Generalized derivation of the added-mass and circulatory forces for viscous flows

The concept of added mass arises from potential flow analysis and is associated with the acceleration of a body in an inviscid irrotational fluid. When shed vorticity is modeled as vortex singularities embedded in this irrotational flow, the associated force can be superimposed onto the added-mass force due to the linearity of the governing Laplace equation. This decomposition of force into added-mass and circulatory components remains common in modern aerodynamic models, but its applicability to viscous separated flows remains unclear. The present work addresses this knowledge gap by presenting a generalized derivation of the added-mass and circulatory force decomposition which is valid for a body of arbitrary shape in an unbounded, incompressible fluid domain, in both two and three dimensions, undergoing arbitrary motions amid continuous distributions of vorticity. From the general expression, the classical added-mass force is rederived for well-known canonical cases and is seen to be additive to the circulatory force for any flow. The formulation is shown to be equivalent to existing theoretical work under the specific conditions and assumptions of previous studies. It is also validated using a numerical simulation of a pitching plate in a steady freestream flow, conducted by Wang and Eldredge [Theor. Comput. Fluid Dyn. 27, 577 (2013)]. In response to persistent confusion in the literature, a discussion of the most appropriate physical interpretation of added mass is included, informed by inspection of the derived equations. The added-mass force is seen to account for the dynamic effect of near-body vorticity and is not (as is commonly claimed) associated with the acceleration of near-body fluid which “must” somehow move with the body. Various other consequences of the derivation are discussed, including a concept which has been labeled the conservation of image-vorticity impulse.

Figure 1

(a) Plate-fixed and plate-centered coordinate systems. (b) Coordinate system in the circle domain, related to the plate-fixed frame of reference via the conformal map given in Eq. (47). Point $A$ in the plate domain maps to point $A$ in the circle domain and the points labeled $B$ likewise map to one another. All angles are treated as counterclockwise positive.

Figure 2

Time history of the plate angle of attack $\alpha$ (in radians) and normalized pitch rate ${\dot{\alpha }}^{*}$ in the simulation of Wang and Eldredge [21].

Figure 3

Normalized vorticity plots at discrete times during the pitch maneuver from $\alpha =0$ to $\pi /4$ rad. The freestream velocity is from left to right.

Figure 4

Comparison of force coefficients calculated from the vorticity field using formulation 1 [the added-mass formulation (95)] and formulation 2 [the no-slip formulation (96)] with the force coefficients reported by Wang and Eldredge [21].

Figure 5

Contributions of individual impulse terms to the (a) plate-normal and (b) plate-aligned force coefficients. The contribution of the shed-vorticity impulse closely follows the force coefficients reported by Wang and Eldredge [21], as shown in Fig. 4. Note the change in the vertical scale between (a) and (b).

Figure 6

The plane $M$ intersects the arbitrary body occupying volume $V_b$, generating the cross section $D$ bounded by the closed curves $\left\{L_i\right\}$. The surfaces $\left\{S_i\right\}\subset S_b$, which exist on one side of $M$, act as capping surfaces on the closed curves $\left\{L_i\right\}$, permitting the application of Stokes's theorem.