2.1 Sheet dynamics

Let $\boldsymbol{x}$ be the position vector of an arbitrary point in three-dimensional space. The sheet is immersed in an irrotational, incompressible fluid with density $\unicode[STIX]{x1D70C}$ and velocity $\boldsymbol{u}(\boldsymbol{x},t)$ , where $t$ is time. A right-handed orthogonal coordinate system ( $s$ , $n$ , $b$ ) is defined on the sheet such that the basis $\hat{\boldsymbol{s}}$ - $\hat{\boldsymbol{b}}$ spans the local tangent manifold that is the sheet and $\hat{\boldsymbol{n}}$ is the corresponding sheet normal vector. The sheet is a two-dimensional surface whose location is specified by $\boldsymbol{x}=\boldsymbol{x}_{s}(s,b,t)$ ; see figure 2.

Figure 2.

Schematic of a vortex-entrainment sheet (shaded surface) embedded in an outer fluid. The ‘intrinsic’ surface flow $\boldsymbol{w}$ is in the tangent space $(s,b)$ of the sheet. The sheet position is given by $\boldsymbol{x}=\boldsymbol{x}_{s}(s,b)$ . The surface velocity of the sheet is $\boldsymbol{v}=\boldsymbol{w}+(\boldsymbol{v}\boldsymbol{\cdot }\hat{\boldsymbol{n}})\hat{\boldsymbol{n}}$ . The sheet has a surface density $\unicode[STIX]{x1D70C}_{s}$ (mass per-unit-area). The outer fluid velocity is $\boldsymbol{u}$ and is discontinuous across the sheet. The fluid density $\unicode[STIX]{x1D70C}$ and pressure $p$ may also be discontinuous.

The jump in a quantity $f$ across the sheet is defined as

(2.1) $$\begin{eqnarray}\unicode[STIX]{x27E6}f\unicode[STIX]{x27E7}=f^{+}-f^{-},\end{eqnarray}$$

where $f^{+}$ and $f^{-}$ are the limiting values of $f$ as approached from the different sides of the sheet with $\hat{\boldsymbol{n}}$ pointing to the $(+)$ side. The surface velocity on the sheet is $\boldsymbol{v}(\boldsymbol{x}_{s},t)$ and given our choice of coordinates this can be decomposed as

(2.2) $$\begin{eqnarray}\boldsymbol{v}=\boldsymbol{w}+(\boldsymbol{v}\boldsymbol{\cdot }\hat{\boldsymbol{n}})\hat{\boldsymbol{n}}=(v_{s}\hat{\boldsymbol{s}}+v_{b}\hat{\boldsymbol{b}})+v_{n}\hat{\boldsymbol{n}}.\end{eqnarray}$$

where $\boldsymbol{w}(\boldsymbol{x}_{s},t)=v_{s}\hat{\boldsymbol{s}}+v_{b}\hat{\boldsymbol{b}}$ is the intrinsic velocity of the flow inside the sheet and $v_{n}(\boldsymbol{x}_{s},t)=\boldsymbol{v}\boldsymbol{\cdot }\hat{\boldsymbol{n}}$ is the normal velocity component. The surface operator $\unicode[STIX]{x1D735}_{s}$ , which only acts in the tangent space of the sheet, is given by:

(2.3) $$\begin{eqnarray}\unicode[STIX]{x1D735}_{s}=\left(\frac{1}{h_{s}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}s}}\right)\hat{\boldsymbol{s}}+\left(\frac{1}{h_{b}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}b}}\right)\hat{\boldsymbol{b}},\end{eqnarray}$$

with $h_{s}$ and $h_{b}$ as the scale factors for the chosen surface coordinate system. The operator giving the surface material derivative of the quantity $f$ on the sheet is defined as:

(2.4) $$\begin{eqnarray}\frac{\text{D}_{s}f}{\text{D}t}={\displaystyle \frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}t}}+\boldsymbol{w}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{s}f.\end{eqnarray}$$

Given our specific choice of the coordinate system we have $(v_{n}\hat{\boldsymbol{n}})\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{s}f=0$ and thus we may write $\boldsymbol{w}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{s}f=\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{s}f$ . The reader is referred to Aris (Reference Aris1962) and Slattery, Sagis & Oh (Reference Slattery, Sagis and Oh2007) for further details regarding these surface operators as well as the sheet conservation equations discussed next.

Now, we define the mass within the sheet by integrating across the layer in the normal direction to give

(2.5) $$\begin{eqnarray}\unicode[STIX]{x1D70C}_{s}(\boldsymbol{x}_{s},t)=\int \unicode[STIX]{x1D70C}(\boldsymbol{x},t)\,\text{d}n.\end{eqnarray}$$

The quantity $\unicode[STIX]{x1D70C}_{s}$ is the sheet mass density and has units of mass per-unit-area of sheet. In this way, the mass that was originally in the viscous layer is preserved by ‘collapsing’ it to the sheet; recall figure 1. Fluid may enter the sheet via entrainment and the mass conservation equation on the sheet is:

(2.6) $$\begin{eqnarray}\frac{\text{D}_{s}\unicode[STIX]{x1D70C}_{s}}{\text{D}t}+\unicode[STIX]{x1D70C}_{s}(\unicode[STIX]{x1D735}_{s}\boldsymbol{\cdot }\boldsymbol{v})=-\unicode[STIX]{x27E6}\unicode[STIX]{x1D70C}(\boldsymbol{u}-\boldsymbol{v})\boldsymbol{\cdot }\hat{\boldsymbol{n}}\unicode[STIX]{x27E7}.\end{eqnarray}$$

The left-hand side has the familiar form of mass conservation in the bulk fluid, but with the understanding of the intrinsic sheet-tangent operators. In contrast, the right side is non-zero with $-\unicode[STIX]{x27E6}\unicode[STIX]{x1D70C}(\boldsymbol{u}-\boldsymbol{v})\boldsymbol{\cdot }\hat{\boldsymbol{n}}\unicode[STIX]{x27E7}$ representing a source of $\unicode[STIX]{x1D70C}_{s}$ due to entrainment from the outer flow. While the fluid velocity may have a discontinuity, the sheet velocity is continuous meaning that $\unicode[STIX]{x27E6}\boldsymbol{v}\unicode[STIX]{x27E7}=0$ .

The conservation of momentum on the sheet also takes on a similar form to the usual equation for the bulk fluid, but again with additional source terms associated with the flux and entrainment of momentum from the flow outside the sheet:

(2.7) $$\begin{eqnarray}\unicode[STIX]{x1D70C}_{s}\frac{\text{D}_{s}\boldsymbol{v}}{\text{D}t}-(\unicode[STIX]{x1D735}_{s}\boldsymbol{\cdot }\boldsymbol{T}_{s}+\unicode[STIX]{x1D70C}_{s}\boldsymbol{f}_{s})=-\unicode[STIX]{x27E6}\unicode[STIX]{x1D70C}(\boldsymbol{u}-\boldsymbol{v})(\boldsymbol{u}-\boldsymbol{v})\boldsymbol{\cdot }\hat{\boldsymbol{n}}\unicode[STIX]{x27E7}-\unicode[STIX]{x27E6}p\unicode[STIX]{x27E7}\hat{\boldsymbol{n}}+\unicode[STIX]{x27E6}\unicode[STIX]{x1D749}\unicode[STIX]{x27E7},\end{eqnarray}$$

where $\unicode[STIX]{x27E6}p\unicode[STIX]{x27E7}$ is the pressure jump across the sheet and $\unicode[STIX]{x1D749}=\unicode[STIX]{x27E6}\boldsymbol{T}_{v}\boldsymbol{\cdot }\hat{\boldsymbol{n}}\unicode[STIX]{x27E7}$ is the shear stress vector with $\boldsymbol{T}_{v}$ as the deviatoric stress tensor outside the sheet. In general, the sheet may also have its own phenomena, such as a surface stress tensor $\boldsymbol{T}_{s}$ (yielding a surface tension force, for example), or a ‘body’ force $\boldsymbol{f}_{s}$ that acts only on the sheet mass density $\unicode[STIX]{x1D70C}_{s}$ , as well as different surface fluid properties (Slattery et al. Reference Slattery, Sagis and Oh2007). For simplicity we shall neglect each of these terms in (2.7). The surface flow $\boldsymbol{v}$ is clearly dynamically coupled to the outer flow $\boldsymbol{u}$ , which must also be solved. The main unknowns to be solved for are $\boldsymbol{u}$ , $\unicode[STIX]{x1D70C}_{s}$ , $\boldsymbol{v}$ and $\unicode[STIX]{x27E6}p\unicode[STIX]{x27E7}$ . For a free sheet we have $\unicode[STIX]{x1D749}^{+}=\unicode[STIX]{x1D749}^{-}=0$ since the outer flow is potential. When the sheet is bound to a surface its normal velocity will be known from the motion $\boldsymbol{U}$ of that surface as $v_{n}=\hat{\boldsymbol{n}}\boldsymbol{\cdot }\boldsymbol{U}$ and the wall shear stress $\unicode[STIX]{x1D749}_{w}=\unicode[STIX]{x27E6}\unicode[STIX]{x1D749}\unicode[STIX]{x27E7}$ is then appended to the list of unknowns.

At this stage it is useful to discuss the occurrence and interpretation of entrainment by examining the jump source term on the right-hand side of (2.6). Using the properties of jumps and averages of products (e.g. Wu et al. Reference Wu, Ma and Zhou2006, p. 34) we may write

(2.8) $$\begin{eqnarray}\unicode[STIX]{x27E6}\unicode[STIX]{x1D70C}(u_{n}-v_{n})\unicode[STIX]{x27E7}=\unicode[STIX]{x27E6}\unicode[STIX]{x1D70C}\unicode[STIX]{x27E7}(\overline{u_{n}}-v_{n})+\overline{\unicode[STIX]{x1D70C}}\unicode[STIX]{x27E6}u_{n}\unicode[STIX]{x27E7},\end{eqnarray}$$

where $\overline{(\cdot )}=[(\cdot )^{+}+(\cdot )^{-}]/2$ is the arithmetic mean of the values of a quantity across the sheet. Throughout, an overline will indicate this type of average unless otherwise noted. The sheet must coincide with the discontinuity $\unicode[STIX]{x27E6}u_{n}\unicode[STIX]{x27E7}\neq 0$ for entrainment to occur meaning that fluid will not pass through the sheet. However, since fluid may be entrained into the sheet, then $v_{n}$ will take a value between $u_{n}^{+}$ and $u_{n}^{-}$ . To elucidate this, consider a weighted average:

(2.9) $$\begin{eqnarray}v_{n}=\left[\frac{1+\unicode[STIX]{x1D6FC}}{2}\right]u_{n}^{+}+\left[\frac{1-\unicode[STIX]{x1D6FC}}{2}\right]u_{n}^{-}=\overline{u_{n}}+\frac{\unicode[STIX]{x1D6FC}}{2}\unicode[STIX]{x27E6}u_{n}\unicode[STIX]{x27E7},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}$ is a weighting parameter with $-1\leqslant \unicode[STIX]{x1D6FC}\leqslant 1$ as similarly used by Baker et al. (Reference Baker, Meiron and Orszag1982). Substituting (2.9) into the preceding equation gives:

(2.10) $$\begin{eqnarray}\unicode[STIX]{x27E6}\unicode[STIX]{x1D70C}(u_{n}-v_{n})\unicode[STIX]{x27E7}=\unicode[STIX]{x27E6}u_{n}\unicode[STIX]{x27E7}\left(\overline{\unicode[STIX]{x1D70C}}-\frac{\unicode[STIX]{x1D6FC}}{2}\unicode[STIX]{x27E6}\unicode[STIX]{x1D70C}\unicode[STIX]{x27E7}\right)=\unicode[STIX]{x27E6}u_{n}\unicode[STIX]{x27E7}\left(\left[\frac{1-\unicode[STIX]{x1D6FC}}{2}\right]\unicode[STIX]{x1D70C}^{+}+\left[\frac{1+\unicode[STIX]{x1D6FC}}{2}\right]\unicode[STIX]{x1D70C}^{-}\right)\end{eqnarray}$$

so that fluid with a weighted average density is entrained at a rate $\unicode[STIX]{x27E6}u_{n}\unicode[STIX]{x27E7}$ . However, notice the weighting is opposite to that of $v_{n}$ : for example, if $\unicode[STIX]{x1D6FC}>0$ then $v_{n}$ moves with a velocity closer to $u_{n}^{+}$ , but in doing so the sheet entrains or ‘consumes’ more fluid with the density $\unicode[STIX]{x1D70C}^{-}$ . In a similar fashion, substitution of (2.9) into the momentum flux term in (2.7) yields:

(2.11) $$\begin{eqnarray}\unicode[STIX]{x27E6}\unicode[STIX]{x1D70C}(\boldsymbol{u}-\boldsymbol{v})(u_{n}-v_{n})\unicode[STIX]{x27E7}=\unicode[STIX]{x27E6}u_{n}\unicode[STIX]{x27E7}(\overline{\unicode[STIX]{x1D70C}(\boldsymbol{u}-\boldsymbol{v})}-\frac{\unicode[STIX]{x1D6FC}}{2}\unicode[STIX]{x27E6}\unicode[STIX]{x1D70C}(\boldsymbol{u}-\boldsymbol{v})\unicode[STIX]{x27E7})\end{eqnarray}$$

so that $\unicode[STIX]{x27E6}u_{n}\unicode[STIX]{x27E7}$ simultaneously entrains a weighted average of the difference momentum $\unicode[STIX]{x1D70C}(\boldsymbol{u}-\boldsymbol{v})$ from either side of the sheet. If there is no mass in a free sheet, $\unicode[STIX]{x1D70C}_{s}=0$ , then (2.6) dictates that $\unicode[STIX]{x27E6}u_{n}\unicode[STIX]{x27E7}=0$ . Then (2.7) gives a zero pressure jump $\unicode[STIX]{x27E6}p\unicode[STIX]{x27E7}=0$ . Of course, these are the kinematical and dynamical features of a conventional vortex sheet. Note that each was consequently obtained from the assumption that $\unicode[STIX]{x1D70C}_{s}=0$ .

2.2 Sheet kinematics

In order to solve the surface flow equations in (2.6)–(2.7) we must also determine the outer flow $\boldsymbol{u}$ . A vector field is determined from its essential characteristics, namely its divergence, curl, discontinuities and boundary values (Phillips Reference Phillips1959). In general, the field can be expressed by the sum of a divergence-free part and a curl-free part via a Helmholtz–Hodge decomposition, and if the full vector is known on the boundary a harmonic part can be distinguished (Bhatia et al. Reference Bhatia, Norgard, Pascucci and Bremer2013). As such, the decomposition can be made unique in several ways depending on the known boundary conditions.

While $\boldsymbol{u}$ is an irrotational, incompressible flow with no distribution of vorticity ( $\unicode[STIX]{x1D74E}$ ) or dilatation rate ( $\unicode[STIX]{x1D6E5}$ ) by definition, it is discontinuous across the sheet. Therefore, the decomposition of $\boldsymbol{u}$ has distinct contributions from its discontinuities. These contributions can be rigorously derived (e.g. Phillips Reference Phillips1959), but for simplicity we employ a straightforward, albeit crude device that yields the same result. If we insist on allowing the fluid to possess a singular curl and divergence we can represent the discontinuity in $\boldsymbol{u}$ across the sheet with a Heaviside function. Then the singular parts of the curl and divergence are (see appendix A):

(2.12a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D74E}=\unicode[STIX]{x1D735}\times \boldsymbol{u}=(\hat{\boldsymbol{n}}\times \unicode[STIX]{x27E6}\boldsymbol{u}\unicode[STIX]{x27E7})\unicode[STIX]{x1D6FF}(n),\quad \unicode[STIX]{x1D6E5}=\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=(\hat{\boldsymbol{n}}\boldsymbol{\cdot }\unicode[STIX]{x27E6}\boldsymbol{u}\unicode[STIX]{x27E7})\unicode[STIX]{x1D6FF}(n), & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}(n)$ is the Dirac delta function and $n$ is the sheet normal coordinate. These components of the curl and contribution to the divergence are those affected by normal derivatives across the sheet. The intrinsic surface operators ‘remove’ these singular parts as each is obtained from the appropriate projection of the full spatial operator onto the sheet. These expressions afford a convenient way to define the strengths of the vortex and entrainment sheets as:

(2.13) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D738}(\boldsymbol{x}_{s},t)=\int \unicode[STIX]{x1D74E}(\boldsymbol{x}_{s},t)\,\text{d}n=\hat{\boldsymbol{n}}\times \unicode[STIX]{x27E6}\boldsymbol{u}\unicode[STIX]{x27E7}=\unicode[STIX]{x27E6}u_{b}\unicode[STIX]{x27E7}\hat{\boldsymbol{s}}+(-\unicode[STIX]{x27E6}u_{s}\unicode[STIX]{x27E7})\hat{\boldsymbol{b}}, & \displaystyle\end{eqnarray}$$

(2.14) $$\begin{eqnarray}\displaystyle & \displaystyle q(\boldsymbol{x}_{s},t)=\int -\unicode[STIX]{x1D6E5}(\boldsymbol{x}_{s},t)\,\text{d}n=-\hat{\boldsymbol{n}}\boldsymbol{\cdot }\unicode[STIX]{x27E6}\boldsymbol{u}\unicode[STIX]{x27E7}=-\unicode[STIX]{x27E6}u_{n}\unicode[STIX]{x27E7}. & \displaystyle\end{eqnarray}$$

The negative sign in the definition of $q$ is so that $q>0$ corresponds to entrainment into the sheet (also see appendix B). Relative to the outer flow this will appear as a sink-like motion. The flux into the vortex-entrainment sheet is what sets it apart from a contact discontinuity (no flux) and from a shock (through flux).

The vectorial vortex sheet strength $\unicode[STIX]{x1D738}=\unicode[STIX]{x1D6FE}_{s}\hat{\boldsymbol{s}}+\unicode[STIX]{x1D6FE}_{b}\hat{\boldsymbol{b}}$ is tangent to the sheet, however, the surface flow may also have finite intrinsic vorticity normal to the sheet: $\unicode[STIX]{x1D714}_{n}\hat{\boldsymbol{n}}=(\unicode[STIX]{x2202}v_{s}/\unicode[STIX]{x2202}b-\unicode[STIX]{x2202}v_{b}/\unicode[STIX]{x2202}s)\hat{\boldsymbol{n}}$ . When (2.12) is substituted into the decomposition of $\boldsymbol{u}$ , the fluid volume integrals reduce to surface integrals over the sheet area $S$ . Then using the definitions (2.13)–(2.14) we can combine the decomposition contributions from the curl/vortex sheet and divergence/entrainment sheet as:

(2.15) $$\begin{eqnarray}\displaystyle \boldsymbol{u}_{\unicode[STIX]{x1D714}}(\boldsymbol{x},t)+\boldsymbol{u}_{\unicode[STIX]{x1D6E5}}(\boldsymbol{x},t)=\frac{1}{4\unicode[STIX]{x03C0}}\int _{S}\frac{\unicode[STIX]{x1D738}(\boldsymbol{x}_{s},t)\times (\boldsymbol{x}-\boldsymbol{x}_{s})-q(\boldsymbol{x}_{s},t)(\boldsymbol{x}-\boldsymbol{x}_{s})}{|\boldsymbol{x}-\boldsymbol{x}_{s}|^{3}}\,\text{d}A, & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{x}_{s}$ is the position of $S$ . The prescribed boundary conditions on a given problem will then determine whether or not these contributions are sufficient to describe $\boldsymbol{u}$ , for example if a harmonic contribution is needed to impose a boundary condition at infinity.

Finally the circulation vector $\unicode[STIX]{x1D71E}$ and net entrainment rate $Q$ are defined as:

(2.16a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D71E}=\unicode[STIX]{x1D6E4}_{s}\hat{\boldsymbol{s}}+\unicode[STIX]{x1D6E4}_{b}\hat{\boldsymbol{b}}=\int \unicode[STIX]{x1D738}\,\text{d}l,\quad Q=\int _{S}q\,\text{d}A, & & \displaystyle\end{eqnarray}$$

where $l(s,b)$ is a prescribed integration path in the sheet.

2.3 Evolution equations for the sheet strengths

Here, the evolution equations for the sheet strengths are given. To this end, the jump operator in (2.1) is applied to the governing Euler equation for the outer flow $\boldsymbol{u}$ . Alben (Reference Alben2009) used this form of the unsteady Bernoulli equation to obtain a scalar vortex sheet evolution equation to study two-dimensional flexible propulsive appendages. The jump Euler equation is:

(2.17) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x27E6}\boldsymbol{u}\unicode[STIX]{x27E7}}{\unicode[STIX]{x2202}t}}=-\left.\unicode[STIX]{x27E6}\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}+\unicode[STIX]{x1D735}\frac{p}{\unicode[STIX]{x1D70C}}\right.\unicode[STIX]{x27E7}\!.\end{eqnarray}$$

Note that the del operator is inside the jump brackets. While the identity $\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}=\unicode[STIX]{x1D735}(1/2)|\boldsymbol{u}|^{2}$ is valid on each side of the sheet, we cannot use it to write $\unicode[STIX]{x27E6}\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\unicode[STIX]{x27E7}=\unicode[STIX]{x1D735}\unicode[STIX]{x27E6}1/2|\boldsymbol{u}|^{2}\unicode[STIX]{x27E7}$ , since normal derivatives across the sheet are not defined. Therefore, we isolate the tangential and normal components of the right side by making use of the surface-tangent operator $\unicode[STIX]{x1D735}_{s}$ in (2.3) to allow the jump operator inside on the tangential part:

(2.18) $$\begin{eqnarray}\displaystyle \left.\unicode[STIX]{x27E6}\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}+\unicode[STIX]{x1D735}\frac{p}{\unicode[STIX]{x1D70C}}\right.\unicode[STIX]{x27E7}=\unicode[STIX]{x1D735}_{s}\left.\unicode[STIX]{x27E6}\frac{1}{2}|\boldsymbol{u}|^{2}+\frac{p}{\unicode[STIX]{x1D70C}}\right.\unicode[STIX]{x27E7}+\left.\unicode[STIX]{x27E6}\left(\unicode[STIX]{x1D735}\boldsymbol{\cdot }(u_{n}\boldsymbol{u})+\hat{\boldsymbol{n}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\frac{p}{\unicode[STIX]{x1D70C}}\right)\right.\unicode[STIX]{x27E7}\hat{\boldsymbol{n}}, & & \displaystyle\end{eqnarray}$$

and we have used the identity $\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}=\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\boldsymbol{u}\otimes \boldsymbol{u})$ for the normal component. We can effectively take the curl and divergence of (2.17) by the cross-product and dot product, respectively, with $\hat{\boldsymbol{n}}$ . Using the sheet strength definitions in (2.13)–(2.14) along with the properties of jumps of product quantities and the triple vector product we have the following identities:

(2.19) $$\begin{eqnarray}\unicode[STIX]{x27E6}{\textstyle \frac{1}{2}}|\boldsymbol{u}|^{2}\unicode[STIX]{x27E7}=\overline{\boldsymbol{u}}\boldsymbol{\cdot }\unicode[STIX]{x27E6}\boldsymbol{u}\unicode[STIX]{x27E7}=\overline{\boldsymbol{u}}\boldsymbol{\cdot }(\unicode[STIX]{x1D738}\times \hat{\boldsymbol{n}}-q\hat{\boldsymbol{n}}).\end{eqnarray}$$

We then obtain:

(2.20) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{\boldsymbol{n}}\times \left.\unicode[STIX]{x27E6}\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}+\unicode[STIX]{x1D735}\frac{p}{\unicode[STIX]{x1D70C}}\right.\unicode[STIX]{x27E7}={\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}b}}\left(\overline{\boldsymbol{u}}\boldsymbol{\cdot }\unicode[STIX]{x27E6}\boldsymbol{u}\unicode[STIX]{x27E7}+\left.\unicode[STIX]{x27E6}\frac{p}{\unicode[STIX]{x1D70C}}\right.\unicode[STIX]{x27E7}\right)\hat{\boldsymbol{s}}-{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}s}}\left(\overline{\boldsymbol{u}}\boldsymbol{\cdot }\unicode[STIX]{x27E6}\boldsymbol{u}\unicode[STIX]{x27E7}+\left.\unicode[STIX]{x27E6}\frac{p}{\unicode[STIX]{x1D70C}}\right.\unicode[STIX]{x27E7}\right)\hat{\boldsymbol{b}}\quad & \displaystyle\end{eqnarray}$$

(2.21) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{\boldsymbol{n}}\boldsymbol{\cdot }\left.\unicode[STIX]{x27E6}\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}+\unicode[STIX]{x1D735}\frac{p}{\unicode[STIX]{x1D70C}}\right.\unicode[STIX]{x27E7}={\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}s}}\unicode[STIX]{x27E6}u_{n}u_{s}\unicode[STIX]{x27E7}+{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}b}}\unicode[STIX]{x27E6}u_{n}u_{b}\unicode[STIX]{x27E7}+\left.\unicode[STIX]{x27E6}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}n}}\left(u_{n}^{2}+\frac{p}{\unicode[STIX]{x1D70C}}\right)\right.\unicode[STIX]{x27E7}. & \displaystyle\end{eqnarray}$$

In the event that there is no entrainment, then (2.21) would be zero since the pressure jump immediately communicates normal momentum from one side of the vortex-entrainment sheet to the other. To see this, consider the normal component of the sheet momentum equation (2.7):

(2.22) $$\begin{eqnarray}\unicode[STIX]{x1D70C}_{s}\frac{\text{D}_{s}v_{n}}{\text{D}t}=-\unicode[STIX]{x27E6}\unicode[STIX]{x1D70C}(u_{n}-v_{n})^{2}\unicode[STIX]{x27E7}-\unicode[STIX]{x27E6}p\unicode[STIX]{x27E7}=-2\unicode[STIX]{x1D70C}(\overline{u_{n}}-v_{n})\unicode[STIX]{x27E6}u_{n}\unicode[STIX]{x27E7}-\unicode[STIX]{x27E6}p\unicode[STIX]{x27E7}.\end{eqnarray}$$

With $q=-\unicode[STIX]{x27E6}u_{n}\unicode[STIX]{x27E7}=0$ then $\unicode[STIX]{x27E6}p\unicode[STIX]{x27E7}$ delivers the acceleration $\text{D}_{s}v_{n}/\text{D}t$ to the sheet mass $\unicode[STIX]{x1D70C}_{s}$ . On the other hand, when $q\neq 0$ then the last term in (2.21) represents a gradient of energy across the sheet meaning that the pressure jump must expend additional energy to also accelerate the newly entrained sheet mass. Obviously we cannot evaluate this gradient directly. However, from what has been said about the $q=0$ case we can conclude that the normal component of the jump in the Euler equation is equal to the amount of normal momentum that is lost or entrained into the sheet. Therefore, after defining

(2.23) $$\begin{eqnarray}\unicode[STIX]{x1D707}\equiv \overline{\boldsymbol{u}}\boldsymbol{\cdot }\unicode[STIX]{x27E6}\boldsymbol{u}\unicode[STIX]{x27E7}+\left.\unicode[STIX]{x27E6}\frac{p}{\unicode[STIX]{x1D70C}}\right.\unicode[STIX]{x27E7}=\overline{\boldsymbol{u}}\boldsymbol{\cdot }(\unicode[STIX]{x1D738}\times \hat{\boldsymbol{n}}-q\hat{\boldsymbol{n}})+\left.\unicode[STIX]{x27E6}\frac{p}{\unicode[STIX]{x1D70C}}\right.\unicode[STIX]{x27E7}\end{eqnarray}$$

we recombine the tangential and normal components to obtain the evolution equation for the vectorial strength $\unicode[STIX]{x1D736}=\unicode[STIX]{x1D738}+q\hat{\boldsymbol{n}}$ of the vortex-entrainment sheet as:

(2.24) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}_{s}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D736}}{\unicode[STIX]{x2202}t}}=(\unicode[STIX]{x1D70C}_{s}\unicode[STIX]{x1D735}_{s}\unicode[STIX]{x1D707})\times \hat{\boldsymbol{n}}+(\unicode[STIX]{x27E6}\unicode[STIX]{x1D70C}(u_{n}-v_{n})^{2}\unicode[STIX]{x27E7})\hat{\boldsymbol{n}}. & & \displaystyle\end{eqnarray}$$

Evidently the evolution of the sheets are coupled to each other through the quantity $\unicode[STIX]{x1D707}$ , which partly represents the jump in dynamic pressure. For the vortex sheet (2.24) represents the familiar result that tangential pressure gradients generate vorticity components bi-normal to the gradient direction (Lighthill Reference Lighthill1963). For the entrainment sheet the pressure jump acts to ‘push’ or entrain fluid into the sheet.

2.4 Physics of inviscid separation: interpretation of the Kutta condition

We now briefly discuss inviscid flow separation at a sharp edge from a physical point of view. Regardless of the shedding problem or its formulation, the requirement of a bounded flow at the sharp edge should be imposed. This constraint is the Kutta condition, which we note has no fundamental basis. However, from (2.21)–(2.22) we see that the condition of a regular flow is ensured by an appropriate balance of normal momentum with the pressure jump. For the conventional vortex sheet with $\unicode[STIX]{x27E6}u_{n}\unicode[STIX]{x27E7}=\unicode[STIX]{x27E6}p\unicode[STIX]{x27E7}=0$ this is trivially satisfied by the condition of mass conservation, namely $\unicode[STIX]{x1D70C}_{s}=0$ .

Next, consider the inviscid mechanism that eliminates the singular pressure gradient, but not the jump, and allows the flow to separate from the surface. To see how this neutralization of the normal pressure gradient to its jump value relates to vorticity, we again employ the simple device of allowing the fluid to have a singularity in the form of a delta function as was done for the curl and divergence in (2.12). Namely, the singular part of the pressure gradient is $\unicode[STIX]{x1D735}(p/\unicode[STIX]{x1D70C})=\hat{\boldsymbol{n}}\unicode[STIX]{x27E6}p/\unicode[STIX]{x1D70C}\unicode[STIX]{x27E7}\unicode[STIX]{x1D6FF}(n)$ . Since $p$ is not $C^{2}$ at the irregular sharp edge point, then $\unicode[STIX]{x1D735}\times \unicode[STIX]{x1D735}(p/\unicode[STIX]{x1D70C})$ is not necessarily zero there (Greenberg Reference Greenberg1998) and thus contributes to the (singular) vorticity equation for $\unicode[STIX]{x1D74E}=\unicode[STIX]{x1D738}\unicode[STIX]{x1D6FF}(n)$ as:

(2.25) $$\begin{eqnarray}\unicode[STIX]{x1D735}\times \unicode[STIX]{x1D735}\frac{p}{\unicode[STIX]{x1D70C}}=\left({\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}b}}\left.\unicode[STIX]{x27E6}\frac{p}{\unicode[STIX]{x1D70C}}\right.\unicode[STIX]{x27E7}\hat{\boldsymbol{s}}-{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}s}}\left.\unicode[STIX]{x27E6}\frac{p}{\unicode[STIX]{x1D70C}}\right.\unicode[STIX]{x27E7}\hat{\boldsymbol{b}}\right)\unicode[STIX]{x1D6FF}(n),\end{eqnarray}$$

which are the pressure terms on the right-hand side of (2.20). Hence, a non-zero pressure jump at the edge generates new vorticity and will result in a loading there (e.g. see Sears Reference Sears1956). For the conventional vortex sheet $\unicode[STIX]{x27E6}p\unicode[STIX]{x27E7}=0$ and while $\unicode[STIX]{x1D74E}$ is still singular the above pressure terms are in fact zero. Therefore, in this case the vorticity in a vortex sheet simply convects off the edge.

Lastly, an interesting physical interpretation of separation arises when a wall-bounded vortex-entrainment sheet is represented as a dipole sheet or ‘double layer’ (Kellogg Reference Kellogg1929). The singularity in the pressure gradient field at the edge can be represented by adding the potential of a doublet to the vector potential decomposing $\unicode[STIX]{x1D735}p$ (Phillips Reference Phillips1959). Hence, the finite strength of the doublet would be equal to the pressure jump. Then the neutralization of the pressure gradient singularity ‘tears apart’ the double layer and sheds one sheet into the fluid and the other sheet inside the wall as its image.

3.1 Blasius boundary layer with and without suction

Here, we consider the intrinsic flow in the vortex-entrainment sheet representing the Blasius boundary layer with $m=0$ and so $U(x)=U_{o}$ . First the case without suction is discussed. Noting that the problem is two-dimensional, steady and $v_{n}=0$ since the sheet is bound to the stationary plate ( $u_{s}^{-}=u_{n}^{-}=0$ ), then the mass and momentum equations in (2.6)–(2.7) reduce to:

(3.4a-c ) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}s}}(\unicode[STIX]{x1D70C}_{s}v_{s})=\unicode[STIX]{x1D70C}q,\quad \unicode[STIX]{x1D70F}_{w}=\unicode[STIX]{x1D70C}q\overline{u_{s}}+\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6FE}\overline{u_{n}}-{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}s}}(\unicode[STIX]{x1D70C}_{s}v_{s}^{2}),\quad \unicode[STIX]{x27E6}p\unicode[STIX]{x27E7}=2\unicode[STIX]{x1D70C}\overline{u_{n}}q.\end{eqnarray}$$

Note that the sheet coordinate is also the plate coordinate $s=x$ . The sheet strengths $\unicode[STIX]{x1D6FE}$ and $q$ are obtained from (3.1)–(3.2) and we can use the plate boundary conditions to give $\overline{u_{s}}=u_{s}^{+}/2=-\unicode[STIX]{x1D6FE}/2$ and $\overline{u_{n}}=u_{n}^{+}/2=-q/2$ . Since the entrainment strength decays significantly for $x\gg 1$ , the normal momentum equation gives $\unicode[STIX]{x27E6}p\unicode[STIX]{x27E7}\approx 0$ which is the boundary layer assumption that the pressure gradient is zero across the layer, or more precisely $O(Re_{x}^{-1/2})$ . Of course this small pressure jump due to entrainment is balanced by an equal and opposite one on the other side of the plate.

Figure 3.

(a) Numerical coefficients representing quantities at the edge of the Falkner–Skan boundary layers as functions of $m$ ; see text for detail. (b) The wall shear stress $\unicode[STIX]{x1D70F}_{w}$ predicted by the vortex-entrainment sheet model compared with the Blasius solution: $\unicode[STIX]{x1D70F}_{w,B}/(\unicode[STIX]{x1D70C}U_{o}^{2})=0.33/\sqrt{Re_{x}}$ . Inset: the per cent relative error on a logarithmic horizontal scale.

For a time-dependent problem, we must be given an initial condition for $\unicode[STIX]{x1D70C}_{s}$ and $v_{s}$ . In this steady case we can obtain the distribution of $\unicode[STIX]{x1D70C}_{s}$ from (2.5) so that $\unicode[STIX]{x1D70C}_{s}=\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6FF}$ . Integrating the mass equation yields $\unicode[STIX]{x1D70C}_{s}(x)v_{s}(x)$ and subsequently we obtain:

(3.5) $$\begin{eqnarray}\displaystyle v_{s}(x)-v_{s}(0)=\frac{1}{\unicode[STIX]{x1D70C}_{s}(x)}\int _{0}^{x}\unicode[STIX]{x1D70C}q\,\text{d}s & = & \displaystyle \frac{1}{c_{\unicode[STIX]{x1D6FF}}\sqrt{Re_{x}}}\int _{0}^{Re_{x}}q\,\text{d}(Re_{s})\nonumber\\ \displaystyle & = & \displaystyle \frac{2U_{o}(c_{\unicode[STIX]{x1D6FF}}c_{m}-c_{v})}{c_{\unicode[STIX]{x1D6FF}}\sqrt{Re_{x}}}[\sqrt{(c_{\unicode[STIX]{x1D6FF}}c_{m})^{2}+Re_{x}}-c_{\unicode[STIX]{x1D6FF}}c_{m}].\end{eqnarray}$$

Although the imposed boundary condition on the flat plate is $v^{-}=0$ , the edge of the boundary layer also exists on the plate at $x=0$ . Hence, the vertical velocity at the edge, $v^{+}(x,\unicode[STIX]{x1D6FF}(x))=c_{v}U_{o}/\sqrt{Re_{x}}$ , also exists at $x=0$ . This would indicate that $v^{+}(0,0)$ becomes infinite. However, the limit of $v^{+}(0,0)$ should be a finite, non-zero value and with a net flux occurring through the leading edge point. Recognizing that the horizontal velocity $U_{o}$ is also incident to the leading edge we arrive at the following form of a mass balance at this point:

(3.6) $$\begin{eqnarray}v_{s}(0)=U_{o}-v^{+}(0,0).\end{eqnarray}$$

If we take $v^{+}(0,0)=c_{v}U_{o}$ , then we obtain $v_{s}(0)=U_{o}(1-c_{v})$ as a boundary condition. This means the horizontal velocity $U_{o}$ is partly ‘deflected’ or displaced into the vertical velocity $c_{v}U_{o}$ , while the remainder is entrained into the leading edge point.

Figure 3(b) plots the computed wall shear stress $\unicode[STIX]{x1D70F}_{w}$ from the vortex-entrainment sheet model and compares favourably with the Blasius result: $\unicode[STIX]{x1D70F}_{w,B}/(\unicode[STIX]{x1D70C}U_{o}^{2})=0.33/\sqrt{Re_{x}}$ . The stress is still singular at $x=0$ since the tangential velocity discontinuity of $u^{-}-u^{+}=0-U_{o}$ remains. The inset plot shows that the relative error between the two calculations is bounded below 2 % for $Re_{x}>100$ , which is consistent with our assumed region of validity $[Re_{x}\approx 100]\gg [(c_{\unicode[STIX]{x1D6FF}}c_{m})^{2}\approx 6]$ .

Next, we present the case with uniform suction at the plate: $-v_{o}<0$ . The viscous solution obtained from the boundary layer equations is also an exact solution of the full Navier–Stokes equations. This flow field is characterized by $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}x=0$ and so the vertical velocity field is constant and equal to the suction value: $v(x,y)=-v_{o}$ (Schlichting Reference Schlichting1955, pp. 230–235). We can then write ( $u_{s}^{+}$ , $u_{n}^{+}$ ) = ( $U_{o}$ , $-v_{o}$ ) and ( $u_{s}^{-}$ , $u_{n}^{-}$ ) = ( $0$ , $-v_{o}$ ) and the quantities relevant to the vortex-entrainment sheet become:

(3.7a-d ) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}=-U_{o},\quad \overline{u_{s}}=\frac{U_{o}}{2},\quad q=0,\quad \overline{u_{n}}=-v_{o}.\end{eqnarray}$$

The sheet governing equations are again those in (3.4), and since $q=0$ we see that the mass flux along the sheet is constant: $\unicode[STIX]{x1D70C}_{s}v_{s}=const$ . This feature represents the constant displacement thickness associated with the viscous solution. Similarly, the momentum thickness is also constant so that $v_{s}=const.$ as well and substitution of all variables into the tangential sheet momentum equation yields

(3.8) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{w}=\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6FE}\overline{u_{n}}=\unicode[STIX]{x1D70C}v_{o}U_{o},\end{eqnarray}$$

which is exactly the wall shear stress obtained from the viscous solution and is notably independent of the viscosity or $Re$ .

4.1 The Laplace equation

The problem is defined by specifying the governing equation and boundary conditions. The conjugacy of the harmonic potential $\unicode[STIX]{x1D719}$ and the streamfunction $\unicode[STIX]{x1D713}$ allows two equivalent formulations of the problem since both functions satisfy a Laplace equation. Here, we move forward with the $\unicode[STIX]{x1D719}$ framework, but provide a summary showing the duality with the $\unicode[STIX]{x1D713}$ framework. Some useful definitions relating $\unicode[STIX]{x1D6E4}$ and $Q$ to $\unicode[STIX]{x1D6FE}$ and $q$ as well as to $\unicode[STIX]{x1D719}$ and $\unicode[STIX]{x1D713}$ may be found in appendix B.

Figure 4.

(a) Definition of the fluid domain $D$ and its boundary $\unicode[STIX]{x2202}D=S_{i}\cup S_{v}\cup C_{\infty }$ . $S_{i}$ : inner boundary, $S_{v}$ : free sheet boundary, $C_{\infty }$ : outer boundary, $\boldsymbol{x}_{o}$ : sharp edge location. $C_{\infty }$ is arbitrary and is not a surface of discontinuity. (b) The contour integral path $C$ laying entirely in the analytic fluid domain. The path is able to cross itself at the irregular point $\boldsymbol{x}_{o}$ to pass on both sides of $S_{i}$ .

Let the open fluid domain be $D$ with boundary/closure defined as the union $\unicode[STIX]{x2202}D=S_{i}\cup S_{v}\cup C_{\infty }$ (figure 4 a), where $S_{i}$ and $S_{v}$ correspond to the vortex-entrainment sheets representing the surface with a sharp edge and the free sheet shed from the edge, respectively, and $C_{\infty }$ is a large arbitrary boundary ‘at infinity’ that does not represent a surface of discontinuity. Usually, the boundary condition on $S_{i}$ and $S_{v}$ is prescribed as continuity of normal velocity, namely $(\unicode[STIX]{x2202}\unicode[STIX]{x1D719}/\unicode[STIX]{x2202}n)^{+}=(\unicode[STIX]{x2202}\unicode[STIX]{x1D719}/\unicode[STIX]{x2202}n)^{-}$ . Under this condition no entrainment occurs, and for a solid surface the normal velocity of that surface, $U_{n}=(\unicode[STIX]{x2202}\unicode[STIX]{x1D719}/\unicode[STIX]{x2202}n)^{\pm }$ , is specified.

However, the vortex-entrainment sheet is characterized by jumps in both the tangential and normal velocity components across the sheet. Hence, the values of $(\unicode[STIX]{x2202}\unicode[STIX]{x1D719}/\unicode[STIX]{x2202}n)^{\pm }$ and thus the entrainment rate $q$ are assumed as given boundary conditions (however, see end of § 4.3) . The special conditions for the sharp edge will be discussed in § 5. With a fluid of infinite extent, the specification of the flow there serves as a boundary condition on $C_{\infty }$ , which may be expressed using an asymptotic form of the potential (Batchelor Reference Batchelor1967, chap. 2.10). The statement of the problem thus far is:

(4.1) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D719}=0\quad \boldsymbol{x}\in D & \displaystyle\end{eqnarray}$$

(4.2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}(\boldsymbol{x})-\unicode[STIX]{x1D719}_{\infty }(\boldsymbol{x})\rightarrow C_{\unicode[STIX]{x1D719}}+\frac{\unicode[STIX]{x1D6E4}_{\infty }\unicode[STIX]{x1D703}}{2\unicode[STIX]{x03C0}}-\frac{Q_{\infty }}{2\unicode[STIX]{x03C0}}\log (r)+O(r^{-1})\quad \boldsymbol{x}\in C_{\infty } & \displaystyle\end{eqnarray}$$

(4.3) $$\begin{eqnarray}\displaystyle & \displaystyle \left.{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}n}}\right|^{\pm }=u_{n}^{\pm }(\boldsymbol{x})\quad \boldsymbol{x}\in S_{i}\cup S_{v}, & \displaystyle\end{eqnarray}$$

where $C_{\unicode[STIX]{x1D719}}$ is a constant, $\unicode[STIX]{x1D719}_{\infty }$ is an external potential representing a non-zero flow at infinity, $\unicode[STIX]{x1D6E4}_{\infty }$ and $Q_{\infty }$ are, respectively, the net circulation along $C_{\infty }$ and the net flux across $C_{\infty }$ into $D$ . In the case of a semi-infinite geometry $S_{i}$ intersects $C_{\infty }$ at infinity where the potential asymptotically becomes compatible with (4.2).

Now consider a contour $C$ residing entirely within the analytic fluid region and bounding the open domain $D$ as defined in figure 4(b). By mass conservation the total flux $Q_{C}$ across $C$ must be zero. This yields a ‘compatibility condition’ required for a solution $\unicode[STIX]{x1D719}$ to this Neumann problem to exist (e.g. Stakgold Reference Stakgold1968). Similarly, by circulation conservation (Kelvin’s theorem) the total circulation $\unicode[STIX]{x1D6E4}_{C}$ around $C$ must also be zero. Since $C=\unicode[STIX]{x2202}D$ , then with $\boldsymbol{u}=\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}$ these conditions can be written as:

(4.4a,b ) $$\begin{eqnarray}Q_{C}=-\oint _{C}\boldsymbol{u}\boldsymbol{\cdot }\hat{\boldsymbol{n}}\text{d}s=-\oint _{C}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}n}}\text{d}s=0,\quad \unicode[STIX]{x1D6E4}_{C}=\oint _{C}\boldsymbol{u}\boldsymbol{\cdot }\hat{\boldsymbol{s}}\text{d}s=\oint _{C}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}s}}\text{d}s=0,\end{eqnarray}$$

where $s$ - $n$ is an orthogonal coordinate system along $C$ . The $Q_{C}$ equation can be derived from the Laplace equation itself upon using the divergence theorem. In general, the integration path is allowed to cross itself at the irregular point $\boldsymbol{x}_{o}$ in order to trace both sides of $S_{i}$ . When $S_{i}$ represents a solid, no-slip surface this is not necessary because $\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}=\boldsymbol{U}$ on the inside of $S_{i}$ . Since $\unicode[STIX]{x2202}\unicode[STIX]{x1D719}/\unicode[STIX]{x2202}n$ is given everywhere on the boundary, then $\boldsymbol{u}(\boldsymbol{x})=\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}$ is uniquely determined throughout the domain $D$ .

Table 1.

Conjugate frameworks for potential flow governed by Laplace equations for the harmonic potential $\unicode[STIX]{x1D719}$ and the streamfunction $\unicode[STIX]{x1D713}$ .

Lastly, in the $\unicode[STIX]{x1D713}$ problem formulation the normal derivative $\unicode[STIX]{x2202}\unicode[STIX]{x1D713}/\unicode[STIX]{x2202}n$ is given everywhere on the boundary and corresponds to the tangential velocity component. The conditions in (4.4) can be written with derivatives of $\unicode[STIX]{x1D713}$ via the Cauchy–Riemann relations. Table 1 summarizes the two different frameworks, which are essentially two different, yet unique decompositions (Bhatia et al. Reference Bhatia, Norgard, Pascucci and Bremer2013).

4.2 Boundary integral formulation

Now we write the outer flow $\boldsymbol{u}$ as a complex conjugate velocity field $w=u-\text{i}v$ and the contribution from the vortex-entrainment sheet is given by a Cauchy-type boundary integral. Jones (Reference Jones2003) obtained an elegant solution in this manner for the case of a moving flat plate with no entrainment by representing both the free and plate-bound vorticity as conventional vortex sheets. Namely, the two vortex sheets on each side of the plate were combined into a single sheet by coupling the plate and fluid velocities via a tangential boundary condition representing no slip.

First, the velocity induced by the vortex-entrainment sheet in the two-dimensional version of (2.15) is expressed in a convenient complex form. Namely, $\boldsymbol{x}$ is replaced with $z=x+\text{i}y$ and the sheet position $\boldsymbol{x}_{s}$ with $\unicode[STIX]{x1D701}=x_{s}(s)+\text{i}y_{s}(s)$ where $s$ is the arclength coordinate. Introducing $\unicode[STIX]{x1D703}(\unicode[STIX]{x1D701})$ as the angle measured from the horizontal to the local unit tangent vector on the sheet we have $\unicode[STIX]{x2202}\unicode[STIX]{x1D701}/\unicode[STIX]{x2202}s=\text{e}^{\text{i}\unicode[STIX]{x1D703}(s)}$ . The conjugate velocity $w(z)=u-iv$ at an analytic fluid point can then be written as:

(4.5) $$\begin{eqnarray}\displaystyle w(z)-w_{\infty }(z)=\frac{1}{2\unicode[STIX]{x03C0}\text{i}}\int _{S}\frac{\unicode[STIX]{x1D6FE}(s)-\text{i}q(s)}{z-\unicode[STIX]{x1D701}(s)}\,\text{d}s=\frac{1}{2\unicode[STIX]{x03C0}\text{i}}\int _{S}\frac{\unicode[STIX]{x1D712}(\unicode[STIX]{x1D701})}{z-\unicode[STIX]{x1D701}}\,\text{d}\unicode[STIX]{x1D701}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D701})=(\unicode[STIX]{x1D6FE}-\text{i}q)\text{e}^{-\text{i}\unicode[STIX]{x1D703}}$ is the complex sheet strength and the integration is over all sheets: $S=S_{i}\cup S_{v}$ . Also, $w_{\infty }(z)=\text{d}\unicode[STIX]{x1D6F7}_{\infty }/\text{d}z$ is the velocity of an external flow that need not decay at infinity. When $z\rightarrow \unicode[STIX]{x1D701}$ this singular integral is a generalized Birkhoff–Rott equation for the vortex-entrainment sheet with complex strength. Following Muskhelishvili (Reference Muskhelishvili1946) the left and right sides of the sheet are relative to an observer traversing the sheet in the positive direction of integration in (4.5) and the limits of any quantity approached from the left and right will be denoted with ( $+$ ) and ( $-$ ) superscripts, respectively. For any point on a sheet besides the sharp edge at $\unicode[STIX]{x1D701}_{o}$ we have the following by the smooth-arc Plemelj formulae (see comment in appendix C):

(4.6) $$\begin{eqnarray}\left.\begin{array}{@{}l@{}}\unicode[STIX]{x1D712}(\unicode[STIX]{x1D701})=(w^{-}-w^{+})=(\unicode[STIX]{x1D6FE}-\text{i}q)\text{e}^{-\text{i}\unicode[STIX]{x1D703}}\\ w(\unicode[STIX]{x1D701})={\textstyle \frac{1}{2}}(w^{+}+w^{-})=(\overline{u_{s}}-\text{i}\overline{u_{n}})\text{e}^{-\text{i}\unicode[STIX]{x1D703}}\end{array}\right\}\quad \text{for }\unicode[STIX]{x1D701}\in S_{i}\cup S_{v}\backslash \unicode[STIX]{x1D701}_{o},\end{eqnarray}$$

where $\overline{u_{s}}$ and $\overline{u_{n}}$ are the averages of the fluid velocity components. With this formulation it is easy to see how the sheet strength $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D701})$ relates to the jump in the complex potential $\unicode[STIX]{x1D6F7}=\unicode[STIX]{x1D719}+\text{i}\unicode[STIX]{x1D713}$ across the sheet (see appendix B):

(4.7) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F7}^{-}-\unicode[STIX]{x1D6F7}^{+}=(\unicode[STIX]{x1D719}^{-}-\unicode[STIX]{x1D719}^{+})+\text{i}(\unicode[STIX]{x1D713}^{-}-\unicode[STIX]{x1D713}^{+})=\unicode[STIX]{x1D6E4}-\text{i}Q & \displaystyle\end{eqnarray}$$

(4.8) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}s}}(\unicode[STIX]{x1D6F7}^{-}-\unicode[STIX]{x1D6F7}^{+})=(u_{s}^{-}-u_{s}^{+})-\text{i}(u_{n}^{-}-u_{n}^{+})=\unicode[STIX]{x1D6FE}-\text{i}q. & \displaystyle\end{eqnarray}$$

Note that $\unicode[STIX]{x2202}\unicode[STIX]{x27E6}\unicode[STIX]{x1D6F7}\unicode[STIX]{x27E7}/\unicode[STIX]{x2202}n=\text{i}(\unicode[STIX]{x2202}\unicode[STIX]{x27E6}\unicode[STIX]{x1D6F7}\unicode[STIX]{x27E7}/\unicode[STIX]{x2202}s)$ , again highlighting the conjugacy of the problem.

Now, the normal boundary condition (4.3) on a sheet $S$ can be expressed by:

(4.9) $$\begin{eqnarray}\text{Re}\left\{{\textstyle \frac{1}{2}}(w^{+}+w^{-})\text{i}\text{e}^{\text{i}\unicode[STIX]{x1D703}}\right\}={\textstyle \frac{1}{2}}(u_{n}^{+}+u_{n}^{-})=\overline{u_{n}},\end{eqnarray}$$

where each of the functions is evaluated at a position $\unicode[STIX]{x1D701}$ on the sheet. Since we have supposed the normal velocities $u_{n}^{\pm }$ to be given, and thus the entrainment strength $q$ to be known, the boundary condition can be rearranged as:

(4.10) $$\begin{eqnarray}\displaystyle f(\unicode[STIX]{x1D701})\equiv \text{Re}\left\{\frac{\text{e}^{\text{i}\unicode[STIX]{x1D703}(\unicode[STIX]{x1D701})}}{2\unicode[STIX]{x03C0}}\int _{S}\frac{\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D709})\text{e}^{-\text{i}\unicode[STIX]{x1D703}(\unicode[STIX]{x1D709})}\,\text{d}\unicode[STIX]{x1D709}}{\unicode[STIX]{x1D701}-\unicode[STIX]{x1D709}}\right\} & = & \displaystyle \overline{u_{n}}(\unicode[STIX]{x1D701})-w_{\infty ,n}(\unicode[STIX]{x1D701})\nonumber\\ \displaystyle & & \displaystyle +\,\text{Re}\left\{\frac{\text{e}^{\text{i}\unicode[STIX]{x1D703}(\unicode[STIX]{x1D701})}}{2\unicode[STIX]{x03C0}}\int _{S}\frac{\text{i}q(\unicode[STIX]{x1D709})\text{e}^{-\text{i}\unicode[STIX]{x1D703}(\unicode[STIX]{x1D709})}\,\text{d}\unicode[STIX]{x1D709}}{\unicode[STIX]{x1D701}-\unicode[STIX]{x1D709}}\right\},\qquad\end{eqnarray}$$

where $\unicode[STIX]{x1D709}\in \mathbb{C}$ is a dummy integration variable along the sheet. This equation is to be solved for $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D701})=\unicode[STIX]{x1D6FE}(s)$ . The inversion formula and subsequent manipulations used by Jones (Reference Jones2003) are specific to the case of a plane boundary with $\unicode[STIX]{x1D703}(\unicode[STIX]{x1D701})$ constant along the sheet. As such, equation (4.10) is more amenable to numerical solution via, for example, expansion of $f(\unicode[STIX]{x1D701})$ as a Chebyshev series, which was also done by Jones as well as others since the series converges rapidly for smooth functions (Alben Reference Alben2008).

4.3 Solution algorithm

We now present a brief solution algorithm for the coupled system of the vortex-entrainment sheet and the outer fluid. First, the outer flow $\boldsymbol{u}$ is solved according to the specified normal boundary conditions with either the Laplace formulation (4.1)–(4.3) or the boundary integral formulation (4.5)–(4.10). For a free sheet we assume that it acquires circulation only as a result of having been shed from a body. Hence, $\unicode[STIX]{x1D6FE}$ and $q$ will be known for these sheets. For a surface-bound sheet, the solution of the Laplace formulation will yield the velocity on the fluid side of the sheet, say $\boldsymbol{u}^{+}$ . Then, the additional boundary conditions on the surface side of the sheet are imposed to yield the sheet strengths. More specifically, we have $\boldsymbol{u}^{-}=\boldsymbol{U}$ with $\boldsymbol{U}$ as the known surface velocity and thus $\boldsymbol{u}^{-}-\boldsymbol{u}^{+}=\unicode[STIX]{x1D6FE}\hat{\boldsymbol{s}}+q\hat{\boldsymbol{n}}$ . The imposed conditions will typically be the no-slip and no-penetration conditions for a solid body, but in general we may model actual fluid slip on or porosity of the surface.

When the boundary integral formulation is used the specification of $u_{n}^{-}=U_{n}$ on the surface side of the sheet will uniquely solve the flow inside that surface. Then the solution for $\unicode[STIX]{x1D6FE}$ will correspond to the $u_{s}^{-}-u_{s}^{+}$ of these two flows. To obtain the desired vortex sheet strength, the tangential boundary condition can be imposed by using the relative velocity $(\overline{u_{s}}-U_{s})$ in the evolution equation (2.24) as was done by Jones (Reference Jones2003). For a solid body, the harmonic potential $\unicode[STIX]{x1D719}_{b}$ inside the body will be such that $\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}_{b}=\boldsymbol{U}$ ; this is most easily seen in the reference frame of the body or for a stationary body where $\unicode[STIX]{x1D719}_{b}=\text{const.}$ and thus the flow inside vanishes (Lamb Reference Lamb1945).

Equipped with $\boldsymbol{u}$ , $\unicode[STIX]{x1D6FE}$ and $q$ the solution for the sheet flow quantities can be obtained. The sheet mass and momentum equations (2.6)–(2.7) along with the evolution equation (2.24) represent 4 equations for the 4 unknowns $\unicode[STIX]{x1D70C}_{s}$ , $\boldsymbol{v}=(v_{s},v_{n})$ and $\unicode[STIX]{x27E6}p\unicode[STIX]{x27E7}$ ; in three dimensions the system is augmented by the third component $v_{b}$ and the corresponding vortex sheet strength component $\unicode[STIX]{x1D6FE}_{s}$ . Again, these equations are coupled through the pressure jump and the entrainment strength. An initial condition at $t_{0}$ is required for the intrinsic flow quantities $\unicode[STIX]{x1D70C}_{s}$ and $\boldsymbol{v}=\boldsymbol{w}+v_{n}\hat{\boldsymbol{n}}$ . In most cases, such as a flow started from rest, these initial conditions will simply be $\unicode[STIX]{x1D70C}_{s}(\boldsymbol{x}_{s},t_{0})=\boldsymbol{w}(\boldsymbol{x}_{s},t_{0})=0$ since no entrainment will have occurred yet anywhere in the domain.

Lastly, we note that it is theoretically possible to obtain both the vortex and entrainment sheet strengths if supplemental conditions are given elsewhere in the domain. In particular, if we are given the full velocity vector on $C_{\infty }$ ‘at infinity,’ recall figure 4(a), then we may remove the requirement of a priori specification of the normal boundary conditions on the sheets. By (4.2) and (4.5) we obtain:

(4.11) $$\begin{eqnarray}w(z)-w_{\infty }(z)=\left(\frac{\unicode[STIX]{x1D6E4}_{\infty }-\text{i}Q_{\infty }}{2\unicode[STIX]{x03C0}\text{i}R}\right)\text{e}^{-\text{i}\unicode[STIX]{x1D703}(z)}=\frac{1}{2\unicode[STIX]{x03C0}\text{i}}\int _{S}\frac{\unicode[STIX]{x1D6FE}(s)-\text{i}q(s)}{z-\unicode[STIX]{x1D701}(s)}\,\text{d}s\end{eqnarray}$$

for $|z|=R\rightarrow \infty$ . The above integral is non-singular since $z\notin S$ and we note that by writing $z-\unicode[STIX]{x1D701}(s)\approx R\text{e}^{\text{i}\unicode[STIX]{x1D703}(z)}$ then:

(4.12) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{\infty }-\text{i}Q_{\infty }=\int _{S}[\unicode[STIX]{x1D6FE}(s)-\text{i}q(s)]\,\text{d}s,\end{eqnarray}$$

which states that the circulation and flux in the sheets are equal to those at infinity or that the total of these quantities in the domain $D$ is zero; recall (4.4). This condition makes the inversion formula of the singular integral equation in (4.10) unique (Jones Reference Jones2003). The boundary conditions at infinity, for example by specification of $\unicode[STIX]{x1D6E4}_{\infty }$ and $Q_{\infty }$ on $C_{\infty }$ would then determine the sheet strengths. However, the determination of $\unicode[STIX]{x1D6FE}-\text{i}q$ from (4.11) is the solution to a class of inverse problems, which are usually ill-conditioned. For this reason we do not pursue this option further.

Figure 5.

The variation of the surface-normal vector for a rounded corner of radius $\unicode[STIX]{x1D700}\ll 1$ where the black dot represents the location of an irregular point. (a) Irregularity inside the surface, (b) irregularity on a smooth boundary and (c) irregularity on a sharp boundary as $\unicode[STIX]{x1D700}\rightarrow 0$ .

5.1 Discussion on special cases of entrainment

In this subsection we discuss some sets of simplified circumstances pertaining to separation at a sharp edge that are of practical interest. When $\unicode[STIX]{x1D6FD}=0$ then (5.6) simply gives $\unicode[STIX]{x1D703}_{v}=0$ irrespective of the intrinsic flow $\boldsymbol{v}$ and thus $\unicode[STIX]{x27E6}p\unicode[STIX]{x27E7}$ . Of course this corresponds to the usual tangential shedding result for a cusped edge.

Now consider zero entrainment everywhere so that there is no intrinsic flow $\boldsymbol{v}=0$ or mass $\unicode[STIX]{x1D70C}_{s}=0$ in the sheets, namely the conventional vortex sheet. Equation (5.6) then becomes indeterminate and a new condition for the shedding angle is required. This is obtained from (5.3), which with each $q_{j}=0$ reduces to:

(5.8) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{1}\sin (\unicode[STIX]{x1D6FD}\unicode[STIX]{x03C0}/2+\unicode[STIX]{x1D703}_{v})=\unicode[STIX]{x1D6FE}_{2}\sin (\unicode[STIX]{x1D6FD}\unicode[STIX]{x03C0}/2-\unicode[STIX]{x1D703}_{v}).\end{eqnarray}$$

There is the trivial solution with $\unicode[STIX]{x1D6FE}_{1}=\unicode[STIX]{x1D6FE}_{2}=0$ meaning no flow at the edge. The only other solutions that put $\unicode[STIX]{x1D703}_{v}$ in a physical range are $\unicode[STIX]{x1D703}_{v}=\pm \unicode[STIX]{x1D6FD}\unicode[STIX]{x03C0}/2$ and with the respective consequences of $\unicode[STIX]{x1D6FE}_{1}=0$ or $\unicode[STIX]{x1D6FE}_{2}=0$ . This corresponds to a tangential shedding angle and that the flow is stagnated on one side of the vortex sheet depending on the sign of shed circulation. Hence, as mentioned in § 2.4 the vortex sheet from one wedge face convects off the surface with no contribution from the opposing face. These features are consistent with the results of Pullin (Reference Pullin1978) in which there was zero entrainment.

Lastly, consider the onset of motion at $t=0$ when no vorticity has been shed from the sharp edge and $S_{v}$ does not yet exist. As used by Rott (Reference Rott1956) and Pullin (Reference Pullin1978) the attached potential flow responsible for separation is obtained as the leading-order term in an expansion of the complex potential near the sharp edge: $\unicode[STIX]{x1D6F7}\sim (z-\unicode[STIX]{x1D701}_{o})^{n}$ where $n=1/(2-\unicode[STIX]{x1D6FD})$ for the wedge of angle $\unicode[STIX]{x1D6FD}\unicode[STIX]{x03C0}$ . The imposed pressure gradient at infinity that drives this flow suggests the windward wedge face will ‘feel’ the pressure at $t=0^{+}$ before the leeward face. Hence, we would expect more significant entrainment to occur on the windward face, say $S_{2}$ , so that $v_{s,2}\gg v_{s,1}$ and (5.6) gives $A\approx 1$ and $\unicode[STIX]{x1D703}_{v}\approx \unicode[STIX]{x1D6FD}\unicode[STIX]{x03C0}/2$ . This is again consistent with the small-time similarity solution. In a real flow the ensuing roll-up would be sure to quickly induce a pressure and so entrainment on the leeward face thus changing $\unicode[STIX]{x1D703}_{v}$ .

6.1 Set-up of computed simulations

First we present the problem set-up for a case with zero entrainment that is given in § 6.2. A case with non-zero entrainment is then considered in § 6.3 and the required modifications to the problem setup are also discussed there. The geometry of each problem is a wedge of infinite extent.

We use a combination of the Laplace and boundary integral formulations. The wedge is stationary and a boundary condition is imposed at infinity to drive the flow in the form of a known harmonic potential function $\unicode[STIX]{x1D719}_{\infty }(r,\unicode[STIX]{x1D703},t)=\text{Re}\{\unicode[STIX]{x1D6F7}_{\infty }(z,t)\}$ that represents the attached flow. Since there is no entrainment $S_{1}$ and $S_{2}$ have no-penetration conditions and $S_{v}$ has a continuity of normal velocity condition. Moreover, by construction the potential $\unicode[STIX]{x1D719}_{\infty }$ satisfies homogeneous Neumann boundary conditions on the wedge faces. Hence, after each new free vortex sheet segment is shed, we need only to account for the normal velocity induced by $S_{v}$ on the walls. We can then arrange the problem for a potential $\unicode[STIX]{x1D719}_{w}$ , which represents the image system of the free vortex sheet inside the wedge. This $\unicode[STIX]{x1D719}_{w}$ satisfies:

(6.2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D719}_{w}=0\quad \text{for }r>0,\quad \unicode[STIX]{x1D703}\in [-\unicode[STIX]{x1D703}_{w},\unicode[STIX]{x1D703}_{w}] & \displaystyle\end{eqnarray}$$

(6.3) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{1}{r}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{w}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}}=-\text{Re}\{\text{i}\text{e}^{\text{i}\unicode[STIX]{x1D703}}w_{v}(r,\unicode[STIX]{x1D703})\}\quad \text{for }r>0,\quad \unicode[STIX]{x1D703}=\pm \unicode[STIX]{x1D703}_{w} & \displaystyle\end{eqnarray}$$

(6.4) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}_{w}\rightarrow 0\quad \text{for }r\rightarrow \infty ,\quad \unicode[STIX]{x1D703}\in [-\unicode[STIX]{x1D703}_{w},\unicode[STIX]{x1D703}_{w}], & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D703}_{w}=\unicode[STIX]{x03C0}(1-\unicode[STIX]{x1D6FD}/2)$ and ( $\pm$ ) corresponds to the angular coordinates of $S_{1}$ and $S_{2}$ , and so $w_{v}(r,\pm \unicode[STIX]{x1D703}_{w})$ is the velocity induced by $S_{v}$ on the wedge faces. For brevity we will write $\unicode[STIX]{x1D701}_{w}=r\text{e}^{\pm \text{i}\unicode[STIX]{x1D703}_{w}}$ for the coordinate on either $S_{1}$ or $S_{2}$ .

In § 5.1 we saw that for zero entrainment the shedding angle equation (5.6) becomes indeterminate and $\unicode[STIX]{x1D703}_{v}$ is instead obtained by (5.3) with $q_{v}=q_{1}=q_{2}=0$ . Specifically $\unicode[STIX]{x1D703}_{v}$ is necessarily tangent to one wedge face and the flow is stagnated at the apex on the other face. This second condition could be imposed by $\unicode[STIX]{x2202}\unicode[STIX]{x1D719}/\unicode[STIX]{x2202}r=0$ on the $S_{1}$ side where $\unicode[STIX]{x1D719}$ is the total potential. However, it is automatically satisfied by imposing $\unicode[STIX]{x1D703}_{v}\equiv \unicode[STIX]{x1D6FD}\unicode[STIX]{x03C0}/2$ . We note that this is a result of having to impose the no through-flow condition on both $S_{1}$ and $S_{2}$ as well as the continuity of normal velocity on each side of $S_{v}$ , all of which exist at the same location. As a consequence of all these constraints, the strength of the new vortex sheet segment of $S_{v}$ is $\unicode[STIX]{x1D6FE}_{v}=\unicode[STIX]{x1D6FE}_{2}$ by (5.2), which represents $S_{2}$ convecting off to form $S_{v}$ .

The calculation procedure is as follows. Given at time $t_{0}=0$ is the vortex sheet strength $\unicode[STIX]{x1D6FE}_{w}$ on $S_{1}$ and $S_{2}$ from $\unicode[STIX]{x1D719}_{\infty }(\unicode[STIX]{x1D701}_{w},t_{0})$ and an initialized position $\unicode[STIX]{x1D701}_{v}$ and vortex sheet strength $\unicode[STIX]{x1D6FE}_{v}$ of the free sheet $S_{v}$ . The flow is initialized by representing the first shed segment of $S_{v}$ as a point vortex as was similarly done by Jones (Reference Jones2003). Then, for each $k\geqslant 1$ time step:

1. (i) Compute new velocity on wedge faces due to $S_{v}$ as

   (6.5) $$\begin{eqnarray}w_{v}(\unicode[STIX]{x1D701}_{w})=\frac{1}{2\unicode[STIX]{x03C0}\text{i}}\int _{S_{v}}\frac{\unicode[STIX]{x1D6FE}_{v}(s)\,\text{d}s}{\unicode[STIX]{x1D701}_{w}-\unicode[STIX]{x1D701}_{v}(s)}.\end{eqnarray}$$

2. (ii) Solve the Laplace equation for $\unicode[STIX]{x1D719}_{w}$ in (6.2) with boundary conditions (6.3)–(6.4).

3. (iii) Compute new $\unicode[STIX]{x1D6FE}_{w}$ along $S_{1}$ and $S_{2}$ , i.e. $\unicode[STIX]{x1D6FE}_{1}(r)$ and $\unicode[STIX]{x1D6FE}_{2}(r)$ , as:

   (6.6) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{w}(\unicode[STIX]{x1D701}_{w})={\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}}(\unicode[STIX]{x1D719}_{w}(\unicode[STIX]{x1D701}_{w})+\unicode[STIX]{x1D719}_{\infty }(\unicode[STIX]{x1D701}_{w}))+\text{Re}\{w_{v}(\unicode[STIX]{x1D701}_{w})\text{e}^{\text{i}\unicode[STIX]{x1D703}(\unicode[STIX]{x1D701}_{w})}\}.\end{eqnarray}$$

4. (iv) Set $\unicode[STIX]{x1D6FE}_{v}=\unicode[STIX]{x1D6FE}_{2}(r=0)$ and $\unicode[STIX]{x1D703}_{v}=\unicode[STIX]{x1D6FD}\unicode[STIX]{x03C0}/2$ of new sheet segment of $S_{v}$ .

5. (v) Compute new total induced velocity on free sheet $S_{v}$ as

   (6.7) $$\begin{eqnarray}w(\unicode[STIX]{x1D701}_{v})={\displaystyle \frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D701}_{v}}}{\unicode[STIX]{x2202}t}}=\frac{\text{d}\unicode[STIX]{x1D6F7}_{\infty }}{\text{d}z}+\frac{1}{2\unicode[STIX]{x03C0}\text{i}}\int _{S_{v}}\frac{\unicode[STIX]{x1D6FE}_{v}(s)\,\text{d}s}{\unicode[STIX]{x1D701}_{v}-\unicode[STIX]{x1D709}_{v}(s)}+\frac{1}{2\unicode[STIX]{x03C0}\text{i}}\int _{S_{1}\cup S_{2}}\frac{\unicode[STIX]{x1D6FE}_{w}(s)\,\text{d}s}{\unicode[STIX]{x1D701}_{v}-\unicode[STIX]{x1D701}_{w}(s)}.\end{eqnarray}$$

6. (vi) Time integrate $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D701}_{v}}/\unicode[STIX]{x2202}t$ to advect free sheet $S_{v}$ to $t_{k+1}$ .

In steps (v) and (vi) $\overline{\unicode[STIX]{x1D701}}_{v}$ is the complex conjugate of the sheet position. Due to the simple geometry we use the method of images that maps the domain outside the wedge to a semi-infinite plane as $z^{\ast }=z^{n}$ . This combines steps (i)–(iii) to obtain $\unicode[STIX]{x1D6FE}_{w}$ directly. The integrals may be computed with a discrete sheet method or as a system of point vortices. Lastly, the free sheet $S_{v}$ is advected with a fourth-order Runge–Kutta scheme.

Figure 7.

(a) Comparison of streamlines (solid) and vortex sheet position (dashed) for the case with $m=0$ and $\unicode[STIX]{x1D6FD}\unicode[STIX]{x03C0}=\unicode[STIX]{x03C0}/2$ ( $=90^{\circ }$ ) at $t=1$ scaled to the similarity space $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D709}+\text{i}\unicode[STIX]{x1D702}$ . (b) The total and rate of shed circulation as a function of time. The inset shows the radial velocity along the upper wall, $u_{r}$ , as a function of distance from the apex normalized by the separation bubble length $L_{b}$ . The flow is stagnated at the apex and the reattachment point $r=L_{b}$ with reversed flow in the separation bubble.

6.2 Starting flow past an infinite wedge

As validation we first reproduce a result from the similarity solutions of Pullin (Reference Pullin1978) for a starting flow over an infinite wedge of interior angle $\unicode[STIX]{x1D6FD}\unicode[STIX]{x03C0}$ . There is zero entrainment everywhere for all time and the attached flow complex potential is $\unicode[STIX]{x1D6F7}_{\infty }(z,t)=-\text{i}at^{m}z^{n}$ where $n=1/(2-\unicode[STIX]{x1D6FD})$ . We simulate a time-dependent flow beginning from $t=0$ .

Figure 7(a) shows a comparison of the streamlines and vortex sheet location for the case with $m=0$ and $\unicode[STIX]{x1D6FD}\unicode[STIX]{x03C0}=\unicode[STIX]{x03C0}/2$ . The computed simulation is at $t=1$ and is scaled to the similarity space $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D709}+\text{i}\unicode[STIX]{x1D702}$ (for more detail see Pullin Reference Pullin1978). There is good agreement between the visual character of the flow field. For a more quantitative comparison figure 7(b) plots the total and rate of shed circulation, again showing excellent agreement with the similarity laws given by Pullin. The inset shows the velocity along the upper wedge surface (i.e. $S_{1}$ ), which stagnates at the leeward side of the apex as expected from the boundary condition for the case of zero entrainment.

6.3 Oscillating flow with estimated entrainment

Technically the entrainment strengths are to be given as boundary conditions. However, we may estimate the entrainment $q_{v}$ of the free sheet by relaxing the condition of stagnated flow on the leeward side of the apex. In other words, a non-zero wall-tangent velocity is allowed just outside both $S_{1}$ and $S_{2}$ at the apex: $\unicode[STIX]{x1D6FE}_{1}\neq 0$ and $\unicode[STIX]{x1D6FE}_{2}\neq 0$ . Each of these velocities then has a normal component relative to the free sheet and must merge to form $S_{v}$ . Hence, the merging of the $S_{1}$ and $S_{2}$ sheets is a form of entrainment. This means a non-tangential shedding angle is possible and this is determined next.

These wall-tangent flows are in the outer potential flow $\boldsymbol{u}$ just upstream of the sharp edge point and thus must be entrained through the sharp edge to become the intrinsic flow $\boldsymbol{v}$ . Therefore, since there cannot be flow through $S_{v}$ the normal impulse relative to $S_{v}$ of these incoming flows must be equal just as was the case for (5.5). Hence, we obtain an analogous equation for the shedding angle:

(6.8a,b ) $$\begin{eqnarray}\tan \unicode[STIX]{x1D703}_{v}=B\tan (\unicode[STIX]{x1D6FD}\unicode[STIX]{x03C0}/2)\quad B=\left(\frac{u_{s,2}^{-}-u_{s,1}^{+}}{u_{s,2}^{-}+u_{s,1}^{+}}\right).\end{eqnarray}$$

Although we have assumed $\unicode[STIX]{x27E6}\unicode[STIX]{x1D70C}\unicode[STIX]{x27E7}=0$ we could accommodate a stratified flow by the substitution $u^{\pm }\rightarrow \unicode[STIX]{x1D70C}^{\pm }u^{\pm }$ . The above equation is equivalent to the one used by Xia & Mohseni (Reference Xia and Mohseni2017); the two relations are related by the trigonometric identity $\arctan (x)=\arccos (1/\sqrt{1+x^{2}})$ . However, there it was not recognized that this necessarily requires entrainment into the shed sheet.

Since $\unicode[STIX]{x1D703}_{v}$ is now determined by (6.8) instead of prescribed as $\unicode[STIX]{x1D6FD}\unicode[STIX]{x03C0}/2$ , then step (iv) of the procedure given in § 6.2 must be updated as:

(iv) $^{\prime }$ Compute $\unicode[STIX]{x1D6FE}_{v}$ and $q_{v}$ from (5.2) and (5.3) from $\unicode[STIX]{x1D6FE}_{1}$ , $\unicode[STIX]{x1D6FE}_{2}$ , $\unicode[STIX]{x1D703}_{v}$ with $q_{1}=q_{2}=0$ .

In the same way that a segment of vortex sheet must be of finite length to possess circulation, $\unicode[STIX]{x1D6E4}=\int \unicode[STIX]{x1D6FE}\,\text{d}s$ , so too must an entrainment sheet segment have a finite length to have an entrainment rate, $Q=\int q\,\text{d}s$ . By confining the entrainment over some small sheet segment $\unicode[STIX]{x0394}s$ to the point at the sharp edge means we are essentially approximating a net entrainment rate $Q$ as

(6.9) $$\begin{eqnarray}Q=\int _{\unicode[STIX]{x0394}s}q_{v}\,\text{d}s\approx q_{v}\unicode[STIX]{x0394}s.\end{eqnarray}$$

Then the amount of mass of fluid with density $\unicode[STIX]{x1D70C}$ that is entrained is $\unicode[STIX]{x0394}m\approx \unicode[STIX]{x1D70C}Q\unicode[STIX]{x0394}t$ . Hence, the sheet mass density $\unicode[STIX]{x1D70C}_{s,v}=\unicode[STIX]{x0394}m/\unicode[STIX]{x0394}s$ put into $S_{v}$ in a given amount of time is:

(6.10) $$\begin{eqnarray}\unicode[STIX]{x1D70C}_{s,v}\approx \unicode[STIX]{x1D70C}q_{v}\unicode[STIX]{x0394}t,\end{eqnarray}$$

which is equivalent to integrating the mass equation in (6.1) over $\unicode[STIX]{x0394}t$ . Therefore, with a constant $\unicode[STIX]{x0394}t$ and $\unicode[STIX]{x1D70C}$ the behaviours of $\unicode[STIX]{x1D70C}_{s,v}$ and $q_{v}$ are synonymous with the understanding that $q_{v}$ exists only at the edge and puts $\unicode[STIX]{x1D70C}_{s,v}$ in the sheet which then convects downstream. In other words, the time trace of $q_{v}$ is analogous to the trace of $\unicode[STIX]{x1D70C}_{s,v}$ along the sheet arclength at a given time.

Having physically explained the estimated entrainment through the sharp edge, we now compute a solution that is similar to the wake created by the trailing edge of an oscillating airfoil. The driving potential flow is given by

(6.11) $$\begin{eqnarray}\unicode[STIX]{x1D6F7}(z,t)=A_{1}(t)(z-\unicode[STIX]{x1D701}_{o})^{n}+A_{2}(t)(z-\unicode[STIX]{x1D701}_{o})^{2n},\end{eqnarray}$$

where $A_{1}(t)=A_{o}\cos (2\unicode[STIX]{x03C0}ft)$ is the oscillating component and $A_{2}=U_{\infty }$ represents the translational velocity of the airfoil. To provide a qualitative comparison we matched the oscillation frequency from one of the dye visualization experiments of Koochesfahani (Reference Koochesfahani1989). The experiments used a NACA 0012 airfoil with chord-based Reynolds number $Re_{c}=11\,400$ and $2^{\circ }$ oscillation amplitude and frequencies $f=4$ , 5 and 6 Hz. We ran a simulation with $f=6$ Hz and where $q_{v}$ was estimated as described above. Due to the more complicated sheet shape, we used a point vortex method to approximate the sheet.

Figure 8.

(a) Wake of an oscillatory trailing edge flow. (Top) Experimental flow visualization of a NACA 0012 airfoil oscillating at $f=6$ Hz from Koochesfahani (Reference Koochesfahani1989). (Bottom) Qualitative comparison of a simulated wake at $t=2$ with wedge angle $16^{\circ }$ to approximate the airfoil trailing edge. (b) Zoomed in view of the sharp edge showing the sheet shapes at different times $t$ with corresponding shedding angles $\unicode[STIX]{x1D703}_{v}(t)$ as labelled with $(t,\unicode[STIX]{x1D703}_{v})$ .

Figure 9.

Computed quantities for the simulated flow in figure 8. (a) The entrainment sheet strength at the sharp edge, (b) shedding angle $\unicode[STIX]{x1D703}_{v}$ where the dashed lines are the tangential limits $\unicode[STIX]{x1D6FD}\unicode[STIX]{x03C0}/2=\pm 8^{\circ }$ , and (c) vortex sheet strength at the edge. All are plotted against time. The solid circles and open triangles respectively mark where the entrainment strength is strongest and weakest.

Figure 8(a) plots a comparison of the simulated wake structure at $t=2$ and shows reasonably good agreement with the experiment. Figure 8(b) shows a zoomed in view of the sharp edge and plots the free sheet shape at several times through a period of oscillation. We can clearly see the smooth variation of the shedding angle $\unicode[STIX]{x1D703}_{v}$ . The computed entrainment sheet strength $q_{v}$ , shedding angle $\unicode[STIX]{x1D703}_{v}$ and vortex sheet strength $\unicode[STIX]{x1D6FE}_{v}$ are the top, middle and bottom plots in figure 9, respectively. First, we see that $q_{v}$ oscillates with twice the frequency as the oscillating flow, but is shifted upward by a positive mean, $\overline{q_{v}}\approx 0.159$ , such that $q_{v}(t)>0$ . This is to be expected since entrainment is a viscous process that cannot be reversed such that the fluid is returned to an irrotational state. The shedding angle $\unicode[STIX]{x1D703}_{v}$ shows the expected oscillatory behaviour within the two tangential shedding limits of $\pm \unicode[STIX]{x1D6FD}\unicode[STIX]{x03C0}/2$ and about a zero mean, namely the wedge bisector and likewise for the vortex sheet strength.

The solid circles mark when $q_{v}$ is strongest and the open triangles mark when $q_{v}$ is weakest. Although we have oscillated the flow, we will now speak as if the trailing edge were oscillating instead. Hence, entrainment is strongest when the trailing edge is passing through the zero incidence position. At these times the edge is moving with its highest velocity and changes sign of acceleration. This deceleration is known to cause an impingement of the flow onto the leeward side of the wedge and a corresponding significant normal force (DeVoria & Ringuette Reference DeVoria and Ringuette2013). This large pressure/normal force on the wedge is then responsible for the strong entrainment. The vortex sheet displays the opposite behaviour showing maxima and minima when the wedge changes sign of velocity at the largest amplitudes of the oscillation. This conjugacy between the vortex sheet and entrainment sheet, or in general tangential and normal quantities, is a feature that we have repeatedly observed throughout this work.