See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/329151912 Numerical solutions of steady incompressible ο¬ow around a circular cylinder up to Reynolds number 500 Article · October 2018 CITATIONS READS 0 53 2 authors, including: Ercan Erturk 33 PUBLICATIONS 891 CITATIONS SEE PROFILE All content following this page was uploaded by Ercan Erturk on 26 November 2018. The user has requested enhancement of the downloaded file. International Journal of Mechanical Engineering and Technology (IJMET) Volume 9, Issue 10, October 2018, pp. 1368–1378, Article ID: IJMET_09_10_140 Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=9&IType=10 ISSN Print: 0976-6340 and ISSN Online: 0976-6359 © IAEME Publication Scopus Indexed NUMERICAL SOLUTIONS OF STEADY INCOMPRESSIBLE FLOW AROUND A CIRCULAR CYLINDER UP TO REYNOLDS NUMBER 500 Ercan Erturk Bahcesehir University, Mechatronics Engineering Department Besiktas, Istanbul, Turkey Orhan Gokcol Bahcesehir University, Computer Education and Instructional Technologies Department Besiktas, Istanbul, Turkey ABSTRACT In this study, steady incompressible viscous flow around a circular cylinder at high Reynolds numbers are solved numerically. Using a very efficient numerical method and a very large mesh, numerical solutions are calculated up to π π=500. It is found that the solution change behavior around π π=100 and π π=300. At the center of the wake bubble, around π π=300 the vorticity starts to increase fast linearly and the width of the wake bubble starts to increase rapidly. Good agreement is found with the results found in the literature. Detailed results are presented. Key words: Steady flow around a circular cylinder, high Reynolds numbers Cite this Article: Ercan Erturk and Orhan Gokcol, Numerical Solutions of Steady Incompressible Flow Around a Circular Cylinder Up To Reynolds Number 500, International Journal of Mechanical Engineering and Technology, 9(10), 2018, pp. 1368–1378. http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=9&IType=10 1. INTRODUCTION The flow around a circular cylinder is one of the classical flow problems in fluid mechanics. It is well known that the two-dimensional incompressible flow around a circular cylinder is steady for Reynolds number up to approximately 40 and beyond that Reynolds number, there appears Karman vortex street at downstream of the cylinder. However, for this flow case, the twodimensional incompressible flow around a cylinder, it is possible to obtain a steady solution mathematically when Re goes to infinity as the limiting solution (Kirchoff–Helmholtz solution, see Schlichting and Gersten [1]). Smith [2] and Peregrine [3] have done detailed mathematical analysis on two-dimensional steady incompressible flow around a circular cylinder at high http://www.iaeme.com/IJMET/index.asp 1368 editor@iaeme.com Ercan Erturk and Orhan Gokcol Reynolds numbers. Apart from these mathematical studies, in the literature there are also computational numerical studies on the same flow problem. Fornberg [4,5], Son and Hanratty [6], Tuann and Olson [7] and Dennis and Chang [8], Christov, Marinova and Marinov [9] have presented numerical solutions of steady flow past a circular cylinder at Re=300, 600, 500, 100, 100 and 200 respectively. Most probably, the studies of Fornberg [4,5] are the most comprehensive studies on the subject of two-dimensional steady incompressible flow around a circular cylinder in the literature. At high Reynolds numbers, for the considered flow problem although it is unstable, steady solution still exists along with the transient time dependent solutions. For the theory of the Navier-Stokes model it is important to study the stationary solution at high Reynolds numbers even when it loses stability. In the literature Fornberg [5] have presented the largest Reynolds number solutions of the steady flow around a circular cylinder and he [5] has presented solutions up to π π=600. The results of Fornberg [5] show that above π π=300 the width of the wake bubble change behavior and starts to increase rapidly. In the literature the numerical results of the steady flow around a circular cylinder at high Reynolds numbers presented by Fornberg [5] have never been verified. The aim of this study is then, to solve the 2-D steady incompressible flow around a circular cylinder. In this study, the efficient numerical method that allows numerical solutions at high Reynolds numbers described in Erturk, Corke and Gökçöl [10] and Erturk, Haddad and Corke [11] is applied to solve to the streamfunction and vorticity form of the Navier-Stokes equations. With this, the steady flow around a circular cylinder is solved up to high Reynolds number of π π=500 using a very large mesh. We compare our results in detail with the numerical study of Fornberg [5]. Detailed quantitative numerical results are presented for future references. 2. PROBLEM FORMULATION AND NUMERICAL METHOD For the numerical solution of steady incompressible viscous flow around a circular cylinder, we use the streamfunction (π) and vorticity (π) formulation of the steady Navier-Stokes equations given as the following π2 π ππ₯ 2 π2 π ππ₯ 2 π2 π + ππ¦ 2 − π2 π + ππ¦ 2 + π = 0 π π ππ ππ 2 ( ππ¦ (1) ππ ππ − ππ₯ ππ₯ ππ¦ )=0 (2) where π₯ and π¦ are the Cartesian coordinates and π π is the Reynolds number based on the ππ diameter of the cylinder which is defined as π π = where π is the diameter, π is the free π stream velocity and π is the kinematic coefficient of viscosity. Following Fornberg [5] we transform the physical domain to the computational domain using the following conformal mapping π = √π + 1 (3) √π where both π and π are complex numbers defined as the following π =π₯+ππ¦ (4) π =π+ππ (5) With this conformal mapping, the governing equations in the complex π-plane in the computational domain become as the following π2 π ππ 2 π2 π ππ 2 π2 π + ππ2 − π2 π + ππ2 + π π½ π π ππ ππ 2 ( ππ ππ =0 − ππ ππ ππ ππ (6) )=0 (7) where π½ is Jacobian of the mapping which is defined as the following http://www.iaeme.com/IJMET/index.asp 1369 editor@iaeme.com Numerical Solutions of Steady Incompressible Flow Around a Circular Cylinder up to Reynolds Number 500 ππ 2 π½ = |ππ| (8) The physical domain and also the computational domain are shown in Figure 1. Figure 1 Schematics of the flow problem and the solution domain, a) physical domain (top), b) computational domain (bottom) 2.1. Boundary Conditions For external flows such as the flow around a cylinder, the boundary conditions are very important for the accuracy of the solution. The boundary conditions used in this study are described briefly below. 2.1.1. Boundary conditions for streamfunction (π) Left Boundary (π = 0 , 0 ≤ π ≤ ππππ₯ ); As seen in Figure 1-a and 1-b, left boundary in the (π, π) corresponds to the symmetry line upstream of the cylinder. The symmetry line itself is a streamline and its value is chosen as zero. Therefore, the value of streamfuction on this symmetry line is π=0 (9) Bottom Boundary (π = 0 , 0 ≤ π ≤ ππππ₯ ); The bottom boundary in Figure 1-b corresponds to the wall of the cylinder and the symmetry line downstream of the cylinder. On this boundary the streamfunction value is the same with that of the symmetry line upstream of the cylinder such that π=0 (10) Top Boundary (π = ππππ₯ , 0 ≤ π ≤ ππππ₯ ); The top boundary corresponds the to the free stream boundary and at sufficiently far away from the cylinder the π’-velocity should be equal to 1 (π’ → 1). This states that at the free stream ππ boundary ππ¦ → 1. Therefore, at the free stream, the value of the streamfunction should approach to π → π¦. Using the π-plane coordinates the value of the streamfunction becomes http://www.iaeme.com/IJMET/index.asp 1370 editor@iaeme.com Ercan Erturk and Orhan Gokcol ππ π → 2ππ. Hence at the free stream boundary ππ → 2π. In external flows such as the flow around a cylinder, the free stream boundary condition is the driving boundary condition of the numerical solution. For this reason, the numerical solution is usually sensitive to the location of the free stream boundary. In order to decrease this sensitivity, the free stream boundary condition must be chosen very away from the cylinder. At the boundaries, one can either use the boundary condition or use both the governing equations and boundary conditions in order to calculate the flow variables. After extensive numerical experimentation on the flow around a cylinder flow problem, we think that the latter approach is less stringent on the numerical solution and therefore the numerical solution is less sensitive to the location of the free stream boundary condition. Thus, at the free stream boundary we use both the streamfunction equation and the boundary condition for streamfunction in order to calculate the streamfunction value. At the free stream boundary, the following equations are valid π2 π ππ 2 π2 π + ππ2 + π π½ =0 and ππ ππ → 2π (11) The discretization of the streamfunction equation requires a ghost point outside the computational domain. The value of this ghost point is calculated using the boundary condition ππ for the streamfunction, i.e. ππ → 2π. We note that, with this approach, at the free stream boundary we satisfy both the boundary condition and also the governing streamfunction equation. Right Boundary (π = ππππ₯ , 0 ≤ π ≤ ππππ₯ ); The right boundary is the outflow boundary. At this outflow boundary we used a nonreflecting boundary condition such that any wave generated in the computational domain could pass through the exit boundary and leave without any reflection back into the computational domain. For details on the subject the reader is referred to the study of Engquist and Majda [12] in which the nonreflecting boundary condition concept is first introduced in the name of “Absorbing Boundary Condition”, and also to Jin and Braza [13] for a review of non-reflecting boundary conditions. In their study Liu and Lin [14] attached a buffer region to the physical domain to damp erroneous numerical fluctuations. In this region they [14] added a buffer function to the streamwise second order derivatives in the momentum equations such that the reflected outgoing waves from an artificially truncated outlet are thus absorbed. This approach for the exit boundary condition has been used in various similar studies [11,15] successfully. Therefore, at the right boundary we solve the streamfunction equation without the elliptic π2 terms, i.e. (ππ2 ) terms, such as π2 π ππ 2 + π π½ =0 (12) In the computational domain near the outflow boundary, in order to have smooth transition π2 π π2 π from the streamfunction equation ( ππ2 + ππ2 + π π½ = 0) used inside the solution domain to the π2 π streamfunction equation without the elliptic terms used at the outflow boundary ( ππ2 + π π½ = 0) π2 we used a buffer zone as it was done in [11,15]. In this buffer zone we kill the elliptic (ππ2 ) terms in the governing equations gradually. To accomplish this, these elliptic terms are multiplied by a weighting factor π . At the beginning of the buffer zone, we set π =1 and at the end of the buffer zone, it is zero, π = 0. In between, the weighting factor π changes as π = tanh(4)+tanh(πππ) 2 tanh(4) (13) where http://www.iaeme.com/IJMET/index.asp 1371 editor@iaeme.com Numerical Solutions of Steady Incompressible Flow Around a Circular Cylinder up to Reynolds Number 500 2(π−πππ’π) πππ = 4 (1 − (ππππ₯−πππ’π)) (14) where π is the numerical streamwise index, ππππ₯ is the numerical index of the last grid point in streamwise direction and πππ’π is the index π of the first grid point at the beginning of the buffer zone. We used 20 grid points in this buffer zone. Therefore, near the outflow boundary in the computational domain, in this buffer zone we basically solve π2 π π2 π π ππ2 + ππ2 + π π½ =0 (15) where π is defined above. The non-reflecting boundary condition together with a buffer zone provides very smooth solutions. This approach also decrease the sensitivity of the numerical solution to the location of the outflow boundary. 2.1.2. Boundary conditions for vorticity (π) Left Boundary (π = 0 , 0 ≤ π ≤ ππππ₯ ); As the same with the streamfunction variable, at the symmetry line the vorticity value is zero. π=0 (16) Bottom Boundary (π = 0 , 0 ≤ π ≤ ππππ₯ ); On the bottom boundary, for vorticity calculations there are two regions. On the cylinder where π = 0 , 0 ≤ π ≤ 2, we used the wall condition for vorticity such as −2π1 2 1 −π0 ) π0 = π½0 (π (17) where 0 denotes the points on the wall and 1 denotes the first grid points adjacent to the wall. Also, at the symmetry line downstream of the cylinder where π = 0 , 2 ≤ π ≤ ππππ₯ , the vorticity value is zero π=0 (18) Top Boundary (π = ππππ₯ , 0 ≤ π ≤ ππππ₯ ); At the free stream the vorticity should be equal to zero (π → 0). Instead of using a Drichlet type boundary conditions for vorticity we decided to use a Neumann type boundary condition. At sufficiently far away from the cylinder since π → 0, the derivative of the vorticity should ππ also be equal to zero ( → 0). As the same done for the streamfunction variable, at the free ππ stream boundary we use both the vorticity equation and the Neumann boundary condition for vorticity in order to calculate the vorticity value. At the free stream boundary, the following equations are valid π2 π ππ 2 π2 π ππ + ππ2 − π π π ππ = 0 where ππ ππ →0 and ππ ππ → 2π (19) The discretization of the above substituted vorticity equation requires a ghost point outside the computational domain. The value of this ghost point is calculated using the boundary ππ condition for the vorticity, i.e. ππ → 0. We note that, with this approach, at the free stream boundary we both satisfy the boundary conditions and also the governing vorticity equation. Right Boundary (π = ππππ₯ , 0 ≤ π ≤ ππππ₯ ); Similar to the explained approach earlier, we used a non-reflecting boundary condition at π2 the outflow boundary and thus we solve the vorticity equation without the elliptic terms (ππ2 ) such as http://www.iaeme.com/IJMET/index.asp 1372 editor@iaeme.com Ercan Erturk and Orhan Gokcol π2 π ππ 2 − π π ππ ππ 2 ( ππ ππ − ππ ππ ππ ππ )=0 (20) We note that the discretization of this equation requires a ghost point outside the computational domain. The value of this ghost point is calculated using the fact that on this boundary π2 π ππ 2 = 0 and π2 π ππ 2 = 0 such that π and π change linearly. As it was done for streamfunction variable, in order to have smooth transition from the π2 π π2 π vorticity equation ( ππ2 + ππ2 − π π ππ ππ 2 ( ππ ππ − ππ ππ ππ ππ π2 π ) = 0) to the vorticity equation without the elliptic terms used at the outflow boundary ( ππ2 − π π ππ ππ 2 ( ππ ππ − ππ ππ ππ ππ ) = 0) we use a buffer zone and in this buffer zone we solve π2 π π2 π π ππ2 + ππ2 − π π ππ ππ 2 ( ππ ππ − ππ ππ ππ ππ )=0 (21) where π is defined the same as explained earlier. We think that the outflow boundary conditions used in this study provides much smooth and accurate solutions compared to the Oseen approximation at the outflow boundary as it was used in Fornberg [5]. 3. RESULTS AND DISCUSSIONS Using the boundary conditions described above, we solved the governing equations using a very efficient numerical method. The numerical method used is described briefly in Erturk, Corke and Gökçöl [10] and Erturk, Haddad and Corke [11]. With this numerical method the governing equations (6) and (7) are solved up to very low residuals. We assume that the convergence is achieved when the streamfunction π and the vorticity π values satisfies the governing equations (6) and (7) with a maximum residual of 10−10 at every grid point inside the computational domain. In order to capture the physical phenomenon more accurately, more grid points were located near the wall in π-direction, and near the cylinder in π-direction. Also, in order to eliminate the effect of the far field and outflow boundary condition on the interior solution, the far field and the outflow boundary should be chosen sufficiently away from the cylinder. This was done using Robert's stretching transformation of the original uniform grid (Anderson, Tannehill and Pletcher [16]). The formula of the transformation is π¦=β Μ (π½+1)−(π½−1)[(π½+1)⁄(π½−1)]1−π¦ Μ +1 [(π½+1)⁄(π½−1)]1−π¦ (22) where π¦Μ represents the original uniformly spaced grid points, π¦ represents the stretched grid points and π½ is the stretching parameter. In order to place the far field boundary and the outflow boundary sufficiently away from the cylinder we used ππππ₯ =20 and ππππ₯ =15 and also we have used 1800 and 300 grid points in π- and π-directions respectively. Figure 2-a shows the grid points used in this study. In this figure since we have used such large number of grid points, we have plotted only one out of every 20 grid points in both π- and π-directions. We note that, as it is seen in this figure, the outflow boundary is almost 600 diameters away from the cylinder. We also note that in this figure the cylinder is located inside the small red rectangle shown in the Figure 2-a around π₯=0 and π¦=0 location. The grid points and also the cylinder inside this red rectangle is shown with magnification in Figure 2-b. In this Figure 2-b again for a clear view one out of every 4 grid points is plotted in both π- and π-directions. We believe that with the use of large number of grid points and with having the free stream boundary and the outflow boundary sufficiently away from the cylinder and using a stretching http://www.iaeme.com/IJMET/index.asp 1373 editor@iaeme.com Numerical Solutions of Steady Incompressible Flow Around a Circular Cylinder up to Reynolds Number 500 function that allows to concentrate more points close to the cylinder, and also the very low residuals used for convergence criteria assure the accuracy of our numerical solutions. a) computational mesh (one out of every 20 grids is shown) b) enlarged view near the cylinder (one out of every 4 grids is shown) Figure 2 Computational mesh used in the numerical solution We numerically solve the steady incompressible viscous flow around a circular cylinder starting from π π=10 up to π π=500. Figure 3 shows the formation and the growth of the wake bubble as the Reynolds number increases up to π π=100. Figure 3 Growth of the wake bubble as the Reynolds number increases In Figure 3, one can notice that at π π=10 the wake bubble is very small and its height and length grows bigger as the Reynolds number increases. Since Reynolds number range considered in this study is very big, we have plotted the streamfunction (π) and vorticity (π) contours in Figure 4 with βπ π=50 increments up to π π=500. http://www.iaeme.com/IJMET/index.asp 1374 editor@iaeme.com Ercan Erturk and Orhan Gokcol a) streamfunction contours b) vorticity contours Figure 4 Streamfunction (π) and vorticity (π) contours as Reynolds number increases In Figure 4 one can easily notice that after π π=300 the wake bubble behave differently and it starts to grow bigger in height faster. Figure 5 shows the schematic view of the wake bubble. Figure 5 Schematic view of the wake bubble (πΉ: front stagnation point, π: separation point, π : rear stagnation point, π: wake stagnation point) As shown in the schematics we will analyze several parameters such as the length of the wake bubble measured from the center of the cylinder (πΏ), the maximum height of the wake bubble (π» max), the separation angle measured from the front stagnation point (ππ ), the π₯- and π¦location of the vortex centers (π₯center , π¦center) and the streamfunction (πcenter) and vorticity (πcenter) values at the vortex center. Table 1 tabulates all these values as a function of the Reynolds number. http://www.iaeme.com/IJMET/index.asp 1375 editor@iaeme.com Numerical Solutions of Steady Incompressible Flow Around a Circular Cylinder up to Reynolds Number 500 Table 1 Solutions at different Reynolds numbers Reynolds π₯center Number (from the center) 10 1.1956 20 1.6771 30 2.0655 40 2.4272 50 2.7807 60 3.1339 70 3.5931 80 4.0611 90 4.5401 100 5.1438 150 8.7585 200 13.9923 250 20.7981 300 26.8167 350 30.2251 400 33.4746 450 38.2300 500 42.7574 π¦center πcenter πcenter ππ πΏ π» max 0.2400 0.4370 0.5203 0.5919 0.6331 0.6841 0.7189 0.7598 0.8054 0.8302 0.9858 1.1060 1.2202 1.4227 1.7976 2.4378 3.5491 4.9593 -3.4341 × 10−4 -8.3209 × 10−3 -2.2279 × 10−2 -3.7126 × 10−2 -5.1189 × 10−2 -6.4214 × 10−2 -7.6210 × 10−2 -8.7411 × 10−2 -9.7946 × 10−2 -1.0803 × 10−1 -1.5470 × 10−1 -2.0158 × 10−1 -2.6237 × 10−1 -3.7410 × 10−1 -5.6899 × 10−1 -9.3001 × 10−1 -1.6384 -2.6442 -0.04271 -0.18848 -0.27752 -0.34596 -0.38477 -0.42071 -0.44045 -0.45822 -0.47409 -0.47513 -0.47286 -0.45285 -0.44952 -0.43156 -0.37248 -0.31852 -0.26638 -0.22572 2.635 2.382 2.276 2.206 2.152 2.109 2.072 2.041 2.013 1.989 1.900 1.841 1.798 1.765 1.737 1.714 1.692 1.672 1.4774 2.8099 4.1465 5.4792 6.8037 8.1168 9.4179 10.7090 11.9926 13.2718 19.7024 26.3917 33.5003 41.1125 49.4163 58.7430 69.0971 78.7269 0.4822 0.7502 0.9129 1.0270 1.1158 1.1911 1.2590 1.3221 1.3819 1.4406 1.7228 2.0100 2.3245 2.7595 3.5564 4.8906 7.1177 9.8780 Fornberg [5] have plotted the length and the maximum height of the wake bubble with respect to the Reynolds number. Following Fornberg [5], we plotted the same figure using the values given in Table 1. Figure 6 shows the variation of the length of the wake bubble measured from the center of the cylinder (πΏ) and also the maximum height of the wake bubble (π»πππ₯ ) as a function of the Reynolds number. Figure 6 also shows the same results of Fornberg [5] for comparison. We note that the values are scanned and digitized from Fornberg [5] and included in Figure 6. In Figure 6 we can see that our results agree with that of Fornberg [5] up to π π=300 very good, however above π π=300 there is a slight difference. We believe that our results are more accurate. b) maximum height of the wake bubble πππ₯ )bubble Figure 6 Length and the maximum height of the(π» wake a) length of the wake bubble (πΏ) As seen in Figure 6-a, the length of the wake bubble increases almost linearly up to π π=300. Above this Reynold number the length starts to increase with a little faster rate. Also, in Figure 6-b, except π π=10 where the wake bubble is very small and developing, while the rate of the increase in the maximum height of the wake bubble is gradual before π π=300, the increase is fast at higher Reynolds numbers. In Figure 7 we plot the variation of the π₯- and π¦location of the vortex centers and also the streamfunction (π) and vorticity (π) values at the vortex center as a function of the Reynolds number. As it can be seen in Figure 7-a, the π₯location of the vortex center change almost linearly in the Reynolds number regions of 0-100, http://www.iaeme.com/IJMET/index.asp 1376 editor@iaeme.com Ercan Erturk and Orhan Gokcol 100-300 and >300. The colored lines in Figure 7 show the linear regression line fitted to the values between the Reynolds number range of 0-100, 100-300 and ≥ 300. The change in the behavior of the solution is best seen in the vorticity value at the vortex center. In Figure 7-d there is a clear change in the behavior of the vorticity at the vortex center at π π=100 and π π=300, where in the range between 100 ≥ π π ≥ 300 and also π π ≥ 300 the vorticity change almost linearly with different rates. a) π₯-location b) π¦-location c) streamfunction value (π) d) vorticity value (π) Figure 7 Values at the vortex center 4. CONCLUSIONS In this study the steady incompressible viscous flow around a circular cylinder is solved numerically at high Reynolds numbers. Using a very efficient numerical method with large number of grid points, highly accurate numerical solutions are obtained with very low residuals. The presented results agree well with the results found in the literature. Our results show that the solution of the steady incompressible flow around a circular cylinder change behavior at π π=100 and π π=300 as the Reynolds number increases. REFERENCES [1] [2] Schlichting H, Gersten K. “Boundary Layer Theory (8th revised and enlarged edn)”. 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