Natural Velocity Decomposition: A Review


A review of the natural velocity decomposition (NVD) formulation for the analysis of Navier–Stokes flows developed by the authors is presented. The decomposition consists in expressing the velocity field as the sum of two terms, one of which is given as the gradient of a potential \(\varphi \), whereas the other, \(\mathbf{w}\), is rotational and is governed by its own evolution equation. Hence, contrary to all the related approaches such as the Helmholtz decomposition, the NVD does not require the evaluation of the vorticity. The theoretical formulation is presented in detail for incompressible viscous flows and briefly outlined for compressible viscous flows. Of particular significance is the relationship between the present approach for thin vortical layers (almost-potential flows) and that for zero-thickness layers (quasi-potential flows), when this is combined with the transpiration-velocity correction by Lighthill. The methodology may also be a valuable computational tool. To assess this, two problems are addressed. The first consists in the planar flow around a disk, for low Reynolds numbers (separated laminar flows). The computational scheme used is discussed thoroughly. The corresponding numerical results compare favorably with data available in the open literature. These include the separation angle and the length of the recirculation bubble. Also, the flow-field results are in very good agreement with experimental flow visualizations available in the open literature. The second problem that we addressed is the flow due to a jet. This is tackled by the almost-potential flow approximation of the NVD formulation. Again, the results are in good agreement with experimental data available in the open literature. Theoretical innovations (and limitations) of the methodology are discussed, along with its advantages and drawbacks.

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  1. 1.

    The Einstein summation convention on repeated indices is used throughout the paper.

  2. 2.

    Latin letters are used to denote Cartesian components, and always range from 1 to 3. On the other hand, Greek letters are used to denote covariant (subscripted) or contravariant (superscripted) components or base vectors. Specifically, the Greek letters \(\alpha \), \(\beta \) and \(\gamma \) (used in Appendix A) range from 1 to 3, whereas the superscript \(\sigma \) is used when the range is from 1 to 2.


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We wish to express our appreciation to the Hariri Center of Boston University, and its Director, Prof. Azer Bestavros, for hosting P.G. and L.M., as well as the Engineering Faculty of Embry-Riddle Aeronautical University and its Dean, Prof. Anastasios Lyrinzis, for hosting M.C. and L.S., in connection with the activity reported here.

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Correspondence to Giovanni Bernardini.

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A: Material Covariant Components of \(\mathbf{w}\)

We have seen that the definition for \(\varpi =\frac{1}{2}\, \Vert \mathbf{v}_{_{^{P}}}\Vert ^2 - \frac{1}{2}\, \Vert \mathbf{w}\Vert ^2\) (Eq. 17) yields an elegant form for both the equation governing \(\mathbf{w}\) (Eq. 20) and for the Bernoulli theorem (Eq. 32). Here, we present one more interesting consequence of such a choice, one which pertains to the material covariant components of \(\mathbf{w}\) (defined below). This result, obtained in Ref. [17], provides another convincing argument in support of our belief that the choice for \(\varpi \) in Eq. 17 is the correct one [Furthermore, as a by product, this result provides the theoretical basis for kinematics of the wake (Eq. 55)].

Let us introduce a set of material curvilinear coordinates, \(\xi ^\alpha \), in a one-to-one correspondence with the material points. Let \(\mathbf{x}(\xi ^\alpha ,t)\) be the corresponding mapping and \(\xi ^\alpha =\xi ^\alpha (\mathbf{x},t)\) its inverse. Next, introduce the material covariant and contravariant base vectors, defined by \(\mathbf{g}_\alpha := \partial {\mathbf{x}} / \partial {\xi ^\alpha }\) and \(\mathbf{g}^\alpha :=\nabla \xi ^\alpha \) (We use the term “material” because the vectors evolve in time with the material points). Specifically, \(\mathbf{g}_\alpha \) is always tangent to the material \(\xi ^\alpha \)-lines, whereas \(\mathbf{g}^\alpha \) is always normal to the material surfaces \(\xi ^\alpha =\) constant.

Next, set

$$\begin{aligned} \mathbf{w}= w_\alpha (\xi ^\gamma ,t) \, \mathbf{g}^\alpha (\xi ^\gamma ,t), \end{aligned}$$

where \(w_\alpha \) are the material covariant components of \(\mathbf{w}\). Note that

$$\begin{aligned} \frac{\mathrm{D}{\mathbf{g}_\alpha }}{\mathrm{D}t}=\frac{\partial {\mathbf{v}}}{\partial {\xi ^\alpha }}. \end{aligned}$$

[Use \({\mathrm{D}{f}}/{\mathrm{D}t}=[\partial {} / \partial {t}+\mathbf{v}\cdot \nabla ]f(\mathbf{x},t)=\partial {f(\xi ^\alpha ,t)} / \partial {t}\) and \(\mathbf{v}={\mathrm{D}{\mathbf{x}}}/{\mathrm{D}t}\) (Serrin [56], p. 129–130); see Ref. [30] for details.] Moreover, from their definitions, we have \(\mathbf{g}^\alpha \cdot \mathbf{g}_\beta =\delta _\beta ^\alpha \). Taking the substantial derivative of this equation, and using Eq. 62, one obtains, as shown in Ref. [17],

$$\begin{aligned} \frac{\mathrm{D}{\mathbf{g}^\alpha }}{\mathrm{D}t} = - \mathbf{Q}^{\textsf {T}}\, \mathbf{g}^\alpha , \end{aligned}$$

where \(\mathbf{Q}=\nabla \mathbf{v}\) (as in Eq. 21). Then, taking the substantial time derivative of Eq. 61 and combining the result with Eq. 63, we have:

$$\begin{aligned} \frac{\mathrm{D}{\mathbf{w}}}{\mathrm{D}t} = \frac{\mathrm{D}{w_\alpha }}{\mathrm{D}t} \, \mathbf{g}^\alpha + w_\alpha \, \frac{\mathrm{D}{\mathbf{g}^\alpha }}{\mathrm{D}t} = \frac{\mathrm{D}{w_\alpha }}{\mathrm{D}t} \, \mathbf{g}^\alpha - w_\alpha \, \mathbf{Q}^{\textsf {T}}\mathbf{g}^\alpha = \frac{\mathrm{D}{w_\alpha }}{\mathrm{D}t} \, \mathbf{g}^\alpha - \mathbf{Q}^{\textsf {T}}\mathbf{w}. ~~ \end{aligned}$$

Substituting into Eq. 20, dotting with \(\mathbf{g}_\beta \) and using \(\mathbf{g}^\alpha \cdot \mathbf{g}_\beta =\delta _\beta ^\alpha \), one obtains

$$\begin{aligned} \frac{\mathrm{D}{w_\beta }}{\mathrm{D}t} = \nu \, \mathbf{g}_\beta \cdot \nabla ^2\mathbf{w}. \end{aligned}$$

For inviscid flows, we have \({\mathrm{D}{w_\beta }}/{\mathrm{D}t} = 0\), that is, \(w_\beta \) remains constant following a material point, namely \(w_\beta (\xi ^\gamma ,t)=w_\beta (\xi ^\gamma ,0)\). On the other hand, for viscous flows, using \({\mathrm{D}{f}}/{\mathrm{D}t}=\partial {f} / \partial {t}\big |_{\xi ^\gamma }\) (Serrin [56], p. 129), Eq. 65 may be integrated to yield

$$\begin{aligned} w_\beta (\xi ^\gamma ,t) = w_\beta (\xi ^\gamma ,0) + \nu \, \int _0^t \mathbf{g}_\beta (\xi ^\gamma ,t) \cdot \nabla ^2\mathbf{w}(\xi ^\gamma ,t) \, \mathrm{d}t. \end{aligned}$$

B: Natural Velocity Decomposition for Compressible Flows

All the material covered in the main body is limited to incompressible fields. This was done for the sake of clarity and also because all the results obtained are limited to such flows. However, we do not want to give the impression that the NVD is subject to such a limitation. Accordingly, here we present the compressible-flow extension. Again, we limit ourselves to fluid dynamics. For aeroacoustics, see Refs. [36] and [46].

Comment The formulation presented here is subject to three limitations, namely: (i) we deal with an ideal gas, (ii) we have zero bulk viscosity, namely \(\lambda =-2\,\mu /3\), and (iii) the Prandtl number \(\Pr :=\mu \,c_p/\kappa \) equals 3/4 (diatomic gas).

Unsteady compressible viscous fields are governed by the continuity, Navier–Stokes and entropy equations (which, for shock-free flows, is equivalent to the energy equation, Serrin [56]). These are given by

$$\begin{aligned}&\frac{\mathrm{D}{{\varrho }}}{\mathrm{D}t} + {\varrho }\, \nabla \cdot \mathbf{v}= 0, \end{aligned}$$
$$\begin{aligned}&\frac{\mathrm{D}{\mathbf{v}}}{\mathrm{D}t} = - \frac{1}{{\varrho }} \, \nabla p + \nu \nabla ^2\mathbf{v}+ \frac{1}{3} \, \nu \, \nabla \nabla \cdot \mathbf{v}, \end{aligned}$$
$$\begin{aligned}&\frac{\mathrm{D}{{S}}}{\mathrm{D}t} = \frac{1}{{\varrho }\, {\vartheta }} \, \Big [ \kappa \, \nabla ^2{\vartheta }+ \mathbf{D}:\mathbf{V}\Big ], \end{aligned}$$

where \({\vartheta }\) denotes the absolute temperature, \(\kappa \) the thermal conductivity, \(\mathbf{D}\) the deformation rate tensor, and \(\mathbf{V}\) the viscous stress tensor. For simplicity, we have assumed \(\lambda =-2\,\mu /3\) (zero bulk viscosity). The boundary conditions on \({\mathcal {S}}_{_{^{\mathrm{B}}}}\) are now \(\mathbf{v}=\mathbf{v}_{_{^{\mathrm{B}}}}\) and \({\vartheta }={\vartheta }_{_{^{\mathrm{B}}}}\). The boundary conditions at infinity are \(\mathbf{v}=\mathbf{0}\), \({\varrho }={\varrho }_\infty \) and \({S}={S}_\infty \). Finally, assuming a start from a situation of dynamic and thermodynamic equilibrium, the initial conditions are \(\mathbf{v}(\mathbf{x},0)=\mathbf{0}\), \({\varrho }(\mathbf{x},0)={\varrho }_\infty \) and \({S}(\mathbf{x},0)={S}_\infty \) for \(\mathbf{x}\in {\mathcal {V}}\).

In this appendix, we assume the air to be an ideal gas, with constant coefficients \(c_v\) and \(c_p\). Then, we have \(p = {\varrho }\,R\,{\vartheta }\) (with \(R=c_p-c_v\)) and \(h=c_p{\vartheta }\), where h is the enthalpy. In addition, setting \(\gamma =c_p/c_v\), we have

$$\begin{aligned} \frac{{\varrho }}{{\varrho }_\infty } = \bigg (\frac{p}{p_\infty }\bigg )^{1/\gamma } \, \mathrm{e}^{-({S}-{S}_\infty )/c_p} = \bigg (\frac{h}{h_\infty }\bigg )^{1/(\gamma -1)} \, \mathrm{e}^{-({S}-{S}_\infty )/R}. \end{aligned}$$

Natural Velocity Decomposition for Compressible Flows

Recalling that \(\mathrm{d}{h}=\mathrm{d}{p}/{\varrho }+{\vartheta }\,\mathrm{d}{S}\), the Navier–Stokes equation (Eq. 68) yields

$$\begin{aligned} \frac{\mathrm{D}{\mathbf{v}}}{\mathrm{D}t} + \nabla h - {\vartheta }\, \nabla {S}= \nu \, \nabla ^2\mathbf{v}+ \frac{1}{3} \, \nu \, \nabla \, \nabla \cdot \mathbf{v}. \end{aligned}$$

Using the NVD (Eq. 1) and \(\varpi =\frac{1}{2}\,\Vert \mathbf{v}_{_{^{P}}}\Vert ^2-\frac{1}{2}\,\Vert \mathbf{w}\Vert ^2\) (Eq. 17), one obtains, in analogy with the incompressible-flow case,

$$\begin{aligned}&\nabla \bigg [ \frac{\mathrm{D}{\varphi }}{\mathrm{D}t} - \frac{1}{2}\, \Vert \mathbf{v}\Vert ^2 + \hat{h} - {\vartheta }_\infty \,\hat{{S}} - \frac{4}{3}\,\nu \,\nabla ^2\varphi \bigg ]\nonumber \\&\quad +\,\bigg [ \frac{\mathrm{D}{\mathbf{w}}}{\mathrm{D}t} + \mathbf{Q}^{\textsf {T}}\mathbf{w}- \hat{{\vartheta }}\,\nabla \hat{{S}} - \nu \,\nabla ^2\mathbf{w}- \frac{1}{3}\,\nu \,\nabla \,\nabla \cdot \mathbf{w}\bigg ] = \mathbf{0}. \end{aligned}$$

where \(\hat{h}=h-h_\infty \), \(\hat{{\vartheta }}={\vartheta }-{\vartheta }_\infty \) and \(\hat{{S}}={S}-{S}_\infty \) (it may be worth noting that setting \({\vartheta }\,\nabla {S}={\vartheta }_\infty \nabla {S}+\hat{{\vartheta }}\nabla {S}\) occurs naturally in the boundary integral formulation of Ref. [36], because of the need to separate linear from nonlinear terms).

Again, we equate to zero individually each of the two terms on the left side of Eq. 72. The first term equated to zero and integrated yields, in the air frame,

$$\begin{aligned}&\frac{\mathrm{D}{\varphi }}{\mathrm{D}t} - \frac{1}{2}\, \Vert \mathbf{v}\Vert ^2 + \hat{h} - {\vartheta }_\infty \, \hat{{S}} - \frac{4}{3} \, \nu \, \nabla ^2\varphi \nonumber \\&\quad = \frac{\partial {\varphi }}{\partial {t}} + \varpi + \hat{h} - {\vartheta }_\infty \, \hat{{S}} - \frac{4}{3} \, \nu \, \nabla ^2\varphi = 0. \end{aligned}$$

This is the generalized Bernoulli theorem for compressible Navier–Stokes fields.

On the other hand, equating to zero the second term in Eq. 72, we have

$$\begin{aligned} \frac{\mathrm{D}{\mathbf{w}}}{\mathrm{D}t} + \mathbf{Q}^{\textsf {T}}\mathbf{w}- \hat{{\vartheta }} \, \nabla {S}- \nu \, \nabla ^2\mathbf{w}- \frac{1}{3} \, \nu \, \nabla \, \nabla \cdot \mathbf{w}= \mathbf{0}. \end{aligned}$$

Akin to the incompressible flows, this equation is coupled with the equation for \(\varphi \), which appears in the velocity \(\mathbf{v}=\nabla \varphi +\mathbf{w}\). In addition, there is a weak coupling with the entropy equation, due to the nonlinear term \(\hat{{\vartheta }}\,\nabla {S}\).

Further elaborations of Eqs. 73 and 74 are presented in Eqs. 79 and 81, after we discuss the entropy equation.

Entropy Evolution Equation

Using \(h=c_p{\vartheta }\), the entropy evolution equation (Eq. 69) yields

$$\begin{aligned} {\vartheta }\, \frac{\mathrm{D}{{S}}}{\mathrm{D}t} = \frac{\kappa }{{\varrho }\, c_p} \, \nabla ^2{h}+ \frac{\mathbf{D}:\mathbf{V}}{{\varrho }} = \frac{\nu }{\Pr } \, \nabla ^2{h}+ \frac{\mathbf{D}:\mathbf{V}}{{\varrho }}, \end{aligned}$$

where \(Pr:=\mu \,c_p/\kappa \) denotes the Prandtl number. We can eliminate \({h}\) using the generalized Bernoulli theorem (Eq. 73). This yields

$$\begin{aligned} {\vartheta }\, \frac{\mathrm{D}{{S}}}{\mathrm{D}t} = - \frac{\nu }{\Pr } \, \nabla ^2\bigg [ \bigg (\frac{\partial {}}{\partial {t}} - \frac{4}{3} \, \nu \, \nabla ^2\bigg ) \varphi + \varpi - {\vartheta }_\infty \, {S}\bigg ] + \frac{\mathbf{D}:\mathbf{V}}{{\varrho }}. \end{aligned}$$

Next, following Pierce [51], let us introduce the assumption that the Prandtl number equals 3/4. [This is the value of Pr for diatomic gases (Serrin [56], p. 239, who also gives the experimental value for the air at the ambient temperature as \(Pr=0.72\)).] Accordingly, the above equation yields

$$\begin{aligned} \bigg (\frac{\partial {}}{\partial {t}} - \frac{4}{3} \, \nu \, \nabla ^2\bigg ) \, {S_*}= \bigg ( 1- \frac{{\vartheta }}{{\vartheta }_\infty } \bigg ) \, \frac{\mathrm{D}{{S}}}{\mathrm{D}t} - \mathbf{v}\cdot \nabla {S}- \frac{4\,\nu }{3\,{\vartheta }_\infty } \nabla ^2\varpi + \frac{\mathbf{D}:\mathbf{V}}{{\varrho }\,{\vartheta }_\infty }, ~~~~ \end{aligned}$$


$$\begin{aligned} {S_*}:= \hat{{S}} + \frac{4\,\nu }{3\,{\vartheta }_\infty }\, \nabla ^2\varphi . \end{aligned}$$

Bernoulli Theorem

We can now exploit Eq. 78 to further elaborate the Bernoulli theorem. Indeed, combining Eqs. 73 and 78, one obtains

$$\begin{aligned} \frac{\mathrm{D}{\varphi }}{\mathrm{D}t} - \frac{1}{2}\, \Vert \mathbf{v}\Vert ^2 + \hat{h} - {\vartheta }_\infty \, {S_*}= 0. \end{aligned}$$

Separating linear and nonlinear terms, Bernoulli’s theorem reads

$$\begin{aligned} \frac{\partial {\varphi }}{\partial {t}} + \hat{h} = \sigma _{_{^{\mathrm{B}}}}, \end{aligned}$$

where \(\sigma _{_{^{\mathrm{B}}}}=-\varpi +{\vartheta }_\infty \,{S_*}\).

Equation for \(\mathbf{w}\)

Moreover, we can again exploit Eq. 78 to further elaborate the differential equation for \(\mathbf{w}\) (Eq. 74) to obtain

$$\begin{aligned} \frac{\mathrm{D}{\mathbf{w}}}{\mathrm{D}t} + \mathbf{Q}^{\textsf {T}}\mathbf{w}+ \frac{4\,\nu }{3\,{\vartheta }_\infty } \, \hat{{\vartheta }} \, \nabla ^2\varphi - \nu \, \nabla ^2\mathbf{w}- \frac{1}{3} \, \nu \, \nabla \, \nabla \cdot \mathbf{w}= \hat{{\vartheta }} \, \nabla {S_*}. \end{aligned}$$

Continuity Equation

Here, we choose as our primary thermodynamics variables \({h}\) and \({S}\). Then, we have

$$\begin{aligned} \frac{\mathrm{D}{{\varrho }}}{\mathrm{D}t} = \frac{\partial {{\varrho }}}{\partial {{h}}}\bigg |_{S}\, \frac{\mathrm{D}{{h}}}{\mathrm{D}t} + \frac{\partial {{\varrho }}}{\partial {{S}}}\bigg |_{h}\, \frac{\mathrm{D}{{S}}}{\mathrm{D}t}. \end{aligned}$$

Next, set (use Eq. 70)

$$\begin{aligned} a^2 := {\varrho }\, \frac{\partial {{h}}}{\partial {{\varrho }}}\bigg |_{S}=\frac{\partial {p}}{\partial {{\varrho }}}\bigg |_{S}= \gamma \, \frac{p}{{\varrho }} = \gamma \, R \, {\vartheta }. \end{aligned}$$

In addition, using again Eq. 70, we have \(\partial {{\varrho }} / \partial {{S}}\big |_h=-{\varrho }/R\). Thus, the continuity equation (Eq. 67) may be written as

$$\begin{aligned} \frac{1}{a^2} \, \frac{\mathrm{D}{{h}}}{\mathrm{D}t} - \frac{1}{R} \, \frac{\mathrm{D}{\hat{{S}}}}{\mathrm{D}t} + \nabla \cdot \mathbf{v}= 0. \end{aligned}$$

Then, using the NVD (Eq. 1), and eliminating \(\hat{{S}}\) using Eq. 78, one obtains

$$\begin{aligned} \frac{1}{a^2} \, \frac{\mathrm{D}{{h}}}{\mathrm{D}t} + \frac{4\,\nu }{3\,R\,{\vartheta }_\infty }\, \frac{\mathrm{D}}{\mathrm{D}t}\nabla ^2\varphi - \frac{1}{R} \, \frac{\mathrm{D}{{S_*'}}}{\mathrm{D}t}+ \nabla ^2\varphi + \nabla \cdot \mathbf{w}= 0. \end{aligned}$$

Finally, separating linear and nonlinear terms, and using \(c^2:=a_\infty ^2=\gamma \,R\,{\vartheta }_\infty \), the continuity equation reads

$$\begin{aligned} \frac{1}{c^2} \, \frac{\partial {{h}}}{\partial {t}} + \frac{4\,\gamma \,\nu }{3\,c^2} \, \nabla ^2\dot{\varphi }+ \nabla ^2\varphi + \nabla \cdot \mathbf{w}= \sigma _{_{^{\mathrm{C}}}}, \end{aligned}$$

where \(\sigma _{_{^{\mathrm{C}}}}\) comprises all the nonlinear terms in Eq. 85. Here and in the remainder of this appendix, we use \(\dot{u}:=\partial {u} / \partial {t}\), for any function \(u(\mathbf{x},t)\).

Equation for \(\varphi \)

Combining Bernoulli’s theorem (Eq. 80) with the continuity equation (Eq. 86), one obtains

$$\begin{aligned} \nabla ^2\varphi - \frac{1}{c^2} \, \big ( \ddot{\varphi } - 2 \, \alpha \, \nabla ^2\dot{\varphi }\big ) = \breve{\sigma }_\varphi , \end{aligned}$$

where \(\alpha = 2\gamma \nu /3\), and \(\breve{\sigma }_\varphi =-\nabla \cdot \mathbf{w}+\sigma _{_{^{\mathrm{C}}}}-\dot{\sigma }_{_{^{\mathrm{B}}}}/c^2\).

The solution of Eq. 87, for \(\mathbf{x}\in {\mathbb {R}}^3\) is given by (see Ref. [36] for details)

$$\begin{aligned} \varphi (\mathbf{x},t) = \int _0^t\int _{{\mathbb {R}}^3}Q(\mathbf{x}-\mathbf{y},t-\tau ) \, \breve{\sigma }_\varphi (\mathbf{y}, \tau ) \,\mathrm{d}{\mathcal {V}}\, \mathrm{d}\tau , \end{aligned}$$

where \(Q\) is the (causal) fundamental solution for the operator in Eq. 87, defined by

$$\begin{aligned} \nabla ^2Q- \frac{1}{c^2} \, \big ( \ddot{Q} - 2 \, \alpha \, \nabla ^2\dot{Q}\big ) = \delta (\mathbf{x}-\mathbf{y}) \, \delta (t-\tau ). \end{aligned}$$

An exact expression for \(Q\) is given in Ref. [52]. However, such an expression is too cumbersome to be used in practice. Alternatively, following Ref. [52] again, one may use the approximate expression \(Q(\mathbf{x},t)\simeq -\delta _\alpha (t,r)/4\pi r\), where \(r=\Vert \mathbf{x}\Vert \), whereas

$$\begin{aligned} \delta _\alpha (t,r) :=c\,\frac{H(t)}{\sqrt{4\pi \alpha t}} \, \big ( \mathrm{e}^{-(ct-r)^2/4\alpha t} - \mathrm{e}^{-(ct+r)^2/4\alpha t} \big ). \end{aligned}$$

Comment It may be noted that the assumption of isentropic flows is sometimes used to study the Navier–Stokes equations (see for instance Lighthill [24]). However, there is an inconsistency: one of the effects of the vorticity is to increase the entropy. One could argue that the effects of the entropy variation are much smaller than those of the viscosity. However, this is not the case, as shown in Ref. [46]. Specifically, this approximation still yields Eq. 87; however it gives the wrong expression for the coefficient \(\alpha \), namely \(\alpha =2\nu /3\), instead of \(\alpha =2\gamma \nu /3\) (see Eqs. 51 and 66 of Ref. [46]). This shows that the effects due to the entropy variations are of the same order of magnitude as those due only to the viscosity. Accordingly, such a hypothesis should not be used.

C: Evaluation of Singular Source and Doublet Integrals

As mentioned in the main body of this work, the integrands in the source and doublet coefficients are singular (see Ref. [39] for a deeper analysis of this point). In this case, we used a special treatment to evaluate these integrals.

Specifically, in the two elements adjacent to the collocation points, we first use a three-point trigonometric interpolation to describe the function, be it \(\chi (\mathbf{y})\) for the sources, or \(\varphi (\mathbf{y})\) for the doublets (Eq. 36). The three points are the three end points of the two adjacent boundary elements. For the geometry, we used the circle determined by the same three points, namely the boundary of the disk for the case under consideration. The result is a locally third-order scheme. Then, we evaluate the resulting integral over the complete circle. This is done analytically as shown in this rest of appendix. However, at this point we must subtract the integral over the portion outside the interval of interest. This is done numerically, element by element, for which the results are already available (Eq. 39).

Accordingly, in this appendix we only have to present: (1) the three-point trigonometric interpolation and (2) the analytic evaluation of the integrals over the complete circle.

The Trigonometric Interpolation

For the sake of simplicity, we present the formulation with nodes uniformly spaced over the circle. This was introduced in Ref. [8] and is used for all the results presented here. For the formulation with non-uniform spacing, see Ref. [57].

Consider three contiguous points on a circle, say at \(\mathbf{x}_0\) at \(\theta =0\) and \(\mathbf{x}_{_{^{\pm \!}}}\) at \(\theta =\pm \varDelta \theta \). Let \(u_0\) and \(u_{_{^{\pm \!}}}\) denote the known values of a generic function \(u(\theta )\) at these three points. We want to interpolate this function and obtain

$$\begin{aligned} u(\theta ) = A_0 + A_1 \cos \theta + A_2 \sin \theta , \end{aligned}$$

with \(A_0\), \(A_1\) and \(A_2\) such that \(u(0)=u_0\) and \(u(\pm \varDelta \theta )=u_{_{^{\pm \!}}}\). These yield \(u_0 = A_0 + A_1\) and \(u_{_{^{\pm \!}}}= A_0 + A_1 \cos \varDelta \theta \pm A_2 \sin \varDelta \theta \), namely

$$\begin{aligned} A_0 = u_0 - A_1, ~~~~ A_1 = \frac{u_{_{^{{+}\!}}}- 2 \, u_0 + u_{_{^{{-}\!}}}}{2\,(\cos \varDelta \theta - 1)}, ~~~~ A_2 = \frac{u_{_{^{{+}\!}}}- u_{_{^{{-}\!}}}}{2\,\sin \varDelta \theta }. ~~~~ \end{aligned}$$

Evaluation of Source and Doublet Integrals

The source and doublet integrals are invariant with respect to changes in frame of reference. Thus, we conveniently choose the origin to coincide with the center of the disk, and the \(x_1\) axis to go through \(\mathbf{x}\). Consider the point \(\mathbf{x}\), \(\mathbf{y}\) and \(\mathbf{y}'\) as shown in Fig. 12.

Fig. 12

The points \(\mathbf{x}\), \(\mathbf{y}\) and \(\mathbf{y}'\)

With these axes, we have

$$\begin{aligned}&x_1 = \rho&x_2 = 0, \nonumber \\&y_1 = \cos \theta&y_2 = \sin \theta , \nonumber \\&y_1' = \rho _1 \cos \theta&y_2' = \rho _1 \sin \theta , \end{aligned}$$

where \(\rho :=\Vert \mathbf{x}\Vert \) and \(\rho _1:=\Vert \mathbf{y}'\Vert \), whereas \(\Vert \mathbf{y}\Vert =1\). Therefore, we have

$$\begin{aligned} r_1^2 := \Vert \mathbf{y}_1-\mathbf{x}\Vert ^2 = (\rho _1 \cos \theta - \rho )^2 + \rho _1^2\sin ^2\theta = \rho ^2 - 2\,\rho \,\rho _1 \cos \theta + \rho _1^2. ~~~~ \end{aligned}$$

Similarly, we have

$$\begin{aligned} r^2 := r^2_1 \Big |_{\rho _1 = 1} = \rho ^2 - 2 \, \rho \, \cos \theta + 1. \end{aligned}$$

Accordingly, on the contour \({\mathcal {C}}_{_{^{\mathrm{B}}}}\) we have

$$\begin{aligned} 4\,\pi \,G(\mathbf{x},\mathbf{y}) = \ln r^2 = \ln \big [\rho ^2 - 2 \rho \cos \theta + 1 \big ]. \end{aligned}$$

Thus, setting \(\chi (\theta ) = A_0^\chi + A_1^\chi \, \cos \theta + A_2^\chi \, \sin \theta \) (Eq. 91), one obtains

$$\begin{aligned} S := \int _{-\pi }^\pi \chi (\theta ) \, G \, \mathrm{d}\theta = A_0^\chi \, S_0 + A_1^\chi \, S_1 + A_2^\chi \, S_2, \end{aligned}$$


$$\begin{aligned} 4 \pi S_k = \int _{-\pi }^\pi \cos ^k\theta \, \ln r^2 \mathrm{d}\theta ~~~ \hbox { and } ~~~ 4 \pi S_2 = \int _{-\pi }^\pi \sin \theta \, \ln r^2 \mathrm{d}\theta , ~~~~ \end{aligned}$$

where \(k=0,1\). On the other hand, on the contour \({\mathcal {C}}_{_{^{\mathrm{B}}}}\) we have

$$\begin{aligned} 4 \, \pi \, \frac{\partial {G}}{\partial {n}} = \bigg [ \frac{\partial {}}{\partial {\rho _1}} \ln r^2_1 \bigg ]_{\rho _1=1} = \frac{2}{r^2} \, (1-\rho \,\cos \theta ). \end{aligned}$$

Thus, setting \(\varphi (\theta ) = A_0^\varphi + A_1^\varphi \, \cos \theta + A_2^\varphi \, \sin \theta \) (Eq. 91) one obtains

$$\begin{aligned} D = - \int _{-\pi }^\pi \varphi (\theta ) \, \frac{\partial {G}}{\partial {n}} \, \mathrm{d}\theta = A_0^\varphi \, D_0 + A_1^\varphi \, D_1 + A_2^\varphi \, D_2, \end{aligned}$$


$$\begin{aligned} 4 \pi D_k = \int _{-\pi }^\pi \cos ^k\theta \, {\check{D}}\, \mathrm{d}\theta ~~~~ \hbox { and } ~~~~ 4 \pi D_2 = \int _{-\pi }^\pi \sin \theta \, {\check{D}}\, \mathrm{d}\theta , ~~~~ \end{aligned}$$

where \({\check{D}}:=4\,\pi \,\partial {G} / \partial {n}=-2(1-\rho \cos \theta )/r^2\), and again \(k=0,1\).

Note that \(S_2=D_2=0\), because \(\sin \theta \) is an odd function of \(\theta \), whereas \(\cos \theta \) is an even function of \(\theta \). Thus, we only need to evaluate \(S_0\), \(S_1\), \(D_0\) and \(D_1\).

Evaluation of \(S_0\)

Recall that \(2\pi S_0=\int _{-\pi }^\pi \ln r\,\mathrm{d}\theta \) (Eq. 98), where \(r^2=\rho ^2-2\rho \cos \theta +1\) (Eq. 95). Note that \(S_0=\phi _{_{^{\mathrm{S}}}}\), where \(\phi _{_{^{\mathrm{S}}}}\) is the potential due to a uniform distribution of sources over the unit circle, with intensity equal to one. The governing equation for this problem is \(\nabla ^2\phi =\delta (\rho -1)\). Using the Gauss theorem over a disk \({\mathcal {A}}_\rho \), having radius \(\rho \) and center O,

$$\begin{aligned} \oint _{{\mathcal {C}}_\rho } \frac{\partial {\phi }}{\partial {n}} \, \mathrm{d}s = \int _{{\mathcal {A}}_\rho } \nabla ^2\phi \, \mathrm{d}{\mathcal {A}}= 2\,\pi \int _0^\rho \delta (\rho '-1) \, \rho ' \mathrm{d}\rho ' = 2 \pi , ~~~~ \end{aligned}$$

(For \(\rho <1\) the integral vanishes). This yields \(\partial {\phi } / \partial {n}=1/\rho \) for \(\rho > 1\), namely \(\phi =\ln \rho \) for \(\rho \ge 1\) (For \(\rho \le 1\), we have \(\phi _{_{^{\mathrm{S}}}}=0\)). The constant of integration vanishes, so as to match the asymptotic behavior of \(S_0\) at infinity. Therefore, for \(\mathbf{x}\) either in the field (\(\rho >1\)), or on the boundary (\(\rho =1\)), we have \(S_0=\ln \rho \) (Note that \(S_0\) is continuous at \(\rho =1\), as expected, because source layers do not cause discontinuities of the function, only of its normal derivative [21]).

Evaluation of \(S_1\)

Recall that \(4\pi S_1= \int _{-\pi }^\pi \cos \theta \, \ln r^2\,\mathrm{d}\theta \) (Eq. 98), where \(r^2=\rho ^2-2\rho \cos \theta +1\) (Eq. 95). Integrating by parts and using \(2\sin ^2\theta =1-\cos 2\theta \), we have

$$\begin{aligned} 4 \pi S_1 = \sin \theta \ln r^2 \Big |_{-\pi }^\pi \!\! - \! \int _{-\pi }^\pi \! \frac{ 2 \rho \sin ^2\theta \, \mathrm{d}\theta }{\rho ^2 - 2 \rho \cos \theta + 1} = - \! \int _{-\pi }^\pi \! \frac{\rho \, (1 - \cos 2\theta ) \, \mathrm{d}\theta }{\rho ^2 - 2 \rho \cos \theta + 1}. ~~~~~~ \end{aligned}$$

Note that setting \(z=\mathrm{e}^{\imath \theta }\), we have \(\mathrm{d}\theta =\mathrm{d}z/\imath z\) and

$$\begin{aligned} \frac{\rho \, \mathrm{d}\theta }{\rho ^2 - 2 \rho \cos \theta + 1} = \frac{-\mathrm{d}z/\imath }{ z^2 - (\rho +1/\rho ) z + 1} = \frac{-\mathrm{d}z/\imath }{(z - z_{_{^{{+}\!}}}) \, (z - z_{_{^{{-}\!}}})}, \end{aligned}$$

where \(z_{_{^{\pm \!}}}\) denotes the roots of \(z^2 - (\rho +1/\rho ) z + 1 = 0\), namely \(z_{_{^{{+}\!}}}=\rho \) and \(z_{_{^{{-}\!}}}=1/\rho \).

Next, let us go back to Eq. 103, and add \(\imath \sin 2\theta \) to \(\cos 2\theta \) (This is legitimate, because the addition to the integrand is an odd function and this does not change the value of the integral). Therefore, we finally have

$$\begin{aligned} 4 \pi S_1 \! =\! -\! \int _{-\pi }^\pi \! \frac{\rho \, \big (1 - \mathrm{e}^{\imath 2 \theta }\big )}{\rho ^2 - 2 \rho \cos \theta + 1} \, \mathrm{d}\theta = \! \frac{1}{\imath } \oint \frac{1-z^2}{(z - z_{_{^{{+}\!}}}) (z - z_{_{^{{-}\!}}})} \, \mathrm{d}z. \end{aligned}$$

For \(\rho >1\), we have that \(z_{_{^{{-}\!}}}=1/\rho \) is inside \({\mathcal {C}}\), whereas \(z_{_{^{{+}\!}}}=\rho \) is inside \({\mathcal {C}}\), for \(\rho <1\). Hence, using the residue theorem, Eq. 105 yields

$$\begin{aligned} 4 \pi S_1 = \frac{2 \pi \imath }{\imath } \, \frac{1 - z^2}{z - z_{_{^{\pm \!}}}} \bigg |_{z=z_{_{^{\mp \!}}}} \quad \hbox {for} \; \rho \gtrless 1, \end{aligned}$$

namely \(S_1 = -1/(2\,\rho )\) for \(\rho > 1\), whereas for \(\rho < 1\), we have \(S_1= - \frac{1}{2}\rho \). For \(\rho =1\), both roots are on the contour. Then, using the Cauchy principal value rule (namely, one half of both residues, Ref. [4]), we obtain \(S_1=-\frac{1}{2}\). Again, we have continuity of \(S_1\) on \({\mathcal {C}}\).

Evaluation of \(D_0\)

Recall that \(4\pi D_0=-2\int _{-\pi }^\pi (1/r^2)\,(1-\rho \,\cos \theta )\,\mathrm{d}\theta \) (Eq. 101), where \(r^2=\rho ^2-2\rho \cos \theta +1\) (Eq. 95). Again, adding \(\imath \sin \theta \) to \(\cos \theta \), setting \(\mathrm{e}^{\imath \theta }=z\), and using Eq. 104, we have

$$\begin{aligned} 4 \pi D_0 = - 2 \int _{-\pi }^\pi \frac{1 - \rho \, \mathrm{e}^{\imath \theta }}{\rho ^2 - 2 \, \rho \, \cos \theta + 1} \, \mathrm{d}\theta = \frac{2}{\imath \,\rho } \oint _{\mathcal {C}}\frac{1 - \rho \, z}{(z-z_{_{^{{+}\!}}}) \, (z-z_{_{^{{-}\!}}})} \, \mathrm{d}z. \end{aligned}$$

Again, \(z_{_{^{{-}\!}}}\) is inside \({\mathcal {C}}\) for \(\rho >1\), whereas \(z_{_{^{{+}\!}}}\) is inside \({\mathcal {C}}\) for \(\rho <1\). Hence, using the residue theorem, one obtains

$$\begin{aligned} D_0 = \frac{2 \pi \imath }{4\pi \imath } \, \frac{2}{\rho } \, \frac{1-\rho \, z}{z-\rho } \bigg |_{z=1/\rho } = 0,&\hbox { for } \; \rho > 1, \end{aligned}$$

whereas for \(\rho < 1\) we have \(D_0=-1\). For \(\mathbf{x}\) on \({\mathcal {C}}\), we have \(\rho = 1\). Again, both roots are on the contour, and the Cauchy principal value rule is used to yield \(D_0 = -1/2\). Note that \(\partial {D_0} / \partial {\rho }\) is continuous at \(\rho =1\), as expected, because doublet layers do not cause discontinuities of the normal derivative, only of the function (Ref. [21]).

Evaluation of \(D_1\)

Following a similar procedure, one obtains \(D_1= 1/(2 \rho )\), for \(\rho >1\), and \(D_1=- \frac{1}{2}\rho \) for \(\rho < 1\), whereas for \(\rho =1\) we have \(D_1=0\) (see Ref. [8] for details).

D: Vortex Rings—Problem and Solution

In Sect. 8.2, we discussed the limits of the NVD, having to do with the initial condition \(\mathbf{w}(\mathbf{x},0)=\mathbf{0}\) used throughout the paper. To illustrate the issue, consider a problem with non-homogeneous initial conditions, specifically a vortex ring that is initially circular, with a circular cross section and uniform \(\Vert \varvec \zeta \Vert \). In this case, if \(\mathbf{w}\) is limited to the vortical region, we obtain that the potential is multi–valued (consider a contour that is wrapped around the vortex ring, and use the Stokes theorem).

To this end, let the (xy)-plane coincide with the midplane of the vortex ring at \(t=0\). Using the direct integration approach, using straight lines parallel, say, to the x-axis, one obtains that, at time \(t=0\), we have \(\mathbf{w}\ne \mathbf{0}\) in the inner region \({\mathcal {R}}\) (locus of the points that lie on lines that intersect the ring and are inside the ring), with \(\partial {\mathbf{w}} / \partial {x}=0\) in such a region (for, the vorticity encountered along the line of integration for \(x>0\) is equal and opposite to that for \(x<0\)). Of course, \(\mathbf{w}=\mathbf{0}\) in the outer region. Of course, as time goes by, the points initially in \({\mathcal {R}}\) follow the motion of the fluid, and get wrapped around the ring, rendering the NVD unpractical.

For this reason, the formulation used by Gradassi in Ref. [17] to address this problem is not the NVD. For the sake of completeness, the approach used (based on the vorticity kinematics of Ref. [30] and the Biot–Savart law for continuous distribution of vorticity) is briefly outlined here and some results are presented. For inviscid flows, the formulation is based upon the fact that the material contravariant components of the vorticity remain constant in time. Indeed, starting from the vorticity equation \({\mathrm{D}{{\varvec{\zeta }}}}/{\mathrm{D}t}={\varvec{\zeta }}\cdot \nabla \mathbf{v}\), setting \({\varvec{\zeta }}=\zeta ^\alpha \,\mathbf{g}_\alpha \) and using \({\mathrm{D}{\mathbf{g}_\alpha }}/{\mathrm{D}t}=\partial {\mathbf{v}} / \partial {\xi ^\alpha }\) (Eq. 62), one obtains \({\mathrm{D}{\zeta ^\beta }}/{\mathrm{D}t}=0\), namely \(\zeta ^\beta \) constant following a material point. Following the grid points (namely integrating the equation \({\mathrm{D}{\mathbf{x}}}/{\mathrm{D}t}=\mathbf{v}\)), one may obtain \(\mathbf{g}_\alpha \), and hence \({\varvec{\zeta }}=\zeta ^\alpha (\xi ^\beta ,0)\,\mathbf{g}_\alpha \). Once \({\varvec{\zeta }}\) has been obtained, the velocity is evaluated by using

$$\begin{aligned} \mathbf{v}(\mathbf{x}) = - \nabla \times \int _{{\mathbb {R}}^3}G(\mathbf{x},\mathbf{y}) \, {\varvec{\zeta }}(\mathbf{y}) \, \mathrm{d}{\mathcal {V}}(\mathbf{y}), \end{aligned}$$

with \(G(\mathbf{x},\mathbf{y})= - 1/[4\pi \Vert \mathbf{x}- \mathbf{y}\Vert \) (Eq. 5). This is the extension (arising from the Helmholtz decomposition) of the Biot–Savart law for the velocity induced by a line vortex. Comment The formulation in terms of material contravariant components of the vorticity was introduced in Ref. [30], which includes viscous flows. For inviscid flows, the result is equivalent to that obtained by Cauchy in 1815 (Serrin [56], p. 152, Eq. 17.5). Still for the limited case of inviscid flows, the formulation in Ref. [30] was arrived at, independently, by Casciola and Piva [5], who present also interesting numerical results.

The results obtained with this approach (Ref. [17] and subsequent activity) are very good. Just to give an example, in Fig. 13 we present an analysis of the convergence of the self-induced velocity \(v_{_{^{\mathrm{S}\mathrm{I}}}}\) for relatively thin vortex rings. The present results compare favorably with the Thomson [60] formula (denoted by \(\blacksquare\) on the ordinate axis), namely

$$\begin{aligned} v_{_{^{\mathrm{S}\mathrm{I}}}}= \frac{\varGamma }{4\pi R} \, \bigg [ \ln \frac{8R}{a} - 1 \bigg ]. \end{aligned}$$

For historical details, see Ref. [27], p. 11, Eq. 5, and comments that follow.

Fig. 13

Convergence analysis

As another example, it is known that vortex rings present instabilities [63,64,65]. Accordingly, a computational analysis (using the above scheme) of the kinematics of a vortex ring subject to an initial perturbation of the type \(\cos n\theta \) was performed. The asymptotic result is presented in Fig. 14. This new configuration is stable.

Fig. 14

Stability analysis

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Bernardini, G., Coderoni, M., Gradassi, P. et al. Natural Velocity Decomposition: A Review. Aerotec. Missili Spaz. 98, 5–29 (2019).

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  • Computational fluid dynamics
  • Viscous flows
  • Potential/vortical-field decomposition