An improved algorithm for the computation of added mass
[Objective]In incompressible flows,the added mass is regarded as an important concept used to represent the hydrodynamics.Unfortunately,the present theory for computing this quantity cannot be readily understood and does not seem compatible with the modern vorticity dynamics.The purpose of this study lies in proposing a more comprehensible theory for computing the added mass so that these two shortcomings can be remedied.Finally,the effect of the dimension,the reference frame,and the reference point of the hydrodynamic moment on added masses are also revealed analytically to further simplify the computation.[Methods]Based on the Newton mechanics,the pressure removal method is used to develop new hydrodynamic formulae.By distinguishing the slip-boundary velocity and the solid-surface velocity,the hydrodynamic formula in non-circulatory flow is recovered.New velocity-vector-based-added-mass formulae are developed from this formula and the panel method is used to compute the velocity.Then,coordinate transformation technology is used to translate added mass between different frames.Finally,the kinematic relation of the velocity on the solid is used to clarify the effect of the dimension and the translational reference point on the added mass.[Results]New hydrodynamic formulae applicable in both non-circulatory flows and viscous flows are proposed.Compared with the classical non-circulatory flow theory,some errors are avoided under the adoption of the pressure removal method.Specifically,the pressure integration on a large far field surface is concluded to be zero in the classical theory and this conclusion conflicts with modern vorticity dynamics.Compared with the modern vorticity dynamics,the new hydrodynamic moment formula considers the effect of the reference-point velocity.This consideration not only generalizes the new formula but also explains why the present hydrodynamic moment formula in vorticity dynamics cannot be reduced to that in the classical non-circulatory flow theory.A new velocity-vector-based-added-mass formula is proposed based on these hydrodynamic formulae.The effect of the reference frame,the translational reference point,and the dimension of the solid on the added mass is clarified and transformation formulae are proposed analytically.Numerical examples are chosen and solved to verify the correctness of the new added-mass formula and the transformation formulae.For ellipses and axisymmetric ellipsoids with a major axis of lengtha and a minor axis of length b,the added mass is computed while fixing b=a/4 and varying a=1,2,3,4,5.Results show that,when the object size is magnified by a factor of x in 2D,elements of the four added mass partitioned matrixes MFv,MFω,MMv,and MMω will increase by a factor of x2,x3,x3,and x4,respectively.In 3D,elements of MFv,MFω,MMv,and MMω will increase by a factor of x3,x4,x4,and x5,respectively,confirming the correctness of our size-related transformation formulae.As the translational reference point is gradually shifted to the left,these theoretical solutions derived from the transformation rules agree with the numerical solutions,thus verifying the correctness of our reference point-related transformation formulae.[Conclusions]New hydrodynamic formulae are developed based on the pressure removal method.A new velocity-vector-based-added-mass computation method is developed according to these hydrodynamic formulae.The effect of the reference frame,the translational reference point,and the dimension of the solid on the added mass is clarified analytically.Numerical examples demonstrate the correctness of these formulae.Compared with the classical non-circulatory theory,the proposed theory exhibits its rigorousness and comprehensiveness.Compared with the modern vorticity dynamics,the proposed hydrodynamic moment formula appears more general because it considers the effect of the translational reference point velocity.Hopefully,the proposed theory that holds the compatibility with the vorticity dynamics can help establish the relationship of hydrodynamics between non-circulatory and viscous flows.Currently,this relationship contributes significantly to the application of the added mass,yet seems to remain vague in the academia.
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