A detailed study of lid-driven cavity flow at moderate Reynolds numbers using Incompressible SPH

Transcription

1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2014; 76: Published online 28 August 2014 in Wiley Online Library (wileyonlinelibrary.com) A detailed study of lid-driven cavity flow at moderate Reynolds numbers using Incompressible SPH Shahab Khorasanizade and Joao M. M. Sousa*, Instituto Superior Técnico, Mechanical Engng. Dept., Av. Rovisco Pais, Lisboa, Portugal SUMMARY Lid-driven cavity flow at moderate Reynolds numbers is studied here, employing a mesh-free method known as Smoothed Particle Hydrodynamics (SPH). In a detailed study of this benchmark, the incompressible SPH approach is applied together with a particle shifting algorithm. Additionally, a new treatment for no-slip boundary conditions is developed and tested. The use of the aforementioned numerical treatment for solid walls leads to significant improvements in the results with respect to other SPH simulations carried out with similar spatial resolution. However, the effect of spatial resolution is not considered in the present study as the number of particles used in each case was kept constant, approximately reproducing the same resolutions employed in reference studies available in the literature as well. Altogether, the detailed comparisons of field variables at discreet points demonstrate the accuracy and robustness of the new SPH method. Copyright 2014 John Wiley & Sons, Ltd. Received 14 February 2014; Revised 11 June 2014; Accepted 26 July 2014 KEY WORDS: incompressible smoothed particle hydrodynamics; mesh-free method; no-slip boundary condition; lid-driven cavity 1. INTRODUCTION Traditional mesh-based numerical methods have been widely applied in different fields of science and engineering, but the growing interest in complex geometry simulations limits the application of this kind of methods. One of the ways to overcome this drawback is to use mesh-less methods instead. Smoothed Particle Hydrodynamics (SPH) is among this category. By nature, it is a Lagrangian method, which discretizes the domain by introducing particles as computational points that move in the flow according to the equation of motion. It was first introduced by Lucy [1] and Gingold and Monaghan [2] for compressible astrophysical flows. The application of SPH to incompressible flows started with the work of Morris et al. [3] and continued to this day. Although most SPH studies have been restricted to free surface flows, thanks to its unique abilities to simulate such flows, several efforts have been made to improve the method for confined flows as well [4 9]. In order to deal with incompressible flows, researchers resort to two different types of approaches. The first scheme is referred to as Weakly Compressible SPH (WCSPH) [4]. It assumes that the fluid has some compressibility, all the flow equations are solved explicitly and an equation of state is used to relate density, and pressure. Compared with the second scheme, which is based on a projection method and often called Incompressible SPH (ISPH), classical WCSPH exhibits a few drawbacks such as reduced accuracy in the calculation of pressure [4, 10]. Moreover, the stability requirement to use considerably smaller increments in time advancement is another limitation faced by WCSPH, although ISPH incurs a supplementary computational cost in order to solve the *Correspondence to: Joao M. M. Sousa, Instituto Superior Técnico, Mechanical Engng. Dept., Av. Rovisco Pais, Lisboa, Portugal. msousa@ist.utl.pt Copyright 2014 John Wiley & Sons, Ltd.

2 654 S. KHORASANIZADE AND J. M. M. SOUSA pressure Poisson equation implicitly. Recently, researchers dedicated their efforts to improve the accuracy of SPH for both algorithms, in a number of studies [5 7,11 13] among others. Within WCSPH, the SPH-ALE algorithm, first introduced by Vila [14], shows great capabilities as pointed out by Marongiu et al. [13]. In addition, δ-sph also allows major improvements over classical WCSPH via the addition of a diffusive term to the equations [11]. The treatment of solid walls in SPH simulations, however, requires great care. A variety of methods has been proposed, such as the use of image particles [10], dummy particles [4], the multiple boundary tangent method [15], and semi-analytical approaches [16], to name a few. All of these methods, except the latter, which is based on a variational formulation, can be used to place the particles inside the solid wall, while the choice of the velocity boundary condition (BC) is less straightforward and has a large effect on the results as investigated by Basa et al. [17]. Some of these features have also been studied by González et al. [18] for various implementations, thus demonstrating that an inconsistent shear force calculation at initial steps can later be balanced by another boundary force appearing when slip velocities are present at the boundary. Anyway, the foregoing issues become of great concern when dealing with confined flow problems such as the lid-driven cavity. The lid-driven cavity problem involves viscous flow inside a square cavity in which the top wall moves with a constant velocity, whereas the other walls remain stationary [19]. It is tackled frequently as a benchmark case in the framework of SPH [20]. Numerical schemes are tested in that context with respect to robustness, accuracy, and efficiency. No analytical solution exists for this flow problem, but it has been extensively studied, employing panoply of methods [19, 21 24]. Erturk [21] divides numerical studies of lid-driven cavity flow into three categories: steady solutions, hydrodynamic stability analyses, and direct numerical simulations of the transition from steady regime to unsteady flow. A number of studies carried out for moderate Reynolds numbers using SPH can also be found in the literature (e.g., [4, 5, 17]). Within the first of the aforementioned categories, there are controversial issues such as the limit of steadiness. Some researchers, such as Ghia et al. [19], showed steady solutions up to a Reynolds number Re = 10000, while others [25] claimed that the flow is steady up to a value as high as Re = According to Erturk [21], one should always remember that steady, twodimensional lid-driven cavity flow does not occur at high Reynolds numbers in reality and such flows are therefore fictitious. Moreover, he pointed out that the second and third categories of studies for this problem contradict the first category in the sense of steadiness [21]. In the present work, we have developed a new solid wall BC treatment via changing the velocity profile for the particles placed inside the wall, with the aim of calculating wall viscous forces more accurately. The implementation is tested on lid-driven cavity flow at four Reynolds numbers, namely Re = 100, 400, 1000, and 3200, and the results are compared with available data in literature [5, 17, 19, 22, 23]. Additionally, a much more detailed flow analysis that those performed in previous SPH studies of this benchmark is provided. In Section 2, we review SPH formulations, governing equations and the ISPH algorithm used in present investigation. Section 3 is devoted to the proposed BC, and its effect on the velocity distribution inside the wall is discussed. In Section 4, the complete SPH method is tested on lid-driven cavity flow at moderate Reynolds numbers, and the results are extensively scrutinized. Major improvements with respect to other SPH simulations previously published were found upon the use of the methodology described herein. 2. SPH METHODOLOGY According to Monaghan [26], SPH is based on the filtering (smoothing) of any generic field f with a convolution integral extended over the fluid domain Ω hf iðþ¼ r Ω f ðr Þwðr r; hþdv ; (1) where w is a weighting function, which has a compact support of size h bounded to Ω, and r represents the spatial coordinates of a generic point. The weighting function is known in the context of SPH as a kernel function, and it is defined as positive and symmetric. Apart from the foregoing properties, this

3 A DETAILED STUDY OF LID-DRIVEN CAVITY FLOW USING INCOMPRESSIBLE SPH 655 function is also expected to exhibit a few other specific characteristics, as described in detail by Monaghan [26] and Liu et al. [27]. Equation (1) can be applied to the gradient of a function as well. After mathematical manipulation, it reads as h f iðþ¼ r Ω f ðr Þ wðr r; hþdv þ Ω f ðr Þwðr rþn ds ; (2) where w stands for the derivative with respect to r, Ω is the boundary of Ω, and n indicates the normal unit vector of the boundary. It can be shown that if the particle s compact support is within the domain of integration, the surface integral in Equation (2) vanishes, whereas in the case of nearboundary particles, kernel truncation occurs. However, the latter case can be remedied by the usage of boundary particles [27]. Based on these equations, the SPH scheme is obtained by substituting continuous fields with discrete Lagrangian particles carrying field values (e.g., velocity and pressure). In order to interpolate a field f at a calculation point i, one uses the following convolution summation, which is the particle representation of Equation (1): m j f i ¼ j f ρ j w ij ; (3) j where i and j denote the target and its neighboring particle, respectively, m and ρ stand for the particle mass and density, respectively, and w ij is the kernel function centered on i as calculated at j. Here, the gradient of a field is given by the formula used in [5], which reads as follows: f i ¼ j m j ρ j f j f i wij : (4) 2.1. Governing equations Two different approaches can be used to simulate incompressible flows in SPH. The first one is called WCSPH, which solves the Navier Stokes (NS) equations in Lagrangian form fully explicitly with the help of an equation of state that relates pressure and density. In this method, fluid is considered to have a small compressibility. The second method, which is the focus of this work, is truly ISPH, thereby solving the Lagrangian NS equations by means of a projection method. Various projection methods exist [5], among which we choose the divergence free method. With respect to WCSPH, this methodology has the advantage of allowing larger time steps, thanks to its implicit nature. Nevertheless, solving the ensuing system of equations may be very costly as well in some situations. The Lagrangian NS equations to be used for ISPH are 8 u ¼ 0 ; >< du dt ¼ p ρ þ ν 2 u þ F ; (5) >: dr dt ¼ u ; where u is the velocity vector, p is the pressure, ρ and ν stand for the density and viscosity of the fluid, respectively, F represents the external forces (e.g., gravity or other driving force), and r is the position vector. In order to simulate fluid flows using the system of Equations (5), it is convenient to split it into a predictor step and a corrector step. Intermediate values (within a time step Δt) are initially calculated at the predictor step and used later, at the corrector step, in the solution of a Poisson equation for the field pressure. In simple versions of the so-called projection method, sometimes referred to as the homogeneous scheme [7], the pressure gradient is omitted in the calculation of the intermediate velocity field. In this case, the predictor step relations are simply as follows: r i ¼ r n i þ Δt u n i ; (6) u i ¼ u n i þ ν 2 u n i þ F n i Δt: (7)

4 656 S. KHORASANIZADE AND J. M. M. SOUSA According to Equations (6) and (7), an intermediate position (r * i ) and velocity (u * i ) for particle i is calculated based on the values of position, velocity, and external force from the previously completed iteration (superscript n). Subsequently, the pressure is used to project the velocity into a divergence free space, thus allowing the calculation of the pressure field in the corrector step, as well as the final velocity field to move the particles to their final position at the end of this time iteration. The corrector step relations are r nþ1 i 1 ρ pnþ1 u nþ1 i ¼ 1 Δt u i ; (8) ¼ u i Δt ρ pnþ1 i ; (9) ¼ r n i þ Δt ui nþ1 þ u n i 2 : (10) In Equations (8) (10), corrected values of the pressure field obtained via the corresponding Poisson equation, velocity, and position for the next time iteration (superscript n + 1) are for particle i. An alternative formulation is followed by Lee et al. [4], calculating the intermediate velocity at the position given by the previous iteration instead. The consequences of this option have already been investigated [5], and the results were found to be identical to those obtained by Cummins and Rudman [10]. In the present study, the so-called negative form of the relations for both the pressure gradient and velocity divergence is used, despite the more widespread use of the positive form in the context of ISPH [4, 10]. Following Basa et al. [17], the discreet form of these equations is p i ¼ j m j ρ j p j p i wij ; (11) m j u i ¼ j u j u i wij ; (12) ρ j where the kernel gradient w ij is calculated at the position of the j th particle with respect to i th spatial coordinates. Various forms of SPH discretization exist for the viscous term in the system of Equations (5). This subject has been extensively studied by Basa et al. [17] in the framework of WCSPH. In addition, in a recent investigation carried out by Shahriari [6], the results of two different approaches were compared in the context of WCSPH as well. There, it is shown that the formula proposed by Morris et al. [3] produces better results in a wider range of Reynolds numbers. Based on these indications, together with those obtained from a few more studies [5, 7], it was decided to use the following relation: ν 2 2m j ν r ij : w ij u i ¼ j ρ j r 2 u ij ; (13) ij where r ij = r i r j and r ij = jr ij j. The solution of Equation (8) requires the assembly of a coefficient matrix and the corresponding right-hand side. The latter is plainly obtained by applying Equation (12), while at least two different procedures [10] may be considered when constructing the matrix. Here, the following expression is used: 1 ρ p 2m j p ij r ij w ij i ¼ j ρ 2 ; (14) where p ij = p i p j. It must be noted that the symbol ρ in Equation (14) does not carry anymore a subscript addressing any particle. This is a consequence of fluid incompressibility, and hence, the fluid density ρ remains a constant. Naturally, this notion should be extended to all equations used in the present study. r 2 ij

5 A DETAILED STUDY OF LID-DRIVEN CAVITY FLOW USING INCOMPRESSIBLE SPH Kernel corrections Simply based on a Taylor series expansion, the interpolation among neighboring particles exhibits second-order accuracy [28]. However, as particles move and change their spatial position, additional numerical errors are introduced. Aiming to improve this problem, a number of remedies have been proposed and analyzed [28 31]. Among these methods, and also following [5, 7], the procedure proposed by Oger et al. [28] is implemented here: Lr ðþ i ¼ 0 w ij j V j x j x i x w ij j V j y j y i x j V j x j x i j V j y j y i 1 w ij y C w ij A y 1 ; (15) ew ij ¼ Lr ðþ i w ij ; (16) where L is the kernel gradient correction operator, x and y denote the horizontal and vertical coordinates of a particle, and V represents the volume of the particle, which is directly calculated here from m/ρ. The corrected values resulting from Equation (16) replace the original kernel gradients in the flow equations Particle shifting algorithm As mentioned earlier in Section 2.2, the numerical accuracy of SPH is compromised once the particles start moving. Monaghan [26] proposed the so-called XSPH variant, which modifies the particle movement based on the local average velocity to attain a more isotropic particle distribution. However, as pointed out by Shahriari et al. [6] and Khorasanizade et al. [8, 32], the use of XSPH does not produce significant improvements, and in some cases, it leads to non-physical results. Another procedure to overcome the aforementioned problem is to use the particle shifting algorithm developed by Xu et al. [5]. This method has been used in other studies as well [7, 33], but without the correction of the variables proposed in the original work [5]. The complete set of relations to be applied for this purpose in the present ISPH method is 8 δr i ¼ CαR i ; >< R i ¼ N r i ij j¼1 r 2 i ; (17) >: r 3 ij r i ¼ 1 N i N i j¼1 r ij ; where δr stands for the shifting vector, C is a constant in the range of [5] and usually taken to be 0.04, α =u max Δt, where u max represents the maximum velocity, and N i is the number of neighboring particles around particle i. As noted before in several studies [8, 32, 33], the use of shifting is mandatory in ISPH to avoid major particle clumping and voids. After the particles have been shifted to their new positions in accordance to the set of Equations (17), the field variables must be corrected. Here, the following relation is applied: ϕ i ¼ ϕ i þ δr ii : ð ϕþ i ; (18) where i and i denote the old and new positions of a particle, respectively, and ϕ canbereplacedby pressure or any of the velocity components. 3. BOUNDARY CONDITIONS The widely used technique of Image Particles [10] is employed to model the presence of solid walls. Throughout the present study, no-slip BC for velocity and a homogenous Neumann condition for pressure are used. The implementation of no-slip BC can be carried out in different ways (see, e.g., [3, 10]). As noted by Basa et al. [17], the no-slip condition obtained by the simple use

6 658 S. KHORASANIZADE AND J. M. M. SOUSA of image particles [10] generally results in the underestimation of wall viscous forces. A correct evaluation of these forces is of great interest, so the foregoing authors suggested the use of a linear extrapolation applied to the tangential velocity component. However, a new proposal still allowing major improvements in the calculation of the wall viscous forces is made here. It is built onthesameideaastheconceptofmorriset al. [3] but with a slightly different formulation. Namely, the present methodology is applied to both velocity components. Similar to [3], a parameter β is defined, but the following relation is used instead: β ¼ max β max ; 1 þ d BC d f ; (19) where β max is an empirical value to be defined for the case of interest. The second term inside the parenthesis corresponds to a linear extrapolation of velocity inside the wall, where d BC and d f denote normal distances to the boundary of the image particle and the fluid particle, respectively. The velocity distribution of image particles is later shaped according to u f u BC ¼ β u f u wall ; (20) where the subscripts f, BC, and wall represent fluid, image particle and wall values, in that order. In order to better understand how this implementation of the BC operates, a sketch of the calculated velocities inside the wall and the velocity differences appearing on the left-hand side of Equation (20) is portrayed in Figure 1. For illustration purposes, a typical velocity distribution near the wall (the vertical dash line) is assumed for the fluid particles, and the resulting calculated velocities inside the wall are shown for different values of β max. It can be seen that with β max = 2, the velocity distribution inside the wall up to the distance of a fluid particle s image exhibits a constant ratio to the target fluid particle, but beyond that location, a linear extrapolation is applied instead. As β max increases, the region of constant velocity inside the wall is extended further, as expected. Another observation drawn from Figure 1 is the drastic increase obtained in the velocity difference term (directly used in the SPH governing equations) as β max is raised, thus contributing to bring the usually underestimated wall viscous forces closer to their physically correct value. Hence, the design idea behind this BC treatment is mainly to improve the calculation of the wall viscous force at the position of the target fluid particle, but still ensuring wall impermeability. Consequently, as the solid wall is approached, higher values of β must be applied, in accordance to Equation (19). However, the present simulations show that there is an upper limit to the value of β max, which depends on the Reynolds number. Three additional notes should also be made at this stage. First, Basa et al. [17] pointed out that the use of (wall) edge particles does not have a significant impact on the results, and therefore, such particles are not used in the present implementation. Second, the first fluid particle from the wall is placed at half the initial particle spacing, a scenario also reproduced in Figure 1. Third, the value of β max is chosen in all calculations considered in the present study by taking the highest value for which the numerical simulation remains stable at each particular Reynolds number. Figure 1. Calculated velocity distributions near the wall (top) for different values of β max and resulting velocity differences (bottom), based on a typical velocity distribution assumed for fluid particles (right). Dash lines indicate the target fluid particle for which each calculation is made.

7 A DETAILED STUDY OF LID-DRIVEN CAVITY FLOW USING INCOMPRESSIBLE SPH TEST CASES Despite the fact that several SPH studies have considered before the lid-driven cavity problem, detailed flow information is usually not provided and mere comparisons of velocity and/or pressure profiles with reference results are made. In the present work, extensive numerical data are collected from ISPH simulations at Reynolds numbers of 100, 400, 1000, and 3200, employing a corresponding spatial resolution similar to the lower of the set used by Ghia et al. [19]. A single exception was made for the last case, for which a higher resolution had to be used as discussed later in this section. The Reynolds number is based on the lid velocity, size of the cavity, and fluid viscosity. From simulation to simulation, the size of cavity (L) and the kinematic viscosity of the fluid remain unchanged, set to 1 m and 10 6 m 2 /s, respectively, whereas the lid velocity (U lid ) varied according to the Reynolds number. A schematic of the flow problem is shown in Figure 2 along with the definition of the various vortices expected to be formed inside the cavity. In all the cases studied, the results are compared in detail with those obtained by Ghia et al. [19] and, when available, also with the corresponding ISPH results of Xu et al. [5] for their L/160 resolution. Contours of normalized stream function (ψ) and normalized vorticity (ω) are always presented, in the following sections, for the levels listed in Table I, which reproduces exactly those used in [19] to facilitate a visual comparison Study at Re = 100 The first ISPH simulation was made here at Re = 100 for which a value of β max = 8 was used. The spatial resolution was chosen to be L/130 in order to approximate that used in [19], as mentioned before, despite that the two computational methodologies differ enormously (a finite difference method applied to a stream function vorticity formulation of the flow equations in the latter case). The filtering in Equation (1) was numerically performed with the quintic kernel [3]. A value of 1.3 was specified for the ratio of the smoothing length to the initial particle spacing and kept constant throughout this work. Profiles of v- and u-velocity along horizontal and vertical geometric centerlines, respectively, are shown in Figure 3. These profiles are in very good agreement with the reference results also presented. Although the present ISPH simulation employed about 1/3 less particles than in the work of Xu et al. [5], the implemented BC treatment allowed us to obtain virtually the same results, as Figure 2. Schematic of the flow problem and definition of vortices.

8 660 S. KHORASANIZADE AND J. M. M. SOUSA Table I. Contour levels used in the graphical representation of stream function and vorticity. Stream function Vorticity Value of ψ Contour label Value of ψ Contour label Value of ω Contour label Figure 3. Velocity profiles along vertical (left) and horizontal (right) centerlines for Re = 100 compared with the results of Xu et al. [5] and Ghia et al. [19]. in this case the remainder of both SPH algorithms coincide. A minor deviation is found, however, in the vertical velocity component next to the right side wall, presumably due to the high velocity gradients above. Basa et al. [17] have computed the value of nondimensional kinetic energy (k) for this flow employing a finite difference method. As shown in Table II, these authors have obtained a value of 0.034, whereas in the current ISPH simulation, it is calculated to be approximately As a curiosity, it should be noted that when the traditional image BC treatment was used instead, this quantity fell sharply to Hence, as expected for this flow problem, the new BC adds energy to the system. This mechanism should also have a beneficial impact on the formation of the flow vortices inside the cavity, as it will be analyzed next. Again, if the traditional image BC is used, the method is not able of capturing any of the bottom vortices (BL and BR, see Figure 2). The change to the new BC results in the appearance of the BR vortex, but the BL vortex is still not clearly observed at this Reynolds number. The values of various flow characteristics are shown in Table II, namely for the primary (P) and BR vortices, together with the maximum and the minima of the velocity along the centerlines. Except for the stream function value at the center of vortex BR, the calculated errors are always below 6%. Contours of vorticity and stream function are depicted in Figure 4. The contour levels are from Table I, as mentioned earlier. Because these two flow quantities are not primitive variables in the ISPH method, post-processing calculations must be performed to compute them from the velocity field, thus introducing additional errors. Because of the nature of the method and the flow problem,

9 A DETAILED STUDY OF LID-DRIVEN CAVITY FLOW USING INCOMPRESSIBLE SPH 661 Table II. Detailed comparison of flow characteristics and relative errors at Re =100. Present Ghia et al. [19] Error (%) k x P y P ψ P ω P x BR y BR ψ BR ω BR u min v min v max Data from Basa et al. [17]. Minima and maximum along geometric centerlines. Figure 4. Field contours of vorticity (left) and stream function (right) for Re = 100. Contour levels in each figure are from Table I. these mainly affect the region adjacent to the walls, where the maximum errors were found. However, despite the inability of these ISPH simulations to clearly capture the BL vortex, a direct comparison of the two flow maps with the corresponding ones published by Ghia et al. [19] further substantiates the quality of the present results, namely bearing in mind that both methods were applied to this flow problem employing basically the same spatial resolution Study at Re = 400 The flow at Re = 400 was studied by Ghia et al. [19] using two different spatial resolutions and also reporting a negligible difference between the corresponding results. Here, a resolution L/130 is used once again in the ISPH method, now with a value of β max = 6.3. Besides the primary (central) vortex and the two secondary vortices (BR and BL) previously mentioned, minute additional vortices have also been observed for this regime [19], but these have not been captured by the ISPH simulation. However, this cannot be judged as surprising because the size of such additional vortices was expected to be comparable with the particle spacing itself. The comparison of centerline velocities with reference data is shown for this case in Figure 5. It is important to notice that published ISPH results [5] employing a spatial resolution L/160 display larger discrepancies with respect to the results of Ghia et al. [19] than the present ISPH simulation using a resolution L/130. The overall agreement obtained at this regime for the flow velocities seems to be slightly worse than for the latter one, but the results are still very good.

10 662 S. KHORASANIZADE AND J. M. M. SOUSA Figure 5. Velocity profiles along vertical (left) and horizontal (right) centerlines for Re = 400 compared with the results of Xu et al. [5] and Ghia et al. [19]. Aiming to investigate the effect of the new BC on the pressure field as well, a profile obtained along the diagonal line connecting the lower left corner and the upper right is also compared with data from Xu et al. [5], which further includes results from a commercial code (Star CD). The pressure profile in Figure 6 shows that improvements were obtained in the region of minimum pressure. However, in the vicinity of the upper right corner of the cavity, larger discrepancies can be seen. This may be due to the numerical treatment of the foregoing corner in this flow problem, which was found mandatory here to prevent fluid particles from escaping the domain. A downward corner point velocity equal to half of the lid velocity was imposed in the present ISPH simulations to smooth out the discontinuity between the high-speed lid and the stationary side wall. Nothing is mentioned about a special corner treatment in the work of Xu et al. [5]. Contours of stream function and vorticity are now depicted in Figure 7. As in the previous case, additional errors were introduced in the near-wall region (mostly visible in the flow streamlines) because of the need for post-processing the raw SPH data. However, good agreement is still observed with the corresponding figures published by Ghia et al. [19]. Detailed flow characteristics Figure 6. Pressure profile along the diagonal line connecting lower left corner and the upper right corner for Re = 400 compared with the results of Xu et al. [5].

11 A DETAILED STUDY OF LID-DRIVEN CAVITY FLOW USING INCOMPRESSIBLE SPH 663 Figure 7. Field contours of vorticity (left) and stream function (right) for Re = 400. Contour levels in each figure are from Table I. Table III. Detailed comparison of flow characteristics and relative errors at Re = 400. Present Ghia et al. [19] Error (%) k x P y P ψ P ω P x BL y BL ψ BL ω BL x BR y BR ψ BR ω BR u min v min v max Minima and maximum along geometric centerlines. are provided for this regime in Table III along with reference data, but only the three vortices that could be clearly captured in the ISPH simulations have been considered. It can be seen that, excluding the weaker BL vortex, the computed errors remained relatively low, even for derived quantities such as vorticity and stream function Study at Re = 1000 Lid-driven cavity flow at Re = 1000 has been extensively studied in many publications. In one of these studies, Erturk [21] has questioned whether the flow at this Reynolds number remains steady and two-dimensional. The present two-dimensional ISPH simulation with a spatial resolution L/130 and β max = 5 produced a solution free of any flow oscillations. Figure 8 shows the stream function and vorticity contours at this regime. Again, only three vortices were clearly captured by the ISPH method. An additional vortex developing on the bottom right corner could not be adequately resolved. As for the previous cases, the flow streamlines exhibit irregularities in the close vicinity of the walls that may be attributed to post-processing. However, in this case, some noise with unknown origin is also detected in vorticity at the central region (level 8).

12 664 S. KHORASANIZADE AND J. M. M. SOUSA Figure 8. Field contours of vorticity (left) and stream function (right) for Re = Contour levels in each figure are from Table I. Xu et al. [5] also studied this flow regime using a similar ISPH algorithm and three different spatial resolutions. In Figure 9, velocity profiles along cavity centerlines generated by the present ISPH method are compared with those obtained by the aforementioned authors employing a resolution L/160, together with data from Ghia et al. [19]. It can be seen that the new BC yields a drastic reduction of the errors in velocity fields obtained with ISPH [4, 5]. Hence, very good agreement with reference results [19] could be obtained also at Re = 1000, using a similar spatial resolution in ISPH. Once more, it is important to scrutinize the effect of the new BC in the pressure field as well. Pressure profiles along the centerlines of the cavity have been reported in the literature by Botella and Peyret [23] for this regime. A direct comparison between the foregoing spectral solution and the present ISPH results is depicted in Figure 10. The general agreement is good, but noticeable discrepancies can still be observed near the four cavity walls, presumably due to the finer mesh spacing used in this region by the spectral method as a consequence of a Chebyshev Gauss Lobatto collocation. A detailed analysis of flow characteristics and a comparison with reference data [4, 5] were also carried out for Re = In this case, flow parameters have been tabulated in Table IV both for the traditional image BC and the new BC. In addition, the effect of varying β max up to the maximum acceptable value for this regime can also be appreciated from the results listed in this table. The continuous increase of β max from zero (linear extrapolation of velocity inside the wall) up to the solution convergence limit strongly improves the results in every aspect. It is therefore reasonable to assume that the previous findings concerning the velocity profiles obtained from the ISPH simulations can be generalized to the more global analysis supported by the parameters in Table IV. Figure 9. Velocity profiles along vertical (left) and horizontal (right) centerlines for Re = 1000 compared with the results of Xu et al. [5] and Ghia et al. [19].

13 A DETAILED STUDY OF LID-DRIVEN CAVITY FLOW USING INCOMPRESSIBLE SPH 665 Figure 10. Pressure profile along vertical (left) and horizontal (right) centerlines for Re = 1000 compared with the results of Botella and Peyret [23]. Table IV. Detailed comparison of flow characteristics at Re =1000 for different BC and various values of β max. Present BC Image BC β max =0 β max =2 β max =3 β max =4 β max = 5 Ghia et al. [19] k x P y P ψ P ω P x BL y BL ψ BL ω BL x BR y BR ψ BR ω BR u min v min v max Data from Bruneau and Saad [22]. Minima and maximum along geometric centerlines. Figure 11. Velocity profiles along vertical (left) and horizontal (right) centerlines for Re = 3200 compared with the results of Ghia et al. [19].

14 666 S. KHORASANIZADE AND J. M. M. SOUSA Unfortunately, to the authors best knowledge, the values of these parameters have never been reported before in the context of SPH simulations Study at Re = 3200 This section is closed with the presentation of the results from the present ISPH method for the liddriven cavity at Re = 3200 with β max = 2.2. In this case, a comparison is made with the reference data of Ghia et al. [19] only. Le Touzé et al. [34] also carried out SPH simulations for this flow regime employing a spatial resolution of L/200. Unfortunately, because of the low quality of the figures in [34], data for a direct comparison could not be obtained. However, those authors concluded that their SPH method, using the resolution above, was not able of adequately reproducing the results of Ghia et al. [19]. This task was eventually achieved in their study, but it required the use of a Finite Volume Particle Method instead. Figure 12. Field contours of vorticity (left) and stream function (right) for Re = Contour levels in each figure are from Table I. Table V. Detailed comparison of flow characteristics and relative errors at Re = Present Ghia et al. [16] Error (%) k x P y P ψ P ω P x BL y BL ψ BL ω BL x BR y BR ψ BR ω BR x TL y TL ψ TL ω TL u min v min v max Minima and maximum along geometric centerlines.

15 A DETAILED STUDY OF LID-DRIVEN CAVITY FLOW USING INCOMPRESSIBLE SPH 667 The velocity profiles along cavity centerlines generated by the present ISPH simulation, employing the same spatial resolution suggested by Le Touzé et al. [34], are shown in Figure 11. It can be seen that these profiles are in very good agreement with the corresponding results of Ghia et al. [19]. A more global view of the flow field is provided in Figure 12, where stream function and vorticity contours are depicted. The formation of a new secondary vortex (TL) on the top of the left wall of the cavity is now evident, as anticipated. The characteristics of this flow structure, along with the primary and other secondary vortices within the cavity, are, once again, compared in detail with the reference data of Ghia et al. [19] in Table V. It must be emphasized that despite the rather high value of the Reynolds number for this regime, the relative errors affecting the parameters listed in the table are always small, although an expected increase has been found for non-primitive variables. 5. CONCLUSIONS The lid-driven cavity problem has been studied for four different Reynolds numbers up to 3200 employing an ISPH method. A new BC treatment for solid walls was also tested in this problem. It was designed to improve the calculation of viscous forces acting on the particles in the vicinity of the walls, still ensuring impermeability conditions. The implementation requires setting a single parameter, β max, which controls the wall velocity difference magnitude used in the SPH governing equations. In the present cases, it was found that there is an optimum value of β max for each Reynolds number, limited by stability constraints of the simulations. To the authors best knowledge, the solutions obtained for this benchmark problem have been analyzed with unprecedented detail in the context of SPH simulations. Close agreement with reference data has been obtained, hence demonstrating the potential capabilities and competitiveness of ISPH methods with respect to more traditional methods in terms of numerical accuracy. Moreover, the application of the new BC treatment yielded significant improvements over other SPH results of the same problem available in the literature. ACKNOWLEDGEMENTS This work has been partially supported by Fundação para a Ciência e a Tecnologia (FCT) via grant PTDC/EME-MFE/103640/2008. The financial support for the PhD studies of Sh. Khorasanizade via FCT scholarship SFRH/BD/75057/2010 is also acknowledged. The authors want also to thank Diogo Chambel Lopes for his contribution regarding computing facilities. REFERENCES 1. Lucy LB. A numerical approach to the testing of the fission hypothesis. The Astronomical Journal 1977; 82: doi: / Gingold RA, Monaghan JJ. Smoothed particle hydrodynamics theory and application to non-spherical stars. Monthly Notices of the Royal Astronomical Society 1977; 181: Morris JP, Fox PJ, Zhu Y. Modeling low Reynolds number incompressible flows using SPH. Journal of Computational Physics 1997; 136: doi: /jcph Lee ES, Moulinec C, Xu R, Violeau D, Laurence D, Stansby P. Comparisons of weakly compressible and truly incompressible algorithms for the SPH mesh free particle method. Journal of Computational Physics 2008; 227: doi: /j.jcp Xu R, Stansby P, Laurence D. Accuracy and stability in incompressible SPH (ISPH) based on the projection method and a new approach. Journal of Computational Physics 2009; 228: doi: /j.jcp Shahriari S, Hassan IG, Kadem L. Modeling unsteady flow characteristics using smoothed particle hydrodynamics. Applied Mathematical Modelling 2013; 37: doi: /j.apm Hosseini SM, Feng JJ. Pressure boundary conditions for computing incompressible flows with SPH. Journal of Computational Physics 2011; 230: doi: /j.jcp Khorasanizade S, Sousa JMM, Pinto JF. On the use of a time-dependent driving force in SPH simulations. Proceedings of 7th International SPHERIC Workshop. Prato, Italy: Shadloo MS, Zainali A, Yildiz M, Suleman A. A robust weakly compressible SPH method and its comparison with an incompressible SPH. International Journal for Numerical Methods in Engineering 2012; 89: doi: /nme Cummins SJ, Rudman M. An SPH Projection method. Journal of Computational Physics 1999; 152: doi: /jcph

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