Numerical Investigations on a Curved Pipe Flow

Transcription

1 Vienna Unversity of Technology Institute of Fluid Mechanics and Heat Transfer Numerical Investigations on a Curved Pipe Flow Bachelor Thesis by Martin Schwab Supervisors Univ.Prof. Dipl.-Phys. Dr.rer.nat. Hendrik C. Kuhlmann Dr. Frank H. Muldoon, M.Sc. Dipl.Ing. Jakob Kühnen November 18, 2011

2 1 Abstract. The aim of this work is the simulation of the three dimensional flow in a fluid filled torus of circular cross section at different Reynolds numbers with the engineering simulation software Fluent. In this way a research program which is currently in progress at the Vienna University of Technology will be supported. A useful and reliable method for simulating a curved pipe flow shall be developed so that different geometries can easily be investigated. The results will be compared to measured data gained from an experiment which has already been set up in the laboratory in order to validate the numerical model. After some explanations on meshing the torus the settings and solution strategies for the three different cases are described. Finally the results and thus the flow structure will be illustrated.

3 2 Contents 1. Introduction Project description Modelling the Torus 3 2. Mesh Mesh dimensions and cell zones Turbulent microscales 6 3. Case Settings Setup for Re = Setup for Re = Setup for Re = Results Re = Re = Re = Outlook 50 Appendix A. Script for mesh generation 51 References 57

4 1. Introduction 1.1. Project description. Currently, a research program concerning the investigation of the flow in curved pipes is in progress at the Vienna University of Technology. An experiment has already been set up in order to gain measured data of the flow s structure. This work will support the research by delivering a reliable numerical model in order to be able to investigate other geometries without the need of an experimental set up. Before describing the details of this work the aims of the reasearch project shall be statet. The following passage is extracted from [Kuhlmann et al., 2011]. The main objective is to reveal the three dimensional structure of the transitional 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 circular pipes. Furthermore the role of Dean-flow instabilities on the transition scenario will be investigated and measurements of the torque required to maintain constant speed shall provide some information on the power dissipation in the torus. Finally, mechanisms which could permit control of the onset of turbulence shall be identified. This investigation is not only of theoretical relevance since most piping systems are curved, be it in the field of engineering or in nature, for example blood vessels Modelling the Torus. In the experiment the torus consisting of plexiglass is filled with refractive index matched water so that optical measurement techniques can be applied. The flow is driven by a box with a metal sphere within. The box exactly fits into a section of the pipe and is driven through a magnet from outside. The schematic setup is shown in figure 1. 3 Figure 1. Torus with box

5 4 For the numerical investigations the test set up is modelled as a torus section with plane surfaces at the ends. These surfaces correspond to the box which drives the flow. Therefore a rotating reference system connected to these surfaces will be 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 written by Marica Vlad (see appendix A, [Marica, 2011]). The whole volume consists of hexahedral elements meshed with a cooper scheme. The geometry of the toroidal pipe section is explained in figure 2. As one can see, the box is modelled as a gap of 15 degrees in the pipe section. Since the geometry of the pipe is the same as in the experiment the results can be directly compared. Figure 2. Mesh geometry. R = 307mm; d = 30.3mm; The rotating reference system is shown in blue. Aditionally the results shall be compared to the ones from other computations 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. As its axis is a circle the curvature κ is constant. (1) κ := 1 R

6 The results in these three articles are presented in a dimensionless form with d/2 as the length scaling variable. Accordingly, the dimensionless curvature is (2) κ := d 2R = The numerical simulation is done with Fluent for Reynolds numbers (see chapter 3.1) of 2000, 3000 and Figure 3 shows the mesh at one end of the pipe. 5 Figure 3. Mesh More details on the mesh and cell dimensions as well as the simulation settings will be given in chapter 3. Finally the results will be presented in chapter 4.

7 6 2. Mesh 2.1. Mesh dimensions and cell zones. Let s take a closer look at the cells of the mesh. Using the settings described in appendix A it consists of cells and nodes. The largest cell dimension is about 3mm. There are several cell zones defined during the mesh generation in Gambit. The two cross sections at the ends of the pipe are defined as boundary zone and called box in the simulation. The surface of the torus is also defined as boundary zone and called wall. Finally the volume is defined as fluid Turbulent microscales. In order to get an overview of what the mesh is able to resolve we can take a look at the turbulent microscales. They will be important 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 length scales where the energy is transported from the largest scales to the smallest ones until it is finally dissipated. This mechanism is known as energy cascade. The occuring length scales can be divided into several areas. The largest scales in the energy spectrum are the integral length scales where the energy is passed down to smaller scales. The Kolmogorov length scales are the smallest scales in the spectrum where the energy is dissipated. Finally there are the Taylor microscales. They mark the border between the integral scales and those where the energy is dissipated (see [Glasgow, 2010] and [Durbin and Pettersson-Reif, 2011]). In order to determine what scales the mesh is able to resolve the Taylor microscale for the torus is calculated. The simulation of turbulent structures will be run at a Reynolds number of Re = 7000, which is defined as follows. (3) Re d := ud ν Here, d describes the diameter of the pipe, ν the kinematic viscosity an u the velocity of the fluid flow. The subscript d shall indicate that the Reynolds number is built using the diameter as characteristic length scale. The Taylor microscale can be determined with the following equation, which is extracted from [Glasgow, 2010]. (4) λ 15 d = A Re d λ represents the Taylor microscale, d is the diameter of the pipe and the constant A 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 notice that the mesh is able to resolve the smallest eddies where energy is not yet dissipated. It should be mentioned that equation 4 can only be applied to isotropic turbulence so the calulated scale should only be seen as a reference. 1 2

8 7 The Kolmogorov microscale can be calculated with the following relations, (5) ɛ A u3 η = ( ν3 d ɛ ) 1 4 where ɛ stands for the dissipation rate per unit mass and η for the Kolmogorov microscale. They can also be found in [Glasgow, 2010]. If the kinematic viscosity of the matched water ν 23 C = m 2 /s is used the dissipation rate yields about 2.76m 2 /s 3 and the Kolmogorov microscale is about 0.04mm. Considering the mesh s cell dimensions it is obvious that the mesh is way too coarse to perform a direct numerical simulation but the resolution will be high enough for the intended calculations. However those length scales will be of importance if a large eddy simulation or even a DNS shall be done because there cannot be seen any sign of turbulence if the mesh is too coarse. There also exists the Kolmogorov time scale which is defined as follows. (6) τ η = ( ν ɛ ) 1 2 In our case τ η = s. Figure 4. A cross section of the torus mesh Figure 4 shows a cross section of the torus mesh. The cells with the largest side length are outside at the perimeter of the profile with a width of 2.97mm.

9 8 Figure 5. Cell geometry of the torus mesh Figure 5 illustrates how the cross section (figure 4) is revolved thorugh the torus volume so that the cells are formed. Every 3mm a new cross section is inserted in the volume.

10 3. Case Settings This chapter is about how the simulation is conducted in Fluent. After the mesh has been created in Gambit with Vlad Marica s script it is imported in Fluent. Now the problem has to be defined and the solution methods have to be set Setup for Re = Before any settings are defined it is interesting to know if the flow is laminar or turbulent. Therefore the Dean number can be calculated. It is defined as 9 (7) De := ud d ν 2R = Re d κ where u is the axial velocity, d the diameter of the pipe and R the radius of curvature. In the case of Re = 2000 the Dean number is De = According to [Dennis and Ng, 1982] the flow can be laminar up to a critical Dean number of De c 956. Laminar flow has also been observed up to a Reynolds number of about Re 3800 in the experiment. Since there is no time dependence in the laminar flow it is possible to calculate a steady solution. Table 1 below lists the settings which are done to define the problem. Problem setup General Table 1. Parameters to define the problem in Fluent Mesh Scale Mesh created in mm Solver Type Pressure - based Time Vel. formulation Steady Absolute Gravity On y = m/s 2 Models Viscous Laminar - Materials Fluid mat. Water 23 C ρ = kg/m 3 µ = P a s Solid Plexiglass ρ = 1190kg/m 3 Cell Zone Cond. Frame motion rot. ref. frame ω = rad/s Boundary Conditions Wall no Slip ω = rad/s Box no Slip ω = 0rad/s

11 10 The angular velocity specified for the boundary conditions is relative to the adjacent cell zone, in our case relative to the rotating reference frame. It is calculated using the Reynods number of Re = 2000 with an average velocity in the middle of 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 angular velocity necessary for achieving a Reynolds number of Re = 2000 can be calculated. (9) ω = Re ν = rad/s R d As can be seen in table 1 the laminar viscous model is selected. This means that the Navier Stokes equations are directly solved without using any turbulence model. The next step is to set the solution methods and other calculation specific settings which are listed in table 2. Solution Methods Table 2. Calculation specific settings in Fluent Press.-vel. coupling Scheme PISO Spatial discret. Gradient Least Sq. cell b. Pressure Momentum Second order 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 Although the coupled scheme for the pressure velocity coupling would be more precise the PISO scheme was selected. It is a good compromise between accuracy and computational effort. The solution is initialized with all velocities and the pressure set to zero. When the scaled residuals of the three velocity componets and the continuity equation are smaller than 10 3 the solution is considered to be converged. During the calculation the scaled residuals of the velocities were below 10 4 and the one of the continuity equation reached convergence at about as can be seen in figure 6.

12 11 Figure 6. Scaled Residuals In order to get an idea of the error in the results one can have a closer look at the definition of the scaled residuals in Fluent. Generally the residual of a function (10) f(x) := b with an aproximation x 0 of x is defined as (11) R := b f(x 0 ) Since the exact solution x is not known we can not calculate the error but in many cases the error is proportional to the size of the residual. This means that the residual can be used as a reference value for the error. Let s consider the residual s definition in Fluent which is extracted from the User s Guide. The conservation equation for a general variable φ at a cell P can be written as (12) a P φ P = nb a nb φ nb + b

13 12 where nb describes the neighboring cells. The unscaled residual R φ computed by Fluent s pressure based solver is the imbalance in equation 12 summed over all cells P. (13) R φ = Finally the scaled residual is (14) R φ sc = cells P cells P nb a nb φ nb + b a P φ P a nb φ nb + b a P φ P nb a P φ P cells P For the momentum equations the term a P φ P in the denominator is replaced by a P u P with u P as the velocity magnitude at cell P. Taking equation 14 the average error of φ at one cell can be estimated. The average azimuthal velocity at a Reynolds number of Re = 2000 is u = 0.125m/s and the scaled residual for the x and z velocity is about as can be seen in figure 6. It can be assumed that the coefficient a P is of the order of O(1). Therefore you get a residual of R u = Taking into account that the residual has the same scale as the error one can claim that there is a variation in the fifth decimal place between the calculated velocity and the exact solution. The simulation was conducted on the CAE Cluster of the Vienna University of Technology using 4 processors. The calculation of 1800 iterations took about one hour Setup for Re = Since there has not been any sign of turbulence at a Reynolds number of Re = 3000 in the experiment we do not expect that it occurs in the simulation. Also the Dean number of De = 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 conditions because the box is rotating faster now. The calculation specific settings haven t changed 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. Problem setup Table 3. Parameters to define the problem in Fluent Cell Zone Cond. Frame motion rot. ref. frame ω = rad/s Boundary Conditions Wall no Slip ω = rad/s Box no Slip ω = 0rad/s

14 13 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 continuity equation defined in the settings has not been reached at all. Its scaled residual is at about whereas the velocities scaled residuals are slightly below According to the considerations about the estimated average error above the residual of R u = for the velocities can be evaluated by taking equation 14 and multiplying with the average velocity of u = ωr = m/s. Therefore it can be assumed that the variation between the calculated and the exact solution in of the order of O(10 5 ). The calculation time per iteration has not changed compared to the case of Re = 2000 since there have not been any major changes Setup for Re = Before considering the simulation settings it is again a good idea to check the expected flow state. The Dean number for the case of Re = 7000 and the torus geometry is De = 1555 which is fairly above the critical value. Also the experiment indicates that the fluid flow is going to be turbulent at this Reynolds number. As a first step we want to get some information about the mean flow. Therefore the k ɛ turbulence model based on the Reynolds averaged Navier Stokes equations (RANS) is chosen. In order to prove if it is applicable in our case the results will be compared to the data gained from the experiment later on. Since this turbulence model does only calculate the mean flow there is still no need to

15 14 compute a transient solution but there are two more equations to be solved besides the continuity and the Navier Stokes equations. These are the transport equation for the turbulent kinetic energy k and one for the rate of dissipation ɛ. 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 the rotating reference frame. Table 4. Parameters to define the problem in Fluent using the k ɛ model Problem setup General Mesh Scale Mesh created in mm Solver Type Pressure - based Time Vel. formulation Steady Absolute Gravity On y = m/s 2 Models Viscous realizeable k ɛ enh. wall treat. Materials C 2ɛ = 1.9 T KE = 1 T DR = 1.2 Fluid mat. Water 23 C ρ = kg/m 3 µ = P a s Solid Plexiglass ρ = 1190kg/m 3 Cell Zone Cond. Frame motion rot. ref. frame ω = rad/s Boundary Conditions Wall no Slip ω = rad/s Box no Slip ω = 0rad/s All calculation specific settings haven t changed so far but there are two new variables, the turbulent kinetic energy and the dissipation rate, which have to be discretized. Table 5 lists the additions compared to chapter 3.1. The initial condition of the dissipation rate is chosen according to chapter 2.2 where its value has been calculated for the average flow speed of u = m/s at a Reynold number of Re = 7000 to ɛ = 2.76m 2 /s 3. As can be see in figure 8 the solution seems to be converged after about 800 iterations. The computation lasted about one hour whereas four processors were used. Since the k ɛ turbulence model does also have some disadvantages the more accurate Reynolds Stress model will be used for a second simulation. This model

16 15 Table 5. Calculation specific settings in Fluent using the k ɛ model Solution Methods Spatial discret. Turb. kin. energy Second ord. up. Diss. rate Second ord. up. Calculation Act. Autosave Every 100 iterations - Solution Init. Standard Absolute - k = 1 ɛ = 2 Run Calulation Iterations Figure 8. Scaled Residuals using the k ɛ model does also calculate the mean flow and therefore solve the RANS equations with the advantage of the ability to render anisotropic turbulence. Six equations have to be solved for the Reynolds stresses and one for the dissipation additionally which increases the computational effort. Thus it is reasonable to use the solution gained from 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 the Reynolds stress model is started.

17 16 Table 6. Changes for setting up the Reynolds Stress model compared to table 4 Problem setup Models Viscous Reynolds Stress quad. pre. strain C mu = 0.09 C 1ɛ = 1.44 C 2ɛ = 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 Fluent using the Reynolds Stress model compared to table 5 Solution Methods Spatial discret. Reynolds stresses Second ord. up. Run Calulation Iterations Figure 9. Scaled Residuals using the Reynolds stress model

18 4. Results After the settings have been discussed, the results are finally presented separately for each case. They will also be compared to data gained from the experiment in order to validate the numerical model. This data consists of velocity profiles along the horizontal axis for the azimuthal and radial velocity components measured at different heights of the cross section. Figure 10 illustrates where the velocity profiles are measured. 17 Figure 10. Exemplary positions where velocity profiles are measured in the experiment The azimuthal velocity component points out of the plane. The y - coordinate describes the different heights where the profiles are obtained. The coordinate system s origin is in the torus center of curvature. As already mentioned in chapter 3 an absolute velocity formulation is used. This means that the velocities in the figures presented in the following chapters are with reference to a stationary system Re = At first we want to get an overview of the flow structure. Therefore a plane in the middle of the pipe, at y = 0mm according to figure 10, is cut out of the torus. It is a good idea to plot the contours of the velocity magnitude in this plane to check if the solution looks reasonable.

19 18 Before having a look at the plot the expected average velocity magnitude can be calculated 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) u av = u max 2 is 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 with laminar flow. Figure 11. Contours of velocity magnitude in m/s at a plane in the middle of the pipe. As can be seen in figure 11 the relation 15 holds true approximately. The box which drives the flow is rotating counter clockwise. One may also notices that the velocity maximum is not in the center of the cross section as it would be for straight pipes. It is shifted towards the outer region due to the centrifugal force. Besides some small perturbations emanating from the box surface, there cannot

20 be observed any dependencies of the flow in the angular direction. For that reason it will be sufficient to consider one cross section at some distance to the box for the further investigation of the flow structure. Everything is what we expect it to be and the solution looks reasonable so far. Let us have a look at a vector plot of the same plane next. Since we have only seen a plot of the velocity magnitude it will be interesting to know the direction of the velocity too. Additionally one can get a first impression of the velocity profile in the horizontal direction. 19 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 point along the azimuthal direction. There is a small velocity gradient from the inner side to the maximum and a big one to the outer side. The velocity at the wall is zero so the no slip boundary condition is fulfilled. It can be seen very clearly that the flow is laminar. Now we will have a closer look at a cross section somewhere in the middle of the pipe with enough distance to the box. This view will reveal the flow structure along the vertical y - coordinate. Another thing what we want to investigate is the occuring secondary flow perpendicular to the mean flow. The pipe s curvature induces a centrifugal force proportional to the flow speed squared. It causes the

21 20 flow being accelerated outwards where its speed is high. Since the continuity has to 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 clearly that the flow moves outwards in the center, then bifurcates at the outer wall and moves back inwards along the circumference. This is exactly what is described above if the velocity magnitude at the different regions is considered. Figure 13. Contours of velocity magnitude in m/s in a cross section of the pipe. The vectors illustrate the secondary flow. The center 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 consideration that there is used a bulk Reynolds number of Re b = 1000 and a curvature of κ = 0.01, where the bulk Reynolds number is defined in the same way as it is for this computation. Only the vector plot of the secondary flow is a little different

22 where the biggest difference is in the region near the upper and lower wall. The pattern of this solution seems to be more realistic since the inwards flow is closer to the wall and therefore at a region of lower axial velocity. The shape of the contour lines changes depending on the curvature. In the case of a straight pipe the pressure is only a function of the coordinate in axial direction. For curved pipes the developing centrifugal force has to be balanced by a pressure gradient prependicular to the pipe axis. This is what figure 14 illustrates. It is quite similar to [Huttl et al., 1999] again. 21 Figure 14. Contours of static pressure in Pascal in a cross section of the pipe. The center of curvature is towards the positive x direction. For the further investigation of the flow we will plot the axial and radial velocity components along horizontal and vertical lines in a cross section. Since a dimensionless description is often used there will be given some popular scaling factors for the velocity in order to make the presented dimensioned profiles comparable.

23 22 One of these scaling factors is the bulk velocity u b. It is the mean flow velocity which is easy to calculate in our case since it is equal to the speed of the rotating box. (16) u b = ωr In equation 16 R represents the radius of the torus axis and ω is the angular velocity of the box. It gives a value of u b = 0.125m/s. Another possible scaling factor is the mean friction velocity u τ which is used in [Huttl et al., 1999]. It is defined as (17) u τ = τw,m ρ τ w,m = 1 2π 2π 0 τ w (θ) dθ where τ w,m is the mean wall shear stress. Figure 15 shows the wall shear stress in dependence of the radial position. It has the same value along the upper and lower 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 τ w,m = P a. Using the density listed in table 1 the mean friction velocity can be calculated. It is u τ = m/s.

24 23 Figure 16. Horizontal profiles of the axial velocity component at different heights of the cross section. The blue line describes the computed values whereas the green crosses represent the PIV measured values from the experiment ([Dusek, 2011]).

25 24 Figure 17. See figure 16 for a description.

26 25 Figure 18. See figure 16 for a description. The figures 16, 17 and 18 show a comparison of the axial horizontal velocity profiles between the computed results of this work (blue line) and measured values through particle imaging velocimetry (green crosses) at different heights of the pipe cross 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 smaller than the computed one in the center of the cross section at almost all heights. The only exception is the near wall region where the measured velocity is larger. There is quite good conformity at y = 2mm and y = 5mm whereas the gap gets a little larger towards the upper wall. A reason for the larger computed velocity in the middle might be the idealizations made in the numerical model. For example the flow cannot pass the box since it is tightly connected to the wall. This is not true in reality because the box will not be able to move then. Therefore there will be a little gap between the box s circumference and the wall where a small mass flow passes which results in a lack of velocity. This circumstance would explain the differences very well considering 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.

27 26 All in all there is a conformity in the axial horizontal velocity profiles between the numerical simulation and the experiment with some minor differences. In order to make the results comparable to other computations the two scaling factors mentioned in equation 16 and 17 can be used to non dimensionalize the velocity. This is illustrated in equation 18 by using the maximum velocity. u max (18) = u b = u max = u τ = In the following the radial velocity component will be plotted along a horizontal line at different heights in the same way as it has been done with the axial component. Figure 19. Horizontal profiles of the radial velocity component at different heights of the cross section. The blue line describes the computed values whereas the green crosses represent the PIV measured values from the experiment ([Dusek, 2011]). Positive values mean a flow direction towards the center of curvature. The velocity s scale is The figures 19 and 20 show a comparison of the computed results to measured values. It can be seen that the values are way smaller than they are for the axial velocity. Therefore it will only be reasonable to do a comparison at regions

28 27 Figure 20. See figure 19 for a description. where the radial velocity is larger. For that reason the plots are made along horizontal lines in the upper region of the cross section at y = 8mm and y = 11mm because the radial velocity component is even smaller in the center. A conformity is basically given but the relative variation is a little larger here, which is again caused due to the assumptions made in the numerical model. Additionally the absolute values are very small which could also be a reason for the larger relative variation. A more precise simulation can only be done by modifying the numerical model so that the leackage through the gap between the box s circumference and the 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 the radial velocity component along a vertical line which cuts the cross section in the middle. This is what is illustrated in figure 21. It can be seen that the flow moves inwards in a small region near the upper and lower wall whereas it is forced outwards in a large region in the center. Since the continuity equation shall be fulfilled the overall mass flow along the radial direction has to be zero. Therefore the flow velocity is higher in those small areas where it moves inwards. This behaviour has already been shown in figure 13 qualitatively.

29 28 Figure 21. Radial velocity component along a vertical line in the middle of the cross section. Positive values for the radial velocity mean a flow direction towards the center of curvature. The last thing which has not been investigated yet is the vertical profile of the axial velocity. As with the horizontal profile its shape has a strong dependence on the curvature so it will not be parabolic like that for straight pipes. Figure 22 shows 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 peaks in the near wall region. The values are slightly above the bulk velocity which is u b = 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 which is proportional to the flow speed squared. Therefore those areas with a higher velocity are forced outwards whereas areas with a smaller velocity stay closer to the center of curvature. This can also be seen in the figures 13 and Re = At this Reynolds number the flow state has not changed very much compared to the case of Re = The Dean number has increased a little bit but the flow is still laminar. Because of this it is reasonable to expect results with an almost similar shape as in chapter 4.1. Therefore the comments on the figures in this chapter will be kept short since the descriptions of the previous

30 29 Figure 22. Axial velocity component along a vertical line in the middle of the cross section. chapter are still valid. In order to visualize the flow structure several contour plots will 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 from the center to the outer wall. The bulk velocity for this case is u b = m/s according to equation 16, where the value of the angular velocity has changed. Considering the maximum velocity magnitude of about u max 0.32 one will get a ratio of u max (19) = 0.32 u b = 1.7 which is a reasonable value for a curved pipe flow. Figure 23 looks almost equal to figure 11 beside different values for the velocities. Figure 24 shows the velocity vectors in a section of the plane of figure 23. There can be seen a small velocity gradient form the inner wall to the maximum velocity and a large one from there to the outer wall. The no slip boundary condition seems to 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 of the pipe. The contours represent the velocity magnitude whereas the vectors are the in plane components of the velocity and thus visualizing the secondary flow.

31 30 Figure 23. Contours of velocity magnitude in m/s at a plane in the middle of the pipe. Figure 24. Velocity vectors colored by velocity magnitude in m/s. The occurence of the secondary flow has already been explained in chapter 4.1 and shall not be repeated here since it is caused by the same mechanisms. There are two recirculating Dean cells again where the flow moves inwards near the upper

32 and lower wall and is forced outwards in the center. This is exactly what we would expect for a curved pipe flow at this Reynolds number. 31 Figure 25. Contours of velocity magnitude in m/s in a cross section of the pipe. The vectors illustrate the secondary flow. The center of curvature is towards the positive x direction. One can get a good imagination of the effect of the centrifugal force by having a look at the distribution of the static pressure in a cross section because they are balancing each other. This is what figure 26 illustrates. Again, it is quite equal to figure 14 but the value of the maximum static pressure is almost twice as much as before. As expected, the results have not changed very much compared to the case of Re = 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 once again.

33 32 Figure 26. Contours of static pressure in Pascal in a cross section of the pipe. The center of curvature is towards the positive x direction. The next step in order to validate the numerical model for this case is a comparison of the computed solution with measured data. This is done by superimposing several velocity profiles for the axial and radial velocity component at different regions of the cross section. Before the presentation of the actual plots the wall shear stress and the mean friction velocity shall be shown so that the profiles are comparable 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 wall shear stress according to equation 17 gives τ w,m = P a and the mean friction veocity is u τ = m/s. Therefore the non dimensional maximum velocity will be

34 33 u max (20) = 0.32 u τ = These scaling factors will also be used later on when the velocity profiles for Re = 2000 and Re = 3000 are compared in order to determine the influence of the Dean 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 measured through particle imaging velocimetry (from [Dusek, 2011]) and the computed ones. They are evaluated at different heights according to figure 10. Anyhow some assumptions have been made which are not valid in reality but it is expected that their effect on the solution will be small (see chapter 4.1). There is a good conformity of the horizontal axial velocity profiles at all heights with some minor differences, similar to the case of Re = It is remarkable that 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 velocity in the experiment as well. However these differences can partially be explained by the assumptions made in the numerical model. The lack of velocity can arise due to 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

35 34 Figure 28. Horizontal profiles of the axial velocity component at different heights of the cross section. The blue line describes the computed values whereas the green crosses represent the PIV measured values from the experiment ([Dusek, 2011]).

36 Figure 29. See figure 28 for a description. 35

37 36 Figure 30. See figure 28 for a description. mesh too. Thus you will probably get a more precise solution by using an even finer grid. Figure 31 shows the horizontal profile of the radial velocity component at two different heights. A positive value of the radial velocity means that the flow moves towards the center of curvature. You can see that the velocity magnitude is about ten times larger at y = 11mm compared to the values at y = 6mm. Therefore one can imagine that the relative variation between the experimental data and the simulation will be higher when the absolute values get very small. Besides those relatively larger variations for small values a qualitative conformity is 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 peaks represent 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 as well. Figure 32 shows the radial component thus representing the secondary flow in another way. Finally figure 33 illustrates the vertical distribution of the axial velocity component. The shape of both plots is almost equal to that of figure 21 and 22.

38 37 Figure 31. Horizontal profiles of the radial velocity component at different heights of the cross section. The blue line describes the computed values whereas the green crosses represent the PIV measured values from the experiment ([Dusek, 2011]).

39 38 Figure 32. Radial velocity component along a vertical line in the middle of the cross section. Positive values for the radial velocity mean a flow direction towards the center of curvature. Since the Dean number is the determining parameter in the system of a curved pipe flow we want to investigate how the velocity profile has changed due to the increase in speed compared to the previous chapter. Let us recall equation 7. De will change its value due to a change of curvature or a change of the Reynolds number, which is what we have done. The Dean number in the case of Re = 2000 is De 1 = whereas it is De 2 = in the case of Re = In order to make the velocity profiles comparable they have to be made non dimensional first. Therefore we will use the bulk velocity u b as the characteristic velocity. This means that the values of the velocities will be divided by u b1 = 0.125m/s and u b2 = m/s. The comparison of the profiles itself can be seen in figure 34 for a horizontal and a vertical profile of the axial velocity. They are plotted 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 the maximum velocity to move outwards and does also reduce it a little bit. Furthermore the bending of the profile has increased too.

40 39 Figure 33. Axial velocity component along a vertical line in the middle of the cross section. The second plot shows that the peaks move outwards with a slightly decreasing maximum too. There is a local minimum developing in the transition area from the linear range in the center to the peaks in the near wall region. This behaviour has also been observed in the computations of [Huttl et al., 1999] Re = As already indicated in chapter 3.3 we expect the flow to be turbulent at this Reynolds number. For that reason two turbulence models have been selected and will be compared to the experimental data in order to prove if they are applicable to this problem. These models calculate the mean flow with consideration of turbulent effects but do not resolve any vortices. Since these effects are only modeled and not directly calculated, like in a direct numerical simulation, some variations between the experiment and the computed solution have to be expected. There are even differences between the results of the two models as illustrated in figure 35. It shows the horizontal profiles of the axial velocity component along a cut at a height of y = 2mm according to figure 10. At this point it is a good idea to reflect which profile is more reasonbale considering a curved pipe flow

41 40 Figure 34. Comparison of the horizontal and vertical profile of the axial velocity for two different Dean numbers.

42 41 Figure 35. Comparison between the results of the k ɛ and the Reynolds 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 the profiles look like there. In this case the Reynolds stress model delivers a more realistic solution which makes sense too having in mind that it can render turbulence more accurate. It seems that the k ɛ model is not able to resolve the effects of curvature completely. For that reason the further investigation of the flow will be done by using the results of the simulation based on the Reynolds stress model. Before the comparison to measured velocity profiles the three dimensional structure of the flow shall be visualized by some plots. Figure 36 shows the contours of the velocity magnitude in a plane at y = 0mm. Although it looks almost similar to the figures 11 and 23 it should be mentioned once again that the mean flow is calculated and therefore no vortices are resolved. Beside some small perturbations close to the box there is no dependence of the solution in the axial direction. The maximum velocity in the non perturbed area is approximately u max 0.614m/s and the bulk velocity gives u b = m/s. This gives a ratio of

43 42 Figure 36. Contours of velocity magnitude in m/s at a plane in the middle of the pipe. (21) u max u b = = 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 the contours has changed quite a lot compared to the laminar case. The peaks near the upper 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 of the recirculating cells has moved a little bit towards the center of curvature. This behaviour 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

44 43 Figure 37. Velocity vectors colored by velocity magnitude in m/s. component look a little different there. They have the shape of the ones calculated for the laminar case. This is a first indication that the turbulence model will not be able to reproduce a completely equal flow field compared to that of the experiment. It seems that the sharp structures are washed out a little bit. However this may be a suitable approximation. At the bottom of figure 38 the contours of static pressure are shown. It can again be observed that there is a pressure gradient in the radial direction which balances the centrifugal force. The shape of the contour lines has changed a little bit here too. It can be seen that there is a buckling in the outer region (left side of the picture) close to the wall. This cannot be observed in the figures 14 and 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 velocity component shall be compared to those from [Dusek, 2011] measured in the experiment. The measured profile of the axial component is qualitatively reproduced up to a height of y = 6.5mm (figures 39 and 40 top) although the values are a little higher in the computation. This could again be caused by the assumptions made in the numerical model which are discussed in chapter 4.1. It seems that the influence of the centrifugal force is still too small in the numerical model because

45 44 Figure 38. Top: Contours of velocity magnitude in m/s in a cross section of the pipe. The vectors illustrate the secondary flow. Bottom: Contours of static pressure in Pascal in a cross section of the pipe. The center of curvature is towards the positive x direction.

46 the conformity would be better if the whole profile was shifted towards the outer wall. 45 Figure 39. Horizontal profiles of the axial velocity component at different heights of the cross section. The blue line describes the computed values whereas the green crosses represent the PIV measured values from the experiment ([Dusek, 2011]). At a height from y = 8.5mm to y = 11mm a step near the inner wall can be found in the measured data which is not resolved in the computation. This correlates with the velocity contours in figure 38 top where their shape is too smooth. All in all it can be stated that the Reynolds stress model is able to simulate the flow field qualitatively in the center whereas there are higher differences closer to the wall. As already mentioned above a reason might be that the influence of the centrifugal force is weakened due to the use of the turbulence model. It has already been observed that there is a correlation between the used model and the curvature effects in the comparison of the k ɛ and the Reynolds stress model in figure 35. There you can see that the k ɛ model can definitely not reproduce the real flow field. Figure 42 shows the horizontal profile of the radial velocity component at a height of y = 8.5mm. It can be seen that there is almost no conformity between the profiles. There is similar disagreement at other heights of the cross section so

47 46 Figure 40. See figure 39 for a description.

48 Figure 41. See figure 39 for a description. 47

49 48 they will not be presented here. The poor reproduction of the secondary flow once again indicates that the curvature effects are not completely resolved. Figure 42. Horizontal profiles of the radial velocity component at a height y = 8.5mm according to Figure 10. The blue line describes the computed values whereas the green crosses represent the PIV measured values from the experiment ([Dusek, 2011]). Now the vertical and horizontal velocity profile of the previous case shall be compared to the current one in order to investigate the influence of the Dean number. In this case its value is De 3 = The profiles are non dimensionalized by dividing them by the particular bulk velocities. This comparison is shown in figure 43. With an increasing Dean number we would expect the horizontal velocity profile to be even more shifted towards the outer wall with a decreasing maximum value according to [Huttl and Friedrich, 2001]. Furthermore the peaks in the vertical profile should also move outwards. However, the computation does not show this effect. Even though the maximum value in horizontal profile is reasonable the effect of the centrifugal force is weakened by the turbulence model. For the sake of complenteness there shall be given the mean friction velocity in order to make the results comparable. The mean wall shear stress is τ w,m = P a and thus the mean friction velocity gives u τ = m/s according to equation 17.

50 Figure 43. Comparison of the horizontal and vertical profile of the axial velocity for two different Dean numbers. 49

51 50 5. Outlook Considering the comparisons of the velocity profiles between the experiment and the computation in the laminar case it can be stated that a useful and reliable method for simulating the laminar flow in a curved pipe has been developed. Those profiles have shown good agreement for the radial and the axial velocity component even though the variations have increased a little bit for the higher Reynolds number. The occuring flow field in different geometries can reliably be computed and thus investigated by using this method. However the results would be even more precise if the numerical model could take the mass flow through the gap between the box and the wall into account. The grid resolution seems to be fine enough although a finer mesh would probably deliver an even more accurate solution especially in the near wall region. The simulation of the turbulent case at a Reynolds number of Re = 7000 has shown qualitative agreement for the axial velocity component wheras the approximation is better in the center area. Near the upper wall the profile gets almost parabolic which is not observed in the experiment. The vertical velocity component can hardly be reproduced although a trend is noticeable. This leads to the assumption that there is a correalation between the effects of curvature and the used turbulence model. It has been illustrated that there is a much weaker influence of the centrifugal force on the solution using the k ɛ model but it seems that it is still not enough in the Reynolds stress model solution. Therefore it can be stated that the solution depends on the used turbulence model. In order to achieve better agreement between computation and experiment in the case of a turbulent flow state a large eddy simulation or the even more precise direct numerical simulation is necessary. These simulations are able to resolve the transient behaviour through a selected range of length scales (LES) or from the smallest Kolmogorov scales up to the largest ones (DNS). The computational effort is 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 calculation time for two seconds of flow time took about 4 hours by using four processors on the CAE cluster. Thus a finer mesh can only be used if the computational domain is reduced accordingly considering the higher computational effort a very fine grid will bring along.