numerical investigations on a curved pipe flowpub.ist.ac.at/~jkuehnen/files/schwab11.pdfvienna...

Vienna Unversity of TechnologyInstitute of Fluid Mechanics and Heat

Transfer

Numerical Investigations on a CurvedPipe Flow

Bachelor Thesis

by

Martin Schwab

SupervisorsUniv.Prof. Dipl.-Phys. Dr.rer.nat. Hendrik C. Kuhlmann

Dr. Frank H. Muldoon, M.Sc.Dipl.Ing. Jakob Kuhnen

November 18, 2011

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Abstract. The aim of this work is the simulation of the three dimensional flowin a fluid filled torus of circular cross section at different Reynolds numbers withthe engineering simulation software Fluent. In this way a research programwhich is currently in progress at the Vienna University of Technology will besupported. A useful and reliable method for simulating a curved pipe flow shallbe developed so that different geometries can easily be investigated. The resultswill be compared to measured data gained from an experiment which has alreadybeen set up in the laboratory in order to validate the numerical model. Aftersome explanations on meshing the torus the settings and solution strategies forthe three different cases are described. Finally the results and thus the flowstructure will be illustrated.

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Contents

1. Introduction 31.1. Project description 31.2. Modelling the Torus 32. Mesh 62.1. Mesh dimensions and cell zones 62.2. Turbulent microscales 63. Case Settings 93.1. Setup for Re = 2000 93.2. Setup for Re = 3000 123.3. Setup for Re = 7000 134. Results 174.1. Re = 2000 174.2. Re = 3000 284.3. Re = 7000 395. Outlook 50Appendix A. Script for mesh generation 51References 57

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1. Introduction

1.1. Project description. Currently, a research program concerning the inves-tigation of the flow in curved pipes is in progress at the Vienna University ofTechnology. An experiment has already been set up in order to gain measureddata of the flow’s structure. This work will support the research by delivering areliable numerical model in order to be able to investigate other geometries withoutthe need of an experimental set up. Before describing the details of this work theaims of the reasearch project shall be statet. The following passage is extractedfrom [Kuhlmann et al., 2011].

The main objective is to reveal the three dimensional structure of the transi-tional flow through a curved pipe represented by a torus of circular cross section.The transition properties of the flow will be compared to those in straight circularpipes. Furthermore the role of Dean-flow instabilities on the transition scenariowill be investigated and measurements of the torque required to maintain constantspeed shall provide some information on the power dissipation in the torus. Fi-nally, mechanisms which could permit control of the onset of turbulence shall beidentified. This investigation is not only of theoretical relevance since most pipingsystems are curved, be it in the field of engineering or in nature, for example bloodvessels.

1.2. Modelling the Torus. In the experiment the torus consisting of plexiglass isfilled with refractive index matched water so that optical measurement techniquescan be applied. The flow is driven by a box with a metal sphere within. The boxexactly fits into a section of the pipe and is driven through a magnet from outside.The schematic setup is shown in figure 1.

Figure 1. Torus with box

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For the numerical investigations the test set up is modelled as a torus sectionwith plane surfaces at the ends. These surfaces correspond to the box which drivesthe flow. Therefore a rotating reference system connected to these surfaces willbe used to describe the flow variables whereas its origin is located in the torus’center of curvature. The torus is meshed in Gambit by using a script writtenby Marica Vlad (see appendix A, [Marica, 2011]). The whole volume consists ofhexahedral elements meshed with a cooper scheme. The geometry of the toroidalpipe section is explained in figure 2. As one can see, the box is modelled as a gapof 15 degrees in the pipe section. Since the geometry of the pipe is the same as inthe experiment the results can be directly compared.

Figure 2. Mesh geometry. R = 307mm; d = 30.3mm; The rotatingreference system is shown in blue.

Aditionally the results shall be compared to the ones from other computa-tions later on, especially to [Huttl et al., 1999], [Huttl and Friedrich, 2000] and[Huttl and Friedrich, 2001]. Therefore the curvature of the torus is of interest. Asits axis is a circle the curvature κ is constant.

(1) κ :=1

R

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The results in these three articles are presented in a dimensionless form withd/2 as the length scaling variable. Accordingly, the dimensionless curvature is

(2) κ :=d

2R= 0.04935

The numerical simulation is done with Fluent for Reynolds numbers (see chap-ter 3.1) of 2000, 3000 and 7000. Figure 3 shows the mesh at one end of the pipe.

Figure 3. Mesh

More details on the mesh and cell dimensions as well as the simulation settingswill be given in chapter 3. Finally the results will be presented in chapter 4.

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2. Mesh

2.1. Mesh dimensions and cell zones. Let’s take a closer look at the cells ofthe mesh. Using the settings described in appendix A it consists of 272832 cellsand 283650 nodes. The largest cell dimension is about 3mm. There are severalcell zones defined during the mesh generation in Gambit. The two cross sectionsat the ends of the pipe are defined as boundary zone and called ”box” in thesimulation. The surface of the torus is also defined as boundary zone and called”wall”. Finally the volume is defined as fluid.

2.2. Turbulent microscales. In order to get an overview of what the mesh isable to resolve we can take a look at the turbulent microscales. They will beimportant to if turbulent structures shall be resolved later on as well.

In a turbulent fluid flow a large number of eddies exist over a wide range of lengthscales where the energy is transported from the largest scales to the smallest onesuntil it is finally dissipated. This mechanism is known as energy cascade. Theoccuring length scales can be divided into several areas. The largest scales in theenergy spectrum are the integral length scales where the energy is passed downto smaller scales. The Kolmogorov length scales are the smallest scales in thespectrum where the energy is dissipated. Finally there are the Taylor microscales.They mark the border between the integral scales and those where the energy isdissipated (see [Glasgow, 2010] and [Durbin and Pettersson-Reif, 2011]).

In order to determine what scales the mesh is able to resolve the Taylor mi-croscale for the torus is calculated. The simulation of turbulent structures will berun at a Reynolds number of Re = 7000, which is defined as follows.

(3) Red :=ud

ν

Here, d describes the diameter of the pipe, ν the kinematic viscosity an u thevelocity of the fluid flow. The subscript d shall indicate that the Reynolds numberis built using the diameter as characteristic length scale.

The Taylor microscale can be determined with the following equation, which isextracted from [Glasgow, 2010].

(4)λ

d=

√15

ARe

− 12

d

λ represents the Taylor microscale, d is the diameter of the pipe and the constantA is of the order of 1. Working this out gives a Taylor length scale of λ = 1.4mm.

Comparing the largest cell dimension to the Taylor length scale one will noticethat the mesh is able to resolve the smallest eddies where energy is not yet dis-sipated. It should be mentioned that equation 4 can only be applied to isotropicturbulence so the calulated scale should only be seen as a reference.

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The Kolmogorov microscale can be calculated with the following relations,

(5) ε ≈ Au3

dη = (

ν3

ε)

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where ε stands for the dissipation rate per unit mass and η for the Kolmogorovmicroscale. They can also be found in [Glasgow, 2010]. If the kinematic viscosityof the matched water ν23◦C = 1.894 · 10−6m2/s is used the dissipation rate yieldsabout 2.76m2/s3 and the Kolmogorov microscale is about 0.04mm. Consideringthe mesh’s cell dimensions it is obvious that the mesh is way too coarse to perform adirect numerical simulation but the resolution will be high enough for the intendedcalculations. However those length scales will be of importance if a large eddysimulation or even a DNS shall be done because there cannot be seen any sign ofturbulence if the mesh is too coarse.

There also exists the Kolmogorov time scale which is defined as follows.

(6) τη = (ν

ε)

12

In our case τη = 0.00083s.

Figure 4. A cross section of the torus mesh

Figure 4 shows a cross section of the torus mesh. The cells with the largest sidelength are outside at the perimeter of the profile with a width of 2.97mm.

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Figure 5. Cell geometry of the torus mesh

Figure 5 illustrates how the cross section (figure 4) is revolved thorugh the torusvolume so that the cells are formed. Every 3mm a new cross section is inserted inthe volume.

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3. Case Settings

This chapter is about how the simulation is conducted in Fluent. After themesh has been created in Gambit with Vlad Marica’s script it is imported inFluent. Now the problem has to be defined and the solution methods have tobe set.

3.1. Setup for Re = 2000. Before any settings are defined it is interesting toknow if the flow is laminar or turbulent. Therefore the Dean number can becalculated. It is defined as

(7) De :=ud

ν

√d

2R= Red

√κ

where u is the axial velocity, d the diameter of the pipe and R the radius ofcurvature. In the case of Re = 2000 the Dean number is De = 444.3. Accordingto [Dennis and Ng, 1982] the flow can be laminar up to a critical Dean numberof Dec ≈ 956. Laminar flow has also been observed up to a Reynolds number ofabout Re ≈ 3800 in the experiment. Since there is no time dependence in thelaminar flow it is possible to calculate a steady solution.

Table 1 below lists the settings which are done to define the problem.

Table 1. Parameters to define the problem in Fluent

Problem setup

General

Mesh Scale Mesh created in mm

Solver Type Pressure - based

Time Steady

Vel. formulation Absolute

Gravity On y = -9.81 m/s2

Models Viscous Laminar -

MaterialsFluid mat. Water 23◦C ρ = 1138.43kg/m3

µ = 2.1562 · 10−3Pa · sSolid Plexiglass ρ = 1190kg/m3

Cell Zone Cond. Frame motion rot. ref. frame ω = 0.4072rad/s

Boundary Conditions Wall no Slip ω = −0.4072rad/s

Box no Slip ω = 0rad/s

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The angular velocity specified for the boundary conditions is relative to the ad-jacent cell zone, in our case relative to the rotating reference frame. It is calculatedusing the Reynods number of Re = 2000 with an average velocity in the middleof the pipe. The angular velocity is defined as

(8) ω := u/R

where R is the radius of the torus. Taking the equations 3 and 8 the angularvelocity necessary for achieving a Reynolds number of Re = 2000 can be calculated.

(9) ω =Re · νR · d

=2000 · 1.894 · 10−6

0.307 · 0.0303≈ 0.4072rad/s

As can be seen in table 1 the laminar viscous model is selected. This meansthat the Navier Stokes equations are directly solved without using any turbulencemodel.

The next step is to set the solution methods and other calculation specific set-tings which are listed in table 2.

Table 2. Calculation specific settings in Fluent

Solution

Methods

Press.-vel. coupling Scheme PISO

Spatial discret. Gradient Least Sq. cell b.

Pressure Second order

Momentum Third order

Monitors Residuals All set to 10−3 Abs. conv. crit.

Calculation Act. Autosave Every 50 iterations -

Solution Init. Standard Absolute All set to 0

Run Calulation Iterations 1800 -

Although the coupled scheme for the pressure velocity coupling would be moreprecise the PISO scheme was selected. It is a good compromise between accuracyand computational effort. The solution is initialized with all velocities and thepressure set to zero.

When the scaled residuals of the three velocity componets and the continuityequation are smaller than 10−3 the solution is considered to be converged. Duringthe calculation the scaled residuals of the velocities were below 10−4 and the oneof the continuity equation reached convergence at about 2 · 10−3 as can be seen infigure 6.

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Figure 6. Scaled Residuals

In order to get an idea of the error in the results one can have a closer lookat the definition of the scaled residuals in Fluent. Generally the residual of afunction

(10) f(x) := b

with an aproximation x0 of x is defined as

(11) R := b− f(x0)

Since the exact solution x is not known we can not calculate the error but inmany cases the error is proportional to the size of the residual. This means thatthe residual can be used as a reference value for the error.

Let’s consider the residual’s definition in Fluent which is extracted from theUser’s Guide. The conservation equation for a general variable φ at a cell P canbe written as

(12) aPφP =∑nb

anbφnb + b

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where nb describes the neighboring cells. The unscaled residual Rφ computedby Fluent’s pressure based solver is the imbalance in equation 12 summed overall cells P.

(13) Rφ =∑cells P

|∑nb

anbφnb + b− aPφP |

Finally the scaled residual is

(14) Rφsc =

∑cells P

|∑nb

anbφnb + b− aPφP |∑cells P

|aPφP |

For the momentum equations the term aPφP in the denominator is replaced byaPuP with uP as the velocity magnitude at cell P.

Taking equation 14 the average error of φ at one cell can be estimated. Theaverage azimuthal velocity at a Reynolds number of Re = 2000 is u = 0.125m/sand the scaled residual for the x and z velocity is about 8 · 10−5 as can be seen infigure 6. It can be assumed that the coefficient aP is of the order ofO(1). Thereforeyou get a residual of Ru = 1 · 10−5. Taking into account that the residual has thesame scale as the error one can claim that there is a variation in the fifth decimalplace between the calculated velocity and the exact solution.

The simulation was conducted on the CAE Cluster of the Vienna University ofTechnology using 4 processors. The calculation of 1800 iterations took about onehour.

3.2. Setup for Re = 3000. Since there has not been any sign of turbulence at aReynolds number of Re = 3000 in the experiment we do not expect that it occursin the simulation. Also the Dean number of De = 666.4 is below the ciritcal value.Due to this reason a steady solution is calculated.

The problem setup has not changed except the cell zone and boundary conditionsbecause the box is rotating faster now. The calculation specific settings haven’tchanged either. Table 3 shows the changed settings compared to the case of Re =2000 above. All other settings can be read in tables 1 and 2.

Table 3. Parameters to define the problem in Fluent

Problem setup




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Figure 7. Scaled Residuals

As can be seen in figure 7 the solution converged after about 1000 iterations.However 3000 iterations were made because the convergence criterion for the con-tinuity equation defined in the settings has not been reached at all. Its scaledresidual is at about 9 · 10−3 whereas the velocities’ scaled residuals are slightlybelow 2 · 10−4.

According to the considerations about the estimated average error above theresidual of Ru = 3.75·10−5 for the velocities can be evaluated by taking equation 14and multiplying with the average velocity of u = ωR = 0.1875m/s. Therefore itcan be assumed that the variation between the calculated and the exact solutionin of the order of O(10−5).

The calculation time per iteration has not changed compared to the case ofRe = 2000 since there have not been any major changes.

3.3. Setup for Re = 7000. Before considering the simulation settings it is againa good idea to check the expected flow state. The Dean number for the case ofRe = 7000 and the torus geometry is De = 1555 which is fairly above the criticalvalue. Also the experiment indicates that the fluid flow is going to be turbulentat this Reynolds number.

As a first step we want to get some information about the mean flow. There-fore the k − ε turbulence model based on the Reynolds averaged Navier Stokesequations (RANS) is chosen. In order to prove if it is applicable in our case theresults will be compared to the data gained from the experiment later on. Sincethis turbulence model does only calculate the mean flow there is still no need to

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compute a transient solution but there are two more equations to be solved besidesthe continuity and the Navier Stokes equations. These are the transport equationfor the turbulent kinetic energy k and one for the rate of dissipation ε.

Table 4 lists the settings for defining the problem in Fluent. As in chapter 3.1,the angular velocities specified for the boundary conditions are relative to therotating reference frame.

Table 4. Parameters to define the problem in Fluent using thek − ε model

Problem setup

General

Mesh Scale Mesh created in mm

Solver Type Pressure - based

Time Steady

Vel. formulation Absolute

Gravity On y = -9.81 m/s2

Models Viscous realizeable k − ε enh. wall treat.

C2ε = 1.9 TKE = 1

TDR = 1.2

MaterialsFluid mat. Water 23◦C ρ = 1138.43kg/m3

µ = 2.1562 · 10−3Pa · sSolid Plexiglass ρ = 1190kg/m3




All calculation specific settings haven’t changed so far but there are two newvariables, the turbulent kinetic energy and the dissipation rate, which have to bediscretized. Table 5 lists the additions compared to chapter 3.1.

The initial condition of the dissipation rate is chosen according to chapter 2.2where its value has been calculated for the average flow speed of u = 0.4376m/sat a Reynold number of Re = 7000 to ε = 2.76m2/s3. As can be see in figure 8the solution seems to be converged after about 800 iterations. The computationlasted about one hour whereas four processors were used.

Since the k − ε turbulence model does also have some disadvantages the moreaccurate Reynolds Stress model will be used for a second simulation. This model

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Table 5. Calculation specific settings in Fluent using the k − ε model

Solution

MethodsSpatial discret. Turb. kin. energy Second ord. up.

Diss. rate Second ord. up.

Calculation Act. Autosave Every 100 iterations -

Solution Init.Standard Absolute -

k = 1 ε = 2


Figure 8. Scaled Residuals using the k − ε model

does also calculate the mean flow and therefore solve the RANS equations with theadvantage of the ability to render anisotropic turbulence. Six equations have tobe solved for the Reynolds stresses and one for the dissipation additionally whichincreases the computational effort. Thus it is reasonable to use the solution gainedfrom the first calculation as initial value in order to achieve faster convergence.The computation lasted about half an hour using 4 processors.

The peak in figure 9 describes the point where the computation using theReynolds stress model is started.

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Table 6. Changes for setting up the Reynolds Stress model com-pared to table 4

Problem setup

Models

Viscous Reynolds Stress quad. pre. strain

Cmu = 0.09 C1ε = 1.44

C2ε = 1.83 C1-SSG-PS = 3.4

C1’-SSG-PS = 1.8 C2-SSG-PS = 0.8

C3-SSG-PS = 0.8 C3’-SSG-PS = 1.3

C4-SSG-PS = 1.25 C5-SSG-PS = 0.4

TKE = 1 TDR = 1.3

Table 7. Changes in the calculation specific settings in Fluentusing the Reynolds Stress model compared to table 5

Solution

Methods Spatial discret. Reynolds stresses Second ord. up.


Figure 9. Scaled Residuals using the Reynolds stress model

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4. Results

After the settings have been discussed, the results are finally presented sepa-rately for each case. They will also be compared to data gained from the exper-iment in order to validate the numerical model. This data consists of velocityprofiles along the horizontal axis for the azimuthal and radial velocity componentsmeasured at different heights of the cross section. Figure 10 illustrates where thevelocity profiles are measured.

Figure 10. Exemplary positions where velocity profiles are mea-sured in the experiment

The azimuthal velocity component points out of the plane. The y - coordinatedescribes the different heights where the profiles are obtained. The coordinatesystem’s origin is in the torus’ center of curvature.

As already mentioned in chapter 3 an absolute velocity formulation is used. Thismeans that the velocities in the figures presented in the following chapters are withreference to a stationary system.

4.1. Re = 2000. At first we want to get an overview of the flow structure. There-fore a plane in the middle of the pipe, at y = 0mm according to figure 10, is cutout of the torus. It is a good idea to plot the contours of the velocity magnitudein this plane to check if the solution looks reasonable.

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Before having a look at the plot the expected average velocity magnitude can becalculated by taking the equations 8 and 9. This yields a value of |u|av = 0.125m/s.Considering a laminar flow in a straight pipe the relation

(15) |uav| =|umax|

2is valid. Having this in mind one will expect to observe a maximum velocity

of a little less than twice as much as the average velocity in a curved pipe withlaminar flow.

Figure 11. Contours of velocity magnitude in m/s at a plane inthe middle of the pipe.

As can be seen in figure 11 the relation 15 holds true approximately. The boxwhich drives the flow is rotating counter clockwise. One may also notices thatthe velocity maximum is not in the center of the cross section as it would be forstraight pipes. It is shifted towards the outer region due to the centrifugal force.Besides some small perturbations emanating from the box surface, there cannot

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be observed any dependencies of the flow in the angular direction. For that reasonit will be sufficient to consider one cross section at some distance to the box forthe further investigation of the flow structure. Everything is what we expect it tobe and the solution looks reasonable so far.

Let us have a look at a vector plot of the same plane next. Since we have onlyseen a plot of the velocity magnitude it will be interesting to know the direction ofthe velocity too. Additionally one can get a first impression of the velocity profilein the horizontal direction.

Figure 12. Velocity vectors colored by velocity magnitude in m/s.

As far as it can be seen in figure 12 the velocity vectors are parallel and pointalong the azimuthal direction. There is a small velocity gradient from the innerside to the maximum and a big one to the outer side. The velocity at the wall iszero so the no slip boundary condition is fulfilled. It can be seen very clearly thatthe flow is laminar.

Now we will have a closer look at a cross section somewhere in the middle ofthe pipe with enough distance to the box. This view will reveal the flow structurealong the vertical y - coordinate. Another thing what we want to investigate isthe occuring secondary flow perpendicular to the mean flow. The pipe’s curvatureinduces a centrifugal force proportional to the flow speed squared. It causes the

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flow being accelerated outwards where its speed is high. Since the continuity hasto be fulfilled the flow is forced inwards where the speed is lower.

Figure 13 shows the behaviour of the secondary flow. It can be seen very clearlythat the flow moves outwards in the center, then bifurcates at the outer wall andmoves back inwards along the circumference. This is exactly what is describedabove if the velocity magnitude at the different regions is considered.

Figure 13. Contours of velocity magnitude in m/s in a cross sec-tion of the pipe. The vectors illustrate the secondary flow. Thecenter of curvature is towards the positive x direction.

The two symmetric recirculation cells are called Dean cells. The results of[Huttl et al., 1999] look quite similar to figure 13 if you take into considerationthat there is used a bulk Reynolds number of Reb = 1000 and a curvature ofκ = 0.01, where the bulk Reynolds number is defined in the same way as it is forthis computation. Only the vector plot of the secondary flow is a little different

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where the biggest difference is in the region near the upper and lower wall. Thepattern of this solution seems to be more realistic since the inwards flow is closerto the wall and therefore at a region of lower axial velocity. The shape of thecontour lines changes depending on the curvature.

In the case of a straight pipe the pressure is only a function of the coordinatein axial direction. For curved pipes the developing centrifugal force has to bebalanced by a pressure gradient prependicular to the pipe axis. This is whatfigure 14 illustrates. It is quite similar to [Huttl et al., 1999] again.

Figure 14. Contours of static pressure in Pascal in a cross sec-tion of the pipe. The center of curvature is towards the positive xdirection.

For the further investigation of the flow we will plot the axial and radial velocitycomponents along horizontal and vertical lines in a cross section. Since a dimen-sionless description is often used there will be given some popular scaling factorsfor the velocity in order to make the presented dimensioned profiles comparable.

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One of these scaling factors is the bulk velocity ub. It is the mean flow velocitywhich is easy to calculate in our case since it is equal to the speed of the rotatingbox.

(16) ub = ωR

In equation 16 R represents the radius of the torus axis and ω is the angularvelocity of the box. It gives a value of ub = 0.125m/s.

Another possible scaling factor is the mean friction velocity uτ which is used in[Huttl et al., 1999]. It is defined as

(17) uτ =

√τw,mρ

τw,m =1

2π

∫ 2π

0

τw(θ) dθ

where τw,m is the mean wall shear stress. Figure 15 shows the wall shear stressin dependence of the radial position. It has the same value along the upper andlower half of the circumference.

Figure 15. Wall shear stress in a cross section of the pipe.

The mean wall shear stress gives a value of τw,m = 0.185454Pa. Using thedensity listed in table 1 the mean friction velocity can be calculated. It is uτ =0.012763m/s.

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Figure 16. Horizontal profiles of the axial velocity component atdifferent heights of the cross section. The blue line describes thecomputed values whereas the green crosses represent the PIV mea-sured values from the experiment ([Dusek, 2011]).

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Figure 17. See figure 16 for a description.

25


The figures 16, 17 and 18 show a comparison of the axial horizontal velocityprofiles between the computed results of this work (blue line) and measured valuesthrough particle imaging velocimetry (green crosses) at different heights of the pipecross section according to figure 10. The measured data has been extracted from[Dusek, 2011]. It can be seen that the measured axial velocity is slightly smallerthan the computed one in the center of the cross section at almost all heights. Theonly exception is the near wall region where the measured velocity is larger. Thereis quite good conformity at y = 2mm and y = 5mm whereas the gap gets a littlelarger towards the upper wall.

A reason for the larger computed velocity in the middle might be the idealiza-tions made in the numerical model. For example the flow cannot pass the boxsince it is tightly connected to the wall. This is not true in reality because the boxwill not be able to move then. Therefore there will be a little gap between thebox’s circumference and the wall where a small mass flow passes which results ina lack of velocity. This circumstance would explain the differences very well con-sidering that this additional mass flow means a loss of energy for the main flow.Nevertheless there is a higher measured velocity close to the upper wall (figure 18)which indicates that there may be some other reasons for the variations too.

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All in all there is a conformity in the axial horizontal velocity profiles between thenumerical simulation and the experiment with some minor differences. In orderto make the results comparable to other computations the two scaling factorsmentioned in equation 16 and 17 can be used to non dimensionalize the velocity.This is illustrated in equation 18 by using the maximum velocity.

(18)umaxub

=0.218

0.125= 1.744

umaxuτ

=0.218

0.012763= 17.08

In the following the radial velocity component will be plotted along a horizon-tal line at different heights in the same way as it has been done with the axialcomponent.

Figure 19. Horizontal profiles of the radial velocity component atdifferent heights of the cross section. The blue line describes thecomputed values whereas the green crosses represent the PIV mea-sured values from the experiment ([Dusek, 2011]). Positive valuesmean a flow direction towards the center of curvature. The veloc-ity’s scale is 10−3.

The figures 19 and 20 show a comparison of the computed results to measuredvalues. It can be seen that the values are way smaller than they are for theaxial velocity. Therefore it will only be reasonable to do a comparison at regions

27


where the radial velocity is larger. For that reason the plots are made alonghorizontal lines in the upper region of the cross section at y = 8mm and y = 11mmbecause the radial velocity component is even smaller in the center. A conformityis basically given but the relative variation is a little larger here, which is againcaused due to the assumptions made in the numerical model. Additionally theabsolute values are very small which could also be a reason for the larger relativevariation. A more precise simulation can only be done by modifying the numericalmodel so that the leackage through the gap between the box’s circumference andthe wall and also the that one of the test set-up is considered.

Another good view for the investigation of the secondary flow is a plot of theradial velocity component along a vertical line which cuts the cross section in themiddle. This is what is illustrated in figure 21. It can be seen that the flowmoves inwards in a small region near the upper and lower wall whereas it is forcedoutwards in a large region in the center. Since the continuity equation shall befulfilled the overall mass flow along the radial direction has to be zero. Thereforethe flow velocity is higher in those small areas where it moves inwards. Thisbehaviour has already been shown in figure 13 qualitatively.

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Figure 21. Radial velocity component along a vertical line in themiddle of the cross section. Positive values for the radial velocitymean a flow direction towards the center of curvature.

The last thing which has not been investigated yet is the vertical profile of theaxial velocity. As with the horizontal profile its shape has a strong dependenceon the curvature so it will not be parabolic like that for straight pipes. Figure 22shows this profile. Its shape is quite equal to the one that has been computed in[Huttl et al., 1999].

It can be seen that there is a constant velocity in the center area and two peaksin the near wall region. The values are slightly above the bulk velocity which isub = 0.125m/s and thus way smaller than they would be for a straight pipe flow.As already explained above this behaviour is caused by the centrifugal force whichis proportional to the flow speed squared. Therefore those areas with a highervelocity are forced outwards whereas areas with a smaller velocity stay closer tothe center of curvature. This can also be seen in the figures 13 and 16.

4.2. Re = 3000. At this Reynolds number the flow state has not changed verymuch compared to the case of Re = 2000. The Dean number has increased alittle bit but the flow is still laminar. Because of this it is reasonable to expectresults with an almost similar shape as in chapter 4.1. Therefore the comments onthe figures in this chapter will be kept short since the descriptions of the previous

29

Figure 22. Axial velocity component along a vertical line in themiddle of the cross section.

chapter are still valid. In order to visualize the flow structure several contour plotswill be presented in the same way as it has been done in the previous section.

As expected the contours of the velocity magnitude in the x-z plane at y = 0mm(figure 23) show a laminar behaviour with a maximum velocity shifted away fromthe center to the outer wall. The bulk velocity for this case is ub = 0.1875m/saccording to equation 16, where the value of the angular velocity has changed.Considering the maximum velocity magnitude of about |umax| ≈ 0.32 one will geta ratio of

(19)umaxub

=0.32

0.1875= 1.7

which is a reasonable value for a curved pipe flow. Figure 23 looks almost equalto figure 11 beside different values for the velocities.

Figure 24 shows the velocity vectors in a section of the plane of figure 23. Therecan be seen a small velocity gradient form the inner wall to the maximum velocityand a large one from there to the outer wall. The no slip boundary condition seemsto be fulfilled too. Furthermore the laminar structure can be seen very clearly.

The next plot (figure 25) illustrates the velocity distribution in a cross section ofthe pipe. The contours represent the velocity magnitude whereas the vectors arethe in plane components of the velocity and thus visualizing the secondary flow.

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The occurence of the secondary flow has already been explained in chapter 4.1 andshall not be repeated here since it is caused by the same mechanisms. There aretwo recirculating Dean cells again where the flow moves inwards near the upper

31

and lower wall and is forced outwards in the center. This is exactly what we wouldexpect for a curved pipe flow at this Reynolds number.

Figure 25. Contours of velocity magnitude in m/s in a cross sec-tion of the pipe. The vectors illustrate the secondary flow. Thecenter of curvature is towards the positive x direction.

One can get a good imagination of the effect of the centrifugal force by havinga look at the distribution of the static pressure in a cross section because they arebalancing each other. This is what figure 26 illustrates. Again, it is quite equal tofigure 14 but the value of the maximum static pressure is almost twice as much asbefore.

As expected, the results have not changed very much compared to the case ofRe = 2000 and also look quite equal to the one presented in [Huttl et al., 1999].Thus it can be claimed that the computed solution is physically reasonable onceagain.

32

Figure 26. Contours of static pressure in Pascal in a cross sec-tion of the pipe. The center of curvature is towards the positive xdirection.

The next step in order to validate the numerical model for this case is a compar-ison of the computed solution with measured data. This is done by superimposingseveral velocity profiles for the axial and radial velocity component at differentregions of the cross section. Before the presentation of the actual plots the wallshear stress and the mean friction velocity shall be shown so that the profiles arecomparable to other computations.

Figure 27 presents the wall shear stress in dependence of the radial position.There is an almost linear increase towards the outermost point. The mean wallshear stress according to equation 17 gives τw,m = 0.326796Pa and the meanfriction veocity is uτ = 0.016943m/s. Therefore the non dimensional maximumvelocity will be

33

(20)umaxuτ

=0.32

0.016943= 18.887

These scaling factors will also be used later on when the velocity profiles forRe = 2000 and Re = 3000 are compared in order to determine the influence of theDean number on their distortion.

Figure 27. Wall shear stress in a cross section of the pipe.

The figures 28, 29 and 30 below show a comparison of velocity profiles measuredthrough particle imaging velocimetry (from [Dusek, 2011]) and the computed ones.They are evaluated at different heights according to figure 10. Anyhow someassumptions have been made which are not valid in reality but it is expected thattheir effect on the solution will be small (see chapter 4.1).

There is a good conformity of the horizontal axial velocity profiles at all heightswith some minor differences, similar to the case of Re = 2000. It is remarkablethat the maximum velocity in the experiment does not reach the computed one.You can see that the buckling near the inner wall in figure 28 is at a lower velocityin the experiment as well. However these differences can partially be explained bythe assumptions made in the numerical model. The lack of velocity can arise dueto additional mass flows because the main flow does not get the same energy then.Nevertheless it shall be mentioned that some inaccuracies can be induced by the

34


35


36


mesh too. Thus you will probably get a more precise solution by using an evenfiner grid.

Figure 31 shows the horizontal profile of the radial velocity component at twodifferent heights. A positive value of the radial velocity means that the flow movestowards the center of curvature. You can see that the velocity magnitude is aboutten times larger at y = 11mm compared to the values at y = 6mm. Thereforeone can imagine that the relative variation between the experimental data and thesimulation will be higher when the absolute values get very small.

Besides those relatively larger variations for small values a qualitative conformityis given. Nevertheless it seems that the peaks near the outer and inner wall(figure 31 below) are not completely resolved in the measured data. These peaksrepresent the small area of the recirculating cell where the flow moves back inwards.

For the sake of completeness there shall be given two vertical velocity profiles aswell. Figure 32 shows the radial component thus representing the secondary flowin another way.

Finally figure 33 illustrates the vertical distribution of the axial velocity com-ponent. The shape of both plots is almost equal to that of figure 21 and 22.

37

Figure 31. Horizontal profiles of the radial velocity componentat different heights of the cross section. The blue line describesthe computed values whereas the green crosses represent the PIVmeasured values from the experiment ([Dusek, 2011]).

38

Figure 32. Radial velocity component along a vertical line in themiddle of the cross section. Positive values for the radial velocitymean a flow direction towards the center of curvature.

Since the Dean number is the determining parameter in the system of a curvedpipe flow we want to investigate how the velocity profile has changed due to theincrease in speed compared to the previous chapter. Let us recall equation 7. Dewill change its value due to a change of curvature or a change of the Reynoldsnumber, which is what we have done. The Dean number in the case of Re = 2000is De1 = 444.3 whereas it is De2 = 666.4 in the case of Re = 3000.

In order to make the velocity profiles comparable they have to be made nondimensional first. Therefore we will use the bulk velocity ub as the characteristicvelocity. This means that the values of the velocities will be divided by ub1 =0.125m/s and ub2 = 0.1875m/s. The comparison of the profiles itself can be seenin figure 34 for a horizontal and a vertical profile of the axial velocity. They areplotted along the horizontal and vertical symmetry lines of the cross section.

Let us consider the upper plot first. The increase of the Dean number forces themaximum velocity to move outwards and does also reduce it a little bit. Further-more the bending of the profile has increased too.

39

Figure 33. Axial velocity component along a vertical line in themiddle of the cross section.

The second plot shows that the peaks move outwards with a slightly decreasingmaximum too. There is a local minimum developing in the transition area fromthe linear range in the center to the peaks in the near wall region. This behaviourhas also been observed in the computations of [Huttl et al., 1999].

4.3. Re = 7000. As already indicated in chapter 3.3 we expect the flow to beturbulent at this Reynolds number. For that reason two turbulence models havebeen selected and will be compared to the experimental data in order to prove ifthey are applicable to this problem. These models calculate the mean flow withconsideration of turbulent effects but do not resolve any vortices. Since theseeffects are only modeled and not directly calculated, like in a direct numericalsimulation, some variations between the experiment and the computed solutionhave to be expected. There are even differences between the results of the twomodels as illustrated in figure 35.

It shows the horizontal profiles of the axial velocity component along a cutat a height of y = 2mm according to figure 10. At this point it is a goodidea to reflect which profile is more reasonbale considering a curved pipe flow

40

Figure 34. Comparison of the horizontal and vertical profile of theaxial velocity for two different Dean numbers.

41

Figure 35. Comparison between the results of the k − ε and theReynolds Stress model.

at this condition. One can have a look at other computations, for example[Huttl and Friedrich, 2000] and [Huttl and Friedrich, 2001] and check what theprofiles look like there. In this case the Reynolds stress model delivers a morerealistic solution which makes sense too having in mind that it can render turbu-lence more accurate. It seems that the k− ε model is not able to resolve the effectsof curvature completely. For that reason the further investigation of the flow willbe done by using the results of the simulation based on the Reynolds stress model.

Before the comparison to measured velocity profiles the three dimensional struc-ture of the flow shall be visualized by some plots. Figure 36 shows the contours ofthe velocity magnitude in a plane at y = 0mm. Although it looks almost similarto the figures 11 and 23 it should be mentioned once again that the mean flow iscalculated and therefore no vortices are resolved.

Beside some small perturbations close to the box there is no dependence of thesolution in the axial direction. The maximum velocity in the non perturbed areais approximately umax ≈ 0.614m/s and the bulk velocity gives ub = 0.4376m/s.This gives a ratio of

42


(21)umaxub

=0.614

0.4376= 1.4

which is fairly below the one of the previous chapters.The velocity vectors in the x-z plane at y = 0mm are illustrated in figure 37.

The boundary condition is fulfilled and the structure looks reasonable so far.Figure 38 top shows the velocity distribution in a cross section of the pipe

superimposed by the velocity vectors of the secondary flow. The shape of thecontours has changed quite a lot compared to the laminar case. The peaks near theupper and lower wall are way smoother now and the gradients are a little smaller.Also the secondary flow’s structure has changed. You can see that the center ofthe recirculating cells has moved a little bit towards the center of curvature. Thisbehaviour can also be found in the solution of a direct numerical simulation in[Huttl and Friedrich, 2001] although the contour lines of the mean axial velocity

43


component look a little different there. They have the shape of the ones calculatedfor the laminar case.

This is a first indication that the turbulence model will not be able to reproducea completely equal flow field compared to that of the experiment. It seems thatthe sharp structures are washed out a little bit. However this may be a suitableapproximation.

At the bottom of figure 38 the contours of static pressure are shown. It canagain be observed that there is a pressure gradient in the radial direction whichbalances the centrifugal force. The shape of the contour lines has changed a littlebit here too. It can be seen that there is a buckling in the outer region (leftside of the picture) close to the wall. This cannot be observed in the figures 14and 26 for the laminar case. Also the contours lines of the mean pressure in[Huttl and Friedrich, 2001] do not show this behaviour.

In the following passage the horizontal profiles of the axial and vertical velocitycomponent shall be compared to those from [Dusek, 2011] measured in the exper-iment. The measured profile of the axial component is qualitatively reproducedup to a height of y = 6.5mm (figures 39 and 40 top) although the values are alittle higher in the computation. This could again be caused by the assumptionsmade in the numerical model which are discussed in chapter 4.1. It seems that theinfluence of the centrifugal force is still too small in the numerical model because

44

Figure 38. Top: Contours of velocity magnitude in m/s in a crosssection of the pipe. The vectors illustrate the secondary flow. Bot-tom: Contours of static pressure in Pascal in a cross section of thepipe. The center of curvature is towards the positive x direction.

45

the conformity would be better if the whole profile was shifted towards the outerwall.


At a height from y = 8.5mm to y = 11mm a step near the inner wall canbe found in the measured data which is not resolved in the computation. Thiscorrelates with the velocity contours in figure 38 top where their shape is toosmooth. All in all it can be stated that the Reynolds stress model is able to simulatethe flow field qualitatively in the center whereas there are higher differences closerto the wall. As already mentioned above a reason might be that the influence ofthe centrifugal force is weakened due to the use of the turbulence model. It hasalready been observed that there is a correlation between the used model and thecurvature effects in the comparison of the k − ε and the Reynolds stress model infigure 35. There you can see that the k− ε model can definitely not reproduce thereal flow field.

Figure 42 shows the horizontal profile of the radial velocity component at aheight of y = 8.5mm. It can be seen that there is almost no conformity betweenthe profiles. There is similar disagreement at other heights of the cross section so

46


47


48

they will not be presented here. The poor reproduction of the secondary flow onceagain indicates that the curvature effects are not completely resolved.

Figure 42. Horizontal profiles of the radial velocity component ata height y = 8.5mm according to Figure 10. The blue line describesthe computed values whereas the green crosses represent the PIVmeasured values from the experiment ([Dusek, 2011]).

Now the vertical and horizontal velocity profile of the previous case shall becompared to the current one in order to investigate the influence of the Deannumber. In this case its value is De3 = 1555. The profiles are non dimensionalizedby dividing them by the particular bulk velocities. This comparison is shownin figure 43. With an increasing Dean number we would expect the horizontalvelocity profile to be even more shifted towards the outer wall with a decreasingmaximum value according to [Huttl and Friedrich, 2001]. Furthermore the peaksin the vertical profile should also move outwards. However, the computation doesnot show this effect. Even though the maximum value in horizontal profile isreasonable the effect of the centrifugal force is weakened by the turbulence model.

For the sake of complenteness there shall be given the mean friction velocityin order to make the results comparable. The mean wall shear stress is τw,m =2.0832Pa and thus the mean friction velocity gives uτ = 0.04278m/s according toequation 17.

49

Figure 43. Comparison of the horizontal and vertical profile of theaxial velocity for two different Dean numbers.

50

5. Outlook

Considering the comparisons of the velocity profiles between the experimentand the computation in the laminar case it can be stated that a useful and reliablemethod for simulating the laminar flow in a curved pipe has been developed.Those profiles have shown good agreement for the radial and the axial velocitycomponent even though the variations have increased a little bit for the higherReynolds number. The occuring flow field in different geometries can reliably becomputed and thus investigated by using this method. However the results wouldbe even more precise if the numerical model could take the mass flow through thegap between the box and the wall into account. The grid resolution seems to befine enough although a finer mesh would probably deliver an even more accuratesolution especially in the near wall region.

The simulation of the turbulent case at a Reynolds number of Re = 7000 hasshown qualitative agreement for the axial velocity component wheras the approx-imation is better in the center area. Near the upper wall the profile gets almostparabolic which is not observed in the experiment. The vertical velocity compo-nent can hardly be reproduced although a trend is noticeable. This leads to theassumption that there is a correalation between the effects of curvature and theused turbulence model. It has been illustrated that there is a much weaker influ-ence of the centrifugal force on the solution using the k − ε model but it seemsthat it is still not enough in the Reynolds stress model solution. Therefore it canbe stated that the solution depends on the used turbulence model.

In order to achieve better agreement between computation and experiment inthe case of a turbulent flow state a large eddy simulation or the even more precisedirect numerical simulation is necessary. These simulations are able to resolve thetransient behaviour through a selected range of length scales (LES) or from thesmallest Kolmogorov scales up to the largest ones (DNS). The computational effortis very large considering the small time steps and the necessary grid resolution.A large eddy simulation was tried but the used mesh seemed to be too coarse,indicated by the relaminarization after some seconds of flow time. The calculationtime for two seconds of flow time took about 4 hours by using four processors onthe CAE cluster. Thus a finer mesh can only be used if the computational domainis reduced accordingly considering the higher computational effort a very fine gridwill bring along.

51

Appendix A. Script for mesh generation

The script which is used for the mesh generation in Gambit can be seen below.It has been developed by Vlad Marica during a project work and can be foundin [Marica, 2011]. The geometry of the toroidal pipe and the mesh density aredefined through a few parameters in this script. After it has been run the volumehas to be meshed and the boundary cell zones must be defined.

$factor =1

$grid_density_factor =1

$grid_density_counter =1

do para "$grid_density_counter" init 1 cond ($grid_density_counter .le. 1) incr

(1)

reset

solver select "FLUENT 5"

$aspect_ratio =0.066006601

$radius_top=$factor/$aspect_ratio

$height =1

$nazimuthal =4

if cond($nazimuthal .eq. 3)

$radius_inside_block1=$radius_top *.35


endif

if cond($nazimuthal .eq. 4)



endif

$grid_density_azimuthal =3* $grid_density_factor

$grid_density_radial =6* $grid_density_factor

$ratio_radial =.60

$grid_density_axial =8* $grid_density_factor

$ratio_axial =.65

vertex create "center" coordinates 0 0 (-$height /2)

$n=1

do para "$n" init 1 cond ($n .le. ($nazimuthal *2))

$angle =($n -1) *360/( $nazimuthal *2) +360/( $nazimuthal *2)

vertex create coordinates ($radius_top*cos($angle)) ($radius_top*sin($angle)) (-

$height /2)

enddo

$n=1

do para "$n" init 1 cond ($n .le. ($nazimuthal *2))

$angle =($n -1) *360/( $nazimuthal *2) +360/( $nazimuthal *2)

if cond(mod($n ,2) .eq. 0.)

52

vertex create coordinates ($radius_inside_block1*cos($angle)) (

$radius_inside_block1*sin($angle)) (-$height /2)

else

vertex create coordinates ($radius_inside_block2*cos($angle)) (

$radius_inside_block2*sin($angle)) (-$height /2)

endif

enddo

do para "$n" init 2 cond ($n .le. ($nazimuthal *2)) incr (1)

edge create center2points "center" ("vertex." + ntos($n)) ("vertex." + ntos($n+1))

minarc arc

enddo

edge create center2points "center" ("vertex." + ntos (2)) ("vertex." + ntos((

$nazimuthal *2+1))) minarc arc

do para "$n" init ($nazimuthal *2+2) cond ($n .le. ($nazimuthal *4))

/edge create center2points "center" ("vertex." + ntos($n)) ("vertex." + ntos($n+1)

) minarc arc

edge create straight ("vertex." + ntos($n)) ("vertex." + ntos($n+1))

enddo

/edge create center2points "center" ("vertex." + ntos($nazimuthal *2+2)) ("vertex."

+ ntos(( $nazimuthal *4+1))) minarc arc

edge create straight ("vertex." + ntos($nazimuthal *2+2)) ("vertex." + ntos((

$nazimuthal *4+1)))

do para "$n" init 2 cond ($n .le. ($nazimuthal *2+1)) incr (1)

edge create straight ("vertex." + ntos($n)) ("vertex." + ntos($nazimuthal *2+$n))

enddo

do para "$n" init ($nazimuthal *2+2) cond ($n .le. ($nazimuthal *4+1)) incr

(1)

if cond(mod($n ,2) .eq. 1.)

edge create straight "center" ("vertex." + ntos($n))

endif

enddo

do para "$n" init 1 cond ($n .le. $nazimuthal *2-1) incr (1)

face create wireframe ("edge." + ntos($n)) ("edge." + ntos($nazimuthal *2+$n)) ("

edge." + ntos($nazimuthal *4+$n)) ("edge." + ntos($nazimuthal *4+$n+1)) real

enddo

face create wireframe ("edge." + ntos($nazimuthal *2)) ("edge." + ntos($nazimuthal

*4)) ("edge." + ntos($nazimuthal *4+1)) ("edge." + ntos($nazimuthal *6)) real

do para "$n" init 1 cond ($n .le. $nazimuthal -1) incr (1)

face create wireframe ("edge." + ntos($nazimuthal *2+$n*2)) ("edge." + ntos(

$nazimuthal *2+$n *2+1)) ("edge." + ntos($nazimuthal *6+$n)) ("edge." + ntos(

$nazimuthal *6+$n+1)) real

53

enddo

face create wireframe ("edge." + ntos($nazimuthal *2+1)) ("edge." + ntos(

$nazimuthal *4)) ("edge." + ntos($nazimuthal *6+1)) ("edge." + ntos($nazimuthal

*7)) real

// --------------------------------meshing --------------------

// do para "$n" init (1) cond ($n .le. ($nazimuthal *4))

//edge mesh ("edge." + ntos($n)) bellshape ratio1 .5 intervals

$grid_density_azimuthal

// enddo

//

// do para "$n" init ($nazimuthal *4+1) cond ($n .le. ($nazimuthal *6))

//edge mesh ("edge." + ntos($n)) exponent ratio1 $ratio_radial intervals

$grid_density_radial

// enddo

//

// do para "$n" init ($nazimuthal *6+1) cond ($n .le. ($nazimuthal *7))

//edge mesh ("edge." + ntos($n)) bellshape ratio1 .5 intervals

$grid_density_azimuthal

// enddo

// --------------------------------meshing --------------------

// do para "$n" init (1) cond ($n .le. ($nazimuthal *3))

//face mesh ("face." + ntos($n)) map size 1

// enddo

// export fluent5 ("/ usr/temp22/engineering -marangoni -flows/sim3/sim1/sim1/sim10/AR

-"+ ntos($aspect_ratio)+"-grid -"+ ntos($grid_density_counter)+".msh")

// $grid_density_factor=$grid_density_factor *2

enddo

// %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

// Begin Vlad’s Script

// %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

// %%%%%%%%%%%%%%

// 2D Geometry

// %%%%%%%%%%%%%%

// Define mesh density factors for the azimuthal and the radial direction

$mesh_density_factor_azimuthal =1

$mesh_density_factor_radial =4

54

// Split each of the 16 azimuthal edges. Parameters ending in "_i" are needed

later. They should have the same value as their root paramter. Don’t forget to

initiate $azimuthal_edges with the same value in the do loop.

$azimuthal_edges =4* $mesh_density_factor_azimuthal

$azimuthal_edges_i =4* $mesh_density_factor_azimuthal

// Running variable for azimuthal edge number

$i=1

do para "$i" init (1) cond ($i .le. 16) incr (1)

do para "$azimuthal_edges" init (4* $mesh_density_factor_azimuthal) cond (

$azimuthal_edges .ge. 2) incr (-1)

edge split ("edge." + ntos($i)) percentarclength (1 -(1/(

$azimuthal_edges))) connected

enddo

enddo

// Split the innermost , orthogonal edges in the same way

$inner_orthogonal_edges =4* $mesh_density_factor_azimuthal

$inner_orthogonal_edges_i =4* $mesh_density_factor_azimuthal

$i=25


do para "$inner_orthogonal_edges" init (4*

$mesh_density_factor_azimuthal) cond ($inner_orthogonal_edges

.ge. 2) incr (-1)

edge split ("edge." + ntos($i)) percentarclength (1 -(1/(

$inner_orthogonal_edges))) connected

enddo

enddo

// Split the outer spokes. Use a gradient which mimics a boundary layer. In order

to take advantage of this , we first mesh the outer spokes , and we then split

them with the mesh nodes.

$outer_spoke_intervals =3* $mesh_density_factor_radial

$outer_spoke_nodes =( $outer_spoke_intervals - 1)

$ratio_outer_spoke_intervals =1.2

// First mesh the outer spokes

$i=0

do para "$i" init 17 cond ($i .le. 24) incr (1)

edge mesh ("edge." + ntos($i)) successive ratio1

$ratio_outer_spoke_intervals intervals $outer_spoke_intervals

enddo

// Split outer spokes only once

55

$i=0


edge split ("edge." + ntos($i+17)) meshnode (($i*(2))+3)

enddo

// Split edges with remaining nodes

$n=0

do para "$n" init (0) cond ($n .le. (( $outer_spoke_nodes -1)*8)) incr (1)

edge split ("v_edge." + ntos(($n*2)+( $azimuthal_edges_i *16)+(

$inner_orthogonal_edges_i *4) +8+2)) meshnode ($n+18)

enddo

// Finishing touches on 2D Geometry

// Transform virtual faces into real faces

face convert "v_face .13" "v_face .14" "v_face .15" "v_face .16" "v_face .17" "v_face

.18" "v_face .19" "v_face .20"

// Move faces so they are on the x-y plane

face move "face.9" "face .10" "face .11" "face .12" "face .13" "face .14" "face .15" "

face .16" "face .17" "face .18" "face .19" "face .20" offset 0 0 0.5

// Translate faces from the origin along the x-axis , so they can be rotated

face move "face.9" "face .10" "face .11" "face .12" "face .13" "face .14" "face .15" "

face .16" "face .17" "face .18" "face .19" "face .20" offset 307 0 0

// Mesh faces

face mesh "face.9" "face .10" "face .11" "face .12" "face .13" "face .14" "face .15" "

face .16" "face .17" "face .18" "face .19" "face .20" map intervals 1

// %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Finish 2D

Geometry

// %%%%%%%%%%%%%%

// 3D Geometry

// %%%%%%%%%%%%%%

// Revolve faces to create torus volume

$torus_angle =345

volume create revolve "face.9" "face .10" "face .11" "face .12" "face .13" "face .14" "

face .15" "face .16" "face .17" "face .18" "face .19" "face .20" dangle $torus_angle

vector 0 1 0 origin 0 0 0 withmesh

56

// Merge the 12 volumes. This deletes all inner faces. Useful for defining

boundary conditions

volume merge "volume .1" "volume .2" "volume .3" "volume .4" "volume .5" "volume .6" "

volume .7" "volume .8" "volume .9" "volume .10" "volume .11" "volume .12" real

// Now mesh opposing slug faces

// Mesh the one created volume. Choose Hex Elements with a Cooper scheme. As

Sources , manually select the torus slug faces. Spacing should not be applied.

// Boundary conditions

// All faces are automatically listed as WALL upon exporting the mesh

// The volume is automatically listed as FLUID upon exporting the mehs

57

References

[Dennis and Ng, 1982] Dennis, S. C. R. and Ng, M. (1982). Dual solutions for steady laminarflow through a curved tube curved tube. Q. J. Mech. Appl. Math., 35:305. 9

[Durbin and Pettersson-Reif, 2011] Durbin, P. A. and Pettersson-Reif, A. (2011). Statistical The-ory and Modeling for Turbulent Flows. John Wiley & Sons Ltd. 6

[Dusek, 2011] Dusek, S. (2011). Bachelor thesis. Institute of Fluid Mechanics and Heat Transfer,Vienna University of Technology. 23, 25, 26, 33, 34, 37, 43, 45, 48

[Glasgow, 2010] Glasgow, L. A. (2010). Transport Phenomena: An Introduction to AdvancedTopics. John Wiley & Sons. Inc., Hoboken, New Jersey. page 75. 6, 7

[Huttl et al., 1999] Huttl, T., Wagner, C., and Friedrich, R. (1999). Navier-stokes solutions oflaminar flows based on orthogonal helical co-ordinates. International Journal for NumericalMethods in Fluids, 29(7):749–763. 4, 20, 21, 22, 28, 31, 39

[Huttl and Friedrich, 2000] Huttl, T. J. and Friedrich, R. (2000). Influence of curvature andtorsion on turbulent flow in helically coiled pipes. Int. J. Heat Fluid Flow, 21:345–353. 4, 41

[Huttl and Friedrich, 2001] Huttl, T. J. and Friedrich, R. (2001). Direct numerical simulation ofturbulent flows in curved and helically coiled pipes. Comput. Fluids, 30:591–605. 4, 41, 42, 43,48

[Kuhlmann et al., 2011] Kuhlmann, H., Kuhnen, J., and Muldoon, F. (2011). Proposal for a fwfresearch project: Investigating transition to turbulence in curved pipes using a torus model.Institute of Fluid Mechanics and Heat Transfer, Vienna University of Technology. 3

[Marica, 2011] Marica, V. (2011). Numerical investigations on curved pipe flow. Institute of FluidMechanics and Heat Transfer, Vienna University of Technology. 4, 51

numerical investigations on a curved pipe flowpub.ist.ac.at/~jkuehnen/files/schwab11.pdfvienna...

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