Artificial Organs 27(10):935–937, Blackwell Publishing, Inc. © 2003 International Society for Artificial Organs
Studies of Turbulence Models in a Computational Fluid Dynamics Model of a Blood Pump *Xinwei Song, *Houston G. Wood, *Steven W. Day, and †Don B. Olsen *Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA; and †Utah Artificial Heart Institute, Salt Lake City, UT, U.S.A.
Abstract: Computational fluid dynamics (CFD) is used widely in design of rotary blood pumps. The choice of turbulence model is not obvious and plays an important role on the accuracy of CFD predictions. TASCflow (ANSYS Inc., Canonsburg, PA, U.S.A.) has been used to perform CFD simulations of blood flow in a centrifugal left ventricular assist device; a k-e model with near-wall functions was used in the initial numerical calculation. To improve the simulation, local grids with special distribution to
ensure the k-w model were used. Iterations have been performed to optimize the grid distribution and turbulence modeling and to predict flow performance more accurately comparing to experimental data. A comparison of k-w model and experimental measurements of the flow field obtained by particle image velocimetry shows better agreement than k-e model does, especially in the near-wall regions. Key Words: Computational fluid dynamics— Turbulence model—Blood pump.
Computational fluid dynamics (CFD) is an important tool that is being used by many investigators to design and analyze the rotary blood pumps. Besides studying overall pump performance, CFD has been used to detect vortex and stagnation points, to predict blood damage induced by high shear stress, and to obtain the hydraulic properties, for example, forces, moments, torque, and efficiency. Due to highly disturbed flow caused by moving blades, flow pattern in most regions inside the blood pump is turbulent. The choice of turbulence models is an important factor for the CFD simulation. However, which turbulence model is appropriate for miniature blood pump is not obvious. The assumptions of turbulence models have to be reevaluated for the specific application of small-size blood pumps. A second concern is the complicated geometry of blood pump and relatively low absolute speed, which makes wall effects more strongly influence the flow field. The different treatments of near-wall regions, their constraint for grid distribution, and their compatibility with turbulence
models become critical for accuracy and feasibility of CFD prediction. TURBULENCE MODELS CFX software (ANSYS Inc., Canonsburg, PA, U.S.A.) has been used in this CFD study. In CFX, the available turbulence models include k-e model and k-w model. Here k, e, w denote the turbulent kinetic energy, turbulent dissipation rate, turbulent frequency, respectively. Both models utilize the eddy viscosity assumption to relate the Reynolds stress and turbulent terms to the mean flow variables. The near-wall treatment includes Log-law wall functions, k-w combined low/high Re wall functions, and a two-layer turbulence model. Log-law wall function employs a logarithmic function to bridge the viscous near-wall layer to eliminate the necessity of numerically resolving the large gradients in the thin nearwall region. The recommended Log-law wall function employed by k-e model is scalable Log-law wall function, which artificially moves the virtual walls to the edge of the viscous sublayer in order to successfully avoid fine grid inconsistence. The two-layer turbulence model is another near-wall treatment developed for k-e model. It implements one equation model to solve the near-wall region and uses k-e model for far-wall region. It needs more
Received June 2003. Address correspondence and reprint requests to Dr. Xinwei Song, 122 Engineer’s Way, Charlottesville, VA 22903, U.S.A. E-mail:
[email protected] Presented in part at the 10th Congress of the International Society for Rotary Blood Pumps, held September 11–14, 2002 in Osaka, Japan.
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FIG. 1. Comparison of CFD and PIV results at mid-height span. Left, k-e model; middle, PIV; right, k-w model.
FIG. 2. Comparison of CFD and PIV results at near hub layer. Left, k-e model; middle, PIV; right, k-w model.
FIG. 3. Comparison of CFD and PIV results at near shroud layer. Left, k-e model; middle, PIV; right, k-w model. Artif Organs, Vol. 27, No. 10, 2003
TURBULENCE MODELS IN A CFD MODEL OF A BLOOD PUMP computational resources and is less computationally stable, therefore is discouraged to use. k-w model is able to provide the analytical expression for w in the viscous sublayer. This advantage is exploited to achieve an automatic switch from the near-wall functions for coarse grids to near-wall formulation for fine grids. This switch always results from the decrease of Reynolds number and corresponding attenuation of viscous sublayer. This feature makes the grid design in k-w model more flexible and robust. In order to compare k-e model and k-w model, two grids have been created. A fine grid system with the first near-wall grid point located y+ < 2, which satisfies the criterion to ensure k-w model activated in both near-wall and far-wall regions, has been generated for k-w model, while a grid system with the first near-wall grid point located y+ = 11, where is assumed to coincide the edge of viscous sublayer, has been developed for k-e model. Grid generation has to be performed before any information on wall shear stress or boundary layer thickness is available. As a preliminary case, a standard k-e model with near-wall function was run to obtain the best guess of the first near-wall grid location, characteristic velocity, and thickness of boundary layer for different regions (1). The centrifugal blood pump prototype is the HeartQuest ventricular assist device (VAD). The boundary layer thickness d is estimated by Eq. 1: d = 0.14L Re 6 7
1 Re L
(1)
ReL is the Reynolds number in term of characteristic length. Equation 2 is employed to decide the first point location: +
y y =m
t wall r ,
(2)
where y is the distance of the first point to wall, twall is the wall shear stress, µ is the viscosity, and r is the density of blood; twall was obtained from the preliminary results. RESULTS Figures 1–3 show the absolute speeds at axially mid-height span, near hub layer, and near shroud
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layer in the HeartQuest VAD. The results according to k-e model with scalable near-wall function, the results obtained by particle image velocimetry (PIV) (2), and k-w model have been plotted from left to right in one row. The CFD boundary conditions are same as the testing condition: the inlet is specified as a constant flow rate with 6 L/min, while the outlet is with constant reference pressure. The comparison exhibits that the better agreement has been obtained for k-w model, especially around the near-wall regions. Also, one can see the prediction of boundary layer thickness around the blades in k-w model is closer to the experiment than k-e model (Fig. 2). CONCLUSIONS CFD is an important approach to study the flow pattern inside the miniature blood pumps, and turbulence model is a key issue for CFD simulation. The assumptions and grid constrains have been indicated for different turbulence models and near-wall treatment. The k-e model with scalable near-wall function and k-w model have been employed in CFD simulations for a centrifugal blood pump. The comparisons with PIV experimental data showed better agreement for k-w model. In the future, different combinations of turbulence model and near-wall treatment will be studied further. Also more experimental comparison will be done upon more available test results. Acknowledgments: The authors wish to acknowledge the financial support for this work provided by MedQuest Products, Inc., Utah Artificial Heart Institute, Department of Health and Human Services, National Institutes of Health, and the National Heart, Lung, and Blood Institute (grant number R01 HL64378-01). REFERENCES 1. Song X, Wood HG, Olsen DB. CFD study of the 4th generation prototype of a continuous flow ventricular assist device. J Biomech Eng (in press). 2. Day SW, McDaniel JC, Wood HG, Allaire PE, Song X, Lemire PP, Miles SD. A prototype HeartQuest ventricular assist device for particular image velocimetry measurements. Artif Organs 2002;26:1002–5.
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