Comparison of some approximation schemes for convective terms for solving gas flow past a square in a micorchannel Kiril S. Shterev and Sofiya Ivanovska Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 4, Sofia 1113, Bulgaria Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Acad. G. Bonchev St., Bl. 25A, Sofia 1113, Bulgaria Abstract. Rapidly emerging micro-electro-mechanical devices create new potential microfluidic applications. A simulation of an internal and external gas flows is important for their design. For small Knudsen number Kn < 0.1 (Kn = l 0 /L, where l 0 is the mean free path of the gas molecules and L is the characteristic length), a continuum approach based on modified Navier-Stokes-Fourier or extended hydrodynamic continuum models with corresponding velocity-slip and temperature-jump boundary conditions is still applicable and, respectively, preferable. We restrict ourself to the use of Navier-Stokes-Fourier continuum model. A development of the algorithm to solve a specific class of problems is closely related to numerical schemes used for approximation of equations terms. Higher-order approximation schemes can reduce the number of mesh nodes and respectively computational time, but it is possible to obtain physical unrealistic results. In this paper we study influence of some approximation schemes for convective terms over the spatial steps. It is compared upwind, central difference and total variation diminishing (TVD) schemes Min-Mod, QUICK and SUPERBEE. A test case is gas flow past a square in a microchannel at subsonic speed (Mach number M = 0.1) and supersonic speed (M = 2.43), available in a literature. Keywords: mesh convergence, TVD schemes, subsonic flow, supersonic flow, SIMPLE-TS, micro-gas flows, continuum model PACS: 02.30.Mv, 47.11.-j, 47.11.Df, 47.40.-x, 47.45.-n, 47.61.Fg, 47.60.Dx INTRODUCTION All devices with character dimensions between 1µm and 1mm are called micro-devices. Micro mechanical devices are rapidly emerging technologies, where new potential applications are continuously being developed. Microchannel fluid flow is currently modelling using either the continuum approach or the non-continuum approach most commonly called molecular approach. The Knudsen number (Kn), a nondimensional parameter, determines the degree of appropriateness of the continuum model. It is defined as the ratio of the mean free path l 0 to the macroscopic length scale of a physical system L. Kn = l 0 (1) L Micro-gas flow simulations should be fast and accurate. Higher order spatial accuracy scheme can reduce required computational resources. Correct calculation of supersonic microflows use first order upwind scheme for approximation of convective terms and density in the middle points requires a very fine mesh and enormous computational resources [1]. On the other hand, higher order schemes reduce the number of nodes in the mesh, but it is possible to be unstable and to obtain physical unrealistic results. TVD schemes are second order spatial accuracy schemes and are designed to prevent undesirable oscillatory behaviour of higher-order schemes. In TVD schemes, the tendency towards oscillation is counteracted by adding an artificial diffusion fragment or by adding a weighting towards upstream contribution. In the literature early schemes based on these ideas were called flux corrected transport (FCT) schemes: see Boris and Book [2, 3]. Further works by Van Leer [4, 5, 6, 7], Harten [8, 9], Sweby [10], Roe [11], Osher and Chakravarthy [12] and many others have contributed to the development of present-day TVD schemes. These schemes are widely used in Computational Fluid Dynamic (CFD) codes and software packages like OpenFOAM [13]. The main target of the present work is to compare mesh-dependent convergence of TVD schemes applied for calculation of gas microflows at different regimes. TVD schemes are implemented in the algorithm SIMPLE-TS [1]. There is comparison between first order upwind scheme and second order TVD schemes: Min-Mod [11], QUICK [14] and SUPERBEE [11]. Flow past a square is used as a test case in a microchannel at different regimes: subsonic (M = 0.1) and supersonic (M = 2.43) speed.
CONTINUUM MODEL EQUATIONS A two dimensional system of equations describing the unsteady flow of viscous, compressible, heat conductive fluid can be expressed in a general form as follows: where: (ρu) t (ρv) t (ρt ) t + (ρuu) x + (ρuv) x + (ρut ) x + (ρvu) + (ρvv) + (ρvt ) [ ( u ) 2 Φ = 2 + x ρ t + (ρu) + (ρv) = 0 x (2) =ρg x A p [ ( x + B Γ u ) Γ u )] x x { ( + B Γ u ) Γ v ) 2 [ ( u Γ x x x 3 x x + v )]} (3) =ρg y A p [ ( + B Γ v ) Γ v )] x x { ( + B Γ v ) Γ u ) 2 [ ( u Γ x 3 x + v )]} (4) [ ( =C T 1 Γ λ T ) Γ λ T )] +C T 2.Γ.Φ +C T 3 Dp x x Dt (5) p = ρt (6) ( ) ] v 2 ( v + x + u ) 2 2 ( u 3 x + v ) 2 (7) u is the horizontal component of velocity, v is the vertical component of velocity, p is pressure, T is temperature, ρ is density, t is time, x and y are coordinates of a Cartesian coordinate system. The reference quantities used to scale the system of equations (2) - (6) will be defined later. The parameters A, B, g x, g y, C T 1, C T 2, C T 3 and diffusion coefficients Γ and Γ λ, given in Eqs. (2)-(6), depend on the gas model and the equation non-dimensional form. The system of equations (2) - (6) is solved by using an algorithm SIMPLE-TS [1]. Second order central difference scheme is employed for the approximation of the diffusion terms. The main topic of this paper is comparison of mesh-dependent convergence of schemes for approximation of convective terms and density in middle points. In following sections are presented details of applications and results. The system of equations (2) - (6) is given in a general form of the Navier-Stokes-Fourier equations. For a gaseous microflow description we use the model of a compressible, viscous hard sphere gas with diffusion coefficients determined by the first approximation of the Chapman-Enskog theory for low Knudsen numbers [15]. For a hardsphere gas, the viscosity coefficient µ and the heat conduction coefficient λ read (first approximations are sufficient for our considerations) as: µ = µ h T, µh = (5/16)ρ 0 l 0 V th π (8) λ = λ h T, λh = (15/32)c p ρ 0 l 0 V th π (9) The Prandtl number is given by Pr = 2/3, γ = c p /c v = 5/3. The dimensionless system of equations (2) - (6) is scaled by the following reference quantities, as given in [15]: molecular thermal velocity V 0 = V th = 2RT 0 for velocity, for length - square size a (Fig. 3), for time - t 0 = a/v 0, the reference pressure (p 0 ) is the pressure at the inflow of the channel, the reference temperature (T 0 ) is equal to the channel walls, reference density (ρ 0 ) is calculated using equation of state (6), the calculated case neglects the influence of gravity field, therefore g x = g y = 0. The corresponding non-dimensional parameters in the equation system (2) - (6) read as follows: A = 0.5, B = 5 π 16 Kn, Γ = Γλ = T, C T 1 = 15 π π 32 Kn, CT 2 = 4 Kn, CT 3 = 2 5 (10)
TVD SCHEMES - ESSENCE AND IMPLEMENTATION Generalization of upwind-biased discretisation schemes is considered in many papers and books (see [16]). Here is presented briefly the application of TVD scheme in algorithm SIMPLE-TS. Consider the standard control volume discretisation of the one dimensional case, Fig. 1. The implementation for the two dimensional case is straightforward. Approximated variable φ is defined in nodes of a mesh, φ i 1, φ i and φ i+1, Fig. 1. TVD scheme is applied to FIGURE 1. 1D Structured mesh approximate variable in points, where the variable is not defined. The variable is not defined on control surfaces, φi CS and φi+1 CS, Fig. 1. The general form of the value φcs i+1 in a descretisation scheme for u > 0, Fig. 1, can be written as: φ CS i+1 = φ i + 1 2 Ψ(r)(φ i+1 φ i ) (11) where r is the ratio of upwind-side gradient to downwind-side gradient r = (φ i φ i 1 )/(φ i+1 φ i ) determines the value of function Ψ and the nature of the scheme. To prevent division by zero, in code is included condition that r = 0, if (φ i+1 φ i )/ < 10 12. In the literature ([8, 9, 10]) the total variation has been considered for transient one-dimensional transport equation and are used explicit TVD schemes. TVD schemes are successfully applied also for calculation of steady problems (see [16]). Here are used implicit TVD schemes for calculation of steady microfluidic problems. The algorithm SIMPLE- TS, with the implemented TVD scheme, is also applicable for calculation of unsteady problems. Sweby [10] has given necessary and sufficient conditions for a scheme to be first- (Fig. 2. (a)) and second-order (a) FIGURE 2. Region in grey for a first-order TVD schemes (a) and second-order TVD schemes (Fig. 2. ) TVD in terms of r Ψ relationship. Fig. 2. (a) confirm that upwind is a first order TVD scheme, while the central difference is second-order TVD scheme for r 0.5, but is not TVD for r < 0.5. The quadratic upstream interpolation for convective kinetics (QUICK) scheme [17], Ψ(r) = (3 + r)/4, also leave the TVD region, Fig. 2. (a). The idea of designing a TVD scheme is to force the r Ψ relationship to remain within the shaded region for all values of r. QUICK schemes with limiters (13), [14], is in a TVD region. In this paper are implemented and used TVD schemes Min-Mod (12), [11], QUICK (13) and SUPERBEE (14), [11]. The Min-Mod limiter function exactly traces the lower limit of the TVD region, whereas SUPERBEE scheme follows the upper limit, (Fig. 2. ). Ψ(r) Min Mod = max(0,min(r,1)) (12) Ψ(r) QUICK = max(0,min(2r,(3 + r)/4,2)) (13) Ψ(r) SUPERBEE = max(0,min(2r,1),min(r,2)) (14) TVD schemes are implemented for approximation of convective terms of equations (3), (4), (5). When density in middle points is approximated, using central differences in algorithm SIMPLE-TS, [1], unphysical oscillations of fluid flow at supersonic speed are obtained. The approximation of upwind first-order scheme is used to prevent these oscillations. Here are applied the same approach: density approximation in middle points is the same as an approximation for convective terms.
PROBLEM FORMULATION As a example we consider a 2D steady-state laminar flow around a small square particle with size a confined in a FIGURE 3. Flow geometry for a square-shaped particle with size a confined in a channel with length L ch and height H ch. plane microchannel (height H ch ) as shown in Fig. 3. The blockage ratio B = a/h ch is equal to B = 10, the inflow length is L a. The problem is considered in a local Cartesian coordinate system, which is moving with the particle. Thus for an observer moving along with the particle the problem is transformed to a consideration of a gas flow past a stationary square confined in a microchannel with moving walls. Reference parameters for this problem are: p 0 = p in, T 0 = T in and L = a. Temperature of the square and channel walls are equal to T 0. On the walls of the channel and the square velocity-slip and temperature-jump boundary conditions are imposed. The velocity-slip BC is given as: v s v w = ζ v n, where v s is velocity of the gas at the solid wall surface, v w is velocity of the wall, s v ζ = 1.1466.Kn local = 1.1466.Kn/ρ local, Kn local is the local Knudsen number, ρ local is the local density, n is the s derivative of velocity normal to the wall surface. The temperature-jump boundary condition is: T s T w = τ T, where s T s is temperature of the gas at the wall surface, T w is temperature of the wall, τ = 2.1904.Kn local = 2.1904.Kn/ρ local, T n is the derivative of temperature normal to the wall surface. The Knudsen number is Kn = 0.05. s Detailed problem formulation and validation of SIMPLE-TS using upwind first-order scheme to approximate convective terms and density in middle points are presented in [1]. Subsonic gas flow In the subsonic case the Mach number is 0.1, L ch = 40, L a = 15.5. The channel walls are moving with a constant velocity equal to u = 0.09129 (according to Mach number 0.1). The inflow boundary conditions (BC in ) are: u in is calculated from the continuity equation for first control volume on OX, similarly to the pressure driven flow in a channel, v/ x = 0, p in = 1 and T in = 1. The outflow boundary conditions (BC out ) are: u out is calculated using the continuity equation defined for the last control volume on OX. Here are compared upwind first-order scheme, results are available in [1], second-order TVD schemes Min-Mod (12) and QUICK (13), and central difference scheme. Mesh-dependent convergence is investigated on spatial steps = base, base /2, base /4 (corresponding uniform mesh: 400x100, 2 2.400x100 and 4 2.400x100), where = x = y, x is step on OX, y is step on OY, base = 0.1 is based spatial step. Here are compared profiles of horizontal velocity and temperature at the centre line along the channel, Fig. 4, obtained at different spatial steps and schemes. The profiles for horizontal velocity, obtained using different spatial steps and approximations are very close to each other, Fig. 4. (a). The profiles of temperature obtained, using different approximations at a given spatial steps are very close to each other, Fig. 4.. The groups of profiles obtained at a different spatial steps are different. On the other hand, the temperature deviations of fluid flow is negligible therefore fluid flow can be considered isothermal. The results show no advantages of second order schemes for a considered case. The velocity of subsonic case is very low (Mach number is 0.1). When Mach number is sufficiently low (close to zero) the fluid flow became creeping (Stokes flow) and convective terms can be neglected in a mathematical model of fluid. The low flow velocity is the most likely reason of independence of the numerical solution of the order of approximation of convective members. n
FIGURE 4. Profiles of horizontal velocity (a) and temperature obtained by upwind first-order, central difference, QUICK and Min-Mod schemes at the centre line along the channel (y = H ch /2), M = 0.1. Supersonic gas flow In the supersonic calculations the Mach number is 2.4261. The channel length is L ch = 50, the distance between the square and the channel inlet is L a = 5.5. The channel walls are moving with a constant velocity u = 2.2147 (according to Mach number 2.4261). The inflow boundary conditions (BC in ) are: u in = 2.2147, v = 0, p in = 1 and T in = 1. The outflow boundary conditions (BC out ) are: u/ x = 0, v/ x = 0, p/ x = 0 and T/ x = 0. Here are compared results, obtained using first-order and second order schemes for calculation of flow past square particle in a microchannel at supersonic speed. The supersonic speed leads to the existence of a bow shock wave, which reflects from the channel walls. As a result, past the particle, we obtain a complex picture of the interaction of reflected shock waves (see Fig. 5). The shock waves have significant gradients of velocities, pressure and temperature. The calculation of this flow using first-order upwind schemes accurately requires a very fine mesh (8000x1600) cells, [1], or adaptive one. Here are compared results obtained using first-order upwind scheme, available in [1] with results obtained using second-order TVD schemes Min-Mod (12), QUICK (13) and SUPERBEE (14). Mesh-dependent convergence is investigated for a range of spatial steps for first- and second order schemes. Range FIGURE 5. Fields of the horizontal component of velocity (left part) and temperature (right part), M = 2.43, obtained using TVD scheme SUPERBEE, spatial step is = base /2 = 0.05. of spacial steps for upwind first-order schemes are = base /2, base /4, base /8, base /16 (corresponding uniform mesh: 2 2.500x100, 4 2.500x100, 8 2.500x100 and 16 2.500x100), where = x = y, base = 0.1 is based spatial step. Range of spacial steps for second-order TVS schemes are = base, base /2, base /4, base /8 (corresponding uniform mesh: 500x100, 2 2.500x100, 4 2.500x100 and 8 2.500x100). It is expected that second-order schemes will reduce computational time in comparison with first-order schemes, but it is possible to bring instability and to obtain physical unrealistic results. Here are compared profiles of horizontal velocity and temperature at the centre line along the channel obtained using first- and second-order schemes for the approximation of convective terms and density in middle points, Fig. 6-10. The second-order Min-Mod scheme mesh convergence is much better than the first-order upwind scheme, Fig. 6. We accepted that the numerical solution obtained using Min-Mod scheme is accurate enough for a spatial step base /4 = 0.025, while for upwind scheme is base /16. Therefore Min-Mod scheme needs 4 times less cells in each spatial direction, which is 4 2 = 16 less cells for the two dimensional case. Mesh convergence of QUICK scheme is also rapider compared to those of upwind scheme, Fig. 7. However mesh convergence of QUICK scheme is slightly better then Min-Mod scheme. The convergence
of SUPERBEE scheme is better than upwind scheme, Fig. 8 and is even better than convergences of Min-Mod and QUICK schemes. SUPERBEE schemes obtain an accurate solution on a spatial step base /2 = 0.05, therefore needs 8 2 = 64 times less cells in the computational domain, then the upwind scheme. The results show many times reduction of the number of cells in a computational domain of TVD schemes, Fig. 10, compared to first-order upwind scheme. On the other hand, investigated TVD schemes required smaller time step and larger number of iterations to be algorithm converge against upwind first-order scheme. TVD scheme with best convergence SUPERBEE obtain small over-shoot in a shock wave in front of the square for spatial steps = base = base /2, Fig. 9. Other investigated TVD schemes Min-Mod and QUICK do not obtain such kind of physical unrealistic results. Nevertheless the requirements for smaller time steps and increase the number of internal iterations compared to first-order upwind scheme TVD schemes reduce many times computational time. Investigated TVD schemes are applied using their standard format and do not obtain a convergent solution for spatial step = base /16 = 0.00625. A round-off error is the most probable reason TVD schemes to be unstable using small spatial steps. The implementation of TVD schemes in algorithm has to be changed so the round-off error to be reduced. Results show that mesh convergence of TVD schemes closer to the upper TVD region (Fig. 2, ) is faster than TVD schemes closer to lower TVD region limit, Fig. 6, Fig. 7 and Fig. 8. On the other hand TVD schemes closer to the upper TVD limit (SUPERBEE) needs smaller time step and more internal iteration to be algorithm SIMPLE-TS converged according to TVD schemes closer to lower TVD limit region (Min-Mod). TVD schemes at the lower limit of the TVD region (Min-Mod) required less time step and more internal iterations then upwind first-order scheme, which correspond to Ψ(r) = 0. This investigation shows that schemes, closer to the upper limit of TVD region have rapider mesh convergence, then schemes, closer to to lower limit of TVD region and first-order upwind scheme (Ψ(r) = 0). On the other hand, schemes, closer to the upper limit of TVD region obtain small over-shoot (SUPERBEE) and are less stable then schemes, closer to to lower limit of TVD region and first-order upwind scheme (Ψ(r) = 0). The results show that use of schemes, closer to the upper TVD region will give us rapider mesh convergence small reducing of stability of the algorithm and possible appearance of small over-shoots. (a) FIGURE 6. Profiles of horizontal velocity (a) and temperature obtained by upwind first-order and Min-Mod schemes at the centre line along the channel (y = H ch /2), M = 2.43. (a) FIGURE 7. Profiles of horizontal velocity (a) and temperature obtained by upwind first-order and QUICK schemes at the centre line along the channel (y = H ch /2), M = 2.43.
(a) FIGURE 8. Profiles of horizontal velocity (a) and temperature obtained by upwind first-order and SUPERBEE schemes at the centre line along the channel (y = H ch /2), M = 2.43. FIGURE 9. Profiles of temperature obtained by upwind first-order and SUPERBEE schemes in front of the square at the centre line along the channel (y = H ch /2), M = 2.43. (a) FIGURE 10. Profiles of horizontal velocity (a) and temperature obtained by upwind first-order and Min-Mod, QUICK and SUPERBEE schemes at the centre line along the channel (y = H ch /2), M = 2.43. CONCLUSIONS It this paper is investigated mesh convergence of first-order and second order schemes for the approximation of convective terms of Navier-Stokes-Fourier system of equations described gas microflow. A test case is flow past a square in a microchannel at subsonic (M = 0.1) and supersonic (M = 2.43) speeds. Results, obtained from subsonic case show no advantages of any of compared schemes (first-order upwind and second order central difference, TVD
Min-Mod and TVD QUICK schemes. Results, obtained from supersonic case is that TVD schemes obtained the same results as first-order upwind schemes with 4 2 = 16 (Min-Mod and QUICK) to 8 2 = 64 (SUPERBEE) times less cells. The disadvantages of TVD schemes then first-order upwind scheme are: the usage of smaller time step and the increase of iterations in internal loops. SUPERBEE scheme, which mesh convergence is rapider according to investigated schemes, obtains small over-shoot in a shock wave in front of the square. Investigated TVD schemes (Min-Mod, QUICK, SUPERBEE) are not converged for small spatial steps = base /16 = 0.00625. Nevertheless the requirements for smaller time steps and increase the number of internal iterations compared to first-order upwind scheme TVD schemes reduce many times resources to calculate supersonic fluid flow and are highly recommended for this flow regime. ACKNOWLEDGMENTS The author appreciates the financial support by the NSF of Bulgaria under Grant No DMU 03/37, since 2011. This work makes use of results produced by the High-Performance Computing Infrastructure for South East Europe s Research Communities (HP-SEE), a project co-funded by the European Commission (under contract number 261499) through the Seventh Framework Programme. HP-SEE involves and addresses specific needs of a number of new multidisciplinary international scientific communities (computational physics, computational chemistry, life sciences, etc.) and thus stimulates the use and expansion of the emerging new regional HPC infrastructure and its services. Full information is available at: http://www.hp-see.eu/ REFERENCES 1. K. S. Shterev, and S. K. Stefanov, Journal of Computational Physics 229, 461 480 (2010), ISSN 0021-9991, URL http://dx.doi.org/10.1016/j.jcp.2009.09.042. 2. J. P. Boris, and D. L. Book, Journal of Computational Physics 11, 38 69 (1973), ISSN 0021-9991, URL http: //www.sciencedirect.com/science/article/pii/0021999173901472. 3. J. P. Boris, and D. L. Book, Solution of continuity equations by the method of flux-corrected transport, 1976, pp. 85 129. 4. B. van Leer, Journal of Computational Physics 14, 361 370 (1974), ISSN 0021-9991, URL http://www. sciencedirect.com/science/article/pii/0021999174900199. 5. B. V. Leer, Journal of Computational Physics 23, 263 275 (1977), ISSN 0021-9991, URL http://www. sciencedirect.com/science/article/pii/0021999177900948. 6. B. V. Leer, Journal of Computational Physics 23, 276 299 (1977), ISSN 0021-9991, URL http://www. sciencedirect.com/science/article/pii/002199917790095x. 7. B. van Leer, Journal of Computational Physics 32, 101 136 (1979), ISSN 0021-9991, URL http://www. sciencedirect.com/science/article/pii/0021999179901451. 8. A. Harten, Journal of Computational Physics 49, 357 393 (1983), ISSN 0021-9991, URL http://www. sciencedirect.com/science/article/pii/0021999183901365. 9. A. Harten, SIAM Journal on Numerical Analysis 21, 1 23 (1984), URL http://epubs.siam.org/doi/abs/10. 1137/0721001, http://epubs.siam.org/doi/pdf/10.1137/0721001. 10. P. K. Sweby, SIAM Journal on Numerical Analysis, vol. 21, no. 5, pp. 995-1011 21, 995 1011 (1984). 11. P.L.Roe, Lectures in Applied Mathematics 22, 163 193 (1985). 12. S. Osher, and S. Chakravarthy, SIAM Journal on Numerical Analysis 21, 955 984 (1984), URL http://epubs.siam. org/doi/abs/10.1137/0721060, http://epubs.siam.org/doi/pdf/10.1137/0721060. 13. Openfoam (????), URL www.openfoam.org. 14. B. P. Leonard, International Journal for Numerical Methods in Fluids 8, 1291 1318 (1988), ISSN 1097-0363, URL http://dx.doi.org/10.1109/84.585795. 15. S. Stefanov, V. Roussinov, and C. Cercignani, Physics of Fluids 14, 2255 2269 (2002). 16. H. K. Versteeg, and W. Malalasekra, An Introduction to Computational Fluid Dynamics: The Finite Volume Method, Prentice Hall, Pearson, 2007, 2nd edn. 17. B. Leonard, Computer Methods in Applied Mechanics and Engineering 19, 59 98 (1979), ISSN 0045-7825, URL http://www.sciencedirect.com/science/article/pii/0045782579900343.