2.2.1. Normal and axial force in the form of streamwise, spanwise and vertical decomposition

In order to get a normal and axial force formula where only near-body vortices affect the force, we introduce the conservation law of vorticity in an infinite flow field (Wu Reference Wu1981)

(2.8)\begin{equation} \iiint_{\Omega } \boldsymbol{\omega }\,\textrm{d}\Omega =0. \end{equation}

For an AOA of $\alpha$ (which may be large), the free stream velocity is $\boldsymbol{V}_{\infty }=\left ( V_{\infty }\cos \alpha ,0,V_{\infty }\sin \alpha \right )$. Adding the identically zero term $\rho \iiint _{\Omega }(0,0,1)\cdot (\boldsymbol{\omega }\times \boldsymbol{V}_{\infty }) \textrm {d}\Omega \ (\equiv 0)$ to the force formula (2.2) with $k=3$ gives the normal force as the sum of three components

(2.9)\begin{equation} \left. \begin{array}{c@{}} \displaystyle F_{3}=F_{3}^{\left( \omega _{x}\right) }+F_{3}^{\left( \omega _{y}\right) }+F_{3}^{\left( \omega _{z}\right) },\\ \displaystyle F_{3}^{\left( \omega _{x}\right) }=-\rho \iiint_{\Omega } \boldsymbol{ \Lambda}_{3,1}\cdot (0,v,w)\omega _{x}\,\textrm{d}\Omega,\\ \displaystyle F_{3}^{\left( \omega _{y}\right)}=-\rho \iiint_{\Omega }\left( \boldsymbol{ \Lambda} _{3,2}\cdot (u,0,w)-V_{\infty }\cos \alpha \right)\omega _{y}\,\textrm{d}\Omega,\\ \displaystyle F_{3}^{\left( \omega _{z}\right) }=-\rho \iiint_{\Omega } \boldsymbol{ \Lambda} _{3,3}\cdot (u,v,0)\omega _{z}\,\textrm{d}\Omega, \end{array} \right\} \end{equation}

where the normal VFM vectors $\boldsymbol{\Lambda}_{3,1}$, $\boldsymbol{\Lambda}_{3,2}$ and $\boldsymbol{\Lambda}_{3,3}$ are given in (2.4a--c) with $k=3$.

Similarly, adding the identically zero term $\rho \iiint _{\Omega }(1,0,0)\cdot \left (\boldsymbol{\omega }\times \boldsymbol{V}_{\infty } \right )\,\textrm {d}\Omega \ ( \equiv 0)$ to the force formula (2.2) with $k=1$ gives the axial force as the sum of three components

(2.10)\begin{equation} \left. \begin{array}{c@{}} \displaystyle F_{1}=F_{1}^{\left( \omega _{x}\right) }+F_{1}^{\left(\omega _{y}\right) }+ F_{1}^{\left( \omega _{z}\right) },\\ \displaystyle F_{1}^{\left( \omega _{x}\right) }=-\rho \iiint_{\Omega} \boldsymbol{ \Lambda}_{1,1}\cdot (0,v,w)\omega _{x}\,\textrm{d}\Omega,\\ \displaystyle F_{1}^{\left( \omega _{y}\right) }=-\rho \iiint_{\Omega }\left( \boldsymbol{ \Lambda} _{1,2}\cdot (u,0,w)+V_{\infty }\sin \alpha \right)\omega _{y}\,\textrm{d}\Omega,\\ \displaystyle F_{1}^{\left( \omega _{z}\right) }=-\rho \iiint_{\Omega } \boldsymbol{\Lambda} _{1,3}\cdot (u,v,0)\omega _{z}\,\textrm{d}\Omega, \end{array} \right\} \end{equation}

where the axial VFM vectors $\boldsymbol{\Lambda}_{1,1}$, $\boldsymbol{\Lambda}_{1,2}$ and $\boldsymbol{\Lambda}_{1,3}$ are given in (2.4a--c) with $k=1$.

2.2.2. Lift and drag in the form of streamwise, spanwise and vertical decomposition

With the normal and axial force given and for AOA being $\alpha$, the lift $L$ and drag $D$, are

(2.11)\begin{equation} \left. \begin{array}{c@{}} L=F_{3}\cos \alpha -F_{1}\sin \alpha, \\ D=F_{3}\sin \alpha +F_{1}\cos \alpha . \end{array} \right \} \end{equation}

Using (2.9) and (2.10), the vortex lift is

(2.12)\begin{equation} \left. \begin{array}{c@{}} \displaystyle L=L^{\left( \omega _{x}\right) }+L^{\left( \omega _{y}\right) }+L^{\left( \omega _{z}\right) },\\ \displaystyle L^{\left( \omega _{x}\right) }=-\rho \iiint_{\Omega } \boldsymbol{\Lambda}_{L,1}\cdot (0,v,w)\omega _{x}\,\textrm{d}\Omega,\\ \displaystyle L^{\left( \omega _{y}\right) }=-\rho \iiint_{\Omega }\left( \boldsymbol{ \Lambda} _{L,2}\cdot (u,0,w)-V_{\infty }\right)\omega _{y}\,\textrm{d}\Omega,\\ \displaystyle L^{\left( \omega _{z}\right) }=-\rho \iiint_{\Omega } \boldsymbol{\Lambda} _{L,3}\cdot (u,v,0)\omega _{z}\,\textrm{d}\Omega, \end{array} \right\} \end{equation}

and the vortex drag is

(2.13)\begin{equation} \left. \begin{array}{c@{}} \displaystyle D=D^{\left( \omega _{x}\right) }+D^{\left( \omega _{y}\right) }+D^{\left( \omega _{z}\right) },\\ \displaystyle D^{\left( \omega _{x}\right) }=-\rho \iiint_{\Omega } \boldsymbol{\Lambda}_{{D,1}}\cdot (0,v,w)\omega _{x}\,\textrm{d}\Omega,\\ \displaystyle D^{\left( \omega _{y}\right) }=-\rho \iiint_{\Omega } \boldsymbol{\Lambda}_{{D,2}}\cdot (u,0,w)\omega _{y}\,\textrm{d}\Omega,\\ \displaystyle D^{\left( \omega _{z}\right) }=-\rho \iiint_{\Omega } \boldsymbol{\Lambda}_{{D,3}}\cdot (u,v,0)\omega _{z}\,\textrm{d}\Omega, \end{array} \right\} \end{equation}

where

(2.14)\begin{equation} \left. \begin{array}{l@{}} \boldsymbol{\Lambda}_{{L,i}}= \boldsymbol{\Lambda}_{{3,i}}\cos \alpha - \boldsymbol{\Lambda}_{{1,i}}\sin \alpha, \\ \boldsymbol{\Lambda}_{{D,i}}= \boldsymbol{\Lambda}_{{3,i}}\sin \alpha + \boldsymbol{\Lambda}_{{1,i}}\cos \alpha, \\ i=1,2\text{ or }3, \end{array} \right \} \end{equation}

are the VFM vectors for lift and drag.

Note that, when a vortex is far from the wing, we have

\[ \left. \left( \frac{\partial \phi _{3}}{\partial x},\frac{\partial \phi _{3} }{\partial y},\frac{\partial \phi _{3}}{\partial z}\right) \right \vert _{x,y,z\rightarrow \infty }=\left( 0,0,1\right) \]

and

\[ \left. \left( \frac{\partial \phi _{1}}{\partial x},\frac{\partial \phi _{1} }{\partial y},\frac{\partial \phi _{1}}{\partial z}\right) \right \vert _{x,y,z\rightarrow \infty }=\left( 1,0,0\right) . \]

Similarly the velocity far from the body is the free stream velocity $(( u,v,w) \vert _{x,y,z\rightarrow \infty } =( V_{\infty }\cos \alpha ,0,V_{\infty }\sin \alpha ))$ and therefore its contribution to lift and drag are both zero in (2.12) and (2.13). Thus the vorticity far from the body doesn't contribute to the aerodynamic force and the present force formulas are compatible with the physical understanding that only near-body vortices induce pressure load on the body and affect lift and drag.

2.2.3. Decomposition of lift and drag, such that the correction for uncaptured vortices is treated separately

For applications where there are vortices very close to or far from the body surface, other unsteady vortex force approaches including the one presented in expression (2.2) fail to provide an accurate estimation of force. In order to give the correction term when applying the proposed force method, (2.12) and (2.13) are rewritten as a decomposed form similar to that in Li & Wu (Reference Li and Wu2018)

(2.15)\begin{align} \left. \begin{array}{c@{}} \displaystyle L=L^{\left( 1\right) }+L^{\left( 2\right) }, \\ \displaystyle L^{\left( 1\right) }=-\rho \iiint_{\Omega }[ \boldsymbol{\Lambda}_{L,1} \cdot (0,v,w)\omega _{x}+ \boldsymbol{\Lambda}_{{L,2}}\cdot (u,0,w)\omega _{y}+ \boldsymbol{\Lambda}_{{L,3}}\cdot (u,v,0)\omega _{z}]\,\textrm{d}\Omega, \\ \displaystyle L^{\left( 2\right) }=\rho V_{\infty }\iiint_{\Omega }\omega _{y}\,\textrm{d}\Omega \end{array} \right\} \end{align}

and

(2.16)\begin{align} \left. \begin{array}{c@{}} \displaystyle D=D^{\left( 1\right) }+D^{\left( 2\right) }, \\ \displaystyle D^{\left( 1\right) }=-\rho \iiint_{\Omega }[ \boldsymbol{\Lambda}_{{D,1}} \cdot (0,v,w)\omega _{x}+ \boldsymbol{\Lambda}_{{D,2}}\cdot (u,0,w)\omega _{y}+ \boldsymbol{\Lambda}_{{D,3}}\cdot (u,v,0)\omega _{z}]\,\textrm{d}\Omega, \\ D^{\left( 2\right) }=0. \end{array} \right\} \end{align}

Here the vortex lift and drag vectors are defined in (2.14).

1. (i) In the limit of vanishing viscosity, the vortices which are in the boundary layers and very close to the body surface become the bound vorticity of the inviscid flow as a sheet of vorticity on the body surface.

   Consider first spanwise vorticity $\omega _{y}$. Since the vortex force results above have been derived for a flow field which starts from rest, the total circulation about the $y$-axis remains equal to zero for all time (by the Helmholtz theorem). Therefore in any given finite sample of the flow field $\Omega$ the part of the bound circulation around the body associated with the spanwise vorticity which has been missed from the sample will have a value

   \[ \Gamma _{\infty }=-\iiint_{\Omega }\omega _{y}\,\textrm{d}\Omega . \]

   This missing circulation will be either due to the excluded vortices too close to the body surface which are not able to be picked up (a typical problem for PIV) or due to the vortices far from the body lying in a uniform velocity field $( V_{\infty },0,0)$ relative to the body. Both of them contribute to the bound circulation $-\Gamma _{\infty }$. Therefore $L^{( 2) }=-\rho V_{\infty }\Gamma _{\infty }$ and $D^{( 2) }=0$ represent the correction to the forces due to the missing vortices.

   In the present case, where the flow is symmetric about the midplane $(x\text {--}z)$ and the mean relative velocity of the body to the fluid is in the $x$-direction, there is no contribution to either $L^{( 2) }$ or $D^{( 2) }$ from $\omega _{x}$ or $\omega _{z}$.

2. (ii) In steady or quasi-steady flows where the vortex structures and the vorticity inside them are treated as fixed and unvarying in the flow, i.e. $(u,v,w)=0$, the terms $L^{( 2) }$ and $D^{( 2) }$ will be the only forces. These terms can be used to help understand the LEV suction analogy for high lift production in some quasi-steady flow problems such as flapping wings (Dickinson & Gotz Reference Dickinson and Gotz1993) and delta wings (Polhamus Reference Polhamus1971), where the concentrated LEV may be stably attached to the wing.

2.3.1. VFMs

Li & Wu (Reference Li and Wu2018) have given a detailed introduction on building a 2-D VFM $( p,q)$. In their work, the vortex force lines were displayed as lines with arrows which were locally parallel to the VFM vectors in the same way as streamlines are locally parallel to the fluid velocity. The magnitude of the vortex force vector was displayed as contour lines in the map. For any given vortex, the force contribution depends on the angle between the vortex force line and the relative velocity streamline at the location of the vortex, as well as its circulation and the magnitude of the vortex force vector.

Here for the 3-D case, a display of the sectional VFMs will be presented. The VFM vectors defined in (2.14) have three components $\boldsymbol{\Lambda}_{{L,i}}$$( i=1,2,3)$ for lift and three components $\boldsymbol{\Lambda}_{{D,i}}$$( i=1,2,3)$ for drag. Thus the 3-D VFMs can be projected into three coordinate planes, i.e. $y\text {--}z,$$x\text {--}z$ and $x\text {--}y$ planes, accordingly. They describe the contribution of streamwise vorticity $(\omega _{x})$, spanwise vorticity $(\omega _{y})$ and vertical vorticity $(\omega _{z})$, respectively, to the total force. Note that the 2-D VFM is a section which only contains the spanwise vorticity $(\omega _{y})$ in the $x\text {--}z$ plane of a 3-D vortex force distribution.

The projected VFM for 3-D flow in each plane is drawn in a similar way as in the 2-D flow. For example, the projected VFM in the $y\text {--}z$ plane contains vortex force lines locally parallel to the vortex force vector $\boldsymbol{\Lambda}_{L,1}\ (\boldsymbol{\Lambda}_{{D,1}})$ and the contour lines for the magnitude of $\boldsymbol{\Lambda}_{L,1}\ ( \boldsymbol{\Lambda}_{{D,1}})$. The VFMs in the $x\text {--}z$ and $x\text {--}y$ planes are similar.

2.3.2. Vortex force analysis

With the VFM vectors $\boldsymbol{\Lambda}_{{L,i}}$ and $\boldsymbol{ \Lambda}_{D,i}$$( i=1,2,3)$ precomputed, and with the flow vorticity $( \omega _{x},\omega _{y},\omega _{z})$ and velocity $( u,v,w)$ computed by CFD or measured in experiments, forces can be directly obtained from the vortex force formulas as follows.

1. (i) Substitute $\boldsymbol{\Lambda}_{{L,i}}$ and $\boldsymbol{ \Lambda}_{D,i}$$( i=1,2,3)$ into (2.12) and (2.13) to get the lift and drag contributed by streamwise, spanwise and vertical vortices.

2. (ii) Substitute $\boldsymbol{\Lambda}_{{L,i}}$ and $\boldsymbol{ \Lambda}_{D,i}$$( i=1,2,3)$ into (2.15) and (2.16) to get the lift and drag $(L^{( 1) },D^{( 1) })$ contributed from the vortices in the sampled field and the correction terms $(L^{( 2) },D^{( 2) })$ from missing vortices.

This VFM method will be implemented for specific flow problems in the next section with more detailed discussions.

3.2.1. VFM for streamwise vortex $(\omega _{x})$

Figures 2(a–c), 4(a–c) and 6(a–c) are the projected VFMs for lift of a delta wing at $\alpha =15,\ 45\text { and }60^{\circ }$, respectively. These maps can be used to analyse the force contributions of the streamwise vortex ($\omega _{x}$) in different transverse cross-sections $(y\text {--}z)$ along the wing axis. Each vortex force line with arrows defines the direction for maximum increase in lift for an anticlockwise rotating LEV or maximum decrease in lift for a clockwise rotating LEV. Figures 3(a–c), 5(a–c) and 7(a–c) are for drag at the same AOAs. Similarly, an anticlockwise (clockwise) rotating LEV provides positive (negative) drag if it has a local velocity aligned with the vortex drag force lines.

Because of the analogy between the normal force of a delta wing and that of a 2-D flat plate whose width is steadily increasing with time, there should also be an analogy between the lift ($L$) of a delta wing and $F_{N}\cos \alpha$ on a corresponding 2-D flat plate, as shown in figure 1$(b)$. Accordingly, correspondence between the $( y\text {--}z)$ sectional VFM for $L$ on the delta wing and that for $F_{N}\cos \alpha$ on the 2-D flat plate are found in figures 2(b,c), 4(b,c) and 6(b,c). Note that although the normal force or lift here is total for the whole delta wing rather than sectional $(y\text {--}z)$, these VFMs for total force can represent those for sectional force if the body is slender and conical. It is shown that in the front part of the delta wing, the VFM for the 2-D flat plate nearly coincides with the VFM for the delta wing. The small offset between them comes from the effect of finite $AR$ and the fact that the flat plate result is for a case with a constant width whereas the analogue of the delta wing is a plate whose width grows with time. In the rear part of the delta wing especially the part close to the TE, the VFMs show bigger differences as expected since the analogy is no longer accurate in this region.

Vortex force lines are nearly horizontal straight lines with some deformation near the body. They converge or diverge at the two edges. This means that in most cases, streamwise vortices moving in the horizontal direction are most likely to contribute (positively or negatively) to the lift or drag. This conclusion is conditionally consistent with Wu's (Reference Wu1981) force formula revealing the fact that the transverse motion of a vortex contributes to the vertical force, which will be discussed in detail in § 4.3.

The lift and drag are proportional to the magnitude of the VFM vectors, which are displayed as contour lines in the VFMs. By symmetry, the contour lines are symmetric about $y=0$, and the regions for large contribution of lift (or drag) lie close to the LEs. The larger the AOA is, the smaller is this region for lift while the larger it is for drag. The VFM function is singular at LEs, and goes to zero at the points in the middle span close to the wing section.

3.2.2. VFM for spanwise vortex $(\omega _{y})$

Figures 2(d–f ), 4(d–f) and 6(d–f) are the projected VFMs for the lift contribution of spanwise vortex ($\omega _{y}$) at $\alpha =15,\ 45\text { and }60^{\circ }$ which are in different axial cross-sections of a delta wing. Figures 3(d–f), 5(d–f) and 7(d–f) are for the drag.

The vortex force lines depend on the AOA according to (2.14). In most regions except the part very close to the wing, the vortex force lines for lift are parallel to the free steam velocity and those for drag are perpendicular to the free stream velocity, which means that vortices with a spanwise component convected with the free stream are more likely to contribute to the lift while less likely to contribute to the drag.

Although the VFMs for the spanwise vortex are distorted by the 3-D effect and they are significantly different at different axial cross-sections, the fact that the critical value $\left ( \vert \boldsymbol{\Lambda}_{{L,2}} \vert =1, \vert \boldsymbol{\Lambda}_{{D,2}} \vert =1\right )$ at infinity divides the whole area into four parts is similar to that in a 2-D VFM for a flat plate. For VFMs for lift, the region above the middle part of the wing section where $\left \vert \boldsymbol{ \Lambda}_{L,2} \right \vert <1$ becomes smaller and moves towards the LE when the AOA gets larger . For VFMs for the drag, a similar region above the TE where $\vert \boldsymbol{\Lambda}_{{D,2}} \vert <1$ moves to the middle when increasing AOA.

3.2.3. VFM for vertical vortex $(\omega _{z})$

Figures 2(g–i), 4(g–i) and 6(g–i) are the projected VFMs for the lift contribution of vertical vortex ($\omega _{z}$) at different AOAs ($15,\ 45\text { and }60^{\circ }$), and at different normal cross-sections of the domain above the wing. While figures 3(g–i), 5(g–i) and 7(g–i) are for the drag.

Vortex force lines are approximately parallel to the $y$-axis with some deformation near the body. For the lift, they bend towards the TE, and the smaller the AOA, the more curved the vortex lift force line. For the drag, they bend towards the LE, and the greater the AOA, the more curved the vortex drag force line.

The contour lines for the magnitude of VFM vectors show that the regions with the largest contribution to forces lie above the LE and the regions with the smallest contribution lie above the TE area in both VFMs for lift and drag.

4.2.1. Evolution with time of the total force and decomposed force

Figure 9 shows the total lift coefficient and the vorticity field (in the form of isosurfaces for $Q$-criteria$ \, =3$) at a series of instants. It is shown that the total lift coefficient $C_{L}$ computed by the VFM method agrees well with that obtained directly from the CFD. The lift coefficient is singular at the initial moment (which is similar to that in the starting flow of a 2-D airfoil), then drops down to a first minimum point at approximately $\tau =1$. After a short period during which the lift increases again from $\tau =1$ to $\tau =2.6$, it begins to drop to its long-term asymptotic value with some small amplitude oscillations. The initial singularity is mainly caused by the inertia effect at $\tau =0$ and the roll-up of the LEVs (Graham Reference Graham1983), while the growth and the forming of stable conical structures of the LEVs are responsible for the rapid reduction and the asymptotic behaviour of the force, respectively.

Figure 9.

Comparison of time-dependent lift coefficients generated by the VFM method and directly from the CFD and the isosurfaces of $Q$-criteria $=3$ at typical instants for a delta wing flow at $\alpha =60^{\circ }$ and $Re=5\times 10^{4}$.

As we know, four regimes can be described for steady flow incident on a delta wing at AOA from $0^{\circ }$ to $60^{\circ }$ (Ayoub & McLachlan Reference Ayoub and McLachlan1987). They are the regime of symmetric pair of LEVs (I), the regime of independent vortex breakdowns (vortex bursting) (II), the regime of interaction between vortex asymmetry and breakdown (III) and the regime of apex vortex (IV). From the isosurfaces for $Q$-criteria in figure 9, we can observe the regime of initial roll-up of the LEVs (O), as well as regimes (I), (II) and (III) during the unsteady starting process considered here. These different vortex flow regimes are closely related to the force behaviour, which will be discussed in detail in § 4.2.2.

Figure 10 shows the three components of lift coefficient contributed by streamwise, spanwise and vertical vorticity ($\omega _{x}$, $\omega _{y}$ and $\omega _{z}$). It can be seen that the lift from $\omega _{x}$ is dominant and its overall trend is similar to that of the total lift. In the asymptotic state, the lift from $\omega _{z}$ oscillates slightly around $0$ and, according to the componential lift from $\omega _{x}$ and $\omega _{y}$, the vorticity direction giving main force contribution can be identified as $( 0.13,0.58,0)$ in the wing plane (i.e. the $x\text {--}y$ plane), at an angle $\delta \approx 12.63^{\circ }$ to the axis of the wing, which is slightly smaller than the semivertex angle of the delta wing ($\beta =\arctan /{4}$). Note that $\omega _{y}$ contains both bound vorticity in the plane of the wing and the component of LEVs in the spanwise direction.

Figure 10.

Time-dependent lift coefficients contributed by vorticity in different directions for a delta wing flow at $\alpha =60^{\circ }$ and $Re=5\times 10^{4}$.

4.2.2. The spatial distribution of vortex force contribution

A quantitative study on how much the vorticity in the flow field contributes to lift is demonstrated here. Drag can be analysed in a similar way. By normalizing equation (2.12), we have

(4.3)\begin{equation} C_{L}=\iiint_{\Omega }\left(\frac{\textrm{d}C_{L,1}}{\textrm{d}\Omega }+\frac{\textrm{d}C_{L,2}}{\textrm{d}\Omega }+ \frac{\textrm{d}C_{L,3}}{\textrm{d}\Omega }\right)\,\textrm{d}\Omega, \end{equation}

where the local lift coefficient intensity contributed by $\omega _{x}$, $\omega _{y}$ and $\omega _{z}$ are defined as

(4.4)\begin{equation} \left. \begin{array}{c@{}} \dfrac{\textrm{d}C_{L,1}}{\textrm{d}\Omega }=-\dfrac{2 \boldsymbol{\Lambda}_{L,1}\cdot (0,v,w)\omega _{x}}{V_{\infty }^{2}S},\\ \dfrac{\textrm{d}C_{L,2}}{\textrm{d}\Omega }=-\dfrac{2\left( \boldsymbol{\Lambda}_{{L,2}} \cdot (u,0,w)-V_{\infty }\right) \omega _{y}}{V_{\infty }^{2}S},\\ \dfrac{\textrm{d}C_{L,3}}{\textrm{d}\Omega }=-\dfrac{2 \boldsymbol{\Lambda}_{{L,3}}\cdot (u,v,0)\omega _{z}}{V_{\infty }^{2}S}. \end{array} \right\} \end{equation}

For four typical instants ($\tau = 0.2,\ 1,\ 6.5$ and 11.5) located in the four different regimes (O, I, II, III), the spatial distribution of ${\textrm {d}C_{L,1}}/{\textrm {d}\Omega }$, ${\textrm {d}C_{L,2}}/{\textrm {d}\Omega }$ and ${ \textrm {d}C_{L,3}}/{\textrm {d}\Omega }$ at different cross-sections are shown in figure 11. The red contours show the positive lift contribution while the blue contours show the negative lift contribution. The lift contribution of each concentrated LEV contains both positive and negative parts. It is shown that the main region to contribute lift is consistent with the region where LEVs are located, which reflects the fact that near-body vortices are more likely to induce pressure (thus lift) variation according to the Biot–Savart law. Except for regime O, the lift due to $\omega _{y}$ or $\omega _{z}$ is much lower than that from $\omega _{x}$.

Figure 11.

Contours of vortex lift coefficient distribution for a delta wing starting flow at $\alpha =60^{\circ }$ and $Re=5\times 10^{4}$ for instants $\tau =0.2$, $\tau =1$, $\tau =2.5$ and $\tau =11.5$. $(a{,}d{,}g{,}j)$ Contours of $\textrm {d}C_{L,1}/\textrm {d}\Omega$ contributed by streamwise vorticity $\omega _{x}$. $(b{,}e{,}h{,}k)$ Contours of $\textrm {d}C_{L,2}/\textrm {d}\Omega$ contributed by spanwise vorticity $\omega _{y}$. $(c{,}f{,}i{,}l)$ Contours of $\textrm {d}C_{L,3}/\textrm {d}\Omega$ contributed by vertical vorticity $\omega _{z}$.

In regime O (e.g. $\tau =0.2$), the LEVs’ area, thus the vortex lift contributing area expands while the total lift decreases as shown in figure 9. This is because the lift contribution is proportional to not only the circulation but also the magnitude of VFM vectors. The circulation of the LEV spreads to a larger area where the magnitude of the VFM vectors decay as $1/r^{3}$ ($r$ is the distance from the LE) according to the properties of the solution for Laplace equation.

In regime I (e.g. $\tau =1$), the LEVs are fully developed in a near-conical area above the wing and stabilized by the axial flow effect. For the streamwise vortex, the outer part contributes positive lift, while the inner part contributes negative lift. The lift contribution from spanwise vorticity is small, and the lift contribution from vertical vorticity mainly locates above the front part of the wing.

Regime II occurs in the rear part of the wing and propagates towards the apex, which can be seen from figure 9. Figure 11$(g)$ shows that in this regime, the negative lift contribution regions for streamwise vortex are squashed to the wing and the lift contribution regions for vertical vortex spread downstream as shown in figure 11$(i)$.

Regime III leads to an asymmetrical vorticity distribution, and then an asymmetrical lift distribution (figure 11j–l). It is worth noting that the main lift contribution comes from streamwise vortex and the main region for positive lift contribution lies in an approximate symmetric region at the outer edge of the concentrated LEV. The flow structure reaches a quasi-steady state since then, leading to an asymptotic value of the total lift coefficient.

4.3.1. For large AOA flow at different instants

Figure 12 shows the lift coefficient distribution in the midchord cross-section ($x=0$) normal to the delta wing at $\alpha =60^{\circ }$ and $Re=5\times 10^{4}$ at four typical instants $( \tau =0.2,1,2.5\text { and }11.5)$. Each instant represents a specific regime. The 2-D streamlines in the same cross-section are also shown.

Figure 12.

Contours of vortex lift coefficient distribution $\textrm {d}C_{L,1}/\textrm {d}\Omega$$(a{,}c{,}e{,}g)$ and contours of streamwise vorticity $\omega _{x}$$(b{,}d{,}f{,}h)$ at a cross-section of $x=0$ of a delta wing, which is starting from rest at $\alpha =60^{\circ }$ and $Re=5\times 10^{4}$, for typical instants $\tau =0.2$ (a,b), $1$ (c,d), $2.5$ (e,f) and $11.5$ (g,h).

According to the streamlines and contours of vorticity in figure 12, a pair of contra-rotating LEVs form and grow symmetrically during regime O, I, II (e.g. $\tau =0.2$, $1$ and $2.5$). Part of the vorticity (shown as red in figure 12$a{,}c{,}e$) in the concentrated LEVs contributes a large positive lift and the other part (shown as blue in figure 12$a{,}c{,}e$) contributes a small negative lift, leading to a positive total lift. The symmetrical vortical structure breaks down in regime III (e.g. $\tau =11.5$) and the contours of vorticity in the vortex sheet connecting to both starboard and port LEs are much denser than in the core region of the LEVs, leading to the corresponding positive lift contributing region shown as red in figure 12$(g)$. This region is relatively stable in the asymptotic state of regime III since the vortex spiral doesn't grow any more and the velocity away from the axis only depends on the lateral flow effect ($V_{\infty }\cos \alpha \tan \beta$).

Moreover, it is not surprising to find that in regimes O and III, the vortex force distribution in figure 12(a,g) are in agreement with the qualitative conclusion apparent from (4.6), i.e. a pair of streamwise vortices contribute a positive lift when separating laterally, and lead to a negative lift when moving transversely towards each other. This is because, in these two regimes, the vorticity reduction effect is weak thus the lift contribution from ${\textrm {d} \boldsymbol{\omega }}/{\textrm {d}t}$ is small. On the contrary, in the regimes I and II, the vorticity reduction effect is strong, thus the first term in (4.6) alone cannot represent the total vortex force contribution accurately and the quantitative results shown in figure 12(c,e) do not agree with the qualitative conclusion mentioned above.

4.3.2. For moderate AOA flow at different cross-sections

For delta wing flow at large AOA, it is surprising to find that, in the asymptotic state, the force comes mainly from the spiral vortex sheet rather than the centre core region of the LEVs. As a comparison, a common case, the quasi-steady delta wing flow at $Re=5\times 10^{4}$ and a moderate AOA ($\alpha =35^{\circ }$) is studied using the VFM method proposed here. The results are shown in figure 13. Without any LEV bursting or interaction, the flow is symmetric. The left halves of figures 13(a)–13(c) show the $y\text {--}z$ plane contours of ${\textrm {d}C_{L,1}}/{\textrm {d}\Omega }$ at three different chordwise cross-sections $x=-0.5$ (a quarter chord), $0.5$ (3/4 quarter chord) and $0.99$ (near the TE), and the right halves show the contours of $\omega _{x}$. Streamlines are also drawn. It is obvious that the magnitude of the vorticity directly determines the intensity of the lift contribution. In all cross-sections, the vorticity of the LEVs are condensed in the vortex core region as well as the spiral vortex sheet region, in which the upper part with a cross-flow $( y\text {--}z)$ velocity towards the axis of symmetry contributes a negative lift (blue) while the lower part with a cross-flow $( y\text {--}z)$ velocity away from the axis of symmetry contributes quasi-steady flow field considered here, the results in figure 13 shows good agreement with the qualitative conclusion drawn from Wu's impulse force formula (4.6).

Figure 13.

The $y\text {--}z$ plane contours of lift coefficient intensity $\textrm {d}C_{L,1}/\textrm {d}\Omega$, contours of the streamwise vorticity $\omega _{x}$ and the streamlines at various chordwise locations for the $AR=1$ delta wing with $\alpha =35^{\circ }$, $Re=5\times 10^{4}$ at asymptotic flow state: $(a)$$x=-0.5$; $(b)$$x=0.5$; $(c)$$x=0.99$.

The drag coefficient can be analysed in a similar way as the lift coefficient, thus we omit the details here.