematicviscosity) 50.000 and 300 . 000. The data were obtainedwith a 2-component laser-Doppler system. The totalvelocity vector and Reynolds stress tensor wereacquired by performing three measurements in each point, with the laser-Doppler system aligned under three different angles within the azimuthal plane, i.e.the plane perpendicular to the pipe axis.In our investigation the experimental data obtainedfor concentrated vortex with initial swirl intensity S  0 ≈  0.18 and Reynolds number 300 . 000 is used toassess the accuracy of numerical results. 3. COMPUTATIONAL DOMAIN, EQUATIONS AND BOUNDARY CONDITIONS3.1. Computational domain The 3D computational domain corresponds to a pipe with constant diameter D. The first section of measurements (x/D=4.3, see Figure 1) is used toimpose the inlet boundary conditions for the com- putation. As a result, throughout the paper the sectionx/D=4.3 is denoted inlet while other seven experimen-tal sections are as follows: 3.4 ( S  1 ), 7.2 ( S2 ), 17 ( S3 ),23.8 ( S4 ), 30.6 ( S5 ), 44.2 ( S6  ) and 71.3 ( S7  ) (diametersdownstream the inlet, see Figure 2).  Proceedings of the Workshop on VORTEX DOMINATED FLOWS. ACHIEVEMENTS AND OPEN PROBLEMS, Timisoara, Romania, June 10-11, 2005  79 3.2. Equations The governing equations for the incompressibleand inviscid flow are: ( ) momentum pt d d continuity ∇−ρ= ρ=⋅∇ gVV 0 (1)Due to the geometry of the problem, a cylindricalcoordinate system is chosen. The axial, radial andcircumferential coordinates are denoted by  x , r   and θ  ,respectively, while U  , V   and W   represent the velocitycomponents in these directions. The coordinate sys-tem and the positive velocity directions are givenin Figure 3. The equations for the mean momentumof an unsteady Euler flow in cylindrical coordinatesystem are presented in Eq. 2a-c.The method developed and presented in this paper is designed to solve the unsteady Euler swirlingflows. All simulations are carried out by applyingthe FLUENT commercial code, [Flu01].  Figure 3. Cylindrical system of coordinates, geometrical definition and velocity components.  x pU r W r U V  xU U t U  ∂∂−=     θ∂∂+∂∂+∂∂+∂∂ρ (2a) r  pr W V r W r V V  xV U t V  ∂∂−=     −θ∂∂+∂∂+∂∂+∂∂ρ (2b) θ∂∂−=     +θ∂∂+∂∂+∂∂+∂∂ρ r  pr VW W r W r W V  xW U t W  (2c) 3.3. Boundary conditions Given that a solution exists, [6] proved that a setof boundary conditions, sufficient to yield a uniquesolution of an initial value problem in a bounded for the time-dependent Euler equations, are nu ⋅ , given onthe entire domain boundary ( n  is the unit outwardnormal), and specification of the complete velocityvector where 0 <⋅ nu  (fluid enters the domain). The boundary conditions for a general  steady  Euler flow,which ensure existence and local uniqueness are notknown, [2].At the inflow section the velocity distribution is prescribed. The distribution of the circumferentialvelocity in a real swirling flow is mainly due to themethod used to generate the swirl. From a mathematical point of view, it is possible to create models to properlyapproximate the behavior of the flow field. A Rankinevortex represents a simply model for rotating flow(see Figure 4). It displays a solid body rotation corefollowed by a 1 − r   decay in the radial direction. Inapplication to a swirling flow, it is worth to note that themodel does not take into account the finite thicknessof the shear layer region at c  Rr   =  where the curvehas a singularity, Figure 4. The characteristic vortexradius c  R  measures the vortex core radial extent (seethe grey region marked in Figure 4).These two parameters define the Rankine vortexcircumferential velocity, ≥Ω≤≤Ω= ccc  Rr r  R Rr r W  ,0, (3)where r   is the radial distance from the vortex axis.This simplified model provides a continuous functionfor )( r W  , but the derivative is discontinuous.A more suitable model for developed swirling flowsis the Batchelor vortex (red solid line in Figure 4).The circumferential velocity field is described via asimilarity solution applied to wakes in the far field: ( ) [ ] 222 exp1 cc  Rr r  RW   −−Ω= .(4)Formula 4 is an exact solution for a viscous vortex produced by radial inflow and axial outflow wherethe conditions at large radial distance are irrotational.It is well known that the Rankine vortex is builtusing the asymptotic behavior of the Burgers' vortexfor large and small radius with respect to the vortexcharacteristic radius c  R , Figure 4.In Figure 5 the data as well as the curves fitted with q  vortex model Eqs. 5a-c introduced by Leibovich[13] for both tangential and axial velocity profilesare presented. The quality of the fit can be assessed byobserving that most of the time the curves approachthe experimental points within the measurement errorsof 2%. The wall boundary layer is not correctly re- produced since the q  model is specifically built for aninviscid   flow.   Table   1   shows   the   q-vortex    parametersfor the swirling flow at two Reynolds numbers 300.000and 50.000. The experimental data is imposed at theinlet section throughout this paper. However, theexperimental data is fitted with q  vortex model. ( ) 2210 /exp c  Rr U U U   −+= (5a) 0 = V  (5b) ( ) [ ] 222 /exp1 cc  Rr r  RW   −−Ω= (5c)  Proceedings of the Workshop on VORTEX DOMINATED FLOWS. ACHIEVEMENTS AND OPEN PROBLEMS, Timisoara, Romania, June 10-11, 2005 80  Figure 4. The circumferential velocity distribution. −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1r/R [−]−1−0.75−0.5−0.2500.250.50.7511.251.5   <   U  > ,  <   W  > exp. [Ste95]fit curve U W   viscouscoreboundarylayer  Figure 5. Swirling flow at pipe inlet section. The points correspond to the experimental data [2 1  ]while the solid lines represent the curve fitted with qvortex model.Table 1 . Parameters for the swirling flow at pipeinlet section at two Reynolds values. Swirl Param.Re = 300.000Re = 50.000 0 U  1.020691.00808 1 U  -0.751712-0.264299 Ω -5.91316-2.91727 c  R 0.1773080.249214The flow configuration is computed at weak swirlnumber S = 0.2 and Reynolds number Re = 300.000with the swirl number S defined as: ∫ ∫  =  R R dr rU  RUWdr r S  0202 where U    is the axial velocity component, W    the circum-ferential velocity component, and R the radius of the pipe. 4. NUMERICAL INVESTIGATIONS4.1. Interpolation schemes In this study, both velocity and pressure interpolationschemes are investigated using FLUENT code [9]. − 1  − 0.75  − 0.5  − 0.2500.250.50.751 r/R [−] − 0.75 − 0.5 − 0.2500.250.50.7511.25   <   U  > ,  <   W  >   [  −   ] <U> exp [Ste95]<W> exp [Ste95]1st order scheme2nd order scheme S291k  Figure 3a. The radial distribution of the axial <U>(  ●  ) and circumferential <W> (  ■  ) mean velocities(± 1 %) at section S2. Comparison experimental data[2 1  ] against 3D inviscid numerical results computed with different order of velocity schemes:  ─     ─    1  st   order and  ──   2 nd   order. − 1  − 0.75  − 0.5  − 0.2500.250.50.751 r/R [ − ] − 0.75 − 0.5 − 0.2500.250.50.7511.25   <   U  > ,  <   W  >   [   −    ] <U> exp [Ste95]<W> exp [Ste95]PRESTOstandard1st order2nd order S2  Figure 3b. The radial distribution of the axial <U>(  ●  ) and circumferential <W> (  ■  ) mean velocities(± 1 %) at section S2. Comparison experimental data[2 1  ] against 3D inviscid numerical results computed with different order of pressure schemes:  ──   PRESTO,  ─     ─   standard,  ─   -  1  st   order, - - - 2 nd   order. Figure 3a shows the mean velocity components onS2 section computed using first (blue long dashed line)and second (red solid line) order velocity interpolationschemes, respectively. One can see a good agreement between numerical velocity components computed withsecond order interpolation scheme against experimentaldata in free vortex region. In centre of the computa-tional domain a significant discrepancy between invis-cid numerical result and experimental data is obtained.The first order interpolation scheme predicts wellthe axial velocity component while it fails to predictthe tangential component (see Figure 3a). The tangential  Proceedings of the Workshop on VORTEX DOMINATED FLOWS. ACHIEVEMENTS AND OPEN PROBLEMS, Timisoara, Romania, June 10-11, 2005 81 distribution remains unchanged even if the 167k or 914k grids and convergence limit (10 -5 ) are considered.Therefore, the velocity first order interpolation schemefails to predict correctly the swirling flow.Further, the pressure interpolation schemes areinvestigated. Four pressure interpolation schemes areavailable when the segregated solver is used: •    standard   scheme interpolates the pressure values atthe faces using momentum equation coefficients [19]; •   first order interpolation scheme (linear scheme la- belled 1  st   order  ) computes the face pressure as theaverage of the pressure values in the adjacent cells; •   second order interpolation scheme (labelled 2 nd  order  ) reconstructs the pressure quantities at cellfaces using a multidimensional linear reconstruc-tion approach [1]; •   PREssure STaggering Option scheme (labelled  PRESTO ) uses the discrete continuity balance for a"staggered" control volume about the face tocompute the "staggered" (i.e., face) pressure [17].All pressure interpolation schemes show the samedistribution of the velocity components in free vortexregion whereas the PRESTO pressure interpolationscheme predict better the experimental data in theviscous core region, see Figure 3b.In summary, only the velocity second order inter- polation scheme predicts correct the velocity field inswirling flow while PRESTO pressure interpolationscheme reproduces better de axial velocity into theviscous core region than other pressure interpolationschemes. As a result, the second order and PRESTOinterpolation schemes are recommended for computingthe velocity and pressure fields in swirling flow. 4.2. Mesh refinement investigation In order to avoid the quadrilateral cells to be forcedtoward triangles close to the axis, a multiblock mappingis considered in the pipe cross-section, see Figure 4a.A 2D hexahedral mesh is built on the pipe cross sectionwith 26 nodes on the square edge a  and 11 nodes onthe segments  s , respectively. In this case a 3D hexa-hedral mesh is generated with 91.000 finite volumescells (labelled 91k), Figure 4b(left).A mesh refinement in the neighborhood of the axiswhere the largest gradients occur is performed. The basic idea of the adaptive mesh is refinement of thecomputational domain in those regions in which highresolution is needed to resolve developing features,while leaving less interesting parts of the domain atlower resolutions. Consequently, two adaptive meshesare generated with one refinement level RefL1 = 0.3D,Figure 4b(middle), and two refinement levels RefL1= 0.3D and RefL2 = 0.5D (see Figure 4b(right)).These grids are used for the 3D Euler simulationsand thus there is no need for grid refinement near thesolid wall since no severe velocity gradients are presentthere in inviscid simulations. The mesh parametersused to generate the adaptive mesh on the pipe cross-section are shown in Table 2.  Figure 4a. The multiblock mapping generated in the pipe cross-section. 91k167k914k  Figure 4b. The mesh generation on the pipe cross section using adaptive mesh refinement  . Table 2. Mesh parameters on the pipe cross section. MeshCross section parameterssizeasRefL1RefL291k00167k0.3D0914K 26110.3D0.5DBased on the algorithm described above two mesheswith 167k and 914k are obtained.First, the mesh refinement along the pipe is inves-tigated. Due to different L/D ratio of the consideredcomputational domain the ratio between discretisationsegment along the pipe length and the pipe diameter,denoted  D x / ∆ , is chosen as refinement parameter.Adequately, only an uniform discretisation along the pipe length is used. Table 3 presents the five valuesof  D x / ∆  ratio considered in this study along withcorrespondent mesh size. Table 3. Mesh refinement along to the pipe length. Mesh size  D x / ∆ 9k3.0018k1.5045k0.6091k0.30182K0.15Although, the  D x / ∆  ratio is modified with oneorder of magnitude from 0.15 to 1.5 no significant im- provement is observed. Apart from this rule 0.3/  =∆  D x is taken. In this case, the tangential component is
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