Estimation of volume flow in curved tubes based on analytical and computational analysis of axial velocity profiles
Authors: Verkaik A. C. , Beulen B. W. A. M. M. , Bogaerds A. C. B. , Rutten M. C. M. , van de Vosse F. N. 
Published: February 2009

To monitor biomechanical parameters related to cardiovascular disease, it is necessary to perform correct volume flow estimations of blood flow in arteries based on local blood velocity measurements. In clinical practice, estimates of flow are currently made using a straight-tube assumption, which may lead to inaccuracies since most arteries are curved. Therefore, this study will focus on the effect of curvature on the axial velocity profile for flow in a curved tube in order to find a new volume flow estimation method. The study is restricted to steady flow, enabling the use of analytical methods. First, analytical approximation methods for steady flow in curved tubes at low Dean numbers (Dn) and low curvature ratios (δ) are investigated. From the results a novel volume flow estimation method, the cos θ-method, is derived. Simulations for curved tube flow in the physiological range (1≤Dn≤1000 and 0.01≤δ≤0.16) are performed with a computational fluid dynamics (CFD) model. The asymmetric axial velocity profiles of the analytical approximation methods are compared with the velocity profiles of the CFD model. Next, the cos θ-method is validated and compared with the currently used Poiseuille method by using the CFD results as input. Comparison of the axial velocity profiles of the CFD model with the approximations derived by Topakoglu [J. Math. Mech. 16, 1321 (1967)] and Siggers and Waters [Phys. Fluids 17, 077102 (2005)] shows that the derived velocity profiles agree very well for Dn≤50 and are fair for 50&lt;Dn≤100, and this result applies for 0.01≤δ≤0.16, while Dean’s [Philos. Mag. 5, 673 (1928)] approximation only coincides for δ=0.01. For higher Dean numbers (Dn&gt;100), no analytical approximation method exists. In the position of the maximum axial velocity, a shift toward the inside of the curve is observed for low Dean numbers, while for high Dean numbers, the position of the maximum velocity is located at the outer curve. When the position of the maximum velocity of the axial velocity profile is given as a function of the Reynolds number, a “zero-shift point” is found at Re=21.3. At this point the shift in the maximum axial velocity to the outside of the curve, caused by the difference in axial pressure gradient, balances the shift to the inside of the curve, caused by the centrifugal forces (radial pressure gradient). Comparison of the volume flow estimation of the cos θ-method with the Poiseuille method shows that for Dn≤100 the Poiseuille method is sufficient, but for Dn≥100 the cos θ-method estimates the volume flow nearly three times better. For δ=0.01 the maximum deviation from the exact flow is 4% for the cos θ-method, while this is 12.7% for the Poiseuille method in the plane of symmetry. The axial velocity profile measured at a certain angle from the symmetry plane results in a maximum estimation error of 6.2% for Dn=1000 and δ=0.16. The results indicate that the estimation of the volume flow through a curved tube from a given asymmetrical axial velocity profile is more precise with the cos θ-method than the Poiseuille method, which is currently used in clinical practice.