AN EFFICIENT MOVING MESH METHOD FOR A MODEL OF TURBULENT FLOW IN CIRCULAR TUBES *

Transcription

1 Journal of Computational Mathematics, Vol.27, No.2-3, 29, AN EFFICIENT MOVING MESH METHOD FOR A MODEL OF TURBULENT FLOW IN CIRCULAR TUBES * Yin Yang Hunan Ke Laborator for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan Universit, Xiangtan 41115, China angintu@tu.edu.cn Yanping Chen School of Mathematical Sciences, South China Normal Universit, Guangzhou, China anpingchen@scnu.edu.cn Yunqing Huang Hunan Ke Laborator for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan Universit, Xiangtan 41115, China huangq@tu.edu.cn Abstract This paper presents an efficient moving mesh method to solve a nonlinear singular problem with an optimal control constrained condition. The phsical problem is governed b a new model of turbulent flow in circular tubes proposed b Luo et al. using Prandtl s miing-length theor. Our algorithm is formed b an outer iterative algorithm for handling the optimal control condition and an inner adaptive mesh redistribution algorithm for solving the singular governing equations. We discretize the nonlinear problem b using a upwinding approach, and the resulting nonlinear equations are solved b using the Newton- Raphson method. The mesh is generated and the grid points are moved b using the arc-length equidistribution principle. The numerical results demonstrate that proposed algorithm is effective in capturing the boundar laers associated with the turbulent model. Mathematics subject classification: 65L1, 65L12. Ke words: Edd viscosit, Turbulent pipe flow, Boundar laer, Optimal control, Moving mesh. 1. Introduction The modeling of turbulent flows still plas an important role in computational fluid dnamics because direct simulation of flows are restricted to ver simple geometries and low Renolds number [6,9,18,19]. Development of turbulence model is therefore still an important task and even some semi-empirical means such as the edd viscosit or Prandtl s miing length are ver helpful to deal with man problems in engineering practice due to their simplicit. Luo et al. [15] established a new model of turbulent flow in circular tubes which is an application and improvement of Prandtl s miing-length theor. The model epresses the single phase flow in circular tubes, which is an optimal parameter control problem governed b a nonlinear singular equation. The model ields man comple mathematical characters such as strong boundar laer. The computational results resulting from the new model are found in good agreement with the eperimental results on fluid velocit distribution, edd viscosit distribution and friction factor. On the mathematical side, the governing equations associate with this model * Received Januar 5, 28 / Revised version received August 5, 28 / Accepted September 5, 28 /

2 Moving Mesh Method for a Model of Turbulent Flow 389 are quite complicated, and an effective numerical scheme for finding numerical approimations seems useful. The difficulties of this problem include the eistence of the boundar laers and an optional control condition enforced on the governing equations. To resolve the laers, numerical simulations require etremel fine meshes on the small localized portions of the phsical domain. It will become ver epensive if a uniform fine mesh is emploed. Recent stud has demonstrated that moving mesh methods are powerful in resolving large solution variations b increasing the solution accurac and decreasing the cost of computations, see, e.g., Adaptive moving mesh methods have important applications in solving partial differential equations (PDEs). Up to now, there have been important progresses [1, 3, 2]. Harten and Hman [11] began the earliest stud of the adaptive methods to improve resolution of discontinuous solutions of hperbolic equations. After their work, man other moving mesh methods on this direction have been proposed based on combining the variational grid methods with high resolution shock capturing methods, including the so-called moving mesh PDE (MMPDE) approach of moving mesh methods of W. Huang [1], moving finite element methods of Miller [17], and moving finite volume methods [22]. Recentl, there have been works on moving mesh methods based on Harmonic maps [8,16,22]. Theoretical results on adaptive mesh arising from equidistribution of a monitor function can be found in [3,4,12 14,21]. In particular, Kopteva [13] derived certain maimum norm a posterior error estimates for one-dimensional singularl perturbed convection-diffusion problems, see also a recent paper [14] for a similar posterior error estimate. The aim of this paper is to present an efficient and fast numerical method for the turbulent model. The proposed numerical algorithm includes two parts: (i) the outer iterative algorithm is used to solve the optimal control condition and (ii) the inner adaptive mesh redistribution algorithm is used to solve the singular problem. We discretize this nonlinear problem b using upwinding scheme. The discretized nonlinear equations is solved b Newton-Raphson method. The arc-length equidistribution principle is used in the part (ii) above. The numerical eamples will be provided to demonstrate the effectiveness of the proposed algorithm. This paper is organized as follows. In Section 2, we briefl review the model of turbulent flow in circular tubes b emploing Prandtl s miing-length theor. In Section 3, we will present the discrete schemes and algorithms. Numerical eperiments will be carried out in Section 4. Some concluding remarks will be presented in the final section. 2. A New Model of Turbulent Flow in Circular Tubes In this section, we briefl review the background of the model of turbulent flow in circular tubes which was proposed b Luo et al. [15]. Moreover, using dimensionless analsis we will derive a complete mathematical description for this model. Note that the shearing stress of Newtonian fluid for turbulent flow can be described b edd viscosit with dimensionless analsis [9]. We then have following epression: dũ dφ = ˆRφ 1 + µ t /µ L, (2.1) where ũ is dimensionless time-smoothed velocit, µ t is edd viscosit, µ L is kinematic viscosit, ˆR is dimensionless radius of a circular pipe and ˆR = ρûr/µ with R is the tube radius, ρ is the liquid densit, û is friction velocit, µ is the molecule viscosit, φ is dimensionless radial position in a circular pipe, φ = is the center of the tube and φ = 1 corresponds to the wall of

3 39 Y. YANG, Y. CHEN AND Y. HUANG the tube. As the velocit is disappeared on the wall, we have the boundar condition ũ(φ) φ=1 =. (2.2) Based on Prandtl s miing-length theor, the edd viscosit can be written as µ t = µ ˆR L L dũ L R R dφ, (2.3) and the constrained condition is 2 1 ũφdφ = ṷ u = Re ˆR, (2.4) where L is the miing length, u is average velocit in cross section of tubes, Re is the Renolds number defined b Re = ρur/µ. Let λ = L/R be the dimensionless miing length. Then (2.3) can be rewritten as µ t = µ ˆRλ L λdũ dφ. (2.5) Coupling (2.1) and (2.5), we have the following equation which is the application of the classical Prandtl s miing-length, dũ dφ = ˆRφ 1 + ˆRλ λ dũ. (2.6) Set ũ (φ) = dũ/dφ. Then (2.6) can be written as ũ (φ) = Using the smmetr of turbulent flow, we have dũ dφ =. φ= dφ ˆRφ 1 + ˆRλ λũ (φ). (2.7) It can be seen from Eq. (2.5) that the edd viscosit µ t at the center of a circular tube will be zero. However, the edd viscosit at the center is non-zero according to either the conception or the eperiment. This inconsistenc ma be caused mainl b the fact that the Prandtl theor onl took the first-order derivative of velocit to approimate the edd velocit (2.5). Thus, Luo et al. [15] introduced a simple wa to modif the Prandtl s miing length theor b utilizing the second-order derivative to approimate the edd velocit. B doing this, Eq. (2.5) can be changed to µ t = µ ˆRλ L λũ (φ) λ2 ũ (φ). (2.8) Combining (2.1) and (2.8) ields ũ (φ) = ˆRφ 1 + ˆRλ λũ (φ) λ2 ũ (φ). (2.9) The dimensionless miing length λ in (2.9) is found to be not a universal constant but a function of position, λ = L/R = kf(φ), f(φ), f() = 1, f(1) =, (2.1)

4 Moving Mesh Method for a Model of Turbulent Flow 391 where k is a dimensionless positive constant. For single phase flow in circular tubes, so ũ and ũ are nonpositive, the absolute sign in (2.9) can be dropped. Then (2.9) can be rewritten as [ ũ (φ) 1 ˆR(kf(φ)) 2 ũ (φ) ˆR(kf(φ)) ] 3 ũ (φ) = ˆRφ. (2.11) Since ( 1 ˆR(kf(φ)) 2 ũ (φ) ˆR(kf(φ)) 3 ũ (φ)) φ= >, and ( ũ () 1 ˆR(kf()) 2 ũ () ˆR(kf()) ) 3 ũ () =, we can derive that (2.11) satisfies the smmetr of the turbulent flow: We then obtain from (2.11) that ũ () =. (2.12) 1 ˆRk 2 f(φ) 2 ũ (φ) ˆRk 3 f(φ) 3 ũ (φ) = ˆRφ ũ, φ (, 1]. (2.13) (φ) Set a(φ) = ˆRk 2 f(φ) 2 and b(φ) = ˆRk 3 f(φ) 3. Letting φ in (2.13) gives Then we obtain that lim (1 φ a(φ)ũ (φ) b(φ)ũ (φ)) = lim ˆRφ φ ũ (φ) = (1 b()ũ ()) = ˆR ũ () ũ () = b() ˆR 2b() (using L Hospital rule). = ˆR 2 k 3 ˆRk 3. (2.14) For the turbulent flow, Re (4, 1). In (2.1), k and f(φ) have man possible choices. Because we are aimed at showing how to solve the non-linear singular model (2.11) with an optimal control condition (2.4), we use the epression proposed b Prandtl. When the Renolds number is sufficientl high, the miing length in the wall turbulence can be written as f(φ) = 1 φ, k =.2. (2.15) Then we change the problem to the following mathematical model. For an given α (4, 1), find constant c such that satisf ( () 1 k 2 c(1 2 ) 2 () 1 ) 2 k3 c(1 2 ) 3 () = c, [, 1] (2.16) and the constrained condition together with three boundar conditions 2 1 d = α c, (2.17) (1) =, () =, () = c 2 k 3 ck 3. (2.18)

5 392 Y. YANG, Y. CHEN AND Y. HUANG It is eas to show that c >. Set = 1 in (2.16), it can be verified that (1) = c. (2.19) Since () and () are nonpositive, k and c are positive, we have (), ( 1 k 2 c(1 2 ) 2 () 1 ) 2 k3 c(1 2 ) 3 (), c. (2.2a) (2.2b) (2.2c) It is then epected that we have the boundar condition () = and () has no other zero point. Moreover, = 1 is the singular point of the equation, which ma ield a boundar laer on the right end. 3. A Moving Mesh Algorithm In this section, we present an efficient and fast approach for the model (2.16)-(2.18), which includes an iterative method and an adaptive method Analsis of the model We consider the model (2.16)-(2.17). The difficult of this problem involves two parts. Firstl, c is unknown as well as ; this problem is an optimal control problem and (2.17) is a control equation. For a given α, we want to obtain the solution and c which not onl satisf the differential equation (2.16) but also the control (2.17). Secondl, (2.16) is not onl a nonlinear differential equation, but also a singular problem. In order to solve the optimal control problem, we use an iterative algorithm to compute and c. For an initial c, solve (2.16) to obtain. Then compute c b solving (2.17) with the obtained. Compare the computed c and the initial c, if the discrepanc satisfies an preassigned tolerance, stop computing, else, the obtained c becomes the initial value and repeat the process. This is an outer iteration for solving the model. A detailed algorithm will be described in the net subsection. Then we consider Eq. (2.16) with boundar conditions in (2.18). As a singular problem, there is a region in which the solution of the differential equation is steep. For (1 2 ) 1 when 1, the solution has a boundar laer near the boundar = 1. It is well known that central or upwinding differential scheme on an uniform mesh will not give a satisfactor numerical solution in this case. To obtain a reliable numerical solution for (2.16) and (2.18), it is better to use a mesh that concentrates nodes in the boundar laer. Ideall, the mesh should be generated b adapting it to the features of the computed solution and this is usuall done b equidistributing a monitor function over the domain of the problem. This adaptive methods can handle not onl boundar laer but also interior laer problems. In fact, this is a moving mesh approach that can capture the laer effectivel Numerical method and algorithm Given a partition of [, 1]: Ω N = { j = < 1 < < N = 1}.

6 Moving Mesh Method for a Model of Turbulent Flow 393 On Ω N, we discretize (2.16) and (2.18) b using a upwinding scheme as follows: D + i (1 a i D + i b i DD i ) = c i for 1 i N 1 (3.1) with N =, where the operators used are given b D v i = v i v i 1 h i, D + v i = v i+1 v i h i+1, Dv i = v i+1 v i h i, h i = i i 1, hi = (h i+1 + h i )/2, a i = k 2 c(1 i 2 ) 2, b i = 1 2 k3 c(1 i 2 ) 3. At =, since () () + () = (), we set () 1 h 1 1 h 1 = 2 1, h 1 where 1 = 1 is used. Consequentl, the discretization of (2.16) at = becomes h ( h 2 1 b 2 1 ) 1 h 2 = c. (3.2) 1 As Eq. (3.1) is a nonlinear equation, we use Newton-Raphson iteration method to solve it. The integration in (2.17) is approimated b the following quadrature formula: 2 1 d N h i ( i i + i 1 i 1 ). (3.3) Denote i () C[, 1] the piecewise linear interpolant through the knots ( i, i ). We choose a monitor function: w() = 1 + i () 2 to equidistribute mesh which is the discrete analogue of the standard arc-length monitor function w() = 1 + () 2. Note that ( i ()) = D i, I i = ( i 1, i ), 1 i N 1. In an other words, we construct a mesh b solving the following equations: ( i+1 i ) 2 + ( i+1 i ) 2 = ( i i 1 ) 2 + ( i i 1 ) 2, 1 i N 1, =, N = 1. (3.4) To equidistribute the mesh and to capture the boundar laer, we move mesh iterativel to generate a fine mesh.

7 394 Y. YANG, Y. CHEN AND Y. HUANG Main Algorithm 1. Initialize c and give mesh Ω N ; 2. Use Newton-Raphson method and call adaptive algorithm to compute the discrete solution { i } b (3.1) on Ω N ; 3. Solve c = α 2 1 d b (3.3), if c c η, goto step 5, otherwise continue; 4. Set c = c c, return to step 2; 5. Set c = c, then stop. Adaptive Algorithm 1. Initialize mesh: Ω () N 2. Solve (3.1) for { (k) i }, let l (k) i = = {() j = () < () 1 < < () N = 1}, k = ; ( (k) ( (k) i ) (k) ( (k) i 1 ))2 + (h (k) i ) 2, L (k) = 3. Test mesh: let τ be a user chosen constant with τ > 1. If mal (k) /L (k) τ/n, then goto step 5. N i=1 l (k) i. 4. Generate new mesh: choose point = (k+1) < (k+1) 1 < < (k+1) N that for each i, the distance from ( (k+1) i 1 L (k) /N. Let k = k + 1, return to step Set i = (k) i, i = (k) i, then stop., (k) ( (k+1) i 1 ) to ( (k+1) i, (k) ( (k+1) i = 1, such )) equals 4. Numerical Results This section considers numerical solutions of the model (2.16)-(2.17) to demonstrate the performance of the iterative method and the moving mesh method proposed in last section. The parameters α in (2.17) are taken as 2, 4, 6, 8, 1 respectivel. For a given α, there is a corresponding c satisfing (2.16)-(2.17). Table 4.1 and Fig. 4.1 show the relation between α and c. From Table 4.1, we conclude that c is monotonicall increasing with respect to α for a fied N and c is convergent with increasing N. Fig. 4.1 shows that the relation of α and c is almost linear. So using least-square methods we can write α and c in the form of c = kα + b with k =.79 and α = Table 4.1: The value of c for different α and different N. N α = 2 α = 4 α = 6 α = 8 α =

8 Moving Mesh Method for a Model of Turbulent Flow c Fig The relation of dimensionless radius c and the Renold number α. In this model () is the velocit distribution. Eq. (2.16) is a nonlinear convection-dominated stationar convection-diffusion problem. The solution () to (2.16) and (2.18) has a boundar laer near = 1. Our numerical solution { i } showing b Fig. 4.2 confirms that { i } is ver steep near = =2, c= =6, c= =1, c= =4, c= =8, c= =5, c= Fig Liquid velocit distribution for different Renold number α.

9 396 Y. YANG, Y. CHEN AND Y. HUANG With the boundar condition () = and since () ma have no other zero point, we change Eq. (2.16) to (1 k 2 c(1 2 ) 2 12 ) k3 c(1 2 ) 3 = c, (, 1]. Set z() = (1 k 2 c(1 2 ) 2 12 ) k3 c(1 2 ) 3 At =, b (2.18), we have z() = c () = k3 c 2. 2 = c, (, 1]. (4.1) Note that z() is the edd viscosit distribution. Let z i be the discrete form of z(). Then we have z = k 3 c 2, 2 z i = c ih i+1, for i = 1,, N 1, i+1 i z N = z(1) = c () = c =1 c = z Fig The edd viscosit distribution z for Renold number α = 6. Denote z i () the piecewise linear interpolation though the knots ( i, z i ). Fig. 4.3 presents the value of z() in the case of α = 6 with N = 124. Fig. 4.4 shows the whole histor of the mesh moving from its initial uniform mesh and the convergence histor of c from its initial value c = 1.. It is clearl demonstrated that the mesh concentrates nodes near the boundar laer. From Fig. 4.4, it is observed that for c = 1. the mesh move from the uniform mesh to the 1st level mesh; for c = 156 the mesh moves from 2nd to 3rd mesh while the 3rd mesh is the final adaptive mesh corresponding to c = 156. The mesh of 6th level is good enough for other

10 Moving Mesh Method for a Model of Turbulent Flow 397 c=66.6, 13 th N=64, k th moving mesh c=66.5, 12 th c=66.23, 11 th c=65.55, 1 th c=63.82, 9 th c=599.45, 8 th c=588.48, 7 th c=561.44, 6 th c=497.8, 5 th c=364.34, 4 th c=156.3, 3 th c=156.3, 2 th c=1., 1 th c=1., th initial mesh: uniform mesh 1 Fig Moving mesh histor. choices of c, so onl outer iterations are needed to compute the convergent c while the mesh does not need to move an more. In our algorithm, for a fied α, c can be convergent from an arbitrar initial value. Taking α = 6 for eample, the following datum shows how c = 1 and c = 1 convergence respectivel with N = 124: c = c = Our outer iterative is mainl to compute c and. Take α = 6 and N = 124 with initial c = 1. as an eample. Fig. 4.5 gives the convergent velocit of c b plotting log( c c ), where c is computed b the present iterative value, c the last iterative value. We can find that its convergence is almost linear. We close this section b making several remarks. First, in the main algorithm, we have another choice of step 4, i.e., set c = c, return to step 2. Secondl, in the Newton-Raphson iteration, we choose the initial guess for () i = and () i = 1; both give the same convergent value. Thirdl, in step 3 of the adaptive algorithm, τ can be chosen that to satisf τ 1. In this paper, we set τ = Conclusions and Future Works In this paper, following a new proposal of Luo et al. [15] the mathematical governing equations for a model for turbulent flow in circular tubes are derived. The mathematical setting involves a nonlinear singular problem with an optimal control condition. An efficient numerical strateg based on the moving mesh method is proposed. The numerical results show that

11 398 Y. YANG, Y. CHEN AND Y. HUANG log( c c ) i th outer iterative Fig Variation of log( c c ) vs. i-th outer iteration. our proposed numerical scheme is efficient in solving the nonlinear singular problem with constraints. In our future work, we will consider the theoretical aspects of the proposed method including issues on stabilit and accurac. Acknowledgments. We thank Professors Hean Luo, Pingle Liu and Xiao Luo for man helpful discussions during the preparation of this work. We would also like to thank the referees for ver constructive comments. This work is supported b NSFC Outstanding Youth Scientist Grant and NSFC grant , the National Basic Research Program of China under the grant 25CB3217(1,3), and Scientific Research Fund of Hunan Provincial Education Department. Special thanks go to Guangdong Provincial Zhujiang Scholar Award Project awarded to the second author. References [1] J.U. Brackbill and J.S. Saltzman, Adaptive gid with directional control, J. Comput. Phs, 18 (1993), [2] M. Beckett and J.A. Mackenzie, Convergence analsis of finite difference approimations on equidistributed grids to a boundar value problem, Appl. Numer. Math., 35 (2), [3] H.D. Ceniceros and T.Y. Hou, An efficient dnamicall adaptive mesh for potentiall singular solutions, J. Comput. Phs., 172 (21), [4] Y. Chen, Uniform pointwise convergence for a songularl perturbed problem using arc-length equidistribution, J. Comput. Appl. Math., 159 (23), [5] Y. Chen, Uniform convergence analsis of finite difference approimations for songular perturbations on an adaptive grid, Adv. Comput. Math., 24 (26), [6] R. Datta, Edd viscosit and velocit distribution in turbulent pipe flow revisited, AICHE J., 39 (1993), [7] S.F. Davis and J.E. Flahert, An adaptive finite element method for initial boundar value problems for partial differential equations, SIAM J. Sci. Comput., 3 (1982), [8] Y. Di, R. Li and T. Tang, A general moving mesh framework in 3D and its application for simulating the miture of multi-phase flows. Commun. Comput. Phs., 3 (28), [9] U. Frisch, Turbulence, Cambridge Universit Press, (1995).

12 Moving Mesh Method for a Model of Turbulent Flow 399 [1] W.Z. Huang, Y. Ren and R. Russell, Moving mesh partial differential equations (MMPDEs) based on the equidistribution principle, SIAM J. Numer Anal., 31 (1994), [11] A. Harten and J.M. Hman, Self-adjusting grid methods for one-dimensional hperbolic conservation laws, J. Comput. Phs., 5 (1983), [12] N. Kopteva, Maimum norm a posteriori error estimates for a one-dimensional convection-diffusion problem, SIAM J. Numer. Anal., 39 (21), pp [13] N. Kopteva and M. Stnes, A robust adaptive method for a quasi-linear one-dimensional convection-diffusion problem, SIAM J. Numer. Anal, 39 (21), [14] T. Linβ, Maimum-norm error analsis of a non-monotone FEM for a singularl perturbed reaction-diffusion problem, BIT, 47 (27), [15] X. Luo, P. Liu and H. Luo, Modeling of turbulent flow in circular tubes the application and improvement of Prandtl s miing-length theor, J. Cent. South Univ. T. (English Edition), submitted. [16] R. Li, T. Tang and P.W. Zhang, Moving mesh methods in multiple dimensions based on Harmonic maps, J. Comput. Phs., 17 (21), [17] K. Miller and R.N. Miller, Moving finite element I, SIAM J. Numer. Anal., 18 (1981), [18] Y.S.M. Morsi, P.G. Holland, B.R. Claton, Prediction of turbulent swirling flows in aismmetric annuli, Appl. Math. Model., 19 (1996), [19] T. Mizushina, F. Ogino, Edd viscosit and universal velocit profile in turbulent flow in a straight pipe, Journal of Chemical Engineering of Japan, 35 (197), [2] Z.-H. Qiao, Numerical investigations of the dnamical behaviors and instabilities for the Gierer- Meinhardt sstem. Commun. Comput. Phs., 3 (28), [21] Y. Qiu, D.M. Sloan and T. Tang, Numerical solution of a singularl perturbed two-point boundar value problem using equidistribution:analsis of convergence, J. Comput. Appl. Math., 116 (2), [22] H.Z. Tang, T. Tang and P.W. Zhang, An adaptive mesh redistribution methods for non-linear Hamilton-Jacobi equations in two- and three-dimensions, J. Comput. Phs., 188 (23),