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A new modification of the immersed boundaries method for fluid-solid flows: moderate Reynolds numbers.

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A new modification of the immersed boundaries method for fluid-solid flows: moderate Reynolds numbers.
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  Short Note A new modification of the immersed boundaries methodfor fluid–solid flows: moderate Reynolds numbers A. Vikhansky * Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge CB2 3RA, UK  Received 22 October 2002; received in revised form 4 June 2003; accepted 11 June 2003 Abstract In this study we consider a combination of an interpolation technique with the D Õ Alambert principle that allows thedirect numerical simulation of fluid-solid flows on a regular rectangular grid. The method models a no-slip boundarywith the second-order accuracy interpolation scheme without explicit calculation of the forces that the particles exert atthe fluid. The method is applicable for a wide range of Reynolds numbers from creeping flows up to Re % 100 : Ó 2003 Elsevier Science B.V. All rights reserved. 1. Introduction In this study we consider a method that allows the direct numerical simulation of fluid–solid flows on aregular rectangular grid without explicit calculation of the forces that the particles exert at the fluid. The no-slip boundary conditions at the solid–fluid interface are satisfied automatically. The combination of aninterpolation technique with the D Õ Alambert principle allows an efficient numerical realization of thesuggested numerical method.Direct numerical simulation of solid–liquid flow is a difficult task since the domain occupied by the fluid isirregular and changes with motion of the particles. Also, the particles are advected by the fluid and exertforces at the fluid, so the body–liquid interaction requires calculation of the fluid stress (i.e., derivatives of theflowfield) at the fluid–solid interface. Such an approach was realized in the arbitrary Lagrangian–Eulerian(ALE) technique (see e.g. [1] and references therein), which is based on automatic generation of an un-structured finite element body-fitted mesh at each time step of the numerical algorithm. Since flow simu-lations on an unstructured mesh are still a challenging problem for today Õ s computers, methods that avoidgeneration of a body-fitted mesh and do not require explicit calculation of a stress tensor are preferred.During the last decade the lattice Boltzmann method has become a useful tool for studying dynamicsof suspensions at low and moderate Reynolds numbers (a comprehensive review is in [2]). Anothernumerical technique that allows simulation of arbitrary Reynolds number flows in complex geometries on Journal of Computational Physics 191 (2003) 328–339www.elsevier.com/locate/jcp * Tel.: +44-1223-334786. E-mail address: av277@cam.ac.uk(A. Vikhansky).0021-9991/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved.doi:10.1016/S0021-9991(03)00313-9  non-body-fitted grids is the immersed boundaries method. It was demonstrated recently that this methodhas lower RAM and CPU-time requirements than more complicated boundary-fitted solvers [3].The general idea of the immersed boundaries method [4] is to solve the equations for liquid velocity bothinside and outside the moving boundaries, while the fluid and particle velocities are matched by forcesdistributed along the boundary (in the most recent variants of this method the forces are distributed overthe entire region occupied by the particle). In the srcinal paper by Peskin [4] (see also [5]) the interactionbetween a fluid and immersed solids is accounted for by a penalty term that makes the equations stiff andimposes restrictions on the time step. Recently, similar forcing was performed using a duality method.Distributed-Lagrange-multiplier (DLM) method [6,7] was developed and extensively applied to differentparticulate flows. In this method the variational equations for a fluid flow are solved on a regular grid, andthe constraints of rigid-body motion of a fluid inside the particles are enforced by distributed Lagrangemultipliers.The similarity between the above two methods is due to the similarity between penalty and dualitymethods. One can minimize a function f  ð ~  x x ) under a constraint g  ð ~  x x Þ ¼ 0 either by penalty or dualitymethods. Penalty method yields the following minimization problem: f  ð ~  x x Þ þ r  ð  g  ð ~  x x ÞÞ 2 ! min and r  f  ð ~  x x Þ þ 2 rg  ð ~  x x Þr  g  ð ~  x x Þ ¼ 0 in the location of the minimum, while the duality technique leads to a modifiedminimization problem f  ð ~  x x Þ þ k  g  ð ~  x x Þ ! min and r  f  ð ~  x x Þ þ k r  g  ð ~  x x Þ ¼ 0. Comparison of the above formulasshows that 2 rg  ð ~  x x Þ ! k as penalty parameter r  goes to infinity, i.e., the penalty method and the method of Lagrange multipliers are different realization of the same forcing scheme.In the present study we suggest a variant of the immersed boundary method that does not require anyexplicit forcing either by the penalty method or by calculation of Lagrange multipliers. The key idea of theproposed method is a combination of the interpolation technique considered in [3,8] with a variationalprinciple. Owing to the special treatment of near-boundary points, the constraints of rigid-body motion of fluid inside the region occupied by the particles are satisfied automatically, while the force interactionbetween the fluid and the immersed particles is captured implicitly by the principle of virtual work.Therefore, the problem is reduced to a sequence of unconstrained minimization problems, while the DLMmethod treats particle–fluid interaction via more computationally expensive constrained variationalequations. 2. Variational formulation of the problem and interpolation scheme Consider for simplicity a single disc C with radius R immersed into a space domain X filled with anincompressible viscous fluid with density q f  and viscosity l (Fig. 1). The disc is made of homogeneousmaterial with density q s , the mass of the disc is M  ¼ q s p  R 2 , and its moment of inertia I  ¼ 1 = 2 q s p  R 4 . The Fig. 1. Schematic view of a two-dimensional cavity with one rigid disc. A. Vikhansky / Journal of Computational Physics 191 (2003) 328–339 329  center of the disc, its axial and angular velocities are ~  x x c ð t  Þ , ~ U U  , and ~ xx , respectively (angular velocity is or-thogonal to the flow). The boundary of the disc is o C ð t  Þ . The acceleration of the external body force actingat the system is ~  g  g  . The starting point of our numerical method is the virtual power principle [6]: Z  X n C q f  o ~ uu o t  " þ ~ uu Á r  ~ uu À ~  g  g   Á d ~ uu þ l r ~ uu : r d ~ uu  À p  r Á d ~ uu # d X þ M  d ~ U U  d t   À ~  g  g   Á d ~ U U  þ I  d ~ xx d t  Á d ~ xx ¼ 0 : ð 1 Þr Á ~ uu ¼ 0 ; r Á d ~ uu ¼ 0 ;~ uu j C ¼ ~ U U  þ ~ xx  ð ~  x x À ~  x x c Þ ; d ~ uu j C ¼ d ~ U U  þ d ~ xx  ð ~  x x À ~  x x c Þ ; ð 2 Þ where d denotes variation of the corresponding variable and ~ uu ð t  ;~  x x Þ and p  ð t  ;~  x x Þ are the velocity and pressureof the fluid, respectively. The first term in Eq. (1) describes inertia of the fluid, the second term accounts forviscous dissipation, and the third is virtual work due to compression/expansion of the fluid. The last twoterms describe linear and angular inertia of the disc.The velocity field inside the disc is ~ uu ¼ ~ U U  þ ~ xx  ð ~  x x À ~  x x c Þ , thus, the disc–inertia term in Eq. (1) can be splitas  M  d ~ U U  d t   À ~  g  g   Á d ~ U U  þ I  d ~ xx d t  Á d ~ xx ¼ 1  À q f  q s  M  d ~ U U  d t  " À ~  g  g   Á d ~ U U  þ I  d ~ xx d t  Á d ~ xx # þ q f  q s Z  X q s o ~ uu o t   þ ~ uu Á r  ~ uu À ~  g  g   Á d ~ uu d X : Since the flow inside the disc is incompressible and does not dissipate energy, insertion of the above formulainto Eq. (1) yields: Z  X q f  o ~ uu o t  " þ ~ uu Á r  ~ uu À ~  g  g   Á d ~ uu þ l r ~ uu : r d ~ uu  À p  r Á d ~ uu # d X þ 1  À q f  q s  M  d ~ U U  d t  " À ~  g  g   Á d ~ U U  þ I  d ~ xx d t  Á d ~ xx # ¼ 0 : ð 1 0 Þ Eq. (1 0 ) after applying Gauss theorem implies Navier–Stokes equations q f  o ~ uu o t  ( þ ~ uu Á r  ~ uu À ~  g  g   þ r  p  À l D ~ uu ) Á d ~ uu ¼ 0 ; for ~  x x 2 X n C ; force balance over the disc Z  C q f  o ~ uu o t  " þ ~ uu Á r  ~ uu À ~  g  g   Á d ~ U U  # d C þ 1  À q f  q s  M  d ~ U U  d t  " À ~  g  g   Á d ~ U U  # þ Z  o C l o ~ uu o n  À p  ~ nn  Á d ~ U U  d o C ð Þ ¼ M  d ~ U U  d t   À ~  g  g   Á d ~ U U  þ Z  o C l o ~ uu o n  À p  ~ nn  Á d ~ U U  d o C ð Þ ¼ 0 ; 330 A. Vikhansky / Journal of Computational Physics 191 (2003) 328–339  and moment balance over the disc Z  C q f  o ~ uu o t  " þ ~ uu Á r  ~ uu  Á d ~ xx  ~  x x  À ~  x x c # d C þ 1  À q f  q s  I  d ~ xx d t  Á d ~ xx "  ~  x x  À ~  x x c # þ Z  o C l o ~ uu o n Á d ~ xx  ~  x x  À ~  x x c  d o C ð Þ ¼ I  d ~ xx d t  Á d ~ xx  ~  x x  À ~  x x c  þ Z  o C l o ~ uu o n Á d ~ xx  ~  x x  À ~  x x c  d o C ð Þ ¼ 0 : In the above formulas we have used the relation that is valid inside the disc o ~ uu = o t  þ ð ~ uu Á rÞ ~ uu ¼ _ ~ U U  ~ U U  þ _ ~ xx ~ xx  ð ~  x x À ~  x x c Þ . Thus, the above variational principle can be applied to simulation of particle flow in-teraction.The suggested numerical procedure is as follows. We calculate the positions of the disc ~  x x c at times t  n ¼ n D t  , while the velocities are calculated at semi-integer moments t  n ¼ ð n þ 1 = 2 Þ D t  . Note that the pro-posed procedure can be easily implemented with the variable time step. First, an intermediate position iscalculated as ~  x x ð t  þ D t  = 2 Þ ¼ ~  x x t  ð Þ þ D t  = 8 7 ~ U U  ð t  h À D t  = 2 Þ À 3 ~ U U  ð t  À 3 D t  = 2 Þ i : Then, for the determined particle positions, Eqs. (1 0 ) and (2) for pressure p and velocities ~ uu ;~ U U  ;~ xx are solvedby operator-splitting method [6]. We used finite-difference method on a staggered five-point stencil asfollows. Variational problem (1 0 ) and (2) is split into the following three steps:(1) The continuity condition is enforced at the pressure-correction step ~ uu à ¼ ~ uu n À D t  ð Þ r  p  n þ 1 ÀÁ : Substituting the latter equation in the first of Eq. (2) yields the Poisson equation for p  n þ 1 : r Á ~ uu à ¼ 0 ¼ r Á ~ uu n À D t  ð Þ D  p  n þ 1 ÀÁ : ð 3 Þ (2) The advection step: ~ uu Ãà À ~ uu à D t  ¼ À ~ uu à Á r  ~ uu à : ð 4 Þ (3) The pseudo-Stokes step that is a solution of a following quadratic minimization problem: U ð ~ uu n þ 1 ;~ U U  n þ 1 ;~ xx n þ 1 Þ ¼ Z  X q f  2 D t  ð ~ uu n þ 1 & À ~ uu Ãà À D t  ~  g  g  Þ 2 þ l 2 r ~ uu n þ 1  2 ' d X þ 1  À q f  q s  M  2 D t  ð ~ U U  n þ 1  À ~ U U  n À D t  ~  g  g  Þ 2 þ I  2 D t  ð ~ xx n þ 1 À ~ xx n Þ 2 ! ! min : ð 5 Þ While in the case q s < q f  the coefficient near ~ U U  and ~ xx in Eq. (5) becomes negative, the functional (5) re-mains positively defined with respect to these variables. Note that the terms that account for the inertia of the particle read: Z  C q f  2 D t  ð ~ uu n þ 1 À ~ uu ÃÃ Þ 2 d C þ 1  À q f  q s  M  2 D t  ð ~ U U  n þ 1  À ~ U U  n Þ 2 þ I  2 D t  ð ~ xx n þ 1 À ~ xx n Þ 2 ! ; A. Vikhansky / Journal of Computational Physics 191 (2003) 328–339 331  where the integration is performed over the region occupied by the particle, and ~ uu n þ 1 ¼ ~ U U  n þ 1 þ ~ xx n þ 1  ð ~  x x À ~  x x c Þ for ~  x x 2 C .The main idea of the suggested variant of the immersed boundary method that is proposed in this studyis illustrated in Fig. 2. At the grid points that are far from the body the central difference approximation of dissipation term in Eq. (5) reads: r u j j 2 d  x d  y  % h 2 2 ð u ij À u i À 1  j Þ 2 h 2  þð u ij À u i þ 1  j Þ 2 h 2 þð u ij À u ij À 1 Þ 2 h 2 þð u ij À u ij þ 1 Þ 2 h 2  ¼ 12 ð u ij  À u i À 1  j Þ 2 þ ð u ij À u i þ 1  j Þ 2 þ ð u ij À u ij À 1 Þ 2 þ ð u ij À u ij þ 1 Þ 2  : Thus, at the points the closest to the boundary, the first term in the above expression is replaced by h 2 g u ij À U  À x ð  x À x c Þð Þ ÀÁ 2 g h ð Þ 2 ¼ u ij À U  À x ð  x À x c Þð Þ ÀÁ 2 g : ð 6 Þ Here h is the step of the grid and g is the dimensionless distance from the i th node to the surface of the disc.Since Eq. (6) is based on linear interpolation of flow field, it requires the mesh to be fine so that the lin-earized velocity is accurate [3]. Thus, the applicability of this method to direct simulation of two phaseturbulent flows should be subject of additional investigations.When g ¼ 1, i.e., the distance from the i th node to the surface is equal to the step of the grid, the dis-sipation rate given by Eq. (6) is identical to that obtained by central difference stencil. The velocities in thenodes that are inside the disc are set equal to the velocity of the body in these points. As one can see, the Fig. 2. Scheme of the immersed boundary method.332 A. Vikhansky / Journal of Computational Physics 191 (2003) 328–339
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