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Wall modeling challenges for the immersed boundary method G. Pascazio Workshop Num. Methods non-body fitted grids - Maratea, May POLITECNICO DI BARI DIMeG & CEMeC Via Re David 200, 70125, Bari, ITALY M. D. de Tullio, P. De Palma, M. Napolitano; G. Iaccarino, R. Verzicco; G. Adriani, P. Decuzzi …….

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OUTLINE Immersed Boundary technique Tagging, Forcing, Near-wall reconstruction High-Re turbulent flows: wall modeling Tables, Analytical, Numerical Preconditioned compressible-flow solver Results Arbitrarily shaped particle transport in an incompressible flow Fluid-structure interaction solver Results

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IMMERSED BOUNDARY TECHNIQUE

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Geometry Cartesian Grid “ ray tracing” TAGGING A special treatment is needed for the cells close to the immersed boundary “fluid” cells “solid” cells interface cells

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FORCING Direct forcing (Mohd-Yusof) - The governing equations are not modified - The boundary conditions are enforced directly - Sharp interface Thus, a local reconstruction of the solution close to the immersed boundary is needed. The boundary condition has to be imposed at the interface cells, which do not coincide with the body. During the computation, the flow variables at the center of the fluid cells are the unknowns, the solid cells do not influence the flow field at all, and at the interface cells the forcing is applied

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Procedure (de Tullio et al., JCP 2007) Generation of a first uniform mesh (Input: X i min, X i max, X i ) Refinement of prescribed selected regions ( Input: X i min, X i max, X i ) Automatic refinement along the immersed surface ( Input: n, t ) Iterative Automatic refinement along prescribed surface ( Input: Dn, Dt ) Iterative Coarsening Locally refined grids: automatic generation

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One-dimensional approach (Fadlun et al., JCP 2000) At each interface cell, a linear interpolation is employed along the Cartesian direction closest to the normal to the immersed surface RECONSTRUCTION

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Multi-dimensional linear recontruction (2D) (e.g., Yang and Balaras, JCP 2006) Dirichlet boundary condition: Neumann bounary condition:

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DISTANCE-WEIGHTED RECONSTRUCTION (LIN) (de Tullio et al., JCP 2007) Dirichlet boundary conditions: Neumann boundary conditions:

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COMPRESSIBLE SOLVER (RANS)

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Pseudo-time derivative term added to the LHS to use a “time marching” approach for steady and unsteady problems (Venkateswaran and Merkle, 1995) Preconditioning matrix Γ to improve the efficiency for a wide range of the Mach number (Merkle, 1995) NUMERICAL METHOD Reynolds Averaged Navier-Stokes equations (RANS) k-ω turbulence model (Wilcox, 1998): Euler implicit scheme discretization in the pseudo-time 2nd order accurate three point backward discretization in the physical time Diagonalization procedure (Pulliam and Chausee, 1981) Factorization of the LHS

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BiCGStab solver to solve the three sparse matrices: NUMERICAL METHOD Colocated cell-centred finite-volume space discretization Convective terms: 1st, 2nd and 3rd order accurate flux difference splitting scheme or 2nd order accurate centred scheme Viscous terms: 2nd order accurate centred scheme Minmod limiter in presence of shocks Semi-structured Cartesian grids

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- For each face, the contributions of the neighbour cells are collected to build the corresponding operators (convective/diffusive) for the cell FLUX EVALUATION

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RESULTS

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NACA-0012 M=0.8, Re=20, angle of attack= 10° Space domain: [-10 c, 11 c] [ -10 c, 10 c] 5 meshes: “Exact” solution obtained by means of a Richardson extrapolation employing the two finest meshes: MESH-REFINEMENT STUDY

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pressuretemperature MESH-REFINEMENT STUDY velocity, u componentvelocity, v component

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Exp: (Wang et al., Phys. Fluids, 2000) - M=0.03; Re=100,120,140; T*=1.0, 1.1, 1.5, 1.8 (T* = Tw/Tinf) - (-10,40) D – (-15, 15) D - Mesh: cells, faces - T (Energy equation) is crucial for T*>1 - unsteady periodic flow Temperature contours (Re=100, T*=1.8) Heated circular cylinder

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Flusso supersonico su cilindro Mach number contours Pressure coefficient - M=1.7; Re=2.e5 - Domain: (-10,15) D – (-10, 10) D Locally refined mesh: cells faces =112° (exp.) C D = 1.41 C D = 1.43 (exp.) Supersonic flow past a cylinder

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Ref.[1]: Blackburn, J.Fluid Mech, Re=500, M=0.003 vorticity (F=0.875) α(t)=y(t)/D Cross-flow oscillating cylinder Imposed cross-flow frequency Natural shedding frequency (fixed cylinder)

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WALL MODELING

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viscous sublayer: logarithmic layer: Wall functions: motivated by the universal nature of the flat plate boundary layer WALL MODELING Linear interpolation is adequate for laminar flows or when the interface point is within the viscous sublayer Brute-force grid refinement is not efficient in a Cartesian grid framework Local grid refinement alleviates the resolution requirements, but still it is not an adequate solution for very high Reynolds number flows

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W I P1 The Navier Stokes equations are solved down to the fluid point P1 Flow variables at the interface point I are imposed solving a two-point boundary value problem: WALL FUNCTIONS Turbulence model equations F1

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It is possible to define a local Reynolds number, based on y and U. The following is a universal function: This function is evaluated once and for all using a wall resolved, grid- converged numerical solution and stored in a table along with its inverse (look-up tables) WALL FUNCTIONS (TAB) Re y y+y+ u+u+ k+k+ ++ (Kalitzin et al. J. Comput. Phys. 2005)

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W F1 Compute friction velocity corresponding to IB surface (W), based on wall model I Extract mean velocity and turbulence quantities in I Compute velocity in F1 F1-W is equal to twice the largest distance from the wall of the interface cells WALL FUNCTIONS (TAB) u F1, y F1, F1 Re F1 Re F1, [tables] y + F1 u y + F1 F1 ) / y F1 u , y I, I y + I y + I, [tables] u + i, k + i, i u , y I, u + i, k + i, + i u i, k i, i

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Molecular and turbulent viscosity variations: WALL FUNCTIONS (ANALYTICAL) viscous sublayer (Craft et al. Int. J. of Heat Fluid Flow, 2002) To simplify integration, rather than a conventional damping function, a shift of the turbulent flow origin from the wall to the edge of the viscous layer is modeled. where: (variation of fluid properties in the viscous sublayer is neglected)

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(Craft et al. Int. J. of Heat Fluid Flow, 2002) Velocity variation in the near-wall region: The equation is integrated separately across the viscous and fully turbulent regions, resulting in analytical formulations for U, given the value of U N : WALL FUNCTIONS (ANALYTICAL) Shear stress:

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TWO-LAYER WALL MODELING W F1 I A virtual refined mesh is embedded between the wall point W and F1 in the normal direction P1 h The Navier Stokes equations are solved down to the fluid point P1 Point F1 is found, along the normal-to-the- wall direction, at twice the largest distance from the wall of the interface cells Simplified turbulent boundary layer equations are solved at the virtual grid points Velocity at F1 is interpolated using the surrounding cells Velocity at the interface point I is interpolated

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y = normal direction x = tangential direction Momentum equation (Balaras et al. 1996; Wang and Moin, 2002) : An iterative procedure has been implemented to solve the equations simultaneously k = 0.4 A=16 Boundary conditions: velocity at point F1 (interpolated from neighbours fluid nodes) velocity at the wall (zero). The eddy viscosity is obtained from a simple mixing length model with near wall damping (Cabot and Moin, FTC 1999) : WALL FUNCTIONS (NUMERICAL, NWF)

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y = normal direction x = tangential direction Momentum equation: WALL FUNCTIONS (THIN BOUNDARY LAYER, TBLE) Turbulence model equations: Boundary conditions: at point F1 (interpolated from neighbours fluid nodes) at the wall (zero velocity and k, and Menter for ). An iterative procedure has been implemented to solve the equations simultaneously

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FLAT PLATE = 1.6 x m 2 /s U inf = 90 m/s Re L=1 = 6 x 10 6

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FLAT PLATE

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RECIRCULATING FLOW Re L = 3.6 x 10 7 L = 6 m A = 0.35 x U inf U inf = 90 m/s x/L = 0.16x/L = 0.58x/L = 0.75

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RECIRCULATING FLOW x/L = 0.16x/L = 0.58 x/L = 0.75

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RECIRCULATING FLOW (A = 0.35 Uinf)

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RECIRCULATING FLOW (A = 0.27 Uinf)

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M 2,is = 0.81, 1.0, 1.1, 1.2 Re = 8.22x10 5, 7.44x10 5, 7.00x10 5, 6.63x10 5 Locally refined mesh: cells wall functions (Tables) Mach number contours M is along the blade M 2,is =1.2 VKI-LS59 Turbine cascade

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RAE-2822 AIRFOIL Local view of the grid Wall resolved reference solution: cells. IB grid: cells;

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Mach number contours (NWF) Pressure coefficient distribution RAE-2822 AIRFOIL

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Conjugate heat transfer: T106 LP turbine Temperature contours

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Study the influence of source terms Investigate in details the robustness and efficiency issues Include an accurate thermal wall model Wall modelling appears to be an efficient tool for computation of high- Re flows; Different approaches have been investigated to model the flow behaviour normal to the wall: (a) look-up tables; (b) analytical wall functions; (c) numerical wall functions (NWF & TBLE); Wall functions provide good results for attached flows; Encouraging results for separated flows; in particular NWF and TBLE with embedded one-dimensional grids CONCLUSIONS (1) High Reynolds number turbulent flows Work in progress and future developments

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Arbitrarily shaped particle transport in an incompressible flow

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MOTIVATION and BACKGROUND Ferrari, Nat Can Rev, 2005 Use of micro/nano-particles for drug delivery and imaging. Properly designed micro/nano- particles, once administered at the systemic level and transported by the blood flow along the circulatory system, are expected to improve the efficiency of molecule-based therapy and imaging by increasing the mass fraction of therapeutic molecules and tracers that are able to reach their targets PEG ligands

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MOTIVATION and BACKGROUND Ferrari, Nat Can Rev, 2005 Particles are transported by the blood flow and interact specifically (ligand- receptor bonds) and non-specifically (e.g., van der Waals, electrostatic interactions) with the blood vessel walls, seeking for their target (diseased endothelium). The intravascular “journey” of the particle can be broken down into two events: margination dynamics and firm adhesion. PEG ligands

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MOTIVATION and BACKGROUND The margination is a well-known term in physiology conventionally used to describe the lateral drift of leukocytes and platelets from the core blood vessel towards the endothelial walls. The observation of inhomogeneous radial distributions of particles in tube flow dates from the work of Poiseuille (1836) who was mainly concerned by the flow of blood and the behavior of the red and white corpuscles it carries. Experimental results (Segré & Silberberg, JFM 1962) show the radial migration develops in a pipe from a uniform concentration at the entrance. Equilibrium position r/R = 0.62

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MOTIVATION and BACKGROUND Experimental distribution of particle position (particle diameter 900 μm) over a cross section of the flow observed for two values of the Reynolds number: Re = 60 (left) and Re = 350 (right). Matas, Morris, Guazzelli, 2004

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Micro/nano-particle with different size: from few tens of nm to few μm composition: gold- and iron-oxide, silicon shape: spherical, conical, discoidal, …. surface physico-chemical properties: charge, ligants

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Design parameters Particle size and shape Reynolds number based on the channel diameter Particle density (particle-fluid density ratio) Number of particles in the bolus An accurate model predicting the behavior of intravascularly injectable particles can lead to a dramatic reduction of the “bench-to-bed” time for the development of innovative MNP-based therapeutic and imaging agents.

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Governing equations Navier-Stokes equations for a 3D unsteady incompressible flow solved on a Cartesian grid: Rigid body dynamic equations:

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Flow solver (Verzicco, Orlandi, J. Comput. Phys., 1996) staggered-grid second-order-accurate space discretization fractional-step method: non-linear terms: explicit Adam-Bashford scheme linear terms: implicit Crank-Nicholson scheme immersed boundary with 1D reconstruction (Fadlun et al., J. Comput. Phys., 2000)

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Implicit coupled approach PREDICTORPREDICTORPREDICTORPREDICTOR Flow equationsF and T exerted by the fluid on the particle Rigid-body dynamic equations New particle configuration Flow equations F and T exerted by the fluid on the particle Rigid-body dynamic equations New particle configuration CORRECTORCORRECTORCORRECTORCORRECTOR YES NEW TIME LEVEL NO

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Predictor-corrector scheme Predictor: second-order-accurate Adam-Bashford scheme Corrector: iterative second-order-accurate implicit scheme with under-relaxation

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W/d = 4 Re = 200 202x1002 cells ρs/ ρf = 1.1 Fr = Yu, Z., and Shao, X., “A direct-forcing fictitious domain method for particulate flows”. Journal of Computational Physics (227), pp. 292–314. Sedimentation of a circular particle in a channel

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W/d = 4 Re = 0.1 202x302 cells ρs/ ρf = 1.2 Fr = 1398 Yu, Z., and Shao, X., “A direct-forcing fictitious domain method for particulate flows”. Journal of Computational Physics (227), pp. 292–314. Sedimentation of a circular particle in a channel

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W/d = 7 Re = 1.5 150x150x192 cells ρs/ ρf = Fr = Sedimentation of a sphere in a channel: settling velocity ten Cate, A., Nieuwstad, C.H., Derksen, J.J., and Van den Akker, H.E.A., “Particle imaging velocimetry experiments and lattice-Boltzmann simulations on a single sphere settling under gravity”. Physics of fluid Vol.14 (11), pp. 4012–4025.

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W/d = 7 Re = 1.5 150x150x192 cells ρs/ ρf = Fr = Sedimentation of a sphere in a channel: sphere trajectory

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W/d = 7 Re = 11.6 150x150x192 cells ρs/ ρf = Fr = ten Cate, A., Nieuwstad, C.H., Derksen, J.J., and Van den Akker, H.E.A., “Particle imaging velocimetry experiments and lattice-Boltzmann simulations on a single sphere settling under gravity”. Physics of fluid Vol.14 (11), pp. 4012–4025. Sedimentation of a sphere in a channel: settling velocity

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ten Cate, A., Nieuwstad, C.H., Derksen, J.J., and Van den Akker, H.E.A., “Particle imaging velocimetry experiments and lattice-Boltzmann simulations on a single sphere settling under gravity”. Physics of fluid Vol.14 (11), pp. 4012–4025. W/d = 7 Re = 11.6 150x150x192 cells ρs/ ρf = Fr = Sedimentation of a sphere in a channel: sphere trajectory

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W/d = 7 Re = 31.9 150x150x192 cells ρs/ ρf = Fr = 8.98 ten Cate, A., Nieuwstad, C.H., Derksen, J.J., and Van den Akker, H.E.A., “Particle imaging velocimetry experiments and lattice-Boltzmann simulations on a single sphere settling under gravity”. Physics of fluid Vol.14 (11), pp. 4012–4025. Sedimentation of a sphere in a channel: settling velocity

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ten Cate, A., Nieuwstad, C.H., Derksen, J.J., and Van den Akker, H.E.A., “Particle imaging velocimetry experiments and lattice-Boltzmann simulations on a single sphere settling under gravity”. Physics of fluid Vol.14 (11), pp. 4012–4025. W/d = 7 Re = 31.9 150x150x192 cells ρs/ ρf = Fr = 8.98 Sedimentation of a sphere in a channel: sphere trajectory

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W/d = 7 Re = 100 300x602 cells ρs/ ρf = 1.5 Fr = 50 Sedimentation of a triangular particle in a channel

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W/d = 4 Re = 12.6 161x402 cells ρs/ ρf = 1.1 Fr = θx = 45°; a/b = 2 Sedimentation of an elliptical particle in a channel

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Xia, Z., W. Connington, K., Rapaka, S., Yue, P., Feng, J. and Chen, S., “Flow patterns in the sedimentation of an elliptical particle”. J. Fluid Mech. Vol.625, pp Sedimentation of an elliptical particle in a channel

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Xia, Z., W. Connington, K., Rapaka, S., Yue, P., Feng, J. and Chen, S., “Flow patterns in the sedimentation of an elliptical particle”. J. Fluid Mech. Vol.625, pp Sedimentation of an elliptical particle in a channel

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Transport dynamics of a triangular particle in a plane Poiseuille flow W/d = 7 Re = 50 161x402 cells ρs/ ρf = 1.1

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Selection of the particle shape for “optimal” margination Interaction models: particle-wall & particle-particle Fluid-structure interaction solver is effective in the simulation of the transport dynamics of particles in an incompressible flow; Particles with arbitrary shape can be handled; Transport of bolus of particles is feasible. CONCLUSIONS (2) Arbitrarily shaped particle transport Work in progress

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Sedimentation of cylindrical and spherical particles

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Pressure gradient influence: VALIDATION

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Conjugate heat transfer: T106 LP turbine cells

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Conjugate heat transfer: T106 LP turbine Mach number contours

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