On the calculation of force from PIV data using the generalized added-mass and circulatory force decomposition

Abstract

To understand the forces generated on an accelerating body in a fluid flow, it is useful to have the same framework for theory and experiment. A candidate for this purpose is the decomposition of fluid-dynamic force into added-mass and circulatory components. In its generalized form applicable to viscous incompressible flows, this formulation, referred to as the GAMC formulation, is applied to planar particle image velocimetry data to calculate instantaneous forces in the present study. These estimates are compared to direct force measurements and to an alternative force formulation from impulse theory, referred to as the standard impulse formulation (SIF). The chosen test case is a nominally two-dimensional circular cylinder towed through quiescent water under three acceleration profiles with peak Reynolds numbers between 5100 and 5150. For all three motion profiles, the measured and filtered drag force is consistently greater than the calculated forces with a bias of 10–20%, but the trends are in close agreement. Inspection of the presented equations reveals that the GAMC is less sensitive to near-body vorticity data than the SIF, which has the following consequences. First, forces calculated using the GAMC formulation are less sensitive to random error in the velocity than the SIF. This benefit comes at the cost of increased sensitivity to errors in cylinder position, but the associated uncertainty is negligible in the present study. Second, the GAMC is much more tolerant to the omission of near-body vorticity data, which is an attractive feature for PIV investigations. Finally, when no data are omitted, the SIF is more sensitive to the force components induced by uncharacterized high-frequency vibrations.

Appendices

Appendix A: Discretization errors in numerical integration of vortical impulse

The shed-vorticity impulse, as defined in Eq. (7), is evaluated using the approximation in (11). For a single elemental area, its contribution to ${\mathbf {P}}_{\text {v}}$ is calculated as

$$\begin{aligned} {\mathbf {P}}_{{\text {v}},j} \approx h^2 \bigg [ {\mathbf {x}}_j \times {\varvec{\omega }}_j \bigg ]_V . \end{aligned}$$

This approximation is equivalent to a one term Taylor series expansion. Setting

$$\begin{aligned} {\mathbf {f}}(x,y) = {\mathbf {x}}\times {\varvec{\omega }}, \end{aligned}$$

(52)

we can express

$$\begin{aligned} {\mathbf {f}}(x,y)&= {\mathbf {f}}(x_0,y_0) + (x-x_0)\frac{\partial {\mathbf {f}}}{\partial x}\bigg |_{(x_0,y_0)} + (y-y_0)\frac{\partial {\mathbf {f}}}{\partial y}\bigg |_{(x_0,y_0)} \nonumber \\&\quad + \frac{1}{2}(x-x_0)^2 \frac{\partial ^2 {\mathbf {f}}}{\partial x^2}\bigg |_{(x_0,y_0)} +\frac{1}{2}(y-y_0)^2 \frac{\partial ^2 {\mathbf {f}}}{\partial y^2}\bigg |_{(x_0,y_0)} +\cdots \end{aligned}$$

(53)

Integrating over a square area of dimensions $h \times h$ centred on $(x_0,y_0)$, we obtain

$$\begin{aligned} \int _{-h/2}^{h/2}\int _{-h/2}^{h/2} {\mathbf {f}}(x,y) {\text {d}}x {\text {d}}y = h^2 {\mathbf {f}}(x_0,y_0) + \frac{h^2}{24} \bigg [\frac{\partial ^2 {\mathbf {f}}}{\partial x^2} + \frac{\partial ^2 {\mathbf {f}}}{\partial y^2} \bigg ] +\cdots . \end{aligned}$$

(54)

The first-order terms in the series do not contribute to the integral, and the leading-order error, $e_n$, introduced by truncating the series to the zeroth order term is of order

$$\begin{aligned} e_n \sim {\mathcal {O}}\bigg (\frac{1}{24}h^4 \nabla ^2({\mathbf {x}}\times {\varvec{\omega }})\bigg ). \end{aligned}$$

(55)

Appendix B: Propagating uncertainty in velocity data to uncertainty in calculated force

If some quantity M is a function of the set of variables $\{L_j\}$, the uncertainty in M can be calculated from the uncertainties in each $L_j$ according to (Moffat 1988):

$$\begin{aligned} \delta M = \Bigg [ \sum _j \bigg (\frac{\partial M}{\partial L_j} \bigg )^2 \delta L_j \Bigg ]^{1/2}. \end{aligned}$$

(56)

This methodology can be applied to the calculation of the vortical impulse terms, the x-components of which are calculated for each formulation using

$$\begin{aligned} P_{{\text {v}}x,{\text {SIF}}}&= h^2 \sum _j y_j \omega _j, \end{aligned}$$

(57)

$$\begin{aligned} P_{{\text {v}}x,{\text {GAMC}}}&= h^2 \sum _j \bigg (1 - \frac{D^2}{4 r_j^2}\bigg )y_j \omega _j. \end{aligned}$$

(58)

Errors in these calculated impulses result from two key sources: (1) uncertainty in the velocity vectors due to error in the PIV correlations, and (2) errors in the estimated cylinder position. Errors of the first kind can be treated as random, and the standard error propagation approach in Eq. (56) can be applied. In this approach, the error in each vorticity data point is treated as an independent random variable. In fact, neighbouring vorticity data points may have correlated errors, but the assumption that these errors are uncorrelated permits tractable analysis. By contrast, uncertainty arising due to errors in the position variable are not properly treated as uncorrelated across the domain of integration. Since the position variable is measured relative to the cylinder centre, we expect the errors in $y_j$ to be dominated by errors in the estimated cylinder location. Thus, at any instant in time, the positional error is treated as a constant offset across the domain.

Let $y_j$ be the measured y-position in the cylinder-fixed frame of reference and let $Y_j$ be the true position such that

$$\begin{aligned} y_j = Y_j + e_{\text {p}}, \end{aligned}$$

(59)

where $e_{\text {p}}$ is the error in the position estimate. The vortical impulse is then calculated by means of the SIF as

$$\begin{aligned} P_{{\text {v}}x,{\text {SIF}}} = h^2 \sum _j Y_j \omega _j + e_{\text {p}} h^2 \sum _j \omega _j, \end{aligned}$$

(60)

where h is the grid spacing in both directions. Vorticity is calculated from velocity using a central differencing scheme:

$$\begin{aligned} \omega _{i,j} = \frac{1}{2h} ( v_{i+1,j} - v_{i-1,j} - u_{i,j+1} + u_{i,j-1} ), \end{aligned}$$

(61)

where u and v are the velocities in the x- and y-directions, i and j are the x- and y-indices of the PIV dataset. The second term on the right-hand side of Eq. (60) represents the impulse error due to cylinder position error, which we will denote as $\delta P_{x,{\text {SIF,p}}}$. We expect the total circulation to be zero in the domain around a surging cylinder, but the sum $\sum _j \omega _j$ may not be evaluated to be identically zero due to errors in the vorticity.

Using Eqs. (56) and (61), the uncertainty in the calculated vorticity can be expressed as

$$\begin{aligned} \delta \omega _{i,j} = \frac{1}{2h} \sqrt{ \delta v_{i+1,j}^2 - \delta v_{i-1,j}^2 - \delta u_{i,j+1}^2 + \delta u_{i,j-1}^2 }, \end{aligned}$$

(62)

where $\delta u$ and $\delta v$ are the velocity uncertainties. Within the DaVis 8.3 software, estimates of $\delta u$ and $\delta v$ based on correlation statistics are furnished by the method of Wieneke (2015) (n.b. herein, the estimates furnished by the DaVis software are multiplied by two to yield uncertainties with a $95\%$ confidence integral, i.e. covering two standard deviations of random scatter). Their method accounts for random errors which tend to blunt the correlation peak to reduce the quality of particle displacement estimates. The resulting component of impulse uncertainty, $\delta P_{x,{\text {SIF,vel}}}$, is determined using Eqs. (56) and (57) to yield

$$\begin{aligned} \delta P_{x,{\text {SIF,vel}}} = h^2 \left[ \sum _j y_j^2 \delta \omega _j^2 \right] ^{1/2}, \end{aligned}$$

(63)

where the sum over the index j includes every data point in the two-dimensional vorticity field, i.e. $\delta \omega _j$ in Eq. (63) is corresponds to $\omega _{i,j}$ in Eq. (61). Let us assume that the errors $\delta P_{x,{\text {SIF,p}}}$ and $\delta P_{x,{\text {SIF,vel}}}$ are additive according to

$$\begin{aligned} \delta P_{{\text {v}}x,{\text {SIF}}}^2&= \delta P_{x,{\text {SIF,p}}}^2 + \delta P_{x,{\text {SIF,vel}}}^2 \nonumber \\&= e_{\text {p}}^2 h^4 \left[ \sum _j \omega _j \right] ^2 + h^4 \left[ \sum _j y_j^2 \delta \omega _j^2 \right] . \end{aligned}$$

(64)

Equation (64) yields an estimate of uncertainty in the x-impulse values calculated by means of the SIF, accounting for both random errors in the PIV correlations and errors in the estimated position of the cylinder. Estimates of y-impulse uncertainty would follow an identical pattern. Let us now repeat a similar analysis for the total vortical impulse uncertainty for the GAMC formulation. For convenience, let us define

$$\begin{aligned} y_j^\prime = \bigg (1 - \frac{D^2}{4 r_j^2}\bigg )y_j, \end{aligned}$$

(65)

where $r_j = |{\mathbf {x}}_j|$, and $r_j^2 = x_j^2 + y_j^2$, such that

$$\begin{aligned} P_{{\text {v}}x,{\text {GAMC}}} = h^2 \sum _j y_j^\prime \omega _j. \end{aligned}$$

(66)

Equation (66) has the same form as (57), and the uncertainty due to random error in the PIV correlations is given by

$$\begin{aligned} \delta P_{x,{\text {GAMC,vel}}}&= h^2 \left[ \sum _j y_j^{\prime 2} \delta \omega _j^2 \right] ^{1/2} \nonumber \\&= h^2 \left[ \sum _j \left( 1 - \frac{D^2}{r_j^2}\right) ^2y_j^2 \delta \omega _j^2 \right] ^{1/2}. \end{aligned}$$

(67)

Comparing Eqs. (67) and (63), it is clear that the GAMC formulation is less sensitive to random error in the PIV correlations because

$$\begin{aligned} 0< \frac{D^2}{4 r_j^2} < 1 \end{aligned}$$

(68)

for all positions in the fluid domain outside the cylinder where $r_j> D/2$. However, the cost paid for this reduced sensitivity to random error is an increased sensitivity to error in the cylinder position estimate. As above, this error is treated by considering an offset to all location estimates in the evaluation of Eq. (67). For reasons that will become apparent, let us consider the magnitude of the total vortical impulse vector rather than solely the x-component:

$$\begin{aligned} |{\mathbf {P}}_{{\text {v}}t,{\text {GAMC}}}|&= h^2 \sum _j \left( 1 - \frac{D^2}{4 r_j^2}\right) r_j \omega _j \nonumber \\&= h^2 \sum _j \left( r_j - \frac{D^2}{4 r_j}\right) \omega _j. \end{aligned}$$

(69)

Positional error will be considered radially; let $r_j$ be the measured radial position, and let $R_j$ be the true radial position. These are related by the position error $e_{\text {p}}$ by

$$\begin{aligned} r_j = R_j + e_{\text {p}}. \end{aligned}$$

(70)

Equation (69) can be rewritten as

$$\begin{aligned} |{\mathbf {P}}_{{\text {v}}t,{\text {GAMC}}}| = h^2 \sum _j \bigg (R_j + e_{\text {p}} - \frac{D^2}{4(R_j + e_{\text {p}})}\bigg ) \omega _j. \end{aligned}$$

(71)

So long as the error $e_{\text {p}}$ is small, the error in impulse magnitude due to radial position error $e_{\text {p}}$ can be estimated as

$$\begin{aligned} \delta |{\mathbf {P}}_{{\text {v}}t,{\text {GAMC}}}|_p \approx \frac{\partial |{\mathbf {P}}_{{\text {v}}t,{\text {GAMC}}}|}{\partial e_{\text {p}}} e_{\text {p}}, \end{aligned}$$

(72)

where

$$\begin{aligned} \frac{\partial |{\mathbf {P}}_{{\text {v}}t,{\text {GAMC}}}|}{\partial e_{\text {p}}}&= h^2 \sum _j \bigg ( 1 + \frac{D^2}{4(R_j + e_{\text {p}})^2} \bigg ) \omega _j \end{aligned}$$

(73)

$$\begin{aligned}&= h^2 \sum _j \bigg ( 1 + \frac{D^2}{4 r_j^2} \bigg ) \omega _j. \end{aligned}$$

(74)

Conservatively, let it be assumed that this magnitude error contributes entirely to the x-direction of impulse, i.e. $\delta P_{x,{\text {GAMC,p}}} \approx \delta |{\mathbf {P}}_{{\text {v}}t,{\text {GAMC}}}|_{\text {p}}$. Finally, we arrive at

$$\begin{aligned} \delta P_{x,{\text {GAMC,p}}} \approx e_{\text {p}} h^2 \sum _j \left( 1 + \frac{D^2}{4 r_j^2} \right) \omega _j. \end{aligned}$$

(75)

As before, we take the position error and random error to be additive according to

$$\begin{aligned} \delta P_{x,{\text {GAMC}}}^2= & {} e_{\text {p}}^2 h^4 \left[ \sum _j \left( 1 + \frac{D^2}{4 r_j^2} \right) \omega _j \right] ^2 \nonumber \\&+ h^4 \left[ \sum _j \left( 1 - \frac{D^2}{4 r_j^2}\right) ^2y_j^2 \delta \omega _j^2 \right] . \end{aligned}$$

(76)

Comparison of Eqs. (64) and (76) affirms the earlier assertion that the GAMC formulation is less sensitive to random error in the PIV data but more sensitive to cylinder position error, with both conclusions resulting from the inequalities in (68).