2 Vortex moment expression for incompressible viscous flows

Consider two-dimensional unsteady viscous flows of constant density $\unicode[STIX]{x1D70C}$ and viscosity $\unicode[STIX]{x1D707}$ around a solid body (e.g. a general airfoil) of volume $\unicode[STIX]{x1D6FA}_{B}$, bounded by a closed curve $l_{B}$. The control volume $\unicode[STIX]{x1D6FA}$ is bounded by $l_{\infty }$ at infinity. In the body-fixed frame $(x,y)$, the free-stream velocity is $V_{\infty }$, incident at an angle $\unicode[STIX]{x1D6FC}$ to the body axis. At any instant, the velocity field of the resulting flow is $\boldsymbol{U}=(u,v)$, and the vorticity $\unicode[STIX]{x1D714}_{z}=(\unicode[STIX]{x2202}v/\unicode[STIX]{x2202}x)-(\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y)$. In a previous work, the two-dimensional VFM method was derived for general airfoils (Li & Wu Reference Li and Wu2018). It was shown that the instantaneous force $F_{k}$ on a two-dimensional body in the $k$th-direction can be expressed as

(2.1)$$\begin{eqnarray}F_{k}=\unicode[STIX]{x1D70C}\iint _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D726}_{k}\boldsymbol{\cdot }\boldsymbol{U}\unicode[STIX]{x1D714}_{z}\,\text{d}\unicode[STIX]{x1D6FA},\end{eqnarray}$$

where the vortex force vector $\unicode[STIX]{x1D726}_{k}=(\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{k}/\unicode[STIX]{x2202}y,-\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{k}/\unicode[STIX]{x2202}x)$ is a function of a hypothetical potential $\unicode[STIX]{x1D719}_{k}$ defined as the velocity potential induced by unit incident velocity of ideal flow in the $k\text{th}$-direction. The vortex force vector is normalized by the free-stream velocity according to its definition and it is a non-dimensional vector coefficient introduced here to calculate the aerodynamic force when the real flow velocity and vorticity are given. It is dependent only on the geometry of the body and not the flow field, which allows for the construction of a flow-independent VFM for a given body.

For the pitching moment $M_{p}$ acting on the body at point $\boldsymbol{x}_{p}$, we can assume a similar form of expression

(2.2)$$\begin{eqnarray}M_{p}=\unicode[STIX]{x1D70C}\unicode[STIX]{x1D701}\iint _{\unicode[STIX]{x1D6FA}}\pmb{\digamma }_{p}\boldsymbol{\cdot }\boldsymbol{U}\unicode[STIX]{x1D714}_{z}\,\text{d}\unicode[STIX]{x1D6FA}.\end{eqnarray}$$

Here, $\unicode[STIX]{x1D701}$ is the characteristic length of the body and the moment is counterclockwise positive (a positive value means a nose-down pitching moment for flow coming from the left). Obtaining the VMM vector $\pmb{\digamma }_{p}$ will allow a similar map to be constructed for the moment $M_{p}$ on the body. Although there is no simple analogue between the vortex force vector $\unicode[STIX]{x1D726}_{k}$ and the VMM vector $\pmb{\digamma }_{p}$, luckily, we can derive the expression for $\pmb{\digamma }_{p}$ from the integral moment theory of Howe (Reference Howe1995), where the moment of a rigid body due to free vortices in the body-fixed frame is

(2.3)$$\begin{eqnarray}M_{p}=\unicode[STIX]{x1D70C}\iint _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D735}\unicode[STIX]{x1D712}_{p}\boldsymbol{\cdot }(\unicode[STIX]{x1D74E}\times \boldsymbol{U})\,\text{d}\unicode[STIX]{x1D6FA}.\end{eqnarray}$$

Here, the hypothetical potential $\unicode[STIX]{x1D712}_{p}$ is defined as the velocity potential for hypothetical fluid motion induced by the rotation of the body $\unicode[STIX]{x1D6FA}_{B}$ at unit angular velocity about an axis that passes through the reference point $\boldsymbol{x}_{p}$ and is perpendicular to the coordinate plane. Since we consider the application of the starting flow problem, the added mass force is zero at any instant after the starting process. The skin friction is also neglected here since the CFD results in the next section show its contribution is very small, even at low Reynolds numbers. By comparing (2.3) with (2.2), the VMM vector $\pmb{\digamma }_{p}$ can be obtained as

(2.4)$$\begin{eqnarray}\pmb{\digamma }_{p}=\frac{1}{\unicode[STIX]{x1D701}}\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D712}_{p}}{\unicode[STIX]{x2202}y},-\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D712}_{p}}{\unicode[STIX]{x2202}x}\right).\end{eqnarray}$$

The VMM vector $\pmb{\digamma }_{p}$ is normalized by a unit angular velocity multiplied by the characteristic length. It is independent of the flow field and only dependent on the geometry of the body. According to the definition of hypothetical potential $\unicode[STIX]{x1D712}_{p}$, it satisfies the following Laplace equation and boundary conditions:

(2.5)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D712}_{p}=0,\\ {\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D712}_{p}}{\unicode[STIX]{x2202}n}}=\left(\boldsymbol{x}-\boldsymbol{x}_{p}\right)\times \boldsymbol{p}\boldsymbol{\cdot }\boldsymbol{n}\quad (x,y)\rightarrow l_{B},\\ \unicode[STIX]{x1D735}\unicode[STIX]{x1D712}_{p}=0\quad (x,y)\rightarrow \infty ,\end{array}\right\}\end{eqnarray}$$

where $\boldsymbol{n}$ is the normal vector pointing inward from the body surface, and the unit vector along the moment axis is denoted as $\boldsymbol{p}$. The hypothetical potential $\unicode[STIX]{x1D712}_{p}$ vanishes at infinity and is made unique by requiring no circulation about any irreducible path. Thus, a closed-form expression for the VMM method around a body is obtained.

The VMM vector $\pmb{\digamma }_{p}$ facilitates the construction of a flow-independent VMM which can be used to analyse the moment contribution of each given vortex in the flow field, as will be shown in § 3. On the other hand, with the real flow velocity and vorticity identified from the velocity field either from the mesh grids on CFD or PIV, the pre-computed VMM vector can be used to calculate the total pitching moment acting on the body. An example of extracting pitching moment from CFD field will be given in § 4.

3 Vortex moment map analysis for a NACA0012 airfoil

In this section, a NACA0012 airfoil is used to demonstrate how VMMs are built and to identify the moment contribution effect of each given vortex according to its position, strength and local velocity.

For the NACA0012 airfoil with a chord length of $c$ aligned with the $x$-axis ($x/c\in [0,1]$), the reference points $\boldsymbol{x}_{p}=(x_{p},0)$ ($p=1,2,3$ and $4$) are chosen as the leading edge (LE, $x_{1}/c=0$), the quarter chord ($x_{2}/c=1/4$, which is the aerodynamic centre for a variety of airfoils including NACA0012 airfoil), the half-chord ($x_{3}/c=1/2$) and the trailing edge (TE, $x_{4}/c=1$), respectively. To obtain the hypothetical potential $\unicode[STIX]{x1D712}_{p}$ ($p=1,2,3$ and $4$), the Laplace equations (2.5) with four different $p$ are solved numerically by using a vortex panel method as suggested by Katz & Plotkin (Reference Katz and Plotkin2001) in solving the steady-state potential flow. This vortex panel method solves the Laplace equation via a superposition of singularity elements on the body surface and enforces the non-penetration boundary condition on the surface. For rotating bodies, the requirement of no circulation about any irreducible path should also be imposed and a uniformly distributed vorticity with strength $-2$ must be deployed to describe the solid-body motion (Koumoutsakos, Leonard & Pepin Reference Koumoutsakos, Leonard and Pepin1994), so that the correct potential solution may be reached. The method has been validated against the analytical solution for a circular cylinder. The VMM vectors $\pmb{\digamma }_{p}$ ($p=1,2,3$ and $4$) are then computed by (2.4).

With the VMM vectors precomputed, the VMMs here are plotted in the two-dimensional plane $(x,y)$ and contain vortex moment lines that are locally parallel to the VMM vector $\pmb{\digamma }_{p}$. Vortex moment lines, independent of specific flow conditions (including Reynolds number), can be obtained through a streamline procedure, with the velocity replaced by the vortex force factors. The moment contribution of any individual vortex can be identified according to its circulation (sign and magnitude), position and direction (the angle between the vortex force line and streamline at the position of the vortex). Meanwhile, the norm of the VMM vector $|\pmb{\digamma }_{p}|$ is also presented in the map as contour lines to analyse the magnitude of the effect on the moment.

Figure 1 shows the VMMs of a NACA0012 airfoil about different reference points, located at LE, $c/4$, $c/2$ and TE along the chordline. On the maps, a negative strength vortex provides a nose-up pitching moment if it moves so as to have a component of motion in the direction of the vortex moment lines, and the reverse is true for positive strength ones.

Figure 1.

Vortex moment maps for NACA0012 airfoil with different reference points: (a) moment map about the LE; (b) moment map about $c/4$; (c) moment map about $c/2$; (d) moment map about the TE. The lines with arrows are vortex moment lines locally parallel to the vector $\pmb{\digamma }_{p}$, and the lines without arrows are contours of magnitude of $\pmb{\digamma }_{p}$.

From the resulting VMMs, the following observations can be made:

1. (I) The magnitude of the VMM vectors ($|\pmb{\digamma }_{p}|$) decreases with the distance from the body and vanishes at infinity, which means the fact that the vorticity far away from the body should have a negligible effect on the pitching moment is satisfied automatically.

2. (II) The vortex moment lines point towards the reference points, which means a vortex with negative strength moving away from the reference point contributes to a nose-down pitching moment, and vice versa.

3. (III) For any reference points on the airfoil except for the LE and TE, the vortex moment lines diverge from both the LE and TE, which means a vortex with negative strength moving away from the LE/TE contributes to a nose-up pitching moment, and vice versa.

4 Vortex moment for viscous flows around an impulsively started NACA0012 airfoil

In this section, the VMM method is applied to an impulsively started flow around the NACA0012 airfoil. Using the velocity field provided by CFD and hence obtaining the vorticity numerically, and with the VMM vector $\pmb{\digamma }_{p}$ precomputed in § 3, the vortex moments are given by (2.2). Here, all of the flow field is assumed to be laminar in the CFD simulation. The theoretical moment results for pressure component $M_{p}$ will be compared to the moment obtained by the integration of the body surface pressure in the CFD code. The skin friction moment results obtained from the CFD code will also be presented to show its negligible effect. Here, the moment results will be represented in the form of non-dimensional coefficients defined as

(4.1)$$\begin{eqnarray}C_{M}=\frac{M}{\frac{1}{2}\unicode[STIX]{x1D70C}V_{\infty }^{2}\unicode[STIX]{x1D701}^{2}}.\end{eqnarray}$$

For the NACA0012 airfoil used here, the characteristic length $\unicode[STIX]{x1D701}=c$. The time-dependent moment will be displayed as a function of the non-dimensional time $\unicode[STIX]{x1D70F}=tV_{\infty }/c$.

In the CFD used in this work, the Navier–Stokes (N–S) equations in unsteady laminar flow are solved numerically, with the options of a second-order upwind SIMPLE (semi-implicit method for pressure-linked equations) pressure–velocity coupling method. The flow is impulsively started at a constant speed from an initially uniform flow. Note that laminar N–S solver is adopted for all of the Reynolds numbers considered here, including the high Reynolds numbers (e.g. $Re=1\times 10^{6}$), where a laminar solver is used purely for the purpose of numerical comparison. The computational domain is $31\times c$ in the horizontal direction and $20\times c$ in the vertical direction. Three different mesh sizes (101 845, 180 470 and 253 300 in total; 205, 430 and 550 on the body surface) are chosen for three different Reynolds numbers ($50$, $1000$ and $1\times 10^{6}$). A minimum of $30$ layers in the laminar boundary are used so that there is enough resolution in the grid size normal to the wall and in the boundary layer.

In order to validate the numerical method used here, selected numerical results for time-averaged moment for NACA 0012 and NACA 0015 airfoils at a series of AoAs from $0^{\circ }$ up to $60^{\circ }$ and for $Re=1{-}8\times 10^{4}$ are compared with those from experiments (Ohtake et al. Reference Ohtake, Nakae and Motohashi2007; Rainbird Reference Rainbird2016) in figure 2. It is shown that the CFD results for the NACA0012 airfoil at $\unicode[STIX]{x1D6FC}<20^{\circ }$ compare well with data from Ohtake et al. (Reference Ohtake, Nakae and Motohashi2007). The CFD results and experimental results (Rainbird Reference Rainbird2016) for NACA0015 at $\unicode[STIX]{x1D6FC}>20^{\circ }$ are also in good agreement. Moreover, the time-averaged moments for the NACA0012 airfoil at $\unicode[STIX]{x1D6FC}>20^{\circ }$ given by CFD are slightly larger than those from experiments for the NACA0015 airfoil.

In order to validate the numerical method used here, numerical results for the time-averaged moment for NACA airfoils at $Re=1{-}8\times 10^{4}$ are compared with those from experiments. The experimental data are collected from Ohtake et al. (Reference Ohtake, Nakae and Motohashi2007) for $0^{\circ }<\unicode[STIX]{x1D6FC}<20^{\circ }$ and Rainbird (Reference Rainbird2016) for $20^{\circ }<\unicode[STIX]{x1D6FC}<60^{\circ }$. The former uses the NACA0012 airfoil while the latter uses the NACA0015 airfoil. We could not find the experimental data for the NACA0012 airfoil with $\unicode[STIX]{x1D6FC}>20^{\circ }$ at such low Reynolds numbers, thus experimental data for the NACA0015 airfoil are used instead for the region of $\unicode[STIX]{x1D6FC}>20^{\circ }$ since both airfoils have similar aerodynamic characteristics. Good agreement between numerical and experimental results are observed in figure 2. We would like to point out that the time-averaged moments for the NACA0012 airfoil given by CFD are slightly larger than those for the NACA0015 airfoil (from both CFD and experiments) as shown in figure 2 (i.e. when $\unicode[STIX]{x1D6FC}>20^{\circ }$). This is due to the slight decrease in the thickness of the airfoil.

Figure 2.

Comparison of numerical results for time-averaged moments of NACA airfoils at different angles of attack with Ohtake et al.’s (Reference Ohtake, Nakae and Motohashi2007) experimental data for NACA0012 at $0^{\circ }<\unicode[STIX]{x1D6FC}<20^{\circ }$, and with Rainbird’s (Reference Rainbird2016) experimental results for the NACA0015 airfoil at $20^{\circ }<\unicode[STIX]{x1D6FC}<60^{\circ }$.

4.1 Vortex moment about different reference points

Applying the VMM method to the NACA0012 airfoil, we find good comparison between the time-dependent moment obtained by the VMM method and CFD about different reference points: LE, $c/4$, $c/2$ and TE at $Re=1\times 10^{6}$, as shown in figure 3. For the CFD results, the moment around any reference point $(x_{p},0)$ can be obtained by

(4.2)$$\begin{eqnarray}C_{M_{p}}=C_{M_{c/4}}+C_{L}\left(x_{p}/c-1/4\right)\cos \unicode[STIX]{x1D6FC}+C_{D}(x_{p}/c-1/4)\sin \unicode[STIX]{x1D6FC}.\end{eqnarray}$$

Here, $C_{L}$, $C_{D}$ and $\unicode[STIX]{x1D6FC}$ are the lift coefficient, the drag coefficient and the AoA, respectively. As mentioned above, there is no direct analogy between the VMM and the VFM, but according to (4.2), VMMs about different reference points are related by a superposition with the appropriate VFMs.

Figure 3.

Comparison between the vortex moment method and CFD for time-dependent moment coefficients for the NACA0012 airfoil about different reference points (LE, $c/4$, $c/2$ and TE) at $Re=1\times 10^{6}$.

It is seen from figure 3 that the pitching moments at the LE and at $c/4$ are positive (nose-down) for the whole time period herein examined ($0<\unicode[STIX]{x1D70F}<15$), while the pitching moment at the TE is always negative (nose-up). The average value of the pitching moment at $c/4$ is higher than its steady-state value ($0.39$) shown in figure 2. This increment in the nose-down moment, as well as the oscillation of the unsteady pitching moment, is closely related to the alternate shedding of the LEVs and TEVs, which will be discussed in detail in § 4.3.

Figure 4.

Comparison between vortex moment method and CFD for time-dependent moment coefficients for NACA0012 at different angles of attack and for different Reynolds numbers at $Re=1\times 10^{6}$.

4.2 Vortex moment at different AoAs and for different Reynolds numbers

Figure 4 shows a good comparison between the VMM and CFD moment about $c/4$ of a NACA0012 airfoil for $\unicode[STIX]{x1D6FC}=20^{\circ }$ and $\unicode[STIX]{x1D6FC}=60^{\circ }$ at different Reynolds numbers ($Re=50$, $1000$ and $1\times 10^{6}$). The friction-induced moments are shown to be very small for all Reynolds numbers presented here.

Figure 4(a,b) shows the effect of Reynolds number on the pitching moment. For a low Reynolds number ($Re=50$), a positive (nose-down) moment decreases from infinity to a relative stable value ($0.5$). For a large Reynolds number ($Re=1\times 10^{6}$), after the initial drop, the moment oscillates substantially with non-dimensional time and its average value is significantly larger than $0.5$. This is because the LEV and TEV in a low Reynolds number case (see figure 4a for the vorticity distribution) are much weaker than those in a high Reynolds number case (see figure 4b) and, are constrained to relatively fixed regions above the airfoil.

The time-dependent pitching moments show a clear periodicity for $Re=1000$ due to vortex shedding (see figure 4c,d). For this specific Reynolds number, with increasing AoA, the average value and the oscillating amplitude of the moment increase while the oscillating frequency deceases. This is consistent with the well-known result that the Strouhal number decreases as the AoA increases due to the wake becoming wider. In general, the AoA has a substantial impact on the pitching moment through changing the vortex shedding pattern, which will be further explored in the next subsection.

Figure 5.

Contours of the vortex moment coefficient per unit area (left) and the vorticity (right) at typical instants: (a) $\unicode[STIX]{x1D70F}=0.5$, (b) $\unicode[STIX]{x1D70F}=1.0$, (c) $\unicode[STIX]{x1D70F}=2.0$, (d) $\unicode[STIX]{x1D70F}=3.5$, with streamlines drawn.