Presentation at 15th OpenFOAM Workshop, June 22-25, 2020, Arlington, VA, USA

Faroughi, S.A.

Fernandes, C.

McKinley, G.H.

Nóbrega, J. M.

English

Universidade do Minho: RepositoriUM

Development of the Drag Coefficient of a Sphere Translating Through a Viscoelastic FluidS.A. Faroughi1, C. Fernandes2, J. Miguel Nóbrega2, G.H. McKinley115th OpenFOAM Workshop, June 22-25, 2020, Arlington, VA, USA1 Hatsopoulos Microfluids Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA(faroughi, gareth)@mit.edu2 Institute for Polymers and Composites/i3N, University of Minho, Campus de Azurém ,4800-058 Guimarães, Portugal(cbpf, mnobrega)@dep.uminho.ptUniversity of MinhoC. Fernandes15th OpenFOAM Workshop, June 22-25, 2020, Arlington, VA, USAOutline 2• Motivation• Objectives• Governing Equations and Numerical Method• Code Verification & Validation• Drag Model• ConclusionC. Fernandes15th OpenFOAM Workshop, June 22-25, 2020, Arlington, VA, USA3Motivation - ApplicationsPolymer CompositesHydraulic FracturingThe flow of particle-laden complex fluids is an ubiquitous problem…C. Fernandes15th OpenFOAM Workshop, June 22-25, 2020, Arlington, VA, USA4Structural length scales Nano Micro                MacroPolymers           Particles              FluidThe fluid is described in an Eulerian frame whereas the particles are handled as discrete points which are tracked individually in a Lagrangian frame using Newton’s second law. Motivation - Numerical approachEulerian-Lagrangian ModelC. Fernandes15th OpenFOAM Workshop, June 22-25, 2020, Arlington, VA, USA5Motivation - Numerical approachStructural length scales Nano Micro                MacroPolymers           Particles              FluidEulerian-Lagrangian ModelSteps+        = Viscoelastic FluidsComplex Suspensions+     = 𝜹𝜹𝜹𝜹 > 𝒅𝒅𝒑𝒑But need to know expressions for 1- Drag force 2- Lift force 3- Hindrance effect (due to presence of other particles)C. Fernandes15th OpenFOAM Workshop, June 22-25, 2020, Arlington, VA, USA6Motivation – Bulk transport modelEulerian-Lagrangian ModelParticle Volume fraction, cDrag  Correction Factor, FHill et al. (2001)Newtonian FluidDrag on a particle when placed in random array of particles *Faroughi S. A. (2016) Theoretical Developments to Model Microstructural Effects on The Rheology of Complex Fluids, PhD Thesis.*Subramaniam S., (2013). Progress in Energy and Combustion Science – Elsevier*Hill, R., Koch, D., & Ladd, A. (2001). Journal of Fluid Mechanics, 448, 243-278C. Fernandes15th OpenFOAM Workshop, June 22-25, 2020, Arlington, VA, USA7ObjectivesParametrize the effects of fluid elasticity, especiallythe relaxation and retardation times, as well asinertia on the drag coefficient of a sphere translatingthrough a viscoelastic fluid described by theOldroyd-B modelC. Fernandes15th OpenFOAM Workshop, June 22-25, 2020, Arlington, VA, USA8Governing Equations and Numerical Method(Pseudo) Transient, incompressible laminar flow of an Oldroyd-B fluidContinuityMomentum balanceConstitutive modelRetardation ratioiBSD** C Fernandes, MSB Araujo, LL Ferrás, JM Nóbrega, Improved both sides diffusion (iBSD): A new and straightforward stabilization approach for viscoelastic fluid flows, JNNFM, 249, 63-78, 2017foam-extend 4.0Finite Volume MethodSIMPLE like algorithmSecond order discretization methodsAbsolute tolerance for u, p, 𝝉𝝉 fields 10-10Stop criterion  4th decimal place of CDUCMC. Fernandes15th OpenFOAM Workshop, June 22-25, 2020, Arlington, VA, USA9Case StudyComputational domain(square and circular ducts)Drag CoefficientC. Fernandes15th OpenFOAM Workshop, June 22-25, 2020, Arlington, VA, USA10Case Study – Mesh/geometry independency• Mesh generated by snappyHexMesh utility• 7 steps mesh refinement towards the sphere • 3 steps mesh refinement towards the wall for bounded domain• Lx =180a• ~2.5M cellsΔx= Δz= Δy=0.5a C. Fernandes15th OpenFOAM Workshop, June 22-25, 2020, Arlington, VA, USA11Code Verification & ValidationNewtonian fluidsC. Fernandes15th OpenFOAM Workshop, June 22-25, 2020, Arlington, VA, USA12Viscoelastic fluidsUCM Oldroyd-BCode Verification & ValidationC. Fernandes15th OpenFOAM Workshop, June 22-25, 2020, Arlington, VA, USA13Drag Model• Square duct• Lx =180a, Ly = Lz =40a• Periodic boundary conditions at the lateral wallsC. Fernandes15th OpenFOAM Workshop, June 22-25, 2020, Arlington, VA, USA14Drag ModelCreeping flow conditionsC. Fernandes15th OpenFOAM Workshop, June 22-25, 2020, Arlington, VA, USA15Drag ModelCreeping flow conditions - Proposed drag model (PDM)C. Fernandes15th OpenFOAM Workshop, June 22-25, 2020, Arlington, VA, USA16Drag ModelCreeping flow conditionsPDMPDMC. Fernandes15th OpenFOAM Workshop, June 22-25, 2020, Arlington, VA, USA17Drag ModelEffect of inertiaProposed drag model just valid for Re ≤ 1 !!! PDM PDM PDMC. Fernandes15th OpenFOAM Workshop, June 22-25, 2020, Arlington, VA, USA18Drag ModelEffect of inertia(Spherical polar coordinates)Re= 0.1Re= 10Re= 100Re= 1C. Fernandes15th OpenFOAM Workshop, June 22-25, 2020, Arlington, VA, USA19Drag ModelEffect of inertiaC. Fernandes15th OpenFOAM Workshop, June 22-25, 2020, Arlington, VA, USA20Conclusion• DNS to study the effects of fluid inertia, elasticity and polymer viscosity ratio, on the overall drag coefficient• A drag model was proposed to fit the data collected on numerical experiments• Drag model valid (95% accuracy) for 0 < ζ < 1, De < 9 and Re ≤ 1C. Fernandes15th OpenFOAM Workshop, June 22-25, 2020, Arlington, VA, USAThank you for your attentionThe authors would like to acknowledge the funding by FEDERfunds through the COMPETE 2020 Programme and NationalFunds through FCT - Portuguese Foundation for Science andTechnology under the projects UID/CTM/50025/2013 and POCI-01-0247-FEDER-017656.

Development of the drag coefficient of a sphere translating through a viscoelastic fluid

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Development of the drag coefficient of a sphere translating through a viscoelastic fluid

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Universidade do Minho: RepositoriUM