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Error propagation dynamics of PIV-based pressure field calculation (3): what is the minimum resolvable pressure in a reconstructed field?

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An analytical framework for the propagation of velocity errors into PIV-based pressure calculation is extended. Based on this framework, the optimal spatial resolution and the corresponding minimum field-wide error level in the calculated pressure field are determined. This minimum error can be viewed as the smallest resolvable pressure. We find that the optimal spatial resolution is a function of the flow features (patterns and length scales), fundamental properties of the flow domain (e.g., geometry of the flow domain and the type of the boundary conditions), in addition to the error in the PIV experiments, and the choice of numerical methods. Making a general statement about pressure sensitivity is difficult. The minimum resolvable pressure depends on competing effects from the experimental error due to PIV and the truncation error from the numerical solver, which is affected by the formulation of the solver. This means that PIV experiments motivated by pressure measurements should be carefully designed so that the optimal resolution (or close to the optimal resolution) is used. Flows (and \(5\times 10^4\)) with exact solutions are used as examples to validate the theoretical predictions of the optimal spatial resolutions and pressure sensitivity. The numerical experimental results agree well with the rigorous analytical predictions. We also propose a posterior method to estimate the contribution of truncation error using Richardson extrapolation and that of PIV error by adding artificially overwhelming noise. We also provide an introductory analysis of the effects of interrogation window overlap in PIV in the context of the pressure calculation.

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  1. We will adopt this terminology in the current paper due to the nature of this study (e.g., Fraenkel 2000.) To prevent any confusion, we will address the experimental data from PIV as “experimental results” or “PIV results.”

  2. This assumption is not necessary, which will be discussed in Sect. 5. Making this assumption is solely to simplify presentation of the core idea.

  3. These non-dimensional variables are different from the original choice from G.I. Taylor’s (1918) similarity solutions, but more commonly used recently (e.g., Trieling and van Heijst 1998), since it conserves unit vorticity at the origin and the velocity peaks when \(\zeta \rightarrow 1\). More specifically, \(\xi _\mathrm{{peak}} \rightarrow \exp (-1/2)/2 \approx 0.3033\) as \(\zeta \rightarrow 1\), and \(p_\mathrm{{peak}} \rightarrow -1/8=-0.125\) as \(\zeta \rightarrow 0\).

  4. Although the influence of the data on the pressure (\(f \rightarrow p\)) through the Poisson equation is a linear process, the nonlinear transformation from the velocity to the data (\({\varvec{u}} \rightarrow f\)) makes the error propagation process nonlinear.

  5. Grid spacing \(h/L_0\) locates in the regime where the boxes with the same color having a slope of \(-2\) (e.g., the red patch in Fig. 7b)

  6. Note this is a result based on the use of the central difference scheme and other numerical implementations of the pressure solver may change this result.

  7. This equation applies when the second-order central finite difference is used to calculate the source, as used throughout this paper. Other differential schemes may have different results.

  8. The goal of making this assumption is to streamline the structure of error estimation. Discussions about nonzero correlation cases can be found in Appendix 2


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We appreciate discussions with Jeff McClure and Dr. Serhiy Yarusevych.

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Appendix 1: Derivations of the error estimation

Consider a large domain in two dimensions (2D) with Dirichlet boundary conditions, which refers to the pressure Poisson equation in Eq. (3). Using a five-point scheme on a structured mesh, a point-wise finite difference approximation of Eq. (3) is

$$\begin{aligned}&\left. \nabla ^2_h p\right| _{i,j} + \left. T_{\nabla ^2 p}\right| _{i,j}+ \cdots \approx \left. f({\varvec{u}})\right| _{i,j} \nonumber \\&\quad = - \left. \nabla \cdot \left( \left( {\varvec{u}} \cdot \nabla \right) {\varvec{u}} \right) \right| _{i,j} \quad \text {in} ~ \varOmega , \end{aligned}$$

where \(\nabla ^2_h\) denotes a numerical Laplacian with grid spacing h. For example, evaluation of \(\nabla ^2_h\) at a grid point (ij) is

$$\begin{aligned} \left. \nabla _h^2 p\right| _{i,j} = \frac{p_{i+1,j}+p_{i-1,j}+p_{i,j+1}+p_{i,j-1}-4p_{i,j} }{h^2}, \end{aligned}$$

and the corresponding leading-order truncation error \(\left. T_{\nabla ^2 p}\right| _{i,j}\) is

$$\begin{aligned} \left. T_{\nabla ^2 p}\right| _{i,j} = - \frac{2}{4!}\left( \left. \frac{ \partial ^4 p}{\partial x^4} \right| _{i,j} + \left. \frac{ \partial ^4 p}{\partial y^4} \right| _{i,j} \right) h^2. \end{aligned}$$

This formulation is ignoring the effects of error in the velocity field. To retain such effects, we recognize that the PIV velocity field (\(\tilde{{\varvec{u}}}\)) contains error (\({\varvec{\epsilon }}_u\)), i.e., \(\tilde{{\varvec{u}}} = {\varvec{u}} + {\varvec{\epsilon }}_u\). This will lead to a reconstructed pressure field (\({\tilde{p}}\)) contaminated by both the experimental noise and truncation numerical error, i.e., \({\tilde{p}} = p + \epsilon _p\), where \(\epsilon _p\) is the error in the calculated pressure field and p is the true value of the pressure field. Implemented numerically, this is:

$$\begin{aligned} \left. \nabla ^2_h {\tilde{p}}\right| _{i,j} = \left. f(\tilde{{\varvec{u}}})\right| _{i,j} \quad \text {in}~ \varOmega . \end{aligned}$$

Taking advantage of linearity of the Poisson operator, Eq. (34) becomes

$$\begin{aligned} \left. f(\tilde{{\varvec{u}}})\right| _{i,j} = \left. \nabla ^2_h p\right| _{i,j} + \left. \nabla ^2_h \epsilon _{p}\right| _{i,j} \quad \text {in} ~ \varOmega . \end{aligned}$$

Comparing Eqs. (31) and (35), we see that the numerically evaluated Laplacian of the calculated pressure error is:

$$\begin{aligned} \left. \nabla ^2_h\epsilon _{p}\right| _{i,j} = \left. T_{\nabla ^2 p}\right| _{i,j} + \left. E_{\nabla ^2 p}\right| _{i,j} \quad \text {in} ~ \varOmega , \end{aligned}$$


$$\begin{aligned} \left. E_{\nabla ^2 p}\right| _{i,j}= & {} \left. f({\varvec{u}})\right| _{i,j} - \left. f(\tilde{{\varvec{u}}})\right| _{i,j}\nonumber \\= & {} \left. \nabla \cdot \left( {{\varvec{\epsilon }}}_{u} \nabla {\varvec{u}} + {\varvec{u}}\nabla {{\varvec{\epsilon }}}_{u} + {{\varvec{\epsilon }}}_{u} \nabla {\varvec{\epsilon }}_u \right) \right| _{i,j}, \end{aligned}$$

is the error induced by the noisy PIV measurements. Equation (36) indicates that the total error in the reconstructed pressure field is influenced by two distinct factors: (i) truncation error due to numerical schemes (\(T_{\nabla ^2 p}\)) and (ii) propagated errors from the velocity field due to noisy PIV experimental measurements (\(E_{\nabla ^2 p}\)), an observation which is consistent with works in the area (e.g., Charonko et al. 2010; McClure and Yarusevych 2017b; Pan 2016). More importantly, this formulation sets up a general framework that enables direct analysis of the contribution of each term.

Now we decouple the contributions from \(T_{\nabla ^2 p}\) and \(E_{\nabla ^2 p}\) by first considering the scaling of each term with respect to the spatial resolution (h). Recalling that a Poisson solver filters out the high-frequency noises (de Kat and Van Oudheusden 2012; Faiella et al. 2021), when the errors from PIV experiments are mainly high-frequency random noise, rather than systematic biases, a major contribution from \(E_{\nabla ^2 p}\) would be the squared terms such as \((\partial \epsilon _u/\partial x)^2\) and \((\partial \epsilon _v/\partial y)^2\), which contribute a positive definite bias over the domain. With this supposition, we estimate \(E_{\nabla ^2 p}\) as

$$\begin{aligned} \left. E_{\nabla ^2 p}\right| _{i,j} \approx \left. \left( \frac{ \partial \epsilon _u}{\partial x} \right) ^2 \right| _{i,j} + \left. \left( \frac{ \partial \epsilon _v}{\partial y} \right) ^2 \right| _{i,j}. \end{aligned}$$

Noting that in Eq. (38), the contribution of the u-component and the v-component are assumed to be decoupled and independent, the following derivation only takes the u-component as an example. If the velocity gradients are computed via a second-order central difference scheme, e.g., \(\left. \frac{\partial {\tilde{u}}}{\partial x}\right| _{i,j} = \frac{{\tilde{u}}_{i+1,j} - {\tilde{u}}_{i-1,j}}{2h}\), then error in the velocity gradient fields is automatically computed in the same way:

$$\begin{aligned} \left. \frac{\partial {\epsilon _u}}{\partial x}\right| _{i,j} = \frac{\epsilon _{u}|_{i+1,j} - \epsilon _{u}|_{i-1,j}}{2h}. \end{aligned}$$

We assume that the noise in the velocity field at each nodal point is a zero-mean random variable with a certain distribution \({\mathcal {D}}(0,\sigma _{u,v}^2),\) e.g., \(\left. \epsilon _u \right| _{i,j} \sim {\mathcal {D}}(0,\sigma _{u}^2)\) and \(\left. \epsilon _v \right| _{i,j} \sim {\mathcal {D}}(0,\sigma _{v}^2)\) for u and v components, respectively. The expected value of the error in the velocity gradient in the u-component is:

$$\begin{aligned} {\mathbb {E}}\left[ \left. \frac{\partial {\epsilon _u}}{\partial x}\right| _{i,j}\right]= & {} {\mathbb {E}}\left[ \frac{\epsilon _{u}|_{i+1,j} - \epsilon _{u}|_{i-1,j}}{2h} \right] \nonumber \\= & {} \frac{{\mathbb {E}}\left[ \epsilon _{u}|_{i+1,j} \right] - {\mathbb {E}}\left[ \epsilon _{u}|_{i-1,j} \right] }{2h} = 0, \end{aligned}$$

meaning that the error in the velocity gradient \(\left. \frac{\partial {\epsilon _u}}{\partial x}\right| _{i,j}\) is also zero mean. Thus, the variance of the error is

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\left[ \left( \left. \frac{\partial {\epsilon _u}}{\partial x}\right| _{i,j}\right) ^2\right]&= {\mathbb {E}}\left[ \left( \frac{\epsilon _{u}|_{i+1,j} - \epsilon _{u}|_{i-1,j}}{2h} \right) ^2 \right] \\&= \frac{{\mathbb {E}}\left[ \epsilon ^2_{u}|_{i+1,j} \right] + {\mathbb {E}}\left[ \epsilon ^2_{u}|_{i-1,j} \right] - 2\rho \left( \epsilon _{u}|_{i+1,j}, \epsilon _{u}|_{i-1,j}\right) \sqrt{{\mathbb {E}}\left[ \epsilon ^2_{u}|_{i+1,j} \right] {\mathbb {E}}\left[ \epsilon ^2_{u}|_{i-1,j}\right] }}{4h^2} \\&= \frac{1 - \rho \left( \epsilon _{u}|_{i+1,j}, \epsilon _{u}|_{i-1,j}\right) }{2h^2} \sigma _u^2. \end{aligned} \end{aligned}$$

The variance of \(\left. \frac{\partial {\epsilon _u}}{\partial x}\right| _{i,j}\) is a function of \(\rho \left( \epsilon _{u}|_{i+1,j}, \epsilon _{u}|_{i-1,j}\right)\), which is the correlation coefficient of \(\epsilon _{u}\) between two “neighboring” grid points (whose spacing is 2h due to the use of central finite difference scheme).Footnote 7

For now, we assume the error in the velocity field at each mesh grid is a point-wise independent random variableFootnote 8, thus, \(\rho \left( \epsilon _{u}|_{i,j}, \epsilon _{u}|_{k,l}\right) = 0\) and the squared error gradient at each grid point has a constant expectation, and Eq. (41) reduces to

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\left[ \left( \left. \frac{\partial {\epsilon _u}}{\partial x}\right| _{i,j}\right) ^2\right] = \frac{\sigma _u^2}{2h^2}. \end{aligned} \end{aligned}$$

Similarly, the same deviation can be applied to the gradient of velocity error in the v-component, which takes the same form, i.e.,

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\left[ \left( \left. \frac{\partial {\epsilon _v}}{\partial y}\right| _{i,j}\right) ^2\right] = \frac{\sigma _v^2}{2h^2}. \end{aligned} \end{aligned}$$

The rub of the matter is that the error introduced by the noise from the experiments will scale as

$$\begin{aligned} E_{\nabla ^2p} \sim O(h^{-2}). \end{aligned}$$

Compared to the contribution from truncation error (see Eq. (33)):

$$\begin{aligned} T_{\nabla ^2p} \sim O(h^{2}), \end{aligned}$$

it is clear that when h is small, the experimental error dominates the error propagation, and the contribution from truncation error vanishes. Similarly, when h is large, the numerical truncation error is dominant, but the impact from experimental error is negligible. Each of these terms is analyzed in more detail below.

For the truncation error, rewriting Eq. (33) leads to a point-wise description over the domain:

$$\begin{aligned}&\left. T_{\nabla ^2 p} \right| _{i,j} = - \frac{2}{4!}\left( \left. \frac{ \partial ^4 p}{\partial x^4} \right| _{i,j} + 2\left. \frac{ \partial ^4 p}{\partial x^2 \partial y^2} \right| _{i,j} + \left. \frac{ \partial ^4 p}{\partial y^4} \right| _{i,j} \right) h^2 \nonumber \\&\quad + 2\frac{2}{4!}\left. \frac{ \partial ^4 p}{\partial x^2 \partial y^2} \right| _{i,j}h^2, \end{aligned}$$

and integrating twice, we have the corresponding truncation error of the pressure field:

$$\begin{aligned} \epsilon _{p,T} = \frac{1}{12}\left( -\nabla ^2 p + 2\nabla ^{-2}\frac{\partial ^{4} p}{\partial ^2x \partial ^2y} \right) h^2, \end{aligned}$$

where \(\nabla ^{-2}\) is the inverse Laplacian which is specifically dependent on the domain and type of boundary conditions. Thus, the total error introduced by the truncation error can be estimated as

$$\begin{aligned} \Vert \epsilon _{p,T} \Vert \lesssim C_1 \left( \left\| \frac{\partial ^2 p}{\partial x^2} \right\| _{L^2(\varOmega )}+ 2\left\| \nabla ^{-2}\frac{\partial ^{4} p}{\partial ^2 x \partial ^2 y} \right\| _{L^2(\varOmega )} + \left\| \frac{\partial ^2 p}{\partial y^2} \right\| _{L^2(\varOmega )}\right) h^2, \end{aligned}$$

where \(C_1=1/12\) is a constant inherited from the Taylor expansion relevant to the specific numerical solution of the Poisson equation.

The error arising in the experimental velocity field when \(h \rightarrow 0\) is represented by point-wise squared error gradients implies that \(E_{\nabla ^2 p}\) can be split into high-frequency random components and a uniform nonzero bias. The error introduced into the data field can be estimated as

$$\begin{aligned} E_{\nabla ^2p} \approx \frac{1}{2h^2} \left( \sigma _u^2 + \sigma _v^2 \right) , \end{aligned}$$

With the approach developed in Pan et al. (2016); Faiella et al. (2021), we can bound the error in the pressure field due to experimental error in the velocity field as

$$\begin{aligned} ||\epsilon _{p,E}||_{L^2(\varOmega )} \lesssim C_0 C_2 \left( \frac{\sigma _u^2 + \sigma _v^2}{2} \right) h^{-2}. \end{aligned}$$

\(C_2\) can be considered as the amplification of the error level in the pressure field (\(\epsilon _p\)) to the error level in the data (\(\epsilon _f\)) when the data f have the “worst” profile. In other words, \(C_2 = \frac{ ||\epsilon _{p}||_{L^2(\varOmega )}}{||\epsilon _f||_{L^2(\varOmega )}}\). Assuming a large square \(L \times L\) domain with Dirichlet boundary conditions, the optimal Poincaré constant is given by \(C_2= \frac{L^2}{2\pi ^2}\). \(C_0\) measures the difference between a uniform error in the data and the “worst” possible error field. For the 2D example with Dirichlet boundary conditions as considered here, this ratio is the square of the 1D case [see Faiella et al. (2021) for greater details];, thus, \(C_0 \approx 0.901^2\).

Combining Eqs. (46) and (48), we have an estimate of the total error in the reconstructed pressure field:

$$\begin{aligned} ||\epsilon _{p}||_{L^2(\varOmega )}= & {} ||\epsilon _{p,T} + \epsilon _{p,E}||_{L^2(\varOmega )} \nonumber \\\lesssim & {} ||\epsilon _{p,T} ||_{L^2(\varOmega )} + ||\epsilon _{p,E}||_{L^2(\varOmega )} \nonumber \\\approx & {} C_1 \left( \left\| \frac{\partial ^2 p}{\partial x^2} \right\| _{L^2(\varOmega )}+ 2\left\| \nabla ^{-2}\frac{\partial ^{4} p}{\partial ^2 x \partial ^2 y} \right\| _{L^2(\varOmega )} + \left\| \frac{\partial ^2 p}{\partial y^2} \right\| _{L^2(\varOmega )}\right) h^2 \nonumber \\&\quad + C_0C_2 \left( \frac{\sigma _u^2 + \sigma _v^2}{2} \right) h^{-2}, \end{aligned}$$

which is Eq. (14) in the body of the paper. The derivation of the error estimation provided here is for a simplified setting, but other cases (e.g., with different boundary conditions, dimensions or numerical schemes) could be determined in a similar manner.

Appendix 2 : Effect of the interrogation overlap ratio and correlated error

Employing overlapped interrogation windows is common practice in PIV; however, a nonzero overlap ratio causes correlated measurement and noise (Wieneke 2017; Howell 2018). In this section, we discuss how the interrogation window overlap ratio, and the corresponding correlation of the error in the PIV results, may affect the error propagation from the velocity field to the constructed pressure field.

Currently, there is no concrete theory that models/explains how the overlap ratio of the interrogation window affects the error correlation. However, some observations showed that the random error of real PIV results have a specific character: the covariance with respect to the distance to nearby vectors show a triangular or bell curve-like profile (Wieneke 2017; Howell 2018) and spatial filters (Wieneke 2017) and Cholesky decomposition (McClure and Yarusevych 2017a) are proposed to generate such results. We next model the correlated random noise in PIV using a “top-hat” kernel as proposed in Wieneke (2017).

1.1 Modeling correlated random noise in PIV

Consider independent zero-mean Gaussian variables \(X_{i,j}\) on an \(M \times N\) structured mesh (\(i = 1, 2, \ldots , M\) and \(j = 1, 2, \ldots , N\)) where each \(X_{i,j}\) is a point-wise independent zero-mean variable: \(X_{i,j} \sim {\mathcal {N}}(0,\sigma ^2)\). We apply a moving average filter (the kernel is \(\frac{1}{4}{\varvec{J}}_{2\times 2}\), where \({\varvec{J}}\) is an all-ones matrix) to \(X_{i,j}\) throughout the domain. At each center of the window, the filtered variable is

$$\begin{aligned} Y_{i,j} = \frac{1}{4}\sum _{i-1}^{i+1} \sum _{j-1}^{j+1} X_{i,j}, \end{aligned}$$

and \(Y_{i,j}\) is also a Gaussian variable:

$$\begin{aligned} Y_{i,j} \sim {\mathcal {N}}\left( 0,\frac{\sigma ^2}{4}\right) . \end{aligned}$$

Note that \(Y_{i,j}\) of course enjoys a smaller variance than \(X_{i,j}\) owing to the filtering.

We now calculate covariance of the spatially filtered variable \(Y_{i,j}\) and its neighbor \(Y_{k,l}\). The value of the covariance \({\mathbb {E}}[Y_{i,j}Y_{k,l}]\) depends on the distance (||(ij), (kl)||) between the two points in the domain, which determines the overlapped area of the neighboring filter kernel windows (see Fig. 10a–d for an illustration):

$$\begin{aligned} {\mathbb {E}}[Y_{i,j}Y_{k,l}] = \frac{\sigma ^2}{16} {\left\{ \begin{array}{ll} 4, &{} ||(i,j),(k,l)|| = 0 \\ 2, &{} ||(i,j),(k,l)|| = h \\ 1, &{} ||(i,j),(k,l)|| = \sqrt{2}h \\ 0, &{} ||(i,j),(k,l)|| \ge 2h. \\ \end{array}\right. } \quad \end{aligned}$$

The orthogonal cases of Eq. (51), corresponding to the use of central finite difference schemes in the PIV-based pressure calculation (e.g., Eq. (39), are illustrated in Fig. 10b, d. Figure 10e shows comparison of Eq. (51) (red curve) against statistics from a numerical test (blue triangles) similar to the “top-hat” filtering used in Wieneke (2017). The covariance linearly decays when the distance of the two variables increases and leads to a triangular covariance profile. Particularly, \(Y_{i,j}\) is a zero-mean Gaussian noise field with nonzero correlation, which is a model of correlated PIV error from processing with a 50% overlap interrogation window. Similar properties and experimental tests can be found in Wieneke (2017) and Howell (2018).

Fig. 10
figure 10

ad Overlapped area of the neighboring \({\varvec{J}}_{2\times 2}\) kernels, which refers to \(\text {ov}=50\%\). The distances of the kernels \(||(i,j)-(k,l)||\) are 0, h, \(\sqrt{2}h\), 2h, respectively. The green and blue boxes are the neighboring kernels. The central crosses and circles represent the filtered “data” by each kernel. The intersection between green area and blue area represents the magnitude of covariance. e A comparison of Eq. (51) against a numerical test. The blue triangles are the calculation results of the numerical test. The orange curve is the prediction given by Eq. (51). f, g The representative overlapped area of the neighboring \({\varvec{J}}_{4\times 4}\) kernels, which refers to \(\text {ov}=75\%\). The distances of the kernels \(||(i,j)-(k,l)||\) are h, 2h, respectively

To evaluate the error propagation of the correlated noises based on the finite difference Poisson solver in this work, we start with calculating the statistics of \(\text {d}Y_{i,j}=Y_{i+1,j} - Y_{i-1,j}.\) It is apparent that the expected value \({\mathbb {E}}[\text {d}Y_{i,j}] = 0\), since \({\mathbb {E}}[Y_{i,j}]=0\). The variance can be calculated as

$$\begin{aligned} {\mathbb {E}}[\text {d}Y_{i,j}^2] = {\mathbb {E}}[(Y_{i+1,j} - Y_{i-1,j})^2]= 2{\mathbb {E}}(Y_{i,j}^2) - 2{\mathbb {E}}[Y_{i+1,j} Y_{i-1,j}]. \end{aligned}$$

Noting that \(||(i+1,j),(i-1,j)|| = 2h\) and applying Eq. (51), the above equation leads to

$$\begin{aligned} {\mathbb {E}}[\text {d}Y_{i,j}^2] = \frac{1}{2}\sigma ^2. \end{aligned}$$

Combining Eqs. (50) and (52), the variance of \(\text {d}Y_{i,j}\) is exactly twice of the variance of \(Y_{i,j}\):

$$\begin{aligned} {\mathbb {E}}[\text {d}Y_{i,j}^2] = 2{\mathbb {E}}[Y_{i,j}^2]. \end{aligned}$$

1.2 In the context of PIV-based pressure reconstruction

The dominant contributor in the source of the pressure Poisson equation is the squared terms such as

$$\begin{aligned} \left. \left( \frac{\partial \epsilon _u}{\partial x}\right) ^2\right| _{i,j} = \left( \frac{\epsilon _{u_{i+1,j}} - \epsilon _{u_{i-1,j}}}{2h}\right) ^2. \end{aligned}$$

We assume that \(\epsilon _{u_{i,j}}\) has point-wise variance \(\sigma ^2_u\) and linearly decaying correlation with neighboring cells no further than two grid spacings (2h) apart. It is important to note that the variance \(\sigma ^2_u\) introduced here is the PIV error after window overlap, which corresponds to \({\mathbb {E}}[Y_{i,j}^2]\) in Eq. (53). Using a central finite difference scheme, the expectation of \(\left( \frac{\partial \epsilon _u}{\partial x}\right) ^2\) is

$$\begin{aligned} {\mathbb {E}}\left[ \left( \frac{\partial \epsilon _u}{\partial x}\right) ^2\right] _{ov=50\%} = \frac{\sigma _u^2}{2h^2}, \end{aligned}$$

which corresponds to PIV data generated on a 50% overlap ratio and the same as the “unfiltered” error field. In fact, for the 50% overlap interpretation window, \(\epsilon _{u_{i+1,j}}\) and \(\epsilon _{u_{i-1,j}}\) are uncorrelated.

The above analysis (from (50) to (53)) gives an example for calculating the expected error for velocity gradients with \(\text {ov}=50\%\). Noting that the filter kernel for generating PIV data is equivalent to the interrogation window in PIV, the analysis can be extended to a general calculation procedure for any interrogation window overlap ratio.

Varying overlap ratio (ov) connects the grid spacing (h) of the PIV data and the size of interrogation window (Sz):

$$\begin{aligned} Sz = \frac{h}{1-ov}, \end{aligned}$$

where Sz is also equivalent to that of the space average kernel. For a given grid spacing h, Sz can be calculated for each ov according to Eq. (56), and similar to Eq. (50), a filter kernel of \(\frac{1}{Sz\times Sz}{\varvec{J}}_{Sz\times Sz}\) can be constructed for each corresponding ov. As central finite difference schemes (i.e., Eq. (39)) are applied to compute the derivatives, only vectors with distance \(||(i,j),(k,l)|| = 2h\) in Eq. (51) affect the computation. When \(||(i-1,j),(i+1,j)|| = 2h \ge Sz\), which is corresponding to \(\text {ov}\le 50\%\) (see Eq. (56)), there is no overlapping area between the two kernels, and the vectors at the two grid points (\(\epsilon _{u_{i-1,j}}\) and \(\epsilon _{u_{i+1,j}}\)) are independent and the covariance \({\mathbb {E}}[Y_{i+1,j},Y_{i-1,j}]\) is zero. When \(50\%< \text {ov} < 100\%\), the covariance is no longer zero and the correlation coefficient can be calculated by evaluating the ratio of the overlapped area to the area of the kernel, i.e.,

$$\begin{aligned} \rho \left( \epsilon _{u}|_{i-1,j}, \epsilon _{u}|_{i+1,j}\right) = \dfrac{\left( Sz-||(i-1,j), (i+1,j)||\right) \times Sz}{Sz\times Sz}, \end{aligned}$$

where \(\rho \left( \epsilon _{u}|_{i-1,j}, \epsilon _{u}|_{i+1,j}\right)\) is the correlation coefficient of \(\epsilon _{u}|_{i-1,j}\) and \(\epsilon _{u}|_{i+1,j}\). The numerator \(\left( Sz-||(i-1,j), (i+1,j)||\right) \times Sz\) represents the overlapped area (see Fig. 10b, d, f, g for illustration), and the denominator \(Sz\times Sz\) is the area of each kernel. Considering \(||(i-1,j), (i+1,j)|| = 2h\) due to the use of central finite difference schemes and recalling Eq. (56), the correlation coefficient can be written as

$$\begin{aligned} \rho \left( \epsilon _{u}|_{i-1,j}, \epsilon _{u}|_{i+1,j}\right) = {\left\{ \begin{array}{ll} 0, &{} \text {ov}\le 50\% \\ 2(1-ov), &{} 50\%< \text {ov} < 100\%. \end{array}\right. } \quad \end{aligned}$$

Then Eq. (41) can be written as:

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\left[ \left( \left. \frac{\partial {\epsilon _u}}{\partial x}\right| _{i,j}\right) ^2\right] = {\left\{ \begin{array}{ll} \dfrac{\sigma _u^2 }{2h^2} , &{}\text {ov} \le 50\% \\ \dfrac{(1 - ov)\sigma _u^2}{h^2}, &{} {50\%<\text {ov}<100\%}. \end{array}\right. } \end{aligned} \end{aligned}$$

Therefore, Eq. (59) gives a framework to calculate the effect of overlap ratio.

For example, the calculation of 75% overlap interrogation window corresponds to a filter kernel of \(\frac{1}{16}{\varvec{J}}_{4\times 4}\) (shown in Fig. 10f, g) and leads to

$$\begin{aligned} {\mathbb {E}}\left[ \left( \frac{\partial \epsilon _u}{\partial x}\right) ^2\right] _{ov=75\%} = \frac{\sigma _u^2}{4h^2}, \end{aligned}$$

which contributes half as much error as uncorrelated noise (compare Eqs. (55) and (60)).

Some typical cases of Eq. (57), which correspond to “data” from central finite difference schemes and their corresponding filter kernel used in the PIV-based pressure calculation, are shown in Fig. 10d, g for \(50\%\) and \(75\%\) cases, respectively. Figure 10 shows comparison of Eq. (59) against statistics from a numerical test, where a moving average filter is used to simulate the smear effect caused by an overlapped interrogation window. The covariance linearly decays when the distance of two variables increases and leads to a triangle covariance profile. Similar properties and experimental tests can be found in Wieneke (2017) and Howell (2018).

The effect of the interrogation window overlap in Eq. (59) is fundamentally the same for the uncertainty quantification for the vorticity calculation in Sciacchitano and Wieneke (2016):

$$\begin{aligned} U_{\omega } = \frac{U}{d}\sqrt{1-\mathbb {\rho }(2d)}, \end{aligned}$$

where U is the uncertainty of velocity, d is the grid spacing and \(\mathbb {\rho }(2d)\) is the correlation coefficient between the grid points whose distance is 2d. The difference between Eqs. (61) and (59) is that the computation of Eq. (59) is quadratic, whereas Eq. (61) arises from the linear combination of velocity gradients. In this work, we focus on the effect of overlap ratio (ov) on the spatial resolution (h), which in turn directly impacts the pressure reconstruction. Thus, ov is treated as an independent variable for a given h, and Sz is determined by ov (see Eq. (56)). For a fixed interrogation area (Sz) as some studies of PIV error propagation adopted (e.g., Sciacchitano and Wieneke 2016), Eq. (59) can be rewritten as

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\left[ \left( \left. \frac{\partial {\epsilon _u}}{\partial x}\right| _{i,j}\right) ^2\right] = {\left\{ \begin{array}{ll} \dfrac{\sigma _u^2 }{2(1 - \text {ov})^2Sz^2} , &{} \text {ov} \le 50\% \\ \dfrac{\sigma _u^2}{(1 -\text {ov})Sz^2}, &{} {50\%<\text {ov}<100\%}, \end{array}\right. } \end{aligned} \end{aligned}$$

by invoking Eq. (56). This indicates that, for a PIV experiment with a fixed interrogation window size, increasing the overlap ratio can improve spatial resolution, but it may introduce more uncertainty to pressure reconstruction.

Appendix 3: Remark on domain size and characteristic length scales

In this work, the choice of the characteristic length (or the reference length scale) may be arbitrary and does not appear in the final error estimates. However, in practice, we recommend paying close attention to two important length scales: (i) the smallest length scale of interest in the flow and (ii) the length scale of the dominant flow structure. As one considers these length scales two important rules of thumb should be remembered.

There is a danger in selecting a characteristic length scale that is much smaller than the given domain as this could lead to relatively large errors (from the large domain) when compared to the pressure changes generated by the small-scale flow structures. In such cases, if the pressure change (true value) induced by the small-scale flow structures of interest is comparable to the average error in the pressure field over the domain (the non-dimensional error level is close to unity), the current pressure reconstruction setup (PIV experimental setup, PIV resolution choice, pressure solver choice, etc.) cannot resolve the pressure field corresponding to this small-scale flow. Thus, for a large domain, the dominant flow structure (usually a larger scale than the small-scale structures) should be used as the dimensional scale.

If the small-scale flow structures and corresponding pressure field must be resolved, the obvious thing to do is optimize the pressure reconstruction setup (e.g., reducing the error in the PIV experiments, adjusting the boundary conditions, optimizing the pressure field, etc.). Another direct solution is to shrink the domain size to a scale that is comparable to the small-scale flow structures. With this adjustment, the non-dimensional domain size will be close to unity, and the small-scale structures will become the dominant features in the domain and the error levels will be smaller than the structures of interest.

Appendix 4: Remark on the three-dimensional flow

In this section, we expand the analysis to three-dimensional scenarios. One will see that the techniques and framework for analysis and the structure of the results are the same as in 2D, and thus, the major conclusions demonstrated in 2D would hold in 3D.

1.1 Error estimation in 3D cases

Continuing from Eq. (36), the total error in pressure reconstruction can be separated to truncation error \(\left. T_{\nabla ^2 p}\right| _{i,j,k}\) and experimental error \(\left. E_{\nabla ^2 p}\right| _{i,j,k}\). For the truncation error in 3D, the evaluation of \(\nabla ^2_h\) at a grid point (ijk) can be written as

$$\begin{aligned} \left. \nabla _h^2 p\right| _{i,j} = \frac{p_{i+1,j,k}+p_{i-1,j,k}+p_{i,j+1,k}+p_{i,j-1,k}+p_{i,j,k+1}+ p_{i,j,k-1}-6p_{i,j,k}}{h^2}, \end{aligned}$$

and the corresponding leading-order truncation error \(\left. T_{\nabla ^2 p}\right| _{i,j,k}\) is

$$\begin{aligned} \left. T_{\nabla ^2 p}\right| _{i,j} = - \frac{2}{4!}\left( \left. \frac{ \partial ^4 p}{\partial x^4} \right| _{i,j,k} + \left. \frac{ \partial ^4 p}{\partial y^4} \right| _{i,j,k} + \left. \frac{ \partial ^4 p}{\partial z^4} \right| _{i,j,k} \right) h^2. \end{aligned}$$

Similar to Eq. (44) and (45), by completing the square and integrating twice, the truncation error of the pressure filed can be derived as

$$\begin{aligned} \epsilon _{p,T} = \frac{1}{12}\left( -\nabla ^2 p + 2\nabla ^{-2}\frac{\partial ^{4} p}{\partial ^2x \partial ^2y} + 2\nabla ^{-2}\frac{\partial ^{4} p}{\partial ^2y \partial ^2z} + 2\nabla ^{-2}\frac{\partial ^{4} p}{\partial ^2x \partial ^2z} \right) h^2, \end{aligned}$$

where \(\nabla ^{-2}\) is the inverse Laplacian. Invoking the triangle inequality, the \(L^2\)-norm of truncation error in the whole field can be simplified as

$$\begin{aligned} \begin{aligned} \Vert \epsilon _{p,T} \Vert&=C_1 \left\| \frac{\partial ^2 p}{\partial x^2} + \frac{\partial ^2 p}{\partial y^2} + \frac{\partial ^2 p}{\partial z^2} + 2\nabla ^{-2}\frac{\partial ^{4} p}{\partial ^2 x \partial ^2 y} + 2\nabla ^{-2}\frac{\partial ^{4} p}{\partial ^2 y \partial ^2 z} + 2\nabla ^{-2}\frac{\partial ^{4} p}{\partial ^2 x \partial ^2 z}\right\| _{L^2(\varOmega )}h^2 \\&\lesssim C_1 \left( \left\| \frac{\partial ^2 p}{\partial x^2} \right\| _{L^2(\varOmega )} + \left\| \frac{\partial ^2 p}{\partial y^2} \right\| _{L^2(\varOmega )} + \left\| \frac{\partial ^2 p}{\partial z^2} \right\| _{L^2(\varOmega )} \right. \\&\left. + 2\left\| \nabla ^{-2}\frac{\partial ^{4} p}{\partial ^2 x \partial ^2 y} \right\| _{L^2(\varOmega )} + 2\left\| \nabla ^{-2}\frac{\partial ^{4} p}{\partial ^2 y \partial ^2 z} \right\| _{L^2(\varOmega )} + 2\left\| \nabla ^{-2}\frac{\partial ^{4} p}{\partial ^2 x \partial ^2 z} \right\| _{L^2(\varOmega )} \right) h^2, \end{aligned} \end{aligned}$$

where \(C_1=1/12\), the same as the coefficient in Eq. (46). Comparing Eq. (66) and Eq. (46), we can see that the structure of the truncation error contribution in 3D similar to that in 2D, however, with more terms.

The experimental error contribution in 3D is similar can be estimated by the same technique used for 2D. Similar to Eq. (38) and (47), the dominant terms in experimental error is the squared terms; thus, the experimental error can be estimated as

$$\begin{aligned} \begin{aligned} \left. E_{\nabla ^2 p}\right| _{i,j}&\approx \left. \left( \frac{ \partial \epsilon _u}{\partial x} \right) ^2 \right| _{i,j,k} + \left. \left( \frac{ \partial \epsilon _v}{\partial y} \right) ^2 \right| _{i,j,k} + \left. \left( \frac{ \partial \epsilon _w}{\partial z} \right) ^2 \right| _{i,j,k} \\&\approx \frac{1}{2h^2} \left( \sigma _u^2 + \sigma _v^2 + \sigma _w^2 \right) . \end{aligned} \end{aligned}$$

Then we can bound the \(L^2\)-norm of experimental error in the whole field as

$$\begin{aligned} ||\epsilon _{p,E}||_{L^2(\varOmega )}\lesssim & {} C_0 C_2 \left( \frac{\sigma _u^2 + \sigma _v^2 + \sigma _w^2}{2}\right) h^{-2} \nonumber \\= & {} K_2 \left( \frac{\sigma _u^2+ \sigma _v^2 + \sigma _w^2}{2}\right) h^{-2}, \end{aligned}$$

where the constants \(C_0C_2=K2\) in 3D are in general larger comparing to those in 2D (see Eq. (48)) due to the additional dimension and heuristically a larger space for integral. If the domain is “regular” (e.g., a box with proper boundary conditions), the constants \(C_0\) and \(C_2\) can be explicitly computed as demonstrated in Pan et al. (2016) and Faiella et al. (2021). \(K_2\) can be determined by adding “overwhelming noise” for fitting, as proposed in Sect. 4.

Combining Eqs. (66) and (68) error estimate of the reconstructed pressure can be developed in 3D. We note that the structure, especially the power of grid spacing h, remains unchanged (i.e., \(h^{2}\) in Eq. (66) and \(h^{-2}\) in Eq. (68)) as the analytical framework demonstrated in 2D can be adopted to 3D without change.

1.2 The trick for more robust pressure reconstruction in 3D

As demonstrated in Sect. 6, the core idea of the simple trick that can improve the robustness of pressure field reconstruction is to eliminate the squared terms in Eq. (3) for an incompressible flow. The fundamental principle is that the squared terms introduce positive definite bias and are proved to be the dominant contributors to the PIV error term such as in Eqs. (38) for 2D and (67) in 3D.

In 3D, Eq. (3) can be expanded as

$$\begin{aligned} \nabla ^2 p= & {} - \left( \left( \frac{\partial u}{\partial x}\right) ^2\ + \left( \frac{\partial v}{\partial y}\right) ^2 \right. \nonumber \\&\quad + \left. \left( \frac{\partial w}{\partial z}\right) ^2 + 2\frac{\partial u}{\partial y} \frac{\partial v}{\partial x} + 2\frac{\partial v}{\partial z} \frac{\partial w}{\partial y} + 2\frac{\partial u}{\partial z}\frac{\partial w}{\partial x}\right) \text {in} ~\varOmega \in {\mathbb {R}}^3. \end{aligned}$$

Applying the divergence-free property of an incompressible flow, i.e., \(\nabla \cdot {\varvec{u}} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0\), similar to Eq. (30), the Poisson equation becomes

$$\begin{aligned} \begin{aligned} \nabla ^2 p&= - \nabla \cdot \left( \left( {\varvec{u}} \cdot \nabla \right) {\varvec{u}} \right) + \left( \nabla \cdot {\varvec{u}}\right) ^2\\&= 2\left( \frac{\partial u}{\partial x} \frac{\partial v}{\partial y} + \frac{\partial v}{\partial y} \frac{\partial w}{\partial z} + \frac{\partial u}{\partial x} \frac{\partial w}{\partial z} - \frac{\partial u}{\partial y} \frac{\partial v}{\partial x} - \frac{\partial v}{\partial z} \frac{\partial w}{\partial y} - \frac{\partial u}{\partial z} \frac{\partial w}{\partial x}\right) \quad \text {in} ~\varOmega \in {\mathbb {R}}^3. \end{aligned} \end{aligned}$$

The form of Eq. (70) is the same as that of Eq. (30), where the squared terms are eliminated. Thus, it is expected that this simple trick will also reduce the error propagation in 3D as long as the error in the three components of velocity is not highly correlated.

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Nie, M., Whitehead, J.P., Richards, G. et al. Error propagation dynamics of PIV-based pressure field calculation (3): what is the minimum resolvable pressure in a reconstructed field?. Exp Fluids 63, 168 (2022).

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