Research Papers: Fundamental Issues and Canonical Flows

Reduced Order Model for a Power-Law Fluid

[+] Author and Article Information
M. Ocana, D. Alonso

Aerospace Propulsion
and Fluid Mechanics Department,
School of Aeronautics,
Universidad Politecnica de Madrid,
Plaza del Cardenal Cisneros 3,
Madrid 28040, Spain

A. Velazquez

Aerospace Propulsion
and Fluid Mechanics Department,
School of Aeronautics,
Universidad Politecnica de Madrid,
Plaza del Cardenal Cisneros 3,
Madrid 28040, Spain
e-mail: angel.velazquez@upm.es

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received June 19, 2013; final manuscript received February 3, 2014; published online May 6, 2014. Assoc. Editor: D. Keith Walters.

J. Fluids Eng 136(7), 071205 (May 06, 2014) (9 pages) Paper No: FE-13-1382; doi: 10.1115/1.4026666 History: Received June 19, 2013; Revised February 03, 2014

This article describes the development of a reduced order model (ROM) based on residual minimization for a generic power-law fluid. The objective of the work is to generate a methodology that allows for the fast and accurate computation of polymeric flow fields in a multiparameter space. It is shown that the ROM allows for the computation of the flow field in a few seconds, as compared with the use of computational fluid dynamics (CFD) methods in which the central processing unit (CPU) time is on the order of hours. The model fluid used in the study is a polymeric fluid characterized by both its power-law consistency index m and its power-law index n. Regarding the ROM development, the main difference between this case and the case of a Newtonian fluid is the order of the nonlinear terms in the viscous stress tensor: In the case of the polymeric fluid these terms are highly nonlinear while they are linear when a Newtonian fluid is considered. After the method is validated and its robustness studied with regard to several parameters, an application case is presented that could be representative of some industrial situations.

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Lieu, T., Farhat, C., and Lesoinne, M., 2006, “Reduced-Order Fluid/Structure Modeling of a Complete Aircraft Configuration,” Comput. Meth. Appl. Mech. Eng., 195, pp. 5730–5742. [CrossRef]
Cizmas, P. G. A., Richardson, B. R., Brenner, T. A., O'Brien, T. J., and Breault, R. W., 2008, “Acceleration Techniques for Reduced-Order Models Based on Proper Orthogonal Decomposition,” J. Comput. Phys., 227, pp. 7791–7812. [CrossRef]
De Lucas, S., Vega, J. M., and Velazquez, A., 2011, “Aeronautic Conceptual Design Optimization Method Based on High-Order Singular Value Decomposition,” AIAA J., 49, pp. 2713–2725. [CrossRef]
Burkardt, J., Gunzburger, M., and Lee, H. C., 2006, “POD and CVT Based Reduced Order Modelling of Navier–Stokes Flows,” Comput. Meth. Appl. Mech. Eng., 196, pp. 337–355. [CrossRef]
Sirisup, S., and Karniadakis, G. E., 2005, “Stability and Accuracy of Periodic Flow Solutions Obtained by POD-Penalty Method,” Physica D, 202, pp. 218–237. [CrossRef]
Sirisup, S., and Karniadakis, G. E., 2004, “A Spectral Viscosity Method for Correcting the Long Term Behaviour of POD Models,” J. Comput. Phys., 194, pp. 92–116. [CrossRef]
Galletti, B., Bruneau, C. H., Zannetti, C., and Lollo, A., 2004, “Low-Order Modelling of Laminar Flow Regimes Past a Confined Square Cylinder,” J. Fluid Mech., 503, pp. 161–170. [CrossRef]
Couplet, M., Basdevant, C., and Sagaut, P., 2003, “Calibrated Reduced-Order POD-Galerkin System for Fluid Flow Modeling,” J. Comput. Phys., 207, pp. 192–220. [CrossRef]
Alonso, D., Velazquez, A., and Vega, J. M., 2009, “Robust Reduced Order Modeling of Heat Transfer in a Back Step Flow,” Int. J. Heat Mass Transfer, 52, pp. 1149–1157. [CrossRef]
Alonso, D., Velazquez, A., and Vega, J. M., 2009, “A Method to Generate Computationally Efficient Reduced Order Models,” Comput. Meth. Appl. Mech. Eng., 198, pp. 2683–2691. [CrossRef]
Bache, E., Vega, J. M., and Velazquez, A., 2010, “Model Reduction in the Back Step Fluid-Thermal Problem With Variable Geometry,” Int. J. Therm. Sci., 49, pp. 2376–2384. [CrossRef]
Boukouvala, F., Gao, Y., Muzzio, F., and Ierapetritou, M. G., 2013, “Reduced Order Discrete Element Method Modeling,” Chem. Eng. Sci., 95, pp. 12–26. [CrossRef]
Chinesta, F., Ammar, A., Leygue, A., and Keunings, R., 2011, “An Overview of the Proper Generalized Decomposition With Applications in Computational Rheology,” J. Non-Newtonian Fluid Mech., 166, pp. 578–592. [CrossRef]
Ammar, A., Pruliere, E., Chinesta, F., and LasoM., 2009, “Reduced Numerical Modeling of Flows Involving Liquid-Crystalline Polymers,” J. Non-Newton. Fluid Mech., 160, pp. 140–156. [CrossRef]
Alonso, D., Vega, J. M., and Velazquez, A., 2010, “Reduced-Order Model for Viscous Aerodynamic Flow Past an Airfoil,” AIAA J., 48, pp. 1946–1958. [CrossRef]
Bache, E., Alonso, D., Velazquez, A., and Vega, J. M., 2012, “A Computationally Efficient Reduced Order Model to Generate Multi-Parameter Fluid-Thermal Databases,” Int. J. Therm. Sci., 52, pp. 145–153. [CrossRef]
Rodd, L. E., Cooper-White, J. J., Boger, D. V., and McKinley, G. H., 2007, “Role of the Elasticity Number in the Entry Flow of Dilute Polymer Solutions in Micro-Fabricated Contraction Geometries,” J. Non-Newton. Fluid Mech., 143, pp. 170–191. [CrossRef]
Poole, R. J., and Escudier, M. P., 2003, “Turbulent Flow of Non-Newtonian Liquids Over a Backward-Facing Step, Part II. Viscoelastic and Shear-Thinning Liquids,” J. Non-Newton. Fluid. Mech., 109, pp. 193–230. [CrossRef]
Nadeem, S., Akram, S., Hayat, T., and Hendi, A. A., 2012, “Peristaltic Flow of a Carreau Fluid in a Rectangular Duct,” ASME J. Fluids Eng., 134(4), p. 041201. [CrossRef]
Yapici, K., Karasozen, B., and Uludag, Y., 2012, “Numerical Analysis of Viscoelastic Fluids in Steady Pressure-Driven Channel Flow,” ASME J. Fluids Eng., 134(5), p. 051206. [CrossRef]
Dapra, I., and Scarpi, G., 2011, “Pulsatile Poiseuille Flow of a Viscoplastic Fluid in the Gap Between Coaxial Cylinders,” ASME J. Fluids Eng., 133(8), p. 081203. [CrossRef]
Tripathi, D., 2011, “Numerical Study on Creeping Flow of Burgers's Fluids Through a Peristaltic Tube,” ASME J. Fluids Eng., 133(12), p. 121104. [CrossRef]
Bird, R. B., Amstrong, R. G., and Assager, O., 1987, Dynamics of Polymeric Liquids, John Wiley and Sons, Hoboken, NJ.
Mendez, B., and Velazquez, A., 2004, “Finite Point Solver for the Simulation of 2-D Laminar Incompressible Unsteady Flow,” Comput. Meth. Appl. Mech. Eng., 193, pp. 825–848. [CrossRef]
Velazquez, A., Arias, J. R., and Mendez, B., 2008, “Laminar Heat Transfer Enhancement Downstream of a Backward Facing Step by Using a Pulsating Flow,” Int. J. Heat Mass Transfer, 51, pp. 2075–2089. [CrossRef]
Velazquez, A., Arias, J. R., and Montanes, J. L., 2009, “Pulsating Flow and Convective Heat Transfer in a Cavity With Inlet and Outlet Sections,” Int. J. Heat Mass Transfer, 52, pp. 647–654. [CrossRef]
Tannehill, J. C., Anderson, D. A., and Fletcher, R. H., 1997, Computational Fluid Mechanics and Heat Transfer, Taylor & Francis, Philadelphia, PA.
Smith, T. R., Moehlis, J., and Holmes, P., 2005, “Low-Dimensional Modelling of Turbulence Using the Proper Orthogonal Decomposition: A Tutorial,” Nonlinear Dynam.41, pp. 275–307. [CrossRef]
Fletcher, R., 2007, Practical Methods of Optimization, John Wiley & Sons, Hoboken, NJ.


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Fig. 1

(a) Sketch of the flow domain. (b) Illustration of the typical flow topology in the central cavity characterized by two recirculation regions. Lengths are dimensionless.

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Fig. 2

Dimensionless u velocity profiles at stations x = 3 and x = 4, for cases: (Re,n) = (20,0.5), (20,1.5), (120,0.5), and (120,1.5)

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Fig. 5

Results of cases (S1,g_3), top, and (S2,g_4), for TP5. Front: streamlines comparison between CFD (solid lines) and ROM (dashed lines). Back: map of pressure differences |Pcfd-Prom|/ΔP.

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Fig. 3

Selected snapshots. Set S1 depicted with shaded black squares. Set S2 depicted with transparent circles. The six test points TP1 to TP6 are plotted with crosses.

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Fig. 4

Visual impression of the areas where sets g_1 to g_6 are located inside the cavity

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Fig. 6

Visual impression of the first three POD modes and the cascade of eigenvalues in logarithmic scale



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