Acta Polytechnica https://doi.org/10.14311/AP.2021.61.0516 Acta Polytechnica 61(4):516–525, 2021 © 2021 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague STATIONARY SOLUTIONS OF INCOMPRESSIBLE VISCOUS FLOW IN A WALL-DRIVEN SEMI-CIRCULAR CAVITY Ercan Erturk Istanbul Medeniyet University, Mechanical Engineering Department, Goztepe Campus, 34700 Istanbul, Turkey correspondence: ercan.erturk@medeniyet.edu.tr Abstract. Stationary numerical solutions of incompressible viscous flow inside a wall-driven semicircular cavity are presented. After a conformal mapping of the geometry, using a body-fitted mesh, the Navier-Stokes equations are solved numerically. The stationary solutions of the flow in a wall-driven semi-circular cavity are computed up to Re = 24000. The present results are in good agreement with the published results found in the literature. Our results show that as the Reynolds number increases, the sizes of the secondary and tertiary vortices increase, whereas the size of the primary vortex decreases. At large Reynolds numbers, the vorticity at the primary vortex centre increases almost linearly stating that Batchelor’s mean-square law is not valid for wall-driven semi-circular cavity flow. Detailed results are presented and also tabulated for future references and benchmark purposes. Keywords: Driven semi-circular cavity flow, large Reynolds number flow, numerical solutions, 2-D incompressible viscous flow. 1. Introduction Flows in enclosures are studied very frequently in Com- putational Fluid Dynamics studies since they retain rich flow physics in rather simple geometries. The flow in a lid-driven square cavity is one example for flows in enclosures and the driven cavity flow is studied extensively in the literature and among a very large number of computational studies, the studies [1],[2] and [3] can be given as examples. Other than the flow in a square cavity, in the literature, there are studies on flows in different cavity geometries as well. For example, the flows in a driven skewed cavity ([4, 5]), in a driven triangular cavity ([6–8]) and also in a driven trapezoidal cavity ([9]) are studied in numerous nu- merical studies in the literature. In these enclosures, the formation of the flow structures as the Reynolds number increases attracts the attention of researchers. Similar to the flows in different enclosures men- tioned above, Glowinski et al. [10] studied the flow in a semi-circular cavity. They ([10]) used an unsteady operator-splitting/finite elements method and solved the governing flow equations on an unstructured mesh and presented detailed stationary solutions of the semi-circular cavity flow. Later, Yang et al. [11, 12] and Ding et al. [13] numerically studied the same flow problem using the Lattice Boltzmann method. Also, Yu et al. [14] solved the Navier-Stokes equations in polar coordinates using a compact difference scheme and simulated the semi-circular cavity flow problem. Mercan and Atalik [15] considered the arc-shaped cavity flow for the semi-circular case numerically using an unsteady finite difference method. They ([15]) used an elliptic grid generator scheme in order to obtain a body-fitted general coordinates and solve the governing streamfunction and vorticity equations. In their study, they ([15]) have also presented variations of the arc shaped cavity flow with different aspect ratios. Migeon et al. [16] carried out experiments to study the flow development inside a semi-circular cavity as well as inside square and rectangular cavities. They ([16]) presented qualitative experimental flow visual- izations of the semi-circular cavity flow. At large Reynolds numbers, the stationary flow loses stability and eventually, a transition to turbulence oc- curs. For the Navier-Stokes equations, the stationary solution still exists at large Reynolds numbers along- side with the transient solutions. As stated in [17] and [18], for the theory of the Navier-Stokes model, it is important to study the stationary solutions at large Reynolds numbers even when the flow loses stabil- ity. In order to illustrate the point, the classical flow problem of the incompressible viscous flow around a circular cylinder can be given as an example. For this flow problem, it is well known that the flow is sta- tionary up to Reynolds number of approximately 40. Above this Reynolds number (Re = 40), the flow over a circular cylinder becomes unsteady and Karman vor- tex street appears in the downstream flow field. In the case of vanishing viscosity (i.e. large Reynolds num- ber), Helmholtz-Kirchhoff solution (see [19]) presents a limiting solution when Re → ∞. In the literature, Smith [20] and Peregrine [21] analysed the laminar incompressible flow past a circular cylinder at large Reynolds numbers mathematically. Also, in the lit- erature, there are numerical studies that analysed the incompressible steady viscous flow past a circu- lar cylinder at large Reynolds numbers (well above Re = 40), such as [17, 18, 22–27]. As Christov et al. [27] stated, when Re → ∞, without dissipation, the limiting solutions of the Navier-Stokes is an im- portant fundamental problem. 516 https://doi.org/10.14311/AP.2021.61.0516 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 61 no. 4/2021 Wall-Driven Semi-Circular Cavity Flow In this study, we numerically investigate the be- havior of the stationary solutions of an incompress- ible viscous flow in a wall-driven semi-circular cav- ity at large Reynolds numbers. In our study, unlike in [10, 11, 13, 15], as discussed in [28], we use a steady approach and solve the governing steady Navier-Stokes equations. We used complex algebra and mapped the considered semi-circular geometry into an infinite half domain conformally in order to analytically obtain a body-fitted mesh. For the solution of the semi- circular cavity flow problem, the streamfunction and vorticity equations are solved iteratively up to large Reynolds numbers. The present numerical results are compared in detail with the results of [10, 12, 14, 15]. For the flows in enclosures, Batchelor’s mean square law ([29]) states that the flow, which is coupled to the solid wall velocities at the boundaries, should have a solid body motion inside the enclosure with a uni- form vorticity at large Reynolds numbers and the flow quantities should converge to the limiting values. We examine whether or not the Batchelor’s model ([29]) is valid for the wall-driven semi-circular cavity flow problem. Detailed results are presented. 2. Numerical method The schematic view of the considered flow problem, i.e., the semi-circular cavity flow, is given in Figure 1. The width of the top moving lid, i.e., the straight top side of the semi-circular cavity, is equal to 1 and the velocity of this lid is also equal to 1. The steady incompressible viscous flow in a wall- driven semi-circular cavity is governed by the Navier- Stokes equation. We consider the governing equations in streamfunction and vorticity formulation as the following ψyωx − ψxωy = 1 Re (ωxx + ωyy) (1) ψxx + ψyy = ω (2) where Re is the Reynolds number based on the velocity of the top lid (U) and the width of the lid (L) which are both equal to unity. TV TV end SV begin SV end begin Lid width=L Figure 1. Schematic view of the considered flow problem, boundary conditions and flow topology of the semi-circular cavity flow. 2.1. Numerical mesh In computational fluid dynamics, it is always preferred to use a body-fitted orthogonal coordinate system for numerical solutions. Using a complex variable analysis, one can obtain that the following complex function w = ln z − 1 2 z + 1 2 (3) conformally maps the considered semi-circular cavity geometry into a body-fitted orthogonal coordinate system. In the mapped complex w-domain, the real and imaginary parts are given as w = ϕ+ iψ where ψ and ϕ are the coordinate axes in the mapped domain. Also, the complex z-domain represents the x-y plane as the following z = x + iy. We note that, one can obtain this transformation also using the potential flow analysis with the superposition of a source located at (− 1 2 ,0) coordinates and a sink located at ( 1 2 ,0) coordinates with equal strengths of 2π as shown in Figure 2. Upon separating the real and imaginary parts, the velocity potential function is the real part where ϕ = ln ∣∣z − 1 2 ∣∣∣∣z + 1 2 ∣∣ = ln √( x− 1 2 )2 + y2√( x+ 1 2 )2 + y2 (4) and also the stream function is the imaginary part where ψ = arctan y x− 1 2 − arctan y x+ 1 2 (5) In this study we used the ψ and ϕ coordinates in order to construct a body-fitted mesh for the semi- circular cavity. For any value of the velocity potential function, such that ϕ = c, we have c = ln √( x− 1 2 )2 + y2√( x+ 1 2 )2 + y2 (6) y x Figure 2. Velocity potential and stream function lines of the potential flow with a source and a sink with equal strengths. 517 Ercan Erturk Acta Polytechnica Figure 3. Body-fitted numerical mesh (1 out of every 10 grids is shown). or by rearranging it, we have x2 + y2 + e2c + 1 e2c − 1x+ 1 4 = 0 (7) We note that the above equation defines a system of coaxial circles with their centres being on the x- axis as shown in Figure 2. The centre coordinates of these coaxial circles are given as x = − 1 2 e2c+1 e2c−1 , y = 0 . Also, the radii of these circles are equal to r = 1 2 √( e2c+1 e2c−1 )2 − 1. We also note that when c = 0, the radius of the circle becomes infinite, which is a perpendicular straight line at the origin. Following the same procedure, for any value of the stream function, such that ψ = c, we have c = arctan y x− 1 2 − arctan y x+ 1 2 (8) By rearranging it, we get x2 + y2 − y tan c − 1 4 = 0 (9) We note that the above equation defines a sys- tem of coaxial circles with their centres being on the y-axis as also shown in Figure 2. The centre coor- dinates of these coaxial circles are given as x = 0, y = 1 2 tan c. Also, the radii of these circles are equal to r = 1 2 √ 1 (tan c)2 + 1. We note that each of these circles passes through the location of the source and the sink, i.e., through the points (− 1 2 ,0) and ( 1 2 ,0). Using the equations given above, we can easily ob- tain the mesh points inside the semi-circular cavity. In order to keep the aspect ratio ( ∆x ∆y ) of the grid points in the mesh more or less around unity in ma- jority of the domain, we used 600×150 grid points in the mesh. We divide the upper lid of the arc shaped cavity into 600 equal points and we calculate the val- ues of the velocity potential function (ϕ = c) at these points. Similarly, we divide the vertical line at x = 0 into 150 equal points and we calculate values of the stream function (ψ = c) at these points. Then, the intersection points of these velocity potential function (ϕ) and stream function (ψ) are obtained as the grid point locations. The body-fitted orthogonal mesh we used is given Figure 3. 2.2. Governing equations in the computational domain Since ϕ and ψ provide a body-fitted orthogonal coor- dinate system for the considered semi-circular cavity geometry, the natural choice is to use ϕ and ψ as the coordinate axes in the mapped domain. However, the value of ϕ at the corners of the arc shaped cavity (i.e., at the location of the source and the sink) is equal to ±∞. For this reason, it was not possible to use ϕ and ψ as coordinate axes. Also, using ϕ and ψ as coordinate axes (except at the the corner points) introduces extra difficulty in finite differencing of the spatial derivatives on a non-uniform mesh. In order to overcome this difficulty, we transform the curvilinear mesh in physical domain to a rectangular uniform mesh in computational domain with ξ and η coordi- nates where the grid spacing is uniform and equal to unity (i.e. ∆ξ = ∆η = 1). The x and y derivatives are transformed to the computational domain as the following fx = yηfξ − yξfη J (10) fy = −xηfξ + xξfη J (11) where f is any differentiable function and J is the Jacobian of the transformation, which is defined as J = xξyη − xηyξ (12) In the computational ξ-η domain, the governing streamfunction and vorticity equations are trans- formed as the following αψξξ − 2βψξη + γψηη + σψη + τψξ J2 = ω (13) ψηωξ − ψξωη = 1 Re αωξξ − 2βωξη + γωηη + σωη + τωξ J (14) where the coefficients in these equations (α, β, γ, σ, τ) are defined as α = x2 η + y2 η β = xξxη + yξyη γ = x2 ξ + y2 ξ σ = yξ (Dx) − xξ (Dy) J τ = xη (Dy) − yη (Dx) J (15) where Dx and Dy are Dx = αxξξ − 2βxξη + γxηη Dy = αyξξ − 2βyξη + γyηη (16) Since we are only interested in the stationary solu- tions rather than the transient solutions in this study, 518 vol. 61 no. 4/2021 Wall-Driven Semi-Circular Cavity Flow (a) . Re = 500 (b) . Re = 1000 (c) . Re = 2000 (d) . Re = 3000 (e) . Re = 4000 (f) . Re = 5000 (g) . Re = 6600 (h) . Re = 8000 (i) . Re = 10000 (j) . Re = 15000 (k) . Re = 20000 (l) . Re = 24000 Figure 4. Streamline contours of semi-circular cavity flow as Reynolds number increases. we used the Successive Over Relaxation (SOR) method in order to solve the governing steady streamfunction and vorticity equations numerically. These governing equations are numerically solved in the computational domain using 3 point second order accurate central differencing, O(∆ξ2,∆η2). 2.3. Mapping transformation metrics We calculate the mapping transformation metrics (xξ, xη, yξ, yη) numerically using finite difference. The mapping transformation metrics appear explicitly in the coefficients (α, β, γ, σ, τ) of the governing stream- function and vorticity equations (13 and 14). In order to minimize the effect of the finite difference approxi- mation errors associated with numerically calculated mapping transformation metrics on the numerical so- lution of the governing flow equations, we decided to use fourth order accurate differencing, O(∆ξ4,∆η4), in calculating the mapping metrics, which is higher than the accuracy of the streamfunction and vorticity finite difference equations. With high order accurate finite difference approximation for the mapping trans- formation metrics, we obtain more accurate numerical solutions for streamfunction and vorticity equations. We note that since we calculate the mapping metrics only once, the increase in the computational effort when using higher accuracy for the mapping metrics is insignificant as compared to the computational ef- fort used for solving the governing streamfunction and vorticity equations. At the interior points of the computational domain, the mapping metrics are calcu- lated using the following 5 point central fourth order finite difference equation (fη)i,j = fi,j−2 − 8fi,j−1 + 8fi,j+1 − fi,j+2 12 (17) where again, f is any differentiable function, which, in our case, can be x or y and the subscripts i and j de- note the grid index in ξ and η directions, respectively. Using the same finite difference equation, the trans- formation metrics for ξ-direction (fξ) are calculated similarly. Near the boundaries of the computational domain, at the first grid points above the wall, the mapping metrics are calculated using the following 5 519 Ercan Erturk Acta Polytechnica (a) . Re = 500 (b) . Re = 1000 (c) . Re = 2000 (d) . Re = 3000 (e) . Re = 4000 (f) . Re = 5000 (g) . Re = 6600 (h) . Re = 8000 (i) . Re = 10000 (j) . Re = 15000 (k) . Re = 20000 (l) . Re = 24000 Figure 5. Vorticity contours of semi-circular cavity flow as Reynolds number increases. point one-sided fourth order finite difference equation (fη)i,1 = −3fi,0 − 10fi,1 + 18fi,2 − 6fi,3 + fi,4 12 (18) where the subscripts 0 refer to the wall grid points and 1, 2, 3 and 4 refer to grid points above the wall in the order. Similarly, on the boundaries of the computational domain at the points on the wall, the mapping metrics are calculated using the following 5 point one-sided fourth order finite difference equation (fη)i,0 = −25fi,0 + 48fi,1 − 36fi,2 + 16fi,3 − 3fi,4 12 (19) Using the above finite difference equations, the trans- formation metrics xξ, xη, yξ, yη are calculated with fourth order accuracy O(∆ξ4,∆η4). 2.4. Wall boundary conditions Using the velocity of the top moving lid and simplify- ing terms, at the top wall, we obtain U = ψy = −xηψξ + xξψη J = xξψη J = 1 (20) Simplifying the streamfunction equation and also substituting the above equation, at the grid points on the top moving wall, the vorticity value is obtained as the following ω(i, 0) = γ J2 ( 2ψ(i, 1) + 2 J xξ ) (21) Similarly, on the bottom curved wall, the vorticity value is obtained as the following ω(i, 0) = 2 γ J2ψ(i, 1) (22) 3. Results and discussion As convergence criteria in our simulations, we use the change in the streamfunction and vorticity variables between two consecutive iterations, which is normal- ized by the previous values defined as the following Residualψ = max (∣∣∣∣∣ψ k+1 i,j − ψki,j ψki,j ∣∣∣∣∣ ) (23) 520 vol. 61 no. 4/2021 Wall-Driven Semi-Circular Cavity Flow 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 u-velocit y y -a xi s Re=500 Re=1000 Re=2000 Re=3000 Re=4000 Re=5000 Re=6600 Re=8000 Re=10000 Re=15000 Re=20000 Re=24000 Figure 6. The u-velocity profiles along the vertical line at x = 0 at various Reynolds numbers, symbols [10], symbols [13] Residualω = max (∣∣∣∣∣ω k+1 i,j − ωki,j ωki,j ∣∣∣∣∣ ) (24) where max denotes the maximum value in the solution domain and the superscript denotes the iteration num- ber. In our numerical solutions at the convergence both Residualψ and Residualω are less than 10−6. This means that at the convergence, the streamfunc- tion and vorticity values change less than one millionth of their values between two consecutive iterations at a grid point in the computational domain as the max- imum in absolute value and even less in the rest of the grid points in the computational domain. For the mesh generation and also for the numer- ical solution of the governing equations, computer codes in C++ programming language are written and using these codes, we numerically solve the incom- Re ψmin (x, y) 1000 present study -0.07808 (0.6193,-0.2030) Glowinski et al. [10] -0,0779 (0.6214,-0.2030) Yang et al. [12] -0.0775 (0.6201, -0.2060) Yu et al. [14] -0.0779 (0.6225, -0.2044) 2000 present study -0.07674 (0.6359,-0.2062) Glowinski et al. [10] -0.0763 (0.6359,-0.2052) Yang et al. [12] -0.076 (0.6376, -0.2061) Yu et al. [14] -0.0764 (0.6367, -0.2046) 3000 present study -0.07461 (0.6551,-0.2020) Glowinski et al. [10] -0.0742 (0.6514,-0.2027) Yang et al. [12] -0.0737 (0.6553, -0.2041) Yu et al. [14] -0.0740 (0.6636, -0.1993) 5000 present study -0.07026 (0.6891,-0.1931) Glowinski et al. [10] -0.07 (0.6833,-0.1936) Yang et al. [12] -0.069 (0.6846, -0.1963) Yu et al. [14] -0.0691 (0.7055, -0.1863) Mercan et al. [15] -0.0694 (0.7132,-0.1825) 6600 present study -0.06718 (0.7096,-0.1868) Glowinski et al. [10] -0.067 (0.7009,-0.1891) Yang et al. [12] -0.0657 (0.7002, -0.1905) Yu et al. [14] -0.0652 (0.7353, -0.1745) Mercan et al. [15] -0.0656 (0.7464,-0.1704) Table 1. Comparison of the minimum streamfunction and location of center of the primary vortex. pressible viscous flow in a semi-circular cavity. We start solving the considered flow for Reynolds number of 500. Then, we progressively increase the Reynolds number by using the previous Reynolds number solu- tion as the initial guess for the next highest Reynolds number. With this approach, we obtain numerical solutions of the wall-driven semi-circular cavity flow up to Reynolds number of 24000. The streamfunction and vorticity contours pre- sented in Figure 4 and Figure 5 show the change in the flow topology in a semi-circular cavity as the Reynolds number increases. From Figure 4 and Figure 5, we can see that between Re = 500 and Re = 1000 and Re = 4000 and Re = 5000, a secondary and a ter- tiary vortex, respectively, appear in the flow field. We can also see that as the Reynolds number increases, the size of both the secondary and the tertiary vor- tex increases where as the size of the primary vortex decreases. Glowinski et al. [10], Yang et al. [12], Yu et al. [14] and Mercan et al. [15] have presented detailed data from the flow field. In Table 1 we compare the present results of the minimum streamfunction value and its location in the semi-circular cavity geometry with the results presented in [10, 12, 14, 15]. In their study, Glowinski et al. [10] also presented the detachment angles of the secondary and tertiary vortex measured counterclockwise from the top lid as shown in Figure 1. On the wall boundary, the skin-friction coefficient is Re θbegin SV θend SV θbegin T V θend T V Present study 1000 20.83 74.83 - - 2000 10.87 97.01 - - 3000 8.38 107.94 - - 5000 6.39 121.23 59.12 76.44 Glowinski et al. [10] 1000 21.42 71.49 - - 2000 10.78 94.34 - - 3000 7.77 104.53 - - 5000 6.70 117.09 57.54 73.84 Table 2. Comparison of the detachment angles of the secondary and tertiary vortex. 521 Ercan Erturk Acta Polytechnica Re y 500 1000 2000 3000 4000 5000 6600 8000 10000 15000 20000 24000 0 1 1 1 1 1 1 1 1 1 1 1 1 -0.0333 0.6094 0.5398 0.4523 0.3901 0.3166 0.2239 0.0453 -0.1074 -0.2528 -0.3103 -0.2870 -0.2660 -0.0667 0.3734 0.3962 0.3932 0.3531 0.2973 0.2314 0.1075 0.0043 -0.0606 -0.0828 -0.1135 -0.1315 -0.1 0.2636 0.3161 0.3005 0.2609 0.2175 0.1745 0.1056 0.0437 -0.0042 -0.0630 -0.0937 -0.1030 -0.1333 0.1729 0.2151 0.1954 0.1631 0.1299 0.0998 0.0590 0.0247 -0.0020 -0.0478 -0.0648 -0.0716 -0.1667 0.0733 0.1081 0.0955 0.0712 0.0453 0.0219 -0.0082 -0.0221 -0.0118 -0.0308 -0.0390 -0.0433 -0.2 -0.0321 0.0040 0.0012 -0.0169 -0.0379 -0.0581 -0.0830 -0.0693 -0.0166 -0.0120 -0.0139 -0.0161 -0.2333 -0.1371 -0.0962 -0.0894 -0.1028 -0.1212 -0.1407 -0.1534 -0.0890 -0.0092 0.0109 0.0120 0.0108 -0.2667 -0.2346 -0.1940 -0.1778 -0.1879 -0.2060 -0.2250 -0.1873 -0.0627 0.0108 0.0383 0.0386 0.0374 -0.3 -0.3130 -0.2902 -0.2665 -0.2746 -0.2922 -0.2897 -0.1400 -0.0128 0.0332 0.0676 0.0650 0.0634 -0.3333 -0.3546 -0.3745 -0.3560 -0.3592 -0.3471 -0.2547 -0.0488 0.0197 0.0393 0.0886 0.0915 0.0884 -0.3667 -0.3433 -0.4120 -0.4148 -0.3813 -0.2693 -0.1121 0.0095 0.0256 0.0256 0.0744 0.1134 0.1144 -0.4 -0.2785 -0.3593 -0.3556 -0.2464 -0.0963 -0.0101 0.0250 0.0167 0.0084 0.0287 0.0827 0.1189 -0.4333 -0.1811 -0.2253 -0.1768 -0.0668 0.0027 0.0228 0.0182 0.0039 -0.0038 0.0000 0.0160 0.0384 -0.4667 -0.0809 -0.0836 -0.0309 0.0122 0.0238 0.0225 0.0072 -0.0055 -0.0087 -0.0058 -0.0060 -0.0046 -0.5 0 0 0 0 0 0 0 0 0 0 0 0 Table 3. Selected u-velocity values along x = 0 line. defined as Cf = τw ρU2 (25) where τw = µ∂u∂y ∣∣∣ w . At the detachment points on the wall, the skin-friction coefficient will be zero. In our study, by calculating the skin-friction coefficient, we have located the detachment points in the lower circular wall of the semi-circular cavity. In Table 2, we compare our results of the detachment angles of the secondary and tertiary vortex with the results of Glowinski et al. [10]. Present results in Table 1 and Table 2 are in good agreement with the results of Glowinski et al. [10], Yang et al. [12], Yu et al. [14] and Mercan et al. [15]. In Figure 6, we plot the computed u-velocity profiles along the vertical line at the mid of the semi-circular cavity, i.e., at x = 0 line, at various Reynolds numbers. In the same figure, the velocity profiles of Glowinski et al. [10] and Ding et al. [13] are also given with red and green symbols, respectively. As seen in Figure 6, at low Reynolds numbers, our computed u-velocity profiles are in good agreement with the u-velocity profiles of Glowinski et al. [10] and Ding et al. [13], where as at large Reynolds numbers, our velocity profiles deviate from those of Glowinski et al. [10] and Ding et al. [13]. We believe that due to the fine mesh used in the present study, our results are more accurate. In Table 3, we tabulate selected values of the u-velocity values along x = 0 line at various Reynolds numbers for future references. In Table 4 and Table 5, we present the streamfunc- tion and vorticity values at the centres of the vortices together with the centre locations and also the de- tachment angles of the secondary and tertiary vortex, respectively, for all the Reynolds numbers considered in this study for benchmark purposes. In order to see the behaviour of the detachment points better, in Figure 7, we plot the detachment angles given in Table 4 as a function of the Reynolds number. From Figure 7, we can see that while θbeginSV seems to converge to a value at large Reynolds num- 0 20 40 60 80 100 120 140 160 180 0 5000 10000 15000 20000 25000 A ng le [d eg ] Reynolds Number θSV begin θSV end θTV begin θTV end Figure 7. Detachment angles of the secondary and tertiary vortex as a function of the Reynolds number. bers, θendSV , θbeginTV and θendTV are still increasing as the Reynolds number increases. This states that the solu- tion of the semi-circular cavity flow is still changing even at the highest Reynolds number considered in the present study (i.e. Re = 24000). At large Reynolds number, the Batchelor’s model ([29]) states that the steady laminar flow with closed streamlines moves like a solid body with a uni- form vorticity at the inviscid vortex. Erturk [2] showed that in the driven square cavity flow at large Reynolds numbers in the primary vortex, the vorticity is almost uniform such that the core fluid rotates like a solid body in agreement with the Batchelor’s model ([29]), also, at the primary vortex centre, the vorticity value converges to the limiting value analytically calculated by Burggraf [30]. In the case of the driven semi- circular cavity flow at large Reynolds numbers, from Figure 4 and Figure 7, we can see that the sizes of the primary, secondary and tertiary vortices are still changing significantly even at the highest considered Reynolds number of Re = 24000, while the size of the primary vortex decreases, the size of the secondary and tertiary vortices increases. This indicates that 522 vol. 61 no. 4/2021 Wall-Driven Semi-Circular Cavity Flow Re ψ ω x y 500 PV -7.6674E-02 4.73550 0.1334 -0.1940 1000 PV -7.8084E-02 4.53861 0.1193 -0.2030 SV 4.1296E-04 -0.58656 -0.3456 -0.2570 2000 PV -7.6735E-02 4.57210 0.1359 -0.2062 SV 3.8975E-03 -1.37969 -0.3250 -0.1817 3000 PV -7.4606E-02 4.70449 0.1551 -0.2020 SV 6.5343E-03 -1.48358 -0.3096 -0.1836 4000 PV -7.2385E-02 4.86208 0.1722 -0.1978 SV 8.6265E-03 -1.47665 -0.2829 -0.1933 5000 PV -7.0263E-02 5.02531 0.1891 -0.1931 SV 1.0289E-02 -1.47578 -0.2609 -0.1986 TV -1.5422E-05 0.13901 -0.1629 -0.4461 6600 PV -6.7180E-02 5.28290 0.2096 -0.1868 SV 1.2349E-02 -1.46225 -0.2396 -0.2052 TV -1.5500E-04 0.37152 -0.0828 -0.4365 8000 PV -6.4793E-02 5.50007 0.2252 -0.1786 SV 1.3747E-02 -1.44986 -0.2236 -0.2086 TV -3.0604E-04 0.47522 -0.0321 -0.4323 10000 PV -6.1823E-02 5.79482 0.2432 -0.1720 SV 1.5309E-02 -1.43214 -0.2037 -0.2128 TV -4.8690E-04 0.56585 0.0230 -0.4261 15000 PV -5.6092E-02 6.45364 0.2757 -0.1561 SV 1.7854E-02 -1.39370 -0.1747 -0.2220 TV -7.8118E-04 0.67408 0.1123 -0.4074 20000 PV -5.1952E-02 7.02646 0.2953 -0.1447 SV 1.9332E-02 -1.36248 -0.1543 -0.2243 TV -9.7761E-04 0.73133 0.1649 -0.3892 24000 PV -4.9336E-02 7.44717 0.3088 -0.1382 SV 2.0103E-02 -1.34078 -0.1425 -0.2272 TV -1.0986E-03 0.75759 0.1919 -0.3758 Table 4. Streamfunction, vorticity at the centres of vortices and (x,y) locations at different Reynolds numbers. even at Re = 24000, the flow is still evolving and the flow quantities are not converging to a value asymp- totically. However, in the case of the square driven cavity in Erturk [2], the change in the the primary vortex’s size or the change in the flow quantities at the centre of the primary vortex becomes very small as the Reynolds number increases to high values converging to a stable value. In order to see the behaviour of the driven semi-circular cavity flow at large Reynolds numbers better, in Figure 8, we plot the vorticity value at the centre of the primary vortex with respect to the Reynolds numbers, which are tabulated in Table 3. From Figure 8, we can see that the vorticity value at the primary vortex centre continuously increases almost linearly with respect to the Reynolds number without converging to a value even at Re = 24000. Present results show that at large Reynolds numbers, the driven semi-circular cavity flow does not agree with the Batchelor’s model ([29]). 4. Conclusion In this study we have presented stationary solutions of an incompressible viscous flow in a wall-driven semi- circular cavity at large Reynolds numbers. The govern- ing equations are numerically solved up to Re = 24000 Re θbegin SV θend SV θbegin T V θend T V 1000 20.83 74.83 - - 2000 10.87 97.01 - - 3000 8.38 107.94 - - 4000 7.16 115.52 - - 5000 6.39 121.23 59.12 76.44 6600 5.61 128.01 59.28 88.75 8000 5.14 132.45 60.38 96.53 10000 4.66 137.28 62.15 105.01 15000 3.90 145.04 65.99 118.75 20000 3.43 149.75 68.79 127.04 24000 3.16 152.44 70.52 131.72 Table 5. Flow detachment angles of the secondary (SV) and tertiary (TV) vortices at different Reynolds numbers. 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 0 5000 10000 15000 20000 25000 Reynolds Number w pr im ar y vo rt e x Figure 8. 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