PIV-based estimation of unsteady loads on a flat plate at high angle of attack using momentum equation approaches

Abstract

This work presents, compares and discusses results obtained with two indirect methods for the calculation of aerodynamic forces and pitching moment from 2D Particle Image Velocimetry (PIV) measurements. Both methodologies are based on the formulations of the momentum balance: the integral Navier–Stokes equations and the “flux equation” proposed by Noca et al. (J Fluids Struct 13(5):551–578, 1999), which has been extended to the computation of moments. The indirect methods are applied to spatio-temporal data for different separated flows around a plate with a $16\mathrm {:}1$ chord-to-thickness ratio. Experimental data are obtained in a water channel for both a plate undergoing a large amplitude imposed pitching motion and a static plate at high angle of attack. In addition to PIV data, direct measurements of aerodynamic loads are carried out to assess the quality of the indirect calculations. It is found that indirect methods are able to compute the mean and the temporal evolution of the loads for two-dimensional flows with a reasonable accuracy. Nonetheless, both methodologies are noise sensitive, and the parameters impacting the computation should thus be chosen carefully. It is also shown that results can be improved through the use of dynamic mode decomposition (DMD) as a pre-processing step.

Appendices

Appendix 1: Extension of the “flux equation” to the calculation of moments

It is possible to extend “flux equation” proposed by Noca et al. (1999) to the calculation of aerodynamic moments about an arbitrary defined origin. Assumed negative according to the right-hand rule, the moments can be calculated using

$$\begin{aligned} \mathbf M= & {} {} \oint _{\mathcal {S}_\infty +\mathcal {S}_b}\!\mathbf n \cdot \varvec{\gamma }^{Mt}\,\mathrm{d}\mathcal {S}- \oint _{\mathcal {S}_b}\!\left( \mathbf u ^b\cdot \mathbf n \right) \left( \rho \mathbf u \times \mathbf x \right) \,\mathrm{d}\mathcal {S}\nonumber \\&{}+\oint _{\mathcal {S}_\infty }\!\mathbf n \cdot \varvec{\gamma }^{Mp}\,\mathrm{d}\mathcal {S}-\oint _{\mathcal {S}_\infty }\!\mathbf n \cdot \left( \left[ \rho \mathbf u \mathbf u -\varvec{\tau }\right] \times \mathbf x \right) \,\mathrm{d}\mathcal {S}, \end{aligned}$$

(14)

with

$$\begin{aligned} \varvec{\gamma }^{Mt}= & {} -\frac{\rho }{2}\Vert \mathbf x \Vert ^2~\partial _t\mathbf u \times \mathbf I +\rho \left( \mathbf u \times \mathbf x \right) \mathbf u \nonumber \\&+\frac{\rho }{2}\Vert \mathbf x \Vert ^2~\mathbf u \varvec{\omega }-\frac{\rho }{2}\Vert \mathbf x \Vert ^2~\varvec{\omega }\mathbf u \nonumber \\&-\frac{\rho }{2}\Vert \mathbf u \Vert ^2~\mathbf x \times \mathbf I +\rho \left( \left[ \mathbf u \mathbf u \right] \cdot \mathbf x \right) \times \mathbf I \nonumber \\&{} + \mu \mathbf x \varvec{\omega }+ \mu \mathcal {N}\mathbf u \times \mathbf I -\frac{\mu }{2}\Vert \mathbf x \Vert ^2\nabla \varvec{\omega }, \end{aligned}$$

(15)

and

$$\begin{aligned} \varvec{\gamma }^{Mp}= & {} {} -\frac{\rho }{2}~\Vert \mathbf u \Vert ^2\left( \mathbf x \times \mathbf I \right) \nonumber \\&{}- \frac{\rho }{\mathcal {N}}~\left( \left[ \partial _t\mathbf u \times \mathbf I \right] \cdot \mathbf x \right) \mathbf x + \frac{\rho }{\mathcal {N}}\Vert \mathbf x \Vert ^2 \left( \partial _t\mathbf u \times \mathbf I \right) \nonumber \\&{}+ \frac{\rho }{\mathcal {N}} ~\left( \left[ \left\{ \mathbf u \times \varvec{\omega }\right\} \times \mathbf I \right] \cdot \mathbf x \right) \mathbf x - \frac{\rho }{\mathcal {N}}\Vert \mathbf x \Vert ^2\left( \left\{ \mathbf u \times \varvec{\omega }\right\} \times \mathbf I \right) \nonumber \\&{}+ \frac{1}{\mathcal {N}} ~\left( \left[ \left\{ \nabla \cdot \varvec{\tau }\right\} \times \mathbf I \right] \cdot \mathbf x \right) \mathbf x - \frac{1}{\mathcal {N}}\Vert \mathbf x \Vert ^2\left( \left\{ \nabla \cdot \varvec{\tau }\right\} \times \mathbf I \right) , \end{aligned}$$

(16)

where $\mathbf x$ is a location vector with respect the origin, $\mathbf u ^b$ is the body velocity and $\mathcal {N}$ the number of dimensions. Note that for the sake of concision, only instantaneous quantities are considered. Statistical mean quantities can be retrieved by averaging equations and using the Reynolds decomposition. The derivation of Eqs. (14)–(16) is similar to what was done by Noca et al. (1999) for the calculation of forces. Starting from the integral Navier–Stokes equations, the moments can be expressed as

$$\begin{aligned} \mathbf M = -\mathrm {d}_t\int _\mathcal {V}\!\rho \mathbf u \times \mathbf r \,\mathrm{d}\mathcal {V}-\oint _{\mathcal {S}_\infty }\!\left( \mathbf n \cdot \left[ p\mathbf I +\rho \mathbf u \mathbf u -\varvec{\tau }\right] \right) \times \mathbf r \,\mathrm{d}\mathcal {S}, \end{aligned}$$

(17)

where $\mathbf r$ is the location vector with respect to the point $\mathrm{R}$ about which the moment is calculated. The derivation is then done in two steps, first the elimination of pressure and then the rewriting of volume integral into surface integrals. Note that it is assumed here that the external surface $\mathcal {S}_\infty$ is static and that there is no flow through the body surface.

1.1 Elimination of pressure

To rewrite the pressure, Noca et al. (1999) uses the so-called Pressure identity. However, it cannot be directly used for the calculation of moments. Instead, the pressure term can be rewritten using the Extended Pressure identity, derived from the Pressure identity and defined as

$$\begin{aligned} -\oint _\mathcal {S}\phi \left( \mathbf n \times \mathbf x \right) \,\mathrm{d}\mathcal {S}= \frac{1}{\mathcal {N}}\oint _\mathcal {S}\mathbf x \times \left[ \mathbf x \times \left( \nabla \phi \times \mathbf n \right) \right] \,\mathrm{d}\mathcal {S}, \end{aligned}$$

(18)

where $\mathbf x$ is a location vector, $\mathcal {N}$ the dimension of space, $\phi$ an arbitrary scalar and $\mathbf n$ the unit normal to the surface $\mathcal {S}$. Note that the domain enclosed by $\mathcal {S}$ can be multiply connected.

The pressure term can be rewritten by assuming $\mathbf r =\mathbf x$ and using Eq. (18) with $\phi =p$. Moreover, the pressure gradient can be expressed as a function of the velocity field using the differential form of the Navier–Stokes equations

$$\begin{aligned} \nabla p= -\rho \partial _t\mathbf u -\nabla \left( \frac{\rho }{2}\Vert \mathbf u \Vert ^2\right) + \rho \mathbf u \times \varvec{\omega }+ \nabla \cdot \varvec{\tau }. \end{aligned}$$

(19)

Finally, using the vector identity

$$\begin{aligned} \mathbf x \times \left( \mathbf x \times \left[ \mathbf a \times \mathbf n \right] \right) = \mathbf n \cdot \left( \left\{ \left[ \mathbf a \times \mathbf I \right] \cdot \mathbf x \right\} \mathbf x -\Vert \mathbf x \Vert ^2\left[ \mathbf a \times \mathbf I \right] \right) , \end{aligned}$$

(20)

the pressure term can be written as

$$\begin{aligned} -\oint _{\mathcal {S}_\infty }\!p\left( \mathbf n \times \mathbf x \right) \,\mathrm{d}\mathcal {S}= \oint _{\mathcal {S}_\infty }\!\mathbf n \cdot \varvec{\gamma }^{Mp}\,\mathrm{d}\mathcal {S}, \end{aligned}$$

(21)

where $\varvec{\gamma }^{Mp}$ is given by Eq. (16). Note that if $\mathcal {N}=2$, the first term on the right-hand side of Eq. (20) vanishes, leading to simplification in Eq. (16).

1.2 Elimination of volume integral

The volume integral appearing in Eq. (17) can be first rewritten using the Reynolds transport theorem. Thus, by considering $\mathbf r =\mathbf x$, this yields to

$$\begin{aligned} -\mathrm {d}_t\int _\mathcal {V}\!\rho \mathbf u \times \mathbf x \,\mathrm{d}\mathcal {V}= & {} \int _\mathcal {V}\!\rho \mathbf x \times \partial _t\mathbf u \,\mathrm{d}\mathcal {V}\nonumber \\&- \oint _{\mathcal {S}_b}\!\left( \mathbf u ^b\cdot \mathbf n \right) \left( \rho \mathbf u \times \mathbf x \right) \,\mathrm{d}\mathcal {S}, \end{aligned}$$

(22)

where $\mathbf u ^b$ is the body velocity. Then, the quantity $\mathbf x \times \partial _t\mathbf u$ is rewritten in terms of field derivatives. This is achieved by starting from

$$\begin{aligned} \mathbf x \times \partial _t\mathbf u = \frac{1}{2}\nabla \times \left( \Vert \mathbf x \Vert ^2\partial _t\mathbf u \right) -\frac{1}{2}\Vert \mathbf x \Vert ^2\partial _t\varvec{\omega }, \end{aligned}$$

(23)

and by taking advantage of the vorticity equation

$$\begin{aligned} \partial _t\varvec{\omega }= \left( \varvec{\omega }\cdot \nabla \right) \mathbf u -\left( \mathbf u \cdot \nabla \right) \varvec{\omega }+\nu \nabla ^2\varvec{\omega }. \end{aligned}$$

(24)

Then, the following relations are used

$$\begin{aligned} \Vert \mathbf x \Vert ^2\left( \mathbf u \cdot \nabla \right) \varvec{\omega }={}&\nabla \cdot \left( \Vert \mathbf x \Vert ^2\mathbf u \varvec{\omega }\right) - 2\left( \mathbf x \cdot \mathbf u \right) \varvec{\omega },\end{aligned}$$

(25)

$$\begin{aligned} \Vert \mathbf x \Vert ^2\left( \varvec{\omega }\cdot \nabla \right) \mathbf u ={}&\nabla \cdot \left( \Vert \mathbf x \Vert ^2\varvec{\omega }\mathbf u \right) - 2\left( \mathbf x \cdot \varvec{\omega }\right) \mathbf u ,\end{aligned}$$

(26)

$$\begin{aligned} \Vert \mathbf x \Vert ^2\nabla ^2\varvec{\omega }={}&\nabla ^2\left( \Vert \mathbf x \Vert ^2\varvec{\omega }\right) - \nabla \cdot \left( 4\mathbf x \varvec{\omega }\right) + 2\mathcal {N}\varvec{\omega },\end{aligned}$$

(27)

$$\begin{aligned} \left( \mathbf x \cdot \mathbf u \right) \varvec{\omega }={}&-\nabla \times \left[ \frac{1}{2}\Vert \mathbf u \Vert ^2\mathbf x - \left( \mathbf u \mathbf u \right) \cdot \mathbf x \right] \nonumber \\ {}&-\nabla \cdot \left[ \left( \mathbf u \times \mathbf x \right) \mathbf u \right] + \left( \mathbf x \cdot \varvec{\omega }\right) \mathbf u , \end{aligned}$$

(28)

to finally obtain

$$\begin{aligned} \mathbf x \times \partial _t\mathbf u {}= & {} \nabla \times \left( \frac{1}{2}\Vert \mathbf x \Vert ^2\partial _t\mathbf u \right) +\nabla \cdot \left( \frac{1}{2}\Vert \mathbf x \Vert ^2\mathbf u \varvec{\omega }\right) \nonumber \\ {}&+\nabla \times \left( \frac{1}{2}\Vert \mathbf u \Vert ^2\mathbf x -\left[ \mathbf u \mathbf u \right] \cdot \mathbf x \right) \nonumber \\ {}&+\nabla \cdot \left( \left[ \mathbf u \times \mathbf x \right] \mathbf u \right) -\nabla \cdot \left( \frac{1}{2}\Vert \mathbf x \Vert ^2\varvec{\omega }\mathbf u \right) \nonumber \\ {}&-\nabla \cdot \left( \nabla \left[ \frac{\nu }{2}\Vert \mathbf x \Vert ^2 \varvec{\omega }\right] \right) +\nabla \cdot \left( 2\nu \mathbf x \varvec{\omega }\right) -\nabla \times \left( \nu \mathcal {N}\mathbf u \right) . \end{aligned}$$

(29)

At last, the Gauss theorem is used to express the volume integral as a surface integral:

$$\begin{aligned} -\mathrm {d}_t\int _\mathcal {V}\!\rho \mathbf u \times \mathbf x \,\mathrm{d}\mathcal {V}= & {} {} \oint _{\mathcal {S}_\infty +\mathcal {S}_b}\!\mathbf n \cdot \varvec{\gamma }^{Mt}\,\mathrm{d}\mathcal {S}\nonumber \\&{}- \oint _{\mathcal {S}_b}\!\left( \mathbf u ^b\cdot \mathbf n \right) \left( \rho \mathbf u \times \mathbf x \right) \,\mathrm{d}\mathcal {S}, \end{aligned}$$

(30)

with $\varvec{\gamma }^{Mt}$ given by Eq. (15). Note that if $\mathcal {N}=2$, the vortex stretching term in Eq. (24) vanishes and several terms in Eq. (15) disappear.

Appendix 2: Convergence study on the number of PIV snapshots needed for averaging

The INSE and NOCA methods have been applied to PIV fields obtained by averaging 50, 100, 150 and 200 snapshots, for the three cases investigated here. The results presented below are obtained from the INSE approach but similar conclusions can be drawn for the NOCA methodology.

For the large amplitude pitching plate, it appears that the load responses are very similar for the four numbers of images considered, as depicted in Fig. 10. Moreover, the statistics calculated from each signal are almost the same, with a maximum difference of $4\%$. Therefore, 50 images would be already enough to obtain a good estimation of the load coefficients. Note that this number differs significantly from the results of Gharali and Johnson (2014) who reported a minimum of 500 images required for a similar case.

Fig. 10

Impact of the number of snapshots used on the evolution of lift and drag coefficients within a pitching period T for large amplitude plate oscillations around a mean angle of attack of $0^\circ$: indirect calculation using INSE (symbols) and direct measurements (thick continuous line). The error bars correspond to the sensitivity of the results to the control surface used in the indirect method

For the static plate, the data based on 50 snapshots lead to a reasonable estimation of the mean coefficients. However, 150 images are needed to obtain a sensitivity to the location of $\mathcal {S}$ similar to the results reported in Table 2. Note that the number of images required for this case increases compared to the large amplitude pitching case. This is probably because the coherence between snapshots decreases.

For the small amplitude pitching plate case, the mean coefficients are similar whether computed with PIV fields obtained from 50, 100, 150 or 200 images. Nonetheless, it seems that increasing the number of snapshots leads to a decrease of the noise in the coefficient responses, as depicted in Fig. 11. Therefore, it could be expected that a higher number of snapshots could further improve the results.

Fig. 11

Impact of the number of snapshots used on the evolution of lift and drag coefficients within a pitching period T for small amplitude plate oscillations around a mean angle of attack of $30^\circ$: indirect calculation using INSE (symbols) and direct measurements (thick continuous line). The error bars correspond to the sensitivity of the results to the control surface used in the indirect method