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**Wall modeling challenges for the immersed boundary method**

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

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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

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**IMMERSED BOUNDARY TECHNIQUE**

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**TAGGING “ray tracing” Cartesian Grid Geometry “fluid” cells**

“solid” cells interface cells A special treatment is needed for the cells close to the immersed boundary

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**FORCING Direct forcing (Mohd-Yusof)**

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 Direct forcing (Mohd-Yusof) - The governing equations are not modified - The boundary conditions are enforced directly Sharp interface The boundary condition has to be imposed at the interface cells, which do not coincide with the body. Thus, a local reconstruction of the solution close to the immersed boundary is needed.

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**Locally refined grids: automatic generation**

Procedure (de Tullio et al., JCP 2007) Generation of a first uniform mesh (Input: Ximin , Ximax , DXi ) Refinement of prescribed selected regions ( Input: Ximin , Ximax , DXi ) Automatic refinement along the immersed surface ( Input: Dn, Dt ) Iterative Automatic refinement along prescribed surface ( Input: Dn, Dt ) Iterative Coarsening

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**One-dimensional approach (Fadlun et al., JCP 2000)**

RECONSTRUCTION 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

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**Multi-dimensional linear recontruction (2D)**

RECONSTRUCTION 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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**NUMERICAL METHOD Reynolds Averaged Navier-Stokes equations (RANS)**

k-ω turbulence model (Wilcox, 1998): 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) 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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**NUMERICAL METHOD BiCGStab solver to solve the three sparse matrices:**

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

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RESULTS

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**MESH-REFINEMENT STUDY**

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:

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**MESH-REFINEMENT STUDY**

pressure temperature velocity, u component velocity, v component

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**Heated circular cylinder**

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) Exp: (Wang et al., Phys. Fluids, 2000)

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**Supersonic flow past a cylinder**

Flusso supersonico su cilindro M=1.7; Re=2.e5 Domain: (-10,15) D – (-10, 10) D Locally refined mesh: cells faces q =113° q =112° (exp.) CD= 1.41 CD= 1.43 (exp.) Mach number contours Pressure coefficient

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**Cross-flow oscillating cylinder**

- Re=500, M=0.003 Imposed cross-flow frequency Natural shedding frequency (fixed cylinder) α(t)=y(t)/D vorticity (F=0.875) Ref.[1]: Blackburn, J.Fluid Mech, 1999

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WALL MODELING

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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 Wall functions: motivated by the universal nature of the flat plate boundary layer viscous sublayer: logarithmic layer:

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WALL FUNCTIONS 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: P1 F1 I Turbulence model equations W

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**WALL FUNCTIONS (TAB) (Kalitzin et al. J. Comput. Phys. 2005)**

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) Rey y+ u+ k+ w+

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**WALL FUNCTIONS (TAB) Compute velocity in F1**

Compute friction velocity corresponding to IB surface (W), based on wall model uF1 , yF1 , nF1 ReF1 ReF1 , [tables] y+F1 ut = (y+F1 nF1) / yF1 F1 Extract mean velocity and turbulence quantities in I I W ut , yI , nI y+I y+I , [tables] u+i , k+i, w+i ut , yI , u+i , k+i, w+i ui , ki, wi F1-W is equal to twice the largest distance from the wall of the interface cells

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**WALL FUNCTIONS (ANALYTICAL)**

(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. Molecular and turbulent viscosity variations: where: viscous sublayer (variation of fluid properties in the viscous sublayer is neglected)

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**WALL FUNCTIONS (ANALYTICAL)**

(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 UN : Shear stress:

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**TWO-LAYER WALL MODELING**

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

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**WALL FUNCTIONS (NUMERICAL, NWF)**

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

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**WALL FUNCTIONS (THIN BOUNDARY LAYER, TBLE)**

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

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FLAT PLATE n = 1.6 x m2/s Uinf = 90 m/s ReL=1 = 6 x 106

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FLAT PLATE

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FLAT PLATE

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**RECIRCULATING FLOW ReL = 3.6 x 107 L = 6 m A = 0.35 x Uinf**

Uinf= 90 m/s x/L = 0.16 x/L = 0.58 x/L = 0.75

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

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**RECIRCULATING FLOW (A = 0.35 Uinf)**

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**RECIRCULATING FLOW (A = 0.35 Uinf)**

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**RECIRCULATING FLOW (A = 0.35 Uinf)**

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**RECIRCULATING FLOW (A = 0.27 Uinf)**

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**VKI-LS59 Turbine cascade**

M2,is= 0.81, 1.0, 1.1, 1.2 Re = 8.22x105, 7.44x105, 7.00x105, 6.63x105 Locally refined mesh: cells wall functions (Tables) M2,is=1.2 M2,is=1.2 Mis along the blade Mach number contours

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**RAE-2822 AIRFOIL Local view of the grid Local view of the grid**

Wall resolved reference solution: cells. IB grid: cells; Local view of the grid Local view of the grid

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**RAE-2822 AIRFOIL Pressure coefficient distribution**

Mach number contours (NWF)

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**Conjugate heat transfer: T106 LP turbine**

Temperature contours

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**High Reynolds number turbulent flows**

CONCLUSIONS (1) High Reynolds number turbulent flows 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 Work in progress and future developments Study the influence of source terms Investigate in details the robustness and efficiency issues Include an accurate thermal wall model

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**Arbitrarily shaped particle transport in an incompressible flow**

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**MOTIVATION and BACKGROUND**

ligands 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 Ferrari, Nat Can Rev, 2005

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**MOTIVATION and BACKGROUND**

ligands 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 Ferrari, Nat Can Rev, 2005

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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**

Matas, Morris, Guazzelli, 2004 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).

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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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**Particle density (particle-fluid density ratio)**

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**

PREDICTOR Flow equations F and T exerted by the fluid on the particle Rigid-body dynamic equations New particle configuration CORRECTOR F and T exerted by the fluid on the particle Flow equations Rigid-body dynamic equations New particle configuration YES NO NEW TIME LEVEL

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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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**Sedimentation of a circular particle in a channel**

W/d = 4 Re = 200 202x1002 cells ρs/ ρf = 1.1 Fr = 6.366 Yu, Z., and Shao, X., “A direct-forcing fictitious domain method for particulate flows”. Journal of Computational Physics (227), pp. 292–314.

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**Sedimentation of a circular particle in a channel**

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.

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**Sedimentation of a sphere in a channel:**

settling velocity W/d = 7 Re = 1.5 150x150x192 cells ρs/ ρf = 1.155 Fr = 101.9 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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**Sedimentation of a sphere in a channel:**

sphere trajectory W/d = 7 Re = 1.5 150x150x192 cells ρs/ ρf = 1.155 Fr = 101.9 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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**Sedimentation of a sphere in a channel:**

settling velocity W/d = 7 Re = 11.6 150x150x192 cells ρs/ ρf = 1.164 Fr = 17.77 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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**Sedimentation of a sphere in a channel:**

sphere trajectory W/d = 7 Re = 11.6 150x150x192 cells ρs/ ρf = 1.164 Fr = 17.77 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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**Sedimentation of a sphere in a channel:**

settling velocity W/d = 7 Re = 31.9 150x150x192 cells ρs/ ρf = 1.167 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.

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**Sedimentation of a sphere in a channel:**

sphere trajectory W/d = 7 Re = 31.9 150x150x192 cells ρs/ ρf = 1.167 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.

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**Sedimentation of a triangular particle in a channel**

W/d = 7 Re = 100 300x602 cells ρs/ ρf = 1.5 Fr = 50

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**Sedimentation of an elliptical particle in a channel**

W/d = 4 Re = 12.6 161x402 cells ρs/ ρf = 1.1 Fr = 62.78 θx = 45°; a/b = 2

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**Sedimentation of an elliptical particle in a channel**

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

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**Sedimentation of an elliptical particle in a channel**

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

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**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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**Arbitrarily shaped particle transport**

CONCLUSIONS (2) Arbitrarily shaped particle transport 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. Work in progress Selection of the particle shape for “optimal” margination Interaction models: particle-wall & particle-particle

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**Sedimentation of cylindrical and spherical particles**

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VALIDATION Pressure gradient influence:

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**Conjugate heat transfer: T106 LP turbine**

33000 cells

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**Conjugate heat transfer: T106 LP turbine**

Mach number contours

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