[Paper Review] Validity of Viscous Core Correction Models for Self-Induced Velocity Calculations

Wim R. M. Van, Hoydonck|arXiv (Cornell University)|Apr 11, 2012

Fluid Dynamics and Vibration Analysis11 references3 citations

TL;DR

This paper identifies a critical flaw in viscous core correction models used in rotorcraft free-wake simulations: the unconditional use of perpendicular distance in Biot-Savart regularization leads to a 44% underprediction of self-induced velocity for viscous vortex rings. The authors propose a modified correction model that switches between perpendicular distance and radial distance to vortex endpoints based on projection position, restoring convergence to the theoretical Biot-Savart result and eliminating the error in both segmented and quadrature-based vortex ring representations.

ABSTRACT

Viscous core correction models are used in free wake simulations to remove the infinite velocities at the vortex centreline. It will be shown that the assumption that these corrections converge to the Biot-Savart law in the far field is not correct for points near the tangent line of a vortex segment. Furthermore, the self-induced velocity of a vortex ring with a viscous core is shown to converge to the wrong value. The source of these errors in the model is identified and an improved model is presented that rectifies the errors. It results in correct values for the self-induced velocity of a viscous vortex ring and induced velocities that converge to the values predicted by the Biot-Savart law for all points in the far field.

Motivation & Objective

• To identify and correct a fundamental error in viscous core correction models used in rotorcraft wake simulations.
• To analyze the convergence behavior of self-induced velocity in viscous vortex rings under standard core correction models.
• To develop a modified correction model that ensures accurate self-induced velocity and proper far-field convergence to Biot-Savart law.
• To validate the improved model using both segmented and quadrature-based representations of a vortex ring.

Proposed method

• Proposes a hybrid distance metric in the viscous core correction: use perpendicular distance only when the projection of the evaluation point lies within the vortex segment; otherwise, use radial distance to the nearest endpoint.
• Modifies Eq. (4) and Eq. (5) to implement the conditional distance rule, ensuring corrections are localized and physically consistent.
• Applies the modified model to both straight-line segment discretization and high-accuracy quadrature-based integration of a parametric vortex ring.
• Uses Saffman's analytical solution for viscous vortex ring self-induced velocity as a reference to validate numerical results.
• Employs Gauss quadrature with 128 abscissae for high-accuracy reference solutions and 360,000 segments for the segmented approach to minimize discretization error.
• Compares results from the original and modified core models against the theoretical value to quantify error reduction.

Experimental results

Research questions

• RQ1 Does the standard viscous core correction model correctly predict the self-induced velocity of a viscous vortex ring?
• RQ2 Why do existing viscous core models fail to converge to the correct Biot-Savart result in the far field for points near the tangent line of a vortex segment?
• RQ3 What causes the 44% underprediction of self-induced velocity in both segmented and quadrature-based vortex ring simulations?
• RQ4 Can a modified distance metric in the core correction restore convergence to the theoretical self-induced velocity and far-field Biot-Savart behavior?

Key findings

• The standard viscous core correction model, which unconditionally uses perpendicular distance, fails to converge to the correct Biot-Savart result for points near the tangent line of a vortex segment.
• The self-induced velocity of a viscous vortex ring is underpredicted by approximately 44% when using the conventional core correction model, regardless of whether the vortex ring is discretized into segments or represented parametrically.
• The proposed modification—switching between perpendicular and radial distance based on the projection of the evaluation point—eliminates the 44% error and restores agreement with the analytical solution.
• The improved model ensures that induced velocities converge to the correct values predicted by the Biot-Savart law for all points in the far field, including near the tangent line of a vortex segment.
• The modified model is effective for both the segmented vortex representation and the quadrature-based method, confirming its robustness across numerical approaches.
• The source of the error is traced to the unphysical extension of the correction region to infinity along the vortex centerline, which is corrected by restricting the influence to the physical proximity of the vortex segment.